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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.

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:


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

T = -D/[Loge {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, RD of an earthquake of magnitude M in D years is given by :

RD = 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                   



1.Housner, G.W.(1975). Strong Ground Motion, Chapter 4 in Earthquake Engineering, R.L. Weigel (Ed.), Perentic Hall, N.J.

2.Lomnitz, C.(1976). Global Tectonics And Earthquake Risk, Developments in Geosciences, Elsevier.

3. Epstein,B. and Lomnitz,C. (1966). A model for the occurrence of large earthquakes, Nature 211,954-955.


Felt Area Of An Earthquake

An earthquake is detected at a location through the movement of the ground, and the detection capability depends on the maximum amplitude of the ground motion and the sensitivity of the detecting agency (a living being, a structure or an instrument). When the amplitude exceeds this level of sensitivity (which may be called detection threshold) the earthquake may be considered as “felt”  at that point, and all such points where the ground motion exceeded the detection threshold lie in the felt area of the earthquake. This perception of the felt area can be enlarged to include a variable detection threshold, so that for any specified level of ground motion amplitude (for acceleration, velocity or displacement) the ‘felt area’ can be defined for an earthquake. A ‘felt area’ table can then be constructed in which variation of the ‘felt area’ is shown as a function of the earthquake magnitude for a fixed value of ground motion amplitude, and for a fixed magnitude as a function of different levels of ground motion. One such ‘felt area’ table is illustrated below.



PGA Value   Radius of  Felt area of the earthquake kms (PGA exceeded)

         Magnitude of the earthquake


        4.0    4.5    5.0    5.5    6.0    6.5    7.0    7.5    8.0    8.5


0.01    50.7   77.1  114.2  168.9  250.0  365.7  533.9  781.1 1140.3 1668.3

0.03     7.8   23.8   40.6   63.4   95.0  140.6  208.5  304.4  445.4  650.2

0.05     0.0    2.0   21.4   38.1   59.3   90.2  133.5  195.3  289.0  421.9

0.07     0.0    0.0    9.4   25.4   42.8   66.7  100.1  146.5  216.7  316.4

0.09     0.0    0.0    0.0   16.7   32.5   52.1   79.1  118.7  173.6  253.4

0.11     0.0    0.0    0.0    8.8   25.0   42.2   66.0   99.0  146.5  213.8

0.13     0.0    0.0    0.0    0.0   19.3   35.2   56.3   84.5  125.0  185.4

0.15     0.0    0.0    0.0    0.0   14.1   30.1   48.8   75.1  111.3  164.8

0.17     0.0    0.0    0.0    0.0    9.9   25.4   42.8   66.7  100.1  146.5

0.19     0.0    0.0    0.0    0.0    2.0   21.4   38.1   59.3   90.2  133.5

0.21     0.0    0.0    0.0    0.0    0.0   18.1   33.8   54.2   82.4  121.9

0.23     0.0    0.0    0.0    0.0    0.0   14.8   30.5   50.1   76.1  112.6

0.25     0.0    0.0    0.0    0.0    0.0   11.7   27.5   45.7   70.3  105.5

Such ‘felt area ‘tables can be generated for any chosen empirical relationships described above for any set of PGA values, and come very handy in probability calculations.  

Earthquake Hazard Zonation

 Zonation is a frequently used term in the context of earthquake hazard. This is used for  demarcation of geographically contiguous area on the Earth’s surface for which certain specified values of a selected set of geological and/or physical and/or socio-economic parameters applies. Zonation. to identify the exposure to earthquake hazard, so that a specified hazard mitigation strategy can be adopted within each zone, is the primary objective of the earthquake hazard zonation. Depending on the detailing of the information (geological, geotechnical and seismological data on the one hand and information on cultural and economic development on the other) the nomenclature (macro, micro or site specific) will vary. On global scale the plate boundaries provide a macrozonation of the Earth’s surface. Zonation based on detailed information of a region leads to microzonation. Concerns of a particular site location are covered under more site specific investigations.

 Microzonation is considered an important tool in  mitigating   the disastrous effects o earthquakes, cyclones, floods, landslides and avalanches. An international conference on this subject was held  in 1972. Since then microzonation has been a subject of several regional and internationa conferences  either arising from or giving rise to many research programs. Microzonation has found an important place during the International Decade of Natural Disaster Reduction (IDNDR) :1990-2000. This is considered a thrust area of research by seismologists and geologists.  Microzonation was  defined far back in 1980 as follows:

“ Microzonation is a process for identifying relevant geological, seismological, hydrological and geotechnical site characteristics in a specific region and incorporating them into land use planning and the design of safe structures in order to reduce damage to human life and property resulting from earthquakes.”

Detailing the information contained in this definition requires that the geological and seismotectonic regimes are understood sufficiently well to forecast possible future earthquake occurrences, and to quantify their effects in the area under microzonation. For a meaningful assessment of the earthquake hazard at a location site-specific information is required on the factors, which affect the ground motion parameters (amplitudes of ground acceleration, velocity and displacement), e.g. information on locations, magnitudes and source mechanisms of earthquakes (which could affect the site in future), local geology and topography and those determining the potential of ground failure due to surface faulting, liquefaction, subsidence, collapse and landslides. A zonation map is expected to provide up-to-date information for assessment and mitigation of the impending earthquake hazard at a specified location. By now it is obvious that:

1.                         Zonation is a multidisciplinary exercise covering geology, geophysics, seismology, geotechnical engineering and landuse planning.

2.                         Zonaton is knowledge based (both information and state of the art) making it a dynamic process requiring revision and upgradation as the knowledge base advances.

3.                         The interpretations and deductions, which enter the zoning maps and are derived therefrom, need to be unambiguous and traceable in the information base.

4.                         Reliability of the information base has to be high.


 Constucting reliable zonation maps may be possible only when sufficient information to identify the locations of future earthquakes has been collected, geological mapping of the area on sufficiently large scale to deliver the required geological properties of the are has been carried out and surface and subsurface investigations on soil cover and the underlying rock beds has been completed. (Though it has been well recognized that earthquakes occur along geological faults, and it is believed that faults can be discovered through planned field investigations, faults are often discovered  only after the occurrence of a major earthquake, making one to believe that the area was not properly investigated at earlier occassions. Existence of a single major fault capable of producing a major earthquake would, thus, render any zonation exercise redundant)

Sherif (1980). Definition of Microzonation, Newsletter, Earthquake Research Institute, Vol. 14, No. 4 p-68


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