**Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments**

Zonghu Liao and Ze'ev Reches

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54889

#### **1. Introduction**

Earthquakes are associated with slip along fault-zones in the crust, and the intensity of dynamic-weakening is one of the central questions of earthquake physics (Dieterich, 1979; Reches and Lockner, 2010). Since it is impossible to determine fault friction with seismological methods (Kanamori and Brodsky, 2004), the study of fault friction and earthquake weakening has been usually addressed with laboratory experiments (Dieterich, 1979) and theoretical models (Ohnaka and Yamashita, 1989).

The experimental analyses of dynamic-weakening were conducted in several experimental configurations: bi-axial direct shear (Dieterich, 1979; Samuelson et al, 2009), tri-axial confined shear (Lockner and Beeler, 2002), and rotary shear apparatus (Tsutsumi and Shimamoto, 1997; Goldsby and Tullis, 2002; Di Toro et al., 2004; Reches and Lockner, 2010). The direct shear apparatus allows high normal stress and controlled pore water pressure (up to ~200 MPa) with limited slip velocity (up to 0.01 m/s) and limited slip distance (~10 mm) (Shimamoto and Logan, 1984). These slip velocities and displacements are significantly smaller than those of typical earthquakes (0.1-10 m/s and up to 5 m, respectively). In order to study high velocity and long slip distance, experiments have been conducted in rotary shear machines.

While many studies indicated a systematic weakening with increasing slip-velocity (Dieterich, 1979; Di Toro et al., 2011), recent experimental observations revealed an opposite trend of dynamic-strengthening particularly under high velocity (Reches and Lockner, 2010; Kuwano and Hatano, 2011). This strengthening was attributed to dehydration of the fault gouge due to frictional heating at elevated velocities (Reches and Lockner, 2010; Sammis et al., 2011). If

© 2013 Liao and Reches; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Liao and Reches; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

similar dynamic-strengthening occurs during earthquakes, it should be incorporated in the analyses of earthquake slip.

(Fig. 1b). Earlier work of Tsutsumi and Shimamoto (1997) also indicated temporal periods of strengthening, in which the friction sharply increased, and visible melting was observed at the

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**Figure 1.** Experimental friction-velocity relations in (a) Sierra White granite (after Reches and Lockner, 2010) and (b)

To further explore the occurrence of dynamic-strengthening, we tested four other rocks under conditions similar to Reches and Lockner (2010) tests. The new experiments were conducted on samples made of Blue quartzite (Fig. 2a), St. Cloud diorite (Fig. 2b), Fredricksburg syenite (Fig. 2c), and Karoo gabbro (Fig. 2d). The diorite (Fig. 2b) and syenite (Fig. 2c) samples

Westerly ganite (after Kuwano and Hatano, 2011

strength peak when the slip rate was increased from 0.55 to 0.73 m/s.

We analyze here of the relations between friction coefficient and slip velocity for steady-state velocity the range of 0.001-1 m/s. These relations are derived for the Sierra White granite (SWG) experiments under normal stress up to 7 MPa. The results are presented by a model that is termed WEST (WEkening-STrengtheing). We first present the main experiment observations, for five rock types, and then we derive the numeric model for Sierra White granite (SWG). Finally, we apply the numerical model to tens of experiments with complex velocity history conducted on the same rock.

#### **2. Experimental observations**

The experiments were conducted with a rotary shear apparatus and cylindrical rock samples (Reches and Lockner, 2010; Chang et al., 2012). The experimental setup and sample configu‐ ration are described in Appendix A. The experimentally monitored parameters include slip distance, slip velocity, fault-normal displacement (dilation), shear stress and normal stress. Sample temperature was measured by thermocouples that are embdded ~3mm away from the slip surfaces. The normal load was maintained constant during a given experiment, and the experiments were performed at room conditions. Fault strength is represented by the friction coefficient, µ = τ/σN (where τ is shear stress and σN the normal stress). The slip velocity was either maintained constant, or increased or decreased in steps. We present results of tests with samples of Sierra White granite (after Reches and Lockner, 2010), and new results for samples of Blue quartzite, St. Cloud diorite, Fredricksburg syenite, and Karoo gabbro. The tests with SWG samples were run under the widest range of conditions and the numeric model was derived only for this rock.

Reches and Lockner (2010) determined the friction-velocity relations of SWG for a velocity range of.0003-1 m/s and normal stress up to 7 MPa. Their results revealed three general regimes of friction-velocity relations (Fig. 1a):


Similar pattern of weakening-strengthening of Westerly granite samples was recently ob‐ served by Kuwano and Hatano (2011) (Fig. 1b). They showed that the friction coefficient dropped in the velocity range of 0.001-0.06 m/s, and rose in the velocity range of 0.06-0.2 m/s (Fig. 1b). Earlier work of Tsutsumi and Shimamoto (1997) also indicated temporal periods of strengthening, in which the friction sharply increased, and visible melting was observed at the strength peak when the slip rate was increased from 0.55 to 0.73 m/s.

similar dynamic-strengthening occurs during earthquakes, it should be incorporated in the

We analyze here of the relations between friction coefficient and slip velocity for steady-state velocity the range of 0.001-1 m/s. These relations are derived for the Sierra White granite (SWG) experiments under normal stress up to 7 MPa. The results are presented by a model that is termed WEST (WEkening-STrengtheing). We first present the main experiment observations, for five rock types, and then we derive the numeric model for Sierra White granite (SWG). Finally, we apply the numerical model to tens of experiments with complex velocity history

The experiments were conducted with a rotary shear apparatus and cylindrical rock samples (Reches and Lockner, 2010; Chang et al., 2012). The experimental setup and sample configu‐ ration are described in Appendix A. The experimentally monitored parameters include slip distance, slip velocity, fault-normal displacement (dilation), shear stress and normal stress. Sample temperature was measured by thermocouples that are embdded ~3mm away from the slip surfaces. The normal load was maintained constant during a given experiment, and the experiments were performed at room conditions. Fault strength is represented by the friction coefficient, µ = τ/σN (where τ is shear stress and σN the normal stress). The slip velocity was either maintained constant, or increased or decreased in steps. We present results of tests with samples of Sierra White granite (after Reches and Lockner, 2010), and new results for samples of Blue quartzite, St. Cloud diorite, Fredricksburg syenite, and Karoo gabbro. The tests with SWG samples were run under the widest range of conditions and the numeric model was

Reches and Lockner (2010) determined the friction-velocity relations of SWG for a velocity range of.0003-1 m/s and normal stress up to 7 MPa. Their results revealed three general regimes

**i.** Dynamic-weakening (drop of 20-60% of static strength) as slip velocity increased

**ii.** Transition to dynamic-strengthening regime in the velocity range of V = 0.06-0.2 m/s, during which the fault strength almost regained its static strength; and

**iii.** Quasi-constant strength for V > 0.2 m/s, with possible further drops as velocity

Similar pattern of weakening-strengthening of Westerly granite samples was recently ob‐ served by Kuwano and Hatano (2011) (Fig. 1b). They showed that the friction coefficient dropped in the velocity range of 0.001-0.06 m/s, and rose in the velocity range of 0.06-0.2 m/s

from ~0.0003 m/s to a critical velocity of Vc ~ 0.03 m/s, during which the friction

approaches ~1 m/s. Only few experiments were conducted in this range due to sample

analyses of earthquake slip.

108 Earthquake Research and Analysis - New Advances in Seismology

conducted on the same rock.

derived only for this rock.

of friction-velocity relations (Fig. 1a):

coefficient was 0.3-0.45.

failure by thermal fracturing.

**2. Experimental observations**

**Figure 1.** Experimental friction-velocity relations in (a) Sierra White granite (after Reches and Lockner, 2010) and (b) Westerly ganite (after Kuwano and Hatano, 2011

To further explore the occurrence of dynamic-strengthening, we tested four other rocks under conditions similar to Reches and Lockner (2010) tests. The new experiments were conducted on samples made of Blue quartzite (Fig. 2a), St. Cloud diorite (Fig. 2b), Fredricksburg syenite (Fig. 2c), and Karoo gabbro (Fig. 2d). The diorite (Fig. 2b) and syenite (Fig. 2c) samples displayed a distinct transition into dynamic-strengthening regime (red arrows). The critical transition velocity, VC, depends on the sample lithology; it is ~0.02 m/s and ~0.01 m/s for the diorite and syenite, respectively. The Blue quartzite tests displayed only negligible weakening and strengthening (Fig. 2a).

velocity. This assumption was employed in many previous models (Dieterich, 1979; Beeler et al, 1994; Tsutsumi and Shimamoto, 1997; Reches and Lockner, 2010; Di Toro et al., 2011). Second, the weakening-strengthening mode reflects a transition between different frictional mechanisms. Reches and Lockner (2010) and Sammis et al. (2011) suggested that the weakening is controlled by gouge powder lubrication due to coating of the powder grains by a thin layer of water that is 2-3 monolayers thick. The dehydration of this layer at elevated temperature under high slip velocity leads to strengthening. It is thus assumed that the weakening and strengthening regimes have different parametric relation between friction and velocity.

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Following the above assumptions, we model the relations of the steady-state friction coeffi‐ cient, µ, and the slip distance, D, and slip-velocity, V (Fig. 3). The steady-state is defined for a given slip velocity during which the weakening (or strengthening) intensifies with increasing slip distance. The model addresses the three major features of the experimental observation: (I) the slip-weakening relation, µ(V, D), represents the drop from static friction coefficient, **μS**, to the kinematic friction coefficient, **μK** during slip to the critical distance, DC, at a constant velocity; (II) the dynamic-friction coefficient, µK(V), at V < VC, under steady-state; and (III) the dynamic-strengthening at high velocity regime of V > VC. We regard the friction coefficient decrease from (1) to (2) (Fig. 3) as the dynamic-weakening, and the friction coefficient increase

**Figure 3.** A schematic presentation of WEST model, with velocity controlled kinematic friction coeefcient.

m m

mentally controlling factors of D and V has the general form of

The dependence of the experimentally monitored friction coefficient, µ, on the two experi‐

The friction coefficient, µ, has two end members, the static friction coefficient, **μS,** which is the friction coefficient during slip initiation, and the kinematic friction coefficient, **μK**, the steady-

= (*D V*, ) (1)

**3.2. Model formulation**

from (2) to (3) as dynamic-strengthening.

**Figure 2.** Friction-velocity relations in shear experiements of Blue quartzite (a), St. Cloud diorite (b), Fredricksburg syenite (c), and Karoo gabbro (d). Connected lines indicated data from step-velocities experiments. Unconnected dots indicate single velocity experiments.Note velocity-weakening and velocity-strengthening stages (marked by red ar‐ rows) in (a)-(c) and no velocity-strengthening in (d).

#### **3. Numerical modeling**

#### **3.1. Approach**

The most striking feature of the relations between the friction coefficient and slip-velocity for the above experiments is the systematic transition from weakening under low velocity to strengthening at higher velocities (Figs. 1, 2). We refer to this feature of **WE**akening-**ST**rengthening as the **WEST** mode, and developed a numerical model to describe its character. The model is based on two central assumptions. First, the friction coefficients, in both weak‐ ening and strengthening regimes, can be presented as dependent of the slip distance, and slip velocity. This assumption was employed in many previous models (Dieterich, 1979; Beeler et al, 1994; Tsutsumi and Shimamoto, 1997; Reches and Lockner, 2010; Di Toro et al., 2011). Second, the weakening-strengthening mode reflects a transition between different frictional mechanisms. Reches and Lockner (2010) and Sammis et al. (2011) suggested that the weakening is controlled by gouge powder lubrication due to coating of the powder grains by a thin layer of water that is 2-3 monolayers thick. The dehydration of this layer at elevated temperature under high slip velocity leads to strengthening. It is thus assumed that the weakening and strengthening regimes have different parametric relation between friction and velocity.

#### **3.2. Model formulation**

displayed a distinct transition into dynamic-strengthening regime (red arrows). The critical transition velocity, VC, depends on the sample lithology; it is ~0.02 m/s and ~0.01 m/s for the diorite and syenite, respectively. The Blue quartzite tests displayed only negligible weakening

**Figure 2.** Friction-velocity relations in shear experiements of Blue quartzite (a), St. Cloud diorite (b), Fredricksburg syenite (c), and Karoo gabbro (d). Connected lines indicated data from step-velocities experiments. Unconnected dots indicate single velocity experiments.Note velocity-weakening and velocity-strengthening stages (marked by red ar‐

The most striking feature of the relations between the friction coefficient and slip-velocity for the above experiments is the systematic transition from weakening under low velocity to strengthening at higher velocities (Figs. 1, 2). We refer to this feature of **WE**akening-**ST**rengthening as the **WEST** mode, and developed a numerical model to describe its character. The model is based on two central assumptions. First, the friction coefficients, in both weak‐ ening and strengthening regimes, can be presented as dependent of the slip distance, and slip

and strengthening (Fig. 2a).

110 Earthquake Research and Analysis - New Advances in Seismology

rows) in (a)-(c) and no velocity-strengthening in (d).

**3. Numerical modeling**

**3.1. Approach**

Following the above assumptions, we model the relations of the steady-state friction coeffi‐ cient, µ, and the slip distance, D, and slip-velocity, V (Fig. 3). The steady-state is defined for a given slip velocity during which the weakening (or strengthening) intensifies with increasing slip distance. The model addresses the three major features of the experimental observation: (I) the slip-weakening relation, µ(V, D), represents the drop from static friction coefficient, **μS**, to the kinematic friction coefficient, **μK** during slip to the critical distance, DC, at a constant velocity; (II) the dynamic-friction coefficient, µK(V), at V < VC, under steady-state; and (III) the dynamic-strengthening at high velocity regime of V > VC. We regard the friction coefficient decrease from (1) to (2) (Fig. 3) as the dynamic-weakening, and the friction coefficient increase from (2) to (3) as dynamic-strengthening.

**Figure 3.** A schematic presentation of WEST model, with velocity controlled kinematic friction coeefcient.

The dependence of the experimentally monitored friction coefficient, µ, on the two experi‐ mentally controlling factors of D and V has the general form of

$$
\mu = \mu \begin{pmatrix} D \ \end{pmatrix} \tag{1}
$$

The friction coefficient, µ, has two end members, the static friction coefficient, **μS,** which is the friction coefficient during slip initiation, and the kinematic friction coefficient, **μK**, the steadystate coefficient after large slip distance under a constant velocity. The transition between the weakening regime and the strengthening regime occurs at the critical slip-velocity, **VC**, that is determined from the general plot of friction coefficient as function of slip-velocity (Fig. 1, 2, 3).

We assume that during the weakening stage and under constant velocity, the drop of µ from **μS** to **μK** is controlled only by the slip distance D,

$$
\mu = \mu \begin{pmatrix} D \end{pmatrix}, \\
\text{when} \\
\mu\_{\text{S}} > \mu > \mu\_{\text{K}} \tag{2}
$$

**5.** Apply the above functional relations to set up the specific WEST model (Equations 2, 4, 5).

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments

The experimental data set has large data scatter at low velocities (Fig.1; Reches and Lockner, 2010). For the range of V < 0.03 m/s (Fig. 4), we found the following relations of µK and slip

The RMS (root mean square) of this solution is 0.83 while the correlation coefficient is 0.73.

**Figure 4.** The selected solution (red curve) for dynamic-weakening of kinetic friction coefficient of SWG for V = 0.0003-0.03m/s by Eureqa. Data for V < 0.003 m/s are from Reches and Lockner (2010) and Kuwano and Hatano

In the strengthening regime of 0.03 m/s < V < 1.0 m/s (Fig. 5), the selected solution for velocity-

<sup>V</sup> )for V>0.03*m* / *s* (7)

0.0275

µK(V)=0.824 exp (-

This simple solution provides reasonable fit to the scattered friction data (Fig. 4).

0.00183 <sup>+</sup> <sup>V</sup> , for V≤0.03 m / s (6)

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113

µK(V)=0.742 - 0.375V

Dynamic-weakening regime

velocity,

(2011).

Dynamic-strengthening regime

controlled friction coefficient is

where,

$$\begin{aligned} \mu &= \mu\_{\mathsf{S}} \text{ for } V = 0 \text{ and } D = 0 \text{ (Slipitiation)}, & \mathbf{a} \\ \text{and} \\ \mu &= \mu\_{\mathsf{K}} \text{ for steady} - \text{state velocity, } V, \text{ and } D > D\_{\mathsf{C}'} & \mathbf{b} \end{aligned} \tag{3}$$

where DC is the critical slip distance.

It is also assumed that **μK** is a function only of slip velocity V,

$$
\mathfrak{u}\_{\mathbf{K}} = \mathfrak{u}\_{\mathbf{K}}(V)\_{\prime} \tag{4}
$$

Similarly, we assume that during the strengthening stage, µ is a function of both slip-velocity and slip distance,

$$
\mu = \mu \Big(D, \, V \big), \text{ for } V \, > \, V\_{\mathbb{C}\_{\prime}} \tag{5}
$$

This approach for the empirical relations is in the spirit of the weakening relations developed by Dieterich (1979) who introduced the rate- and state- friction law.

#### **3.3. Model parameterization**

General

Solutions for the model functional relations were searched in the following steps.


**5.** Apply the above functional relations to set up the specific WEST model (Equations 2, 4, 5).

Dynamic-weakening regime

state coefficient after large slip distance under a constant velocity. The transition between the weakening regime and the strengthening regime occurs at the critical slip-velocity, **VC**, that is determined from the general plot of friction coefficient as function of slip-velocity (Fig. 1, 2, 3). We assume that during the weakening stage and under constant velocity, the drop of µ from

m

0 0 , a ( )

, , , b *<sup>C</sup>*

Similarly, we assume that during the strengthening stage, µ is a function of both slip-velocity

( ) , , , *D V for V VC*

This approach for the empirical relations is in the spirit of the weakening relations developed

**3.** Search for a suitable functional fit between slip velocities, friction coefficient, and slip distance. For this search, we chose the simplest functional relation that provided good fit.

Solutions for the model functional relations were searched in the following steps.

**μ μ S K,** (2)

= (*V* ), **μ μ K K** (4)

= > (5)

(3)

= (*D when* ), > >

*for V and D Slip initiation*

=- >

*for steady state velocity V and D D*

**μS** to **μK** is controlled only by the slip distance D,

112 Earthquake Research and Analysis - New Advances in Seismology

and

**S**

**μ**

**μ**

where DC is the critical slip distance.

**K**

m

m

and slip distance,

General

**3.3. Model parameterization**

where,

m m

===

It is also assumed that **μK** is a function only of slip velocity V,

m m

by Dieterich (1979) who introduced the rate- and state- friction law.

**1.** Prepare a table of the experimental velocity-friction relation; **2.** Select experiments with representative weakening stage;

The search revealed the relations between the **μK** and velocity. **4.** Integrate the above relations for the effect of distance and velocity. The experimental data set has large data scatter at low velocities (Fig.1; Reches and Lockner, 2010). For the range of V < 0.03 m/s (Fig. 4), we found the following relations of µK and slip velocity,

$$
\mu\_{\rm K} \text{(V)} = 0.742 \text{ - } \frac{0.375 \text{V}}{0.00183 \text{ + V}} \text{ } \text{ for V} \le 0.03 \text{ m/s} \tag{6}
$$

The RMS (root mean square) of this solution is 0.83 while the correlation coefficient is 0.73. This simple solution provides reasonable fit to the scattered friction data (Fig. 4).

**Figure 4.** The selected solution (red curve) for dynamic-weakening of kinetic friction coefficient of SWG for V = 0.0003-0.03m/s by Eureqa. Data for V < 0.003 m/s are from Reches and Lockner (2010) and Kuwano and Hatano (2011).

Dynamic-strengthening regime

In the strengthening regime of 0.03 m/s < V < 1.0 m/s (Fig. 5), the selected solution for velocitycontrolled friction coefficient is

$$
\mu\_{\rm K} \text{(V)} = 0.824 \exp\left(\cdot \frac{0.0275}{\nabla}\right) \text{for V} > 0.03 m \left/ s \right.\tag{7}
$$

The RMS of this solution is 0.91 and the correlation coefficient is 0.74. The trend of this exponential relation provide good fit for SWG (Reches and Lockner, 2010) in the strengthening trend (Fig. 5), and this trend generally fit the experimental results of Kuwano and Hatano (2011). Note: Friction during 0.305-0.4 m has been locally adjusted to comply with Fig. 4.

**Figure 6.** a) The best fit solution (red curve) for slip distance weakening in ten experiments. SWG 222, 223, 224 were run at velocity 0.024 m/s; SWG 226-232 were run at 0.072 m/s; σ<sup>N</sup> = 1.1 MPa. The solution is μ= 0.2663(±0.1) exp (-D)

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which combines Eq. (2, 4, 5). This simple exponential function characterized the weakening process with relating small deviation of the data. A summary of WEST equations is listed in

**Terms Expression Eq. Number**

0.0275

General function μ =μ(*D*, *V* ) (1) Slip-weakening μ(D, V)=0.2663 exp (-D) + μK(V) (9)

µ(D, V)=0.2663∙exp (-D) + µK(V) (9)

0.00183 <sup>+</sup> <sup>V</sup> , for V≤0.03 m / s (6)

<sup>V</sup> ), for V>0.03*m* / *s* (7)

+0.4061(±0.15). The constants can be adjusted for specific case

Dynamic-weakening <sup>μ</sup>K(V)=0.742 - 0.375V

Dynamic-strengthening μK(V)=0.824 exp (-

**Table 1.** A summary of WEST equations.

By substitute the 0.4061 (Eq. 8) by Eq. (7), Eq. (8) is generalized into,

Model synthesis

Table 1.

**Figure 5.** a) The selected solution (red curve) for dynamic-strengthening of SWG for V = 0.03-0.3m/s by Eureqa. Data for V > 0.003 m/s are from Reches and Lockner (2010) and Kuwano and Hatano (2011)

#### Slip-weakening relations

For the ten experimental results (Fig. 6), we selected the slip-weakening function as,

$$
\mu = \mu(\text{D}) = 0.2663 \left( \pm 0.1 \right) \exp \left( -\text{D} \right) + 0.4061 \left( \pm 0.15 \right) \tag{8}
$$

For D=0, which is slip initiation, the calculated µ = µS = 0.6724. For D > Dc (e.g., 3.0 m), the calculated µ= µK = 0.41.

**Figure 6.** a) The best fit solution (red curve) for slip distance weakening in ten experiments. SWG 222, 223, 224 were run at velocity 0.024 m/s; SWG 226-232 were run at 0.072 m/s; σ<sup>N</sup> = 1.1 MPa. The solution is μ= 0.2663(±0.1) exp (-D) +0.4061(±0.15). The constants can be adjusted for specific case

#### Model synthesis

The RMS of this solution is 0.91 and the correlation coefficient is 0.74. The trend of this exponential relation provide good fit for SWG (Reches and Lockner, 2010) in the strengthening trend (Fig. 5), and this trend generally fit the experimental results of Kuwano and Hatano (2011). Note: Friction during 0.305-0.4 m has been locally adjusted to comply with Fig. 4.

114 Earthquake Research and Analysis - New Advances in Seismology

**Figure 5.** a) The selected solution (red curve) for dynamic-strengthening of SWG for V = 0.03-0.3m/s by Eureqa. Data

For D=0, which is slip initiation, the calculated µ = µS = 0.6724. For D > Dc (e.g., 3.0 m), the

µ=µ(D)=0.2663 ( ± 0.1)exp (-D) + 0.4061( ± 0.15) (8)

For the ten experimental results (Fig. 6), we selected the slip-weakening function as,

for V > 0.003 m/s are from Reches and Lockner (2010) and Kuwano and Hatano (2011)

Slip-weakening relations

calculated µ= µK = 0.41.

By substitute the 0.4061 (Eq. 8) by Eq. (7), Eq. (8) is generalized into,

$$
\mu\text{(D\_\text{.} V)} = 0.2663 \bullet \exp\left(\text{-D}\right) + \mu\_\text{k}\text{(V)}\tag{9}
$$

which combines Eq. (2, 4, 5). This simple exponential function characterized the weakening process with relating small deviation of the data. A summary of WEST equations is listed in Table 1.


**Table 1.** A summary of WEST equations.

#### **4. Model application and analysis**

#### **4.1. General**

We now apply the WEST model to simulate the friction-velocity-distance evolution in a set of experiments with Sierra White granite. In these simulations, we use the experimental slip distance, D, and slip velocity, V, which were controlled by the operator, as input parameters in Equations (6, 7, 9) and Table 1. This substitution predicts the evolution of the friction coefficient during the experiments. The predicted friction coefficient is then plotted with the experimentally observed equivalent. The simulations were done on four types of velocity histories; none of the simulated experiment was used for the derivation of the model.

**4.3. Rise and drop**

We now examine two groups of experiments, each with five runs of the same loading conditions (σN = 5.0MPa). Each group has three velocity stages: low, high, and back to low. In the first group, the three steps were 0.00075, 0.0075, and 0.00075 m/s. The initial friction coefficient ranged 0.4-0.7 for the different runs in the group, and during the first step, the friction coeffi‐ cient was gradually decreased by ~0.1. During the second stage of V = 0.0075 m/s, which lasted 10 s, the friction coefficient reduced drastically, with larger reduction of the runs that started at higher friction coefficient. During the final, low velocity, the sample was slightly strengthened.

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In Fig. 8, this group of experiments was performed on the same sample under σN = 5.0MPa. The different initial friction coefficients (e.g., runs 1, 2 have higher friction coefficient than run 3, 4, 5) were attributed to the holding-times between experiments. The weakening was well noticed when slip velocity was raised and then the friction reached a relative steady value. The fit line shows the weakening immediately after the velocity has been raised. When the velocity

drops, the friction coefficient is raised higher in the fit line than the data.

**Figure 8.** WEST model simulation and experimental results for SWG\_603\_Run\_(1-5), SWG, σN = 5.0 MPa

In the second group (Fig. 9), the scenario was repeated but at velocities which were higher by an order of magnitude: 0.0075 m/s (first and last stages) and 0.075 m/s (second stage). This time, the friction coefficient in the second stage was increased from µ=~0.3 to the highest µ = 0.7. Then, µ gradually reduced to 0.3-0.4 during the final, low velocity stage. In Fig. 9, the friction coefficient of the fit results rises and drops earlier than the experimental data (running from 0.25 m to 0.75 m). The rest of the running, at the beginning low velocity and the final velocity, the friction coefficient between the model and experiments fit well each other.

#### **4.2. Rising velocity steps**

We start with three runs under same σΝ = 5.0MPa, and each with three upward stepping velocities, 0.0025, 0.025 and 0.047 m/s. During these runs the cumulative slip was about 9 m (Fig. 7). These velocities are in the weakening regime, and they display intense friction drops that reflect both the rising velocity and increase of slip distance. The friction coefficient was 0.7-0.9 at the first step of V = 0.0025 m/s, and it decreased to µ ≅ 0.5 in the second velocity step of V = 0.025 m/s. A minimum of friction coefficient of µ = 0.4 developed under V = 0.05 m/s. This experimental evolution fits well the predictions by WEST model.

**Figure 7.** WEST model simulation and experimental results for SWG experiments 244, 245, SWG, σN = 5.0 MPa

#### **4.3. Rise and drop**

**4. Model application and analysis**

116 Earthquake Research and Analysis - New Advances in Seismology

We now apply the WEST model to simulate the friction-velocity-distance evolution in a set of experiments with Sierra White granite. In these simulations, we use the experimental slip distance, D, and slip velocity, V, which were controlled by the operator, as input parameters in Equations (6, 7, 9) and Table 1. This substitution predicts the evolution of the friction coefficient during the experiments. The predicted friction coefficient is then plotted with the experimentally observed equivalent. The simulations were done on four types of velocity

histories; none of the simulated experiment was used for the derivation of the model.

**Figure 7.** WEST model simulation and experimental results for SWG experiments 244, 245, SWG, σN = 5.0 MPa

This experimental evolution fits well the predictions by WEST model.

We start with three runs under same σΝ = 5.0MPa, and each with three upward stepping velocities, 0.0025, 0.025 and 0.047 m/s. During these runs the cumulative slip was about 9 m (Fig. 7). These velocities are in the weakening regime, and they display intense friction drops that reflect both the rising velocity and increase of slip distance. The friction coefficient was 0.7-0.9 at the first step of V = 0.0025 m/s, and it decreased to µ ≅ 0.5 in the second velocity step of V = 0.025 m/s. A minimum of friction coefficient of µ = 0.4 developed under V = 0.05 m/s.

**4.1. General**

**4.2. Rising velocity steps**

We now examine two groups of experiments, each with five runs of the same loading conditions (σN = 5.0MPa). Each group has three velocity stages: low, high, and back to low. In the first group, the three steps were 0.00075, 0.0075, and 0.00075 m/s. The initial friction coefficient ranged 0.4-0.7 for the different runs in the group, and during the first step, the friction coeffi‐ cient was gradually decreased by ~0.1. During the second stage of V = 0.0075 m/s, which lasted 10 s, the friction coefficient reduced drastically, with larger reduction of the runs that started at higher friction coefficient. During the final, low velocity, the sample was slightly strengthened.

In Fig. 8, this group of experiments was performed on the same sample under σN = 5.0MPa. The different initial friction coefficients (e.g., runs 1, 2 have higher friction coefficient than run 3, 4, 5) were attributed to the holding-times between experiments. The weakening was well noticed when slip velocity was raised and then the friction reached a relative steady value. The fit line shows the weakening immediately after the velocity has been raised. When the velocity drops, the friction coefficient is raised higher in the fit line than the data.

**Figure 8.** WEST model simulation and experimental results for SWG\_603\_Run\_(1-5), SWG, σN = 5.0 MPa

In the second group (Fig. 9), the scenario was repeated but at velocities which were higher by an order of magnitude: 0.0075 m/s (first and last stages) and 0.075 m/s (second stage). This time, the friction coefficient in the second stage was increased from µ=~0.3 to the highest µ = 0.7. Then, µ gradually reduced to 0.3-0.4 during the final, low velocity stage. In Fig. 9, the friction coefficient of the fit results rises and drops earlier than the experimental data (running from 0.25 m to 0.75 m). The rest of the running, at the beginning low velocity and the final velocity, the friction coefficient between the model and experiments fit well each other. Comparing the two groups, the different friction responses provide clear evidence for dynamic-strengthening at slip velocity ~ 0.07 m/s.

**Figure 10.** WEST model simulation and experimental results for SWG 240, SWG, σN = 1.1 MPa

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments

http://dx.doi.org/10.5772/54889

119

**Figure 11.** WEST model simulation and experimental results for SWG 660, SWG, σ<sup>N</sup> = 0.5MPa. The rising of friction

coefficient indicated by blue arrow is NOT dynamic-strengthening

**Figure 9.** WEST model simulation and experimental results for SWG\_614\_Run\_(1-5), SWG, σN = 5.0 MPa

The WEST simulations were also performed to investigate if the model can capture the dynamic-strengthening at the high velocity regime. Fig. 10 is an experiment with five alter‐ nating steps of velocity that again indicates the dynamic-strengthening at velocity 0.075 m/s (above the critical value VC). The friction coefficient dropped from 0.7 to 0.5 at velocity of 0.0025 m/s and 0.025 m/s, then dynamically-strengthened to µ = 0.7 at V = 0.75 m/s, and finally slightly dropped to 0.55 with a slip distance of ~4.2 m (from 1.7 m to ~5.9 m) as the velocity decreased. Similarly to the case of rise-drop experiment (Fig. 8), the observed friction coefficient changed only slightly in the end at dropping velocity (from 5.9 m to 7.6 m). The simulation illustrates the dynamic-strengthening during the three-step sliding.

#### **4.4. Drop and rise during long distance slip experiments**

In Fig. 11, the simulation presents the experiment behavior that involved three steps: an ini‐ tial weakening at the higher slip velocity of 0.045 m/s, regaining strengthen at a lower speed of 0.0018 m/s, and a final weakening at high-speed step of 0.045 m/s after healing for about 2,000 seconds. The healing refers to the regaining of strength of the fault during hold time. During the simulation of the second step, the friction coefficient was manually shifted to static value due to healing (indicated by blue arrow, Fig. 11). The increase of friction here is related to healing during hold time, which differs from dynamic-strengthening. The simula‐ tion results show smooth lines instead of noisy curves of the experiment.

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments http://dx.doi.org/10.5772/54889 119

**Figure 10.** WEST model simulation and experimental results for SWG 240, SWG, σN = 1.1 MPa

Comparing the two groups, the different friction responses provide clear evidence for

**Figure 9.** WEST model simulation and experimental results for SWG\_614\_Run\_(1-5), SWG, σN = 5.0 MPa

the dynamic-strengthening during the three-step sliding.

**4.4. Drop and rise during long distance slip experiments**

tion results show smooth lines instead of noisy curves of the experiment.

The WEST simulations were also performed to investigate if the model can capture the dynamic-strengthening at the high velocity regime. Fig. 10 is an experiment with five alter‐ nating steps of velocity that again indicates the dynamic-strengthening at velocity 0.075 m/s (above the critical value VC). The friction coefficient dropped from 0.7 to 0.5 at velocity of 0.0025 m/s and 0.025 m/s, then dynamically-strengthened to µ = 0.7 at V = 0.75 m/s, and finally slightly dropped to 0.55 with a slip distance of ~4.2 m (from 1.7 m to ~5.9 m) as the velocity decreased. Similarly to the case of rise-drop experiment (Fig. 8), the observed friction coefficient changed only slightly in the end at dropping velocity (from 5.9 m to 7.6 m). The simulation illustrates

In Fig. 11, the simulation presents the experiment behavior that involved three steps: an ini‐ tial weakening at the higher slip velocity of 0.045 m/s, regaining strengthen at a lower speed of 0.0018 m/s, and a final weakening at high-speed step of 0.045 m/s after healing for about 2,000 seconds. The healing refers to the regaining of strength of the fault during hold time. During the simulation of the second step, the friction coefficient was manually shifted to static value due to healing (indicated by blue arrow, Fig. 11). The increase of friction here is related to healing during hold time, which differs from dynamic-strengthening. The simula‐

dynamic-strengthening at slip velocity ~ 0.07 m/s.

118 Earthquake Research and Analysis - New Advances in Seismology

**Figure 11.** WEST model simulation and experimental results for SWG 660, SWG, σ<sup>N</sup> = 0.5MPa. The rising of friction coefficient indicated by blue arrow is NOT dynamic-strengthening

#### **4.5. Wide velocity range**

Figs. 12 and 13 display the friction evolution in two experiments with wide velocity range. Fig. 12 displays a slide-hold-slide run under σΝ = 1.1 MPa and hold times of 10 s between the multiple velocity steps. Under this short hold time, the friction coefficient curve displays quasicontinuous trend. There is a marked weakening as the velocity was increased to ~ 0.04 m/s, followed by a gentle strengthening at higher velocities. The modeling results were lower at beginning than expected and strengthening occurred earlier. Fig. 13 displays two major features: an initial gradual weakening in the slip velocity range of ~0.0003 m/s to a critical velocity of ~0.03 m/s, and a fast strengthening at velocities from ~0.03 m/s to 0.2 m/s. In the final stage, the friction reaches ~0.8. The modeling simulates the weakening-strengthening pattern, but it fails to follow the experimental results in the region faster than the critical velocity. In the experiment, the friction coefficient remained relatively low from ~3 m to ~11 m, where the simulated results predicted earlier strengthening. Also, an abrupt friction rise was observed in the experiment (at ~ 11 m) whereas the model indicated smooth and contin‐ uous strengthening.

**Figure 13.** WEST model simulation and experimental results for SWG 616, SWG, σ<sup>N</sup> = 5.0 MPa. Friction coefficient is

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments

http://dx.doi.org/10.5772/54889

121

The WEST model was developed for the experimental results of Sierra White granite for which we have multiple runs that well bound the solutions and simulations. The WEST relations (Eqn. 6, 7, 9) represent the empirical frictional history of SWG in the slip-veloci‐ ty range of 10-4-1m/s. The application of the model to the experimental results shows that friction history can be captured by a numerical combination of the slip velocity and slip distance for both the weakening and strengthening stages. Specifically, in the dy‐ namic-strengthening regime has the following properties: 1) It is a high-velocity regime, V > VC, e.g. V ~ 0.03-0.6 m/s for Sierra White granite; 2) The WEST model uses a kine‐ matic friction coefficient for the strengthening stage, which is derived independently of the weakening stage; 3) The friction coefficient history is well simulated by combining

The WEST model predicts the rate-dependence of friction over the full range of observed natural seismic slip-rates, and in this sense it is applicable to earthquake simulations, and modeling of earthquake ruptures. Accumulating evidence supports the presence of dynamicstrengthening. Kaneko et al (2008) stated that "the velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could

shown for a full-velocity continuously in sliding distance together with a best fit modeling result

the kinematic friction coefficient for strengthening and the slip distance.

**5. Discussion**

**5.1. Field applications**

**Figure 12.** WEST model simulation and experimental results for SWG 531, SWG, σN = 1.1MPa. Friction coefficient is shown for a full-velocity by interval holding in sliding distance together with a best fit modeling result

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments http://dx.doi.org/10.5772/54889 121

**Figure 13.** WEST model simulation and experimental results for SWG 616, SWG, σ<sup>N</sup> = 5.0 MPa. Friction coefficient is shown for a full-velocity continuously in sliding distance together with a best fit modeling result

#### **5. Discussion**

**4.5. Wide velocity range**

120 Earthquake Research and Analysis - New Advances in Seismology

uous strengthening.

