EI and Magnitude

1. Bulletin of the Seismological Society of America, Vol. 72, No. 3, pp. 981-986, June 1982
ON THE GENERALIZED EXPONENTIAL DISTRIBUTION OF
EARTHQUAKE INTENSITY AND MAGNITUDE
BY BERISLAV MAKJANId
ABSTRACT
In continuation of a previous paper, a procedure to obtain the maximum
likelihood values of the parameters for a generalized exponential distribution is
described. The formula for the limits of confidence is also given. As an example,
the maximum likelihood values of parameters for the initial and extremal distri-
butions of Zagreb earthquakes and the limits of confidence for the largest
possible intensity are presented.
INTRODUCTION
In a previous paper (Makjanid, 1980), it was proposed to generalize the Gutenberg-
Richter (1944) formula of frequency distribution of earthquake intensities and
magnitudes by the distribution named their generalized exponential F(x')
F(x') = 1 - e -y' (y' _-_0) (1)
with
1 [ k'(x'-xo') 1
Y'= k'ln 1 a' . (2)
Here x' stands for intensity I or magnitude M; a', k', and Xo' are parameters of the
distribution, and y' is a reduced variate. The primes are here to distinguish this
distribution from the extremal one.
The Gutenberg-Richter formula in its form which is being used in seismology
reads
log N -- a - bM. (3)
If one writes the formula (3) in the form which is analogous to (1), one obtains for
the relation between the reduced variate y' and magnitude M
where
y' = A + BM (4)
A = In ~ N- aln 10
B = b In 10.
Here is ~N average number of all earthquakes in 1 yr.
This shows that the Gutenberg-Richter formula is represented by the expression
for the exponential distribution only if the relationship between magnitude (or
intensity) and the reduced variate y' is linear. The relationship between intensity
and reduced variate in (2) is more adaptable since it allows departure from linearity.
981

2. 982 BERISLAV MAKJANIC
This departure depends on the parameter k'. If one writes the relation (2) as
1 - e -k'y'
X' -~ Xo' + O~~ k' (5)
it is easily seen that in the limit for k' ---*0, the relation (2) becomes linear
x' = xo' + a'y'. (6)
In this sense, distribution represented by the formulas (1) and (2) can be called
generalized exponential distribution. Of course this distribution is equivalent to the
distribution called limited by Gumbel (1958, p. 157). The alternative name for this
distribution could be the generalized Gutenberg-Richter distribution. If we retain
(1) as the form of distribution and (2) as the relationship between intensity (or
magnitude) and the reduced variate, we enjoy advantage of simplicity of the relation
(1) and have to concentrate our attention on the relation y' -- y'(x').
One is lead to distribution (1) + (2) by the fact that a linear relationship between
intensity (and magnitude) and reduced variate overestimates the frequency of strong
earthquakes. The generalized distribution (1) + (2) makes it possible to reduce
frequency of strong earthquakes leaving at the same time almost unaffected fre-
quency of weak and moderate ones.
The fundamental assumption which leads to this generalization is that there are
no seismic events which would release an infinitelylarge amount of energy, i.e., that
there must be an upper limit to the magnitude and intensity of earthquakes. If one
adopts this view, the second step is to require that distribution of all earthquakes
[called generally an initial distribution by Gumbel (1958, p. 157)] be compatible with
distribution of maximum earthquakes, where one takes the greatest event from
everyone of several equal time intervals [called here, extremal distribution to shorten
the lengthy expression "the asymptotic probability of largest values" used by
Gumbel (1958, p. 157)].
The extremal distributions for all three Gumbel's cases (the three "asymptotes")
can very conveniently be written in Jenkinson's form (Jenkinson, 1955)
with
¢(x) = exp(--e -y) (7)
y=_kln[ 1 k(x-xO)]a" (8)
The maximum values and their return periods found for Zagreb and Dubrovnik
(Makjani6, 1972, 1978) by means of this formula show good agreement with the data.
Although much more testing of the above formula should be done, we can, for the
time being, postulate that the Jenkinson's formula (7) + (8) with k > 0 gives
distribution of maximum events, i.e., that this is extremal distribution of earthquake
intensity and magnitude.
Gumbel (1958) has given the method to determine asymptotic pobability of the
largest value or extremal distribution from the initial one by means of the so-called
characteristic largest value Un, which for distribution (1), is equal Un = Inn. Gumbel's

