**3. The employment of seismic hazard analysis for the assessment of seismic risk**

If the uncertainties in the seismic hazard assessment of a specific site could be avoided, a deterministic approach could provide an easy and rational method for this purpose. However, the nature of a seismic event is such that it usually involves various uncertainty sources, such as the location of the source, the faulting mechanism and the magnitude of the event, etc. The probabilistic seismic hazard analysis offers a useful tool for the assessment of annual norms of seismic loss and risk. [27]

#### **3.1. Probabilistic seismic hazard analysis**

In an active area source, k, with a similar seismicity all across it, the seismicity data gives the maximum magnitude of muk and a minimum of mlk and the frequency of the occurrence of

vk. Similar assumptions can be extended for a line source from which the Probability Density Function (PDF) of magnitude for a site, fMk(mk), can be constructed, as is schematically demonstrated by Figure 9.a [27].

Seismic Risk of Structures and Earthquake Economic Issues 15

*a*

PA(*a|m,x*)

relations may be modelled by a probability density function f(am, x) which shows the distribution density function of intensity *a* if a seismic event with a magnitude of *m* occurs at a distance x from the site. Figure 11 shows how f(am, x) changes when an intensity

**Figure 11.** Probability of exceedance from a specific intensity using a probability density function

Intensity (PGA, …)

According to the above-mentioned collected data, the annual rate of earthquakes with an intensity (acceleration) larger than *a*, *v(a)* can be calculated from the following equation:

> , d <sup>d</sup> *uk k k lk*

*<sup>a</sup> k A kkM kX k k k <sup>m</sup> <sup>k</sup> <sup>x</sup>*

Where, PA(amk, xu) stands for the probability of occurrence of an earthquake with an

Poison process is usually employed to model the rate of the occurrence of earthquakes within specific duration. For an earthquake with an annual probability of occurrence of (a), the probability of the occurrence of *n* earthquakes of intensity greater than *a* within *t* years is

> *n vat vat*

Meanwhile, the annual probability of exceedance from the intensity *a*, *P*(*a*) can be expressed

The time interval of earthquakes with an intensity exceeding *a* is called the return period and is shown as *Ta*. The parameter can be calculated first knowing that the probability of *T* is

*n*

*P am x f m f x m x* (12)

(13)

*Pa P a va* 1 0,1, 1 exp (14)

*k*

exp ( ,,) !

*Pnta*

 

fA(*a|m,x*)

PDF for variations of

attenuation function

given by:

as:

longer than t:

*m*

intensity larger than *a* at a site with an attenuation relation of fA(*am,x*).

measure *a* varies.

**Figure 9.** Variability of seismic intensity as a function of magnitude and distance

if in the active zone under study, an area or line source can be assumed as a point, the probability density function of the focal distance of the site, x, fXk (xk) can be developed, as schematically demonstrated in Figure 9.b.

#### *3.1.1. Ground motion prediction models*

Ground motion prediction models - or attenuation functions - include the gradual degradation of seismic energy passing through a medium of ground up to site. The ground motion prediction models, schematically shown in Figure 10, have been provided according to the statistical data, characteristics of the ground, seismic intensity and distance, etc.

**Figure 10.** a) Schematic ground motion prediction models for a site

The ground motion prediction models are usually empirical relations, which do not match the real data exactly. The dispersion between the real data and the empirical attenuation relations may be modelled by a probability density function f(am, x) which shows the distribution density function of intensity *a* if a seismic event with a magnitude of *m* occurs at a distance x from the site. Figure 11 shows how f(am, x) changes when an intensity measure *a* varies.

14 Earthquake Engineering

fmk(*mk*)

*0*

schematically demonstrated by Figure 9.a [27].

schematically demonstrated in Figure 9.b.