Figs. 12 and 13 display the friction evolution in two experiments with wide velocity range. Fig. 12 displays a slide-hold-slide run under σΝ = 1.1 MPa and hold times of 10 s between the multiple velocity steps. Under this short hold time, the friction coefficient curve displays quasicontinuous trend. There is a marked weakening as the velocity was increased to ~ 0.04 m/s, followed by a gentle strengthening at higher velocities. The modeling results were lower at beginning than expected and strengthening occurred earlier. Fig. 13 displays two major features: an initial gradual weakening in the slip velocity range of ~0.0003 m/s to a critical velocity of ~0.03 m/s, and a fast strengthening at velocities from ~0.03 m/s to 0.2 m/s. In the final stage, the friction reaches ~0.8. The modeling simulates the weakening-strengthening pattern, but it fails to follow the experimental results in the region faster than the critical velocity. In the experiment, the friction coefficient remained relatively low from ~3 m to ~11 m, where the simulated results predicted earlier strengthening. Also, an abrupt friction rise was observed in the experiment (at ~ 11 m) whereas the model indicated smooth and contin‐

**Figure 12.** WEST model simulation and experimental results for SWG 531, SWG, σN = 1.1MPa. Friction coefficient is

shown for a full-velocity by interval holding in sliding distance together with a best fit modeling result

#### **5.1. Field applications**

The WEST model was developed for the experimental results of Sierra White granite for which we have multiple runs that well bound the solutions and simulations. The WEST relations (Eqn. 6, 7, 9) represent the empirical frictional history of SWG in the slip-veloci‐ ty range of 10-4-1m/s. The application of the model to the experimental results shows that friction history can be captured by a numerical combination of the slip velocity and slip distance for both the weakening and strengthening stages. Specifically, in the dy‐ namic-strengthening regime has the following properties: 1) It is a high-velocity regime, V > VC, e.g. V ~ 0.03-0.6 m/s for Sierra White granite; 2) The WEST model uses a kine‐ matic friction coefficient for the strengthening stage, which is derived independently of the weakening stage; 3) The friction coefficient history is well simulated by combining the kinematic friction coefficient for strengthening and the slip distance.

The WEST model predicts the rate-dependence of friction over the full range of observed natural seismic slip-rates, and in this sense it is applicable to earthquake simulations, and modeling of earthquake ruptures. Accumulating evidence supports the presence of dynamicstrengthening. Kaneko et al (2008) stated that "the velocity-strengthening region suppresses supershear propagation at the free surface occurring in the absence of such region, which could explain the lack of universally observed supershear rupture near the free surface". Hence, it is important to understand how the dynamic friction at seismic rates affects earthquake rupture dynamics by adopting new friction model in the numerical simulations. Although this is beyond the scope of the present study, the abundance of efforts on experimental studies on dynamic friction provides a realistic velocity-friction relation to explore the ground motions during earthquake rupture.

#### **6. Summary**

The analysis examines the frictional strength of igneous rocks including granite, diorite, syenite, gabbro, and quartzite under slip-velocity approaching seismic-rates. The experimental observations confirm that increasing slip velocity leads to dynamic-weakening followed by **dynamic-strengthening** only in igneous rock samples that contain quartz (Figs. 1, 2). The weakening-strengthening transition occurs at a critical velocity, VC, which depends on the fault lithology (Figs. 1, 2).

**Figure 14.** The Rotary Shear Apparatus with builder Joel Young. The sample block assembled in the loading frame

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments

http://dx.doi.org/10.5772/54889

123

Discussions and suggestions by David Lockner and Yuval Boneh contributed to the analysis. Usage of the 'Eureqa Formulize' program helped in the model computation. Partial funding support was provided by grants 1045414 of NSF Geosciences, Geophysics, G11AP20008 of

[1] Beeler, N. M., T. E. Tullis, and J. D. Weeks (1994), The roles of time and displacement in the evolution effect in rock friction, *Geophys. Res. Lett.*, 21(18), 1987-1990.

[2] Boneh, Y. (2012), Wear and gouge along faults: experimental and mechanical analysis, M.S. thesis, Dep. of Geol. and Geophys., Univ. of Oklahoma, Norman, OK, USA.

School of Geology and Geophysics, University of Oklahoma, Norman, USA

(after Reches and Lockner (2010)

**Acknowledgements**

**Author details**

**References**

Zonghu Liao and Ze'ev Reches

DOI-USG-NEHRP2011, and SCEC 2012.

The present study provides a numerical framework for the experimental observations of dynamic-strengthening at high velocity along rock faults. The model is empirical in nature, and it succeeds to simulate the friction coefficient history in a wide range of experimental loading. We envision that it is a promising tool to analyze rock friction during earthquakes. In future research, several aspects of the sliding mechanism should be linked and interpreted in terms of the WEST model: 1) Powder lubrication by Reches and Lockner (2010) showed that the newly formed gouge organizes itself into a thin deforming layer that changes the fault strength; 2) Fault-strengthening during hold time; and 3) Predicting the effect of lithological compositions on the frictional resistance (Di Toro et al., 2004).

#### **Appendix A: Experiment setup**

The analyzed experiments were conducted on the ROGA in University of Oklahoma. The following description of the apparatus is taken from Reches and Lockner (2010) and related lab proposals. The ROGA system (Rotary Gouge Apparatus, Fig. 14) satisfies the following conditions: (1) normal stress of tens to hundreds of MPa; (2) slip velocity of ~1 m/s; (3) risetime of less than 1 s; and (4) unlimited slip distances.

In the experiments, the fault is composed of solid blocks of Sierra White granite. Each sample includes two cylindrical blocks, diameter = 101.6 mm, height = 50.8 mm. For ~uniform velocity, the upper block has a raised ring with ID = 63.2 mm and OD = 82.3 mm; the blocks are pressed across this raised ring. Thermocouples are cemented into holes drilled 3 mm and 6 mm away from the sliding surfaces (Fig. 14). Each pair of blocks wore to form gouge in between at different slip velocities of 0.001-1 m/s. A large set of experiments will be executed and a large quantity of data will be collected by LabView at frequency of ~100-1000 Hz.

**Figure 14.** The Rotary Shear Apparatus with builder Joel Young. The sample block assembled in the loading frame (after Reches and Lockner (2010)

#### **Acknowledgements**

explain the lack of universally observed supershear rupture near the free surface". Hence, it is important to understand how the dynamic friction at seismic rates affects earthquake rupture dynamics by adopting new friction model in the numerical simulations. Although this is beyond the scope of the present study, the abundance of efforts on experimental studies on dynamic friction provides a realistic velocity-friction relation to explore the ground motions

The analysis examines the frictional strength of igneous rocks including granite, diorite, syenite, gabbro, and quartzite under slip-velocity approaching seismic-rates. The experimental observations confirm that increasing slip velocity leads to dynamic-weakening followed by **dynamic-strengthening** only in igneous rock samples that contain quartz (Figs. 1, 2). The weakening-strengthening transition occurs at a critical velocity, VC, which depends on the fault

The present study provides a numerical framework for the experimental observations of dynamic-strengthening at high velocity along rock faults. The model is empirical in nature, and it succeeds to simulate the friction coefficient history in a wide range of experimental loading. We envision that it is a promising tool to analyze rock friction during earthquakes. In future research, several aspects of the sliding mechanism should be linked and interpreted in terms of the WEST model: 1) Powder lubrication by Reches and Lockner (2010) showed that the newly formed gouge organizes itself into a thin deforming layer that changes the fault strength; 2) Fault-strengthening during hold time; and 3) Predicting the effect of lithological

The analyzed experiments were conducted on the ROGA in University of Oklahoma. The following description of the apparatus is taken from Reches and Lockner (2010) and related lab proposals. The ROGA system (Rotary Gouge Apparatus, Fig. 14) satisfies the following conditions: (1) normal stress of tens to hundreds of MPa; (2) slip velocity of ~1 m/s; (3) rise-

In the experiments, the fault is composed of solid blocks of Sierra White granite. Each sample includes two cylindrical blocks, diameter = 101.6 mm, height = 50.8 mm. For ~uniform velocity, the upper block has a raised ring with ID = 63.2 mm and OD = 82.3 mm; the blocks are pressed across this raised ring. Thermocouples are cemented into holes drilled 3 mm and 6 mm away from the sliding surfaces (Fig. 14). Each pair of blocks wore to form gouge in between at different slip velocities of 0.001-1 m/s. A large set of experiments will be executed and a large

quantity of data will be collected by LabView at frequency of ~100-1000 Hz.

compositions on the frictional resistance (Di Toro et al., 2004).

time of less than 1 s; and (4) unlimited slip distances.

**Appendix A: Experiment setup**

during earthquake rupture.

122 Earthquake Research and Analysis - New Advances in Seismology

**6. Summary**

lithology (Figs. 1, 2).

Discussions and suggestions by David Lockner and Yuval Boneh contributed to the analysis. Usage of the 'Eureqa Formulize' program helped in the model computation. Partial funding support was provided by grants 1045414 of NSF Geosciences, Geophysics, G11AP20008 of DOI-USG-NEHRP2011, and SCEC 2012.

#### **Author details**

Zonghu Liao and Ze'ev Reches

School of Geology and Geophysics, University of Oklahoma, Norman, USA

#### **References**


[3] Chang, J. C., Lockner, D. A., and Z. Reches (2012), Rapid acceleration leads to rapid weakening in earthquake-like laboratory experiments, *Science*, 338(6103), 101-105.

[18] Samuelson, J., D. Elsworth, and C. Marone (2009), Shear-induced dilatancy of fluidsaturated faults: experiment and theory, *J. Geophys. Res.*, 114, B12404, doi:

Modeling Dynamic-Weakening and Dynamic-Strengthening of Granite in High-Velocity Slip Experiments

http://dx.doi.org/10.5772/54889

125

[19] Schmidt, M., and H. Lipson (2009), Distilling free-form natural laws from experimental

[20] Shimamoto, T., and J. M. Logan (1984), Laboratory friction experiments and natural earthquakes: An argument for long teren tests, *Technophysics*, 109, 165-175.

[21] Tsutsumi, A., and T. Shimamoto (1997), High-velocity frictional properties of gabbro,

10.1029/2008JB006273.

data, *Science*, 324, 81-85.

*Geophys. Res. Lett.*, 24(6), 699-702.


[18] Samuelson, J., D. Elsworth, and C. Marone (2009), Shear-induced dilatancy of fluidsaturated faults: experiment and theory, *J. Geophys. Res.*, 114, B12404, doi: 10.1029/2008JB006273.

[3] Chang, J. C., Lockner, D. A., and Z. Reches (2012), Rapid acceleration leads to rapid weakening in earthquake-like laboratory experiments, *Science*, 338(6103), 101-105. [4] Di Toro, G., D. L. Goldsby DL, and T. E. Tullis (2004), Friction falls towards zero in

[5] Di Toro, G., R. Han, T. Hirose, N. De Paola, S. Nielsen, K. Mizoguchi, F. Ferri, M. Cocco, and T. Shimamoto (2011), Fault lubrication during earthquakes, *Nature*, 471, 494-498.

[6] Dieterich, J. H. (1979), Modeling of rock friction: 1. Experimental results and constitu‐

[7] Goldsby, D. L., and T. E. Tullis (2003), Flash heating/melting phenomena for crustal rocks at (nearly) seismic slip rates, SCEC Annual Meeting Proceedings and Abstracts,

[8] Kanamori, H., and E. E. Brodsky (2004), The physics of earthquakes, Reports on *Progress*

[9] Kaneko, Y., N. Lapusta, and J.-P. Ampuero (2008), Spectral element modeling of spontaneous earthquake rupture on rate and state faults: Effect of velocity-strength‐ ening friction at shallow depths, *Journal of Geophysical Research*, 113, B09317, doi:

[10] Kuwano, O., and T. Hatano (2011), Flash weakening is limited by grannular dynamics,

[11] Liao, Z. (2011), Dynamic strengthening at high-velocity shear experiments, MS thesis,

[12] Liao, Z., and Z. Reches (2012), Experiment-based model for granite dynamic strength in slip-velocity range of 0.001-1.0 m/s, AGU Annual Meeting, San Francisco, 2012,

[13] Lockner, D. A., and N. M. Beeler (2003), Stress-induced anisotropic poroelasticity response in sandstone, Electronic Proc. 16th ASCE Engin. Mech. Conf., Univ. of

[14] Marone, C. (1998), Laboratory-derived friction laws and their application to seismic

[15] Ohnaka, M., and T. Yamashita (1989) A cohesive zone model for dynamic shear faulting based on experimentally inferred constitutive relation and strong motion source

[16] Reches, Z., and D. A. Lockner (2010), Fault weakening and earthquake instability by

[17] Sammis, C. G., D. A. Lockner, and Z. Reches (2011), The role of adsorbed water on the friction of a layer of submicron particles, *Pure and Applied Geophysics*, doi: 10.1007/

powder lubrication, *Nature*, 467(7314), 452-455, doi:10.1038/nature09348.

quartz rock as slip velocity approaches seismic rates, *Nature*, 427, 436-439.

tive equations*, J. Geophys. Res*., 84(5), 2161-2168.

*in Physics*, 67, 1429 – 1496, doi: 10./1088/0034-4885/67/8/R03.

*Geophys. Res. Lett.*, 38, L17305, doi:10.1029/2011GL048530.

University of Oklahoma, Norman, 54 pp.

faulting, *Annu. Rev. Earth Planet Sci.,* 26, 643-696.

parameters, *Journal of Geophysical Research*, 94 (B4), 4089-4104.

Palm Springs, California.

124 Earthquake Research and Analysis - New Advances in Seismology

10.1029/2007JB005553.

T13E-2656, ID: 1488537.

Washington, Seattle, WA.

s00024-01-0324-0.


**Chapter 6**

**Characterizing the Noise for Seismic Arrays:**

Additional information is available at the end of the chapter

Sebastiano D'Amico

**1. Introduction**

http://dx.doi.org/10.5772/54387

**Case of Study for the Alice Springs ARray (ASAR)**

A seismic array is defined as a suite of seismometers with similar characteristics. Seismic ar‐ ray were originally built to detect and identify nuclear explosions. Since their development all over the world, seismic arrays have contributed to study interior of volcanoes, continen‐ tal crust and lithosphere, determination of core-mantle boundary and the structure of inner core. Seismic arrays have been used to perform many regional tomographic studies (e.g., Achauer and the KRISP Working Group, 1994; Ritter et al., 1998, 2001); they helped to re‐ solve fine-scale structure well below the resolution level of global seismology in many dif‐ ferent places in the Earth, from the crust using body waves (e.g., Rothert and Ritter, 2001) and surface waves (e.g., Pavlis and Mahdi, 1996; Cotte et al., 2000), the upper mantle (e.g., Rost and Weber, 2001), the lower mantle (e.g., Castle and Creager, 1999), the core-mantle boundary (e.g., Thomas et al., 1999; Rost and Revenaugh, 2001), and the inner core (e.g., Vi‐ dale et al., 2000; Vidale and Earle, 2000; Helffrich et al., 2002). A different branch of seismol‐ ogy that benefited from arrays is "forensic seismology" (Koper et al., 1999; 2001; Koper and Wallace 2003). Studied have been also carried out to track the rapture propagation of large and moderate earthquakes (Goldstein and Archuleta 1991a,b: Spudich and Cranswick 1984; Huang 2001; D'Amico et al. 2010; Sufri et al. 2012; Koper et al. 2012), studies related to the seismic noise have been also developed using seismic arrays (Koper and Fathei, 2007; Ger‐ stoft et al. 2006; D'Amico et al. 2008; Schulte-Pelkum et al., 2004). For example Gerstoft et al. (2006) used beamforming of seismic noise recorded on California Seismic Network to identi‐ fy body and surface waves generated by the Hurricane Katrina. Schulte-Pelkum et al., (2004) measured direction and amplitude of ocean-generated seismic noise in the western United States. Koper and Fatehi (2007) used 950, randomly chose, 4-sec long time windows from 1996 to 2004 at the CMAR array located in Thailand. In their work they found, around 1Hz,

> © 2013 D'Amico; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 D'Amico; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

### **Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR)**

Sebastiano D'Amico

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54387

#### **1. Introduction**

A seismic array is defined as a suite of seismometers with similar characteristics. Seismic ar‐ ray were originally built to detect and identify nuclear explosions. Since their development all over the world, seismic arrays have contributed to study interior of volcanoes, continen‐ tal crust and lithosphere, determination of core-mantle boundary and the structure of inner core. Seismic arrays have been used to perform many regional tomographic studies (e.g., Achauer and the KRISP Working Group, 1994; Ritter et al., 1998, 2001); they helped to re‐ solve fine-scale structure well below the resolution level of global seismology in many dif‐ ferent places in the Earth, from the crust using body waves (e.g., Rothert and Ritter, 2001) and surface waves (e.g., Pavlis and Mahdi, 1996; Cotte et al., 2000), the upper mantle (e.g., Rost and Weber, 2001), the lower mantle (e.g., Castle and Creager, 1999), the core-mantle boundary (e.g., Thomas et al., 1999; Rost and Revenaugh, 2001), and the inner core (e.g., Vi‐ dale et al., 2000; Vidale and Earle, 2000; Helffrich et al., 2002). A different branch of seismol‐ ogy that benefited from arrays is "forensic seismology" (Koper et al., 1999; 2001; Koper and Wallace 2003). Studied have been also carried out to track the rapture propagation of large and moderate earthquakes (Goldstein and Archuleta 1991a,b: Spudich and Cranswick 1984; Huang 2001; D'Amico et al. 2010; Sufri et al. 2012; Koper et al. 2012), studies related to the seismic noise have been also developed using seismic arrays (Koper and Fathei, 2007; Ger‐ stoft et al. 2006; D'Amico et al. 2008; Schulte-Pelkum et al., 2004). For example Gerstoft et al. (2006) used beamforming of seismic noise recorded on California Seismic Network to identi‐ fy body and surface waves generated by the Hurricane Katrina. Schulte-Pelkum et al., (2004) measured direction and amplitude of ocean-generated seismic noise in the western United States. Koper and Fatehi (2007) used 950, randomly chose, 4-sec long time windows from 1996 to 2004 at the CMAR array located in Thailand. In their work they found, around 1Hz,

© 2013 D'Amico; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 D'Amico; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

a large noise peak coming from southwest near 220 degrees and an apparent velocity of 3.5-4.0km/s. Their results are robust from year-to-year and are also consistent from season to season. Two lesser noise peaks show probably a seasonal dependence, being much stronger in the fall and winter than in the summer and spring. Neither peak is sensitive to the "hourto-hour" analysis meaning they are uncorrelated to anthropical noise. Koper and De Foy (2008) showed that the seismic noise recorded at the CMAR array during 1995-2004 can be strongly correlated with the ocean wave's heights. They carried out this information by us‐ ing data from TOPEX/POSEIDON satellite tracks and explained them by the local monsoondriven climate. For all this different purpose a lot of different arrays techniques and methods have been developed (for reviews see: Rost and Thomas, 2002; Filson, 1975) and applied to a wide number of high-quality data set.

of approach is also known in literature as frequency-wavenumber (f-k) analysis; it offers the opportunity to determine the back-azimuth and the slowness of coherent seismic waves with a high resolution. Furthermore, it has the possibility to detect and discriminate simulta‐ neously several microseismic sources. Each trace was examined to eliminate those with spu‐ rious transients or glitches, null traces and those contain obvious earthquake energy. After this selection the original dataset was reduced about of the 5%; each time window is 5 mi‐ nutes long. Figure 3 shows a schematic diagram of the method applied in this study. The analyses are performed at different frequency bands (around 0.4Hz, 0.6Hz, 0.8 Hz, 1.0Hz,

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129

**Figure 1.** Alice Springs Array (ASAR) location and array geometry. The white triangle in the top panel represents the

reference element, while the dark gray triangles are the other 18 elements of the array

2.0Hz and 4.0Hz).

The main goal of this chapter is to highlight the main characteristics of noise for the Alice Springs ARray (Australia). Furthermore detecting the noise we would like, if it exists, try to found the large peak noise, the predominant direction and estimate the optimal phase veloc‐ ity and eventual time dependence. This kind of study could play a key role in for the isola‐ tion of the seismic noise in designing new arrays or particular instruments such as the construction of gravitational wave detectors (Hoffmann et al.,2002 and reference in therein). Theoretically knowing the seismic noise features and source it will be possible to subtract its effect from the data.

#### **2. Data set and processing**

Alice Springs Array is located in Australia and it is made by 19 vertical component short period seismographs deployed with an effective aperture of about 10 km (Fig.1). We ignored elevation differences among the array elements and considered only 2D wavenumber vec‐ tors. This is a reasonable since the ASAR array is relatively flat.

Continuous data were available from 1994 to 2004 and it was possible to get them by using the "autodrm request" of the U.S. Army Space and missile Defense Command monitoring research program (www.rdss.info, last access in 2009). It supports different researches relat‐ ed to the nuclear explosions monitoring. Time series containing randomly noise recorded for each station in all the time period where recordings are available. In the present paper data from 1999 to 2001 are used. We extracted several minutes of continuous data once a week for the selected time period, making sure to vary the time of day and the day of the week (Fig. 2). We used the Generic Array Processing software (GAP; Koper 2005), a set of freely C programs for processing seismic array data. These programs operate on binary SAC files and output GMT (Wessel and Smith, 1991) scripts for visualizing the results; they were developed to work both with small aperture array and other type of array as well. In present paper we used in particular the program called "capon.c" that performs the signal process‐ ing following the maximum likelihood Capon (1969) method; the idea is to use a spectral density function that provides the information concerning the power as a function of fre‐ quency, this function also provides the vector velocities of the propagating waves. This kind of approach is also known in literature as frequency-wavenumber (f-k) analysis; it offers the opportunity to determine the back-azimuth and the slowness of coherent seismic waves with a high resolution. Furthermore, it has the possibility to detect and discriminate simulta‐ neously several microseismic sources. Each trace was examined to eliminate those with spu‐ rious transients or glitches, null traces and those contain obvious earthquake energy. After this selection the original dataset was reduced about of the 5%; each time window is 5 mi‐ nutes long. Figure 3 shows a schematic diagram of the method applied in this study. The analyses are performed at different frequency bands (around 0.4Hz, 0.6Hz, 0.8 Hz, 1.0Hz, 2.0Hz and 4.0Hz).

a large noise peak coming from southwest near 220 degrees and an apparent velocity of 3.5-4.0km/s. Their results are robust from year-to-year and are also consistent from season to season. Two lesser noise peaks show probably a seasonal dependence, being much stronger in the fall and winter than in the summer and spring. Neither peak is sensitive to the "hourto-hour" analysis meaning they are uncorrelated to anthropical noise. Koper and De Foy (2008) showed that the seismic noise recorded at the CMAR array during 1995-2004 can be strongly correlated with the ocean wave's heights. They carried out this information by us‐ ing data from TOPEX/POSEIDON satellite tracks and explained them by the local monsoondriven climate. For all this different purpose a lot of different arrays techniques and methods have been developed (for reviews see: Rost and Thomas, 2002; Filson, 1975) and

The main goal of this chapter is to highlight the main characteristics of noise for the Alice Springs ARray (Australia). Furthermore detecting the noise we would like, if it exists, try to found the large peak noise, the predominant direction and estimate the optimal phase veloc‐ ity and eventual time dependence. This kind of study could play a key role in for the isola‐ tion of the seismic noise in designing new arrays or particular instruments such as the construction of gravitational wave detectors (Hoffmann et al.,2002 and reference in therein). Theoretically knowing the seismic noise features and source it will be possible to subtract its

Alice Springs Array is located in Australia and it is made by 19 vertical component short period seismographs deployed with an effective aperture of about 10 km (Fig.1). We ignored elevation differences among the array elements and considered only 2D wavenumber vec‐

Continuous data were available from 1994 to 2004 and it was possible to get them by using the "autodrm request" of the U.S. Army Space and missile Defense Command monitoring research program (www.rdss.info, last access in 2009). It supports different researches relat‐ ed to the nuclear explosions monitoring. Time series containing randomly noise recorded for each station in all the time period where recordings are available. In the present paper data from 1999 to 2001 are used. We extracted several minutes of continuous data once a week for the selected time period, making sure to vary the time of day and the day of the week (Fig. 2). We used the Generic Array Processing software (GAP; Koper 2005), a set of freely C programs for processing seismic array data. These programs operate on binary SAC files and output GMT (Wessel and Smith, 1991) scripts for visualizing the results; they were developed to work both with small aperture array and other type of array as well. In present paper we used in particular the program called "capon.c" that performs the signal process‐ ing following the maximum likelihood Capon (1969) method; the idea is to use a spectral density function that provides the information concerning the power as a function of fre‐ quency, this function also provides the vector velocities of the propagating waves. This kind

tors. This is a reasonable since the ASAR array is relatively flat.

applied to a wide number of high-quality data set.

128 Earthquake Research and Analysis - New Advances in Seismology

effect from the data.

**2. Data set and processing**

**Figure 1.** Alice Springs Array (ASAR) location and array geometry. The white triangle in the top panel represents the reference element, while the dark gray triangles are the other 18 elements of the array

**Figure 3.** Example of used time windows and schematic representation of the procedure applied in the present study

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131

Figures 4 shows the number of recording as a function of the optimal ray parameter for the frequencies of 0.4, 0.6, 0.8, and 1.0 Hz. Figures 5, 6, 7, 8, 9, 11, and 12 show the time-averaged ambient noise field at ASAR array averaged per each year and the average for the three-year period and the three-year period respectively, binned according to month at the frequency of 0.4Hz, 0.6Hz, 0.8 Hz, 1.0Hz. Figure 13 reports the results obtained for 2.0 and 4.0Hz re‐ spectively and showing the average per each year and the average for the three-year period. Figure 14 plots the local maxima (from 0.4 to 1.0 Hz) computed using the Capon (1969) method; red dots represent all the local maxima while blue are the maxima having a relative power greater than 5db. We observed, for the frequencies of 0.4, 0.6, 0.8 and 1.0 Hz, the most prominent pick coming from the S-W direction with an optimal backazimuth around 190-200 degrees and an apparent velocity of about 3-4km/s indicative of higher mode Ray‐ leigh waves. This energy is probably generated as waves from the interaction of oceanic waves with the coast in the Australian Bight. Because of the high attenuation of short period Rayleigh waves, it is really unlikely that the noise is generated further away from the ASAR array. It is also possible to highlight a possible correlation between noise peaks and the dis‐ tance of the array to the coast line. In fact, according the plot in figure 14 for each different frequency it is possible to notice that the largest number of peak having a relative power greater than 5db is coming from the S-W direction; the second large number of peak is com‐ ing from the N-E direction and a very few are coming from the S-E part that is the largest distance from the coast. An other important noise peak shown in figure 14 occurs in the cen‐ ter of the plot, indicating that energy is coming almost with a vertical incidence on the array. There is not any peak for the high-frequencies (f=2.0; 4.0 Hz); that is probably due to the lo‐ cation of the array. Furthermore we can also point our attention on the amplitude as a func‐

**3. Results and discussion**

**Figure 2.** Characteristics of our data set of seismic noise recorded by ASAR. (a) number of recording as function of year, month (b) and hours (UTC) (c)

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR) http://dx.doi.org/10.5772/54387 131

**Figure 3.** Example of used time windows and schematic representation of the procedure applied in the present study

#### **3. Results and discussion**

**Figure 2.** Characteristics of our data set of seismic noise recorded by ASAR. (a) number of recording as function of

year, month (b) and hours (UTC) (c)

130 Earthquake Research and Analysis - New Advances in Seismology

Figures 4 shows the number of recording as a function of the optimal ray parameter for the frequencies of 0.4, 0.6, 0.8, and 1.0 Hz. Figures 5, 6, 7, 8, 9, 11, and 12 show the time-averaged ambient noise field at ASAR array averaged per each year and the average for the three-year period and the three-year period respectively, binned according to month at the frequency of 0.4Hz, 0.6Hz, 0.8 Hz, 1.0Hz. Figure 13 reports the results obtained for 2.0 and 4.0Hz re‐ spectively and showing the average per each year and the average for the three-year period. Figure 14 plots the local maxima (from 0.4 to 1.0 Hz) computed using the Capon (1969) method; red dots represent all the local maxima while blue are the maxima having a relative power greater than 5db. We observed, for the frequencies of 0.4, 0.6, 0.8 and 1.0 Hz, the most prominent pick coming from the S-W direction with an optimal backazimuth around 190-200 degrees and an apparent velocity of about 3-4km/s indicative of higher mode Ray‐ leigh waves. This energy is probably generated as waves from the interaction of oceanic waves with the coast in the Australian Bight. Because of the high attenuation of short period Rayleigh waves, it is really unlikely that the noise is generated further away from the ASAR array. It is also possible to highlight a possible correlation between noise peaks and the dis‐ tance of the array to the coast line. In fact, according the plot in figure 14 for each different frequency it is possible to notice that the largest number of peak having a relative power greater than 5db is coming from the S-W direction; the second large number of peak is com‐ ing from the N-E direction and a very few are coming from the S-E part that is the largest distance from the coast. An other important noise peak shown in figure 14 occurs in the cen‐ ter of the plot, indicating that energy is coming almost with a vertical incidence on the array. There is not any peak for the high-frequencies (f=2.0; 4.0 Hz); that is probably due to the lo‐ cation of the array. Furthermore we can also point our attention on the amplitude as a func‐ tion of time (fig. 15). It seems there are some seasonal patterns, in fact, the maximum peaks occur in the winter time while the minimum values are during the summer time (please re‐ member that the array is located in the Southern Hemisphere).

**Figure 4.** Number of recordings as a function of the optimal ray parameters for different frequency

**Figure 5.** Average of the relative power per year at 0.4 Hz

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR)

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133

**Figure 5.** Average of the relative power per year at 0.4 Hz

tion of time (fig. 15). It seems there are some seasonal patterns, in fact, the maximum peaks occur in the winter time while the minimum values are during the summer time (please re‐

member that the array is located in the Southern Hemisphere).

132 Earthquake Research and Analysis - New Advances in Seismology

**Figure 4.** Number of recordings as a function of the optimal ray parameters for different frequency

**Figure 7.** Average of the relative power per year at 0.6 Hz

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**Figure 6.** Average of the relative power per month at 0.4 Hz

**Figure 7.** Average of the relative power per year at 0.6 Hz

**Figure 6.** Average of the relative power per month at 0.4 Hz

134 Earthquake Research and Analysis - New Advances in Seismology

**Figure 9.** Average of the relative power per year at 0.8 Hz

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR)

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137

**Figure 8.** Average of the relative power per month at 0.6 Hz

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR) http://dx.doi.org/10.5772/54387 137

**Figure 9.** Average of the relative power per year at 0.8 Hz

**Figure 8.** Average of the relative power per month at 0.6 Hz

136 Earthquake Research and Analysis - New Advances in Seismology

138 Earthquake Research and Analysis - New Advances in Seismology

**Figure 11.** Average of the relative power per year at 1 Hz

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR)

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139

**Figure 10.** Average of the relative power per month at 0.8 Hz

**Figure 11.** Average of the relative power per year at 1 Hz

**Figure 10.** Average of the relative power per month at 0.8 Hz

138 Earthquake Research and Analysis - New Advances in Seismology

**Figure 13.** Average of the relative power per year at 2 Hz (a) and 4 Hz (b)

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR)

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141

**Figure 12.** Average of the relative power per month at 1 Hz

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR) http://dx.doi.org/10.5772/54387 141

**Figure 13.** Average of the relative power per year at 2 Hz (a) and 4 Hz (b)

**Figure 12.** Average of the relative power per month at 1 Hz

140 Earthquake Research and Analysis - New Advances in Seismology

**Figure 15.** Maximum amplitude versus time

Seismic array have contributed to develop different studies to investigate the interior of the Earth. In this paper we used some array techniques in order to highlight the charac‐ teristic of noise for a relative small aperture array (about 10 km): the ASAR array locat‐ ed in central Australia. We used waveforms from 1999 to 2001 choosing the data in order to cover each year, month day and part of it. We used the Capon (1969) method and we performed the analysis at different frequencies (0.4, 0.6, 0.8, 1.0, 2.0 and 4.0 Hz). For each frequency the optimal ray parameter, the optimal phase velocity and the opti‐ mal backazimuth were calculated. Results show that there is a consistent peak for the optimal backazimuth around 190-200 degrees for the frequency ranged from 0.4 to 1.0 Hz; the maximum peak disappears for the 2.0 and 4.0Hz analysis. The predominant peak in the S-W direction could be interpreted as ocean waves interacting with the coast in the Australian Bight. The absence of peaks for the analysis above the 2.0 Hz confirm that there is no evidence of anthropical noise, that is probably due to the location of the ar‐

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR)

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143

**4. Concluding remarks**

**Figure 14.** Local maxima computed using the Capon (1969) method having a relative power greater than 5db for dif‐ ferent frequencies It is possible to notice a relationship between the maxima in the S-W and N-E direction; they seem be quite spread in the N-W and S-E directions; perhaps due to the distance between the array and the coast lines.

Characterizing the Noise for Seismic Arrays: Case of Study for the Alice Springs ARray (ASAR) http://dx.doi.org/10.5772/54387 143

**Figure 15.** Maximum amplitude versus time

#### **4. Concluding remarks**

**Figure 14.** Local maxima computed using the Capon (1969) method having a relative power greater than 5db for dif‐ ferent frequencies It is possible to notice a relationship between the maxima in the S-W and N-E direction; they seem be quite spread in the N-W and S-E directions; perhaps due to the distance between the array and the coast lines.

142 Earthquake Research and Analysis - New Advances in Seismology

Seismic array have contributed to develop different studies to investigate the interior of the Earth. In this paper we used some array techniques in order to highlight the charac‐ teristic of noise for a relative small aperture array (about 10 km): the ASAR array locat‐ ed in central Australia. We used waveforms from 1999 to 2001 choosing the data in order to cover each year, month day and part of it. We used the Capon (1969) method and we performed the analysis at different frequencies (0.4, 0.6, 0.8, 1.0, 2.0 and 4.0 Hz). For each frequency the optimal ray parameter, the optimal phase velocity and the opti‐ mal backazimuth were calculated. Results show that there is a consistent peak for the optimal backazimuth around 190-200 degrees for the frequency ranged from 0.4 to 1.0 Hz; the maximum peak disappears for the 2.0 and 4.0Hz analysis. The predominant peak in the S-W direction could be interpreted as ocean waves interacting with the coast in the Australian Bight. The absence of peaks for the analysis above the 2.0 Hz confirm that there is no evidence of anthropical noise, that is probably due to the location of the ar‐ ray. We found a maximum peak around 3-4Km/s for the phase velocity indicative of higher-mode Rayleigh waves. Some dispersion is evident in the phase velocity peaks, and the large noise peak to the southwest is consistent from season to season, suggesting that there are some seasonal patterns as well. In some of the f-k spectra it is possible to notice a double peak, in which there appears to be a body-wave component to the noise.

[8] Gerstoft P.M., Fehler M.C., Sabra K.G., 2006. When Katrina hit California; *Geophys*

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145

[9] Goldstein and Archuleta, 1991a. Deterministic Frequency Wavenumber methods and direct measurements of rapture propagation during earthquakes using a dense array:

[10] Goldstein and Archuleta, 1991b. Deterministic Frequency Wavenumber methods and direct measurements of rapture propagation during earthquakes using a dense array:

[11] Hoffmann, H., Winterflood, J., Cheng Y., Blair D. G., 2002. Cross-correlation studies

[12] Huang B-S., 2001. Evidence for azimuthal and temporal variations of the rapture propagation of the 1999 Chi-Chi, Taiwan earthquake from dense seismic array obser‐

[13] Helffrich, G., S. Kaneshima, and J.-M. Kendall, 2002. A local, crossing-path study of

[14] Koper K., 2005, The Generic Array Processing (GAP) Software Package, SSA meeting web: http://www.eas.slu.edu/People/KKoper/Free/index.html (last access

[15] Koper K and Fathei A. (2007). Modeling P wave multipathing at regional distances in

[16] Koper, K.D., T.C. Wallace, S.R. Taylor, and H.E. Hartse, 2001. Forensic seismology

[17] Koper, K.D., T.C. Wallace, and D. Hollnack, 1999. Seismic analysis of the 7 August 1998 truck-bomb blast at the American Embassy in Nairobi, Kenya, *Seismol. Res. Lett.,*

[18] Koper, K.D., T.C. Wallace, R.C. 2003. Aster, Seismic recordings of the Carlsbad, New Mexico pipeline explosion of 19 August 2000, *Bull. Seism. Soc. Am., 93*, 1427-1432 [19] Koper, K.D. and B. de Foy (2008), Seasonal anisotropy in short-period seismic noise

[20] Koper, K. D., A. R. Hutko, T. Lay, and O. Sufri (2012), Imaging short-period seismic radiation from the 27 February 2010 Chile (Mw 8.8) earthquake by back-projection of P, PP, and PKIKP waves, J. Geophys. Res., 117, B02308, doi:10.1029/2011JB008576. [21] Kraft T., Wassermann J., Schmedes E., Igel H., 2006. Metereological triggering of earthquake swarms at Mt. Hochstaufen, SE-Germany, *Tectonophysycs*, 424, 245-258. [22] Pavlis, G. L., and H. Mahdi, 1996. Surface wave propagation in central Asia: Obser‐ vations of scattering and multipathing with the Kyrgyzstan broadband array, *J. Geo‐*

attenuation and anisotropy of the inner core, *Geophys. Res. Lett.,* 29, 1568.

theory and methods. *Journal of Geophysical Research*, 98, 6173-6185.

data analysis. *Journal of Geophysical Research*, 98, 6187-6198.

with seismic noise, *Class. Qaunt. Grav.*, 19, 1709-1716

southeast Asia. *Final Scientific report # FA8718-06-C-003.*

and the sinking of the Kursk, *EOS Trans., AGU, 82*, pp. 37,45-46

recorded in South Asia, Bull. Seismol. Soc. Am., 98, 3033-3045.

vations, *Geophys. Res. Lett.*, 28, 17 3377-3380.