3. PARAMETERS FOR A GENERALIZED EXPONENTIAL DISTRIBUTION
procedure applied to (1) gives extremal probability (I)(x) as
O(x)= exp[-e -(y'-u:)]
and this is possible to write in the form (7) + (8) if
983
(9)
k = k p
v¢ = o~'n -k"
01'
Xo = Xo' + k-7 (1 - n-k'). (i0)
The parameters x0 and x0' have different meanings. For initial distribution, Xo' is
its lower limit; the interval of its variate x' is from x0' to Xo'+ a'/k'. Extremal
distribution has only an upper limit and x = x0 is such that • (x0) = e -1. However,
probability of negative values of x is very small.
The maximum value of x which can be attained in this distribution x.... is given
by
!
X.... = Xo' + ~ = Xo + ~. (11)
ESTIMATION OF THE PARAMETERS OF INITIAL AND EXTREMAL DISTRIBUTIONS
The parameters of initial distribution, a', k', and x0' and those of extremal
distribution, a, k, and xo, can be determined only from the sample, i.e., from the
record of intensities or magnitudes for sufficiently long periods. The first estimates
of these parameters can be conveniently obtained by means of the least-squares
method (Makjanid, 1980). It is necessary however to obtain better estimates of these
parameters and they can be calculated by means of the maximum likelihood method.
This method for the extremal parameters was described by Jenkinson (1969). This
is an iterative procedure which must be used because the unknown parameters come
under the summation sign in the maximum likelihood equation. For the same
reason, iterative procedure has to be used also with the initial distribution.
In order to find maximum likelihood estimates for the parameters of an initial
distribution (1) + (2), we must suppose the value of one of them as known. For the
sample of intensities, x0' is known and equal one. This is also what one obtains with
the above-mentioned least-squares method. The likelihood function for the sample
of N observations from distribution (1) + (2) is
N
-L = Nln [a' I + (1 - k') ~ yi'. (12)
i=l
According to the maximum likelihood principle, the most probable values of a'
and k' are those which minimize the function -L. The maximum likelihood values
which we denote by ~' and/~' are solutions of the equations
O(-L) O(-L)
- - - 0 and - - - 0 . (13)
Oa' Ok'

4. 984 BERISLAV MAKJANIC
We put
~t ---- /~1t + O~t(1)
/~' = kl' + k '(1) (14)
where al' and kl' are the first estimates (obtained, e.g., by means of the least-squares
method) and a '(1) and k '(1) are differences between the known first estimates and
unknown maximum likelihood values. The values of derivatives (13) for a' = a~' and
k' = kl' are known and it is possible to develop these derivatives in power series
about a~' and k~'. In this development, only linear terms are retained. We write for
brevity
L (~', re) = £
and since 0/~/0a' = 0 and of_,/Ok' = 0, we obtain the system of two linear algebraic
equations for the first differences a '(1) and k '(1)
02(--/-.~) 02(--£) OL(al', k,')
a,O) }- k m) _ _ _--
Oa'2 Oa'Ok' Oa'
at(1 ) 02(--/~) 02(--/~) OL(al', kl')
Oa'Ok---'--'--";-+ k'(" Ok'-----~- Ok' (15)
Since the elements of the system matrix are not known, they have to be replaced
with their expected values which are
[ a2(-L ) 1 N 1
E[ ~ J = a '21-2k'
E[O2(-L)I N 1
L ~ J = - a' (1 - k')(1 - 2k')
E[a2(-L) ] 1
[. ~ = N (1 - k')(1 - 2k')" (16)
Inserting (16) into system (15) and denoting the right-hand side terms of (15) by S
and T, where
N
S= IN-(1-k') ~ e k'~i]
i~1
T = - k--7 (N- yi + S), (17)
i=1
we can write the solution of (15) as
OL!
a '(1) = ~ (1 - k')(2S + T)
1
k '(1) = ~ (1 - k')[S + (1 - k')T]. (18)

