ELSEVIER Tectonophysics 258 (1996) 71-83

TECTONOPHYSICS

Estimation of the Gutenberg-Richter relation allowing for individual earthquake magnitude uncertainties

D.A. Rhoades

New Zealand Institute.fi~r Industrial Research and Del'elopment, Lower Hutt, New Zealand

Received 2 November 1994: accepted I November 1995

Abstract

For the estimation of seismicity parameters in the Gutenberg-Richter relation, a method is presented which takes account of individual earthquake magnitude uncertainties and rounding intervals. The magnitude of each earthquake is regarded as a continuous distribution. The method corrects bias due to magnitude uncertainties in estimates of the a-value and due to variation in magnitude uncertainties in estimates of the b-value. It extends a method presented by Tinti and Mulargia fbr constant magnitude variance. The method is demonstrated on simulated data and is used to examine activity rates and b-values estimated from the New Zealand earthquake catalogue for the period 1987-1992.

1. Introduct ion

Uncertainties in data are increasingly being formally considered in estimation of seismic hazard, especially where such uncertainties may be sources of bias (Tinti and Mulargia, 1985a,b; Kijko, 1988; Rhoades et al., 1994). Uncertainty in estimating earthquake magnitudes causes bias in standard estimates of the activity parameter a in the Gutenberg-Rich te r (GR) f requency-magni tude law log N = a - b M (Gutenberg and Richter, 1944). This is because, with a negative exponential distribution of magnitudes and symmetric observation errors, more earthquakes with true magnitude less than a given magnitude m are expected to have observed magnitudes greater than m than vice versa. The greater the uncertainty of observed magnitudes is, the greater the bias is.

This effect was analysed by Tinti and Mulargia (1985a), who calculated the probability density for the observed magnitude given normal estimation errors with a common standard deviation or. They gave the formula a = a~r - cr 2/9 ~-logt0(e)/2 for correcting the bias in estimates of the a-value, where a¢; r is the biased estimate obtained from the observed magnitudes and /3 = blogc(10). As they pointed out, observation errors do not cause any bias in the estimation of the b-value of the f requency-magni tude law, if the same ~r applies to all magnitudes.

In practice a seismograph network cannot achieve uniform quality for all magnitudes and locations covered, and so o- will vary across the earthquakes in a catalogue. Some catalogues now give an individual standard error for each earthquake. Also the seismograph network is itself subject to changes over time, so that often there are eras and spatial regions of the catalogue corresponding to different networks of seismographs where a

0040 1951/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 4 0 1 9 5 1 ( 9 5 ) 0 0 1 8 2 - 4

72 D.A. Rhoade.s / Tectonophysics 258 (1996) 71 83

different o-, even if never formally estimated, would apply to each era and region. Also different rounding procedures may have been adopted during different eras. If, as one might expect, the precision of magnitude estimates has been generally improving over time because of higher-quality instrumentation, more accurate measurement of amplitudes and periods, and denser seismograph networks, then a false impression could be created that seismicity has been decreasing.

When observation errors differ between earthquakes the a-value correction of Tinti and Mulargia is not adequate. It is also necessary to consider the possibility that the standard estimates of b-value may be biased.

Kijko (1988) considered the b-value estimation problem allowing for individual magnitude uncertainties expressed as intervals, such as those that arise from rounded data. The formulation here allows both for individual normally distributed magnitude observation errors and for rounding of each magnitude. The magnitude is treated as a continuous distribution. The estimation of both the a-value and b-value is considered and possible issues are analysed.

It is usually accepted (e.g., Freedman, 1967; Ringdal, 1975) that individual station magnitude estimates are approximately normally distributed. This would imply that estimates from a seismograph network, determined as the mean of individual station estimates as for New Zealand local magnitudes, are also approximately normally distributed. In any case, the central limit theorem (e.g., Chung, 1968) ensures that such mean magnitudes are approximately normally distributed regardless of the distribution of individual station magnitude estimates.