*mlk muk*

*3.1.1. Ground motion prediction models* 

vk. Similar assumptions can be extended for a line source from which the Probability Density Function (PDF) of magnitude for a site, fMk(mk), can be constructed, as is

fxk(*xk*)

*0*

if in the active zone under study, an area or line source can be assumed as a point, the probability density function of the focal distance of the site, x, fXk (xk) can be developed, as

(a) (b)

*mk*

Ground motion prediction models - or attenuation functions - include the gradual degradation of seismic energy passing through a medium of ground up to site. The ground motion prediction models, schematically shown in Figure 10, have been provided according to the statistical data, characteristics of the ground, seismic intensity and distance, etc.

*M=7*

*M=6*

*M=5*

The ground motion prediction models are usually empirical relations, which do not match the real data exactly. The dispersion between the real data and the empirical attenuation

Distance from epicenter

*xk*

*x*

**Figure 9.** Variability of seismic intensity as a function of magnitude and distance

**Figure 10.** a) Schematic ground motion prediction models for a site

*a*

Seismic Intensity

in site

**Figure 11.** Probability of exceedance from a specific intensity using a probability density function

According to the above-mentioned collected data, the annual rate of earthquakes with an intensity (acceleration) larger than *a*, *v(a)* can be calculated from the following equation:

$$\boldsymbol{\nu}\_{\left(a\right)} = \sum\_{k} \boldsymbol{\nu}\_{k} \int\_{\boldsymbol{m}\_{\boldsymbol{k}}} \int\_{\boldsymbol{m}\_{\boldsymbol{k}}}^{\boldsymbol{m}\_{\boldsymbol{m}}} \boldsymbol{P}\_{\boldsymbol{A}} \left(\boldsymbol{a} \middle| \boldsymbol{m}\_{\boldsymbol{k}'} \boldsymbol{x}\_{k}\right) \boldsymbol{f}\_{\boldsymbol{M}\_{\boldsymbol{k}}} \left(\boldsymbol{m}\_{\boldsymbol{k}}\right) \boldsymbol{f}\_{\boldsymbol{X}\_{k}} \left(\boldsymbol{x}\_{k}\right) \mathrm{d}m\_{\boldsymbol{k}} \mathrm{d}\boldsymbol{x}\_{k} \tag{12}$$

Where, PA(amk, xu) stands for the probability of occurrence of an earthquake with an intensity larger than *a* at a site with an attenuation relation of fA(*am,x*).

Poison process is usually employed to model the rate of the occurrence of earthquakes within specific duration. For an earthquake with an annual probability of occurrence of (a), the probability of the occurrence of *n* earthquakes of intensity greater than *a* within *t* years is given by:

$$P(n,t,a) = \frac{\left(v\left(a\right)t\right)^{n}\exp\left(-v\left(a\right)t\right)}{n!} \tag{13}$$

Meanwhile, the annual probability of exceedance from the intensity *a*, *P*(*a*) can be expressed as:

$$P(a) = 1 - P(0, 1, a) = 1 - \exp\left(-v\left(a\right)\right) \tag{14}$$

The time interval of earthquakes with an intensity exceeding *a* is called the return period and is shown as *Ta*. The parameter can be calculated first knowing that the probability of *T* is longer than t:

$$\mathbf{P}\left(T \ge t\right) = \mathbf{P}\left(0, t, a\right) = \exp\left(-\upsilon\left(a\right)t\right) \tag{15}$$

Seismic Risk of Structures and Earthquake Economic Issues 17

0 0.5 1 1.5 2

**Seismic Intensity, (PGA(g))**

∞

� ��

A probabilistic hazard analysis for a site has resulted in the following plots of a probability

 (a) (b)

0 0.2 0.4 0.6 0.8 1 1.2

**Commulative Probability** 

**distribution function, P(a)**

**Figure 13.** Seismic hazard data, a) PDF of intensity, b) cumulative probability of occurrence

0 0.2 0.4 0.6 0.8 1 1.2

��, and c) annual probability of exceedance, where the seismic hazard curve = � ����� <sup>+</sup>

(c)

0 0.5 1 1.5 2

**Intensity (PGA(g))**

By applying the data available from seismic hazard and loss curves, an annual seismic risk

A seismic loss curve is a useful tool for comparing the seismic capacity of different facilities. Seismic hazard and loss curves with basic information about the site and facility play a key role in the evaluation of seismic risk assessment and management procedures. The "annual seismic risk density" and "seismic risk" curves constitute two important measures which can be derived from the above data. The steps to obtain annual seismic risk density curves are shown in Figure 14. The probability density function for seismic intensity (e.g., PGA) is found using a seismic hazard curve using equations 18-20. Accordingly, the annual seismic risk density is derived by multiplying this result with the corresponding loss values, as

density function and accumulative distribution.

