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Parametric-historicProcedureforProbabilisticSeismicHazardAnalysisPartI:EstimationofMaximumRegionalMagnitudemmax

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Pure appl. geophys. 152 (1998) 413–4420033–4553/98/030413–30 $ 1.50+0.20/0

Parametric-historic Procedure for Probabilistic SeismicHazard Analysis

Part I: Estimation of Maximum Regional Magnitude mmax

ANDRZEJ KIJKO1 and GERHARD GRAHAM1

Abstract—A new methodology for probabilistic seismic hazard analysis (PSHA) is described. Theapproach combines the best features of the ‘‘deductive’’ (CORNELL, 1968) and ‘‘historical’’ (VENEZIANO

et al., 1984) procedures. It can be called a ‘‘parametric-historic’’ procedure.The maximum regional magnitude mmax is of paramount importance in this approach and Part I of

our work presents some of the statistical techniques which can be used for its evaluation. The work isan analysis of parametric procedures for the evaluation of mmax, when the form of the magnitudedistribution is specified. For each of the formulae given there are notes on its origin, assumptions madeof its derivation, and some comparisons. The statistical concepts of bias and variance are considered foreach formula, and appropriate expressions for these are also given. Also, following KNOPOFF andKAGAN (1977), we shall demonstrate why there must be a finite upper bound to the largest seismic eventif the Gutenberg-Richter frequency-magnitude relation is accepted.

Key words: Seismic hazard, maximum regional earthquake magnitude mmax.

1. Introduction

This introduction serves for both Part I (this paper) and Part II ‘‘Assessment ofSeismic Hazard at Specified Site.’’

Following MCGUIRE (1993), existing procedures of probabilistic seismic hazardanalysis (PSHA) fall into two main categories: deducti6e and historic.2 The name of

1 Council for Geoscience, Geological Survey of South Africa, Private Bag X112, Pretoria 0001,South Africa. Tel: +27 12 8411180, +27 12 8411201. Fax: +27 12 8411221. E-mail:kijko@geoscience.org.za; gerhardg@geoscience.org.za

2 It must be noted that in addition to these two categories, alternative approaches are occasionallyused. These procedures attempt to assess temporal or temporal and spatial dependence on seismicity. Inorder to incorporate memory of past events, they use the formalism of non-Poisson distribution orMarkov chains. In this approach seismogenic zones that recently produced strong earthquakes becomeless hazardous than others that did not rupture in recent history. Clearly such models may result in amore realistic PSHA, but they are nevertheless still only research tools and have not yet reached the levelof development required by engineering applications. An excellent review of such procedures is given byCORNELL and TORO (1970). Other recent treatises of the subject may be found e.g., in MUIR-WOOD

(1993), and BOSCHI et al. (1996).

Andrzej Kijko and Gerhard Graham414 Pure appl. geophys.,

the first category (deductive) comes from the fact that by applying the procedure,we deduce what the causative sources, characteristics, and ground motions forfuture earthquakes are. The theoretical base for the deductive method was formu-lated 30 years ago by C. A. Cornell (CORNELL, 1968). The approach permits theincorporation of geological and geophysical evidence to supplement the seismicevent catalogues. Application of the procedure includes several steps. The initialstep requires definition of potential seismic sources which usually are associatedwith geological or tectonic features (as e.g. faults) and delineation of potentiallyactive regions (zones) over which all available information is averaged. In the nextstep, for each seismogenic source zone, seismicity parameters are determined.Following the most common assumptions made in engineering seismology viz. thatearthquakes are described by a Poisson process and that earthquake magnitudesfollow a Gutenberg-Richter doubly truncated distribution, then, for each seis-mogenic source zone the parameters are: mean seismic activity rate l (with is theparameter of the Poisson distribution), the level of completeness of the earthquakecatalogue mmin, the maximum earthquake magnitude mmax, and the Gutenberg-Richter parameter b. Assessment of the above parameters requires a seismic eventcatalogue containing origin times, size of seismic events (in terms of magnitude orintensity) and spatial location. This then allows the calculation of a probabilitydensity function (PDF) of distance to the specified site. In the ne next step aground-motion relation is selected, giving the cumulative distribution function(CDF) for a required ground-motion parameter such as peak ground velocity, peakground acceleration (PGA), or even amplitude of velocity or acceleration forspecified values from an entire spectrum of frequencies. The final step requires theintegration of individual contributions from each seismogenic zone into a site-spe-cific distribution. The procedure of integration is straightforward and is performedby application of the total probability theorem. There is no doubt that deductive(or deductive-type) procedures of PSHA are dominant and remain the method mostcommonly used worldwide.

Probably the strongest point of any deductive-type procedure of PSHA is itsability to account for all sorts of deviations from the ‘‘standard’’ model, i.e., itaccounts for phenomena such as migration of seismicity, seismic ‘‘gaps’’ or, ingeneral, any nonstationary properties of seismicity. This is possible because theprocedure is parametric by nature.

Unfortunately, the deductive procedure also has significant weak points. Themajor disadvantage stems from the requirement of specifying the seismogenicsource zones. Often, a different seismogenic zone specification leads to significantlydifferent assessments of hazard. In addition, the procedure requires for each zonea knowledge of the model parameters (in the simplest case, the Gutenberg-Richterparameter b, the level of completeness mmin, the mean activity rate l, and mmax),which cannot be determined reliably for areas that are small or have a veryincomplete seismic history.