2. Distribution of an individual earthquake magnitude

Let us first consider an individual earthquake with observed magnitude x, in which the uncertainty is normally distributed with standard deviation o-. It is assumed that any systematic biases, due to station or regional effects or nondetection of the earthquake by some stations, are either negligible or have already been removed (Ringdal, 1975, 1976). Then, if m is the true magnitude, the conditional distribution of x given m and o- is given by:

1 [ , 1 . f ( x l m , o ' ) - ~ exp - ~ ( x - m) 2 (1)

The true magnitude is unknowable, but its prior conditional distribution, given /3, is known from the GR frequency-magnitude law to be:

f ( m l / 3 ) (x e x p ( - / 3 m ) (2)

Let us assume for the moment that /3 is unknown. Then the conditional distribution for m given x, o- and /3 is (e.g., DeGroot, 1970):

f ( mJx,o',/3 ) cx f ( xlm,o- ) f ( ml /3 )

1

[ ' ] ~ exp - -~-2~2(x 2 + m 2 - 2xm + 2o'~/3m) (3)

[ ' 1 ~ exp 2 o- ~ ( - - 7 m X q- O" 2/3 ) 2

Thus the conditional distribution for m, given x, o- and /3, is normal with mean x - cr2/3 and standard deviation or.

D.A. Rhoades / Tectonophysics 258 (1996) 71-83

3. Distribution of magnitudes of earthquakes in a catalogue

73

The values appearing in an earthquake catalogue are rounded, usually to one decimal place in modem catalogues. If only the catalogue magnitude y is known and the half-width of the rounding interval is 6, the observed magnitude x may be regarded as uniformly distributed between y - 6 and v + 6. The true magnitude m may therefore be regarded as the sum of a uniform [ - 6 , 6 ] random variable and a normal random variable with mean v - cr 2/3 and variance o- 2. This has conditional probability density .f(m[ y, cr. ~,/3) given by:

f(m v,~r.~,fl ) = £ ,~ 2 - ~ ~P o- du

= _ _ q ~ " _ q ~ " __ 26 ~r o-

(4)

where ~ is the standard normal probability density, and @ its integral. Suppose there are n earthquakes in a catalogue with magnitude densities and cumulative distributions f1(m)

and ~ (m) , respectively, j = 1 . . . . . n, where now fj(m) denotes either .~(mlxi,o'#,/3) or .~(ml31i,cri,/3). The number of earthquakes exceeding magnitude m, # M > m, is the number of successes in n binomial trials where the jth trial has probability 1 - ~ ( m ) of success. It thus has expected value:

E ( # M > m ) = L [1 - F . / ( m ) ] (5) , j= 1

and variance:

i i

V a r ( # M > m ) : ~ ~ ( m ) [ l - ~ ( m ) ] j= 1

(6)

The central limit theorem (e.g., Chung, 1968, theorem 7.2.1) then allows the distribution of # M > m to be approximated by a normal distribution with mean E(#M > m) and variance Var (#M > m). This approximation could be used to construct a confidence interval for the true value of # M > m given the biased catalogue magnitudes.

4. Confidence interval for the activity rate

Let us now consider the question of placing confidence limits on the long-term activity rate A(m), i.e., the rate of occurrence (assumed constant) of earthquakes with magnitude exceeding m, given only ~ (m), j = I, . . . . n and without using the GR relation. This is needed to estimate the a-value of the GR relation which satisfies A(m0)T= 10", where m o is the lower cut-off magnitude and the catalogue covers a period of length T. If the usual Poisson assumption is adopted, then the number of earthquakes in the catalogue exceeding a given magnitude m is Poisson with mean A(m)T [and variance A(m)T also]. Hence A can be estimated by:

A(m) E(#M>m) (7) T

The Poisson assumption may be reasonable when T is long enough or the region covered by the catalogue

74 D.A. Rhocuh's / TeclOnol)hysics 258 ( / 996) 71 --83

large enough so that the short-term clustering associated with aftershock sequences and other multiple events does not strongly influence the statistics. The variance of A is given by:

1 Var[ ~(m)] - T2

1 T 2

1 ( S )

T 2

1 T 2

where in the second-to-last step the reasonable assumption is made that the two terms in square brackets are uncorrelated. Hence, using Eqs. (5) and (5), we have:

_1 } Var[3.(m)] T2

/= 1 , / = 1 ( 9 )

1 _ 6 ( m ) 2 ]

j = 1

This may be used to construct approximate confidence intervals for A or to compare estimates of A based upon different periods of a catalogue as a means of testing whether A appears to be changing with time.