0 0.5 1 1.5 2

**Seismic Intensity, (PGA(g))**

**Annual Probability of** 

**Exceedance**

� ����� � −∞

0 0.05 0.1 0.15 0.2

**PDF f(a)** 

**3.2. Annual seismic loss and risk** 

shown in Figure 14.d below [27].

density and seismic risk curve can be estimated.

then the probability distribution function of *Ta* becomes:

$$\mathbf{F}\_{\rm T}\left(t\right) = 1 - \mathbf{P}\left(T\_a \ge t\right) = 1 - \exp\left(-v\left(a\right)t\right) \tag{16}$$

Accordingly, the probability density function of T, fT, is derived by taking a derivation of the above FT function:

$$f\_T(t) = v(a) \exp\left(-v(a)t\right) \tag{17}$$

The return period is known as the mean value of *T* and can be calculated as:

$$T\_a = E\left(t\right) = \int t f\_{\big|\_{T\_a}}\left(t\right) dt = 1/\upsilon\left(a\right) \tag{18}$$

The probability density function, fA(*a*), the accumulative probability, *FA(a)*, and the annual probability of exceedance function, *P(a)*, for intensity *a* (for example PGA), are related to each other, as shown below:

$$F\_A\left(a\right) = \int\_{-\infty}^{a} f\_A\left(a\right) da\tag{19}$$

$$P\left(a\right) = \int\_{a}^{x} f\_A\left(a\right) da\tag{20}$$

$$P\left(a\right) = 1 - F\_A\left(a\right) \tag{21}$$

A hazard curve, as shown below, refers to a curve which relates the annual probability of exceedance of an intensity *a*, *P*(*a*), to the intensity value *a*. Two seismic hazard curves were employed in Figure 12 to schematically demonstrate two sites with relatively low and high seismic hazard.

**Figure 12.** Seismic hazard curve. A demonstration of relatively low and high seismic hazard by means of seismic hazard curves

A probabilistic hazard analysis for a site has resulted in the following plots of a probability density function and accumulative distribution.

**Figure 13.** Seismic hazard data, a) PDF of intensity, b) cumulative probability of occurrence � ����� � −∞ ��, and c) annual probability of exceedance, where the seismic hazard curve = � ����� <sup>+</sup>∞ � ��

#### **3.2. Annual seismic loss and risk**

16 Earthquake Engineering

above FT function:

each other, as shown below:

seismic hazard.

P(*a*)

Annul probability

of exceedance

of seismic hazard curves

then the probability distribution function of *Ta* becomes:

*f t va vat*

The return period is known as the mean value of *T* and can be calculated as:

P P 0, , exp *T t ta vat* (15)

F 1 P 1 exp <sup>T</sup> *t T t vat <sup>a</sup>* (16)

*<sup>T</sup>* exp (17)

*A A F a f a da* (19)

(20)

*Pa F a* 1 *<sup>A</sup>* (21)

*a*

High seismic hazard

*<sup>a</sup> <sup>a</sup> <sup>T</sup> T E t tf t dt v a* (18)

Accordingly, the probability density function of T, fT, is derived by taking a derivation of the

1 /

The probability density function, fA(*a*), the accumulative probability, *FA(a)*, and the annual probability of exceedance function, *P(a)*, for intensity *a* (for example PGA), are related to

*<sup>a</sup>*

*<sup>A</sup> <sup>a</sup> P a f a da*

A hazard curve, as shown below, refers to a curve which relates the annual probability of exceedance of an intensity *a*, *P*(*a*), to the intensity value *a*. Two seismic hazard curves were employed in Figure 12 to schematically demonstrate two sites with relatively low and high

**Figure 12.** Seismic hazard curve. A demonstration of relatively low and high seismic hazard by means

Intensity (PGA, …)

Low seismic hazard

By applying the data available from seismic hazard and loss curves, an annual seismic risk density and seismic risk curve can be estimated.