Parametric-historic Hazard Analysis 415Vol. 152, 1998

The second category of PSHA consists of the so-called ‘‘historic’’ methods(VENEZIANO et al., 1984), which, in their original form, are nonparametric. Thesemethods require as input data, information about past seismicity only, and do notrequire specification of seismogenic zones. For each historic earthquake, theempirical distribution of the required seismic hazard parameter is estimated byusing its value of magnitude, distance and assumed ground-motion attenuationrelation. By normalizing this distribution for the duration of the seismic eventcatalogue, one obtains an annual rate of the exceedence of the required hazardparameter. The major advantage of the method is that specification of seismogenicsource zones is not needed. Furthermore the approach does not require designationof the model, and this can be an advantage. By its nature, the historic methodworks well in areas of frequent occurrence of strong seismic events, when the recordof past seismicity is ‘‘reasonably’’ complete (BOSCHI et al., 1996).

At the same time, the nonparametric historic approach has significant weakpoints. Its primary disadvantage is probably its poor reliability in estimating smallprobabilities for areas of low seismicity. The procedure is not recommended for anarea where the seismic event catalogues are highly incomplete. In addition, in itspresent form, the procedure is not capable of making use of any additionalgeophysical or geological information to supplement the pure seismological data.

Bearing in mind both the weak and strong points of the above two approaches,we propose an alternative procedure which, following McGuire’s scheme, could beclassified as a parametric-historic approach. The new approach combines the bestfrom the deductive and nonparametric historic procedures, and, in many cases, isfree from the basic disadvantages characteristic of each of the procedures.

The proposed procedure is parametric and its application consists essentially oftwo steps. The first step is applicable to the area in the vicinity of the site for whichknowledge of the seismic hazard is required. In this respect the procedure is similarto the deductive approach and requires estimation of area-specific parameters. Theparameters depend on the selected PSHA model, which in our case are thearea-specific mean seismic activity rate l, the Gutenberg-Richter parameter b andthe maximum regional magnitude mmax. The approach is also conceptually open toany alternative parameterizations.

The second step is applicable to a specified site and consists of an assessment ofthe parameters of distribution of amplitude of the selected ground-motionparameter.

Since in each step, parameters are estimated by the maximum likelihoodprocedure, by applying the Bayesian formalism, any additional geological orgeophysical information (as well as all kinds of uncertainties) can be easilyincorporated. The new procedure is consequently capable of giving a realisticassessment of seismic hazard in areas of both low and high seismicity, including thecase where the catalogues are incomplete.

In the present form, the procedure allows assessment of seismic hazard in termsof peak ground acceleration, peak ground velocity or peak ground displacement.

Andrzej Kijko and Gerhard Graham416 Pure appl. geophys.,

Extension of the procedure and assessment of the whole spectrum of ground motionis straightforward.

If the procedure is applied to all grid points of and around a region, then a mapof seismic hazard for the entire region can be obtained.

The need for the development of the new approach to PSHA was dictated byincompleteness of seismic catalogues and the difficulty in identifying seismogenicsource zones. The approach closest in conception to ours that we have seen is thatof FERNANDEZ and GUZMAN (1979), in which seismic hazard is mapped for SouthernAfrica. Another similar approach is that of FRANKEL (1995), who has performedseismic hazard analysis for the central and easterm United States. The Frankelapproach has been taken up by LAPAJNE et al. (1997) in modeling seismic hazardin Slovenia.

This paper concentrates on the problem of statistical assessment of maximumregional magnitude mmax.

2. Consideration of the Necessity of a Maximum Regional Magnitude mmax

The belief that there must be an upper limit to earthquake magnitude has beenexpressed by many seismologists. RICHTER (1958), who analyzed the observedfrequency-magnitude relations, expressed the opinion that ‘‘A physical upper limit tothe largest possible magnitude must be set by the strength of the crustal rocks, in termsof the maximum strain which they are competent to support without yielding.’’YEGULALP and KUO (1974) assert, ‘‘It is apparent on physical grounds that there mustexist an upper limit to the occurrence of a maximum magnitude earthquake in eachregion. Such a limit is likely to be a function of maximum source size in the Earth ’scrust and the upper mantle.’’ WEICHERT (1980) affirms that ‘‘A realistic risk analysismust admit a regional maximum possible magnitude, e6en though it may not yet bepossible to estimate this magnitude reliably.’’

KNOPOFF and KAGAN (1977) have demonstrated that if the frequency-magnitudeGutenberg-Richter relation

log N=a−bM (1)

is valid, where N is the number of earthquakes with magnitude M and larger, anda and b are parameters, then an upper bound to the magnitude M must be introduced.In fact relation (1) was first discovered by ISHIMOTO and IIDA in 1939, prior to thepublication of the Gutenberg-Richter formula in 1954. Nevertheless, we shall alsofollow the common practice of calling it the Gutenberg-Richter relation.

Assuming that the seismic energy-magnitude relation is of the form

log E=c+dm, (2)

where c and d are constants, KNOPOFF and KAGAN (1977) showed that the totalamount of energy released by earthquakes in a unit time is

Parametric-historic Hazard Analysis 417Vol. 152, 1998

ETOTAL=& Emax

E min

E dn=const. E1−b/d�EmaxE min

. (3)

For typical values of b$1 and d$1.5 and for Emax��, the total amount ofreleased seismic energy ETOTAL, therefore, tends to infinity. KNOPOFF and KAGAN

called this result the ‘‘Emax catastrophe.’’ Thus it is clear that an upper bound forEmax (or equivalently mmax) must be introduced if the Gutenberg-Richter frequency-magnitude relation is to be applied in a realistic way.

It should be noted that one can avoid the ‘‘Emax catastrophe’’ by introducing adifferent form of the frequency-magnitude relation or by accepting the dependenceof parameter b on the magnitude value. If we allow parameter b to increase withincreasing magnitude, it is possible for the energy integral (3) to remain finite. It canbe shown that if an alternative model of frequency-magnitude is accepted, as e.g.,double exponential (LOMNITZ-ADLER and LOMINITZ, 1979), or one following fromthe Kulbak principle of maximum entropy (MAIN and BURTON, 1984; KAGAN,1991, 1994), then the presence of an upper limit in the value of the magnitude is notrequired (KIJKO, 1982).

