Investigations of earthquake probabilities in the Indian Ocean seismic belts

Tectonophysics,

115 (1985) 335-343

Elsevier Science Publishers B.V.. Amsterdam

335 - Printed in The Netherlands

INVESTIGATIONS OF EARTHQUAKE OCEAN SEISMIC BELTS

PROBABILITIES

B. RAMALINGESWARA

RAO ’

RAO ’ and P. SITAPATHI

IN THE

INDIAN

’ National Geophysical Research Institute, Hyderabad (India) ’ Geologv Department, (Received

Alfateh University, Tripoli (Libya)

August 30, 1984; revised version accepted November

6, 1984)

ABSTRACT

Ramalingeswara

Rao, B. and Sitapathi Rao, P., 1985. Investigations

of earthquake

probabilities

in the

Indian Ocean seismic belts. Tecronophysics, 115: 335-343. Gumbel’s extreme-value

theory is used to estimate the probability

of occurrence

and average return

periods for earthquakes in the Indian Ocean seismic belts. The nature of seismic activity, and annual and 50 year maximum autocorrelation

magnitudes

of earthquakes

are also discussed.

lends support for the periodicity

percentage probability

The earthquake

of the most probable

earthquake

occurrence

model of

in these belts. The

of recurrence of earthquakes of magnitude 8 and above has been estimated for the

region mentioned.

INTRODUCTION

At the time that statistical theories and models were in increasing use in all the fields of geophysics, Court (1952) brought out a full review of the theory of extremes and its importance in solving some problems in geophysics. In the last two decades, the extreme-value theory has been developed and applied by Gayskiy and Katok (1965) in the Soviet Union and by Epstein and Lomnitz (1966) in the United States of America. Dick (1965) used it for the analysis of New Zealand earthquakes, Milne and Davenport (1966) and Schenkova and Kamik (1970) for European earthquakes. Makjanic (1972) worked on the same principle on Zagreb earthquakes. For the north circum-Pacific area, Shake1 and Willis (1972) have applied this theory and estimated the earthquake return periods and probabilities without considering the parent distribution of the earthquake population. The extreme-value methods, their limitations and the earthquake risk involved have been detailed in two articles by Lomnitz (1974). Kamik and Schenkova (1974), working on different regions of the Balkan area, concluded that this method gives an average characteristic of return periods in the 0040-1951/85/$03.30

0 1985 Elsevier Science Publishers B.V.

336

given region which can be used for the classification of future activity. Sitapathi Rao and Ramalingeswara Rao (1979) reported preliminary analysis and results obtained over

northeast

India,

Assam

technique and Ramalingeswara Indian sub-continent. Prakasa for earthquakes THEORETICAL

and

the Andaman-Nicobar

Islands

by using

this

Rao (1979) applied the method to the regions of the Rao et al. (1981, 1983a, 1983b) applied the method

in Chile, Iraq and Iran. MODELS

(1) The theoretical model originally proposed by Gumbe of flood data is based on the random variable function

(1941) for the analysis G(x,t). To fulfill the

function, n independent observations are required collected continuously over an appreciably long time which should be amenable for division into N independent sets each having an equal time-length. The N sets obtained include N extremes, each set contributes invariably a largest extreme. The parent population must follow, as proposed, a known statistical distribution such as the normal, exponential, chi-square or gamma distribution. Based on this, the earthquake data may be modelled into N sets and the largest magnitude earthquake may be picked up from each of the N sets. The earthquakes with the largest magnitude, yr, y2.. . yN taken from the N sets, are arranged in increasing order of magnitude. From the fundamental theorem of the theory of extremes, it is known that as both n and N grow large, the cumulative probability G(y), that any of these N extremes will be less than any chosen quantity, reaches the double exponential expression. The frequency of each yi in the ordered set of extremes and the cumulative probability may be represented by:

G(Y)= (N:

(1)

1>

G(Y)= Q(Y)= exd-exp - P(y -

~11

where

of an event

Q(r)

probability

is the non-occurrence of occurrence

(2)

P = 1 - Q = 1 - G(y).

in a single Expression

trial

and

hence

the

(2) gives the probabil-

ity of non-occurrence of an event y in a single trial. Consequently, the return of extremes can be obtained for values equal to or exceeding y from:

period

1 (3)

?=l-G(v) Further, the variation of the probability eqn. (2) which may be written as: G’(y)=pexp[-p(r-u)lG(y)

of y can be obtained

by differentiation

of

(4)

