UP

  Probability Of Earthquake Occurrence

Earthquakes and earthquake effects are controlled by many unknown factors, not always allowing applications of deterministic methods of analysis. Some of these can be treated statistically. The probability of an area being struck by an earthquake is often referred to in engineering applications. The concept is illustrated below[1].

Suppose an earthquake occurs in an area A. Consider a small area a in A.  Let the area a be, further, made of very small areas Da in it, so that SDa = a. Let us call a specified location hit by an earthquake if the peak value of the ground acceleration there exceeds a certain specified value.  Let the probability of none of the areas Da being hit by the earthquake be written as p (0,a).

The probability of another smaller area Da outside not being hit by the earthquake may be written as: 

p (0, a+Da) = p(0,a).p(0, Da)

                  = p(0,a).(1- Da/A)

or  (1/ Da ).[p(0,a+Da)-p(0,a)] +(1/A).p(0,a)=0

Letting Da -> 0 this equation becomes

dp/da + (1/A).p =0

The solution of this equation is given by:   p(0,a) = e ^–a/A

The probability that the area a will be hit by the earthquake is, then,  given by

p(1,a) = 1 -  e ^–a/A

 This is the expression for probability in space. A similar expression for probability can be derived, by analogy, in the time domain. The probability of the earthquake hitting during a time interval of D years (where D years may be the life of the engineering structures in the area) may be written as:

p(1,D)=1-e^-D/T

Equating the two expressions of probability one gets the return period of the event[2], having a probability of occurence, p(1,a), during an interval of  D years :

T = -D/[Log_e {1 – p (1,a)}]

In statistical analysis it is often assumed that the number of earthquakes in year is a Poisson variable and the earthquake magnitude, x, is a random variable with cumulative density function F (x) = Pr (X £ x) =1 –e ^-bx; x ³0. In this model the probability, R_D of an earthquake of magnitude M in D years is given by [3]:

R_D = 1 –Exp. (-aDe^-bM)

Here a and b are constants related to the seismicity parameters specified by the earthquake magnitude frequency relationship, namely:Log ₁₀ N (M) = a-bM

Where N (M) is the number of earthquakes occurring annually in the region and having magnitudes equal to or greater than M, and a = Log ₁₀ N (0) and b the constant b determines the distribution of magnitudes in the earthquake population.  (see Frequencies of Earthquake occurrence ^4.3 ).

a = Exp. [a Log _e10] and b = b Log _e 10                   

Results of a typical calculation are shown in the Figure below.

Figure showing Probabilities of Earthquakes of Different Magnitude in an area of Moderate Seismicity (a=4.0, b=1.0 for a 300 kilometers radius) over a period of 100 years.