Statistics of Indian earthquakes-Frequency energy distribution

— Frequency-magnitude and energy-magnitude distribution of the Indian earthquakes have been studied for both shallow and intermediate earthquakes using data from 1910 to 1969. Secular energy released by the Indian earthquakes has been studied by considering cumulative energy release and it is found that these results may be used to estimate the size of largest conceivable earthquakes for shallow and intermediate focal depths.

is only 11.0 X 10 20 ergs while the average between 1906-1952 is 12.4 x 10 26 ergs, using energy magnitude relation log E = 12.0 + 1.8 M [1] where E is the energy in ergs and M is the Richter's magnitude.Intermediate shocks show that the maximum energy was released in 1910 and 19.11 owing to two large earthquakes in these years.
The ratio of average annual energy release in shallow, intermediate and deep earthquakes is 31:4:1 according to Gutenberg and Richter ( 5 ).But in the year 1920 this ratio was 265:2:1 i.e. about 98% of the entire energy released in shallow earthquakes.
Similarly in the year 1969, Chouhan and Das ( 3 ) reported that about 98% of the energy was released by shallow earthquakes.
The energy released in different magnitude ranges for shallow, intermediate and deep shocks is found to decrease with the value of magnitude.
The number of shocks decreases with depth.According to Gutenberg and Richter ( 5 ) the number of shocks decreases to a minimum at a depth of about 450 km, there being a clear rise to a minor maximum at a depth of about 600 km, beyond which the number falls off very rapidly.
Ritsema ( 8 ) while considering the distribution of intermediate and deep shocks of varying magnitudes with depth, found that the curves showing variation of earthquake number show maxima at depths of 80 to 90 kms and 220 to 280 kms with a minor maxima at a depth of about 180 kms.These general results can be confirmed by use of small agreegate numbers in the range of times and magnitudes considered statistically.
Thus, there is no general agreement on the distribution of earthquakes with depth, however, these results are expected to be different in different geographic regions of the world.

-OBSERVATIONAL DATA
For the frequency-energy distribution of shallow and intermediate earthquakes of India, the data has been extracted from Gutenberg and Richter ( 5 ), Seismological Bulletins of India Meteorological Department, I.S.S. bulletins and U.S.C.G.S. bulletins in addition to other sources; the span of time being sixty years (from 1910 to 1969).The magnitudes used here are all Richter's magnitudes 31 ^ 5.
For calculations of energy from the magnitude values M, the following relation of Gutenberg and Eichter (") have been used.