3. Formulation of the Problem

Although a knowledge of the value of the maximum regional magnitude mmax isrequired in many engineering applications, it is remarkable how little has been donein developing appropriate techniques for an estimation of this parameter.

At present there is no generally accepted method for estimating the value ofmaximum regional magnitude mmax. The methods for evaluating mmax fall into twomain categories: deterministic and probabilistic.

The deterministic procedure most often applied is based on the empiricalrelationships between the magnitude and various tectonic and fault parameters.There are several papers devoted to such relationships. The relationships arevariously developed for different seismic areas and different types of faults (e.g.,SMITH, 1976; WYSS, 1979; SINGH et al., 1980; SCHWARTZ et al., 1984; WELLS andCOPPERSMITH, 1994; ANDERSON et al., 1996, and references therein). Alternatively,similar relations can be established by computer simulations of faults of variousstrength and shapes (WARD, 1997). Another class of deterministic procedures forthe assessment of maximum regional magnitude is based on extrapolating thefrequency-magnitude curves. As an alternative to the techniques above, researchersoften try to relate the value of mmax to the strain rate or the rate of seismic-momentrelease (e.g., PAPASTAMATIOU, 1980; ANDERSON and LUCO, 1983). Such anapproach has also been applied by MCGARR (1984) to evaluate the maximumpossible magnitude of seismic events induced by mining. An interesting procedurefor estimation of the maximum earthquake magnitude was developed by JIN andAKI (1988). In their work of mapping of coda Q at 1 Hz for China, a relation that

Andrzej Kijko and Gerhard Graham418 Pure appl. geophys.,

was remarkably linear was established between the logarithm of coda Q0 and thelargest observed earthquake magnitude. It was found that if in a certain regionQ0 is 1000, the largest observed magnitude in the past 400 years in that region isabout 4, whereas if Q0 in another region is, say 100, then the magnitude is about8. The authors postulate, consequently, that if the largest earthquake magnitudeobserved during the last 400 years is the maximum possible magnitude mmax, theestablished relation will give a spatial mapping of mmax.

In most cases, unfortunately, the value of the parameter mmax determined bymeans of any deterministic procedure is very uncertain; the uncertainty can reacha value of up to one unit on the Richter scale.

The value of mmax can also be estimated purely on the basis of the seismolog-ical history of the area, viz. by utilizing seismic event catalogues and appropriatestatistical estimation procedures. The statistical techniques falling into this cate-gory form an important class of probability problems dealing with extreme val-ues of random variables and estimation of the end-point of a distributionfunction. The statistical theory of extreme values was already known and well-de-veloped in the fifties (e.g., GUMBEL, 1958), and was probably applied for the firsttime in seismology by NORDQUIST (1945), who demonstrated that the largestearthquakes in California are in good agreement with Gumbel’s Type I asymp-totic distribution of extremes. A major breakthrough in the seismological applica-tions of extreme-value statistics was made by EPSTEIN and LOMNITZ (1966), whoproved that the Gumbel I asymptote can be derived directly from assumptionsthat seismic events are generated by a simple Poisson process and that theyfollow the Gutenberg-Richter frequency-magnitude distribution. Statistical toolsrequired for the estimation of the end-point of distribution functions were devel-oped later (e.g., TATE, 1959; ROBSON and WHITLOCK, 1964; COOKE, 1979) andused in the estimation of maximum regional magnitude only recently (DARGAHI-NOUBARY, 1983; KIJKO and DESSOKEY, 1987; KIJKO and SELLEVOLL, 1989,1992; PISARENKO, 1991; PISARENKO et al., 1996). The available statistical toolssuitable for the estimation of maximum regional magnitude vary significantly.Essentially, they differ in:(i) assumptions regarding the properties of functional representations of the

magnitude distribution (especially in the behavior of the tail for large values),(ii) the procedures applied for the estimation of the end-point of the distribution.In this work we present two statistical procedures which can be used for theevaluation of the maximum regional magnitude mmax. Broadly speaking, the firstprocedure is more ‘‘straightforward,’’ while the second one is more ‘‘advanced’’and requires more complex calculations. It is assumed that for each of theprocedures both the analytical form and the parameters of the distribution func-tions of earthquake magnitude are known. This knowledge can be very limitedand very approximate, but must be available.

Parametric-historic Hazard Analysis 419Vol. 152, 1998

4. Assumptions

Suppose that in the area of concern, within a specified time interval T, there aren main seismic events with magnitudes M1, M2, . . . , Mn. Each magnitude M1]mmin (i=1, . . . , n), where mmin is a known threshold of completeness (i.e., all eventshaving magnitude greater than or equal to mmin are recorded). We assume furtherthat the seismic event magnitudes are independent, identically distributed, randomvalues with PDF, fM (m � mmax) and CDF, FM (m � mmax), respectively. Parametermmax is the upper limit of the range of magnitude and thus denotes the unknownmaximum regional magnitude, which is to be estimated.

It should be noted that the approach offered here is not limited to the case whenthe size of the seismic event M is magnitude. All the results are also valid when thesize of the earthquake is described by energy, seismic moment or seismic intensity.