- - - E [ ,~( r e ) T - A( m)T] 2

- - - E { [ E( # M > m) - # M > m] + [ # M > m - A ( m ) T ] } 2

- - - { E l E ( # M > m) - # M > m] 2 + E [ # M > m - A(m)T] 2}

- - - [ V a r ( # M > m) + A(m)T]

4.1. A simulated example

To illustrate the techniques described above, a simulated catalogue of 5000 earthquakes with true magnitude > 2.7 was generated with a b-value of 1. To simulate the errors that arise in practice, the jth observed magnitude was obtained from the jth true magnitude by adding to it a N(0,o} 2) pseudo random variable with

= 0.1(1 + uim j) where u i was sampled from a uniform (0,1) distribution. The observed magnitude x i was rounded to one decimal place to give the rounded catalogue magnitude yj.

The weak but generally positive correlation between standard deviation and rounded magnitude produced by such a simulation is shown in Fig. 1. The fitted curve in Fig. 1 is a robust smooth trend line calculated by the local regression function " lowess" as implemented in the S-Plus statistical package (Chambers and Hastie, 1992).

The results are given in the magnitude-frequency of exceedance plot, Fig. 2, which shows the distribution of the true, observed and rounded magnitudes, and curves representing E ( # M > m) and approximate 95% confidence limits for # M > m calculated from the rounded magnitudes and known standard deviations with /3 = Ioge(10) and 8J = 0.05. In this case the positive bias in the frequency-magnitude distribution of the observed magnitudes is appreciable and there is also a further small positive bias due to rounding. It can be seen that the procedure described above effectively corrects these biases in the calculation of E ( # M > m) for values of m > 4, since the confidence limits enclose the true magnitude distribution over this range. This was to be expected since the simulated catalogue is essentially complete for rounded magnitudes which could result from true magnitudes > 4 but increasingly incomplete as m decreases below 4, given the distribution of the cri's.

Also shown in Fig. 2 is the distribution that arises if each magnitude Yi is replaced by the mean of the conditional distribution of m i given Yi, o) and /3, i.e., by 3 ) - cr92/3, which is here called the recentred magnitude. The distribution of recentred magnitudes lies below the distribution of true magnitudes, just as those of the observed and rounded magnitudes lie above it. Thus the simple and, at first sight, reasonable procedure of

D.A. Rhoades / Tectonophysics 258 (1996) 71 83

S i m u l a t e d c a t a l o g u e

75

' ' ' " I ; " ! ; "

•

. . . . . , ,

o 0 ° . ~ : . • II •

i l e e 0 0 • l

L_ I I r ;

2 3 4 5 6 7

R o u n d e d m a g n i t u d e

Fig. 1. Plot of standard deviation versus rounded magnitude and robust smooth trend line for one catalogue simulation of 5000 earthquakes with true magnitude _> 2.7 observed with normally distributed errors and rounded to 1 decimal place. The jth magnitude has standard deviation (r / - 0.1( l + uim:), where u i ~ U (0,1).

recentring the magnitudes to correct the bias brings about an over-correction. This highlights the subtlety of the

role of uncertainty in biasing the magnitude-frequency distribution.

In Fig. 3 an approximate 95% confidence interval for A(m)T is compared with the GR frequency-magnitude

relation from which the data were generated, using the same simulated data. Again the confidence interval

includes the true GR relation for magnitudes m > 4. The lower cut-off magnitude used in an analysis must be chosen carefully when applying the methods

described here. As usual the linearity of the relationship between magnitude and the logarithm of frequency is a

good guide to completeness, but the relevant frequencies are those indicated by E ( # M > m), rather than the

rounded magnitudes.

5. C o r r e c t i o n o f b i a s in b - v a l u e e s t i m a t e

It has been assumed above that /3 is known. The observation errors do not bias estimates of /3, as long as (r is constant, as pointed out by Tint• and Mulargia (1985a). So for the case of constant cr we can use a standard procedure appropriate to the data in hand to determine the maximum likelihood estimate of /3 without regard to estimation errors, such as the maximum likelihood equation of Aki (1965):

1 = k5 - m o (10)

76 D.A. Rhoades / Tectonophysics 258 (1996) 71-83

o 0 o o

o O o

t~ 0

8

o

i i

A true data ~ ~ , L ~ o + observed data /

' " ~ ' ~ t- . • rounded data I

...... , × " % ~ , " . . × ",#.

