# Description Of An Area Source Essay

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2.3.1. Characterize Area source
For instance, an area source produces random earthquakes and the probability is equal anywhere within 100 km of the site.
The exampled area source is shown in Figure 2.3.

Figure 2. 3: Map of an area source.
Since the probability of occurrence of location of epicenter is same everywhere, the result of probabilistic results of the distances from earthquake sources to site can be calculated for this example.

F_R (r)=P(R≤r)={█(0 if rx├|m,r┤)=1-Φ((ln⁡x-(ln⁡PGA ) ̅)/σ_ln⁡PGA ) (2.10)

In equation 2.10, it is assumed that the distribution of natural logarithm is normal. Φ((ln⁡x-(ln⁡PGA ) ̅)/σ_ln⁡PGA ) is the standard normal cumulative distribution function (CDF).
The prediction model for the log mean of peak ground acceleration which proposed by Cornell et al.(1979) is shown below

(ln⁡PGA ) ̅=-0.152+0.859M-1.803 ln⁡〖(R+25)〗 (2.11)

In this model, the standard deviation of lnIM is 0.57. This value is constant for all possible earthquake magnitudes and source-to-site distances.
Probability density function for peak ground acceleration is more practical in engineering sometimes. So the cumulative distribution function can be rewritten as below

P(PGA>x│m,r)=∫_x^∞▒〖f_PGA (u)du〗 (2.12)

Where f_PGA (u) is the probability density function for PGA (given magnitude m and distance r).
Equation 2.12 can be also written analytically, shown below

P(PGA>x│m,r)=∫_x^∞▒〖1/(σ_lnPGA √2π) exp⁡(-1/2 〖((lnu-(lnPGA) ̅)/σ_lnPGA )〗^2)du〗 (2.13)

For instance, given an earthquake with magnitude of 6.5, the probability of PGA>1g can be calculated. Equation 2.11 gives predicted mean of -0.5765 for distance of 3 km, -0.9788 for dis...

... middle of paper ...

...3 shows more results of probability for PGA>1g (the same method above).

Table 2. 3: Some results related with λ(PGA>1g)

m_j P(M=m_j) P(PGA>1├|m_j,10┤) P(PGA>1├|m_j,10┤)P(M=m_j)
5.00 0.4381 0 0
5.25 0.2464 0.0002 0.00005
5.50 0.1385 0.0006 0.0001
5.75 0.0779 0.0022 0.0002
6.00 0.0438 0.0067 0.0003
6.25 0.0246 0.0181 0.0004
6.50 0.0139 0.0430 0.0006
6.75 0.0078 0.0901 0.0007
7.00 0.0044 0.1676 0.0007
7.25 0.0025 0.2786 0.0007
7.50 0.0014 0.4168 0.0006
7.75 0.0008 0.5662 0.0004
Sum=0.0048
According to the given condition, λ(M_i>m_min )=0.02. The summation of
P(PGA>1├|m_j,10┤)P(M=m_j) is 0.0048, so the final probability of PGA exceeding 1g is λ(PGA>1g)= 0.02(0.0048) = 〖9.6∙10〗^(-5).
Then we can plot ground motion hazard curve by computing on all PGA levels.

Figure 2. 11: PGA hazard curve for the example case (single source with varying magnitudes).

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