For the classical Gutenberg-Richter relation, which is a frequency-magnituderelation that is unbounded from above (i.e., mmax��), the PDF and CDFfM (m) fM (m ��) and FM (m)=FM (m ��) are continuous and equal to (AKI,1965):

fM (m)=!b exp[−b(m−mmin)],

0,for m]mmin,for mBmmin,

(4)

FM (m)=!1−exp[−b(m−mmin)],

0,for m]mmin

for mBmmin

. (5)

The respective PDF and CDF of earthquake magnitude, which are bounded fromabove by mmax, are (PAGE, 1968; COSENTINO et al., 1977):

fM (m � mmax)=ÍÃ

Ã

Á

Ä

b exp[−b(m−mmin)]1−exp[−b(mmax−mmin)]

,

0,

for mmin5m5mmax

for mBmmin, m\mmax,(6)

and

0, for mBmmin,

FM (m � mmax)=ÍÃ

Ã

Ã

Ã

Á

Ä

1−exp[−b(m−mmin)]1−exp[−b(mmax−mmin)]

, for mmin5m5mmax, (7)

1, for m\mmax’

where b=b ln(10), and b is the b parameter of the Gutenberg-Richter relation (1).Both distribution functions are obtained as a result of truncation of unboundeddistributions (4) and (5) at the point mmax, that is

Andrzej Kijko and Gerhard Graham420 Pure appl. geophys.,

fM (m � mmax)=!fM (m)/FM (mmax),

0,for mmin5m5mmax

for mBmmin, m\mmax,(8)

and

FM (m � mmax)=>0,

FM (m)/FM (mmax),1,

for mBmmin,for mmin5m5mmax,for m\mmax,

(9)

where

FM (m)=& m

m min

fM (j) dj. (10)

The above classical formulation of the problem has significant shortcomings.The main ones are:

(i) Earthquakes are often missing from catalogues, especially lower magnitudeearthquakes from older catalogues. Consequently the most suitable methods foranalyzing old and incomplete data sets are those which require knowledge of thestrongest events only, rather than complete data sets (e.g., BURTON, 1979; YEGU-

LALP and KUO, 1974). If the largest events are used then in all the equations theoriginal magnitude distribution FM (m) must be replaced by its extreme-valuecounterpart Fmax

M (m).(ii) The sizes of earthquakes listed in catalogues might require adjustment or

conversion to a different magnitude scale. Conversion of one type of magnitude toa single measure common to the whole span of the catalogue requires conversion bymeans of empirical relations. As has been pointed out many times (e.g., CHUNG

and BERNEURER, 1981; BENDER, 1993; MCGUIRE, 1993), such a procedure is notnecessarily valid. In addition, a change in the characteristics of seismic sensors cancause systematic errors in the conversion of magnitudes (see, for example, the caseof magnitude conversion for the eastern and western United States, NUTTLI andHERRMANN, 1982). Hence, a catalogue that contains earthquakes with magnitudesadjusted or converted is heterogeneous and requires appropriate techniques fordealing with it. Probably the most efficient technique for estimating the parametersin the magnitude distributions using both complete catalogues as well as cataloguescontaining only the largest events, and which, furthermore, takes into account theuncertainty of earthquake magnitude, was that developed by TINTI and MULARGIA

(1985a,b), in which they introduced the concept of ‘‘apparent magnitude.’’ TheTinti-Mulargia approach was recently extended by RHOADES (1995), by allowingdifferent uncertainties of magnitude for individual earthquakes. Since the presenceof uncertainty of magnitude can also affect the estimation of the maximum regionalmagnitude mmax, this question will be discussed further in the following section.

(iii) The choice of the model for the distribution of earthquake magnitudes cansignificantly influence the results of the estimation of seismic hazard and maximumregional magnitude. Is the simplest model of all, based on the Gutenberg-Richter

Parametric-historic Hazard Analysis 421Vol. 152, 1998

relation, enough? In most engineering applications, for magnitudes in the middle ofthe range of distribution, the choice of the model is relatively unimportant. Themost important consideration is the upper tail of the distribution, which determinesthe value of mmax. Hence, for example, the consideration of the occurrence ofcharacteristic earthquakes (SCHWARTZ and COPPERSMITH, 1984) can drasticallyinfluence the tail of the magnitude distribution, and so influences the value of theestimated mmax.

Even if the concept of the existence of characteristic earthquakes were notapplicable, one would be unable to neglect the effect of multimodality, or ingeneral, nonlinearity in frequency-magnitude relationships. There are, by way ofillustration, some well-documented cases including the natural seismicity in Alaska(DEVISON and SCHOLZ, 1984), Italy (MOLCHAN et al., 1997), Mexico (SINGH et al.,1983), Japan (WESNOUSKY et al., 1983), and the New Madrid Zone of the UnitedStates (MAIN and BURTON, 1984), as well as mine-induced seismicity in Poland andSouth Africa (FINNIE, 1994; GIBOWICZ and KIJKO, 1994).

In addition, a significant shortcoming is our implicit assumption that seismicityparameters such as the b value of the Gutenberg-Richter relation and the meanactivity rate l remain constant in time. Temporal variations of seismic activity havebeen reported and described many times in the literature and a complete list ofthese reports would be very long. We will mention only a few of them, and onlythose that are well documented. Significant fluctuations in seismic activity havebeen reported in, e.g., Kamchatka and the Kuril Islands (FEDOTOV, 1968), Califor-nia, and, notably, Parkfield (BAKUN and MCEVILLY, 1984), China (MCGUIRE andBARNHARD, 1981), the New Madrid Zone, U.S.A. (MENTO et al., 1986), Greece(PAPADOPOULOS and VOIDOMATIS, 1987) and the North Sea (LINDHOLM et al.,1990). For other areas, the seismic activity shows temporal variations but it is notclear whether these changes are periodic, e.g., southern Italy (BOTTARI and NERI,1983), New Zealand (VERE-JONES and DAVIS, 1966), the Alpine-Himalayan belt(RAO and KALIA, 1986), Japan (SHIBUTANI and OIKE, 1989) and all subductionzones of the Circum-Pacific Belt (LAY et al., 1989). Global seismicity also showstemporal variation but with no clear periodicity (e.g., KANAMORI, 1981;SHIMSHONI, 1984).