Differentiation of the same eqn. (2) a second time gives the maximum density of probability at y = u which means that u is theoretically the most frequent value (mode) of the set of extremes under consideration. Theoretically p is considered to

331

be a measure

of concentration

least squares

method

By assuming

about

from the sample

the mode, but practically

it is obtained

In (Y= pu, eqn. (2) can be written

as:

(5)

G(y) = exd--(y exd-IQ)] By taking logarithms ln[-lnG(y)]

by the

data.

twice on both sides, eqn. (5) can be written

=lna-By

as: (6)

The cumulative probability WY (k) in this case, when an extreme value equaling or exceeding y occurs before or along the kth observation, may be given as: W, (k) = [I-

G(Y)‘]

=l-exp[-akexp(-By)]

(7)

A similar model, but with the assumption that the number of earthquakes in a set (per year) has to follow a Poisson distribution, was proposed by Epstein and Lomnitz (1966) for the occurrence of large earthquakes. A case with a poor Poisson distribution suggests that the region under investigation is inhomogeneous and a best fit indicates the homogeneity of the region. Departing from the Epstein and Lomnitz condition of a Poisson distribution for the parent population, Gumbel’s model suggests that a test of the best fit of the actual extreme values, picked up from each set, may be obtained by plotting the confidence bands or control curves to the eqn. (6) on a specially prepared extremal probability paper which has a double logarithmic abscissa and a linear ordinate. The return periods on the upper scale and the probabilities on the lower scale for the given extreme values yi could be described by the confidence level yi k a( yi) within the frequency u(y,)

= 1

range of 0.15 to 0.85 of G(y),

l/k+) -11’ bfi ln G(y)

where:

(8)

Beyond the limit of 0.85, a(y) assumes a value of 1.14078//I at G(yN) and 0.75469/p at G(Y~_~). Gumbel envisages that, if about 2/3 of the observed extremes fall within the confidence bands, the theoretical distribution is considered a good fit to the true distribution. (2) A solution to the problem of earthquake prediction is sought by modelling the earthquake data, based on the time series analysis, involving the autocorrelation functions for the determination of the periodicity of occurrence of earthquakes. This approach is an extension of the analysis of the Fourier spectrum of time series. According to Box and Jenkinson (1970), the efficiency of prediction depends on the degree of autocorrelation which is specified in terms of its coefficients. The series in terms of magnitude m,, t = 1, 2.. , N with time lag k is described function Fk, fully by its mean value Z, the variance am’ and the autocorrelation

338

which are defined

respectively

as: (9)

am ‘=$g(m,-Z)

(10)

Fk = k=0,1,2...N-k

(11)

The coefficients

of autocorrelation

functions

regardless of the absolute magnitude reported the use of the autocorrelation Balkan region. DISCUSSION

will be obtained

in arbitrary

units

m,. Schenk and Schenkova (1978) have functions for earthquake prediction in the

OF THE RESULTS

For the analysis the Indian Oceanic Ridge system with its branches is considered here as a single block, shown in Fig. 1. The pioneering work of Tams (1931) and Rotht (1954) have shown that the rift valleys of the ridge systems in the Atlantic Ocean and Indian Ocean are primarily associated with seismic activity, as the distribution of earthquake epicenters coincided with the rift valleys. However, in the case of the Indian Ocean, the areas beyond the ridge are aseismic. A complete physiographic diagram of the Indian Ocean is given by Heezen and Tharp (1964, 1967). The seismicity of the region has been worked out by Stover (1966) in considerable of earthquakes

detail and Banghar in the Indian

and Sykes (1969) have studied

ocean

and adjacent

regions.

the focal mechanism

Barazangi

and Dorman

(1969) have identified the regions of seismic activity along the east coast of Malagasy. McKenzie and Sclater (1971) have dealt exhaustively with the highly complex nature of the tectonic activity and evolution of the Indian Oceanic Ridge system. According to Lomnitz (1974), the highly complex configuration of the Indian Oceanic Ridge system shows substantially higher seismicity than that of the Atlantic Ocean Ridge system. The Indian Oceanic Ridge system, selected for the present analysis, has epicenters of shallow focus earthquakes distributed on the ridge as shown in Fig. 1. The data used cover a period of 40 years from 1925 to 1964. The line of expected extremes (LEE) as obtained for this block is represented by the following equation: ln[ -In

G(y)]

= 11.80 - 1.85~

This is represented Various parameters Tables 1 and 2.