*Res. Lett.,* 33, L17308.

Dec-17-2007)

70, 512-521.

*phys. Res.,* 101, 8437–8455.

The author thanks Dr. Keith Koper (University of Utah, USA) for providing the Generic Ar‐ ray Processing software. Data where obtained using the "autodrm request" of the U.S. Army Space and missile Defense Command monitoring research program (www.rdss.info, last ac‐ cess in 2009). The author is also very thankful to Ms. Silvia Vlase for her support.

#### **Author details**

Sebastiano D'Amico

Address all correspondence to: sebdamico@gmail.com

Depaertment of Physics, University of Malta, Msida, Malta

#### **References**


[8] Gerstoft P.M., Fehler M.C., Sabra K.G., 2006. When Katrina hit California; *Geophys Res. Lett.,* 33, L17308.

ray. We found a maximum peak around 3-4Km/s for the phase velocity indicative of higher-mode Rayleigh waves. Some dispersion is evident in the phase velocity peaks, and the large noise peak to the southwest is consistent from season to season, suggesting that there are some seasonal patterns as well. In some of the f-k spectra it is possible to notice a double peak, in which there appears to be a body-wave component to the noise.

The author thanks Dr. Keith Koper (University of Utah, USA) for providing the Generic Ar‐ ray Processing software. Data where obtained using the "autodrm request" of the U.S. Army Space and missile Defense Command monitoring research program (www.rdss.info, last ac‐

[1] Achauer, U., and the KRISP Working Group, 1994. New ideas on the Kenya rift based on the onversion of the combined dataset of the 1985 and 1989/90 seismic to‐

[2] Capon J., 1969. High resolution frequency-wavenumber spectrum analysis, Procced‐

[3] Castle, J. C., and K. C. Creager, 1999. A steeply dipping discontinuity in the lower

[4] Cotte, N., H. A. Pedersen, M. Campillo, V. Farra, and Y. Cansi, 2000. Of great-circle propagation of intermediate-period surface waves observed on a dense array in the

[5] D'Amico S., Koper K.. D., Herrmann R B., 2008. Array analysis of short-period seis‐ mic noise recorded in central Australia*. Seismological Research Letters*, vol. 79, 2, 293

[6] D'Amico S., Koper K. D., Herrmann R. B., Akinci A., Malagnini L., 2010. Imaging the rupture of the Mw 6.3 April 6, 2009 L'Aquila, Italy earthquake using back-projection of teleseismic P-waves. *Geophysical Research Letters*, 37, L03301, doi:

[7] Filson J., 1975. Array seismology. *Annual Reviews, earth Planet. Sci*., 3 157-181.

cess in 2009). The author is also very thankful to Ms. Silvia Vlase for her support.

Address all correspondence to: sebdamico@gmail.com

144 Earthquake Research and Analysis - New Advances in Seismology

ing of the IEEE, 57, 8, 1408-1418

10.1029/2009GL042156

French alps, *Geophys. J. Int.,* 142, 825–840.

Depaertment of Physics, University of Malta, Msida, Malta

mography experiments, *Tectonophysics,* 236, 305–329.

mantle beneath Izu-Bonin, *J. Geophys. Res.,* 104, 7279–7292.

**Author details**

Sebastiano D'Amico

**References**


[23] Ritter, J. R. R., U. R. Christensen, U. Achauer, K. Bahr, and M. Weber, 1998. Search for a mantle plume under central Europe, *Eos Trans. AGU,* 79, 420.

**Chapter 7**

**Parameters Identification of Stochastic Nonstationary**

The design of structures resistant to seismic events is an important field in the structural engineering, because it reduces both the loss of lives and the economic damages that earthquakes can produce. The accuracy and the robustness of the design of structures resistant to seismic events are still not completely guaranteed. In order to define rules in the design codes to design earthquake-resistant structures, several scholars have investigated the probability of a seismic event to occur in a specific location and its characteristics, like the intensity and the return time (e.g. frequency). Indeed, the return time and the characteristics of the earthquakes occurring in a given area determine the dynamic loads exciting a structure built in that area for its whole lifetime. The structural response to ground motion is function of the seismological parameters of the area where the earthquake occurs and the structure is built, in addition to the kind of structure. The earthquake characteristics related to the seismological parameters that strongly influence the structural response are the earthquake intensity, the rupture type and the epicentral distance. This leads to define the seismic dynamic loads exciting a structure as function of these seismological parameters. Unfortunately, the seismological parameters are not very useful in structural design. Instead, peak amplitude, frequency content, energy content and duration of the event are the characteristics of the

To design strategic or complex structures and infrastructures resistant to earthquakes, the analysis of the dynamic time-history response of the structure to earthquake records is preferred to the response spectrum analysis. Indeed, the dynamic time-history response provides temporal information of the structural response that is essential in non-linear analysis of some kind of structures to estimate their level of damage. Some design codes indicate the

> © 2013 Marano et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Marano et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Process Used in Earthquake Modelling**

Giuseppe Carlo Marano, Mariantonietta Morga and

Additional information is available at the end of the chapter

Sara Sgobba

**1. Introduction**

http://dx.doi.org/10.5772/54891

earthquakes useful to structural design.


### **Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling**

Giuseppe Carlo Marano, Mariantonietta Morga and Sara Sgobba

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54891

#### **1. Introduction**

[23] Ritter, J. R. R., U. R. Christensen, U. Achauer, K. Bahr, and M. Weber, 1998. Search for

[24] Ritter, J. R. R., M. Jordan, U. Christensen, and U. Achauer, 2001.A mantle plume be‐

[25] Rost S., C. Thomas, 2002. Array seismology: methods and applications. *Reviews of Ge‐*

[26] Rost, S., and J. Revenaugh, 2001. Seismic detection of rigid zones at the top of the

[27] Rost, S., and M. Weber, 2001. A reflector at 200 km depth beneath the NW Pacific,

[28] Rothert, E., and J. R. R. Ritter, 2001. Small-scale heterogeneities below the Gra°fen‐ berg array, Germany, from seismic wave-field fluctuations of Hindu Kush events,

[29] Schulte-Pelkum V., Earle P.S., Vernon F.L., 2004. Strong directivity of ocean-generat‐

[30] Spudich P., E. Cranswick, 1894. Direct observation of rapture propagation during the 1979 imperial valley earthquake using a short baseline accelerometer array. *Bull. Se‐*

[31] Sufri, O., K. D. Koper, and T. Lay (2012), Along-dip seismic radiation segmentation during the 2007 Mw 8.0 Pisco, Peru earthquake, Geophys. Res. Lett., 39, L08311, doi:

[32] Thomas, C., M. Weber, C. Wicks, and F. Scherbaum, 1999. Small scatterers in the low‐ er mantle observed at German broadband arrays, *J. Geophys. Res.,* 104, 15,073–15,088.

[33] Vidale, J. E., and P. S. Earle, 2000. Fine-scale heterogeneity in the Earth's inner core,

[34] Vidale, J. E., D. A. Dodge, and P. S. Earle, 2000. Slow differential rotation of the Earth's inner core indicated by temporal changes in scattering, *Nature*, 405, 445–448.

[35] Wessel P. and Smith W. H. F., 1991. Free software helps map and display data, *Eos*

ed seismic noise; *Geochem. Geophys. Geosyst.,* 5, Q03004.

low the Eifel volcanic fields, Germany, *Earth Planet. Sci. Lett.,* 186, 7–14.

a mantle plume under central Europe, *Eos Trans. AGU,* 79, 420.

*ophys.*, 40, 3

core, *Science,* 294, 1911–1914.

146 Earthquake Research and Analysis - New Advances in Seismology

*Geophys. J. Int.,* 147, 12–28.

*Geophys. J. Int.,* 140, 175–184.

*ism. Soc. Am*., 74, 6, 2083-2114.

10.1029/2012GL051316.

*Nature,* 404, 273–275.

*Trans.*, AGU, 72, 441

The design of structures resistant to seismic events is an important field in the structural engineering, because it reduces both the loss of lives and the economic damages that earthquakes can produce. The accuracy and the robustness of the design of structures resistant to seismic events are still not completely guaranteed. In order to define rules in the design codes to design earthquake-resistant structures, several scholars have investigated the probability of a seismic event to occur in a specific location and its characteristics, like the intensity and the return time (e.g. frequency). Indeed, the return time and the characteristics of the earthquakes occurring in a given area determine the dynamic loads exciting a structure built in that area for its whole lifetime. The structural response to ground motion is function of the seismological parameters of the area where the earthquake occurs and the structure is built, in addition to the kind of structure. The earthquake characteristics related to the seismological parameters that strongly influence the structural response are the earthquake intensity, the rupture type and the epicentral distance. This leads to define the seismic dynamic loads exciting a structure as function of these seismological parameters. Unfortunately, the seismological parameters are not very useful in structural design. Instead, peak amplitude, frequency content, energy content and duration of the event are the characteristics of the earthquakes useful to structural design.

To design strategic or complex structures and infrastructures resistant to earthquakes, the analysis of the dynamic time-history response of the structure to earthquake records is preferred to the response spectrum analysis. Indeed, the dynamic time-history response provides temporal information of the structural response that is essential in non-linear analysis of some kind of structures to estimate their level of damage. Some design codes indicate the

© 2013 Marano et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Marano et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

use of real records of earthquake ground motions as input of the dynamic time-history analysis of the structural response. Unfortunately, the selection of natural earthquake accelerograms that adhere to some criteria, such as the response spectrum for some design scenarios, is difficult. Indeed, the number of occurred earthquakes recorded in a specific area or with some characteristics is often not sufficient, because a wide set of accelograms is required for the design of earthquakes-resistant structures. To overcome this difficulty some design codes allow the use of modified natural records (with changes either in the time domain or in frequency domain) or synthetic accelerograms in the dynamic time history-analysis of the structural response. Unlikely, other design codes preclude the use of artificial accelerograms for the dynamic time-history analysis of the structural response because of the difficulty to generate accelograms that adhere to criteria for some design scenarios [1]. The approach based on the natural accelerograms is prevailing, since a real recorded accelerogram properly processed is undeniably a realistic representation of the ground shaking that is occurred in a particular seismological scenario. On the other hand, the recorded accelerogram represents a past seismic event occurred in a specific area and not a future event that will occur in that area and cannot be predicted because of stochastic nature of the seismic ground motions. This is a further reason to generate artificial accelerograms for the structural design on the basis of a stochastic model.

ars have proposed non-stationary filtered white noise models to simulate the seismic ground motions. This kind of earthquake models is obtained from the product of a filtered stationary White Noise process and an envelope function dependent on the time (figure 1). In literature there are several different envelope functions: the research of a reliable envelope function to model the ground motion intensity has been the goal of many studies. Some of these functions are simple and deterministic, like the one proposed by Bolotin [1], others are complex. Jangid [8] has given an overview of different envelope functions. The main feature that distinguishes the envelope functions proposed in literature is its shape: it describes the temporal evolution of the amplitude of the ground shake (trapezoidal, double exponen‐ tial, log-normal, etc.). The envelope functions have simple parametric forms and the values of the parameters are estimated from some characteristics of the earthquake records available for a specific area, like the duration of the strong ground motion, the energy of the seismic event and the kind of soil. Some studies have proposed envelope functions correlated with seismological parameters [9, 10, 11]. Unfortunately, these parameters are not significant and useful for the structural design. Previously Baker [12] has proposed a correlation of the ground motion intensity parameters used to predict the structural and

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling

http://dx.doi.org/10.5772/54891

149

In order to reproduce the temporal variation of the frequencies of the seismic input shaking the structure, evolutionary non-stationary stochastic model have been proposed. In these complex models the parameters of the stationary filtered stochastic process have a temporal variation. The temporal evolution of the frequency content of the accelograms is due to the different velocity of the P waves, S waves and surface waves that are released in the epicentre

This study presented in this chapter proposes a new simple and effective deterministic envelope function that correlates the temporal variation of the amplitude of the seismic records to the most significant seismological parameters of the ones used in structural design: the PGA and the kind of soil. The shape of the proposed envelope function is based on the Saragoni and Hart's (SH) exponential function [14] with three parameters determined through an energetic criterion. This shape of the envelope function gives a very good agreement with the selected time-histories, as a numerical analysis shows hereafter. The proposed envelope function is calculated through a new procedure composed by two stages. In the first stage a deterministic pre-envelope mean function that is the real envelope of a set of selected earthquake records is estimated. The values of the two parameters of the envelope function for each selected accelerograms are estimated through an identification procedure. In the second stage a regression law for each parameter is estimated to generalize the results and to obtain values

The identification procedure of the parameters of the Modified Saragoni and Hart's function proposed here is based on the continuous energy release of the earthquake measured through

of the parameters of the envelope function useful in the seismic engineering.

geotechnical response.

of the earthquake [13].

the Arias Intensity (AI).

Several scholars have proposed different methods to generate the synthetic accelograms, but nowadays no model is indicated in the design codes to generate the artificial records of seismic ground motions. Moreover, the design codes that allow the use of artificial accelograms prescribe that the mean response spectrum of the synthetic earthquake records has to match a given response spectrum within a given tolerance.

The complex nature of the release of seismic wave, their propagation in soil and the unpre‐ dictability of the earthquake occurrence make the stochastic-based approach the most suitable to model the earthquake ground motion. In that sense the earthquake occurring in a specific area is modelled as a stochastic process, so each recorded seismic ground motion is defined as a sample function of that stochastic process. The artificial accelerograms are also sample functions of the stochastic process modelling the earthquake occurring in that area: they represent the possible future seismic events. For this reason in the design phase of a complex structure the structural response to these artificial accelerograms is calculated.

The stochastic process modelling the seismic events occurring in an area is defined through the characteristics of the strong ground motions recorded in that area. Several scholars have presented methods to define the stochastic model to describe the seismic ground motion and simulate artificial earthquake records. Firstly, stationary filtered white noise model have been proposed to describe and simulate earthquakes [2, 3]. The most known of these models is the Kanai–Tajimi model [4, 5]. Some scholars have modified this model [6] or have proposed stationary multi-filtered white noise models, as the Clough-Penzien model [7]. The stationary filtered white noise models catch only the main frequency of the seismic waves that excite the structure and the bandwidth of the stochastic process. The station‐ ary stochastic models generate artificial accelograms with constant amplitude, while the amplitude of the real accelograms is time-varying. To overcome this limit, several schol‐ ars have proposed non-stationary filtered white noise models to simulate the seismic ground motions. This kind of earthquake models is obtained from the product of a filtered stationary White Noise process and an envelope function dependent on the time (figure 1). In literature there are several different envelope functions: the research of a reliable envelope function to model the ground motion intensity has been the goal of many studies. Some of these functions are simple and deterministic, like the one proposed by Bolotin [1], others are complex. Jangid [8] has given an overview of different envelope functions. The main feature that distinguishes the envelope functions proposed in literature is its shape: it describes the temporal evolution of the amplitude of the ground shake (trapezoidal, double exponen‐ tial, log-normal, etc.). The envelope functions have simple parametric forms and the values of the parameters are estimated from some characteristics of the earthquake records available for a specific area, like the duration of the strong ground motion, the energy of the seismic event and the kind of soil. Some studies have proposed envelope functions correlated with seismological parameters [9, 10, 11]. Unfortunately, these parameters are not significant and useful for the structural design. Previously Baker [12] has proposed a correlation of the ground motion intensity parameters used to predict the structural and geotechnical response.

use of real records of earthquake ground motions as input of the dynamic time-history analysis of the structural response. Unfortunately, the selection of natural earthquake accelerograms that adhere to some criteria, such as the response spectrum for some design scenarios, is difficult. Indeed, the number of occurred earthquakes recorded in a specific area or with some characteristics is often not sufficient, because a wide set of accelograms is required for the design of earthquakes-resistant structures. To overcome this difficulty some design codes allow the use of modified natural records (with changes either in the time domain or in frequency domain) or synthetic accelerograms in the dynamic time history-analysis of the structural response. Unlikely, other design codes preclude the use of artificial accelerograms for the dynamic time-history analysis of the structural response because of the difficulty to generate accelograms that adhere to criteria for some design scenarios [1]. The approach based on the natural accelerograms is prevailing, since a real recorded accelerogram properly processed is undeniably a realistic representation of the ground shaking that is occurred in a particular seismological scenario. On the other hand, the recorded accelerogram represents a past seismic event occurred in a specific area and not a future event that will occur in that area and cannot be predicted because of stochastic nature of the seismic ground motions. This is a further reason to generate artificial accelerograms for the structural design on the basis of a

Several scholars have proposed different methods to generate the synthetic accelograms, but nowadays no model is indicated in the design codes to generate the artificial records of seismic ground motions. Moreover, the design codes that allow the use of artificial accelograms prescribe that the mean response spectrum of the synthetic earthquake records has to match

The complex nature of the release of seismic wave, their propagation in soil and the unpre‐ dictability of the earthquake occurrence make the stochastic-based approach the most suitable to model the earthquake ground motion. In that sense the earthquake occurring in a specific area is modelled as a stochastic process, so each recorded seismic ground motion is defined as a sample function of that stochastic process. The artificial accelerograms are also sample functions of the stochastic process modelling the earthquake occurring in that area: they represent the possible future seismic events. For this reason in the design phase of a complex

The stochastic process modelling the seismic events occurring in an area is defined through the characteristics of the strong ground motions recorded in that area. Several scholars have presented methods to define the stochastic model to describe the seismic ground motion and simulate artificial earthquake records. Firstly, stationary filtered white noise model have been proposed to describe and simulate earthquakes [2, 3]. The most known of these models is the Kanai–Tajimi model [4, 5]. Some scholars have modified this model [6] or have proposed stationary multi-filtered white noise models, as the Clough-Penzien model [7]. The stationary filtered white noise models catch only the main frequency of the seismic waves that excite the structure and the bandwidth of the stochastic process. The station‐ ary stochastic models generate artificial accelograms with constant amplitude, while the amplitude of the real accelograms is time-varying. To overcome this limit, several schol‐

structure the structural response to these artificial accelerograms is calculated.

stochastic model.

a given response spectrum within a given tolerance.

148 Earthquake Research and Analysis - New Advances in Seismology

In order to reproduce the temporal variation of the frequencies of the seismic input shaking the structure, evolutionary non-stationary stochastic model have been proposed. In these complex models the parameters of the stationary filtered stochastic process have a temporal variation. The temporal evolution of the frequency content of the accelograms is due to the different velocity of the P waves, S waves and surface waves that are released in the epicentre of the earthquake [13].

This study presented in this chapter proposes a new simple and effective deterministic envelope function that correlates the temporal variation of the amplitude of the seismic records to the most significant seismological parameters of the ones used in structural design: the PGA and the kind of soil. The shape of the proposed envelope function is based on the Saragoni and Hart's (SH) exponential function [14] with three parameters determined through an energetic criterion. This shape of the envelope function gives a very good agreement with the selected time-histories, as a numerical analysis shows hereafter. The proposed envelope function is calculated through a new procedure composed by two stages. In the first stage a deterministic pre-envelope mean function that is the real envelope of a set of selected earthquake records is estimated. The values of the two parameters of the envelope function for each selected accelerograms are estimated through an identification procedure. In the second stage a regression law for each parameter is estimated to generalize the results and to obtain values of the parameters of the envelope function useful in the seismic engineering.

The identification procedure of the parameters of the Modified Saragoni and Hart's function proposed here is based on the continuous energy release of the earthquake measured through the Arias Intensity (AI).

**Figure 1.** Scheme to simulate a stochastic ground motion

#### **2. Stationary filtered stochastic process modelling earthquakes**

As said in the introduction, a filtered stationary White Noise (WN) process *w(t)* is the simplest model of the ones proposed to represent the stochastic seismic acceleration process. In this kind of models the WN process models the acceleration at the bedrock. One or more filters model the action of the soil between the epicentre and the basement of the structure. Indeed the soil filters the seismic waves and the resulting filtered signal has the mean frequency content of the earthquake acceleration recorded on the ground surface. The most famous model of filtered WN process that defines the ground motion acceleration is the Kanai-Tajimi (K-T) model. In this model the filtering effect of the soil is defined by a SDoF system, characterized by two parameters: the damping ratio *ξg* and the circular frequency *ωg*. The ground acceleration *ast* exiting the structure is the absolute acceleration of the filter. Therefore, the differential equations of the K-T model are:

$$
\ddot{\ddot{\alpha}} + 2\xi\_g \omega\_g \dot{\varkappa} + \omega\_g^2 \varkappa = \text{ - } w(t) \tag{1}
$$

{

where *xf*

damping ratio *ξ<sup>f</sup>*

*ast* (*t*)= *x*

*x* ¨ *<sup>p</sup>*(*t*) <sup>+</sup> *<sup>ω</sup><sup>p</sup>* 2

*x*

**3. Envelope function definition**

A Stationary stochastic process *x*

intensity:

¨ *<sup>p</sup>*(*t*)= - *<sup>ω</sup><sup>p</sup>*

¨ *<sup>f</sup>* (*t*) <sup>+</sup> <sup>2</sup>*<sup>ξ</sup> <sup>f</sup> <sup>ω</sup> <sup>f</sup> <sup>x</sup>*˙ *<sup>f</sup>* (*t*) <sup>+</sup> *<sup>ω</sup> <sup>f</sup>*

2

*xp*(*t*) + 2*ξpω<sup>p</sup> x*˙ *<sup>p</sup>*(*t*)=*ω <sup>f</sup>*

*xp*(*t*) - 2*ξpω<sup>p</sup> x*˙ *<sup>p</sup>*(*t*) + *ω <sup>f</sup>*

<sup>2</sup> *<sup>x</sup> <sup>f</sup>* <sup>=</sup>*w*(*t*)

ground acceleration coincides with the acceleration of the second filter output: *ast*

*(t)* is the response of the first filter characterized by the circular frequency *ω<sup>f</sup>*

*ωp* and the damping ratio *ξf* and *w(t)* is the exciting WN process. In this model the stationary

literature more complex models characterized by Multi-Degree of Freedom (MDoF) filters have be proposed to model seismic ground acceleration in a specific area. These models are defined by a larger number of parameters than the K-T and C-P models, so they model the soil filtering effect of the seismic waves better than the K-T and C-P models. On the other hand, these models with MDoF filters have a higher computational effort than the one of the K-T and C-P models. Some scholars [13, 16] have proposed non-stationary filtered WN models to take into account the temporal variation of the intensity of the acceleration records of the real earthquakes. The envelope function proposed in this study can be applied to modulate the amplitude in time of the stationary filtered White Noise obtained from a C-P model. Indeed, that model has four parameters correlated with the soil diffusion effect and reaches a good compromise between accuracy and computational effort to define seismic ground motion.

envelope function *E*(*t*) to obtain a non-stationary filtered WN process with modulated

Each sample function of this non-stationary stochastic process is a synthetic accelerograms. The envelope function *E*(*t*) proposed here is achieved from a complex procedure in two stages. A pre-envelope function defined from a set of selected recorded real accelerograms is estimated in the first step of the procedure. This pre-envelope function has the shape similar to the one developed by Saragoni and Hart [14] that is the most suitable for the set of the real records used. Indeed, this exponential deterministic envelope function has been chosen because it simulates better the strong ground motion than the other ones. It is continuous while other ones imply an arbitrary division into segments and the abrupt change in the frequency content

*a*(*t*)= *x*

of each segment. The Saragoni and Hart's (SH) function is

<sup>2</sup> *<sup>x</sup> <sup>f</sup>* <sup>+</sup> <sup>2</sup>*<sup>ξ</sup> <sup>f</sup> <sup>ω</sup> <sup>f</sup> <sup>x</sup>*˙ *<sup>f</sup>* (*t*)

¨ *<sup>p</sup>* estimated through the C-P model is multiplied by an

*φ*(*t*)=*αt <sup>η</sup>e*-*β<sup>t</sup> α*, *β*, *η* >0, (6)

¨ *<sup>p</sup>*(*t*)*E*(*t*) (5)

(4)

151

and the

¨ *<sup>p</sup>*(*t*). In

(*t*)= *x*

http://dx.doi.org/10.5772/54891

<sup>2</sup> *<sup>x</sup> <sup>f</sup>* <sup>+</sup> <sup>2</sup>*<sup>ξ</sup> <sup>f</sup> <sup>ω</sup> <sup>f</sup> <sup>x</sup>*˙ *<sup>f</sup>* (*t*)

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling

, *xp(t)* is the response of the second filter characterized by the circular frequency

$$\mathbf{a}\_{\rm st} = \ddot{\mathbf{x}} + w(t) = -\left(2\xi\_{\rm g}\omega\_{\rm g}\dot{\mathbf{x}} + \omega\_{\rm g}^2 \mathbf{x}\right). \tag{2}$$

This filter is a linear second order one, so the Power Spectral Density (PSD) function of the filtered WN is

$$\mathcal{S}\left(\omega\right) = \mathcal{S}\_0 \frac{\left[1 + \xi\_{\frac{\omega}{\mathcal{S}}}^2 (\omega / \omega\_{\frac{\omega}{\mathcal{S}}})^2\right]}{\left[1 - (\omega / \omega\_{\frac{\omega}{\mathcal{S}}})^2\right]^2 + 4\xi\_{\frac{\omega}{\mathcal{S}}}^2 (\omega / \omega\_{\frac{\omega}{\mathcal{S}}})^2} \tag{3}$$

where *S0* is the PSD of the WN process [4, 5, 15]. This model has some limits: the amplitude of the ground acceleration is constant and the frequency content is constant in time. The K-T model fails in simulating earthquake ground motion characterized by medium and long duration, because it is described by only two filter parameters, the damping ratio and the circular frequency. In order to overcome this limit, Clough and Penzien [7] have proposed a model characterized by a double filtered WN process. The two filters that define the effect of the soil between the bedrock and the surface are linear. The differential equations of the C-P model are:

$$\begin{aligned} \left| a\_{\rm si}(t) \right\rangle &= \stackrel{\cdots}{\mathbf{x}}\_{p}(t) = \cdot \omega\_{p}^{2} \mathbf{x}\_{p}(t) \cdot 2\xi\_{p}\omega\_{p} \dot{\mathbf{x}}\_{p}(t) + \omega\_{f}^{2} \mathbf{x}\_{f} + 2\xi\_{f}\omega\_{f} \dot{\mathbf{x}}\_{f}(t) \\ \stackrel{\leftarrow}{\mathbf{x}}\_{p}(t) &+ \omega\_{p}^{2} \mathbf{x}\_{p}(t) + 2\xi\_{p}\omega\_{p} \dot{\mathbf{x}}\_{p}(t) = \omega\_{f}^{2} \mathbf{x}\_{f} + 2\xi\_{f}\omega\_{f} \dot{\mathbf{x}}\_{f}(t) \\ \stackrel{\leftarrow}{\mathbf{x}}\_{f}(t) &+ 2\xi\_{f}\omega\_{f} \dot{\mathbf{x}}\_{f}(t) + \omega\_{f}^{2} \mathbf{x}\_{f} = \mathbf{w}(t) \end{aligned} \tag{4}$$

where *xf (t)* is the response of the first filter characterized by the circular frequency *ω<sup>f</sup>* and the damping ratio *ξ<sup>f</sup>* , *xp(t)* is the response of the second filter characterized by the circular frequency *ωp* and the damping ratio *ξf* and *w(t)* is the exciting WN process. In this model the stationary ground acceleration coincides with the acceleration of the second filter output: *ast* (*t*)= *x* ¨ *<sup>p</sup>*(*t*). In literature more complex models characterized by Multi-Degree of Freedom (MDoF) filters have be proposed to model seismic ground acceleration in a specific area. These models are defined by a larger number of parameters than the K-T and C-P models, so they model the soil filtering effect of the seismic waves better than the K-T and C-P models. On the other hand, these models with MDoF filters have a higher computational effort than the one of the K-T and C-P models. Some scholars [13, 16] have proposed non-stationary filtered WN models to take into account the temporal variation of the intensity of the acceleration records of the real earthquakes. The envelope function proposed in this study can be applied to modulate the amplitude in time of the stationary filtered White Noise obtained from a C-P model. Indeed, that model has four parameters correlated with the soil diffusion effect and reaches a good compromise between accuracy and computational effort to define seismic ground motion.

#### **3. Envelope function definition**

**Figure 1.** Scheme to simulate a stochastic ground motion

150 Earthquake Research and Analysis - New Advances in Seismology

differential equations of the K-T model are:

filtered WN is

model are:

*x*

*ast* = *x*

*S*(*ω*)=*S*<sup>0</sup>

¨ <sup>+</sup> <sup>2</sup>*ξgω<sup>g</sup> <sup>x</sup>*˙ <sup>+</sup> *<sup>ω</sup><sup>g</sup>*

2

This filter is a linear second order one, so the Power Spectral Density (PSD) function of the

2

¨ <sup>+</sup> *<sup>w</sup>*(*t*)= - (2*ξgω<sup>g</sup> <sup>x</sup>*˙ <sup>+</sup> *<sup>ω</sup><sup>g</sup>*

1 + *ξ<sup>g</sup>* 2 (*ω* / *ω<sup>g</sup>* )2

<sup>1</sup> - (*<sup>ω</sup>* / *<sup>ω</sup><sup>g</sup>* )2 <sup>2</sup> <sup>+</sup> <sup>4</sup>*ξ<sup>g</sup>*

where *S0* is the PSD of the WN process [4, 5, 15]. This model has some limits: the amplitude of the ground acceleration is constant and the frequency content is constant in time. The K-T model fails in simulating earthquake ground motion characterized by medium and long duration, because it is described by only two filter parameters, the damping ratio and the circular frequency. In order to overcome this limit, Clough and Penzien [7] have proposed a model characterized by a double filtered WN process. The two filters that define the effect of the soil between the bedrock and the surface are linear. The differential equations of the C-P

2

*x* = - *w*(*t*) (1)

*x*). (2)

(*<sup>ω</sup>* / *<sup>ω</sup><sup>g</sup>* )2 (3)

**2. Stationary filtered stochastic process modelling earthquakes**

As said in the introduction, a filtered stationary White Noise (WN) process *w(t)* is the simplest model of the ones proposed to represent the stochastic seismic acceleration process. In this kind of models the WN process models the acceleration at the bedrock. One or more filters model the action of the soil between the epicentre and the basement of the structure. Indeed the soil filters the seismic waves and the resulting filtered signal has the mean frequency content of the earthquake acceleration recorded on the ground surface. The most famous model of filtered WN process that defines the ground motion acceleration is the Kanai-Tajimi (K-T) model. In this model the filtering effect of the soil is defined by a SDoF system, characterized by two parameters: the damping ratio *ξg* and the circular frequency *ωg*. The ground acceleration *ast* exiting the structure is the absolute acceleration of the filter. Therefore, the

> A Stationary stochastic process *x* ¨ *<sup>p</sup>* estimated through the C-P model is multiplied by an envelope function *E*(*t*) to obtain a non-stationary filtered WN process with modulated intensity:

$$
\hat{\mathbf{x}}(t) = \stackrel{\cdots}{\mathbf{x}}\_p(t)E\{t\} \tag{5}
$$

Each sample function of this non-stationary stochastic process is a synthetic accelerograms. The envelope function *E*(*t*) proposed here is achieved from a complex procedure in two stages. A pre-envelope function defined from a set of selected recorded real accelerograms is estimated in the first step of the procedure. This pre-envelope function has the shape similar to the one developed by Saragoni and Hart [14] that is the most suitable for the set of the real records used. Indeed, this exponential deterministic envelope function has been chosen because it simulates better the strong ground motion than the other ones. It is continuous while other ones imply an arbitrary division into segments and the abrupt change in the frequency content of each segment. The Saragoni and Hart's (SH) function is

$$
\varphi(t) = at\,\,^\eta e^{-\beta t} \qquad \qquad \alpha,\,\,\beta,\,\,\eta \ge 0,\tag{6}
$$

where *α*, *β* and *η* are three calibration parameters. In order to reduce the number of the parameters of the SH function from three to two, in this study the parameters *α* and *β* are expressed as functions of the unknown time *tm* in which the SH function has its maximum value. The maximum value of the modulation function is estimated from the system of equations:

$$\begin{cases} \varphi \left( t\_m \right) = 1 \\ \dot{\varphi} \left( t\_m \right) = 0 \end{cases} \tag{7}$$

From the solution of the eq. (7) the parameters *α* and *β* are valuated as function of the other two parameters *η* and *tm*:

$$
\beta = \frac{\eta}{t\_m} \tag{8}
$$

$$
\alpha = \left(\frac{e}{t\_m}\right)^{\eta} \tag{9}
$$

**Figure 2.** Sensitivity of the MSH function to the parameter η

0

j r h

while *Γ(z)* is the Euler Gamma function

The dimensionless ratio *ψ<sup>a</sup>*

t

Replacing the envelope function (10) in the expression (13), that expression becomes

(,) 2 2

*<sup>n</sup> d* = *z <sup>n</sup>*-1

event and it is plotted in the figure 3 for different value of the parameter η. This ratio is a function of the parameters τ and η. The parameter η determines the velocity of the energy release during a seismic event, while the parameter τ describes the energy release during the duration of the event with respect to the time of maximum amplitude of the accelerograms. The values of the parameters *τ* and *η* are estimated through an identification procedure.

Γ(*z*)=*∫* 0 ∞ *z*-1 *e*-

 ht

*<sup>e</sup> d E*

æ ö æ ö é ù <sup>=</sup> ç ÷ ç ÷ G - ë û ç ÷ è ø è ø

h

2 2 2 2 2

h

2

h

 r t

wher*e En(z)* is the generalised exponential integral function

*En*(*z*)=*∫* 1 <sup>∞</sup> *<sup>e</sup>* -*<sup>z</sup>* ( ) ( ) ( )

ò (14)


Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling

http://dx.doi.org/10.5772/54891

153

 h h t

 h

2


(*τ*, *η*) / *ψa*(*τ<sup>f</sup>* , *η*) describes the energy release during earthquake

h  ht

Γ(1 - *n*, *z*), (15)

*d*. (16)

Replacing the parameters *α* and *β* in the SH function (6) with the expressions (8) and (9), the SH function becomes

$$
\varphi(\tau,\ \eta) = \tau^{\eta} e^{\eta(1-\tau)} \tag{10}
$$

where *η* =*t* / *tm* and the independent variables are only *η* and *tm*. In the follow this expression will be called Modified Saragoni and Hart (MSH) function. The intensity modulation of the earthquake ground motion is described not only by their peak value (Peak Ground Acceleration PGA), but also by the energy content or other quantities related to the energy content, such as the Arias Intensity (AI). The AI is defined as:

$$I\_a = \frac{\pi}{2g} \zeta\_0^{t\_f} \stackrel{\cdots}{\propto}\_g 2 \{ t \} dt \tag{11}$$

and its mean value in stochastic terms is

$$\mu \| I\_a \| = \frac{\pi}{2g} \zeta\_0^{t\_f} \left\{ \stackrel{\cdot}{\chi}\_g^{-2}(t) \right\} dt = \frac{\pi}{2g} \sigma\_{\stackrel{\cdot}{\chi}\_g}^2 \psi\_a(t\_f) \tag{12}$$

where *tf* is total duration of the earthquake, *x* ¨ *<sup>g</sup>*(*t*) is the ground acceleration at time *t* in one of the two horizontal directions, *g* is acceleration due to gravity and the term *ψa*(*t <sup>f</sup>* ) is defined by the expression

$$\psi(a) = \int\_0^t \varphi^2(\rho) d\rho. \tag{13}$$

**Figure 2.** Sensitivity of the MSH function to the parameter η

where *α*, *β* and *η* are three calibration parameters. In order to reduce the number of the parameters of the SH function from three to two, in this study the parameters *α* and *β* are expressed as functions of the unknown time *tm* in which the SH function has its maximum value. The maximum value of the modulation function is estimated from the system of

{

*φ*(*tm*) =1

*<sup>β</sup>* <sup>=</sup> *<sup>η</sup> tm*

*<sup>α</sup>* =( *<sup>e</sup> tm*

content, such as the Arias Intensity (AI). The AI is defined as:

*<sup>μ</sup> Ia* <sup>=</sup> *<sup>π</sup>* <sup>2</sup>*<sup>g</sup> ∫* 0 *<sup>t</sup> <sup>f</sup> x* ¨ *g*

and its mean value in stochastic terms is

where *tf* is total duration of the earthquake, *x*

*Ia* <sup>=</sup> *<sup>π</sup>* <sup>2</sup>*<sup>g</sup> ∫* 0 *<sup>t</sup> <sup>f</sup> x* ¨ *g*

From the solution of the eq. (7) the parameters *α* and *β* are valuated as function of the other

Replacing the parameters *α* and *β* in the SH function (6) with the expressions (8) and (9), the

where *η* =*t* / *tm* and the independent variables are only *η* and *tm*. In the follow this expression will be called Modified Saragoni and Hart (MSH) function. The intensity modulation of the earthquake ground motion is described not only by their peak value (Peak Ground Acceleration PGA), but also by the energy content or other quantities related to the energy

2(*t*) *dt* <sup>=</sup> *<sup>π</sup>*

¨

2 0 () ( ) . *t*

 jrr

y

the two horizontal directions, *g* is acceleration due to gravity and the term *ψa*(*t <sup>f</sup>* ) is defined by

<sup>2</sup>*<sup>g</sup> σx*¨ *<sup>g</sup>*

*<sup>φ</sup>*˙ (*tm*) =0 (7)

)*<sup>η</sup>* (9)

2(*t*)*dt* (11)

*<sup>g</sup>*(*t*) is the ground acceleration at time *t* in one of

*a d* <sup>=</sup> ò (13)

<sup>2</sup> *<sup>ψ</sup>a*(*<sup>t</sup> <sup>f</sup>* ) (12)

*φ*(*τ*, *η*)=*τ <sup>η</sup>e <sup>η</sup>*(1-*τ*) (10)

(8)

equations:

two parameters *η* and *tm*:

152 Earthquake Research and Analysis - New Advances in Seismology

SH function becomes

the expression

Replacing the envelope function (10) in the expression (13), that expression becomes

$$\int\_0^\tau \left[\rho(\rho,\eta)\right]^2 d\rho = \tau^{2\eta} \left[ \left(\frac{e}{2}\right)^{2\eta} \left(\eta\tau\right)^{-2\eta} \Gamma\left(2\eta\right) - \eta^{2\eta} \tau E\_{-2\eta}\left(2\eta\tau\right) \right] \tag{14}$$

wher*e En(z)* is the generalised exponential integral function

$$E\_n(z) = \text{f}\_1^{\circ \circ} \frac{e^{-z}}{\pi} d = z^{\circ -1} \Gamma(1 - n, \ z), \tag{15}$$

while *Γ(z)* is the Euler Gamma function

$$
\Gamma(z) = \int\_0^\infty e^{z \cdot 1} e^{\cdot} \, d\,. \tag{16}
$$

The dimensionless ratio *ψ<sup>a</sup>* (*τ*, *η*) / *ψa*(*τ<sup>f</sup>* , *η*) describes the energy release during earthquake event and it is plotted in the figure 3 for different value of the parameter η. This ratio is a function of the parameters τ and η. The parameter η determines the velocity of the energy release during a seismic event, while the parameter τ describes the energy release during the duration of the event with respect to the time of maximum amplitude of the accelerograms. The values of the parameters *τ* and *η* are estimated through an identification procedure.

