**5. Modeling of PGV and travel-time data**

## **5.1 PGV - distance decay**

We use PGV recorded by MSS in our network area and hypocentre coordinates from the ZAMG bulletin. We model PGV caused by earthquake "e" and observed at station "i" by a power law and station amplification factors. The power law considers geometric spreading and damping of the maximum amplitude seismic waves, the station amplification factors the local geological and technical conditions.

$$\text{PGV}\_{\text{e,i}} = \text{AO}\_{\text{e}} \ast r\_{\text{e,i}}^{\text{n}} \ast \text{SA}\_{\text{i}} \tag{1}$$

**5.2 P-wave travel time**

*MSS\_M = 3 and r = 10 km.*

**Figure 7.**

of Eq. 5.

**45**

worthwhile to analyze and evaluate these data.

T0e … focal time of earthquake "e" Vp … constant P-wave velocity

time of the earthquake e by equation Eq. 7:

**5.3 P-wave to S-wave arrival time difference**

SDpi ... P-wave station delay at station "i"

The acquisition, processing and analysis of ground motion amplitudes is the main task of the MSS network. However, the seismic traces show clear first P-wave arrivals and frequently very distinct S-wave phases (see **Figure 3a**). Therefore, it is

*(a) PGV and PGV/SA reduced to MSS\_M = 3 over log10(r); (b) histograms of PGV and PGV/SA reduced to*

*Seismological Data Acquisition and Analysis within the Scope of Citizen Science*

*DOI: http://dx.doi.org/10.5772/intechopen.95273*

We model the P-wave arrival times Tpi using the following linear relation:

To calculate the station delays SDp, we eliminate T0 by considering differences

Eq. 6 has the same form as the logarithm of Eq. 2. Therefore we proceed analogue to the calculation of the station amplification factors SA and solve the equation system by least squares and the condition SUM(SDpi) = 0. We apply SDp to Tp and find the minimum standard deviation of (Tpobserved – Tpcalculated) with Vp = 5700 m/s. With Vp and the station delays SDp given we calculate the focal

T0e ¼ MEAN Tpe,i � re,i*=*Vp � SDpi

tions at specific stations, we expect an error of about Vp\*σ1 � 0.9 km.

**Figure 8a** and **b** show Tp and (Tp -SDp) reduced to T0 = 0 versus log10(r) and the histograms of Tp and (Tp - SDp) reduced to T0 = 0 and r = 10 km. The STDs derived from the histogram data are σ0 = 0.235 s for Tp and σ1 = 0.161 s for (Tp - SDp). In case we want to calculate the hypocentral distances "r" from Tp observa-

Now, we focus on the P-wave to S-wave arrival time difference Tps. This value is insensitive to the time drift of the MSS, which could occasionally occur if the

SDpi � SDpj <sup>¼</sup> MEAN Tpe,i � Tpe,j– re,i–re,j *<sup>=</sup>*Vp (6)

Tpe,i ¼ T0e þ re,i*=*Vp þ SDpi (5)

(7)

A0e... source strength of earthquake "e".

re,i.... hypocentral distance of earthquake "e" to station "i". n .....constant.

SAi … ..amplification factor of station "i".

We calculate the station amplification factors SA according to [10, 11]. The unknown source strength is eliminated by the ratio of SA at stations "i" and "j". We calculate the mean ratio derived from the different earthquakes:

$$\rm{SA}\_i/\rm{SA}\_j = \rm{MEAN} \left( \rm{PGV\_{e,i} / PGV\_{e,j} \* \left(r\_{e,j} / r\_{e,i}\right)^n} \right) \tag{2}$$

The logarithm of Eq. 2 forms a linear equation system for log(SA), which is solved using least squares and the additional condition the geometrical mean of all SA is unity. We vary the exponent "n" to minimize the standard deviation of (PGVobserved – PGVcalculated). So far, the optimum exponent is n = �2.2 based on the data available.