on specially

(12) prepared

and other values involved

extremal

paper

as shown

in the calculations

in Fig. 2.

are presented

in

339

For this block, a remarkably good agreement is obtained between the theory of extremes and the observed extremes as indicated by the confidence limits. From the j? and In (Y,it can be said (of the ridge system) that earthquakes with magnitudes between 5.0 and 5.5 occur once each year at one place or another throughout the length of the ridge system. The average return period of the largest earthquake of magnitude A4 2 8.5 is 48 years with 26% probability as obtained from the line of expected extremes. The periodicities observed from Fig. 3 are 3 and 26 years. These are compared with the average return periods of extreme-value methods as shown in Table 3. Thus, the expected maximum magnitudes can be obtained with this periodicity from the line of expected extremes.

Fig. 1. Epicentral distribution

of earthquakes

on the Indian Oceanic Ridge system.

340 RETURN

PERIOD

IllIlL 1001 ,010

I

,100

zigzag fashion).

0.0 extremes

600

900

t

I

I

2.0

3.0

(LEE) is shown

R = - In[ - In G(y)]

990

I.

995

9975

I

1

I

4.0

5.0

6.0

as a straight

probability

and other values involved

is shown in the lower scale.

in the calculations

Symbol

Name of the parameters

and

value 40

1

N

sample size

2

mean magnitude

3

P a

4

M InaX

max. magnitude

5

M expc

6.04

model magnitude

(annual)

expected

5.75

in 50 years

with 26%

8.5 magnitude

least squares

in 20 years 7.5 *

with 83% B

Estimated

other values

method

1.876 i 0.087

7 l

ln (Y MexpaEted is greater

than 7.5 magnitude

least squares method on Richter’s

scale.

11.08

999

1

7.0

paper (drawn

line. Confidence

1

No.

6

II

Ill1

980

REDUCED VARIATE (RI on the specially prepared extremal

extremes

variate

I IllI

950

I.0 are plotted

The line of expected

for each yi. The reduced

Parameters Sample

700

I

-1.0

Fig. 2. The observed

TABLE

600

I

R -2.0

drawn

300

Yy (in YearS)

bands

in are

341

TABLE

2

Frequency

G(y)

for different

Magnitude G( y )

Frequency

SC 1

magnitudes

to obtain

the line of expected

extremes

5.50

6.00

6.50

7.00

7.5

0.19

0.53

0.79

0.91

0.96

3 YClrs

[ ear6

0.5

9r

0.4

0.2,-

0.2IF, 0.1

,

,PO

,,I \

*_I

0.1 I-

,

22,24;26 / i, ;

20

30

32

34‘ ?

SIt”\,~ ‘,* .,’

42

44

46 46

50

52 54

i./’

0.2 I-

Time shifts

L.’

0.3 ,-

Fig. 3. The autocorrelation

function

left

to right

of the time series of the annual

Oceanic

Ridge system. The insert in the figure is the observed

TABLE

3

Comparison Oceanic

of results

from

autocorrelation

and

magnitudes maximum

the extreme-value

method

on the Indian magnitudes.

for the Indian

Ridge system

Periodicities

(in yrs)

from autocorrelation

3.0

obtained

maximum

series of annual

Average

return

Probable

size of

period (in yrs)

the earthquake

from extreme-

magnitude

value method

Richter’s

1.0

6.0

3.5

7.0

of

M on

Years of Occurrences

of

earthquakes

scale 1929,1930,1931, 1934,1935,1937 1925,1926,1928, 1933,1939,1944,1947, 1949,1953,1954,1955, 1964,1968

26.0

20.0

8.0

1948

342

The results of the average return the catalogue tained

from the line of expected

probable

maximum

occurrence

periods

which give satisfactory magnitudes

extremes.

for different

with average

From

return

is 7.5 to 7.9 with an average return

events in

periods

the above observations,

are 7.5, 8.0 and 8.2, with percentage

of 83, 53, and 46 respectively.

The authors are grateful Andhra University, Waltair

are back-checked

coincidence

The expected

period of 20-26

ob-

the most

probabilities

maximum

magnitude

of

range

years.

to Prof. V. Bhaskara Rao, Department for his suggestions and encouragement

of Geophysics, in carrying out

this work: They also thank the Director of the N.G.R.I., Hyderabad for permission to publish this paper. Mr. V. Sri Ramulu assisted in typing the manuscript. REFERENCES Banghar,

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