I F I I I

2 3 4 5 6 7

Magnitude

Fig. 2. Distribution of #M > m for true, observed, rounded and recentred magnitudes for a simulated catalogue generated as in Fig. 1. The solid line is the estimated exceedance frequency E ( # M > m) calculated from the rounded magnitudes and known standard deviations. The dotted lines are approximate 95% confidence limits for #M > m.

o.

0

"6

U_

O 0 O 0

O Q Q O

0

" " " " • • • rounded data ~ ~ _ ~ • - . . . . true GR line

estimated exceedance frequency ' • o , • . ~ • .......... approx. 95 Y• confidence hmnts

" ,~. '~ ~ • . . . .

I I [ I

2 3 4 5 6 7

Magnitude

Fig. 3. Estimated exceedance frequency E(#M > m) = ,~(rn)T and approximate 95% confidence limits for A(.,n)T compared to the GR relation used to generate the data for the simulated catalogue of Fig. 2. The frequency-magnitude distribution of the rounded data is also shown.

D.A. Rhoades / Tectonophysics 258 (1996) 71-83 77

for the simple case of exponentially distributed magnitudes in a catalogue which is homogeneous for m > m 0, where ~ is mean magnitude of all earthquakes exceeding magnitude m 0, or the formula of Utsu (1966):

- Y h - m 0 ( l l ) /3 tanh( /33)

which corrects for bias due to rounding. No consideration is given here to the problem of imposing maximum magnitudes as has been done by Weichert (t980), Tinti and Mulargia (1985a) and Kijko and Sellevoll (1989). The maximum magnitude has an important influence on the hazard at high magnitudes, but is not so influential in the estimation of b-value.

When o" varies between earthquakes in a catalogue, the initial estimate of /3 calculated by one of the methods mentioned above may be biased but an iterative backfitting procedure can be used to obtain both an optimal estimate of /3 and frequency-magnitude distributions which are consistent with it.

The backfitting procedure works as follows. An initial estimate o f /3 is used to generate an initial estimate of .~(m] x j, %,/3) or ~.(ml y/, o'/, 6j,/3), j = 1 . . . n. Then a new estimate o f /3 is made, using the method described in the next paragraph. The next estimate o f /3 is used to update f/(ml x j, o-j,~3) or ~(rn] ~), o'j, 6:,/3), j = 1 . . . n, and so on. The procedure continues until the successive estimates of /3 converge, usually after only a few iterations. This is an example of the use of the EM algorithm (Dempster et al., 1977).

The estimation of /3 is carried out by replacing ~ in Aki's maximum likelihood formula by the appropriate generalisation when magnitudes are represented by distributions rather than point estimates. That is:

1 -7 = E( MIM > mo) -- m o (12) /3

where the expected value E(MIM > m 0) given by: 11

E p~ E( M/I M~ > m o )

E(MIM>mo)= J=' (13)

j - I

where p~ = Prob(M/> m o) = 1 - ~(mo) . Noting that:

fm~mfi(m) dm E(MiIM i > m0) = (14)

Pi we have:

f~mfj(m) dm

E(MIM>mo )= j=l~ (15)

~ [ l - ~ ( m o ) ] j = 1

In the above .~(m) may represent fj(ml x j, o'j,/3) or fi(ml yj, o-j, 6/,/3) depending on whether rounding error is neglected or not. If rounding error is neglected, then since z<b(z) = - & ' ( z ) the terms in the numerator of the right hand side of Eq. (15) are given by: )[ (m0,+..)]

f , lfnfj(m) dm = ( x j - 0)2/3 1 - qb + ~05 (16) o-j

78 D.A. Rhoades / Tectonophysics 258 (I 996) 71- ,~¢3

If rounding error is taken into account, using f@(z )dz = z@(z) - 4~(z) and f z @ ( z ) d z = ½[(z 2 - 1)@(z) + zd~(z)] the following is obtained after a calculation:

f . m t ) ( m ) d m = x ' + 2 + J- + 7 5 @ ( 1 - , 3 ) - - - + 1 o

+ + 2,3 4 (1- a ) - + 4 (1)

where x' = Y.i - ~2~ _ 6 ,A = 2 6 / % , and l = (m o - x ' ) / ~ .