A seismic loss curve is a useful tool for comparing the seismic capacity of different facilities. Seismic hazard and loss curves with basic information about the site and facility play a key role in the evaluation of seismic risk assessment and management procedures. The "annual seismic risk density" and "seismic risk" curves constitute two important measures which can be derived from the above data. The steps to obtain annual seismic risk density curves are shown in Figure 14. The probability density function for seismic intensity (e.g., PGA) is found using a seismic hazard curve using equations 18-20. Accordingly, the annual seismic risk density is derived by multiplying this result with the corresponding loss values, as shown in Figure 14.d below [27].

Seismic Risk of Structures and Earthquake Economic Issues 19

Perhaps the most widely used modelling framework is the Input-Output model. The method has been extensively discussed in the literature (for example, in [28-30]). The method is a linear model, which includes purchase and sales between sectors of an economy based on technical relations of production. The method specially focuses on the production interdependencies among the elements and, therefore, is applicable for efficiently exploring how damage in a party or sector may affect the output of the others. HAZUS has employed

Computable General Equilibrium (CGE) offers a multi-market simulation model based on the simultaneous optimization of behaviour of individual consumers and firms in response to price signals, subject to economic account balances and resource constraints. The nonlinear approach retains many of the advantages of the linear I-O methods and

As the third alternative, econometric models are statistically estimated as simultaneous equation representations of the aggregate workings of an economy. A huge data collection is

As another approach, Social Accounting Matrices (SAMs) have been utilized to examine the higher-order effects across different socio-economic agents, activities and factors. Cole, in [34-36], studied the subject using one of the variants of SAM. The SAM approach, like I-O models, has rigid coefficients and tends to provide upper bounds for estimates. On the other hand, the framework can derive the distributional impacts of a disaster in order to evaluate equity considerations for public policies against disasters. A summary of the advantages

The economic consequences of earthquakes due to the intensity of the event and the characteristics of the affected structures may be influential on a large-scale economy. As an example, the loss flowing from the March 2011 earthquake and tsunami in east Japan could amount to as much as \$235 billion and the effects of the disaster will be felt in economies across East Asia [3]. To study how the damage to an economic sector of society may ripple into other sectors, regional economic models are employed. Several spatial economic models have been applied to study the impacts of disasters. Okuyama and Chang, in [30], summarized the experiences about the applications of the three main models - namely Input-Output, Social Accounting and Compatible General Equilibrium - to handle the impact of disaster on socio-economic systems, and comprehensively portrayed both their merits and drawbacks. However, they are based on a number of assumptions that are

Studies have been recommended to address issues such as double-counting, the response of households and the evaluation of financial situations. According to the National Research Council, 'the core of the problem with the statistically based regional models is that the historical relationship, embodied in these models, is likely to be disrupted in a natural disaster. In short, regional economic models have been developed over time primarily to

required for the model and the computation process is usually costly [33].

and disadvantages of the models mentioned has been presented in Table 3 [37].

**4. Regional economic models** 

the model in its indirect loss estimation module [31].

overcomes most of its disadvantages [32].

questionable in, for example, seismic catastrophes.

**Figure 14.** Generating the annual seismic risk density from seismic hazard and loss curves, a) seismic hazard curve, b) probability density function, c) seismic loss curve and d) annual seismic risk density.

**Figure 15.** Seismic risk curve

The seismic risk curve, as shown in Figure 15, is calculated using seismic hazard probability and loss values corresponding to similar intensities.

The seismic risk and annual risk density contain helpful information for risk management efforts. As an example, insurance premiums are calculated using this data for various seismic loss limits which can be decided by the client and insurance company.