The values of parameters *η* and *tm* of the pre-envelope function are obtained by minimizing the difference of the ratios of the mean AI and the ratio of the term *ψ<sup>a</sup>* describing the energy

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling

**Figure 4.** Record 00163 (kind of soil D) of the database and its AI. The blue lines indicates the L component and the red lines indicates the T component of the real record of the seismic event and the green lines indicates the analytical

The ratios of the equation to minimize are functions of the total duration time of the earthquake

total energy of the seismic signal is released. It is evaluated as the time interval between the 1% and 99 % of the AI of the seismic record. The total duration time is calculated for each selected earthquake record of the database. The values of the parameters *η* and *tm* that describe the pre-envelope function for each real seismic event selected from the database seismic are achieved by means of an identification procedure formulated as an optimization problem. The

. The total duration time is defined as the time during which the 98% of the

Areas Intensity

accelerogram *Tf*

Objective Function (OF) to minimize is

*<sup>ψ</sup>a*(*<sup>T</sup> <sup>f</sup>* , *tm*, *<sup>η</sup>*) . (17)

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155

release estimated for analytical expression and real earthquake record:

*μ I <sup>a</sup>*(*t*)

*<sup>μ</sup> <sup>I</sup> <sup>a</sup>*(*<sup>T</sup> <sup>f</sup>* ) - *<sup>ψ</sup>a*(*t*, *tm*, *<sup>η</sup>*)

**Figure 3.** Sensitivity of the dimensionless ratio ψ*<sup>a</sup>* (τ, η) ψ*a*(τ *<sup>f</sup>* , <sup>η</sup>) to the parameter <sup>η</sup>

### **4. Numerical procedure to evaluate the parameters of the pre-envelope functions**

This section presents the identification procedure used to estimate the values of the parameters η and tm that better characterize each of the selected real accelerograms of the PEER Next Generation Attenuation database.

The ground motion records of the PEER Next Generation Attenuation database that have been used in this study match the following criterion: the site where the seismic event is recorded has an average shear wave velocity in the top 30 meters comprised in four ranges according to the EC8 (B-C classes) and the NERPH classification (C-D classes) corresponding to stiff and soft soil respectively. The ground motion records of the PEER Next Generation Attenuation database are more than 7000 and half of them match this criterion. Further, the ground acceleration records of both the horizontal directions are used: for each selected accelogram of the database the weighted squared root of the sum of the squared east–west and north– south components is calculated and after it is used to estimate the pre-envelope mean function and the PGA used in the procedure proposed here.

The values of parameters *η* and *tm* of the pre-envelope function are obtained by minimizing the difference of the ratios of the mean AI and the ratio of the term *ψ<sup>a</sup>* describing the energy release estimated for analytical expression and real earthquake record:

$$\frac{\mu \prod\_{a} I\_{a}(t\_{f})}{\mu \prod\_{a} I\_{a}(T\_{f})} - \frac{\psi\_{a}(t\_{f}, t\_{m}, \eta)}{\psi\_{a}(T\_{f}, t\_{m}, \eta)}. \tag{17}$$

**Figure 3.** Sensitivity of the dimensionless ratio ψ*<sup>a</sup>*

154 Earthquake Research and Analysis - New Advances in Seismology

Generation Attenuation database.

and the PGA used in the procedure proposed here.

**functions**

(τ, η) ψ*a*(τ

*<sup>f</sup>* , <sup>η</sup>) to the parameter <sup>η</sup>

**4. Numerical procedure to evaluate the parameters of the pre-envelope**

This section presents the identification procedure used to estimate the values of the parameters η and tm that better characterize each of the selected real accelerograms of the PEER Next

The ground motion records of the PEER Next Generation Attenuation database that have been used in this study match the following criterion: the site where the seismic event is recorded has an average shear wave velocity in the top 30 meters comprised in four ranges according to the EC8 (B-C classes) and the NERPH classification (C-D classes) corresponding to stiff and soft soil respectively. The ground motion records of the PEER Next Generation Attenuation database are more than 7000 and half of them match this criterion. Further, the ground acceleration records of both the horizontal directions are used: for each selected accelogram of the database the weighted squared root of the sum of the squared east–west and north– south components is calculated and after it is used to estimate the pre-envelope mean function

**Figure 4.** Record 00163 (kind of soil D) of the database and its AI. The blue lines indicates the L component and the red lines indicates the T component of the real record of the seismic event and the green lines indicates the analytical Areas Intensity

The ratios of the equation to minimize are functions of the total duration time of the earthquake accelerogram *Tf* . The total duration time is defined as the time during which the 98% of the total energy of the seismic signal is released. It is evaluated as the time interval between the 1% and 99 % of the AI of the seismic record. The total duration time is calculated for each selected earthquake record of the database. The values of the parameters *η* and *tm* that describe the pre-envelope function for each real seismic event selected from the database seismic are achieved by means of an identification procedure formulated as an optimization problem. The Objective Function (OF) to minimize is

$$\text{GOF}\begin{pmatrix}\text{-}\\b\end{pmatrix} = \frac{1}{T\_{f}}\int\_{0}^{T} \text{'}\begin{pmatrix}\frac{I\_{a}\{\pi\}}{I\_{a}\{T\_{f}\}} \cdot \frac{\psi\_{a}\begin{pmatrix}\cdot\\\pi\_{r}\end{pmatrix}}{\psi\_{a}\begin{pmatrix}\cdot\\T\_{f}\end{pmatrix}}\end{pmatrix}\,d\pi\_{r} \tag{18}$$

*<sup>μ</sup> Ia*(*t*) <sup>=</sup> *<sup>π</sup>*

maximum amplitude of the accelerogram *tm*:

where

where *Tf*

introduced:

**5. Regression laws**

seismic event selected from the database are

*f f T PGA T*

The definition of the PGA is

<sup>2</sup>*<sup>g</sup> <sup>σ</sup>x*¨ *<sup>g</sup> st*

*PGA*=*κσx*¨ *<sup>g</sup>*

2*g*

¨

*max*(*x* ¨ *g*

*Ie*(*<sup>T</sup> <sup>f</sup>* ) <sup>=</sup> *PGA*

*<sup>g</sup> ∫* 0 *<sup>T</sup> <sup>f</sup> ag*

is the total duration of the record. The normalized value of the PGA is

From the evaluation of (19), a new intensity measure *Ie* called Envelope Intensity (EI) is

In the second stage of the procedure to evaluate the envelope function described by the PGA and the kind of soil the regression laws that relate the parameters of the proposed envelope function with the PGA are extracted. The parameters of the envelope function to be identi‐ fied are the total duration time *Tf* , the AI *Ia*, the maximum envelope time *tm*, *η* and *κ*. The regression laws of the envelope function obtained from the pre-envelope functions of the

( ) ( ) ( ) ( ) ( ) <sup>1</sup> (2) <sup>0</sup>

e e *<sup>f</sup> <sup>f</sup> T PGA T PGA*

æ ö = + ç ÷ è ø

*ψa*(*tm*)

A linear function is used to correlate the PGA with its stationary variance *σx*¨ *<sup>g</sup>*

*<sup>κ</sup>* <sup>=</sup>*PGA <sup>π</sup>*

*PGA*=*max*(*x*

<sup>2</sup> *<sup>ψ</sup>a*(*t*, *tm*, *<sup>η</sup>*). (19)

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling

*st* (20)

*<sup>μ</sup> <sup>I</sup> <sup>a</sup>*(*tm*) . (21)

*<sup>g</sup>*(*τ*)|*τ* 0, *T <sup>f</sup>* ). (22)

*<sup>N</sup>* ) =1, (23)

*<sup>N</sup>* (*τ*)*dτ*. (24)

*st* at the time of

157

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(25)

where *b* = *tm*, *η* is the vector of the variables. The optimization problem is solved by means of the Genetic Algorithm (GA) implemented in Matlab. To check the quality of this identification procedure of the parameters *η* and *tm* for the pre-envelope function estimated for each earthquake record in the figures 4 and 5 the authors of this study have plotted a comparison between the AI of a real earthquake record and the analytical AI calculated using the parameters *η* and *tm* estimated through identification procedure proposed. These two figures (4 and 5) show the comparison for two different earthquake records of the selected ones of the PEER Next Generation Attenuation database.

**Figure 5.** Record 00225 (kind of soil D) of the database and its AI. The blue lines indicates the L component and the red lines indicates the T component of the real record of the seismic event and the green lines indicates the analytical Areas Intensity

The identification of the parameters of the pre-envelope function is applied to each selected earthquake record of the PEER Next Generation Attenuation database. The selected earthquakes records of the PEER Next Generation Attenuation database and their identified parameters *tm* and *η* are grouped according to four types of soil, as afore said. After the identification of the parameters *η* and *tm* the mean AI is calculated:

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling http://dx.doi.org/10.5772/54891 157

$$
\mu \boxplus I\_a(t) \Big| = \frac{\pi}{2g} \sigma\_{\check{\mathbf{x}}\_{\check{\mathbf{x}}\_{\check{\mathbf{x}}}}^2}^2 \psi\_a(t, \ t\_{m'}, \eta). \tag{19}
$$

A linear function is used to correlate the PGA with its stationary variance *σx*¨ *<sup>g</sup> st* at the time of maximum amplitude of the accelerogram *tm*:

$$PGA = \kappa \sigma\_{\ddot{x}\_{\ddot{g}}^{st}} \tag{20}$$

where

*OF* (*<sup>b</sup>* - ) <sup>=</sup> <sup>1</sup> *<sup>T</sup> <sup>f</sup> ∫* 0 *T f* ( *<sup>I</sup> <sup>a</sup>*(*τ*) *<sup>I</sup> <sup>a</sup>*(*<sup>T</sup> <sup>f</sup>* ) - *<sup>ψ</sup><sup>a</sup>*

156 Earthquake Research and Analysis - New Advances in Seismology

PEER Next Generation Attenuation database.

Areas Intensity

(*τ*, *<sup>b</sup>* - ) 2

*dτ*, (18)

*ψa* (*<sup>T</sup> <sup>f</sup>* , *<sup>b</sup>* - ) )

where *b* = *tm*, *η* is the vector of the variables. The optimization problem is solved by means of the Genetic Algorithm (GA) implemented in Matlab. To check the quality of this identification procedure of the parameters *η* and *tm* for the pre-envelope function estimated for each earthquake record in the figures 4 and 5 the authors of this study have plotted a comparison between the AI of a real earthquake record and the analytical AI calculated using the parameters *η* and *tm* estimated through identification procedure proposed. These two figures (4 and 5) show the comparison for two different earthquake records of the selected ones of the

**Figure 5.** Record 00225 (kind of soil D) of the database and its AI. The blue lines indicates the L component and the red lines indicates the T component of the real record of the seismic event and the green lines indicates the analytical

The identification of the parameters of the pre-envelope function is applied to each selected earthquake record of the PEER Next Generation Attenuation database. The selected earthquakes records of the PEER Next Generation Attenuation database and their identified parameters *tm* and *η* are grouped according to four types of soil, as afore said. After the

identification of the parameters *η* and *tm* the mean AI is calculated:

$$\kappa = PGA\sqrt{\frac{\pi}{2g}\frac{\psi\_s(t\_m)}{\mu\prod\_s(t\_n)}}.\tag{21}$$

The definition of the PGA is

$$PGA = \max\left(\stackrel{\smile}{\times}\_{\mathcal{E}}\{\pi\}\middle|\,\pi\{\!0,\!\!/\!T\!\_{f}\}\right). \tag{22}$$

where *Tf* is the total duration of the record. The normalized value of the PGA is

$$\max\left(\overset{\cdots}{\underset{\mathcal{S}}{\times}}\right) = \mathbf{1},\tag{23}$$

From the evaluation of (19), a new intensity measure *Ie* called Envelope Intensity (EI) is introduced:

$$I\_e \{ T\_{\,\_f} \} = \frac{PGA}{\mathcal{S}} \mathfrak{f}\_0^{T\_{\,\_f}} a\_{\mathcal{S}}^N \{ \tau \} d\tau. \tag{24}$$

#### **5. Regression laws**

In the second stage of the procedure to evaluate the envelope function described by the PGA and the kind of soil the regression laws that relate the parameters of the proposed envelope function with the PGA are extracted. The parameters of the envelope function to be identi‐ fied are the total duration time *Tf* , the AI *Ia*, the maximum envelope time *tm*, *η* and *κ*. The regression laws of the envelope function obtained from the pre-envelope functions of the seismic event selected from the database are

$$T\_f\left(PGA\right) = T\_f^{(0)}\left(\mathbf{e}^{\left(T\_f^{(1)}PGA\right)} + \mathbf{e}^{\left(T\_f^{(2)}PGA\right)}\right) \tag{25}$$

**Figure 6.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil A.

$$I\_a\left(PGA\right) = \exp\left(I\_a^0 + I\_a^1 \log\left(PGA\right)\right) \tag{26}$$

$$t\_m \left( PGA \right) = t\_m^0 e^{\left(l\_m^1 PGA\right)} \tag{27}$$

valuated from the real data. The figures 6, 7, 8, 9, 10, 11, 12 and 13 show that the curves of the regression laws for the other parameters do not fit perfectly the numerical values of these parameters estimated for the selected accelograms of the database. The regression laws achieve one purpose of this study: the definition of analytical relations to estimate the most important parameters for different kinds of soil that characterize the amplitude modulation of earthquake records and the energy release of seismic events. The numerical values of these parameters for the four kinds of soil are collected in the table 1. These results can be used to calculate the envelope function that modulates the amplitude intensity of stationary filtered WN process to

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159

**Figure 7.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events

selected from the database and occurred in sites characterized by kind of soil B.

generate artificial accelerograms typical of a certain kind of the soil.

$$
\eta \left( PGA \right) = \eta^0 + \alpha\_\eta PGA \tag{28}
$$

$$
\kappa \left( PGA \right) = \kappa^0 + \alpha\_\kappa PGA. \tag{29}
$$

In these equations the PGA is express in g (9.81 m/sec2 ). In the figure 6, 7, 8 and 9 it is fair that the curves of the regression law of the AI (eq. (26)) matches perfectly the trend of the AI valuated from the real data. The figures 6, 7, 8, 9, 10, 11, 12 and 13 show that the curves of the regression laws for the other parameters do not fit perfectly the numerical values of these parameters estimated for the selected accelograms of the database. The regression laws achieve one purpose of this study: the definition of analytical relations to estimate the most important parameters for different kinds of soil that characterize the amplitude modulation of earthquake records and the energy release of seismic events. The numerical values of these parameters for the four kinds of soil are collected in the table 1. These results can be used to calculate the envelope function that modulates the amplitude intensity of stationary filtered WN process to generate artificial accelerograms typical of a certain kind of the soil.

( ) ( ( )) 0 1 exp log *<sup>a</sup> a a I PGA I I PGA* = + (26)

*m m t PGA t e* = (27)

= + (28)

= + (29)

). In the figure 6, 7, 8 and 9 it is fair that

( ) <sup>1</sup> <sup>0</sup> ( ) *mt PGA*

**Figure 6.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events

( ) <sup>0</sup> *PGA PGA*

( ) <sup>0</sup> *PGA PGA*.

 ka

 ha

h

selected from the database and occurred in sites characterized by kind of soil A.

158 Earthquake Research and Analysis - New Advances in Seismology

k

In these equations the PGA is express in g (9.81 m/sec2

h

k

the curves of the regression law of the AI (eq. (26)) matches perfectly the trend of the AI

**Figure 7.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil B.


**Table 1.** Parameters values of the regression laws estimated for all the four kinds of soil.

**Figure 8.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling

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161

selected from the database and occurred in sites characterized by kind of soil C

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling http://dx.doi.org/10.5772/54891 161

**Parameter Soil type A Soil type B Soil type C Soil type D**

16.12 (14.03, 18.2)



1.28

1.407 (1.247, 1.567)

0 [sec] 11.37 15.56 5.379 6.412

1 [sec] -1.281 -6.58 -1.511 -2.185

η <sup>0</sup> 1.798 3.4483 1.8058 1.8473

αη 0.874 3.4498 4.4542 1.9887

κ <sup>0</sup> 1.093 1.0446 1.6106 1.5914

ακ 0.652 2.0301 0.6132 0.5678

**Table 1.** Parameters values of the regression laws estimated for all the four kinds of soil.

(1.092, 1.469)

9.7



1.543 (1.467, 1.618)

1.568 (1.418, 1.717)

(8.189, 11.21)

14.76 (12.13, 17.4)



1.463 (1.351, 1.575)

1.424 (1.312, 1.537)

(-1.841, -0.2726)

*T f* 0

*T f* 1

*T f* 2

*Ia* 0

*Ia* 1

*tm*

*tm*

95% confidence bounds

95% confidence bounds

95% confidence bounds

95% confidence bounds

95% confidence bounds

[sec] 16.73

160 Earthquake Research and Analysis - New Advances in Seismology

[sec] -0.582

[sec] -6.367

(9.371, 24.09)

(-1.496, 0.332)

(-19.57, 6.831)

1.668 (1.556, 1.78)

1.164 (0.9277, 1.4)

**Figure 8.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil C

**Figure 9.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil D.

**Figure 11.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events

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163

**Figure 12.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events

selected from the database and occurred in sites characterized by kind of soil B.

selected from the database and occurred in sites characterized by kind of soil C.

**Figure 10.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil A.

Parameters Identification of Stochastic Nonstationary Process Used in Earthquake Modelling http://dx.doi.org/10.5772/54891 163

**Figure 11.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil B.

**Figure 9.** Values of *Ia* and *Tf* defined as function of the PGA. The value are related to the set of the seismic events

**Figure 10.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events

selected from the database and occurred in sites characterized by kind of soil D.

162 Earthquake Research and Analysis - New Advances in Seismology

selected from the database and occurred in sites characterized by kind of soil A.

**Figure 12.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil C.

In order to estimate the values of the parameters of the envelope function for a class of soil other analytical results are obtained. Relations of the parameters of the MSH envelope function with other characteristics of the earthquake ground motions and the AI are imposed (eqs. (13), (19), (20) and (21)), so analytical formulae to estimate other characteristics of the seismic events in term of PGA and kind of soil are obtained from the regression analysis (eqs. (25),

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The envelope function here presented and the method to calculate its parameters produce a temporal modulation of the amplitude in the synthetic accelograms in term of the most

Finally, the numerical values of the characteristics of the earthquake ground motion obtained from the regression analysis are collected in table to be used in future applications of the

and Sara Sgobba1

significant quantities used in structural engineering: the PGA and the kind of soil.

, Mariantonietta Morga2

the Engineering Mechanics Division, 1964; 90: 113–150.

Bulletin of the Earthquake Research Institute, 1957; 35: 309–325.

ing - Tokio, volume 2, pages 781–798. Science Council of Japan, 1960.

logical Society of America, 1947; 37(1) 19–31.

1 Department of Civil and Architectural Science, Technical University of Bari, Bari, Italy

2 Mobility Department – Transportation Infrastructure Technologies, Austrian Institute of

[1] Bolotin VV. Statistical theory of the aseismic design of structures. In: 2nd World Con‐ ference on Earthquake Engineering - Tokio, volume 2, pages 1365-1374. Science

[2] Housner GW and Jennings PC. Generation of artificial earthquakes. ASCE, Journal of

[3] Housner GW. Characteristics of strong-motion earthquakes. Bulletin of the Seismo‐

[4] Kanai K. Semi-empirical formula for the seismic characteristics of the ground motion.

[5] Tajimi H. A statistical method of determining the maximum response of a building structure during an earthquake. In: 2nd World Conference on Earthquake Engineer‐

(26), (27), (28), (29)).

earthquake engineering.

Giuseppe Carlo Marano1

Technologies GmbH, Vienna, Austria

Council of Japan, 1960.

**Author details**

**References**

**Figure 13.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events selected from the database and occurred in sites characterized by kind of soil D.

#### **6. Conclusions**

One of the main problems of earthquake engineering is the proper estimation of the charac‐ teristics of future earthquakes that will affect new and existing structures. This is a non-trivial problem because of the inner unpredictable nature of earthquakes. Due to this nature of earthquakes, stochastic models have been proposed to generate synthetic future seismic accelerograms to use in the structural design. Some of the stochastic models already proposed relate the stochastic ground motion process to seismological parameters that are not meaningful in structural engineering. The study here proposed overcomes this limit: it presents a model that describes the earthquake ground motion in term of parameters useful in the structural engineering. This model is a non-stationary stochastic one based on the stationary C-P model and characterized by the temporal modulation of the amplitude. The amplitude modulation is produced by a new envelope function that has the same shape of the SH function, but it is described by only two parameters. In order to obtain the values of these parameters of the envelope function a complex procedure is used. The procedure has two stages:


In order to estimate the values of the parameters of the envelope function for a class of soil other analytical results are obtained. Relations of the parameters of the MSH envelope function with other characteristics of the earthquake ground motions and the AI are imposed (eqs. (13), (19), (20) and (21)), so analytical formulae to estimate other characteristics of the seismic events in term of PGA and kind of soil are obtained from the regression analysis (eqs. (25), (26), (27), (28), (29)).

The envelope function here presented and the method to calculate its parameters produce a temporal modulation of the amplitude in the synthetic accelograms in term of the most significant quantities used in structural engineering: the PGA and the kind of soil.

Finally, the numerical values of the characteristics of the earthquake ground motion obtained from the regression analysis are collected in table to be used in future applications of the earthquake engineering.

### **Author details**

**Figure 13.** Values of *tm*, η and κ defined as function of the PGA. The value are related to the set of the seismic events

One of the main problems of earthquake engineering is the proper estimation of the charac‐ teristics of future earthquakes that will affect new and existing structures. This is a non-trivial problem because of the inner unpredictable nature of earthquakes. Due to this nature of earthquakes, stochastic models have been proposed to generate synthetic future seismic accelerograms to use in the structural design. Some of the stochastic models already proposed relate the stochastic ground motion process to seismological parameters that are not meaningful in structural engineering. The study here proposed overcomes this limit: it presents a model that describes the earthquake ground motion in term of parameters useful in the structural engineering. This model is a non-stationary stochastic one based on the stationary C-P model and characterized by the temporal modulation of the amplitude. The amplitude modulation is produced by a new envelope function that has the same shape of the SH function, but it is described by only two parameters. In order to obtain the values of these parameters of the envelope function a complex procedure is used. The procedure has two

**1.** The estimation of parameters for each of selected accelerograms of the PEER Next Generation Attenuation database to generate a pre-envelope function of each accelogram.

**2.** The regression analysis of the values of these parameters to obtain their mean values for

selected from the database and occurred in sites characterized by kind of soil D.

164 Earthquake Research and Analysis - New Advances in Seismology

**6. Conclusions**

stages:

a class of soil.

Giuseppe Carlo Marano1 , Mariantonietta Morga2 and Sara Sgobba1

1 Department of Civil and Architectural Science, Technical University of Bari, Bari, Italy

2 Mobility Department – Transportation Infrastructure Technologies, Austrian Institute of Technologies GmbH, Vienna, Austria

#### **References**


[6] Sgobba S, Marano GC, Stafford PJ, and Greco R. New Trends in Seismic Design of Structures, chapter: Seismologically consistent stochastic spectra. Saxe-Coburg Pub‐ lisher, 2009.

**Chapter 8**

**Provisional chapter**

**Daily Variation in Earthquake Detection Capability: A**

Evaluating the capability for detection of an earthquake catalogue is the first step in statistical seismicity analysis. In recent years, many studies have focused on ways to assess the completeness magnitude (*Mc*), the lowest magnitude level at which all earthquakes are recorded and there are no missing earthquakes, in a global earthquake catalogue [1, 2], regional catalogues [3–6], and small-scale observations such as underground mines [7]. We cannot make full use of the available information in an earthquake catalogue if we

[8] is a representative example showing that an accurate choice of *Mc* is vital in a seismicity analysis. This study examined the global earthquake catalogue provided by the National Oceanic and Atmospheric Administration (NOAA), United States Department of Commerce, and reported that the number of earthquakes during nighttime was significantly higher than that during daytime. Some studies [9, 10] stated that the daily variation in seismic activity [8] had found was just a bias; [8] did not take into account the daily change in detection capability, and earthquakes with small magnitudes — for which records were incomplete —

Empirically, it is well known that the level of cultural noise has daytime/nighttime variations and that this causes a daily variation in the detection capability. Nonetheless, in many studies, the estimation of *Mc* or the evaluation of the detection capability is carried out for a catalogue ranging over several days, weeks, months or years without taking short-time variation into consideration; *Mc* is underestimated in daytime and overestimated

For an appropriate and precise evaluation of the detection capability, it is vital to quantify this temporal change. In this chapter, we propose a statistical method for a quantitative evaluation of the daily variation in the detection capability and also present an example of

> ©2012 Iwata, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the © 2013 Iwata; licensee InTech. This is an open access article distributed under the terms of the Creative original work is properly cited. Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Iwata; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

overestimate *Mc*, and underestimation of *Mc* yields incorrect or biased results.

**Daily Variation in Earthquake Detection Capability:**

**Quantitative Evaluation**

**A Quantitative Evaluation**

http://dx.doi.org/10.5772/54890

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Takaki Iwata

**1. Introduction**

were used in the analysis.

in nighttime, compared with the true value of *Mc*.

Takaki Iwata


**Provisional chapter**

### **Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation A Quantitative Evaluation**

**Daily Variation in Earthquake Detection Capability:**

Takaki Iwata Additional information is available at the end of the chapter

Takaki Iwata

[6] Sgobba S, Marano GC, Stafford PJ, and Greco R. New Trends in Seismic Design of Structures, chapter: Seismologically consistent stochastic spectra. Saxe-Coburg Pub‐

[8] Jangid RS. Response of SDoF system to non-stationary earthquake excitation. Earth‐

[9] Campbell KW. Prediction of strong ground motion using the hybrid empirical meth‐ od: example application to eastern-north America. Bulletin of the Seismological Soci‐

[10] Cua G. Creating the Virtual Seismologist: Developments in Ground Motion Charac‐ terization and Seismic Early Warning. PhD thesis, California Institute of Technology,

[11] Stafford PJ, Sgobba S, and Marano GC. An energy-based envelope function for the stochastic simulation of earthquake accelerograms. Soil Dynamics and Earthquake

[12] Baker JW. Correlation of ground motion intensity parameters used for predicting structural and geotechnical response. In: ICASP10 - 10th International Conference on Applications of Statistics and Probability in Civil Engineering, Tokyo, Japan, 2007. [13] Conte JP and Peng BF. Fully nonstationary analytical earthquake ground-motion

[14] Saragoni GR and Hart GC. Simulation of artificial earthquake. Earthquake Engineer‐

[15] Marano GC, Morga M and Sgobba S. Modelling of stochastic process for earthquake representation as alternative way for structural seismic analysis: past, present and fu‐ ture. In: EQADS 2011 - International Conference on Earthquake Analysis and Design of Structures - Department of Civil Engineering, PSG College of Technology, Coim‐

[16] Amin M and Ang AHS. Nonstationary stochastic model of earthquake motions.

ASCE, Journal of the Engineering Mechanics Division, 1968; 94: 559–583.

model. ASCE, Journal of Engineering Mechanics, 1997; 12: 15–24.

ing and Structural Dynamics, 1974; 2: 249–267.

batore, Tamilnadu, India, December 1-3 2011.

[7] Clough RW and Penzien J. Dynamics of structures, Mc Graw Hill; 1975.

quake Engineering & Structural Dynamics, 2004; 33(15) 1417–1428.

ety of America, 2002; 93(3) 1012–1033.

166 Earthquake Research and Analysis - New Advances in Seismology

Engineering, 2009; 29(7) 1123–1133.

lisher, 2009.

2005.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54890

#### **1. Introduction**

Evaluating the capability for detection of an earthquake catalogue is the first step in statistical seismicity analysis. In recent years, many studies have focused on ways to assess the completeness magnitude (*Mc*), the lowest magnitude level at which all earthquakes are recorded and there are no missing earthquakes, in a global earthquake catalogue [1, 2], regional catalogues [3–6], and small-scale observations such as underground mines [7]. We cannot make full use of the available information in an earthquake catalogue if we overestimate *Mc*, and underestimation of *Mc* yields incorrect or biased results.

[8] is a representative example showing that an accurate choice of *Mc* is vital in a seismicity analysis. This study examined the global earthquake catalogue provided by the National Oceanic and Atmospheric Administration (NOAA), United States Department of Commerce, and reported that the number of earthquakes during nighttime was significantly higher than that during daytime. Some studies [9, 10] stated that the daily variation in seismic activity [8] had found was just a bias; [8] did not take into account the daily change in detection capability, and earthquakes with small magnitudes — for which records were incomplete were used in the analysis.

Empirically, it is well known that the level of cultural noise has daytime/nighttime variations and that this causes a daily variation in the detection capability. Nonetheless, in many studies, the estimation of *Mc* or the evaluation of the detection capability is carried out for a catalogue ranging over several days, weeks, months or years without taking short-time variation into consideration; *Mc* is underestimated in daytime and overestimated in nighttime, compared with the true value of *Mc*.

For an appropriate and precise evaluation of the detection capability, it is vital to quantify this temporal change. In this chapter, we propose a statistical method for a quantitative evaluation of the daily variation in the detection capability and also present an example of

©2012 Iwata, licensee InTech. This is an open access chapter distributed under the terms of the Creative

the method in real earthquake sequences. A short summary of this chapter has already been presented in a non-peer reviewed article [11], and this chapter focuses on showing the details of the approach and results that appeared in that article.

**3. Statistical method to evaluate the daily variation in detection**

**3.1. Evaluation of the detection capability by using a statistical model to**

[15] proposed a probability density function *f*(*M*) for an observed magnitude-frequency distribution of earthquakes over all magnitude range. The probability density function is

The first comes from the Gutenberg-Richter (GR) law [16], which is equivalent to an

where the parameter *β* is related to the *b*-value of the GR-law and their relationship is

The second is a detection rate function *q*(*M*) showing the proportion of detected earthquakes to all earthquakes at magnitude *M*. Following the proposal by [17], the cumulative distribution function of a normal distribution is used as the detection probability function:

> 1 <sup>√</sup>2*πσ*

We normalize the product of the two functions, and then we obtain the target probability

<sup>−</sup><sup>∞</sup> *<sup>w</sup>*(*M*|*β*)*q*(*M*|*µ*, *<sup>σ</sup>*)*dM* <sup>=</sup> exp(−*βM*)*q*(*M*|*µ*, *<sup>σ</sup>*) · *<sup>β</sup>* exp

Of the three parameters (*µ*, *β*, and *σ*) in this statistical model, the parameter *µ* has the closest connection with the detection capability; *µ* indicates the magnitude at which the detection probability is 50 %. This means that the detection capability is better as the value of *µ* is

The values of the three parameters are estimated by the maximum likelihood method. The

*N* ∑ *i*

exp

<sup>−</sup> (*<sup>x</sup>* <sup>−</sup> *<sup>µ</sup>*)<sup>2</sup> 2*σ*<sup>2</sup>

*βµ* <sup>−</sup> *<sup>β</sup>*2*σ*<sup>2</sup> 2 

ln *f*(*Mi*|*β*, *µ*, *σ*), (4)

*dx* (2)

http://dx.doi.org/10.5772/54890

169

. (3)

*w*(*M*|*β*) = *β* exp(−*βM*), (1)

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

**represent an observed magnitude-frequency distribution**

assumed to be the product of two probability distributions.

*<sup>q</sup>*(*M*|*µ*, *<sup>σ</sup>*) = *<sup>M</sup>*

−<sup>∞</sup>

*<sup>f</sup>*(*M*|*µ*, *<sup>β</sup>*, *<sup>σ</sup>*) = *<sup>w</sup>*(*M*|*β*)*q*(*M*|*µ*, *<sup>σ</sup>*) ∞

ln *L*(*β*, *µ*, *σ*) =

**capability of earthquakes**

exponential distribution:

described by *β* = *b* ln 10.

smaller.

density function as follows (Figure 2):

log-likelihood function (ln *L*) is given by

#### **2. Data**

The earthquake data used in this study were retrieved from the Japan Meteorological Agency (JMA) catalogue from January 2006 to December of 2010, with focal depths shallower than 30 km. The magnitude scale used in the JMA catalogue has been termed the "JMA magnitude", which is determined on the basis of velocity and displacement seismograms [12, 13].

The detection capability of earthquakes has regional variations, and in particular there is clear difference between inland and offshore regions. Because the main interest of this study is the temporal variation in the detection capability, to mitigate the influence of the regional differences, this study used the events occurring within the "Mainland" area defined by [14], which covers the inland and coastal regions of Japan (Figure 1). The total number of the events is 331,537.

**Figure 1.** "Mainland" defined by [14], which covers the inland and coastal regions of Japan (after [14]).

Because the main object of the present study is to investigate the daily variation in earthquake detection capability, the data were divided into periods of one day, and the one-day sequences were stacked. The technical limitations of computer memory space and computing time make it difficult to handle hundreds of thousands events simultaneously in a Bayesian analysis, as described in the next section. Thus, the stacked sequences were constructed for each of the five years; five datasets were analyzed to examine the daily change in the detection capability.

#### **3. Statistical method to evaluate the daily variation in detection capability of earthquakes**

2 Earthquake Research and Analysis / Book 2

**2. Data**

events is 331,537.

detection capability.

of the approach and results that appeared in that article.

the method in real earthquake sequences. A short summary of this chapter has already been presented in a non-peer reviewed article [11], and this chapter focuses on showing the details

The earthquake data used in this study were retrieved from the Japan Meteorological Agency (JMA) catalogue from January 2006 to December of 2010, with focal depths shallower than 30 km. The magnitude scale used in the JMA catalogue has been termed the "JMA magnitude", which is determined on the basis of velocity and displacement seismograms [12, 13].

The detection capability of earthquakes has regional variations, and in particular there is clear difference between inland and offshore regions. Because the main interest of this study is the temporal variation in the detection capability, to mitigate the influence of the regional differences, this study used the events occurring within the "Mainland" area defined by [14], which covers the inland and coastal regions of Japan (Figure 1). The total number of the

**Figure 1.** "Mainland" defined by [14], which covers the inland and coastal regions of Japan (after [14]).

Because the main object of the present study is to investigate the daily variation in earthquake detection capability, the data were divided into periods of one day, and the one-day sequences were stacked. The technical limitations of computer memory space and computing time make it difficult to handle hundreds of thousands events simultaneously in a Bayesian analysis, as described in the next section. Thus, the stacked sequences were constructed for each of the five years; five datasets were analyzed to examine the daily change in the

#### **3.1. Evaluation of the detection capability by using a statistical model to represent an observed magnitude-frequency distribution**

[15] proposed a probability density function *f*(*M*) for an observed magnitude-frequency distribution of earthquakes over all magnitude range. The probability density function is assumed to be the product of two probability distributions.

The first comes from the Gutenberg-Richter (GR) law [16], which is equivalent to an exponential distribution:

$$w(M|\beta) = \beta \exp(-\beta M),\tag{1}$$

where the parameter *β* is related to the *b*-value of the GR-law and their relationship is described by *β* = *b* ln 10.

The second is a detection rate function *q*(*M*) showing the proportion of detected earthquakes to all earthquakes at magnitude *M*. Following the proposal by [17], the cumulative distribution function of a normal distribution is used as the detection probability function:

$$q(M|\mu,\sigma) = \int\_{-\infty}^{M} \frac{1}{\sqrt{2\pi}\sigma} \exp\left[-\frac{(\mathbf{x}-\boldsymbol{\mu})^2}{2\sigma^2}\right] d\mathbf{x} \tag{2}$$

We normalize the product of the two functions, and then we obtain the target probability density function as follows (Figure 2):

$$\begin{split} f(M|\mu,\beta,\sigma) &= \frac{w(M|\beta)q(M|\mu,\sigma)}{\int\_{-\infty}^{\infty} w(M|\beta)q(M|\mu,\sigma)dM} \\ &= \exp(-\beta M)q(M|\mu,\sigma) \cdot \beta \exp\left[\beta\mu - \frac{\beta^2 \sigma^2}{2}\right]. \tag{3} \end{split} \tag{3}$$

Of the three parameters (*µ*, *β*, and *σ*) in this statistical model, the parameter *µ* has the closest connection with the detection capability; *µ* indicates the magnitude at which the detection probability is 50 %. This means that the detection capability is better as the value of *µ* is smaller.

