Öncel Akademi: İstatistiksel Sismoloji

Bulletin of the Seismological Society of America, Vol. 76, No. 3, pp. 889-895, June 1986
ON ESTIMATING VARYING b VALUES
By PETER GUTTORP AND DEBORAH HOPKINS
Estimation of the b value in the frequency-magnitude relation has been the
subject of several papers and research efforts since Gutenberg and Richter first put
forth the equation in 1954. In this note, we review estimation procedures for the b
value and discuss in detail the case where the b value may be varying slowly in time.
We apply the proposed method to 20 yrs of central California data. There does not
appear to be any substantial variation in the b value for these data.
BACKGROUND
Gutenberg and Richter (1954) observed a linear relation between the annual
frequencies of earthquakes and surface wave magnitude, with a slope depending on
the depth and region of the event. The relation
logloN(m) = a - bm (1)
where a and b are constants, and N(m) is the number of earthquakes of magnitude
exceeding m, is frequently used by researchers to describe earthquake activity. It
has also been applied to microearthquake studies (e.g., Majer and McEvilly, 1979)
and to acoustic emissions in metals (Pollock, 1980). The value of the slope in (1),
the b value, has emerged as an important variable in seismic risk analysis. It has
also been related to the state of stress in rocks in laboratory experiments (Mogi,
1962; Scholz, 1968).
Gutenberg and Richter used the method of least squares to estimate the param-
eters in their frequency-magnitude relation. Weichert (1980) points out that the
basic assumption of independent, identically (i.i.d.) distributed random variables
usually made in least-squares analysis is violated for the cumulative event counts
[N(m) in (1)]. A discussion of generalized least-squares estimation of b values,
based on equation (1), can be found in Guttorp (1984).
Utsu (1965) proposed a different estimate of the b value, which was shown by Aki
(1965) to be a maximum likelihood estimate (m.l.e.) for a particular exponential
model that can be derived from (1). If we regard the magnitudes as observations of
i.i.d, random variables, then the Gutenberg-Richter relation states that the magni-
tudes (when suitably shifted to the lower limit of complete detectability) are
exponentially distributed. The properties of the m.l.e, of b have been studied by
several authors (e.g., Shi and Bolt, 1982).
In an important paper, Bender (1983) points out that the use of a continuous
model for observations that really are rounded leads to biased estimates. Her main
analytical technique was computer simulation. Shi and Bolt (1982) discussed the
estimation of uncertainties in the determination of b values. In their paper, they
regarded the b value as a random variable. It is customary in statistics, unless the
distribution of a parameter is known a priori, to regard the parameter as a fixed but
unknown number. That is the approach we will take in this paper. Thus, we will
study the variability in estimates of the b value, both in the case where the b value
stays constant (see "The Discretized Exponential Model"), and when it is assumed
to vary over time (see "Varying b Values"). We apply the proposed method for
varying b values to a set of California earthquakes, consisting of 2049 events with
889

890 LETTERS TO THE EDITOR
magnitudes between 2.5 and 5.8, listed in the U.C. Bulletin of the Seismographic
Station from a 40,000 km2 region of the central coast ranges in California during
the two decade interval from 1 January 1962 to 31 December 1981. This data set is
described in more detail in McKenzie et al. (1982).
THE DISCRETIZED EXPONENTIAL MODEL
In current practice, magnitudes are only computed to one decimal place. In
historical data, the roundoff is usually even more severe. In this section, we will
study the statistical effect of this discretization.
The simple exponential model mentioned in the first section assumes that the
magnitudes have density function
[M(X) = fl exp(-/~(x - Mo)), x _ Mo, ~ > 0, (2)
where B = b log 10. More realistically, we may assume that all observations are of
the form
Mi -- Mo* -k kiA
where ki are integers, and A is the accuracy of the data (usually 0.1 in modern
magnitude measurements). Mo* is the lower bound on observed magnitudes in
rounded measurement, so Mo* = Mo + ½A. Furthermore, we assume that the Mi are
rounded observations from the exponential distribution (2), i.e.,
f( (k+l/2)A
Pr{M = Mo* + kA} = fie -mx+A/2) dx
(k--1/2)A
= e-kaY(1 _ e-a~). (3)
In other words, ki has a geometric distribution with parameter 0 = 1 - e-~. The
frequency-magnitude relation holds true also in the discretized model
lOgloN(m) ~ k logloO + loglo(1 - O) = a - bm
where a = Mo*b + loglo(1 - ea~) and k = (m - Mo*)/A.
The m.l.e, of 0 is
t~= 1
l+k-
where E = Y~ kiln, whence the maximum likelihood estimate of/~ is
1 1 (/J=-~log(1-O)=~log 1+.~-- .
A Taylor expansion shows that the difference between the exponential maximum
likelihood estimate and the estimate based on the discretized model is of the order
of A2.
The exact distribution of fl is, in principle, derived in Bender (1983), but is not