Next, we consider a single earthquake and omit the index "e". Given the exponent "n", the hypocentral distance "ri" and the station amplification factor SAi, the logarithm of the source strength related to station i can be estimated by Eq. 3. The average of log10(A0i) over all stations defines the magnitude MSS\_M of the earthquake (Eq. 4).

$$\log 10(\text{A0}\_i) = \log 10(\text{PGV}\_i) - \log 10(\text{SA}\_i) - \text{n} \ast \log 10(\text{r}\_i) \tag{3}$$

$$\text{MSS\\_M} = \text{MEAN}(\log \mathbf{10}(\text{A0}\_i))\tag{4}$$

The units are PGV [nm/s] and r [deg] according to the definition of ML (local magnitude).

**Figure 7** shows a) log10(PGV) and log10(PGV/SA) reduced to MSS\_M = 3 versus log10(r), and b) the histograms of log10(PGV) and log10(PGV/SA) reduced to MSS\_M = 3 and r = 10 km. The standard deviation (STD) derived from the histogram data are σ0 = 0.332 and σ1 = 0.221, respectively. The STD σ1 quantifies the accuracy of log10 (PGV/SA) calculated by Eq. 1, given A0, the exponent "n", and the hypocentral distance "r". We calculate log10(r) from Eq. 3, given "A0", "n", and PGV/SA, the STD σ1/n = 0.1 quantifies the uncertainty. The accuracy of the hypocentral distance "r" calculation is therefore about 25% or +/� 5 km at r � 20 km.

*Seismological Data Acquisition and Analysis within the Scope of Citizen Science DOI: http://dx.doi.org/10.5772/intechopen.95273*

**Figure 7.**

Service for the shake maps (http:\\shakemapa.ethz.ch; visited on 23th October 2020) mimics ours. PGV = 0.8 mm/s, 3 mm/s and 9 mm/s corresponds to intensities

We use PGV recorded by MSS in our network area and hypocentre coordinates from the ZAMG bulletin. We model PGV caused by earthquake "e" and observed at station "i" by a power law and station amplification factors. The power law considers geometric spreading and damping of the maximum amplitude seismic waves, the station amplification factors the local geological and technical conditions.

n

<sup>e</sup>*;*<sup>i</sup> ∗ SAi (1)

<sup>n</sup> (2)

PGVe*;*<sup>i</sup> ¼ A0e ∗ *r*

We calculate the station amplification factors SA according to [10, 11]. The unknown source strength is eliminated by the ratio of SA at stations "i" and "j". We

SAi*=*SAj ¼ MEAN PGVe,i*=*PGVe,j ∗ re,j*=*re,i

The logarithm of Eq. 2 forms a linear equation system for log(SA), which is solved using least squares and the additional condition the geometrical mean of all SA is unity. We vary the exponent "n" to minimize the standard deviation of (PGVobserved – PGVcalculated). So far, the optimum exponent is n = �2.2 based on the

Next, we consider a single earthquake and omit the index "e". Given the exponent "n", the hypocentral distance "ri" and the station amplification factor SAi, the logarithm of the source strength related to station i can be estimated by Eq. 3. The average of log10(A0i) over all stations defines the magnitude MSS\_M of the

The units are PGV [nm/s] and r [deg] according to the definition of ML (local

**Figure 7** shows a) log10(PGV) and log10(PGV/SA) reduced to MSS\_M = 3 versus log10(r), and b) the histograms of log10(PGV) and log10(PGV/SA) reduced to MSS\_M = 3 and r = 10 km. The standard deviation (STD) derived from the histogram data are σ0 = 0.332 and σ1 = 0.221, respectively. The STD σ1 quantifies the accuracy of log10 (PGV/SA) calculated by Eq. 1, given A0, the exponent "n", and the hypocentral distance "r". We calculate log10(r) from Eq. 3, given "A0", "n", and PGV/SA, the STD σ1/n = 0.1 quantifies the uncertainty. The accuracy of the hypocentral distance "r" calculation is therefore about 25% or +/� 5 km at

log 10 A0 ð Þ¼<sup>i</sup> log 10 PGV ð Þ<sup>i</sup> – log 10 SA ð Þ�<sup>i</sup> n ∗ log 10 rð Þ<sup>i</sup> (3)

MSS\_M ¼ MEAN log 10 A0 ð Þ ð Þ<sup>i</sup> (4)

re,i.... hypocentral distance of earthquake "e" to station "i".

calculate the mean ratio derived from the different earthquakes:

II – III, IV, and V. PGV < 0.2 mm/s are classified as 'not felt'.