The values of the three parameters are estimated by the maximum likelihood method. The log-likelihood function (ln *L*) is given by

$$\ln L(\beta, \mu, \sigma) = \sum\_{i}^{N} \ln f(M\_i | \beta, \mu, \sigma), \tag{4}$$

As a preliminary analysis to examine the daily variation, this model was applied to the datasets of nighttime (0.0–0.2 days) and daytime (0.4–0.6 days) earthquakes; we found a clear

> 0.0 - 0.2 days β = 1.91 (*b* = 0.83)

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

http://dx.doi.org/10.5772/54890

171

0.4 - 0.6 days β = 1.78 (*b* = 0.77)

µ = 0.32 σ = 0.36

µ = 0.50 σ = 0.37


**Figure 4.** Observed (open circles) and estimated (solid line) probability density functions of the magnitude of nighttime (0.0–0.2 days, red) and daytime (0.4–0.6 days, blue) earthquakes. The dotted black line indicates the estimated probability density

**3.2. Evaluation of the daily variation of the detection capability in a Bayesian**

To quantify the daily variation of the detection capability, we estimated the temporal change in the three parameters *µ*, *β*, and *σ* that appeared in the model. The procedure of the

To represent the temporal variations in the parameters, we introduced a piecewise linear function or linear spline [19]. The nodal point of the spline were taken at each of the occurrence times of the earthquakes in a stacked sequence. For the sake of brevity, we

of the *i*-th earthquake. Hence, the temporal variations of *µ*(*t*), *β*(*t*), and *σ*(*t*) were represented

*<sup>i</sup>* as the values of *β*, *µ*, and *σ*, respectively, at the occurrence time *ti*

shift of the detection capability between the two datasets (Figure 4).

10-6

estimation is similar to that used in [2, 18].

(3)

**framework**

defined *θ*

by

(1) *<sup>i</sup>* , *θ* (2) *<sup>i</sup>* , and *θ*

function for all the shallow earthquakes used in this study, as shown in Figure 3.

10-5

10-4

10-3

Probability density

10-2

10-1

**Figure 2.** Schematic diagram showing the construction of the probability density function for an observed magnitude-frequency distribution of earthquakes over all magnitude range, as proposed by [15].

where *N* and *Mi* denote the number of the analyzed earthquakes and the magnitude of the *i*-th earthquake, respectively. The set of the values of the three parameters maximizing the log-likelihood function is the best estimate.

To observe the performance of this statistical model, we applied it to the magnitude-frequency distribution of the earthquakes taken from the JMA catalogue described in the previous section. As observed in Figure 3, the estimated curve of the statistical model fits well with the data, suggesting that this model is appropriate to describe the magnitude-frequency distribution on the JMA magnitude scale.

**Figure 3.** Observed (open circles) and estimated (solid line) probability density functions of the magnitudes of all the shallow (depth ≤ 30 km) earthquakes within the "Mainland" area [14], from January 2006 to December 2010. The best estimates of *β* (or *b*-value), *µ*, and *σ* are shown in the top right corner.

As a preliminary analysis to examine the daily variation, this model was applied to the datasets of nighttime (0.0–0.2 days) and daytime (0.4–0.6 days) earthquakes; we found a clear shift of the detection capability between the two datasets (Figure 4).

4 Earthquake Research and Analysis / Book 2

4 5 6 7 8 9 Magnitude

Log-frequency

the magnitude-frequency distribution on the JMA magnitude scale.

distribution of earthquakes over all magnitude range, as proposed by [15].

log-likelihood function is the best estimate.

10-6

(or *b*-value), *µ*, and *σ* are shown in the top right corner.

10-5

10-4

10-3

Probability density

10-2

10-1

4 5 6 7 8 9 Magnitude

**Figure 2.** Schematic diagram showing the construction of the probability density function for an observed magnitude-frequency

where *N* and *Mi* denote the number of the analyzed earthquakes and the magnitude of the *i*-th earthquake, respectively. The set of the values of the three parameters maximizing the

To observe the performance of this statistical model, we applied it to the magnitude-frequency distribution of the earthquakes taken from the JMA catalogue described in the previous section. As observed in Figure 3, the estimated curve of the statistical model fits well with the data, suggesting that this model is appropriate to describe

> -1 0 1 2 3 4 5 6 7 Magnitude

**Figure 3.** Observed (open circles) and estimated (solid line) probability density functions of the magnitudes of all the shallow (depth ≤ 30 km) earthquakes within the "Mainland" area [14], from January 2006 to December 2010. The best estimates of *β*

∝ x

*f*(*M* | β, µ*,* σ) GR-law : *w*(*M* | β) Detection rate function *q*(*M* | µ*,* σ)

Log-frequency

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

β = 1.89 (*b* = 0.82)

µ = 0.42 σ = 0.37 4 5 6 7 8 9 Magnitude µ

Detection rate

**Figure 4.** Observed (open circles) and estimated (solid line) probability density functions of the magnitude of nighttime (0.0–0.2 days, red) and daytime (0.4–0.6 days, blue) earthquakes. The dotted black line indicates the estimated probability density function for all the shallow earthquakes used in this study, as shown in Figure 3.

#### **3.2. Evaluation of the daily variation of the detection capability in a Bayesian framework**

To quantify the daily variation of the detection capability, we estimated the temporal change in the three parameters *µ*, *β*, and *σ* that appeared in the model. The procedure of the estimation is similar to that used in [2, 18].

To represent the temporal variations in the parameters, we introduced a piecewise linear function or linear spline [19]. The nodal point of the spline were taken at each of the occurrence times of the earthquakes in a stacked sequence. For the sake of brevity, we defined *θ* (1) *<sup>i</sup>* , *θ* (2) *<sup>i</sup>* , and *θ* (3) *<sup>i</sup>* as the values of *β*, *µ*, and *σ*, respectively, at the occurrence time *ti* of the *i*-th earthquake. Hence, the temporal variations of *µ*(*t*), *β*(*t*), and *σ*(*t*) were represented by

$$\begin{aligned} \boldsymbol{\phi}\_{1}(t) &= \boldsymbol{\beta}(t) = \frac{\boldsymbol{\theta}\_{i+1}^{(1)} - \boldsymbol{\theta}\_{i}^{(1)}}{t\_{i+1} - t\_{i}} (t - t\_{i}) + \boldsymbol{\theta}\_{i}^{(1)} \; \prime \\ \boldsymbol{\phi}\_{2}(t) &= \boldsymbol{\mu}(t) = \frac{\boldsymbol{\theta}\_{i+1}^{(2)} - \boldsymbol{\theta}\_{i}^{(2)}}{t\_{i+1} - t\_{i}} (t - t\_{i}) + \boldsymbol{\theta}\_{i}^{(2)} \; \prime \\ \boldsymbol{\phi}\_{3}(t) &= \boldsymbol{\sigma}(t) = \frac{\boldsymbol{\theta}\_{i+1}^{(3)} - \boldsymbol{\theta}\_{i}^{(3)}}{t\_{i+1} - t\_{i}} (t - t\_{i}) + \boldsymbol{\theta}\_{i}^{(3)} \quad \text{ for } t\_{i} \le t < t\_{i+1} . \end{aligned} \tag{5}$$

Then, we introduced a penalized log-likelihood function [20, 21]

*<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3)

� *vj*

To deal with this computational problem we isolated *θ*

(1) −*N*; *<sup>θ</sup>* (2) −*N*; *<sup>θ</sup>*(3) −*N*)

*<sup>θ</sup>*−*<sup>N</sup>* = (*<sup>θ</sup>*

(1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , *θ* (3)

(1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , *θ* (3) *<sup>N</sup>* ) = �

where <sup>Θ</sup> denotes the parameter space of *<sup>θ</sup>*−*N*.

We intended to find the set of the values of *v*1, *v*2, *v*3, *θ*

distribution *<sup>π</sup>*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

L(*v*1, *v*2, *v*3, *θ*

= (*θ* (1) <sup>1</sup> ,..., *θ*

� *vj*

*N*−1 ∏ *i*=1

= 3 ∏ *j*=1

and is a so-called improper prior.

integral of *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) over

*<sup>π</sup>*−*N*(*θN*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

·

the values of *vj*(*j* = 1, 2, 3).

*<sup>Q</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) = ln *<sup>L</sup>*(*θ*) <sup>−</sup> <sup>Φ</sup>(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3), (10)

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

*<sup>i</sup>*+<sup>1</sup> <sup>−</sup> *<sup>φ</sup>*(*j*) *<sup>i</sup>* )<sup>2</sup>  

*<sup>N</sup>* (*<sup>j</sup>* <sup>=</sup> 1, 2, 3) from *<sup>θ</sup>*, because the

*<sup>N</sup>*−1). (12)

*<sup>N</sup>* )*dθ*−*N*, (13)

*<sup>N</sup>* which maximizes the

(3)

*<sup>N</sup>* ) and the likelihood function *<sup>L</sup>*(*θ*) over *<sup>θ</sup>*−*N*:

(1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , *θ* (3)

(3)

. (11)

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173

*ti*+<sup>1</sup> − *ti*

<sup>1</sup> <sup>−</sup> *<sup>φ</sup>*(*j*) *<sup>N</sup>* )<sup>2</sup>

*t*<sup>1</sup> − *Ts* + *Te* − *tN*

<sup>−</sup> *<sup>v</sup>*(*φ*(*j*)

(*j*)

(2) *<sup>N</sup>*−1; *<sup>θ</sup>* (3) <sup>1</sup> ,..., *θ*

*<sup>N</sup>* ) which is a proper prior with respect to *<sup>θ</sup>*−*N*.

*<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , and *θ*

and the maximization of *<sup>Q</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) provides the best estimate of *<sup>θ</sup>*, and it depends on

A Bayesian framework with the type II maximum likelihood approach [22] or the maximization of marginal likelihood [23] enables us to determine the values of *vj*'s objectively. We supposed that the prior distribution of *θ* corresponding to the smoothness constraint is proportional to exp[−Φ(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3)]; the equation (9) and the consideration of

> <sup>−</sup> *<sup>v</sup>*(*φ*(*j*)

To find the marginal likelihood [24], we need to integrate the product of the prior distribution *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) and the likelihood function *<sup>L</sup>*(*θ*) shown in the equation (7) over *<sup>θ</sup>*, but this integration is unachievable. This is because the integral of *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) over *<sup>θ</sup>* is infinite

is finite; we rewrote the original prior distribution *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) as

Then, to obtain the marginal likelihood L, we integrated out the product of the prior

marginal likelihood, because such a set is the best estimate of the six parameters [22, 23].

the normalizing constant give us the prior distribution *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) as follows:

*<sup>π</sup>*(*ti*+<sup>1</sup> <sup>−</sup> *ti*) exp

*<sup>π</sup>*(*t*<sup>1</sup> <sup>−</sup> *Ts* <sup>+</sup> *Te* <sup>−</sup> *tN*) exp

(1) *<sup>N</sup>*−1; *<sup>θ</sup>* (2) <sup>1</sup> ,..., *θ*

(1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , *θ* (3)

Θ

The goal of the estimation is the optimization of the following parameters:

$$\begin{split} \boldsymbol{\theta} &= (\boldsymbol{\theta}^{(1)}; \boldsymbol{\theta}^{(2)}; \boldsymbol{\theta}^{(3)}) \\ &= (\boldsymbol{\theta}\_1^{(1)}, \dots, \boldsymbol{\theta}\_N^{(1)}; \boldsymbol{\theta}\_1^{(2)}, \dots, \boldsymbol{\theta}\_N^{(2)}; \boldsymbol{\theta}\_1^{(3)}, \dots, \boldsymbol{\theta}\_N^{(3)}). \end{split} \tag{6}$$

The simultaneous optimization of such a large number (= 3*N*) of parameters is, however, an unstable process; we incorporated a smoothness constraint or roughness penalty on *φi*(*t*)(*i* = 1, 2, 3) to enhance the stability of the optimization.

Following the equation (4), the log-likelihood function in this case is given by

$$\ln L(\boldsymbol{\theta}) = \sum\_{i}^{N} \ln f(M\_i | \boldsymbol{\theta}\_i^{(1)}, \boldsymbol{\theta}\_i^{(2)}, \boldsymbol{\theta}\_i^{(3)}),\tag{7}$$

and the smoothness constraint is quantified by the following equation:

$$\Phi(\boldsymbol{\theta}|\boldsymbol{v}\_{1},\boldsymbol{v}\_{2},\boldsymbol{v}\_{3}) = \sum\_{j=1}^{3} \boldsymbol{v}\_{j} \int\_{T\_{s}}^{T\_{\ell}} \left[\frac{\partial}{\partial t} \boldsymbol{\phi}\_{j}(t)\right]^{2} dt,\tag{8}$$

where *vj* is the parameter controlling the trade-off between the goodness-of-fit of the statistical model to the data and the smoothness constraint, and [*Ts*, *Te*] denotes the domain of the analyzed time period.

In the analysis of an ordinary earthquake sequence such as [2, 18], the start and end points (*Ts* and *Te*) are different but in the present case they are the same, midnight because the stacked one-day data are analyzed. Thus, the values of *β*, *µ*, and *σ* at *Ts* and *Te* should connect smoothly. Considering this property and the equation (5) we rewrite the equation (8) as follows:

$$\Phi(\boldsymbol{\theta}|\boldsymbol{v}\_{1},\boldsymbol{v}\_{2},\boldsymbol{v}\_{3}) = \sum\_{j=1}^{3} v\_{j} \left[ \sum\_{i=1}^{N-1} \frac{(\boldsymbol{\theta}\_{i+1}^{(j)} - \boldsymbol{\mu}\_{i}^{(j)})^{2}}{t\_{i+1} - t\_{i}} + \frac{(\boldsymbol{\theta}\_{1}^{(j)} - \boldsymbol{\theta}\_{N}^{(j)})^{2}}{(t\_{1} - T\_{s}) + (T\_{\varepsilon} - t\_{N})} \right] \tag{9}$$

Then, we introduced a penalized log-likelihood function [20, 21]

6 Earthquake Research and Analysis / Book 2

*φ*1(*t*) = *β*(*t*) =

*φ*2(*t*) = *µ*(*t*) =

*φ*3(*t*) = *σ*(*t*) =

*θ* = (*θ*(1)

= (*θ* (1) <sup>1</sup> ,..., *θ*

1, 2, 3) to enhance the stability of the optimization.

of the analyzed time period.

<sup>Φ</sup>(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) =

(8) as follows:

*θ* (1) *<sup>i</sup>*+<sup>1</sup> − *θ*

*θ* (2) *<sup>i</sup>*+<sup>1</sup> − *θ*

*θ* (3) *<sup>i</sup>*+<sup>1</sup> − *θ*

The goal of the estimation is the optimization of the following parameters:

(1) *<sup>N</sup>* ; *θ* (2) <sup>1</sup> ,..., *θ*

Following the equation (4), the log-likelihood function in this case is given by

*N* ∑ *i*

ln *L*(*θ*) =

and the smoothness constraint is quantified by the following equation:

<sup>Φ</sup>(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) =

3 ∑ *j*=1 *vj N*−1 ∑ *i*=1

; *θ*(2) ; *θ*(3) )

(1) *i ti*+<sup>1</sup> − *ti*

(2) *i ti*<sup>+</sup><sup>1</sup> − *ti*

(3) *i ti*<sup>+</sup><sup>1</sup> − *ti*

(*t* − *ti*) + *θ*

(*t* − *ti*) + *θ*

(*t* − *ti*) + *θ*

The simultaneous optimization of such a large number (= 3*N*) of parameters is, however, an unstable process; we incorporated a smoothness constraint or roughness penalty on *φi*(*t*)(*i* =

ln *f*(*Mi*|*θ*

3 ∑ *j*=1 *vj* � *Te Ts*

where *vj* is the parameter controlling the trade-off between the goodness-of-fit of the statistical model to the data and the smoothness constraint, and [*Ts*, *Te*] denotes the domain

In the analysis of an ordinary earthquake sequence such as [2, 18], the start and end points (*Ts* and *Te*) are different but in the present case they are the same, midnight because the stacked one-day data are analyzed. Thus, the values of *β*, *µ*, and *σ* at *Ts* and *Te* should connect smoothly. Considering this property and the equation (5) we rewrite the equation

*ti*<sup>+</sup><sup>1</sup> − *ti*

<sup>+</sup> (*<sup>θ</sup>*

(*j*) <sup>1</sup> − *θ* (*j*) *<sup>N</sup>* )<sup>2</sup>

(*t*<sup>1</sup> − *Ts*)+(*Te* − *tN*)

(*θ* (*j*) *<sup>i</sup>*+<sup>1</sup> <sup>−</sup> *<sup>µ</sup>*(*j*) *<sup>i</sup>* )<sup>2</sup>

(1) *<sup>i</sup>* , *θ* (2) *<sup>i</sup>* , *θ* (3)

> � *∂ ∂t φj*(*t*) �2

(1) *<sup>i</sup>* ,

(2) *<sup>i</sup>* ,

(3)

(2) *<sup>N</sup>* ; *θ* (3) <sup>1</sup> ,..., *θ*

*<sup>i</sup>* for *ti* <sup>≤</sup> *<sup>t</sup>* <sup>&</sup>lt; *ti*+1. (5)

*<sup>N</sup>* ). (6)

*<sup>i</sup>* ), (7)

*dt*, (8)

(9)

(3)

$$Q(\theta|v\_1, v\_2, v\_3) = \ln L(\theta) - \Phi(\theta|v\_1, v\_2, v\_3), \tag{10}$$

and the maximization of *<sup>Q</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) provides the best estimate of *<sup>θ</sup>*, and it depends on the values of *vj*(*j* = 1, 2, 3).

A Bayesian framework with the type II maximum likelihood approach [22] or the maximization of marginal likelihood [23] enables us to determine the values of *vj*'s objectively. We supposed that the prior distribution of *θ* corresponding to the smoothness constraint is proportional to exp[−Φ(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3)]; the equation (9) and the consideration of the normalizing constant give us the prior distribution *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) as follows:

$$\begin{aligned} &\pi(\boldsymbol{\theta}|\boldsymbol{v}\_{1},\boldsymbol{v}\_{2},\boldsymbol{v}\_{3})\\ &=\prod\_{j=1}^{3}\prod\_{i=1}^{N-1}\sqrt{\frac{\boldsymbol{v}\_{j}}{\pi(t\_{i+1}-t\_{i})}}\exp\left[-\frac{\boldsymbol{v}(\boldsymbol{\phi}\_{i+1}^{(j)}-\boldsymbol{\phi}\_{i}^{(j)})^{2}}{t\_{i+1}-t\_{i}}\right] \\ &\cdot\sqrt{\frac{\boldsymbol{v}\_{j}}{\pi(t\_{1}-T\_{s}+T\_{\varepsilon}-t\_{N})}}\exp\left[-\frac{\boldsymbol{v}(\boldsymbol{\phi}\_{1}^{(j)}-\boldsymbol{\phi}\_{N}^{(j)})^{2}}{t\_{1}-T\_{s}+T\_{\varepsilon}-t\_{N}}\right]. \end{aligned} \tag{11}$$

To find the marginal likelihood [24], we need to integrate the product of the prior distribution *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) and the likelihood function *<sup>L</sup>*(*θ*) shown in the equation (7) over *<sup>θ</sup>*, but this integration is unachievable. This is because the integral of *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) over *<sup>θ</sup>* is infinite and is a so-called improper prior.

To deal with this computational problem we isolated *θ* (*j*) *<sup>N</sup>* (*<sup>j</sup>* <sup>=</sup> 1, 2, 3) from *<sup>θ</sup>*, because the integral of *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) over

$$\begin{split} \boldsymbol{\theta}\_{-N} &= (\boldsymbol{\theta}\_{-N'}^{(1)}; \boldsymbol{\theta}\_{-N'}^{(2)}; \boldsymbol{\theta}\_{-N}^{(3)}) \\ &= (\boldsymbol{\theta}\_{1}^{(1)}, \dots, \boldsymbol{\theta}\_{N-1}^{(1)}; \boldsymbol{\theta}\_{1}^{(2)}, \dots, \boldsymbol{\theta}\_{N-1}^{(2)}; \boldsymbol{\theta}\_{1}^{(3)}, \dots, \boldsymbol{\theta}\_{N-1}^{(3)}) . \end{split} \tag{12}$$

is finite; we rewrote the original prior distribution *<sup>π</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) as *<sup>π</sup>*−*N*(*θN*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , *θ* (3) *<sup>N</sup>* ) which is a proper prior with respect to *<sup>θ</sup>*−*N*.

Then, to obtain the marginal likelihood L, we integrated out the product of the prior distribution *<sup>π</sup>*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , *θ* (3) *<sup>N</sup>* ) and the likelihood function *<sup>L</sup>*(*θ*) over *<sup>θ</sup>*−*N*:

$$L(v\_1, v\_2, v\_3, \theta\_N^{(1)}, \theta\_N^{(2)}, \theta\_N^{(3)}) = \int\_{\Theta} L(\theta) \pi\_{-N}(\theta\_{-N}|v\_1, v\_2, v\_3, \theta\_N^{(1)}, \theta\_N^{(2)}, \theta\_N^{(3)}) d\theta\_{-N\prime} \tag{13}$$

where <sup>Θ</sup> denotes the parameter space of *<sup>θ</sup>*−*N*.

We intended to find the set of the values of *v*1, *v*2, *v*3, *θ* (1) *<sup>N</sup>* , *θ* (2) *<sup>N</sup>* , and *θ* (3) *<sup>N</sup>* which maximizes the marginal likelihood, because such a set is the best estimate of the six parameters [22, 23]. In this hierarchical Bayesian scheme, the six parameters are often called as hyperparemeters which govern the prior distribution.

The maximization of the marginal likelihood <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) was achieved in the following way. In the first step, we intend to maximize the integrand ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) in the equation (13) with respect to *<sup>θ</sup>*−*N*, which is equivalent to the maximization of the penalized log-likelihood function *<sup>Q</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) appeared in the equation (10).

The logarithm of the integrand in the equation (13) is approximated by a quadratic form at the initial value of *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>*−*N*<sup>0</sup> (and *<sup>θ</sup>* <sup>=</sup> *<sup>θ</sup>*0):

$$\begin{split} & \ln L(\boldsymbol{\theta}) \pi\_{-N}(\boldsymbol{\theta}\_{-N} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) \\ & \approx \ln L(\boldsymbol{\theta}\_{0}) \pi\_{-N}(\boldsymbol{\theta}\_{-N0} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) \\ & \quad + \mathbf{g}(\boldsymbol{\theta}\_{-N0} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) \cdot (\boldsymbol{\theta}\_{-N} - \boldsymbol{\theta}\_{-N0}) \\ & \quad - \frac{1}{2} (\boldsymbol{\theta}\_{-N} - \boldsymbol{\theta}\_{-N0}) H(\boldsymbol{\theta}\_{-N0} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) (\boldsymbol{\theta}\_{-N} - \boldsymbol{\theta}\_{-N0})^{T} . \end{split} \tag{14}$$

ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

<sup>2</sup> ln det*H*(*θ***<sup>ˆ</sup>**

[*<sup>i</sup>* <sup>+</sup> *<sup>j</sup>*(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>)]-th diagonal component of *<sup>H</sup>*(*θ***ˆ**−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

−*N*)*π*(*θ***<sup>ˆ</sup>**

<sup>≈</sup> ln *<sup>L</sup>*(*θ***<sup>ˆ</sup>**

−1

where *n* is the number of parameters included in *θ***ˆ**

maximize the value of ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

[27] and ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3)

temporal variations of the three parameters.

procedure following the equation.

the hyperparameters and *<sup>θ</sup>*−*N*.

<sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(*j*) *i* .

**3.3. Model comparison**

−*N*)*H*(*θ***<sup>ˆ</sup>**

<sup>0</sup>)*π*−*N*(*θ***<sup>ˆ</sup>**

(*θ*−*<sup>N</sup>* <sup>−</sup> *<sup>θ</sup>***<sup>ˆ</sup>**

<sup>≈</sup> ln *<sup>L</sup>*(*θ***<sup>ˆ</sup>**

−1 2

from the equation (14).

is given by

*H*(*θ***ˆ**

error of *θ*

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

<sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

Then, in the second step, the integral in the equation (13) is computed by the Laplace

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

> (1) *N* , *θ* (2) *N* , *θ* (3)

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) + *<sup>n</sup>*

*<sup>N</sup>* )(*θ*−*<sup>N</sup>* <sup>−</sup> *<sup>θ</sup>***<sup>ˆ</sup>**

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

<sup>−</sup>*<sup>N</sup>* and *<sup>n</sup>* <sup>=</sup> <sup>3</sup>(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>) in this case.

*<sup>N</sup>* ). Eventually, we can find the optima of

*<sup>N</sup>* ) is approximately a multidimensional

*<sup>N</sup>* )−<sup>1</sup> gives the estimation errors of the parameters [15]; the

(1) *N* , *θ* (2) *N* , *θ* (3) <sup>−</sup>*N*)*<sup>T</sup>* (17)

http://dx.doi.org/10.5772/54890

175

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

<sup>2</sup> ln 2*π*, (18)

*<sup>N</sup>* )−<sup>1</sup> is the standard

<sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

approximation [26]. Consequently, the log marginal likelihood ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

<sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

<sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3)

(1) *N* , *θ* (2) *N* , *θ* (3)

By repeating the two steps, we attempt to find the values of the six hyperparameters which

With a sufficiently large *n*, the quadratic and Laplace approximations are good

Gaussian distribution. In this case, the inverse of the negative of the Hessian matrix

So far, we assumed that each of the parameters *β*, *µ*, and *σ* has daily variations. For more appropriate statistical modelling, however, it is necessary to examine the significance of the

To apply the model where at least one of the three parameters *β*, *µ*, and *σ* is not assumed to have the daily variation to data, we fix *vj*('s) at 0 where *j* takes the value(s) corresponding to the parameter(s) without the temporal variation. Then the procedure described in section 3.2 is conducted, but we exclude *<sup>θ</sup>*(*j*) in the equation (6) and consider this exclusion in the

where *<sup>g</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) and *<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) are the gradient vector and the negative of the Hessian matrix (second partial derivatives) of the integrand at *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>*−*<sup>N</sup>*0, respectively. The symbol *<sup>T</sup>* denotes the transpose of a vector (or a matrix).

If the choice of the initial value *<sup>θ</sup>*−*N*<sup>0</sup> is appropriate and is not far from *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *θ***ˆ** <sup>−</sup>*<sup>N</sup>* which maximizes the integrand, the quadratic form is an upward convex and *<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is a positive definite. Hence, Cholesky decomposition [25] enables us to factorize the matrix *<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) into the following form:

$$H(\boldsymbol{\theta}\_{-N0}|\boldsymbol{v}\_{1\prime}\boldsymbol{v}\_{2\prime}\boldsymbol{v}\_{3\prime}\boldsymbol{\theta}\_{N\prime}^{(1)}, \boldsymbol{\theta}\_{N\prime}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) = A\boldsymbol{A}^{T}\tag{15}$$

where *A* is a lower triangular matrix.

The quadratic form is maximized at *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>*′ <sup>−</sup>*N*<sup>0</sup> which satisfies

$$A^T(\boldsymbol{\theta}\_{-N} - \boldsymbol{\theta}\_{-N0}) = A^{-1} \mathbf{g}(\boldsymbol{\theta}\_{-N0} | \boldsymbol{v}\_{1\prime} \boldsymbol{v}\_{2\prime} \boldsymbol{v}\_{3\prime} \boldsymbol{\theta}\_{N\prime}^{(1)} \boldsymbol{\theta}\_{N\prime}^{(2)} \boldsymbol{\theta}\_{N}^{(3)}) \tag{16}$$

through the Newton method. We replace *<sup>θ</sup>*−*N*<sup>0</sup> by *<sup>θ</sup>*′ <sup>−</sup>*N*<sup>0</sup> and iterate the procedure until *<sup>θ</sup>*−*<sup>N</sup>* converges to find *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>***<sup>ˆ</sup>** <sup>−</sup>*N*.

At *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>***<sup>ˆ</sup>** <sup>−</sup>*<sup>N</sup>* where the integrand ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is maximized, the second term of the equation (14) vanishes because *<sup>g</sup>*(*θ***<sup>ˆ</sup>** <sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is a zero vector. Hence, around *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>***ˆ**−*N*, the integrand is approximated by

$$\begin{split} & \ln L(\boldsymbol{\theta}) \pi\_{-N}(\boldsymbol{\theta}\_{-N} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) \\ & \approx \ln L(\boldsymbol{\hat{\theta}}\_{0}) \pi\_{-N}(\boldsymbol{\hat{\theta}}\_{-N} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) \\ & \quad \quad - \frac{1}{2} (\boldsymbol{\theta}\_{-N} - \boldsymbol{\hat{\theta}}\_{-N}) H(\boldsymbol{\hat{\theta}}\_{-N} | \boldsymbol{v}\_{1}, \boldsymbol{v}\_{2}, \boldsymbol{v}\_{3}, \boldsymbol{\theta}\_{N}^{(1)}, \boldsymbol{\theta}\_{N}^{(2)}, \boldsymbol{\theta}\_{N}^{(3)}) (\boldsymbol{\theta}\_{-N} - \boldsymbol{\hat{\theta}}\_{-N})^{T} \end{split} \tag{17}$$

from the equation (14).

8 Earthquake Research and Analysis / Book 2

which govern the prior distribution.

ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

appeared in the equation (10).

−1 2

> (1) *N* , *θ* (2) *N* , *θ* (3)

where *A* is a lower triangular matrix.

converges to find *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>***<sup>ˆ</sup>**

At *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>***<sup>ˆ</sup>**

The quadratic form is maximized at *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>*′

through the Newton method. We replace *<sup>θ</sup>*−*N*<sup>0</sup> by *<sup>θ</sup>*′

<sup>−</sup>*N*.

the second term of the equation (14) vanishes because *<sup>g</sup>*(*θ***<sup>ˆ</sup>**

where *<sup>g</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

*<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

*θ***ˆ**

the initial value of *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>*−*N*<sup>0</sup> (and *<sup>θ</sup>* <sup>=</sup> *<sup>θ</sup>*0):

The maximization of the marginal likelihood <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3)

ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

<sup>≈</sup> ln *<sup>L</sup>*(*θ*0)*π*−*N*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3)

(*θ*−*<sup>N</sup>* <sup>−</sup> *<sup>θ</sup>*−*N*0)*H*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

*<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

*<sup>A</sup>T*(*θ*−*<sup>N</sup>* <sup>−</sup> *<sup>θ</sup>*−*N*0) = *<sup>A</sup>*−1*g*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

<sup>−</sup>*<sup>N</sup>* where the integrand ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

zero vector. Hence, around *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>***ˆ**−*N*, the integrand is approximated by

<sup>+</sup>*g*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3)

enables us to factorize the matrix *<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

In this hierarchical Bayesian scheme, the six parameters are often called as hyperparemeters

in the following way. In the first step, we intend to maximize the integrand

is equivalent to the maximization of the penalized log-likelihood function *<sup>Q</sup>*(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3)

The logarithm of the integrand in the equation (13) is approximated by a quadratic form at

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

> (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

*<sup>N</sup>* ) · (*θ*−*<sup>N</sup>* <sup>−</sup> *<sup>θ</sup>*−*N*0)

(1) *N* , *θ* (2) *N* , *θ* (3)

<sup>−</sup>*N*<sup>0</sup> which satisfies

(1) *N* , *θ* (2) *N* , *θ* (3)

*<sup>N</sup>* ) and *<sup>H</sup>*(*θ*−*N*0|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

vector and the negative of the Hessian matrix (second partial derivatives) of the integrand at *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup> *<sup>θ</sup>*−*<sup>N</sup>*0, respectively. The symbol *<sup>T</sup>* denotes the transpose of a vector (or a matrix).

If the choice of the initial value *<sup>θ</sup>*−*N*<sup>0</sup> is appropriate and is not far from *<sup>θ</sup>*−*<sup>N</sup>* <sup>=</sup>

(1) *N* , *θ* (2) *N* , *θ* (3)

<sup>−</sup>*<sup>N</sup>* which maximizes the integrand, the quadratic form is an upward convex and

(1) *N* , *θ* (2) *N* , *θ* (3)

*<sup>N</sup>* ) in the equation (13) with respect to *<sup>θ</sup>*−*N*, which

(1) *N* , *θ* (2) *N* , *θ* (3)

*<sup>N</sup>* ) is a positive definite. Hence, Cholesky decomposition [25]

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) was achieved

*<sup>N</sup>* )(*θ*−*<sup>N</sup>* <sup>−</sup> *<sup>θ</sup>*−*N*0)*T*, (14)

*<sup>N</sup>* ) into the following form:

*<sup>N</sup>* ) (16)

*<sup>N</sup>* ) is maximized,

*<sup>N</sup>* ) = *AAT*, (15)

<sup>−</sup>*N*<sup>0</sup> and iterate the procedure until *<sup>θ</sup>*−*<sup>N</sup>*

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is a

(1) *N* , *θ* (2) *N* , *θ* (3)

<sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

*<sup>N</sup>* ) are the gradient

Then, in the second step, the integral in the equation (13) is computed by the Laplace approximation [26]. Consequently, the log marginal likelihood ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is given by

$$\begin{split} & \ln \mathcal{L}(v\_1, v\_2, v\_3, \theta\_N^{(1)}, \theta\_N^{(2)}, \theta\_N^{(3)}) \\ & \approx \ln L(\hat{\theta}\_{-N}) \pi(\hat{\theta}\_{-N} |v\_1, v\_2, v\_3, \theta\_N^{(1)}, \theta\_N^{(2)}, \theta\_N^{(3)}) \\ & \quad - \frac{1}{2} \ln \det H(\hat{\theta}\_{-N} |v\_1, v\_2, v\_3, \theta\_N^{(1)}, \theta\_N^{(2)}, \theta\_N^{(3)}) + \frac{n}{2} \ln 2\pi, \end{split} \tag{18}$$

where *n* is the number of parameters included in *θ***ˆ** <sup>−</sup>*<sup>N</sup>* and *<sup>n</sup>* <sup>=</sup> <sup>3</sup>(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>) in this case.

By repeating the two steps, we attempt to find the values of the six hyperparameters which maximize the value of ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ). Eventually, we can find the optima of the hyperparameters and *<sup>θ</sup>*−*N*.

With a sufficiently large *n*, the quadratic and Laplace approximations are good [27] and ln *<sup>L</sup>*(*θ*)*π*−*N*(*θ*−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is approximately a multidimensional Gaussian distribution. In this case, the inverse of the negative of the Hessian matrix *H*(*θ***ˆ** <sup>−</sup>*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )−<sup>1</sup> gives the estimation errors of the parameters [15]; the [*<sup>i</sup>* <sup>+</sup> *<sup>j</sup>*(*<sup>N</sup>* <sup>−</sup> <sup>1</sup>)]-th diagonal component of *<sup>H</sup>*(*θ***ˆ**−*N*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )−<sup>1</sup> is the standard error of *θ* (*j*) *i* .

#### **3.3. Model comparison**

So far, we assumed that each of the parameters *β*, *µ*, and *σ* has daily variations. For more appropriate statistical modelling, however, it is necessary to examine the significance of the temporal variations of the three parameters.

To apply the model where at least one of the three parameters *β*, *µ*, and *σ* is not assumed to have the daily variation to data, we fix *vj*('s) at 0 where *j* takes the value(s) corresponding to the parameter(s) without the temporal variation. Then the procedure described in section 3.2 is conducted, but we exclude *<sup>θ</sup>*(*j*) in the equation (6) and consider this exclusion in the procedure following the equation.

The possbile combination of the allowance or non-allowance of the temporal variation in *β*, *µ*, and *σ* yields eight cases in total. We introduce Akaike's Bayesian Information Criterion ABIC [23] in the comparison of the goodness-of-fit of the eight cases to data:

$$\begin{aligned} \text{ABIC} &= -2(\text{maximum In } \mathcal{L}(v\_1, v\_2, v\_3, \theta\_N^{(1)}, \theta\_N^{(2)}, \theta\_N^{(3)}) \\ &+ 2(\text{number of non-fixed hyperparameters}), \end{aligned} \tag{19}$$

1.4

0

0.0 0.2 0.4 0.6 0.8 1.0 Time since midnight [day]

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

http://dx.doi.org/10.5772/54890

177

**Figure 5.** (a) Estimated daily variation of *β* for the stacked sequence for 2007 (bold line), and its two standard error bands (thin lines), as a function of the elapsed time since midnight. The two dotted lines indicate the occurrence times of the Noto Hanto

years, the same as that for 2007 (Table 2). As previously mentioned, in the original sequence for 2009, the case where only the daily variation of *µ* is allowed has been chosen as the best

Figure 6 depicts the daily variations of *µ* estimated for the sequences obtained in each of the five years with the exclusion of the significant activities listed in Table 1 (except 2009). There exist minor differences in these five profiles of *µ*, but they follow the almost the same pattern. First, during the time period between 0.0 and 0.2 days (i.e., between 0 a.m. and 5 a.m.) *µ* takes the smallest value and the detection capability is the best. Then the value of *µ* gradually increases as the time elapses, and it has a local peak around 0.4 days. Following the local peak *µ* shows a transient decrease corresponding to the recovery of the detection capability during lunchtime. At around 0.6 days (between 2 p.m. and 3 p.m.) it has a local peak again, and then it becomes smaller as the time gets closer to midnight. The differences

one. This is probably because no significant seismic activity occurred in that year.