~(12-2,3 l)]@(l) (17)

5. L Simulations of b-L, alue estimation

The proposed method was tested by applying it to 200 simulated catalogues with "true magnitudes" conforming exactly to the GR frequency-magnitude law with b-value 1, generated as described above and illustrated in Figs. 1-3.

From each catalogue of rounded magnitudes and known standard deviations the b-value was estimated using the lower magnitude limit m 0 = 4.0 by the standard maximum likelihood method of Aki (Eq. (10)) which ignores magnitude uncertainty, the method of Utsu (Eq. (11)) which corrects Aki's method for rounding errors, and the method of this paper (Eq. (12), Eqs. (15) and (17) with backfitting) which corrects for both magnitude uncertainties and rounding errors using the known o)'s and ¢5 i = 0.05. In Fig. 4 the distributions of the results over 200 simulations are shown in the form of (a) box plots which highlight the median, quartiles, approximate range, and outliers of the distribution (e.g., Tukey, 1977) and (b) approximate 95% confidence limits for the mean b-value from 200 simulations. It can be seen that the first two methods give estimates of b-value which

(a) Results of 200 simulations

.6

m

i i i

Aki (1965) Utsu (1966) This paper

(b) Confidence intervals from 200 simulations

(::5

g (:5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t o

+ +

Aki (1965) U t s u ( 1 9 6 6 ) This paper

Fig. 4. Estimates of b-value from 200 simulations of catalogues with magnitude uncertainties positively correlated with magnitude (see Fig. 1) using standard maximum likelihood (Aki, 1965), maximum likelihood corrected lbr rounding (Utsu, 1966) and maximum likelihood corrected for magnitude uncertainties and rounding (this paper), The true b-value is 1. The estimates are presented as (a) box plots of all 200 simulations and (b) the mean and _+ 2 standard error confidence limits from 200 simulations.

D.A. Rhoades / Tectonophysics 258 (1996) 71 83 79

are significantly biased downwards by about 0.05. Such a bias would result in a 40% over-estimate of the hazard rate for M > 7.0 using the GR frequency-magnitude law with m 0 = 4.0. The bias in b-value is downwards in this case because of the positive correlation of cr and M. It can be seen from Fig. 4 that the method of this paper corrects the bias to such an extent that the mean b-value estimated from 200 simulations is indistinguishable from the true b-value at the 5% level of significance.

6. Example of New Zealand earthquakes 1987-1992

In simulations standard deviations are exactly specified but in practice only estimated by standard errors. Since 1987 standard errors have been presented for the magnitudes of earthquakes in the New Zealand earthquake catalogue published in the annual New Zealand Seismological Reports (e.g., Anon, 1991). The earthquakes from 1987 to 1992 with epicentres from 37 to 47°S are considered here. In a small proportion of cases no standard error estimate is available because the earthquake was observed at an insufficient number of stations. In those cases, which tend to be for low magnitudes and hence have a negligible impact on the calculation of b-value described below, a value of o" = 0.3 has been assumed. A plot of standard error against magnitude for earthquakes with catalogue magnitudes of M L > 2.0 (Fig. 5) shows that the standard error is quite variable but on average increases with magnitude as shown by the fitted " lowess" trend line. Thus there is the possibility of bias in estimated b-values if magnitude uncertainty is neglected.

In Table 1 we again compare b-value estimates computed by the methods of Aki, Utsu. and this paper, now using estimated standard errors in the place of the o'/s. The b-value estimates are presented for each of the years 1987-1992 and are calculated using a lower magnitude limit m 0 = 4.0.

As for the simulated catalogues, the b-values corrected for magnitude uncertainties tend to be higher than

NZ earthquakes 1987-92

O

I O •

I

I

D

8:

D

h I

e l

l | l | ' s l l . . : . . , :

i l l ! ? l u i ! . , , . . . - - I I I I I

2 3 4 5 6

I

Catalogue magnitude

Fig. 5. Plot of standard error versus catalogued magnitude for New Zealand earthquakes 37-47°S, 1987-1992, and robust smooth trend line.