**5. Modeling of PGV and travel-time data**

*Earthquakes - From Tectonics to Buildings*

A0e... source strength of earthquake "e".

SAi … ..amplification factor of station "i".

**5.1 PGV - distance decay**

n .....constant.

data available.

earthquake (Eq. 4).

magnitude).

r � 20 km.

**44**

*(a) PGV and PGV/SA reduced to MSS\_M = 3 over log10(r); (b) histograms of PGV and PGV/SA reduced to MSS\_M = 3 and r = 10 km.*

### **5.2 P-wave travel time**

The acquisition, processing and analysis of ground motion amplitudes is the main task of the MSS network. However, the seismic traces show clear first P-wave arrivals and frequently very distinct S-wave phases (see **Figure 3a**). Therefore, it is worthwhile to analyze and evaluate these data.

We model the P-wave arrival times Tpi using the following linear relation:

$$\mathbf{T}\mathbf{p}\_{\mathbf{e},i} = \mathbf{T}\mathbf{0}\_{\mathbf{e}} + \mathbf{r}\_{\mathbf{e},i}/\mathbf{V}\mathbf{p} + \mathbf{S}\mathbf{D}\mathbf{p}\_{i} \tag{5}$$

T0e … focal time of earthquake "e"

Vp … constant P-wave velocity

SDpi ... P-wave station delay at station "i"

To calculate the station delays SDp, we eliminate T0 by considering differences of Eq. 5.

$$\rm{SDp}\_i - \rm{SDp}\_j = \rm{MEAN} \left( \rm{Tp}\_{e,i} - \rm{Tp}\_{e,j} - (\rm{r}\_{e,i} - \rm{r}\_{e,j}) / \rm{Vp} \right) \tag{6}$$

Eq. 6 has the same form as the logarithm of Eq. 2. Therefore we proceed analogue to the calculation of the station amplification factors SA and solve the equation system by least squares and the condition SUM(SDpi) = 0. We apply SDp to Tp and find the minimum standard deviation of (Tpobserved – Tpcalculated) with Vp = 5700 m/s. With Vp and the station delays SDp given we calculate the focal time of the earthquake e by equation Eq. 7:

$$\mathbf{T}\mathbf{O}\_{\mathbf{e}} = \text{MEAN}\left(\mathbf{T}\mathbf{p}\_{\mathbf{e},i} - \mathbf{r}\_{\mathbf{e},i}/\mathbf{V}\mathbf{p} - \mathbf{SDp}\_{i}\right) \tag{7}$$

**Figure 8a** and **b** show Tp and (Tp -SDp) reduced to T0 = 0 versus log10(r) and the histograms of Tp and (Tp - SDp) reduced to T0 = 0 and r = 10 km. The STDs derived from the histogram data are σ0 = 0.235 s for Tp and σ1 = 0.161 s for (Tp - SDp). In case we want to calculate the hypocentral distances "r" from Tp observations at specific stations, we expect an error of about Vp\*σ1 � 0.9 km.