2

4

Magnitude

6

**(b)**

and Chuuetsu-oki earthquakes. (b) Magnitude-time plot of the stacked sequence for 2007.

of the maximum and minimum values of *µ* are 0.21–0.24.

1.6

β

1.8

2.0

**(a)**

where ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>* (1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* ) is the marginal likelihood given by the equation (18). The model with smaller ABIC value is considered to have a better fit to data.

#### **4. Results of the analysis of the JMA catalogue**

On the basis of the comparison of the ABIC values for the eight models, we found that not only *µ* but also *β* and/or *σ* have the significant daily changes in the four stacked sequences except in the sequence for 2009.

Figure 5 shows the estimated daily variation of *β* (or *b*-value) found in the sequence for 2007. In this sequence the value of *β* drastically decreases at around 0.4 days (i.e., approximately 9:30 a.m.). On 27 March 2007 and 16 July 2007, there were two major earthquakes, the Noto Hanto earthquake with *Mw* = 6.9 and the Chuuetsu-oki earthquake with *Mw* = 6.6. Their occurrence times are coincidentally close to 10 a.m. (9:41 a.m. and 10:13 a.m. for the Noto Hanto and Chuuetsu-oki earthquakes, respectively), and these two major earthquakes are followed by active aftershock sequences (see Figure 5b). Thus, it is suspicious that the drastic decrease of *β* is the daily variation of our interest in this study and the two aftershock sequences would cause the decrease of *β*. To confirm this point the aftershock sequences following the Noto Hanto and Chuuetsu-oki earthquakes were excluded, and then the same analysis was applied; the model where only the daily variation of *µ* is allowed was chosen as the best model.


**Table 1.** Active sequences which may affect the examination of the daily variation in *β*, *µ*, and *σ*. The corresponding areas are shown in parentheses.

In similar to this case, from the sequences for 2006, 2008, and 2010, the active sequences listed in Table 1 were excluded and we re-analyzed them after the exclusion. Consequently, the case with only the daily variation of *µ* was considered as the best case for these three

<sup>176</sup> Earthquake Research and Analysis - New Advances in Seismology Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation 11 Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation http://dx.doi.org/10.5772/54890 177

10 Earthquake Research and Analysis / Book 2

where ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

as the best model.

shown in parentheses.

except in the sequence for 2009.

(1) *N* , *θ* (2) *N* , *θ* (3)

**4. Results of the analysis of the JMA catalogue**

The possbile combination of the allowance or non-allowance of the temporal variation in *β*, *µ*, and *σ* yields eight cases in total. We introduce Akaike's Bayesian Information Criterion

On the basis of the comparison of the ABIC values for the eight models, we found that not only *µ* but also *β* and/or *σ* have the significant daily changes in the four stacked sequences

Figure 5 shows the estimated daily variation of *β* (or *b*-value) found in the sequence for 2007. In this sequence the value of *β* drastically decreases at around 0.4 days (i.e., approximately 9:30 a.m.). On 27 March 2007 and 16 July 2007, there were two major earthquakes, the Noto Hanto earthquake with *Mw* = 6.9 and the Chuuetsu-oki earthquake with *Mw* = 6.6. Their occurrence times are coincidentally close to 10 a.m. (9:41 a.m. and 10:13 a.m. for the Noto Hanto and Chuuetsu-oki earthquakes, respectively), and these two major earthquakes are followed by active aftershock sequences (see Figure 5b). Thus, it is suspicious that the drastic decrease of *β* is the daily variation of our interest in this study and the two aftershock sequences would cause the decrease of *β*. To confirm this point the aftershock sequences following the Noto Hanto and Chuuetsu-oki earthquakes were excluded, and then the same analysis was applied; the model where only the daily variation of *µ* is allowed was chosen

(139.00-139.45◦E, 34.65-35.15◦N)

(136.60-137.00◦E, 37.00-37.50◦N) The Chuuetsu-oki earthquake and its aftershock sequence (138.30-138.80◦E, 37.30-37.65◦N) 2008 The Iwate-Miyagi Nairiku earthquake and its aftershock sequence (140.50-141.10◦E, 38.70-39.30◦N)

(139.95-140.10◦E, 37.20-37.35◦N)

2007 The Noto Hanto earthquake and its aftershock sequence

2010 Earthquake sequence in Nakadoori, Fukushima Prefecture

**Table 1.** Active sequences which may affect the examination of the daily variation in *β*, *µ*, and *σ*. The corresponding areas are

In similar to this case, from the sequences for 2006, 2008, and 2010, the active sequences listed in Table 1 were excluded and we re-analyzed them after the exclusion. Consequently, the case with only the daily variation of *µ* was considered as the best case for these three

(1) *N* , *θ* (2) *N* , *θ* (3) *<sup>N</sup>* )

+2(number of non-fixed hyperparameters), (19)

*<sup>N</sup>* ) is the marginal likelihood given by the equation (18). The

ABIC [23] in the comparison of the goodness-of-fit of the eight cases to data:

ABIC <sup>=</sup> <sup>−</sup>2(maximum ln <sup>L</sup>(*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3, *<sup>θ</sup>*

model with smaller ABIC value is considered to have a better fit to data.

**Year Excluded sequence and area** 2006 Earthquake swarm in Izu Peninsula

**Figure 5.** (a) Estimated daily variation of *β* for the stacked sequence for 2007 (bold line), and its two standard error bands (thin lines), as a function of the elapsed time since midnight. The two dotted lines indicate the occurrence times of the Noto Hanto and Chuuetsu-oki earthquakes. (b) Magnitude-time plot of the stacked sequence for 2007.

years, the same as that for 2007 (Table 2). As previously mentioned, in the original sequence for 2009, the case where only the daily variation of *µ* is allowed has been chosen as the best one. This is probably because no significant seismic activity occurred in that year.

Figure 6 depicts the daily variations of *µ* estimated for the sequences obtained in each of the five years with the exclusion of the significant activities listed in Table 1 (except 2009). There exist minor differences in these five profiles of *µ*, but they follow the almost the same pattern. First, during the time period between 0.0 and 0.2 days (i.e., between 0 a.m. and 5 a.m.) *µ* takes the smallest value and the detection capability is the best. Then the value of *µ* gradually increases as the time elapses, and it has a local peak around 0.4 days. Following the local peak *µ* shows a transient decrease corresponding to the recovery of the detection capability during lunchtime. At around 0.6 days (between 2 p.m. and 3 p.m.) it has a local peak again, and then it becomes smaller as the time gets closer to midnight. The differences of the maximum and minimum values of *µ* are 0.21–0.24.


**Table 2.** Model comparison through ABIC values. The differences in ABIC values, ∆ABIC, from cases where only the daily variation in *<sup>µ</sup>* is allowed are shown. The mark √ indicates the allowance of the daily variation in *<sup>β</sup>*, *<sup>µ</sup>*, and *<sup>σ</sup>*.

#### **5. Discussions**

To some extent, we can figure out the temporal pattern of the detection capability through the magnitude-time plots in Figure 6. In these plots there are some absences of small earthquakes with *<sup>M</sup>* <sup>=</sup> <sup>−</sup>0.5 or below during daytime. These plots also show a short recovery of the detection capability around noon. [28] has found a similar pattern of the detection capability in California through the hourly distribution of the number of reported earthquakes in the Advanced National Seismic Systems (ANSS) catalogue. Note that, however, both magnitude-time plots and hourly distributions provides us with only a qualitative pattern of the detection capability. For an understanding of the precise characteristics of an earthquake catalogue it is necessary to evaluate quantitatively the temporal change of the detection capability, and the statistical approach shown in this chapter is effective for such a purpose.

Of the three parameters *β* (or *b*-value), *µ*, and *σ* in the statistical model to describe an observed magnitude-frequency distribution, only *µ* shows the significant daily variation whereas *β* and *σ* do not show any variation. The temporal changes of *b*-value have been reported in some studies [15, 29, 30], and laboratory experiments [31–33], depth dependence [34–36], and faulting-style dependence of the *b*-value [37, 38] suggest a relation between stress state and *b*-value. We could consider air temperature and atmospheric pressure as physical factors which may cause the diurnal cycle of the stress changes in the Earth's crust. However, a constant value of *β* implies that the diurnal stress changes following the daily change of the temperature and/or pressure are insufficient to affect the magnitude-frequency distribution of earthquakes.

It is difficult to arrive at a plausible interpretation for a constant value of *σ*, because the factors controlling the value of *σ* still remain unknown. It has been suggested that *σ* is related to the spatial distribution of seismograph stations [2]. Following this suggestion the result of the unchanged *σ* is reasonable, because the spatial distribution of seismograph stations does not have daily variation.

0.6 0.5 0.4 0.3

2006

2007

2008

2009

2010

µ

2

1

**Figure 6.** (Top) Estimated daily variations of *µ* in the stacked sequences of each of the five years (bold line) with the exclusion of the significant activities listed in Table 1, and their two standard error bands (thin lines), as a function of the elapsed time since midnight. The horizontal dotted lines shows the best estimate of *µ* in the case where we do not consider the temporal variation of *<sup>µ</sup>*. (Bottom) Magnitude-time plot of the stacked sequences for 2007. Note that only earthquakes with *<sup>M</sup>* <sup>≤</sup> 2.5 are plotted to clarify the absences of small earthquakes during daytime but earthquakes with *M* > 2.5 are used in the analysis.

0 Magnitude -1

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Time since midnight [day]

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179

Time since midnight [day]

Time since midnight [day]

Time since midnight [day]

Time since midnight [day]

As briefly described in section 4, the differences between the maximum and minimum values of *µ* are less than 0.25. In Figure 6, the values of *µ* in the case where we do not consider the daily variation of *µ* are indicated by the dotted lines; the difference between the maximum value of *µ* with the daily variation and that without the variation is approximately 0.15.

12 Earthquake Research and Analysis / Book 2

**5. Discussions**

distribution of earthquakes.

have daily variation.

**Models** ∆**ABIC**

*βµ σ* 2006 2007 2008 2009 2010

<sup>√</sup> 266.7 320.3 236.5 495.7 383.5 √ 0.0 0.0 0.0 0.0 0.0

√ √ 2.0 1.8 2.0 2.0 2.0 √ √ 127.4 139.4 103.8 183.93 208.6 √ √ 2.0 2.0 2.0 1.1 1.1 √√ √ 3.9 4.0 4.0 1.4 1.5

**Table 2.** Model comparison through ABIC values. The differences in ABIC values, ∆ABIC, from cases where only the daily

To some extent, we can figure out the temporal pattern of the detection capability through the magnitude-time plots in Figure 6. In these plots there are some absences of small earthquakes with *<sup>M</sup>* <sup>=</sup> <sup>−</sup>0.5 or below during daytime. These plots also show a short recovery of the detection capability around noon. [28] has found a similar pattern of the detection capability in California through the hourly distribution of the number of reported earthquakes in the Advanced National Seismic Systems (ANSS) catalogue. Note that, however, both magnitude-time plots and hourly distributions provides us with only a qualitative pattern of the detection capability. For an understanding of the precise characteristics of an earthquake catalogue it is necessary to evaluate quantitatively the temporal change of the detection capability, and the statistical approach shown in this chapter is effective for such a purpose. Of the three parameters *β* (or *b*-value), *µ*, and *σ* in the statistical model to describe an observed magnitude-frequency distribution, only *µ* shows the significant daily variation whereas *β* and *σ* do not show any variation. The temporal changes of *b*-value have been reported in some studies [15, 29, 30], and laboratory experiments [31–33], depth dependence [34–36], and faulting-style dependence of the *b*-value [37, 38] suggest a relation between stress state and *b*-value. We could consider air temperature and atmospheric pressure as physical factors which may cause the diurnal cycle of the stress changes in the Earth's crust. However, a constant value of *β* implies that the diurnal stress changes following the daily change of the temperature and/or pressure are insufficient to affect the magnitude-frequency

It is difficult to arrive at a plausible interpretation for a constant value of *σ*, because the factors controlling the value of *σ* still remain unknown. It has been suggested that *σ* is related to the spatial distribution of seismograph stations [2]. Following this suggestion the result of the unchanged *σ* is reasonable, because the spatial distribution of seismograph stations does not

As briefly described in section 4, the differences between the maximum and minimum values of *µ* are less than 0.25. In Figure 6, the values of *µ* in the case where we do not consider the daily variation of *µ* are indicated by the dotted lines; the difference between the maximum value of *µ* with the daily variation and that without the variation is approximately 0.15.

variation in *<sup>µ</sup>* is allowed are shown. The mark √ indicates the allowance of the daily variation in *<sup>β</sup>*, *<sup>µ</sup>*, and *<sup>σ</sup>*.

1160.9 1125.1 1092.6 1405.3 1448.9

√ 389.8 352.2 336.1 450.91 546.6

temporal variation Year

**Figure 6.** (Top) Estimated daily variations of *µ* in the stacked sequences of each of the five years (bold line) with the exclusion of the significant activities listed in Table 1, and their two standard error bands (thin lines), as a function of the elapsed time since midnight. The horizontal dotted lines shows the best estimate of *µ* in the case where we do not consider the temporal variation of *<sup>µ</sup>*. (Bottom) Magnitude-time plot of the stacked sequences for 2007. Note that only earthquakes with *<sup>M</sup>* <sup>≤</sup> 2.5 are plotted to clarify the absences of small earthquakes during daytime but earthquakes with *M* > 2.5 are used in the analysis.

This suggests that, if we determine *Mc* without taking into consideration the daily variation of the detection capability, *Mc* − 0.2 would be appropriate as the completeness magnitude to avoid any unexpected biases in our seismicity analysis.

In the present study, we use the smoothness constraint, as shown in the equation (9), which reflects the cyclic property of data. Suppose that we use a different smoothness constraint

$$\Phi(\boldsymbol{\theta}|\boldsymbol{v}\_{1},\boldsymbol{v}\_{2},\boldsymbol{v}\_{3}) = \sum\_{j=1}^{3} \boldsymbol{v}\_{j} \left[ \sum\_{i=1}^{N-1} \frac{(\boldsymbol{\theta}\_{i+1}^{(j)} - \boldsymbol{\mu}\_{i}^{(j)})^{2}}{t\_{i+1} - t\_{i}} \right],\tag{20}$$

For a small dataset, however, it is important to use the appropriate smoothness constraint. As a demonstration of the importance, a dataset containing 300 earthquakes distributed locally in a northern part of Japan (Figure 7a) was analyzed using the two constraints. The estimated daily variation of *µ* is shown in Figure 7b; we can observe systematic deviation between the two cases. The ABIC value with the smoothness constraint of the equation (9) is smaller than that corresponding to the equation (20), and the difference between the two ABIC values is 2.5, which is statistically significant. This demonstration implies that an analysis with an unreasonable constraint or prior distribution, which does not involve the essential characteristic of data and is statistically invalid, may leads us to an incorrect

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

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181

This chapter provided a statistical technique to make a quantitative evaluation of the daily variation of earthquake detection capability. As an example of its application to actual data, the datasets taken from the recent JMA catalogue were analyzed, and a guideline for the

It should be noted that the results shown in this chapter were derived from an analysis where the earthquake sequence over an wide area was analyzed simultaneously. The daily variation of cultural noise, however, must have location dependency. Thus, in a manner similar to the analysis shown in Figure 7, we should examine regional earthquake sequences comprehensively to deal with an earthquake catalogue in a more sophisticated manner.

In this chapter we considered only the daily variation of detection capability, but human activities also have a weekly periodicity [28]. Additionally, we often observe seasonal (annual) variations in seismic noise level [39, 40], which are mainly caused by meteorological factors. Thus, the development of an appropriate method to handle such multiple periodic variations of earthquake detection capability is a necessary challenge to determine the

This study was partially supported by the Grants-in-Aid 23240039 for Scientific Research (A) by the Ministry of Education, Culture, Sports, Science and Technology, Japan, and by the ISM Cooperative Research Program (2011-ISM·CRP-2007). To obtain the results shown in this chapter, the author used the supercomputer system of the Institute of Statistical Mathematics,

choice of completeness magnitude in the JMA catalogue was shown.

assessment of detection capability.

completeness magnitude more accurately.

Japan. Figures were generated using the GMT software [41].

The Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan

**Acknowledgments**

**Author details**

Takaki Iwata

**6. Concluding remarks**

where we do not impose a smoothness constraint between the start and end points on the values of *β*, *µ*, and *σ*. If we analyze a sufficiently large dataset such as the five sequences used in this study, the introduction of the constraint without considering the cyclic property is not problematic. This is because a sufficiently large sample size reasonably reduce random fluctuations or estimation errors, and therefore the estimated values of the parameters at the start and end points connects smoothly without the smoothness constraint. In actual, the analyses with the constraints given by the equations (20) and (9) gives almost identical temporal profiles of *µ* for the five sequences.

**Figure 7.** (a) Map showing the 300 earthquakes (green circles) used in the demonstrative analysis. (b) (top) Estimated daily variations of *µ* using the constraint following the equation (9) (red) and those following the equation (20). (bottom) Magnitude-time plot of the one-day stacked sequence of the 300 earthquakes.

For a small dataset, however, it is important to use the appropriate smoothness constraint. As a demonstration of the importance, a dataset containing 300 earthquakes distributed locally in a northern part of Japan (Figure 7a) was analyzed using the two constraints. The estimated daily variation of *µ* is shown in Figure 7b; we can observe systematic deviation between the two cases. The ABIC value with the smoothness constraint of the equation (9) is smaller than that corresponding to the equation (20), and the difference between the two ABIC values is 2.5, which is statistically significant. This demonstration implies that an analysis with an unreasonable constraint or prior distribution, which does not involve the essential characteristic of data and is statistically invalid, may leads us to an incorrect assessment of detection capability.

#### **6. Concluding remarks**

14 Earthquake Research and Analysis / Book 2

avoid any unexpected biases in our seismicity analysis.

temporal profiles of *µ* for the five sequences.

**(a)**

140˚E 141˚E

Magnitude-time plot of the one-day stacked sequence of the 300 earthquakes.

40˚N

41˚N

<sup>Φ</sup>(*θ*|*v*1, *<sup>v</sup>*2, *<sup>v</sup>*3) =

This suggests that, if we determine *Mc* without taking into consideration the daily variation of the detection capability, *Mc* − 0.2 would be appropriate as the completeness magnitude to

In the present study, we use the smoothness constraint, as shown in the equation (9), which reflects the cyclic property of data. Suppose that we use a different smoothness constraint

where we do not impose a smoothness constraint between the start and end points on the values of *β*, *µ*, and *σ*. If we analyze a sufficiently large dataset such as the five sequences used in this study, the introduction of the constraint without considering the cyclic property is not problematic. This is because a sufficiently large sample size reasonably reduce random fluctuations or estimation errors, and therefore the estimated values of the parameters at the start and end points connects smoothly without the smoothness constraint. In actual, the analyses with the constraints given by the equations (20) and (9) gives almost identical

0.5

0

1

Magnitude

**Figure 7.** (a) Map showing the 300 earthquakes (green circles) used in the demonstrative analysis. (b) (top) Estimated daily variations of *µ* using the constraint following the equation (9) (red) and those following the equation (20). (bottom)

2

1.0 µ

**(b)**

(*θ* (*j*) *<sup>i</sup>*+<sup>1</sup> <sup>−</sup> *<sup>µ</sup>*(*j*) *<sup>i</sup>* )<sup>2</sup>

*ti*<sup>+</sup><sup>1</sup> − *ti*

, (20)

cyclic non-cyclic

0.0 0.2 0.4 0.6 0.8 1.0 Time since midnight [day]

3 ∑ *j*=1 *vj N*−1 ∑ *i*=1

> This chapter provided a statistical technique to make a quantitative evaluation of the daily variation of earthquake detection capability. As an example of its application to actual data, the datasets taken from the recent JMA catalogue were analyzed, and a guideline for the choice of completeness magnitude in the JMA catalogue was shown.

> It should be noted that the results shown in this chapter were derived from an analysis where the earthquake sequence over an wide area was analyzed simultaneously. The daily variation of cultural noise, however, must have location dependency. Thus, in a manner similar to the analysis shown in Figure 7, we should examine regional earthquake sequences comprehensively to deal with an earthquake catalogue in a more sophisticated manner.

> In this chapter we considered only the daily variation of detection capability, but human activities also have a weekly periodicity [28]. Additionally, we often observe seasonal (annual) variations in seismic noise level [39, 40], which are mainly caused by meteorological factors. Thus, the development of an appropriate method to handle such multiple periodic variations of earthquake detection capability is a necessary challenge to determine the completeness magnitude more accurately.

#### **Acknowledgments**

This study was partially supported by the Grants-in-Aid 23240039 for Scientific Research (A) by the Ministry of Education, Culture, Sports, Science and Technology, Japan, and by the ISM Cooperative Research Program (2011-ISM·CRP-2007). To obtain the results shown in this chapter, the author used the supercomputer system of the Institute of Statistical Mathematics, Japan. Figures were generated using the GMT software [41].

#### **Author details**

Takaki Iwata The Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan

#### **References**

[1] Woessner J, Wiemer S. Assessing the quality of earthquake catalogues: Estimating the magnitude of completeness and its uncertainty. Bulletin of the Seismological Society of America 2005;95(2):684-698.

[15] Ogata Y, Katsura K. Analysis of Temporal and Spatial Heterogeneity of Magnitude Frequency-Distribution Inferred from Earthquake Catalogs. Geophysical Journal

Daily Variation in Earthquake Detection Capability: A Quantitative Evaluation

http://dx.doi.org/10.5772/54890

183

[16] Gutenberg B, Richter CF. Frequency of earthquakes in California. Bulletin of the

[17] Ringdal F. On the estimation of seismic detection thresholds. Bulletin of the

[18] Iwata, T. Revisiting the global detection capability of earthquakes during the period immediately after a large earthquake: considering the influence of intermediate-depth

[19] Powell MJD. Approximation theory and methods. New York: Cambridge University

[20] Good IJ, Gaskins RA. Nonparametric Roughness Penalties for Probability Densities.

[21] Eggermon PPB, LaRiccia VN. Maximum Penalized Likelihood Estimation, vol I: Density

[23] Akaike H. Likelihood and Bayes Procedure. In:Bernardo JE. et al. (eds.) Bayesian

[24] Kass RE, Raftery AE. Bayes factors. Journal of the American Statistical Association

[25] Gill, P. E., Murray, W., Wright, MH. Numerical linear algebra and optimization, vol. I.

[26] Tierney L, Kadane JB. Accurate Approximations for Posterior Moments and Marginal Densities. Journal of the American Statistical Association 1986;81(393):82-86.

[27] Konishi, S., Kitagawa, G. Information Criteria and Statistical Modeling Springer, New

[28] Atef AH, Liu KH, Gao SS. Apparent Weekly and Daily Earthquake Periodicities in the Western United States. Bulletin of the Seismological Society of America

[29] Iwata T, Young RP. Tidal stress/strain and the b-values of acoustic emissions at the Underground Research Laboratory, Canada. Pure and Applied Geophysics

[30] Nanjo KZ, Hirata, N, Obara, K., Kasahara, K. Decade-scale decrease in *b* value prior to the *M*9-class 2011 Tohoku and 2004 Sumatra quakes. Geophysical Research Letters

[22] Good IJ. The estimation of probabilities. Cambridge: The MIT Press; 1965.

International 1993;113(3):727-738.

Biometrika 1971;58(2):255-277.

1995;90(430):773-795.

2009;99(4):2273-2279.

2005;162(6-7):1291-1308.

York; 2008.

Estimation. New York: Springer; 2001.

Redwood City: Addison-Wesley; 1991.

2012;39(20):L20304;doi:10.1029/2012GL052997.

Statistics. Valencia: University Press;1980. p143-166.

Press; 1981.

Seismological Society of America 1944;34(4):185-188.

Seismological Society of America 1975;656:1631–42.

and deep earthquakes. Research in Geophysics 2012;2(1):24–28.


[15] Ogata Y, Katsura K. Analysis of Temporal and Spatial Heterogeneity of Magnitude Frequency-Distribution Inferred from Earthquake Catalogs. Geophysical Journal International 1993;113(3):727-738.

16 Earthquake Research and Analysis / Book 2

America 2005;95(2):684-698.

International 2008;174(3):849-856.

Seismological Society of America 2008;98(5):2103-2117.

Seismological Society of America 2010;100(6):3261-3268.

of the Royal Astronomical Society 1971;24(1):97-99.

2012;Chikyu Monthly;34(9):504-508 (in Japanese).

2010;115B4:B04308;doi10.1029/2008JB006097.

Society of America 2011;101(3):1371-1385.

Society 1972;283:311-313.

abstract and figure captions).

[1] Woessner J, Wiemer S. Assessing the quality of earthquake catalogues: Estimating the magnitude of completeness and its uncertainty. Bulletin of the Seismological Society of

[2] Iwata T. Low detection capability of global earthquakes after the occurrence of large earthquakes: investigation of the Harvard CMT catalogue. Geophysical Journal

[3] Schorlemmer D, Woessner J. Probability of detecting an earthquake. Bulletin of the

[4] Schorlemmer D, Mele F, Marzocchi W. A completeness analysis of the National Seismic Network of Italy. Journal of Geophysical Research-Solid Earth

[5] Nanjo KZ, Ishibe T, Tsuruoka H, Schorlemmer D, Ishigaki Y, Hirata N. Analysis of the completeness magnitude and seismic network coverage of Japan. Bulletin of the

[6] Mignan A, Werner MJ, Wiemer S, Chen CC, Wu YM. Bayesian Estimation of the Spatially Varying Completeness Magnitude of Earthquake Catalogs. Bulletin of the Seismological

[7] Plenkers K, Schorlemmer D, Kwiatek G. On the Probability of Detecting Picoseismicity.

[8] Shimshoni.M. Evidence for Higher Seismic Activity During Night. Geophysical Journal

[9] Flinn EA, Blandford.RR, Mack H. Evidence for Higher Seismic Activity During Night.

[10] Knopoff L, Gardner JK. Higher seismic activity during local night on the raw worldwide earthquake catalogue. Geophysical Journal of the Royal Astronomical

[11] Iwata, T. Quantitative analysis of the daily variation of earthquake detection capability.

[12] Funasaki, J., Earthquake Prediction Information Division, Seismological and Volcanological Department, Japan Meteorological Agency. Revision of the JMA Velocity Magnitude. Quarterly Journal of Seismology 2004;67(1-4):11-20(in Japanese with English

[13] Katsumata, A. (2004) Revision of the JMA Displacement Magnitude. Quarterly Journal of Seismology 2004;67(1-4):1-10(in Japanese with English abstract and figure captions).

[14] Nanjo KZ, Tsuruoka H, Hirata N, Jordan TH. Overview of the first earthquake forecast

testing experiment in Japan. Earth Planets and Space 2011;63(3):159-169.

Bulletin of the Seismological Society of America 2011;101(6):2579-2591.

Geophysical Journal of the Royal Astronomical Society 1972;28(3):307-309.

**References**


[31] Scholz, C. The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes 1968;58(1):399-415.

**Chapter 9**

**Damage Estimation Improvement of Electric Power**

Electric power distribution equipment has a high damage potential due to disasters and requires a long time for restoration. This is because a huge number of electric power distribu‐ tion equipment is installed under various ground and regional conditions and is located near vulnerable trees and old residential buildings. Thus, during the restoration work after a large scale earthquake, it takes a long time to collect reliable disaster information. It is also even difficult to accurately estimate the damage degree of electric power distribution equipment. Therefore, in Japan, electric power companies pay particular attention to technologies associated with a quickly understanding and estimating the damage degree of the entire

On the other hand, with the progress in information technologies, practical disaster informa‐ tion services are increasing in Japan. For example, the Japan Metrological Agency has started a general delivery service of real-time earthquake information since October, 2007[1]. More‐ over, in recent years, remote sensing images, such as satellite, aero and synthetic aperture radar (SAR) images, are now available and open to the public after a large scale disaster [2]. On the basis of such information technologies, more reasonable ways to support the restoration work

Based on this background, our research team has developed an sequentially updated damage estimation system called RAMPEr, which stands for "*Risk Assessment and Management System for Power lifeline Earthquake real time*" [3][4]. RAMPEr enables us to provide the damage estimation results of electric power distribution equipment during the emergency restoration

> © 2013 Shumuta; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Shumuta; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

electric power distribution system during the emergency restorations.

**Distribution Equipment Using Multiple Disaster**

**Information**

Yoshiharu Shumuta

**1. Introduction**

http://dx.doi.org/10.5772/55207

for utility lifelines can be developed.

process against an earthquake.

Additional information is available at the end of the chapter


## **Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information**

Yoshiharu Shumuta

18 Earthquake Research and Analysis / Book 2

2010;180(1):347-360.

EOS 1998;79:579.

to earthquakes 1968;58(1):399-415.

184 Earthquake Research and Analysis - New Advances in Seismology

[31] Scholz, C. The frequency-magnitude relation of microfracturing in rock and its relation

[32] Amitrano D. Brittle-ductile transition and associated seismicity: Experimental and numerical studies and relationship with the *b* value. Journal of Geophysical

[33] Lei XL. How do asperities fracture? An experimental study of unbroken asperities.

[34] Mori J, Abercrombie RE. Depth dependence of earthquake frequency-magnitude distributions in California: Implications for rupture initiation. Journal of Geophysical

[35] Gerstenberger M, Wiemer S, Giardini D. A systematic test of the hypothesis that the b value varies with depth in California. Geophysical Research Letters 2001;28(1):57-60.

[36] Amorèse, D., Grasso JR, Rydelek PA. On varying *b*-values with depth: results from computer-intensive tests for Southern California. Geophysical Journal International

[37] Schorlemmer D, Wiemer S, Wyss M. Variations in earthquake-size distribution across

[38] Narteau C, Byrdina S, Shebalin P, Schorlemmer D. Common dependence on stress for the two fundamental laws of statistical seismology. Nature 2009;462(7273):642-645.

[39] Fyen J. Diurnal and Seasonal-Variations in the Microseismic Noise-Level Observed at the Noress Array. Physics of the Earth and Planetary Interiors 1990;63(3-4):252-268.

[40] Hillers G, Ben-Zion Y. Seasonal variations of observed noise amplitudes at 2-18 Hz in southern California. Geophysical Journal International 2011;184(2):860-868.

[41] Wessel P., Smith, WHF. New, improved version of the Generic mapping Tools released.

Research-Solid Earth 2003;108(B1):2044;doi:10.1029/2001JB000680.

Earth and Planetary Science Letters 2003;213(3-4):347-359.

different stress regimes. Nature 2005;437(7058):539-542.

Research-Solid Earth 1997;102(B7):15081-15090.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55207

#### **1. Introduction**

Electric power distribution equipment has a high damage potential due to disasters and requires a long time for restoration. This is because a huge number of electric power distribu‐ tion equipment is installed under various ground and regional conditions and is located near vulnerable trees and old residential buildings. Thus, during the restoration work after a large scale earthquake, it takes a long time to collect reliable disaster information. It is also even difficult to accurately estimate the damage degree of electric power distribution equipment. Therefore, in Japan, electric power companies pay particular attention to technologies associated with a quickly understanding and estimating the damage degree of the entire electric power distribution system during the emergency restorations.

On the other hand, with the progress in information technologies, practical disaster informa‐ tion services are increasing in Japan. For example, the Japan Metrological Agency has started a general delivery service of real-time earthquake information since October, 2007[1]. More‐ over, in recent years, remote sensing images, such as satellite, aero and synthetic aperture radar (SAR) images, are now available and open to the public after a large scale disaster [2]. On the basis of such information technologies, more reasonable ways to support the restoration work for utility lifelines can be developed.

Based on this background, our research team has developed an sequentially updated damage estimation system called RAMPEr, which stands for "*Risk Assessment and Management System for Power lifeline Earthquake real time*" [3][4]. RAMPEr enables us to provide the damage estimation results of electric power distribution equipment during the emergency restoration process against an earthquake.

© 2013 Shumuta; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Shumuta; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reference [3] proposed a damage estimation function installed in RAMPEr which used the earthquake ground motion intensity as an input parameter, and applied the proposed function to actual electric power distribution equipment damaged by the 2007 Niigata-ken Chuetsu Oki earthquake to clarify the estimation accuracy of the proposed function.

During the limited damage information condition, RAMPEr becomes an effective tool to support some decision makings. RAMPEr is a seismic damage estimation system whose basic concept is a sequential updated damage estimation based on real-time hazard and damage information. RAMPEr was used by Tohoku Electric Power Co., Inc., to support its actual initial

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

http://dx.doi.org/10.5772/55207

187

restoration work due to the 311 earthquake [10].

**Figure 1.** Differences in required information according to restoration stages

**Figure 2.** Time required for the damage detection of electric power distribution equipment [9]

This paper focuses on the updated damage estimation process of RAMPEr. RAMPEr enables us to improve the damage estimation accuracy using sequentially updated disaster informa‐ tion which a power company can collect during the emergency restoration period against a large-scale earthquake. Chapter 2 introduces the necessary disaster information for three emergency restoration stages and emphasizes the significance of RAMPEr. Chapter 3 describes the formulation of the proposed model. Chapter 4 discusses the advantage and limitation of the proposed model using the actual damage records due to the 2007 Niigata-Ken Chuetsu-Oki earthquake.

#### **2. Information required for the emergency restoration work**

Fig.1 shows the differences in the information required for the efficient restoration of a seismic damaged electric power distribution system. The restoration process is generally divided into the initial, emergency, and permanent restoration periods. During the initial restoration period, the information associated with the damage degree of the entire electric power system is needed to judge how many staff members should be dispatched for the restoration. During the emergency restoration period, the information associated with the damage point and mode to judge whether restoration staff members can immediately restore the power is needed. During the permanent restoration process, the information associated with all damaged equipment to be physically repaired is needed.

In order to collect this information, the power company dispatches inspection teams. In addition, some power companies have tried to apply remote sensing technologies, including helicopters and satellites, to quickly collect damage information. However, at the current time, the inspection teams and remote sensing technologies are not very effective to quickly collect seismic damage information especially for the restoration resource allocation during the initial restoration period.

Fig.2 shows the restoration process of an electric power distribution system located in the Tohoku region just after the 2011 earthquake off the Pacific coast of the Tohoku (the 311 earthquake) occurrence. The horizontal axis and the vertical axis show the elapsed time (days) and the number of inspected damaged equipment (%), respectively. Fig.2 indicates that the damage information had not been effectively collected, especially during the initial restoration period. This is because the Tohoku area had frequent aftershocks, much debris, and some coastal regions which were not able to be entered for the restoration analysis. This result illustrates that when a large-scale earthquake occurs, there is the possibility that the restoration work, including damage information collection, is highly limited by the damage related to residential buildings and other infrastructures.

During the limited damage information condition, RAMPEr becomes an effective tool to support some decision makings. RAMPEr is a seismic damage estimation system whose basic concept is a sequential updated damage estimation based on real-time hazard and damage information. RAMPEr was used by Tohoku Electric Power Co., Inc., to support its actual initial restoration work due to the 311 earthquake [10].

**Figure 1.** Differences in required information according to restoration stages

Reference [3] proposed a damage estimation function installed in RAMPEr which used the earthquake ground motion intensity as an input parameter, and applied the proposed function to actual electric power distribution equipment damaged by the 2007 Niigata-ken Chuetsu Oki

This paper focuses on the updated damage estimation process of RAMPEr. RAMPEr enables us to improve the damage estimation accuracy using sequentially updated disaster informa‐ tion which a power company can collect during the emergency restoration period against a large-scale earthquake. Chapter 2 introduces the necessary disaster information for three emergency restoration stages and emphasizes the significance of RAMPEr. Chapter 3 describes the formulation of the proposed model. Chapter 4 discusses the advantage and limitation of the proposed model using the actual damage records due to the 2007 Niigata-Ken Chuetsu-

Fig.1 shows the differences in the information required for the efficient restoration of a seismic damaged electric power distribution system. The restoration process is generally divided into the initial, emergency, and permanent restoration periods. During the initial restoration period, the information associated with the damage degree of the entire electric power system is needed to judge how many staff members should be dispatched for the restoration. During the emergency restoration period, the information associated with the damage point and mode to judge whether restoration staff members can immediately restore the power is needed. During the permanent restoration process, the information associated with all damaged

In order to collect this information, the power company dispatches inspection teams. In addition, some power companies have tried to apply remote sensing technologies, including helicopters and satellites, to quickly collect damage information. However, at the current time, the inspection teams and remote sensing technologies are not very effective to quickly collect seismic damage information especially for the restoration resource allocation during the initial

Fig.2 shows the restoration process of an electric power distribution system located in the Tohoku region just after the 2011 earthquake off the Pacific coast of the Tohoku (the 311 earthquake) occurrence. The horizontal axis and the vertical axis show the elapsed time (days) and the number of inspected damaged equipment (%), respectively. Fig.2 indicates that the damage information had not been effectively collected, especially during the initial restoration period. This is because the Tohoku area had frequent aftershocks, much debris, and some coastal regions which were not able to be entered for the restoration analysis. This result illustrates that when a large-scale earthquake occurs, there is the possibility that the restoration work, including damage information collection, is highly limited by the damage related to

earthquake to clarify the estimation accuracy of the proposed function.

186 Earthquake Research and Analysis - New Advances in Seismology

**2. Information required for the emergency restoration work**

equipment to be physically repaired is needed.

residential buildings and other infrastructures.

Oki earthquake.

restoration period.