5. PARAMETERS FOR A GENERALIZED EXPONENTIAL DISTRIBUTION 985
This way, we have obtained the second approximation
0~2p ~- 0~1p ~- 0Lp(1)
k~' -- kl' + k 'm. (19)
We take the second approximation as starting values and obtain the third, the
fourth, etc., approximations. We stop when difference between two successive
approximations falls under the desired limit.
THE CONFIDENCE LIMITS
We are looking for the variance s~, of x' for given values of F. The details of
similar calculation for extremal distribution can be found in Jenkinson (1969). If we
put for brevity
X p ~ Xo~ + oz/W p
1 - e -k'~'
W / -
k ~
we can write the expression for variance as
s 2 =2 (1-k') 1+ -- + (1-k') (20)
x, ok' ok'
APPLICATIONS TO THE DATA OF ZAGREB EARTHQUAKES
The basic frequency tables for Zagreb earthquakes were given in Makjanid (1980,
Table 1). There, no effort was made to give maximum likelihood estimates of
parameters but instead a preliminary procedure was proposed. Now we are able to
abandon that ad hoc procedure and give the maximum likelihood values for
parameters of both initial and extremal distributions.
Starting from the first estimates calculated by the least-squares method, we
obtain after eight iterations for the initial distribution,
k' = 0.23820
a' = 1.9387
x0' = 1
and after 13 iterations for the extremal one
k = 0.313816
= 1.8985
Xo = 2.8798.
All these computations were done on a programmable desk calculator.
These values of parameters give the following upper limit for the intensity of
earthquakes in Zagreb
from the initial distribution
from the extremal distribution
x~,ax = Imax= 9.1 ~ 9 MCS
x.... = /max = 8.9 -~ 9 MCS.

6. 986 BERISLAV MAKJANI(~
Intensity in grades of Mercalli-Cancani-Sieberg scale has to be given in integer
numbers. However, when we calculate these numbers we also obtain decimals. With
these arithmetically more correct values, we find that the difference between these
two results (9.1383 and 8.9295) amounts to 2.3 per cent of the mean value of the
upper limit 9.
The standard deviation of initial intensities has been calculated by the formula
(20) and that of extremal intensities after Jenkinson (1969). The upper limit Imaxhas
the largest standard deviation. We obtain for the initial distribution S1'(Imax = 9.1)
= 0.5, and for the extremal distribution S~(Imax = 8.9) -- 0.7.
CONCLUSIONS
Procedures described in the previous paper (Makjani~, 1980) and in this one make
it possible to calculate maximum likelihood estimates of the parameters of both
'initial and extremal distributions of earthquakes.
The above reported agreement in the upper limit for this particular case can be
considered encouraging and should stimulate further research.
The fact that, in the case presented here, the upper limit comes out with very
small uncertainty and that, at the same time, the values of parameters k' and k are
surely different, also shows that further research of the relationship between an
initial and its extremal distributions is necessary.
REFERENCES
Gumbel, E. J. (1958).Statistics of Extremes, ColumbiaUniversityPress, New York,2nd printing, 1960.
Gutenberg, B. and C. F. Richter (1944).Frequency of earthquakes in California, Bull. Seism. Soc. Am.
34, 185-188.
Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) values of
meteorologicalelements, Q. J. R. Met. Soc. 87, 158-171.
Jenkinson, A. F. (1969).Statistics of extremes in estimation of maximumfloods,Technical Note no. 98,
WMO,WMO-No.233,T. P. 126,Geneva,183-227.
Makjanid, B. (1972).A contribution to the statistical analysisof Zagreb earthquakes in the period 1869-
1968,Pure Appl. Geophys. 95, 80-88.
Makjani~, B. (1978).On maximum annual earthquake at Dubrovnik in Sixth European Conferenceon
Earthquake Engineering,September 18-22,1978,Dubrovnik,Yugoslavia, 1 Strong Ground Motion,
YugoslavAssociationfor Earthquake Engineering,Ljubljana, 25-31.
Makjani~, B. (1980).On the frequency distribution of earthquake magnitude and intensity, Bull. Seism.
Soc. Am. 70, 2253-2260.
GEOPHYSICAL INSTITUTE
FACULTY OF SCIENCES AND MATHEMATICS
UNIVERSITY OF ZAGREB
GRI~3
41001 ZAGREB,YUGOSLAVIA
Manuscript received20 May 1981