It should, of course, be clear that neglecting the uncertainty resulting from aselection of a wrong model of magnitude distribution and/or temporal variation ofseismicity can lead to significantly biased estimates of seismic hazard, and alsobiased estimates of mmax.

Therefore, the approach, where the model parameters are treated as randomvariables, provides the most appropriate tool for handling the uncertainties consid-ered above.

In the following two sections we will derive two procedures in which theuncertainty of seismic hazard parameters can be incorporated and in whichBayesian-based, analytical (or semi-analytical) equations can be obtained. We shall

Andrzej Kijko and Gerhard Graham422 Pure appl. geophys.,

think of these as the ‘‘Straightforward Procedure ’’ and the ‘‘Ad6anced Procedure,’’respectively. It is assumed that for each of the procedures both the analytical formsand the parameters of the distribution functions are known. If they are not knownprecisely, they must be known at least approximately. In both the proceduresconsidered the largest observed magnitude mobs

max plays a crucial role.

5. Procedure I (‘‘Straightforward ’’)

This procedure is very straightforward and does not require extensive calcula-tions. It can be shown that the procedure attempts to correct the bias of theclassical maximum likelihood estimator m/ max=mobs

max (PISARENKO et al., 1996) butfails to provide an estimator having a smaller mean-squared error. The sameformulae can be obtained by applying at least three different approaches, namely:

(i) by using a purely intuitive criterion and employing well-known properties ofthe simple uniform distribution, or,

(ii) by applying a formal statistical technique of reducing the bias of the estimator,as developed e.g., by QUENOUILLE (1956), or

(iii) by expressing parameter mmax by means of unknown functions in the integralequations of a convolution type, and then obtaining mmax by an integraltransform method (TATE, 1959).

In our derivation we choose the first approach, as described by KIJKO (1997). Ingeneral we are following the convention in which a capital letter is used for arandom variable, and the same letter, in lower case, represents the values which itmay assume.

5.1. Deri6ation of Estimator I

The estimator derived here is based on the ‘‘order statistics’’ of earthquakemagnitudes M15M25 · · ·5Mn−15Mn, where the unordered Mi are independentrandom values and identically distributed according to CDF FM (m � mmax). Inaddition it is assumed that FM (m � mmax) belongs to the class of functions whichallow a Taylor Series expansion about the point mmax. We observe that after thetransformation Y1=FM (M1 � mmax), Y2=FM (M2 � mmax), . . . , Yn=FM (Mn � mmax),the values Y1, . . . , Yn form an ordered data set Y15Y25 · · ·5Yn−15Yn, andeach of them are uniformly distributed, that is

FY (y)=>0,

y,1,

for yB0,for 05y51,for y\1.

(11)

The CDF of the largest among (Y1, . . . , Yn ), that is Yn, is

Parametric-historic Hazard Analysis 423Vol. 152, 1998

FYn Pr [Yn5y ]

=Pr [Y15y, Y25y, . . . , Yn5y ]

= [FY (y)]n

=yn. (12)

The PDF of variable Yn is given by

fYn(y)

dFYn(y)

dy=>0,

nyn−1,n,

for yB0,for 05y51,for y\1,

(13)

and its expected value is

E(Yn )=& 1

j=0

jfYn(j) dj=n

& 1

j=0

jn dj=n

n+1. (14)

One possibility for obtaining the estimator m/ max is to introduce the condition

E(Yn )=yn, (15)

where yn is calculated from the relation yn=FM (mobsmax � mmax), and mobs

max is thelargest obser6ed magnitude.

Bearing in mind that E(Yn )=n/(1+n) we obtain the equation

FM (mobsmax � m/ max)=

nn+1

. (16)

Thus the estimator of mmax becomes a function of the known observations mobsmax

and n, and is obtained as a root of the equation (16). The above result is valid forany of the magnitude CDF FM (m � mmax), (describing the complete as well asextreme events), and does not require the fulfillment of the truncation condition(8)–(10). It can be shown that the estimator of parameter mmax as defined above,belongs to the class of so-called ‘‘moment estimators’’ (KENDALL and STUART,1967).

One of the simplest ways to assess the properties of the above estimator is to usethe Taylor Series expansion of an inverse of CDF F−1

M (Yn � mmax). The expansion ofthe function F−1

M (Yn � mmax) in a Taylor Series about the point Yn=1, yields

Mn=F−1M (1 � mmax)−

dF−1M (Yn, mmax)

dYn

)Yn=1

(1−Yn )+ · · · . (17)

Since F−1M (1 � mmax)=mmax and E(1−Yn )=1−n/(n+1)=1/(n+1),

dF−1M (Yn, mmax)

dYn

)Y max=1

=1

dFM (m, mmax)dm

)m max

=1

fM (mmax � mmax). (18)

Andrzej Kijko and Gerhard Graham424 Pure appl. geophys.,

For n large, E(1−Yn )$1/n, and fM (mmax � mmax)$ fM (mobsmax � mobs

max). After replac-ing 1−Yn by its expected value, viz. 1/n, the estimator (16) takes the simple form

m/ max=mobsmax+

1nfM (mobs

max � mobsmax)

. (19)

The end-pont estimator (19) was probably first derived by TATE (1959). It was usedby PISARENKO et al. (1996), who quoted it without deriving it, after KENDALL andSTUART (1967), and applied it for estimating maximum regional magnitude mmax.For this formula we have given the simple derivation above, since Tate’s originalderivation is very complex, and understanding it requires an advanced backgroundin theoretical statistics.