**Figure 2.** Time required for the damage detection of electric power distribution equipment [9]

### **3. Consecutive integrated process of multiple disaster information**

#### **3.1. Multiple disaster information used by RAMPEr**

Disaster information needed by an electric power company can be usually collected after an earthquake occurrence includes four categories; (1) Earthquake, (2) Power outage, (3) Damage inspection, and (4) Damaged area image by remote sensing.

As for the earthquake information, RAMPEr, which has been already installed in some electric power companies, is supposed to automatically receive the earthquake information through the Internet including the epicenter, magnitude, and seismic ground motion intensities recorded on seismographs from the Japan Metrological Agency within several minutes just after the earthquake occurrence. Based on the received earthquake information, RAMPEr evaluates the seismic ground motion intensity distribution. As other sources, the National Research Institute for Earth Science and Disaster Prevention (NIED) opens seismic ground motion records including K-NET and KiK-net[5]. RAMPEr collects Instrumental Seismic Intensity (ISI), Peak Ground Velocity (PGV), and Peak Ground Acceleration (PGA) recorded from K-NET and KiK-net to improve the evaluation accuracy of the seismic ground motion distribution.

causality of both nodes is defined as a Conditional Probability Table (CPT). The CPT defines the causal relationship between *X1* as a parent node and *X2* as a child node in the Bayesian network. The arrow between the nodes defines the causal relationship between the parent node and child node. The conditional damage probability of the child node *P*(*X2/X1*) can be

4 This paper focuses on a Bayesian network as a basic model to integrate the multiple disaster 5 information. Details of the Bayesian network can be found in reference [6]. This chapter only 6 introduces the basic concept of the Bayesian network for a better understanding of the following

Fig.3 shows a simple Bayesian network. It describes the relationships between variables *X1* and *X2* 8 . The *X1* and *X2* 9 variables are binary (taking a value of either 0: false or 1: true). The causality of both 10 nodes is defined as a Conditional Probability Table (CPT). The CPT defines the causal relationship between *X1* as a parent node and *X2* 11 as a child node in the Bayesian network. The arrow between the 12 nodes defines the causal relationship between the parent node and child node. The conditional

4 Book Title

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

For example, according to the marginalization [6], the probability *P*(*X2*= 1) can be estimated as

2 11

( 1/ ) ( )

*PX X PX*

21 1

( / )( )

18 When variable *X1*=1 on the parent node is given, the probability *P*(*X2*= 1/ *X1*=1) can be estimated

= ×

( 1) 0.305

16 (1)

2 11

14 For example, according to the marginalization [6], the probability *P*(*X2*= 1) can be estimated as

( 1 / 1) ( 1) ( 1 / 1) 0.6

= =× = = == <sup>=</sup>

22 On the contrary, when variable *X2*=1 on the child node is given, the probability *P*(*X1*=1/*X2*=1)

= = =

*PX X PX*

( 1/ ) ( )

*PX X PX*

21 1

×

(1)

( / )( )

21 1

*PX X PX*

21 1

( / 1) ( 1)

å =× = (2)

(2)

*P*(*X2/X1*) : Conditional probability of *X2* assuming that *X2* occurs .

*0: false* 0.99 0.01 *1:true* 0.4 0.6

*<sup>P</sup>* (X2*/X <sup>1</sup>* ) *<sup>X</sup> <sup>1</sup>*

Conditional Probability Table(CPT)

*X <sup>1</sup> =0: false X <sup>1</sup> =1:true* 0.5 0.5

*P(X <sup>1</sup> )*

http://dx.doi.org/10.5772/55207

189

*X <sup>2</sup> =0: false X <sup>2</sup> =1: true*

å = × (3)

*PX X PX*

When variable *X1*=1 on the parent node is given, the probability *P*(*X2*= 1/ *X1*=1) can be estimated as

On the contrary, when variable *X2*=1 on the child node is given, the probability *P*(*X1*=1/*X2*=1)

21 1

( 1 / 1) ( 1) ( 1 / 1) 0.984

= =× = = == <sup>=</sup>

2 11

( 1/ ) ( )

*PX X PX*

1

*X*

0 0

1 2

*X X*

1

*X*

0 2 1 1

1

2 1 1

2

*PX X PX PX X*

*X*

0

1 2 1

*P X*

2 1 1

*P X*

1

2 **Fig.3.** A basic element of a Bayesian network

*X2*

*X1*

3 **3.2. Basic idea of damage information integration** 

å

1 2

*X X*

( 1/ 1) ( 1) ( 1/ 1) 0.6

 

0 0

( 1) 0.305

 

damage probability of the child node *P*(*X2/X1* 13 ) can be determined by the *CPT* in Fig.3.

= =

2

1

*X*

*PX X PX PX X*

0

=

*PX X PX PX X*

*X*

0

21 1

*PX X PX*

21 1

( / 1) ( 1)

=

å å

0 2 1 1

=

determined by the *CPT* in Fig.3.

7 chapters.

*X1*

: : Parent Node

*X2* : Child node

**Figure 3.** A basic element of a Bayesian network

1

15

17

20

21

24

19 as

can be estimated as

23 can be estimated as

Power outages and damage inspection information are usually confidential information that the power company collects. The power outage information is usually collected for every high voltage distribution line (feeder) with power outages obtained from an online business support system maintained by the electric power company. The inspection information includes damaged equipment information including distribution poles, distribution lines, transform‐ ers, and switches. The inspection information is collected by portable transceivers, mobile phones and Personal Digital Assistances (PDA) as offline information. RAMPEr uses this confidential information to improve the damage estimation accuracy [4].

On the other hand, a seismic damage area image provided by remote sensing devices, such as satellites, is also one of the effective resources to understand the damage degree of the earthquake stricken area. However, as mentioned in Chapter 2, at the current technology stage, because it usually takes a long time to take and provide the images, it is difficult for RAMPEr to get the satellite image during an emergency restoration period. Thus, this paper focuses on (1) Earthquake information, (2) Power outage information, and (3) Damage inspection information to formulate the damage information integration as follows.

#### **3.2. Basic idea of damage information integration**

This paper focuses on a Bayesian network as a basic model to integrate the multiple disaster information. Details of the Bayesian network can be found in reference [6]. This chapter only introduces the basic concept of the Bayesian network for a better understanding of the following chapters.

Fig.3 shows a simple Bayesian network. It describes the relationships between variables *X1* and *X2*. The *X1* and *X2* variables are binary (taking a value of either 0: false or 1: true). The 4 Book Title

2 **Fig.3.** A basic element of a Bayesian network **Figure 3.** A basic element of a Bayesian network

3 **3.2. Basic idea of damage information integration** 

2 1 1

1

15

17

21

24

**3. Consecutive integrated process of multiple disaster information**

Disaster information needed by an electric power company can be usually collected after an earthquake occurrence includes four categories; (1) Earthquake, (2) Power outage, (3) Damage

As for the earthquake information, RAMPEr, which has been already installed in some electric power companies, is supposed to automatically receive the earthquake information through the Internet including the epicenter, magnitude, and seismic ground motion intensities recorded on seismographs from the Japan Metrological Agency within several minutes just after the earthquake occurrence. Based on the received earthquake information, RAMPEr evaluates the seismic ground motion intensity distribution. As other sources, the National Research Institute for Earth Science and Disaster Prevention (NIED) opens seismic ground motion records including K-NET and KiK-net[5]. RAMPEr collects Instrumental Seismic Intensity (ISI), Peak Ground Velocity (PGV), and Peak Ground Acceleration (PGA) recorded from K-NET and KiK-net to improve the evaluation accuracy of the seismic ground motion

Power outages and damage inspection information are usually confidential information that the power company collects. The power outage information is usually collected for every high voltage distribution line (feeder) with power outages obtained from an online business support system maintained by the electric power company. The inspection information includes damaged equipment information including distribution poles, distribution lines, transform‐ ers, and switches. The inspection information is collected by portable transceivers, mobile phones and Personal Digital Assistances (PDA) as offline information. RAMPEr uses this

On the other hand, a seismic damage area image provided by remote sensing devices, such as satellites, is also one of the effective resources to understand the damage degree of the earthquake stricken area. However, as mentioned in Chapter 2, at the current technology stage, because it usually takes a long time to take and provide the images, it is difficult for RAMPEr to get the satellite image during an emergency restoration period. Thus, this paper focuses on (1) Earthquake information, (2) Power outage information, and (3) Damage inspection

This paper focuses on a Bayesian network as a basic model to integrate the multiple disaster information. Details of the Bayesian network can be found in reference [6]. This chapter only introduces the basic concept of the Bayesian network for a better understanding of the

Fig.3 shows a simple Bayesian network. It describes the relationships between variables *X1* and *X2*. The *X1* and *X2* variables are binary (taking a value of either 0: false or 1: true). The

confidential information to improve the damage estimation accuracy [4].

information to formulate the damage information integration as follows.

**3.2. Basic idea of damage information integration**

**3.1. Multiple disaster information used by RAMPEr**

188 Earthquake Research and Analysis - New Advances in Seismology

distribution.

following chapters.

inspection, and (4) Damaged area image by remote sensing.

causality of both nodes is defined as a Conditional Probability Table (CPT). The CPT defines the causal relationship between *X1* as a parent node and *X2* as a child node in the Bayesian network. The arrow between the nodes defines the causal relationship between the parent node and child node. The conditional damage probability of the child node *P*(*X2/X1*) can be determined by the *CPT* in Fig.3. 4 This paper focuses on a Bayesian network as a basic model to integrate the multiple disaster 5 information. Details of the Bayesian network can be found in reference [6]. This chapter only 6 introduces the basic concept of the Bayesian network for a better understanding of the following 7 chapters. Fig.3 shows a simple Bayesian network. It describes the relationships between variables *X1* and *X2* 8 . The *X1* and *X2* 9 variables are binary (taking a value of either 0: false or 1: true). The causality of both 10 nodes is defined as a Conditional Probability Table (CPT). The CPT defines the causal relationship between *X1* as a parent node and *X2* 11 as a child node in the Bayesian network. The arrow between the

For example, according to the marginalization [6], the probability *P*(*X2*= 1) can be estimated as 12 nodes defines the causal relationship between the parent node and child node. The conditional damage probability of the child node *P*(*X2/X1* 13 ) can be determined by the *CPT* in Fig.3.

14 For example, according to the marginalization [6], the probability *P*(*X2*= 1) can be estimated as

$$P(X\_2 = 1) = \frac{\sum\_{X\_1=0}^{1} P(X\_2 = 1 \mid X\_1) \cdot P(X\_1)}{\sum\_{1}^{1} \sum\_{X\_2=0}^{1} P(X\_2 \mid X\_1) \cdot P(X\_1)} = 0.305\tag{1}$$

When variable *X1*=1 on the parent node is given, the probability *P*(*X2*= 1/ *X1*=1) can be estimated as 19 as 21 1 ( 1/ 1) ( 1) ( 1/ 1) 0.6 *PX X PX PX X* 20

18 When variable *X1*=1 on the parent node is given, the probability *P*(*X2*= 1/ *X1*=1) can be estimated

21 1

*PX X PX*

( / 1) ( 1)

$$P(X\_2 = 1 \mid X\_1 = 1) = \frac{P(X\_2 = 1 \mid X\_1 = 1) \cdot P(X\_1 = 1)}{\sum\_{X\_2 = 0}^{1} P(X\_2 \mid X\_1 = 1) \cdot P(X\_1 = 1)} = 0.6\tag{2}$$

On the contrary, when variable *X2*=1 on the child node is given, the probability *P*(*X1*=1/*X2*=1) can be estimated as

$$P(X\_1 = 1 \mid X\_2 = 1) = \frac{P(X\_2 = 1 \mid X\_1 = 1) \cdot P(X\_1 = 1)}{\sum\_{X\_1 = 0}^{1} P(X\_2 = 1 \mid X\_1) \cdot P(X\_1)} = 0.984\tag{3}$$

If the causal relationships among disaster events can be defined by a Bayesian network with two or more nodes, which includes the 2 nodes of Fig.3 as a minimum unit, the conditional probability of a target node can be improved with an increase in the observed information of the parent or child nodes.

*EGM*(*a*)*=*1 indicates that the information of the maximum seismic ground motion intensity *a* for every target equipment is given. *EGM*(*a*) = *0* indicates that no earthquake ground motion information is given. *EPDE*=1 and *EPDE*=0 indicate the damage and no damage that occurs to

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

that when an earthquake occurs, the maximum ground motion *a* is always given for every target equipment within several minutes. Thus, in Table 1, *Pi* (*EPDE=*1*/EGM*(*a*)=1) and *Pi*

(*a*) and 1-*Pi*

**CPT(A)** *Pi*

*EGM*(*a*) No Damage

**Table 1.** Conditional probability table between the earthquake intensity and the damage of electric power

**b.** Causal relationship between the electric power distribution damage and the power outage

Table 2 shows the CPT which defines the causal relationship between the electric power distribution damage and the power outage shown as CPT(B) in Fig.4. According to CPT(B), when *EPDE =* 1 is given, which indicates that equipment *i* is damaged, the conditional no power outage probability of equipment *i, Pi* (*PO=0/EPDE=1*), and the conditional power outage

On the other hand, when *EPDE=*0 is given, which indicates that equipment *i* is not damaged, the conditional power outage probability of equipment *i*, *Pi* (*PO=0/EPDE=0*), and no power

( 0 / 0) (1 ( ))

( 1 / 0) 1 (1 ( ))

where *Ne* indicates the total number of equipment connected to the same distribution line (the same feeder). Equation (4) and Equation (5) assumes that when equipment is damaged, a power outage occurs to all equipment connected to the distribution line of the damaged

*i i*

*P PO EPDE P a*

*i i*

*P PO EPDE P a*

1


*Ne*

*i*

=

1

*Ne*

*i*

=

1


1

1:maximum seismic ground motion *a* occurs 1-*Pi*

(*a*) indicates the estimated damage rate of equipment *i* with the

*EPDE*=0

(*PO=1/EPDE=1*), are assumed to be 0 and 1, respectively.

(*PO=1/EPDE= 0*), are, respectively, estimated as

= = = - Õ (4)

= = =- - Õ (5)

(*a*) is estimated by appendix[A]. This paper assumes

*(EPDE/EGM(a))*

(a) *Pi*

(*EPDE*/*EGM*(*a*)=0)

http://dx.doi.org/10.5772/55207

191

Damage *EPDE*=1

(a)

(*a*), respectively, and *Pi*

equipment, respectively. *Pi*

is neglected in Table 1.

distribution equipment (CPT(A))

probability of equipment *i*, *Pi*

outage probability of equipment *i*, *Pi*

(CPT(B))

equipment.

maximum seismic ground motion *a. Pi*

(*EPDE*=0*/EGM*(*a*)=1) are equivalent to *Pi*

#### **3.3. Improvement of damage probability using Bayesian network**

Fig.4 shows a proposed Bayesian network. The proposed Bayesian network consists of 4 nodes; (1) Earthquake Ground Motion (*EGM*(*a*)), (2)Electric Power Distribution Equipment damage (*EPDE*),(3) Damage Inspection(*DI*),and (4) Power Outage(*PO*). The CPT of the proposed Bayesian network is defined as follows.

**Figure 4.** The proposed Bayesian network model

**a.** Causal relationship between earthquake ground motion and electric power distribution damage (CPT(A))

Table 1 shows the CPT which defines the casual relationship between earthquake ground motion and electric power distribution equipment damage shown as CPT(A) in Fig.4. *EGM*(*a*)*=*1 indicates that the information of the maximum seismic ground motion intensity *a* for every target equipment is given. *EGM*(*a*) = *0* indicates that no earthquake ground motion information is given. *EPDE*=1 and *EPDE*=0 indicate the damage and no damage that occurs to equipment, respectively. *Pi* (*a*) indicates the estimated damage rate of equipment *i* with the maximum seismic ground motion *a. Pi* (*a*) is estimated by appendix[A]. This paper assumes that when an earthquake occurs, the maximum ground motion *a* is always given for every target equipment within several minutes. Thus, in Table 1, *Pi* (*EPDE=*1*/EGM*(*a*)=1) and *Pi* (*EPDE*=0*/EGM*(*a*)=1) are equivalent to *Pi* (*a*) and 1-*Pi* (*a*), respectively, and *Pi* (*EPDE*/*EGM*(*a*)=0) is neglected in Table 1.

If the causal relationships among disaster events can be defined by a Bayesian network with two or more nodes, which includes the 2 nodes of Fig.3 as a minimum unit, the conditional probability of a target node can be improved with an increase in the observed information of

Fig.4 shows a proposed Bayesian network. The proposed Bayesian network consists of 4 nodes; (1) Earthquake Ground Motion (*EGM*(*a*)), (2)Electric Power Distribution Equipment damage (*EPDE*),(3) Damage Inspection(*DI*),and (4) Power Outage(*PO*). The CPT of the proposed

CPT(C)

**a.** Causal relationship between earthquake ground motion and electric power distribution

Table 1 shows the CPT which defines the casual relationship between earthquake ground motion and electric power distribution equipment damage shown as CPT(A) in Fig.4.

Damage Inspection information (DI)

**3.3. Improvement of damage probability using Bayesian network**

Earthquake Ground Motion information (EGM(a))

Electric Power Distribution Equipment damage (EPDE)

CPT(A)

Power Outage information (PO)

**Figure 4.** The proposed Bayesian network model

damage (CPT(A))

CPT(B)

the parent or child nodes.

Bayesian network is defined as follows.

190 Earthquake Research and Analysis - New Advances in Seismology


**Table 1.** Conditional probability table between the earthquake intensity and the damage of electric power distribution equipment (CPT(A))

#### **b.** Causal relationship between the electric power distribution damage and the power outage (CPT(B))

Table 2 shows the CPT which defines the causal relationship between the electric power distribution damage and the power outage shown as CPT(B) in Fig.4. According to CPT(B), when *EPDE =* 1 is given, which indicates that equipment *i* is damaged, the conditional no power outage probability of equipment *i, Pi* (*PO=0/EPDE=1*), and the conditional power outage probability of equipment *i*, *Pi* (*PO=1/EPDE=1*), are assumed to be 0 and 1, respectively.

On the other hand, when *EPDE=*0 is given, which indicates that equipment *i* is not damaged, the conditional power outage probability of equipment *i*, *Pi* (*PO=0/EPDE=0*), and no power outage probability of equipment *i*, *Pi* (*PO=1/EPDE= 0*), are, respectively, estimated as

$$P\_i(PO = 0 \; / \; EPDE = 0) = \prod\_{i=1}^{Ne-1} \left( 1 - P\_i(a) \right) \tag{4}$$

$$P\_i(PO = 1 / \, EPDE = 0) = 1 - \prod\_{i=1}^{Ne-1} (1 - P\_i(a)) \tag{5}$$

where *Ne* indicates the total number of equipment connected to the same distribution line (the same feeder). Equation (4) and Equation (5) assumes that when equipment is damaged, a power outage occurs to all equipment connected to the distribution line of the damaged equipment.


1 () () ( ) 2

m

*i*

probability of equipment with the same attribute as equipment *I*, *p*<sup>i</sup>

the beta distribution based on the theory of Bayesian statistics [7].

**CPT(C)** *Pi*

DI No Damage

0: No inspection 1-*Pi*

**d.** Combination of the joint occurrence probability

*i pa*

m

*n*

where, *μpi*

(*a*)

distribution equipment

of equipment *i*, *Pi*

equipment *i*, *Pi*

and *σpi*

damage probability of equipment *i*, *Pi*

the definitive value instead of that by Table 3.

(*a*)

( )

ï ï î þ

the maximum ground motion intensity *a* is given, respectively. For this formulation, it is assumed that the estimated damage probabilities of equipment with the same attribute *i* follow

Note that when the damage inspection information of equipment *i* is given, the conditional

EPDE=0

1: Inspection Equation (7) Equation (6)

Based on Table1, Table2, and Table 3, Table 4 shows the combination of all the joint occurrence probabilities in Fig.4. Based on Table 4, all the conditional damage probabilities of the equipment can be evaluated. For example, when the earthquake ground motion information, *EGM*(*a*)=1 and power outage information, *PO*=1, are given, the conditional damage probability

On the other hand, when the earthquake ground motion information, *EGM*(*a*)=1 and the damage inspection information, *DI*=1, are given, the conditional damage probability of

[1] [5] ( 1 / ( ) 1, 1) [1] [3] [5] [7] *<sup>i</sup> P EPDE EGM a PO* <sup>+</sup> = = == +++ (10)

[1] [2] ( 1 / ( ) 1, 1) [1] [2] [3] [4] *<sup>i</sup> P EPDE EGM a DI* <sup>+</sup> = = == +++ (11)

**Table 3.** Conditional probability table between the inspection information and the damage of electric power

(*EPDE*=1/*EGM*(a)=1,*PO*=1), is evaluated as

(*EPDE*=1/*EGM*(a)=1,*DI*=1), is evaluated as

 m

*i*

s

*p a*

*pa pa*

ì ü - ï ï <sup>=</sup> í ý - -

(1 ) 1 1 *i i*

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

are the average and standard deviation of the estimated damage

(*EPDE/DI=*1), becomes 1 (damage) or 0 (no damage) as

*(EPDE/DI)*

(*a*) *Pi*

(9)

193

(*a*), on the condition that

http://dx.doi.org/10.5772/55207

Damage EPDE=1

(*a*)

**Table 2.** Conditional probability table between the damage to electric power distribution equipment and the power outage (CPT(B))

**c.** Causal relationship between damage inspection and electric power distribution equip‐ ment damage (CPT(C))

Table 3 shows the CPT which defines the causal relationship between the damage inspection and the electric power distribution equipment damage shown as CPT(C) in Fig.4. The damage inspection information indicates the inspection result of equipment with the same attribute as the target equipment *i*, which includes the number of damaged and non-damaged equipment on the condition that the target equipment *i* has no inspection. According to CPT(C), when *DI=*0 is given, which indicates that equipment *i* has no inspection information, the conditional damage probability of equipment *i*, *Pi* (*EPDE=*1*/DI=*0), and the conditional no damage probability of equipment *i*, *Pi* (*EPDE=*0*/DI=*0), are equivalent to *Pi* (*a*) and 1- *Pi* (*a*), respectively.

On the other hand, when *DI=*1 is given, which indicates that equipment *i* has inspection information, the conditional damage probability of equipment *i*, *Pi* (*EPDE=*1*/DI=*1), and the conditional no damage probability of equipment *i*, *Pi* (*EPDE=* 0*/DI=*1), are evaluated as

$$P\_i(EPDE = 1 / DI = 1) = \frac{n\_i^{0} + n\_i^{1} + 1}{M\_i^{0} + M\_i^{1} + 2} \tag{6}$$

$$P\_i(EPDE = 0 \; / \; DI = 1) = 1 - \frac{n\_i^0 + n\_i^1 + 1}{M\_i^0 + M\_i^1 + 2} \tag{7}$$

where *Mi <sup>0</sup>* is the total number of inspected equipment with the same attribute as equipment *i. ni 0* is the total number of damaged equipment with the same attribute as equipment *i*. On the other hand, *Mi 1* and *ni 1* are, respectively, evaluated as

$$M\_i^1 = \frac{\mu\_{p\_i(a)}(1 - \mu\_{p\_i(a)})}{\sigma\_{p\_i(a)}^2} - 3 \tag{8}$$

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information http://dx.doi.org/10.5772/55207 193

$$m\_i^1 = \mu\_{p\_i(a)} \left\{ \frac{\mu\_{p\_i(a)} (1 - \mu\_{p\_i(a)})}{\sigma\_{p\_i(a)}^2} - 1 \right\} - 1 \tag{9}$$

where, *μpi* (*a*) and *σpi* (*a*) are the average and standard deviation of the estimated damage probability of equipment with the same attribute as equipment *I*, *p*<sup>i</sup> (*a*), on the condition that the maximum ground motion intensity *a* is given, respectively. For this formulation, it is assumed that the estimated damage probabilities of equipment with the same attribute *i* follow the beta distribution based on the theory of Bayesian statistics [7].

Note that when the damage inspection information of equipment *i* is given, the conditional damage probability of equipment *i*, *Pi* (*EPDE/DI=*1), becomes 1 (damage) or 0 (no damage) as the definitive value instead of that by Table 3.


**Table 3.** Conditional probability table between the inspection information and the damage of electric power distribution equipment

#### **d.** Combination of the joint occurrence probability

**c.** Causal relationship between damage inspection and electric power distribution equip‐

**Table 2.** Conditional probability table between the damage to electric power distribution equipment and the power

*PO=* 0

0: No damage Equation (4) Equation (5)

1: Damage 0 1

**CPT(B)** *Pi*

*EPDE* No Power Outage

192 Earthquake Research and Analysis - New Advances in Seismology

Table 3 shows the CPT which defines the causal relationship between the damage inspection and the electric power distribution equipment damage shown as CPT(C) in Fig.4. The damage inspection information indicates the inspection result of equipment with the same attribute as the target equipment *i*, which includes the number of damaged and non-damaged equipment on the condition that the target equipment *i* has no inspection. According to CPT(C), when *DI=*0 is given, which indicates that equipment *i* has no inspection information, the conditional damage probability of equipment *i*, *Pi* (*EPDE=*1*/DI=*0), and the conditional no damage

(*EPDE=*0*/DI=*0), are equivalent to *Pi*

On the other hand, when *DI=*1 is given, which indicates that equipment *i* has inspection information, the conditional damage probability of equipment *i*, *Pi* (*EPDE=*1*/DI=*1), and the

<sup>1</sup> ( 1 / 1) <sup>2</sup>

<sup>1</sup> ( 0 / 1) 1 <sup>2</sup>

+ + = == + +

+ + = = = - + +

is the total number of damaged equipment with the same attribute as equipment *i*. On the

2 ( ) (1 ) <sup>3</sup> *i i*

 m

*n n P EPDE DI*

*n n P EPDE DI*

are, respectively, evaluated as

m

*i*

*M*

1 () ()

s

*i*

*p a*

*pa pa*

0 1 0 1

*i i*

*M M*

*i i*

0 1 0 1

*i i*

*M M*

*<sup>0</sup>* is the total number of inspected equipment with the same attribute as equipment *i.*

*i i*


(*a*) and 1- *Pi*

*(PO/EPDE)*

Power Outage *PO*=1

(*EPDE=* 0*/DI=*1), are evaluated as

(*a*), respectively.

(6)

(7)

ment damage (CPT(C))

outage (CPT(B))

probability of equipment *i*, *Pi*

where *Mi*

other hand, *Mi*

*1* and *ni 1*

*ni 0*

conditional no damage probability of equipment *i*, *Pi*

*i*

*i*

Based on Table1, Table2, and Table 3, Table 4 shows the combination of all the joint occurrence probabilities in Fig.4. Based on Table 4, all the conditional damage probabilities of the equipment can be evaluated. For example, when the earthquake ground motion information, *EGM*(*a*)=1 and power outage information, *PO*=1, are given, the conditional damage probability of equipment *i*, *Pi* (*EPDE*=1/*EGM*(a)=1,*PO*=1), is evaluated as

$$P\_i(EPDE = 1 / EGM(a) = 1, PO = 1) = \frac{[1] + [5]}{[1] + [3] + [5] + [7]} \tag{10}$$

On the other hand, when the earthquake ground motion information, *EGM*(*a*)=1 and the damage inspection information, *DI*=1, are given, the conditional damage probability of equipment *i*, *Pi* (*EPDE*=1/*EGM*(a)=1,*DI*=1), is evaluated as

$$P\_i(EPDE = 1 / EGM(a) = 1, DI = 1) = \frac{[1] + [2]}{[1] + [2] + [3] + [4]} \tag{11}$$


**2.** The earthquake ground motion and the damage inspection information are partially given

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

**Figure 5.** The distribution of the seismic intensity scale of the Japan Meteorological Agency (JMA) due to the Chuetsu-

**Figure 6.** The observed power outage of high voltage distribution lines due to the Chuetsu-Oki earthquake (treated as

**Figure 7.** The observed points of the damaged poles due to the Chuetsu-Oki earthquake (treated as DI in Fig.4)

JMA seismic intensity scale

http://dx.doi.org/10.5772/55207

195

High voltage distribution lines with power outage

(discussed in 4.3).

Oki earthquake (treated as EGM(a) in Fig.4)

PO in Fig.4)

**Table 4.** Combination of the joint probability (CPT(A)+CPT(B)+CPT(C))

The following positive analyses discuss the estimation accuracy for the situations of Equation (10) and Equation (11).

#### **4. Positive analyses based on the 2007 Niigata-Ken Chuetsu Oki earthquake**

#### **4.1. Precondition of positive analyses**

This chapterdiscusses the effectiveness oftheproposedmodel basedonanactualpower outage and damage records of an electric power distribution system struck by the 2007 NiigatakenkenChuetsu-Okiearthquake(hereafter,calledtheChuestsu-Okiearthquake).Thetargetelectric power distribution system consists of 32,295 poles including 18,474 high voltage electric power distribution poles and 63 feeders, which indicates a high voltage distribution line.

Fig.5 shows the distribution of the seismic intensity scale of the Japan Meteorological Agency (JMA) due to the Chuetsu-Oki earthquake. The Chuetsu-Oki earthquake caused the 6 upper on the seismic intensity scale of the Japan Meteorological Agency (JMA 6+) as the maximum ground motion intensity to the struck area. In the analysis, it is assumed that the earthquake ground motion information related to Fig.5 has already been given as EGM(a) in Fig.4.

Fig.6 shows the power outage area caused by the Chestsu-Oki earthquake. Power outages also occurred to 15,074 high voltage poles, which is about 80 % of the total number of high voltage poles in the target system. The power outage information is given as the power outage information (PO) in Fig.4.

Fig.7 shows the observed points of the damaged poles due to the Chuetsu-Oki earthquake. The pole damages mainly consisted of two damage modes; i.e., breakage and inclination [8]. The damaged pole information is given as the damage inspection information (DI) in Fig.4.

In order to discuss the effectiveness of the proposed model, two situations are assumed in the following positive analysis;

**1.** The earthquake ground motion andpower outage information are given (discussedin 4.2).

**2.** The earthquake ground motion and the damage inspection information are partially given (discussed in 4.3).

**Combination No EGM(a) DI EPDE PO Joint occurrence probability** [1] 1 1 1 1 Equation(6) [2] 1 1 1 0 0

[5] 1 0 1 1 Pi

[6] 1 0 1 0 0 [7] 1 0 0 1 (1-Pi

[8] 1 0 0 0 (1-Pi

**Table 4.** Combination of the joint probability (CPT(A)+CPT(B)+CPT(C))

194 Earthquake Research and Analysis - New Advances in Seismology

(10) and Equation (11).

information (PO) in Fig.4.

following positive analysis;

**4.1. Precondition of positive analyses**

[3] 1 1 0 1 (1-Equation(6)) (1-Π(1-Equation(6)) [4] 1 1 0 0 (1-Equation(7)) Π(1-Equation(7))

The following positive analyses discuss the estimation accuracy for the situations of Equation

**4. Positive analyses based on the 2007 Niigata-Ken Chuetsu Oki earthquake**

This chapterdiscusses the effectiveness oftheproposedmodel basedonanactualpower outage and damage records of an electric power distribution system struck by the 2007 NiigatakenkenChuetsu-Okiearthquake(hereafter,calledtheChuestsu-Okiearthquake).Thetargetelectric power distribution system consists of 32,295 poles including 18,474 high voltage electric power

Fig.5 shows the distribution of the seismic intensity scale of the Japan Meteorological Agency (JMA) due to the Chuetsu-Oki earthquake. The Chuetsu-Oki earthquake caused the 6 upper on the seismic intensity scale of the Japan Meteorological Agency (JMA 6+) as the maximum ground motion intensity to the struck area. In the analysis, it is assumed that the earthquake ground motion information related to Fig.5 has already been given as EGM(a) in Fig.4.

Fig.6 shows the power outage area caused by the Chestsu-Oki earthquake. Power outages also occurred to 15,074 high voltage poles, which is about 80 % of the total number of high voltage poles in the target system. The power outage information is given as the power outage

Fig.7 shows the observed points of the damaged poles due to the Chuetsu-Oki earthquake. The pole damages mainly consisted of two damage modes; i.e., breakage and inclination [8]. The damaged pole information is given as the damage inspection information (DI) in Fig.4.

In order to discuss the effectiveness of the proposed model, two situations are assumed in the

**1.** The earthquake ground motion andpower outage information are given (discussedin 4.2).

distribution poles and 63 feeders, which indicates a high voltage distribution line.

(*a*)

(*a*)) (1-Π(1-P<sup>i</sup>

(*a*)) Π(1-P<sup>i</sup>

(*a*))

(*a*))

**Figure 5.** The distribution of the seismic intensity scale of the Japan Meteorological Agency (JMA) due to the Chuetsu-Oki earthquake (treated as EGM(a) in Fig.4)

**Figure 6.** The observed power outage of high voltage distribution lines due to the Chuetsu-Oki earthquake (treated as PO in Fig.4)

#### **4.2. The effect on accuracy improvement of power outage information**

Fig.8 shows a comparison between the observed and estimated number of damaged high voltage poles, which are normalized by the total number of observed damage poles. The caption [Observed] indicates the total number of observed poles damaged by the Chuetsu-Oki earthquake. The caption [without POI] indicates the total number of estimated damaged poles based on the causal relationship defined by Table 1, which is CPT (A) in Fig.4. The damage probability for pole *i*, which indicates the estimated damage number of pole *i*, is estimated as *Pi* (*EPDE= 1/EGM*(*a*)=*1*). On the other hand, the caption [with POI] indicates the total number of estimated damage poles based on the two causal relationships including CPT(A) and CPT(B) in Fig.4 evaluated by Equation (10).

0.001 to 0.01, there are significant differences in the damage probability with POI. On the other hand, when *P*(*a*) exceeds 0.03,whose earthquake ground motion intensity becomes over 6+, there is a limited effect of power outage information on improving the damage estimation

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197

This result suggests that the power outage information is usually effective for improving the estimation accuracy. However, when one feeder, which is a unit to identify the power outage range, has over 50 high voltage poles, and over 6+ of the earthquake ground motion level strike target feeder, there is a possibility that the effect of the power outage information only slightly

POI: Power Outage Information

**Figure 9.** Comparison of the number of damaged poles for every third mesh (1km×1km) among [Observed], [with

improves the damage estimation accuracy of the target equipment.

**Figure 8.** Comparison of total number of damaged high voltage poles

accuracy.

POI], and [without POI].

Fig.8 indicates that the normalized damaged number of [with POI], 1.06, is closer to that of the [Observed], 1.00, than that of [without POI], 1.25. This result suggests that the proposed model using the power outage information can effectively improve the damage estimation accuracy of the electric power distribution poles.

Fig.9, on the other hand, shows a comparison of the number of damaged poles for every third mesh (1km×1km) among [Observed], [with POI], and [without POI]. R indicates the correlation coefficients between [Observed] and [with POI], and between [Observed] and [without POI]. Fig.9 indicates that the correlation coefficient R between [Observed] and [with POI] is slightly higher than that between [Observed] and [without POI].

In order to discuss the improved effect of the power outage information on the damage estimation accuracy, Fig.10 shows the relationship among the damage probability of a pole without power outage information, the damage probability of a pole with power outage information, and the total number of poles connected to the same feeder. The horizontal axis indicates the damage probability of a pole without power outage information (PO), which is estimated as *Pi* (*EPDE= 1/EGM* (*a*)=*1*) based on Table 1. The vertical axis indicates that with POI, which is estimated as *Pi* (*EPDE=1/POI=*1) based on Equation (10). *Ne*, the total number of poles on the same feeder, imitates the actually installed feeder conditions of the target electric power distribution system.

Fig.10 illustrates that when the damage probability of a pole without POI is 0.001, POI improves the damage probability to 0.5, 0.1, 0.02 and 0.0015 on the condition that *Ne*=2, *Ne*=10, *Ne*= 50, *Ne*=100, and *Ne*=1000, respectively. This result suggests that the power outage information becomes more effective along with a decrease in the number of poles connected to the same feeder. In this paper, though it is assumed that a power company can identify a power outage range for every feeder, some power companies can identify the power outage within a subdivided range using a switch. In such a situation, the power outage information is more useful to improve the damage estimation accuracy.

Fig.10 also shows that the improvement effect based on the power outage information depends on the earthquake ground motion intensity level. For example, when the earthquake ground motion intensity under a target pole becomes about 6- to 6+ on the JMA seismic intensity scale, it is usually estimated that the damage probability without POI, which is evaluated by *P*(*a*), *Pi* (*EPDE=1/EGM* (*a*)=*1*), becomes 0.001 to 0.01. When the damage probability without POI is from 0.001 to 0.01, there are significant differences in the damage probability with POI. On the other hand, when *P*(*a*) exceeds 0.03,whose earthquake ground motion intensity becomes over 6+, there is a limited effect of power outage information on improving the damage estimation accuracy.

This result suggests that the power outage information is usually effective for improving the estimation accuracy. However, when one feeder, which is a unit to identify the power outage range, has over 50 high voltage poles, and over 6+ of the earthquake ground motion level strike target feeder, there is a possibility that the effect of the power outage information only slightly improves the damage estimation accuracy of the target equipment.

POI: Power Outage Information

**Figure 8.** Comparison of total number of damaged high voltage poles

**4.2. The effect on accuracy improvement of power outage information**

*Pi*

in Fig.4 evaluated by Equation (10).

196 Earthquake Research and Analysis - New Advances in Seismology

of the electric power distribution poles.

POI, which is estimated as *Pi*

power distribution system.

higher than that between [Observed] and [without POI].

useful to improve the damage estimation accuracy.