80 D.A. Rhoades / Tectonophysics 258 (1996) 71-83

Table 1 Estimates of b-values for New Zealand earthquakes 37-47°S using catalogued magnitudes > 2.0 and m 0 = 4.0, by standard maximum likelihood (Aki, 1965), maximum likelihood corrected for rounding (Utsu, 1966) and maximum likelihood corrected for magnitude uncertainties and rounding (this paper)

Year(s) Method

Aki (1965) Utsu (1966) This paper

1987 1.124 1.118 1.131 1988 1.116 1.122 1.184 1989 1.115 1.121 1.184 1990 1.086 1.092 1.139 1991 1.097 1.103 1.077 1992 1.171 1.178 1.175 1987-1992 1.117 1.124 1.146

those calculated by the other two methods. This is because in both cases the errors are positively correlated with magnitude. The greatest difference observed between the estimates using Akrs formula and the method presented in this paper (0.069 in 1989) is equivalent to a 60% difference in estimated hazard rates for M > 7.

In Fig. 6 the curve of estimated exceedance rate k(m) is plotted for each of the years 1987-1992, corrected for magnitude uncertainties and compared to the GR relation estimated from the combined data from all six years with m 0 = 4.0. A comparison of such curves, in the range of magnitudes for which the catalogue can be considered complete, is useful to examine the effect of long-range clustering in the catalogue. In the presence of long-range clustering one might expect the estimated exceedance rate for a given magnitude to vary signifi- cantly from year to year. The curves in Fig. 6 appear to be rather similar between magnitudes 4 and 5 but not for m < 4 o r m > 5 .

q

O

.......... 1987

.......... ~ % ~ . - ~ . - . . . . . 1988 - - - 1 9 8 9 ' ~ - ~ "'-'~"~-" "

o ~ ........ ~ \ - - - - - - 1 9 9 0

' " ~'-~..--~ 1 9 9 1

ua "~---~'\.\. \.

o ,'~ \\ , \ I I l I I E

2 3 4 5 6 7

M a g n i t u d e

Fig. 6. Estimated exceedance rate as a function of magnitude Ior New Zealand earthquakes 37-47°S in each of the years 1987-1992, corrected for magnitude uncertainties and rounding. The solid line is the GR relation estimated from combined data using m 0 = 4.0.

D.A. R h o a d e s / T e c t o n o p h y s i c s 258 (1996) 7 1 - 8 3 81

o

o. 1 2 ~ , 1987 . . . . 1988

. . . . - - 1989 oo , ~ . ~ .... - - - 1990

"" , '~ '~ ,Q" ...... - . . . . 1992 \ , \ " . ~ . ~ . . ": . . . . . - - _ _

q '\ , , \ ' , - , ' .

i ' . t\ ,,~?,,,,'// \ \ I I ! i \ // o ~: i /1~ ,k / \\.// ~t ""

', i \ \ ii : I J

i I I I I ] - -

2 3 4 5 6 7

Magnitude

Fig. 7. Deviation of exceedance rate for New Zealand earthquakes in each of the six years 1987 1992, corrected for magnitude uncertainty

and rounding, from combined estimate of GR relation with m 0 = 4.0. The deviation is expressed in terms of the error ratio, the ratio of

absolute deviation to the standard error.

One can use Fig. 7 to decide on the statistical significance of the apparent divergences in Fig. 6. The error ratio plotted there is the ratio of the absolute deviation of each curve from the combined GR relation to the

/- ^

standard error liVar[ A(m)] (see Eq. (9)), a value greater than 2 indicating statistical significance at the 5c~

level. The error ratio is generally less than 2 for all years for m > 4 except for 1992 where it is above 2 between 4.0 and 4.2. The exceedance rates for the six years are thus statistically almost indistinguishable at m >_ 4, a result which provides little support for long-range clustering. The deviations could be due either to changing seismicity or variation over time in the completeness of the catalogue at m < 4. The latter cause seems most likely, given the curvature of the exceedance rate curves below magnitude 4. The catalogue appears to have gradually become more complete for m < 4 over the period 1987-1992.