It is easy to extend this approach and assess approximate variance of theestimator (19). By applying the relation between the derivative of a continuous,strictly monotonic function and its inverse function (e.g., from APOSTOL, 1961), theapproximate variance of m/ max can be written as

Var(m/ max)=�dFM (mobs

max � mmax)dmmax

n−2)m max=m/ max

Var(y/ n ). (20)

Following (13) and (14) we obtain

E(Y2n)=

& 1

j=0

j2fYn(j) dj=n

& 1

j=0

jn+1 dj=n

n+2, (21)

and

Var(Yn )=E(Y2n)− [E(Yn )]2=

n(n+2)(n+1)2. (22)

We also obtain

dFM (mobsmax � mmax)

dmmax

=−FM (mobsmax � mmax)fM (mmax � mmax) (23)

and after replacing [FM (mobsmax � m/ max)]2 by its expected value E(Y2

n), for large n, weobtain

Var(m/ max)=1

n2f 2M(mobs

max � mobsmax)

. (24)

The above equation describes the variance of the m/ max, estimated according to theformula (19).

5.2. Uncertainty in the Determination of mmax: What Contributes to the Uncertaintyand How?

Formula (24) quantifies the uncertainty of maximum magnitude determination,an uncertainty which has its source in the randomness of the earthquake generation

Parametric-historic Hazard Analysis 425Vol. 152, 1998

process. This variability, known as aleatory variability (PANEL OF SEISMIC HAZ-

ARD . . . , 1997; TORO et al., 1997) is inherent in natural processes and must beclearly distinguished from another type of uncertainty, which has its source in theapplication of the wrong mathematical model of the process [for example aninadequate CDF FM (m)], or wrong values of the model parameters (as e.g.,Gutenberg-Richter parameter b). Often such an uncertainty is known as epistemicvariability (PANEL OF SEISMIC HAZARD . . . , 1997; TORO et al., 1997).

It is important to distinguish between these two types of uncertainties, sinceoften they require entirely different treatments. In this section we will take intoaccount two points which contribute to uncertainty in the estimation of theparameter mmax:

(i) The number of observed earthquakes n, within the time span of thecatalogue T, is a random number. Hence, uncertainty in the number of theearthquake occurrences belongs to the class of aleatory variability, since therandom nature of the number of earthquake occurrences is inherent in theearthquake generation process.

(ii) The observed (apparent) earthquake magnitude m is a true magnitudedistorted by a random observation of magnitude error, o. The concept of apparentmagnitude was introduced by TINTI and MULARGIA (1985a) and the effect ofuncertainty of magnitude on the assessment of seismic hazard has been studiedextensively and is well understood. In practice, two distributions for the error o areconsidered: normal (e.g., TINTI and MULARGIA, 1985a; BENDER, 1987; KIJKO andSELLEVOLL, 1992), and uniform (TINTI and MULARGIA, 1985b). Sometimes themagnitude uncertainties are expressed in terms of intervals, such as those that arisefrom rounded data (KIJKO, 1988). An alternative treatment of the problem wasgiven by RHOADES (1995), who provides both for normally distributed magnitudeobservation errors and for errors arising from rounding-off. Following the classifi-cation we have adopted, this uncertainty in earthquake magnitude determination isto be classified under epistemic variability.The approximate contributions of both of the above uncertainties to the variance ofmmax estimation can easily be taken into account by applying the law of propaga-tion of errors.

Let us assume that n obeys the Poisson distribution having parameter l, themean activity rate. Then, after replacement of n by lT, TATE’s (1959) estimator(19), is

m/ max=mobsmax+

1lTfM (mobs

max � mobsmax)

. (25)

If we consider that, by the definition of the Poisson process, the variance of n isequal to lT, the contribution of the randomness in the number of earthquakeoccurrences to the variance of m/ max is approximately equal to (lT)−3 f2

M ×

Andrzej Kijko and Gerhard Graham426 Pure appl. geophys.,

(mobsmax � mobs

max). Hence, formula (24) should be extended to the form

Var(m/ max)$1

(lT)2f2M (mobs

max � mobsmax)

+1

(lT)3f2M (mobs

max � mobsmax)

. (26)

Clearly, the second term in equation (26) is responsible for the uncertainty in thenumber of earthquake occurrences and is lT times less than the first one. Therefore,for an area with high mean activity rate l, and for a long period of observations (Tlarge), its contribution may be neglected.

Finally, let us assess the effect of uncertainty in earthquake magnitude determi-nation. Relation (17), which was employed as a base and starting point for derivingthe mmax estimator, becomes

mobsmax=F1(1 � m/ max)−

dF1(yn � m/ max)dyn

)yn=1

(1−yn )+ · · ·+o, (27)

where o is a random error in the determination of the largest observed magnitudemobs

max. Assuming that the variance of o is known and equal to Var(o)=s2M, then

the approximate variance of m/ max estimated according to Procedure I is

Var (m/ max)$s2M+

1f 2

M(mobsmax � mobs

max)�lT+1

(lT)3

n. (28)

The contribution of the term s2M to the total variance of m/ max can be significant,

especially when magnitude mobsmax is recovered from historical records. In such cases

the uncertainty in the determination of mobsmax can reach a value of half a unit of

earthquake magnitude (sM=0.5) or even higher.It should be noted that our approach has several limitations. One of these is the

assumption that parameters of the magnitude CDF FM (m � mmax) are knownwithout error. In the following section, we consider the case in which in addition tothe above uncertainties, parameter uncertainties of CDF of earthquake magnitudeare considered. A similar line was taken by PISARENKO et al. (1996). The maindifference between Pisarenko’s approach and ours is that the former requiresnumerical integration, while ours provides an analytical or semi-analytical solution.