LETTERS TO THE EDITOR 891
very illuminating. However, asymptotic properties are straightforward. We see that
/ I (et_- 1)!~*
- AsNfl, A2 nea~ ] .
An approximate confidence band for fl therefore is given by
( z~ z, )- + + + (4)
and a confidence band for b is obtained by dividing the bounds in (4) by log 10.
The effect of using the exponential model m.l.e, flexp= 1/{114- Mo*) to estimate
when the data really are discrete is not very severe when A = 0.1. For example,
suppose we use a confidence band derived from the exponential model with dis-
crete data. The coverage probability of this band is given in Table 1 for the cases
n = 400, ~ = 2, A = 0.1, or 0.5 and a = 0.90, 0.95, or 0.99. The details of the
computation can be found in the Appendix.
We see that the coverage probability is correct for A = 0.1, but there are serious
problems for A = 0.5. It appears important to use the discretized model for historical
data, whereas the continuous model can be used for modern data.
TABLE 1
EFFECT OF DISCRETIZATION ON CONFIDENCE LEVEL
A NominalLevel ActualLevel
0.1 0.900 0.902
0.950 0.951
0.990 0.990
0.5 0.900 0.563
0.950 0.706
0.990 0.903
VARYING b VALUES
It may not alwasy be reasonable to assume that the b value stays constant in
time, although the change may be quite gradual. In this section, we will modify the
arguments to deal with the case of a slowly varying/~ =/~(t). The method obtained
is only approximate, since we do not make any assumptions about the specific
functional form of fl(t).
Assume now that we observe earthquakes at times tl, t2, ..., tN, where Nt =
#{tj E (0, t]}. Also write N(A) for the number of events in the set A, so that
N(0, t] = Nt. Let K be a small number, chosen so that /~ can be considered
approximately constant over periods of length 2K. Define
/~{t) = h(~It - Mo*)
where
1
Mj
N(t - K, t + K) ~e(t-~,t+K)
* X, ~ AsN(~n, a, 2) means that (Xn - ~,)/~, has a standard normal asymptotic distribution.

and h(x) = 1/A log(1 + A/x). The estimator is undefined at t if N(t - K,
t+K) =0.
Assume now for simplicity that Nt is a Poisson process of rate p, and that the
magnitudes are independent of Nt. This is not an unreasonable assumption for the
California data. A bivariate Taylor expansion of h(x/y) shows, when/3 and/3' both
are bounded functions, that
E~(t)=~(t)+ O(~p)+ O(K)
and
VarY(t, =( U2 / +o
Estimates of #(t) more than 2K time units apart are independent. Asymptotic
properties of this estimate can be obtained by letting K --~ 0 and p -~ oo in such a
way that Kp ---, oo. It is seen to be asymptotically unbiased, consistent, and
asymptotically normally distributed. Figure 1 depicts running b values for the
Q
0
>
I
J]
(D
c~
IgB2 1 9 6 4 I966 1 9 6 8 1 9 7 0 I972 Ig7A 1 9 7 6 1 9 7 8 1 9 8 0 1982
TimQ
FIG. 1. Running b values with K = 107.
California data with K = 107. This value was chosen to coincide with an average
30-event interval length to facilitate comparison with Figure 2 of Shi and Bolt. This
amount of smoothing has reduced some of the high peaks in their estimates. The
bars are one standard error on either side of the estimate. These are approximate
two-thirds confidence intervals for individual estimates.
An approximate variance stabilizing transformation is given by
log tanh --~, (5)
i.e., if this function of the estimate ~(t) is computed, the variance is approximately
constantly one. This transformation makes it easier to see how much the b values
are varying, since each point has the same standard error on the transformed scale.
To first order, this variance stabilizing transformation is the same as the logarithmic
transformation suggested by Shi and Bolt. In Figure 2, the variance-stabilized
estimates are shown. There is little evidence of variation in b values, since the
estimates on the transformed scale all fall within two standard errors (the standard
error being the unit of the transformed scale).

The difference between this approach and the approach in Shi and Bolt is that
our averaging is over a fixed size time window, whereas they use a window with a
fixed number of points in it. Thus, their averaging is over time periods of differing
length, and they must assume that ~ (t) is constant over all these different intervals.
In other words, they assume that ~ stays constant for a longer time period whenever
] I i I i i i I
1962 I964 1986 1968 1970 1972 1974 1978 1978 I980 1982
TImQ
FIG. 2. Running b values on the transformed scale.
~f
~d
.3
O
,.w
QO
d
tn
d I I * I I I
O. 1 0.2 0.3 0.4 0.5 0.8 0.7
Intensity
FIG. 3. Intensity estimates, based on 90 events, plotted against the corresponding 107-day b values.
the intensity (number of events per unit time) of earthquake events is low, and that
it varies more rapidly when this intensity is high. Unless a physical model for this
relation is suggested, the Shi and Bolt approach seems less plausible than our
assumption. This is illustrated in Figure 3, where we show estimated intensity
against estimated b value. The intensity is derived from contiguous intervals of 90
events, whereas the b values are based on 107 days of data. There is no obvious
relation between the two variables.
In some cases, particularly when looking at highly clustered events such as
microearthquakes, it may be important to take into account the statistical properties
of the process of quakes, rather than doing the analysis conditional on the number