Fig.8 shows a comparison between the observed and estimated number of damaged high voltage poles, which are normalized by the total number of observed damage poles. The caption [Observed] indicates the total number of observed poles damaged by the Chuetsu-Oki earthquake. The caption [without POI] indicates the total number of estimated damaged poles based on the causal relationship defined by Table 1, which is CPT (A) in Fig.4. The damage probability for pole *i*, which indicates the estimated damage number of pole *i*, is estimated as

 (*EPDE= 1/EGM*(*a*)=*1*). On the other hand, the caption [with POI] indicates the total number of estimated damage poles based on the two causal relationships including CPT(A) and CPT(B)

Fig.8 indicates that the normalized damaged number of [with POI], 1.06, is closer to that of the [Observed], 1.00, than that of [without POI], 1.25. This result suggests that the proposed model using the power outage information can effectively improve the damage estimation accuracy

Fig.9, on the other hand, shows a comparison of the number of damaged poles for every third mesh (1km×1km) among [Observed], [with POI], and [without POI]. R indicates the correlation coefficients between [Observed] and [with POI], and between [Observed] and [without POI]. Fig.9 indicates that the correlation coefficient R between [Observed] and [with POI] is slightly

In order to discuss the improved effect of the power outage information on the damage estimation accuracy, Fig.10 shows the relationship among the damage probability of a pole without power outage information, the damage probability of a pole with power outage information, and the total number of poles connected to the same feeder. The horizontal axis indicates the damage probability of a pole without power outage information (PO), which is estimated as *Pi* (*EPDE= 1/EGM* (*a*)=*1*) based on Table 1. The vertical axis indicates that with

poles on the same feeder, imitates the actually installed feeder conditions of the target electric

Fig.10 illustrates that when the damage probability of a pole without POI is 0.001, POI improves the damage probability to 0.5, 0.1, 0.02 and 0.0015 on the condition that *Ne*=2, *Ne*=10, *Ne*= 50, *Ne*=100, and *Ne*=1000, respectively. This result suggests that the power outage information becomes more effective along with a decrease in the number of poles connected to the same feeder. In this paper, though it is assumed that a power company can identify a power outage range for every feeder, some power companies can identify the power outage within a subdivided range using a switch. In such a situation, the power outage information is more

Fig.10 also shows that the improvement effect based on the power outage information depends on the earthquake ground motion intensity level. For example, when the earthquake ground motion intensity under a target pole becomes about 6- to 6+ on the JMA seismic intensity scale, it is usually estimated that the damage probability without POI, which is evaluated by *P*(*a*), *Pi* (*EPDE=1/EGM* (*a*)=*1*), becomes 0.001 to 0.01. When the damage probability without POI is from

(*EPDE=1/POI=*1) based on Equation (10). *Ne*, the total number of

**Figure 9.** Comparison of the number of damaged poles for every third mesh (1km×1km) among [Observed], [with POI], and [without POI].

damages for every third mesh based on the conditional damage probabilities of all poles,

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199

it is assumed that actual damaged poles by the Chuetsu-Oki earthquake shown in Fig.7 are inspected based on an inspection

3. The conditional damage probabilities, *Pi*(*EPDE*=1/*EGM*(*a*)=1), for all poles are estimated based on table 1.

4. The averages of the estimated damage probabilities for every third mesh and for every fifth mesh are evaluated.

Fig.12 compares the number of estimated damaged poles evaluated by Equation (11) to that of the observed damage poles for every third mesh. Fig.13 also shows a comparison of the number of estimated damaged poles to that of the observed damage poles for every third mesh. The estimated values consist of two factors; i.e., the number of estimated damaged poles with 29% inspection information and that without any inspection information. The correlation coefficient R between the number of estimated damaged poles with 29 % inspection informa‐ tion and that of the observed one is 0.89 while R between that without inspection information and the observed one is 0.59. This result suggests that the estimation accuracy is highly

Fig.14 shows a sensitivity analysis between the inspection rate and the correlation coefficient. Fig.14 indicates that when the inspection rate exceeds 0.35, the correlation coefficient R becomes greater than 0.9. This result suggests that when a target correlation coefficient is assumed, the districts to be inspected for obtaining target damage estimation accuracy could be automatically determined in a target electric power distribution system. The proposed model enables us to rationalize the inspection during the initial and emergency restoration

5. The differences in the average of all the estimated damage probabilities between the third meshes and the 16 fifth meshes

6. The 16 fifth meshes are put in order based on the inspection priority index value. The smaller the priority index, the

Fig.11 shows the allocated inspection districts of fifth meshes with a 29% inspection rate. The red square point shows the inspection point which is determined by the inspection priority index. The background chart with allocated third meshes is the estimated number of pole damages for every third mesh based on the conditional damage probabilities of all poles, *Pi*(*EPDE*=1/*EGM*(*a*)=1).

Fig.12 compares the number of estimated damaged poles evaluated by Equation (11) to that of the observed damage poles for every third mesh. Fig.13 also shows a comparison of the number of estimated damaged poles to that of the observed damage poles for every third mesh. The estimated values consist of two factors; i.e., the number of estimated damaged poles with 29% inspection information and that without any inspection information. The correlation coefficient R between the number of estimated damaged poles with 29 % inspection information and that of the observed one is 0.89 while R between that without inspection information and the observed one is 0.59. This result suggests that the estimation accuracy is highly improved by the 29% inspection

Fig.14 shows a sensitivity analysis between the inspection rate and the correlation coefficient. Fig.14 indicates that when the inspection rate exceeds 0.35, the correlation coefficient R becomes greater than 0.9. This result suggests that when a target correlation coefficient is assumed, the districts to be inspected for obtaining target damage estimation accuracy could be automatically determined in a target electric power distribution system. The proposed model enables us to rationalize the

Inspection district (250m×250m)

Number of damages

*Pi*

periods.

information.

higher the priority level.

(*EPDE*=1/*EGM*(*a*)=1).

priority. The inspection priority is determined as follows:

1. The target area is divided into the third mesh (1km×1km)

improved by the 29% inspection information.

of the same third mesh are evaluated as an inspection priority index.

inspection during the initial and emergency restoration periods.

Figure 11.Effective inspection points for the damage estimation of a whole target area

**Figure 11.** Effective inspection points for the damage estimation of a whole target area

2. The third mesh (1km×1km) is also divided into the 16 fifth meshes (250m×250m).

**Figure 10.** Effect of power outage information on the conditional damage probability of electric power distribution equipment

#### **4.3. The effect on accuracy improvement by the damage inspection information**

This section discusses the effect of the damage inspection information to improve the damage probability of the electric power distribution poles. In order to understand the effect of the inspection information on the improvement of the damage estimation accuracy, the damage probability of poles, *Pi* (*EPDE=1/EGM* (*a*)=*1*) is updated based on the different inspection rates. The inspection rate is evaluated as the number of inspected poles divided by the total number of poles (32,295 poles). In this simulation, it is assumed that actual damaged poles by the Chuetsu-Oki earthquake shown in Fig.7 are inspected based on an inspection priority. The inspection priority is determined as follows:


Fig.11 shows the allocated inspection districts of fifth meshes with a 29% inspection rate. The red square point shows the inspection point which is determined by the inspection priority index. The background chart with allocated third meshes is the estimated number of pole

damages for every third mesh based on the conditional damage probabilities of all poles, *Pi* (*EPDE*=1/*EGM*(*a*)=1). it is assumed that actual damaged poles by the Chuetsu-Oki earthquake shown in Fig.7 are inspected based on an inspection

priority. The inspection priority is determined as follows:

inspection during the initial and emergency restoration periods.

information.

Fig.12 compares the number of estimated damaged poles evaluated by Equation (11) to that of the observed damage poles for every third mesh. Fig.13 also shows a comparison of the number of estimated damaged poles to that of the observed damage poles for every third mesh. The estimated values consist of two factors; i.e., the number of estimated damaged poles with 29% inspection information and that without any inspection information. The correlation coefficient R between the number of estimated damaged poles with 29 % inspection informa‐ tion and that of the observed one is 0.89 while R between that without inspection information and the observed one is 0.59. This result suggests that the estimation accuracy is highly improved by the 29% inspection information. 1. The target area is divided into the third mesh (1km×1km) 2. The third mesh (1km×1km) is also divided into the 16 fifth meshes (250m×250m). 3. The conditional damage probabilities, *Pi*(*EPDE*=1/*EGM*(*a*)=1), for all poles are estimated based on table 1. 4. The averages of the estimated damage probabilities for every third mesh and for every fifth mesh are evaluated. 5. The differences in the average of all the estimated damage probabilities between the third meshes and the 16 fifth meshes of the same third mesh are evaluated as an inspection priority index.

Fig.14 shows a sensitivity analysis between the inspection rate and the correlation coefficient. Fig.14 indicates that when the inspection rate exceeds 0.35, the correlation coefficient R becomes greater than 0.9. This result suggests that when a target correlation coefficient is assumed, the districts to be inspected for obtaining target damage estimation accuracy could be automatically determined in a target electric power distribution system. The proposed model enables us to rationalize the inspection during the initial and emergency restoration periods. 6. The 16 fifth meshes are put in order based on the inspection priority index value. The smaller the priority index, the higher the priority level. Fig.11 shows the allocated inspection districts of fifth meshes with a 29% inspection rate. The red square point shows the inspection point which is determined by the inspection priority index. The background chart with allocated third meshes is the estimated number of pole damages for every third mesh based on the conditional damage probabilities of all poles, *Pi*(*EPDE*=1/*EGM*(*a*)=1). Fig.12 compares the number of estimated damaged poles evaluated by Equation (11) to that of the observed damage poles for every third mesh. Fig.13 also shows a comparison of the number of estimated damaged poles to that of the observed damage poles for every third mesh. The estimated values consist of two factors; i.e., the number of estimated damaged poles with 29% inspection

information and that without any inspection information. The correlation coefficient R between the number of estimated damaged poles with 29 % inspection information and that of the observed one is 0.89 while R between that without inspection information and the observed one is 0.59. This result suggests that the estimation accuracy is highly improved by the 29% inspection

Fig.14 shows a sensitivity analysis between the inspection rate and the correlation coefficient. Fig.14 indicates that when the inspection rate exceeds 0.35, the correlation coefficient R becomes greater than 0.9. This result suggests that when a target correlation coefficient is assumed, the districts to be inspected for obtaining target damage estimation accuracy could be automatically determined in a target electric power distribution system. The proposed model enables us to rationalize the

**Figure 10.** Effect of power outage information on the conditional damage probability of electric power distribution

This section discusses the effect of the damage inspection information to improve the damage probability of the electric power distribution poles. In order to understand the effect of the inspection information on the improvement of the damage estimation accuracy, the damage probability of poles, *Pi* (*EPDE=1/EGM* (*a*)=*1*) is updated based on the different inspection rates. The inspection rate is evaluated as the number of inspected poles divided by the total number of poles (32,295 poles). In this simulation, it is assumed that actual damaged poles by the Chuetsu-Oki earthquake shown in Fig.7 are inspected based on an inspection priority. The

**4.3. The effect on accuracy improvement by the damage inspection information**

**2.** The third mesh (1km×1km) is also divided into the 16 fifth meshes (250m×250m).

**4.** The averages of the estimated damage probabilities for every third mesh and for every

**5.** The differences in the average of all the estimated damage probabilities between the third meshes and the 16 fifth meshes of the same third mesh are evaluated as an inspection

**6.** The 16 fifth meshes are put in order based on the inspection priority index value. The

Fig.11 shows the allocated inspection districts of fifth meshes with a 29% inspection rate. The red square point shows the inspection point which is determined by the inspection priority index. The background chart with allocated third meshes is the estimated number of pole

(*EPDE*=1/*EGM*(*a*)=1), for all poles are estimated

inspection priority is determined as follows:

198 Earthquake Research and Analysis - New Advances in Seismology

**3.** The conditional damage probabilities, *Pi*

based on table 1.

priority index.

fifth mesh are evaluated.

**1.** The target area is divided into the third mesh (1km×1km)

smaller the priority index, the higher the priority level.

equipment

**Figure 11.** Effective inspection points for the damage estimation of a whole target area

Figure 11.Effective inspection points for the damage estimation of a whole target area

**Figure 12.** Comparison of estimation accuracies between the damage estimations with 29 % inspection information and without inspection information

(b) Estimated inspection information

ship between the in n of damaged pole

nspection rate and

es

1 2 14

**Fig. 14.** Relations estimation

29 % inspection inform f damaged poles mation with

Bo

ook Title

**Figure 14.** Relationship between the inspection rate and the correlation coefficient associated with the observed and

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201

This paper proposed a model to integrate multiple disaster information. The proposed model enables us to improve the damage estimation accuracy of electric power distribution equip‐ ment during the initial and emergency restoration periods after an earthquake. The research

The restoration process for an electric power distribution system after an earthquake is divided into three periods; i.e., initial, emergency, and permanent restoration periods. The necessary information and the information that was able to be collected within the three restoration periods were elucidated. As a result, it was clarified that the damage estimation technologies are very useful for actual restoration work under limited disaster information circumstances while the application of a seismic damage estimation system for electric power distribution equipment (RAMPEr) to the actual restoration work after the 2011 earthquake off the Pacific

**2.** Formulation of a sequential updated model for electric power distribution system

A basic model to integrate the sequentially updated disaster information was proposed based on a Bayesian network. The proposed model can effectively integrate multidimensional

estimation of damaged poles

results are summarized as follows.

coast of the Tōhoku is described.

**1.** Information required for the emergency restoration work

**5. Conclusion**

d the correlation co

oefficient associat

ted with the observ

ved and

**Figure 14.** Relationship between the inspection rate and the correlation coefficient associated with the observed and estimation of damaged poles

#### **5. Conclusion**

**Figure 12.** Comparison of estimation accuracies between the damage estimations with 29 % inspection information

number of damage

d poles

Bo

ook Title

h that of observed

d damage poles fo

or every

with

ved and

f damaged poles

mation

ted with the observ

mated number of inspection inform

oefficient associat

amaged poles with

**Figure 13.** Comparison of the number of estimated damaged poles to that of observed damaged poles for every third

t (c) Estim 29 %

d the correlation co

and without inspection information

14

1 2

mesh.

**Fig. 13.** Compari third mes

ison of the numbe

number of damag information

Number of damag

200 Earthquake Research and Analysis - New Advances in Seismology

(a) Observed n

ged poles

er of estimated da

ged poles without

nspection rate and

es

ship between the in n of damaged pole

sh.

(b) Estimated inspection

**Fig. 14.** Relations estimation

This paper proposed a model to integrate multiple disaster information. The proposed model enables us to improve the damage estimation accuracy of electric power distribution equip‐ ment during the initial and emergency restoration periods after an earthquake. The research results are summarized as follows.

**1.** Information required for the emergency restoration work

The restoration process for an electric power distribution system after an earthquake is divided into three periods; i.e., initial, emergency, and permanent restoration periods. The necessary information and the information that was able to be collected within the three restoration periods were elucidated. As a result, it was clarified that the damage estimation technologies are very useful for actual restoration work under limited disaster information circumstances while the application of a seismic damage estimation system for electric power distribution equipment (RAMPEr) to the actual restoration work after the 2011 earthquake off the Pacific coast of the Tōhoku is described.

**2.** Formulation of a sequential updated model for electric power distribution system

A basic model to integrate the sequentially updated disaster information was proposed based on a Bayesian network. The proposed model can effectively integrate multidimensional disaster information, including earthquake ground motion, power outage and damage inspection information, to improve the estimation accuracy of seismic damaged electric power distribution poles.

where *fc*(*z*(*a*)) is the seismic damage ratio of equipment *i* with seismic countermeasure *C* and seismic performance *z*(*a*) assuming that the maximum ground motion *a* affects the target equipment. The seismic performance *z*(*a*) is the seismic safety margin evaluated by the bending moment of the ground surface of the electric power distribution poles caused by the maximum ground motion *a*. *Sl* is the modification coefficient evaluated by the line connected type *l.Tk* is

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

the microtopography division *j. Ci* is the modification coefficient for the seismic countermeas‐ ure of equipment *i. L <sup>m</sup>* is the modification coefficient for local region *m*. *m* is an electric power

where *z0*(*a*) is the safety margin relative to the maximum surface ground acceleration *a* (*m*/*s2*

which defined as the ratio to the static earthquake force of the design collapse load (*N*). The static force is converted into the top concentration load of a distribution pole from the

According to the Japan Electric Technical Standards and Codes Committee (2007), *Z0*(*a*) is

0.3 ( ) (with pole anchor) ( ) ( ')

*K is* a soil coefficient defined by the Japan Electric Technical Standards and Codes Committee

includes hardened soil, sand, gravel, and soil with small stones. The standard soil [B] is defined

*D*0is the diameter on the ground surface of the distribution pole (*m*). *t* is the penetration depth of the distribution pole into the ground (*m*). *H* is the concentration load height from the ground

*P*(*a*) is the concentration load (*kN*) converted from the maximum surface ground acceleration

1 1 ( ) - 0.25 *<sup>a</sup> P a W l*

) , which includes softer soil than [A]. Poor soil [C] is defined as 2.0×107 (N/*m4*

*H*= ´ (15)

ì ü × × ï ï × + <sup>=</sup> í ý ï ï ×× + × + î þ

Seismic performance *z*(*a*) relative to the earthquake intensity *a* is evaluated as

4 0

*K D Q t AJ Pe H t*

0 4

0

*KD t Pe H t*

2 0

(without pole anchor) 120 ( ) ( ) ( )

2 0

which includes a kind of silt without soil. Poor soil [D] is defined as 0.8×107 (N/*m4*

(2007). *K* is divided into four types. The standard soil [A] is defined as 3.9×107

is the modification coefficient for

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),

203

(14)

), which

) , which

) ,

(N/*m4*

<sup>0</sup> *za k k z a* () 1 2 () =×× (13)

the modification coefficient for the land use condition *k.Bj*

supply area covered by a business branch office.

maximum surface ground acceleration.

*Z e*

includes moist clay and humid soil.

). *P*(*a*) is evaluated as

evaluated as

as 2.9×107 (N/*m4*

surface (*m*).

*a* (*m*/*s2*

**3.** Positive analysis based on the 2007 Niigata-Ken Chuetsu-Oki earthquake

The proposed model was applied to an actual electric power distribution system struck by the 2007 Niigataken Chuetsu-oki Earthquake, and the effect of the power outage and damage inspection information for improvement of the damage estimation accuracy was verified. As for the power outage information, it was clarified that under the installed conditions of the actual electric power distribution system, the damage estimation accuracy with power outage information was higher than that without power outage information. It was also realized that in order to effectively utilize the power outage information by the proposed model, the size of one feeder, which was related to a unit to identify the power outage range, and the earth‐ quake ground motion level, which determined the damage probability level, were important parameters.

On the other hand, as for the inspection information, in order to effectively select the damage inspection point, the inspection priority of the actual electric power distribution poles was proposed. Based on the proposed inspection priority, the relationship between the inspection rate and the damage estimation accuracy was analyzed. As a result, it was also clarified that under the installed conditions of the actual electric power distribution system, the estimation accuracy is highly improved only by the 29% inspection information and when the inspection rate exceeds 0.35, the correlation coefficient R between the number of observed damaged poles and that of the estimated one becomes greater than 0.9.

In Japan, the occurrence of the Nankai Trough earthquake is feared. As mentioned in Chapter 2, the damage estimation system, RAMPEr, in which the proposed model has already been installed, is operating in some areas that could be highly affected by the Nankai Trough earthquake. In such a high seismic area, it is expected that RAMPEr will become a useful tool to support the restoration work after the earthquake. As future subjects, in order to improve the damage estimation accuracy of the proposed model, some remote sensing images will be integrated into the proposed model and the damage records due to the 2011 earthquake off the Pacific coast of Tohoku will be analyzed.

### **Apendix [A]**

Equipment damage estimation model [4]

In this paper, based on reference [4], the seismic damage probability with the maximum earthquake ground motion *a* (Pi (*a*)) is evaluated as

$$P\_i(\mathfrak{a}) = L\_m \cdot \mathbb{C}\_i \cdot \mathcal{B}\_j \cdot T\_k \cdot \mathbb{S}\_l \cdot f\_c(\mathfrak{z}(\mathfrak{a})) \tag{12}$$

where *fc*(*z*(*a*)) is the seismic damage ratio of equipment *i* with seismic countermeasure *C* and seismic performance *z*(*a*) assuming that the maximum ground motion *a* affects the target equipment. The seismic performance *z*(*a*) is the seismic safety margin evaluated by the bending moment of the ground surface of the electric power distribution poles caused by the maximum ground motion *a*. *Sl* is the modification coefficient evaluated by the line connected type *l.Tk* is the modification coefficient for the land use condition *k.Bj* is the modification coefficient for the microtopography division *j. Ci* is the modification coefficient for the seismic countermeas‐ ure of equipment *i. L <sup>m</sup>* is the modification coefficient for local region *m*. *m* is an electric power supply area covered by a business branch office.

Seismic performance *z*(*a*) relative to the earthquake intensity *a* is evaluated as

disaster information, including earthquake ground motion, power outage and damage inspection information, to improve the estimation accuracy of seismic damaged electric power

The proposed model was applied to an actual electric power distribution system struck by the 2007 Niigataken Chuetsu-oki Earthquake, and the effect of the power outage and damage inspection information for improvement of the damage estimation accuracy was verified. As for the power outage information, it was clarified that under the installed conditions of the actual electric power distribution system, the damage estimation accuracy with power outage information was higher than that without power outage information. It was also realized that in order to effectively utilize the power outage information by the proposed model, the size of one feeder, which was related to a unit to identify the power outage range, and the earth‐ quake ground motion level, which determined the damage probability level, were important

On the other hand, as for the inspection information, in order to effectively select the damage inspection point, the inspection priority of the actual electric power distribution poles was proposed. Based on the proposed inspection priority, the relationship between the inspection rate and the damage estimation accuracy was analyzed. As a result, it was also clarified that under the installed conditions of the actual electric power distribution system, the estimation accuracy is highly improved only by the 29% inspection information and when the inspection rate exceeds 0.35, the correlation coefficient R between the number of observed damaged poles

In Japan, the occurrence of the Nankai Trough earthquake is feared. As mentioned in Chapter 2, the damage estimation system, RAMPEr, in which the proposed model has already been installed, is operating in some areas that could be highly affected by the Nankai Trough earthquake. In such a high seismic area, it is expected that RAMPEr will become a useful tool to support the restoration work after the earthquake. As future subjects, in order to improve the damage estimation accuracy of the proposed model, some remote sensing images will be integrated into the proposed model and the damage records due to the 2011 earthquake off

In this paper, based on reference [4], the seismic damage probability with the maximum

( ) ( ( )) *i m i jklc P a L C B T S f za* = × × × ×× (12)

(*a*)) is evaluated as

**3.** Positive analysis based on the 2007 Niigata-Ken Chuetsu-Oki earthquake

and that of the estimated one becomes greater than 0.9.

the Pacific coast of Tohoku will be analyzed.

Equipment damage estimation model [4]

earthquake ground motion *a* (Pi

distribution poles.

202 Earthquake Research and Analysis - New Advances in Seismology

parameters.

**Apendix [A]**

$$z(\mathfrak{a}) = k1 \cdot k2 \cdot z\_0(\mathfrak{a}) \tag{13}$$

where *z0*(*a*) is the safety margin relative to the maximum surface ground acceleration *a* (*m*/*s2* ), which defined as the ratio to the static earthquake force of the design collapse load (*N*). The static force is converted into the top concentration load of a distribution pole from the maximum surface ground acceleration.

According to the Japan Electric Technical Standards and Codes Committee (2007), *Z0*(*a*) is evaluated as

$$Z\_0(e) = \begin{cases} \frac{K \cdot D\_0 \cdot t^4}{120 P(e) \cdot \left(H + t\_0\right)^2} & \text{(without pole anchor)}\\ \frac{0.3 K \left(D\_0 \cdot Q \cdot t^4 + A\right)}{P(e) \cdot \left(H + t\_0\right)^2} & \text{(with pole anchor)} \end{cases} \tag{14}$$

*K is* a soil coefficient defined by the Japan Electric Technical Standards and Codes Committee (2007). *K* is divided into four types. The standard soil [A] is defined as 3.9×107 (N/*m4* ), which includes hardened soil, sand, gravel, and soil with small stones. The standard soil [B] is defined as 2.9×107 (N/*m4* ) , which includes softer soil than [A]. Poor soil [C] is defined as 2.0×107 (N/*m4* ) , which includes a kind of silt without soil. Poor soil [D] is defined as 0.8×107 (N/*m4* ) , which includes moist clay and humid soil.

*D*0is the diameter on the ground surface of the distribution pole (*m*). *t* is the penetration depth of the distribution pole into the ground (*m*). *H* is the concentration load height from the ground surface (*m*).

*P*(*a*) is the concentration load (*kN*) converted from the maximum surface ground acceleration *a* (*m*/*s2* ). *P*(*a*) is evaluated as

$$P(a) = \frac{a}{H \cdot 0.25} \times W\_1 l\_1 \tag{15}$$

where *H* is the height of the distribution pole from the ground surface (*m*), *W*1is mass of the upper ground part of the distribution pole (*kg*), *l*1is the height of the gravity center of the upper part of the distribution pole (*m*).

*t0* is the depth of the gyration center of the distribution pole from the ground surface, which is evaluated as

$$t\_0 = \frac{2}{3}t \quad \text{(without pole anchor)}\tag{16}$$

$$t\_0 \stackrel{\cdot}{=} \frac{2}{3} (\frac{t^2 + 3nt\_c^{\prime}}{t + 2nt\_c}) \text{ (with pole anchor)}\tag{17}$$

$$m = \frac{A}{A\_1} \tag{18}$$

Based on a preliminary analysis, *k*1 is evaluated as

*i*=3: overhead transformer; *i=*4: joint use wire; *wi*

In Equation (A.1), the damage ratio is evaluated as

*c xx*

l x

*z a*

*<sup>x</sup>* ={ *<sup>d</sup>* (damaged equipment) *all* (all equipment) }

with seismic countermeasure c. *λx*=*all*

*c*

bution pole from the ground surface.

equipment is more than four).

4

<sup>1</sup> <sup>1</sup>

*k*

=

1

1 1 *i i i*

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

where *a1*and *b1* are recurrent coefficients evaluated by a preliminary analysis, and these values are assumed to be *a1*is 0.000428, *b1*is 1.0. Note that *i*=1: high voltage wire; *i*=2: low voltage wire;

*k2 is* a modification coefficient considering adjacent distribution poles. *k2* is assumed to be

ln ( ( ) / , ) ( ( )) ln ( ( ) / , ) *c c dd*

<sup>2</sup> 1 1 ln( ( ))) ln ( ( ) / , ) exp ( ) 2 () 2

*z a*

*DP z a f za TP z a*

p x

associated with the damaged equipment, respectively. When x is all, ln*<sup>l</sup>*

for all equipment with seismic countermeasure c, respectively.

*c c all all*

l x

= × (24)

*z a*

x

é ù - <sup>=</sup> × -× ê ú × × ë û (25)

*x*

(*z*(*a*));*λx*=*all*

are the mean and standard deviation of ln(z(a))

, *ζx*=*all* ) is

l

l x

*x x*

where, *f*c(z(*a*)) is the seismic damage ratio function of equipment with seismic countermeasure *c* and equipment performance *z*(*a*) assuming that the maximum ground surface acceleration *a. DPc* is the total number of actual seismic damaged equipment with seismic countermeasure *c*, *TPc* is the total number of equipment with seismic countermeasure *c.* When x is *d,* ln*c*(*z*(*a*))/ *λx*=*<sup>d</sup>* , *ζx*=*<sup>d</sup>* ) is the log normal probability density function of the performance value z(a) associated with damaged equipment with seismic countermeasure c due to a target earthquake. When x is d, *λx*=*d* and *ζx*=*d* are the mean and standard deviation of ln(z(a))

the log normal probability density function of the performance value z(a) of all equipment

and *ζx*=*all*

*k2*=1.0 (in the case where adjacent distribution poles have no overhead equipment), *k2=*0.9 (in the case where the number of adjacent distribution poles with overhead equipment is less than three), *k2 =*0.85 (in the case where the number of adjacent distribution poles with overhead

´ + å (23)

is the mass of *I*; *Li* is the height of the distri‐

http://dx.doi.org/10.5772/55207

205

*a wL b* =

where *A is* the area of the pole anchor. *A*<sup>1</sup> is the area of the base part of the distribution pole. A and *A*1 are evaluated as

$$A = (L - D\_0)d\tag{19}$$

$$A\_1 = D\_0 t \tag{20}$$

where *L* is the length of the pole anchor (*m*), *d* is the width of the pole anchor (*m*).

*Q and J* are evaluated as

$$Q = \frac{1}{12}(6m^2 - 8m + 3), m = \frac{t\_0}{t}^{'}\tag{21}$$

$$J = (t\_0 \text{'} - t\_c)^2 t\_c \tag{22}$$

where *tc* is the depth of the center of the pole anchor from the ground surface (*m*).

*k*1 is a modification coefficient considering the effects of the overhead wire including strung and joint use wires, and overhead equipment such as an overhead transformer.

Based on a preliminary analysis, *k*1 is evaluated as

where *H* is the height of the distribution pole from the ground surface (*m*), *W*1is mass of the upper ground part of the distribution pole (*kg*), *l*1is the height of the gravity center of the upper

*t0* is the depth of the gyration center of the distribution pole from the ground surface, which

*t t* = chor) (16)

<sup>+</sup> <sup>=</sup> <sup>+</sup> (17)

*<sup>A</sup>* <sup>=</sup> (18)

<sup>0</sup> *A L Dd* = - ( ) (19)

*A Dt* 1 0 = (20)

= -+ = (21)

<sup>0</sup> (' )*c c J t tt* = - (22)

<sup>0</sup> without pole an <sup>2</sup> ( <sup>3</sup>

<sup>0</sup> with pole an <sup>2</sup> <sup>3</sup> ( )( chor) 3 2 *c c*

> 1 *A n*

where *A is* the area of the pole anchor. *A*<sup>1</sup> is the area of the base part of the distribution pole.

where *L* is the length of the pole anchor (*m*), *d* is the width of the pole anchor (*m*).

where *tc* is the depth of the center of the pole anchor from the ground surface (*m*).

and joint use wires, and overhead equipment such as an overhead transformer.

<sup>2</sup> <sup>0</sup> <sup>1</sup> ' (6 8 3), <sup>12</sup> *<sup>t</sup> Q mm m <sup>t</sup>*

2

*k*1 is a modification coefficient considering the effects of the overhead wire including strung

2 2

*t nt*

*t nt*

'

*t*

part of the distribution pole (*m*).

204 Earthquake Research and Analysis - New Advances in Seismology

A and *A*1 are evaluated as

*Q and J* are evaluated as

is evaluated as

$$k1 = \frac{1}{a1 \times \sum\_{i=1}^{4} w\_i L\_i + b1} \tag{23}$$

where *a1*and *b1* are recurrent coefficients evaluated by a preliminary analysis, and these values are assumed to be *a1*is 0.000428, *b1*is 1.0. Note that *i*=1: high voltage wire; *i*=2: low voltage wire; *i*=3: overhead transformer; *i=*4: joint use wire; *wi* is the mass of *I*; *Li* is the height of the distri‐ bution pole from the ground surface.

*k2 is* a modification coefficient considering adjacent distribution poles. *k2* is assumed to be

*k2*=1.0 (in the case where adjacent distribution poles have no overhead equipment), *k2=*0.9 (in the case where the number of adjacent distribution poles with overhead equipment is less than three), *k2 =*0.85 (in the case where the number of adjacent distribution poles with overhead equipment is more than four).

In Equation (A.1), the damage ratio is evaluated as

$$\ln f\_c(\mathbf{z}(a)) = \frac{DP\_c}{TP\_c} \cdot \frac{\ln\_c(\mathbf{z}(a) / \lambda\_{d'} \tilde{\xi}\_d)}{\ln\_c(\mathbf{z}(a) / \lambda\_{\text{all}} \cdot \tilde{\xi}\_{\text{all}})} \tag{24}$$

$$\ln\_{\boldsymbol{c}}(\boldsymbol{z}(\boldsymbol{a}) / \boldsymbol{\lambda}\_{\boldsymbol{x}}, \boldsymbol{\xi}\_{\boldsymbol{x}}) = \frac{1}{\sqrt{2\pi} \cdot \boldsymbol{\xi}\_{\boldsymbol{x}} \cdot \boldsymbol{z}(\boldsymbol{a})} \cdot \exp\left[ -\frac{1}{2} \cdot (\frac{\ln(\boldsymbol{z}(\boldsymbol{a}))) - \boldsymbol{\lambda}\_{\boldsymbol{x}}}{\boldsymbol{\xi}\_{\boldsymbol{x}}} \boldsymbol{\lambda}^{2} \right] \tag{25}$$

*<sup>x</sup>* ={ *<sup>d</sup>* (damaged equipment) *all* (all equipment) }

where, *f*c(z(*a*)) is the seismic damage ratio function of equipment with seismic countermeasure *c* and equipment performance *z*(*a*) assuming that the maximum ground surface acceleration *a. DPc* is the total number of actual seismic damaged equipment with seismic countermeasure *c*, *TPc* is the total number of equipment with seismic countermeasure *c.* When x is *d,* ln*c*(*z*(*a*))/ *λx*=*<sup>d</sup>* , *ζx*=*<sup>d</sup>* ) is the log normal probability density function of the performance value z(a) associated with damaged equipment with seismic countermeasure c due to a target earthquake. When x is d, *λx*=*d* and *ζx*=*d* are the mean and standard deviation of ln(z(a)) associated with the damaged equipment, respectively. When x is all, ln*<sup>l</sup>* (*z*(*a*));*λx*=*all* , *ζx*=*all* ) is the log normal probability density function of the performance value z(a) of all equipment with seismic countermeasure c. *λx*=*all* and *ζx*=*all* are the mean and standard deviation of ln(z(a)) for all equipment with seismic countermeasure c, respectively.

#### **Acknowledgements**

The views and actual damage records expressed herein are based on research supported by several electric power companies including the Tohoku Electric Power Co., Inc.

[9] Tohoku Electric Power Co. Inc. "Power outage information associated with the 2012 earthquake off the Pacific coast of Tōhoku", http://www.tohoku-epco.co.jp/informa‐

Damage Estimation Improvement of Electric Power Distribution Equipment Using Multiple Disaster Information

http://dx.doi.org/10.5772/55207

207

[10] Miyako,T., and Todou, T,. Development of seismic damage estimation system for electric power equipment, *Electrical Review*, 2011; No.2011. 10, 76-78 (in Japanese).

tion/1182212\_821.html (accessed November 21, 2012).

#### **Author details**

#### Yoshiharu Shumuta

Civil Engineering Laboratory, Central Research Institute of Electric Power Industry, Chiba, Japan

#### **References**


[9] Tohoku Electric Power Co. Inc. "Power outage information associated with the 2012 earthquake off the Pacific coast of Tōhoku", http://www.tohoku-epco.co.jp/informa‐ tion/1182212\_821.html (accessed November 21, 2012).

**Acknowledgements**

206 Earthquake Research and Analysis - New Advances in Seismology

**Author details**

Yoshiharu Shumuta

Japan

**References**

November 2012).

November 2012).

man & Hall/CRC,2004.

Wiley, 2009.

ing(CASP11),2011;MS\_218,2026-2033.

Civil Engineering, 2004 (in Japanese).

nese).

The views and actual damage records expressed herein are based on research supported by

Civil Engineering Laboratory, Central Research Institute of Electric Power Industry, Chiba,

[1] Japan Meteorological Agency: What is an Earthquake Early Warning ?, http:// www.jma.go.jp/jma/en/Activities/eew1.html/ (accessed 21 November 2012).

[2] Sentinel Asia: https://sentinel.tksc.jaxa.jp/sentinel2/topControl.action/ (accessed 21

[3] Shumuta,Y., Todou, T., Takahashi, K., and Ishikawa, T. Development of a Damage Estimation Method for Electric Power Distribution Equipment, *The Journal of the Insti‐ tute of Electrical Engineers in Japan*, Volume 130-C Number 7, 2010;1253-1261 (in Japa‐

[4] Shumuta,Y. Masukawa,K. Todou,T. and Ishikawa T. Proceedings of the 11th Interna‐ tional Conference on Applications of Statistics and Probability in Civil Engineer‐

[5] Strong-motion Seismograph Networks (K-NET, KiK-net), National Research Institute for Earth Science and Disaster Prevention, http://www.k-net.bosai.go.jp/ (accessed 21

[6] Koski, T., and Noble, J. Bayesian Networks, Wiley series in probability and statistics:

[7] Gelman,A.,Caelin, J.B.,Stern, H.S.,and Rubin, D.,B. Bayesian Data Analysis: Chap‐

[8] The Niigata-Ken Chuetsu Earthquake Investigation Committee, Report of the 2004 Niigata-Ken Chuetsu-Oki earthquake damage investigation report, Japan Society of

several electric power companies including the Tohoku Electric Power Co., Inc.

[10] Miyako,T., and Todou, T,. Development of seismic damage estimation system for electric power equipment, *Electrical Review*, 2011; No.2011. 10, 76-78 (in Japanese).

*Edited by Sebastiano D'Amico*

The mitigation of earthquake-related hazards represents a key role in the modern society. The main goal of this book is to present 9 scientific papers focusing on new research and results on earthquake seismology. Chapters of this book focus on several aspect of seismology ranging from historical earthquake analysis, seismotectonics, and damage estimation of critical facilities.

Earthquake Research and Analysis - New Advances in Seismology

Earthquake Research

and Analysis

New Advances in Seismology

*Edited by Sebastiano D'Amico*

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