-FREQUENCY-ENERGY DISTRIBUTION
Annual frequencies of all the earthquakes of different magnitudes M in the range of 5 to 8.6 have been calculated using all the earthquakes that have occurred since 1910; for magnitudes 5 to 5.2 the time span is 13 years.Thus, the frequencies of shallow (N.) and intermediate earthquakes (Nt) have been determined and given in Table 1.Annual energy released by shallow (Es) and intermediate (Et) earthquakes of different magnitudes has been calculated using equation [2] and is also tabulated in Table 1.
Figures 1 and 2 show the frequency-energy distribution of earthquakes against magnitude, the general shape of the figures being approximately parallelogram defining the boundaries of the frequency and energy of different magnitudes in the range of 5.0 to 8.6.These figures clearly show that the frequency of earthquakes is maximum when the energy release is minimum or vice versa, a well known fact in earthquake seismology.In seismology these facts are expressed by representing logarithmic dependence of E the seismic energy and N the earthquake frequency, on magnitude M.
These relations are known as energy-magnitude (Eq.2) and frequency-magnitude relations as given by Gutenberg and Eichter ( 5 ) in the form log N = a -bM [3] where a and b are constants.
However in the present analysis the energy magnitude relation has been assumed logarithmic and then the energy values are calculated as given in the Table 1.The slope of the two sides of the approximate parallelogram (not shown in the figures 1 and 2) may take any shape depending upon the values of B and b.It has been observed by Utsu ( 10 ), Chouhan ( 4 ), Page ( 7 ) that the value of b varies from 0.4 to 1.5 and Bath ( l ) has also shown that the value of B may be taken as 1.44 also, apart from Gutenberg-Eichter's value of 1.5.Thus we can say that the slope of E changes from 1.44 to 1.50 while the slope of N changes from -0.4 to -1.5.Hence, when the values of B are 1.4 (approximately) to 1.5 and b 1.4 to 1.5 then the shape of figures 1 and 2 becomes parallelogram.For all other values of b the general shape of figures 1 and 2 is quadrilateral tending to a triangle for small values of b.These observations may be interpreted in support of energy magnitude relations of Gutenberg-Ricliter (°) and Bath where they have used the upper limit of the slope of frequency magnitude relation.
These relations can be expressed mathematically by combining the equations [2] and [3] in the form where N(E) dE is the number of earthquakes with energy between E and E + dE.
Here it may be mentioned that in U.S.S.B, the frequency-energy relation is used in the form log n(K) -G -v K [5] where 5X = (3.56t + 2.56) 1023 ergs [6] where 2-Es is cumulative energy for shallow earthquakes, and for intermediate earthquakes S-Bi = (2.23 t + 0.85) IO 22 ergs [7] where is the cumulative sum of energy for intermediate earthquakes.
In equations [6] and [7] t is counted from 1910.The distance between the two parallel lines in fig. 3 corresponds to about 2 x 10 24 ergs which is very close to the energy of the largest conceivable earthquakes.Similarly in fig. 4 the distance between the two parallel lines corresponds to 1.81 x 10 23 ergs which is very close to the largest intermediate shocks observed in the past 60 years.The slope in equations [6] and [7] gives approximately the rate of accumulation of energy in the crust and in the upper mantle upto a depth of about 300 km, respectively.Similar study carried out by Tsuboi ( 9 ) shows that the rate of accumulation of energy in Japan is 2.24 x 10 23 ergs per year and this rate of energy accumulation is 2.5 times higher than the rate of accumulation arrived at here which, after normalising in area, comes to 0.89 x 10 23 ergs per year.These interpretation of secular seismic energy release is quite in line with the observations of Bath and Duda ( 2 ).The rate of energy accumulation for shallow and intermediate earthquakes is always greater than the average energy release in these shocks and at the most they may be equal.The distribution of earthquakes does show a maximum at a depth of 200-250 kms which is in agreement with Bitsema's ( 8 ) second maximum and also agrees with Gutenberg-Richter's minimum in the depth range of 250-300 kms.2) Secular studies of energy release clearly show that the average energy release is in general less than rate of accumulation of energy.It also follows from figures 3 and 4 that the maximum size of conceivable shallow earthquake is M ~ 8.6 and of intermediate earthquake is M ~ 7.6.
3) Distribution of number of earthquakes with depth shows a maxima at a depth of 200 to 250 kms and a minima in the depth range of 250 to 300 kms.

Fig. 3 -
Fig. 3 -Cumulative sum of annual energy plotted against time for the same period of figs 1, 2. The figure is a plot of shallow earthquakes.

Fig. 4 -
Fig. 4 -See the caption of fig. 3. The plot is for intermediate earthquakes.
-magnitude and energy-magnitude distribution of shallow and intermediate earthquakes shown in figures 1 and 2 show logarithmic dependence of N and E on M and relation between N and E may be written in the form N (E) = G.E. -<w«>-1

Table 1 -
ANNUAL FREQUENCY AND ANNUAL ENERGY RELEASED BY SHALLOW AND INTERMEDIATE EARTHQUAKES OF DIFFERENT MAGNITUDES DURING THE PERIOD 1910 TO 1969 WITH M ^ 5.
Fig. 1 -Frequency-energy distribution of Indian earthquakes against magnitude M ^ 5 for a period of 60 years, from 1910 to 1969.The figure shows the distribution for shallow earthquakes.
Distribution of number of earthquakes having M > 5.3 with depth is shown below in TABLE 2.