of quakes. The Poisson process assumption used here is not crucial to our compu-
tations. In fact, if Nt is a stationary point process with rate p and the kth product
moment pk(ul, • • • uh-1) [cf. Brillinger (1975) for an explanation of point process
terminology], it is straight forward to verify that the mean and the variance are the
same as in the Poisson case (to the approximation considered), but now estimates
at different times are dependent, with
Cov(~(t), ~(t + u))
_(.1-~a~(t))(1-e-a~(~+u) ( ~1
The choice of K determines the smoothness of the estimated b value function
~(t). If K is small, the curve is jagged, and the variance at each point is large. A
larger value of K yields a smoother function and smaller variance. One strives for a
balance between stability of estimates and the appropriate degree of smoothness. A
technique called cross-validation may be used to select a value of K based on the
data (Efron, 1982, chapter 7).
ACKNOWLEDGMENTS
We are grateful to B. A. Bolt and R. A. Urhammer for access to the data used in this paper.
This research had partial support of the National Science Foundation under Grants MCS-8205991 and
MCS-8302573.
REFERENCES
Aki, K. (1965). Maximum likelihood estimate of b in the formula log N = a - bM and is confidence
limits, Bull. Earthquake Res. Inst., Tokyo Univ. 43, 237-239.
Bender, B. (1983). Maximum likelihood estimation of b-values for magnitude grouped data, Bull. Seism.
Sac. Am. 73,831-851.
Bickel, P. J. and K. A. Doksum (1977).Mathematical Statistics: Basic Ideas and Selected Topics, Holden-
Day, San Francisco, California.
Brillinger, D. R. (1975). Statistical inference for stationary point processes, in Stochastic Processes and
Related Topics, M. L. Purl, Editor, AcademicPress, New York, 55-59.
Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans, CBMS-NSF Reg.Conf.Ser.
Appl. Math. 38, SIAM,Philadelphia, Pennsylvania.
Gutenberg, B. and C. F. Richter (1954). Seismicity of the Earth and Associated Phenomena, 2nd ed.,
Princeton University Press, Princeton, New Jersey.
Guttorp, P. (1984). On least squares estimation of b-values {submitted for publication).
Majer, E. J. and T. V. McEvilly (1979). Seismologicalinvestigations at The Geysers geothermal field,
Geophysics 44, 246-269.
McKenzie, M. R., R. D. Miller, and R. A. Urhammer (1982). Local earthquakes in Northern California,
Bull. Seism. Stations 52, 1-92.
Mogi,K. (1962).Study ofelastic shocks caused bythe fracture ofheterogeneous materials and its relation
to earthquake phenomena, Bull Earthquake Res. Inst., Tokyo Univ. 40, 125-173.
Pollock, A. A. (1980}. Physical interpretation of ae/ma signal processing, Proc. 2nd Conf. on Acoustic
Emission/Microseismic Activity in Geological Structures and Materials, Pennsylvania State Univer-
sity, November 13-15, 1978,Trans Tech Publ., Clausthal, Germany.
Scholz, C. H. (1968).Microfractures, aftershocks, and seismicity, Bull. Seism. Soc. Am. 58, 1117-1130.
Shi, Y. and B. A. Bolt (1982). The standard error of the magnitude-frequency b-value,Bull. Seism. Soc.
Am. 72, 1677-1687.
Utsu, T. (1965). A method for determining the value of b in a formula log N = a - bM showing the
magnitude-frequency relation for earthquakes, Geophys. Bull. Hokkaido Univ. 13, 99-103 (in
Japanese with English summary).

Weichert, D. H. (1980). Estimation of the earthquake recurrence parameters for unequal observation
periods for different magnitudes, Bull. Seism. Soc. Am. 70, 1337-1346.
DEPARTMENTOF STATISTICS,GN-22
UNIVERSITYOF WASHINGTON
SEATTLE, WASHINGTON98195 (P.G.)
LAWRENCEBERKELEYLABORATORY
1 CYCLOTRONROAD
UNIVERSITYOF CALIFORNIA
BERKELEY,CALIFORNIA94720(D.H.)
Manuscript received 8 July 1985
APPENDIX
In this Appendix, we derive the asymptotic properties of the exponential model
estimate ~exp= 1/(h~ - Mo), when the true underlying model is the discretized
model (2). Using standard theory for asymptotic distributions [e.g., Bickel and
Doksum (1977), sections A14 and A15], we see that
)1 +e ~' n (1 +e~) ' "
As an illustration, suppose we use the exponential model asymptotic confidence
band [cf. equation (14) in Shi and Bolt (1982)]
i ~n + zl-~/2)log,oe' (~n + z~/2)logloeJ
where z~ is the ~-quantile of the normal distribution, with rounded data, when the^
sample size is large. Using the asymptotic distribution of ~exp, we see that the
coverage probability of this band is about
I( 1 + --~n--n] ---~---1 +e~/
A ~n (1 + e~) 2 /
B(1 _ _ __
~n/ A l+ea~ I
e"~-=_1 e'W2---- ~- - _ -~
T~ ~ ~e~-~ /
where • is the normal cumulative distribution function.

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