7. Implications for estimates of ground motion exceedance probabilities

Estimates of exceedance probabilities of a given ground motion parameter, such as acceleration or Modified Mercalli intensity, during a specified period of time, involve the GR relation together with the spatial distribution of sources and the attenuation law. In deriving such estimates, it is desirable to treat magnitude uncertainties in a consistent way in the GR relation and the attenuation law. The earthquakes used to estimate attenuation laws are usually quite a different set from those used for the GR relation. They are a restricted set of higher-magnitude events which may not necessarily be from the same region. Tinti and Mulargia (1985a) showed that, for a simple point source and an attenuation law linear in magnitude, biases due to magnitude uncertainties in the estimation of exceedance probabilities of a given level of ground motion cancel out if the data sets used to estimate the GR relation and attenuation law have the same b-value and the same (constant) magnitude standard deviation.

82 D.A. Rhoades / Tectonophysi¢s 258 ~ 1996) 71-83

In deriving attenuation laws for use in conjunction with GR parameters estimated as described here, the effect of each individual magnitude uncertainty must be considered. A first-order correction to the usual analysis is to remove the bias in the magnitudes by replacing each catalogued magnitude Yi by its corrected value y j - ~213 in the attenuation data set. Exceedance probabilities could then be obtained approximately using the formula of Bender (1984) which allows for ground motion variability. A more rigorous treatment would involve careful modifications to the one-step maximum likelihood method or the two step regression method (Joyner and Boore, 1993) to explicitly allow for the variance of each corrected magnitude. Such modifications are the subject of further research.

8. Conclusion

The method presented here overcomes the deficiency in estimating the parameters in the GR relation by allowing for the uncertainties in estimated magnitudes. It is seen to be effective in correcting the biases due to magnitude uncertainties. The results of applying the method to real examples indicate that the biases implied in other data bases are likely to be large enough to be important for hazard estimation purposes.

The same method could be used to compare seismicity between eras of a catalogue when different seismograph networks were in place, if the magnitude uncertainties applicable to each era could be quantified.

Acknowledgements

The author is grateful to T.H. Webb for providing data on New Zealand earthquake magnitudes and standard errors from the files of the New Zealand Seismological Observatory, and to F.F. Evison, D.J. Dowrick, W.D. Smith, R.A. Boyles, F. Mulargia and two anonymous referees for constructive comments on an earlier version of the paper.

References

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Anon, 1991. New Zealand Seismological Report 1987. DSIR Geol. Geophys. Wellington, Seismol. Obs. Bull. E-171, 141 pp. Bender, B., 1984. Incorporating acceleration variability into seismic hazard analysis. Bull. Seismoh Soc. Am., 74: 1451-1462. Chambers, J.M. and Hastie, T.J., 1992. Statistical Models in S. Wadsworth and Brooks/Cole, Pacific Grove, CA. 608 pp. Chung, K.L., 1968. A Course in Probability Theory. Harcourt, Brace and World, New York, NY, 331 pp. DeGroot, M.H., 1970. Optimal Statistical Decisions. McGraw-Hill, New York, NY, 489 pp. Dempster, A.P., Laird, N.M. and Rubin, D.B.. 1977. Maximum likelihood from incomplete data via the EM algorithm (with discussion). J.

R. Star. Soc. B, 39: 1-38. Freedman, H.W., 1967. Estimating earthquake magnitude. Bull. Seismol. Soc. Am., 57: 747-760. Gutenberg, B. and Richter, C.F., 1944. Frequency of earthquakes in California. Bull. Seismol. Soc. Am.. 34: 185-188. Joyner, W.B. and Boore, D.M., 1993. Methods for regression analysis of strong-motion data. Bull. Seismol. Soc. Am., 83: 469-487. Kijko, A., 1988. Maximum likelihood estimation of Gutenberg-Richter b parameter for uncertain magnitude values. Pure Apph Geophys.,

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and incomplete catalogs with different threshold magnitudes. Bull. Seismol. Soc. Am.. 79: 645-654. Rhoades, D.A., Van Dissen, R.J. and Dowrick, D.J., 1994. On the handling of uncertainties in estimating the hazard of rupture on a fault

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