5.3. Account of Parameter Uncertainties in the Distribution Function ofEarthquake Magnitude

The treatment of uncertainties in our model can be effected in different ways.The most efficient is probably the Bayesian approach, which takes into accountuncertainty in the parameters by regarding them as random variables (e.g., DE-

GROOT, 1970).In order to consider the most general case, we have applied the formalism of

compound distributions. Compound distributions arise when parameters of the

Parametric-historic Hazard Analysis 427Vol. 152, 1998

distribution of a random variable are themselves treated as random variables. Letvector p denote the parameters of the model which are known with some degree ofuncertainty. One parameter, for example, could be the b value in the Gutenberg-Richter-based CDF (7). In general, if a random variable M has a CDF FM (m)which depends upon continuous random parameters P [which might be written asFM (m ; p)3], then the CDF

FM (m)=&

p

FM (m ; p)fP (p) dp, (29)

where fP (p) is a PDF of parameters P, is known as a compound (Bayesian)distribution of a random variable M. The compound CDF is therefore the weightedaverage of the CDF of M for each value of P.

An application of the above formalism to the uncertainty of the Gutenberg-Richter parameter b is shown in Section 7.

6. Procedure II (‘‘Ad6anced ’’)

This procedure is based on an equation that compares the largest observedmagnitude mobs

max, and the maximum expected magnitude E(Mn � T) during a spe-cified time interval T.

It is shown that the procedure provides an estimator of the upper end of thedistribution, which in terms of mean-squared error is substantially better than therespective TATE (1959) estimator given by Procedure I. The drawback of theprocedure is that it requires integration, which, for some distribution functions, canbe performed only numerically. Fortunately, for the CDF of Gutenberg-Richter,analytical formulae are available.

6.1. Deri6ation of Estimator II

Let us accept the assumption previously made that the earthquake magnitudesM1, M2, . . . , Mn occurring within a specified time interval T, are independent,random variables, each with CDF FM (m � mmax), where m belongs to the magnitudeinterval [mmin, mmax]. As before, let us assume that the magnitudes are ordered inascending order, i.e., M15M25 · · ·5Mn.

Following the same technique as was used in the derivation of the CDF of thelargest of the variables Y1, . . . , Yn, the largest magnitude Mn has a CDF

3 It must be noted that since the mmax is not included in the distribution parameters P, it would bemore correct to introduce the notation FM (m ; p � mmax). Please note once again that where feasible, byusing capital letter and the same letter in lower case, we distinguish between a random variable and thevalues which it may assume.

Andrzej Kijko and Gerhard Graham428 Pure appl. geophys.,

FMn(m � mmax)=

>0,[FM (m � mmax)]n,1,

for mBmmin,for mmin5m5mmax,for m\mmax.

(30)

After integrating by parts, the expected value of Mn, E(Mn ), is

E(Mn )=& mmax

m min

m dFMn(m � mmax)=mmax−

& mmax

m min

FMn(m � mmax) dm. (31)

Hence

mmax=E(Mn )+& mmax

m min

[FM (m � mmax)]n dm. (32)

This expression, after replacement of the expected value of the largest observedmagnitude E(Mn ) by the largest magnitude observed, mobs

max, suggests an estimatorof mmax of the form:

m/ max=mobsmax+

& mobsmax

m min

[FM (m � mobsmax)]n dm. (33)

This equation follows from the condition E(Mn )=mobsmax, and so, again, the value of

mmax thus obtained belongs to the class of moment estimators mentioned above(KENDALL and STUART, 1967). This estimator is valid for each CDF FM (m � mmax),and does not require fulfillment of the truncation condition (8)–(10). It is alsoimportant to note that since the value of the integral mobs

maxm min

[FM (m � mobsmax)]n dm is

never negative, it provides a value of mmax which is never less than the largestmagnitude mobs

max observed. The drawback of the formula is that it requires integra-tion. For some of the magnitude distribution functions, the analytical expressionfor the integral does not exist, or even if it does exist, it requires awkwardcalculations.

COOKE (1979) was probably the first to obtain estimator (33) as above. Analternative and independent derivation was given by KIJKO (1983), who applied theformalism of moment generating functions.

Further modifications of estimator (33) are straightforward. For example,following the assumption that the number of earthquakes occurring in unit timewithin a specified area obeys the Poisson distribution with parameter l, Cooke’sestimator (33) becomes

m/ max=mobsmax+

& mobsmax

m min

[FM (m � mobsmax)]lT dm. (34)

Again, different approaches can be used in the estimation of higher moments (inparticular, the variance) of estimator (34). It is clear that for catalogues that arelong enough, the main contribution to the uncertainty in the estimation ofparameter m/ max comes from the uncertainty of the largest observed magnitudemobs

max. As in Procedure I, this uncertainty has two components: aleatory and

Parametric-historic Hazard Analysis 429Vol. 152, 1998

epistemic. Elementary computations show that the approximation of aleatoryuncertainty is of the order of the value of the integral in formula (34), that is,

Var(m/ max)$!& mobs

max

m min

[FM (m � mobsmax)]lT dm

"2

. (35)

The effect of randomness in the number of earthquake occurrences and the effectof random error, o, in the determination of the largest observed magnitude mobs

max

can be calculated through the same technique as in Procedure I. If we assume thatthe variance of o is known, and is equal to s2

M, then the approximate variance ofestimator of m/ max (34) is given by

Var(m/ max)$s2M+

!& mobsmax

m min

[FM (m � mobsmax)]lT dm

"2

×lT!& mobs

max

m min

ln[FM (m � mobsmax)][FM (m � mobs

max)lT dm"2

. (36)

For an area with a high mean activity rate l and for a long period of observation,the third term in equation (36), which is responsible for uncertainty in the numberof earthquake occurrences, is significantly less than the first two, and its contribu-tion may be neglected.

7. Application to the Gutenberg-Richter Magnitude Distribution

In this section we will demonstrate how to apply the above formalism to one ofthe most often used frequency-magnitude relationships—the one known as theGutenberg-Richter magnitude distribution.