#### **5.3 P-wave to S-wave arrival time difference**

Now, we focus on the P-wave to S-wave arrival time difference Tps. This value is insensitive to the time drift of the MSS, which could occasionally occur if the

corrections (SA, SDp, SDsp). We search for the hypocenter within a 3D grid, centered at the maximum PGV or minimum travel time station. In our special case the grid extends from �20 km to +20 km in W-E and S-N with a grid spacing of 0.5 km around this center. We consider 17 hypocenter depth levels from �16 km to 0 km. The indices of the grid in the east, north and upward

*Seismological Data Acquisition and Analysis within the Scope of Citizen Science*

These methods share the definition of a proper cost function followed by the search of its minimum within the 3D volume of a grid with the indexes (k, l, m) in X (east), Y (north), and Z (upward) directions. The computational complexity comprises four nested loops, three over the grid dimensions (k, l, m) and one fourth over the number of stations that recorded particular data. We present three cost function methods based on each of the three data sets (PGV, Tp, Tps) described before. We take the liberty to name the various methods after scientists who

Hiroo Kanamori [12] introduced earthquake locating based on amplitudes with application to real-time seismology. He identified the optimum epi- or hypocenter with the location of the minimum standard deviation of the magnitudes calculated from the amplitudes recorded at the different seismic stations. He implemented an empirical 1D model for the amplitude (acceleration) decay with distance. Thus, we take PGV/SA and the amplitude – distance power law according to Eq. 1 and choose

cost function ¼ STD log 10 A0 ð Þ ð Þ<sup>i</sup> *:* (10)

the following cost function according to Eq. 3 and equivalent to Kanamori's

In 1848, William Hopkins [14] identified two types of seismic waves traveling with different velocities. He pointed out the relation between the corresponding arrival time difference and the distance from the point of observation to the origin of seismic waves. At that time the great advantage of using time differences derived from single recordings was the insensitivity to clock reading errors. The definition of the time difference Tps Eq. 8 and the station delays SDps Eq. 9 implies that (Tps - SDpsi) is zero at the origin. Hence, we define the following

Ludwig Geiger [13] was the first to present a method to locate an earthquake through minimization of the differences between observed and calculated arrival times. The corresponding cost function for P-wave arrival times according our

cost function <sup>¼</sup> STD T0 ð Þ¼<sup>i</sup> STD Tpi � SDpi � ri*=*Vp (11)

cost function <sup>¼</sup> MEAN ABS Tpsi � SDpsi � ri*=*Vps (12)

directions are (k, l, m).

*6.1.1 Kanamori*

principle:

*6.1.2 Geiger*

notation Eq. 5 is:

*6.1.3 Hopkins*

cost function:

**47**

**6.1 Cost function methods**

*DOI: http://dx.doi.org/10.5772/intechopen.95273*

defined the principles of the relevant cost functions.

**Figure 8.**

*(a) Tp and Tp - SD reduced to T0 = 0 over log10(r); (b) histograms of T and Tp - SD reduced to T0 = 0 and r = 10 km.*

**Figure 9.** *(a) Tsp and Tsp - SDsp over hypocenter distance); (b) histograms of Tsp and Tsp - SDsp.*

connection to NTP servers is interrupted. We assume that P- and S-waves are generated simultaneously and the focal time difference Tps(r = 0) is zero. We model Tps as follows:

$$\mathbf{Tps}\_{\mathbf{e},\mathbf{i}} = \mathbf{r}\_{\mathbf{e},\mathbf{i}} / \mathbf{Vps} + \mathbf{SDps}\_{\mathbf{i}} \tag{8}$$

Vps = (1/Vs – 1/Vp)-1 … ..difference velocity

SDpsi .....P- wave to S-wave station delay at station "i"

The assumption Tps(r = 0) makes the calculation of the station delays SDps very simple:

$$\text{SDps}\_{i} = \text{MEAN}\left(\text{Tps}\_{\text{e,i}} - \text{r}\_{\text{e,i}} / \text{Vps}\right) \tag{9}$$

As before with Tp, we apply the station delays SDps to Tps and find the minimum standard deviation of (Tpsobserved - Tpscalculated) with Vps = 7300 m/s. **Figure 9** shows a) Tps and (Tps -SDps) versus log10(r) and b) the histograms of Tps and (Tps - SDps) reduced to r = 10 km. The STDs derived from the histogram data are σ0 = 0.212 s for Tps and σ1 = 0.183 s for (Tps – SDps). Consequently the error of hypocenter distances r derived from Tps observations is about Vps\*σ1 � 1.3 km.