7.1. Application of the Estimator I to the Gutenberg-Richter MagnitudeDistribution

By the original definition of this procedure, the estimation of mmax is obtainedas a root of equation (16) where the Gutenberg-Richter CDF of earthquakemagnitude FM (m � mmax) is defined by equation (7). Hence

1−b exp[−b(m−mmin)]1−b exp[−b(m/ max−mmin)]

=n

n+1, (37)

from which the required estimator of mmax is obtained as

m/ max=−1b

ln!

exp(−bmmin)− [exp(−bmmin)−exp(−bmobsmax)]

n+1n

". (38)

Andrzej Kijko and Gerhard Graham430 Pure appl. geophys.,

Here, mobsmax is the largest observed magnitude in the catalogue, for the time span T,

and n is the number of earthquakes occurring within T, with magnitude equal to orexceeding the level of completeness, mmin. The above estimator was first used byGIBOWICZ and KIJKO (1994) for the assessment of the magnitude of the maximumpossible seismic events in the Klerksdorp gold mining district in South Africa.

Such a straightforward solution as given by formula (38) is unfortunately notalways possible—the RHS is not defined if the argument in brackets resultsnegative. It is therefore preferable to use TATE’s (1959) formula (19), which, for theGutenberg-Richter PDF takes the simple form

m/ max=mobsmax+

1n

1−exp[−b(mobsmax−mmin)]

b exp[−b(mobsmax−mmin)]

. (39)

Equation (39) describes TATE’s (1959) estimator, as applied to the Gutenberg-Richter magnitude distribution. If the number of earthquakes n that has occurredis not known, although the mean activity rate l of earthquake occurrence isavailable, equation (39) can be used after replacement of n by lT. Equation (39) isin good agreement with our intuitive expectations: for given values of b and mmin,the larger n is, or the longer the period of observation T, the less the estimatedmaximum regional magnitude m/ max deviates from the largest observed magnitudemobs

max. The estimation of maximum regional magnitude mmax by application ofTATE’s (1959) formula (19) was first used by PISARENKO et al. (1996). We shalldenote the estimator of equation (39) as the Tate-Pisarenko estimator of mmax, orin short as T-P.

Following (28), the approximate variance of the T-P estimator of mmax for theGutenberg-Richter CDF of earthquake magnitude is given by

Var(m/ max)$s2M+

n+1n3

{1−exp[−b(mobsmax−mmin)]}2

b2 exp[−2b(mobsmax−mmin)]

. (40)

Again, if the number of earthquakes n is not known, equations (39) and (40) can beused after replacement of n by lT.

7.2. Application of Estimator II to the Gutenberg-Richter Magnitude Distribution

Following (33), estimator II requires calculation of the integralmobs

maxm min

[FM (m � mobsmax]n dm, where for the Gutenberg-Richter magnitude distribution,

the CDF FM (m � mmax) is described by equation (7). Fortunately, for n large, aftersubstitution of n by lT, the expression [FM (m � mmax)]n can be replaced by itsCramer’s approximation (CRAMER, 1948) as:

[FM (m � mmax)]n$exp!−lT

�exp[−b(m−mmin)]−exp[−b(mmax−mmin)]1−exp[−b(mmax−mmin)]

n", (41)

and the integral in equation (33) (KIJKO and SELLEVOLL, 1989) is equal to

Parametric-historic Hazard Analysis 431Vol. 152, 1998

& mobsmax

m min

[FM (m � mobsmax]n dm=

E1(Tz2)−E1(Tz1)b exp(−Tz2)

+mmin exp(−lT), (42)

where z1=−lA1/(A2−A1), z2=−lA2/(A2−A1), A1=exp(−bmmin), A2=exp(−bmobs

max), and E1( · ) denotes an exponential integral function (ABRAMOWITZ

and STEGUN, 1970)

E1(z)=&�

z

exp(−z)�z dz. (43)

The function E1(z) can be conveniently approximated as

E1(z)=1z

exp(−z)z2+a1z+a2

z2+b1z+b2

, (44)

where a1=2.334733, a2=0.250621, b1=3.330657, and b2=1.681534. Formula (44)is an approximation of the exponential integral function with a maximum error of5 · 10−5 over the interval 15z5�.

Hence, following a general solution (equation 33) for the Gutenberg-Richter-based magnitude CDF, the estimator of mmax is

m/ max=mobsmax+

E1(Tz2)−E1(Tz1)b exp(−Tz2)

+mmin exp(−lT). (45)

The above estimator of mmax for the doubly truncated Gutenberg-Richter relationwas first obtained by KIJKO (1983), who was inspired by discussions with M. A.Sellevoll of Bergen University. Equation (45) has subsequently been used forestimation of the maximum regional earthquake magnitude in more than 30 seismicareas of the world. We shall denote equation (45) as the Kijko-Sellevoll estimatoror, in short, K-S.

From equations (36) and (42) the approximate variance of the maximumregional magnitude m/ max, estimated according to the K-S procedure, is

Var(m/ max)=s2M+

�E1(Tz2)−E1(Tz1)b exp(−Tz2)

+mmin exp(−lT)n2

. (46)

It should be noted that this formula neglects the third term of equation (36), whichis responsible for the uncertainty in the occurrence of a number of earthquakes.Therefore it can be applied only for an area with high seismicity (viz. large activityrate l) and/or for a long period of observation.

7.3. Treatment of Uncertainty in the b Value of Gutenberg-Richter

Let us derive the Gutenberg-Richter-based compound CDF of earthquakemagnitude. In order to do this we must specify the form of the PDF fP (p) (equation29), which characterizes the uncertainty of the Gutenberg-Richter parameter b. Oneof the best candidates for such a choice is the gamma distribution, since it is flexible

Andrzej Kijko and Gerhard Graham432 Pure appl. geophys.,

and can fit a large variety of shapes (e.g., DEGROOT, 1970). Following (29) and byassuming that the variation of the b value in the Gutenberg-Richter-based CDF (7)may be represented by a gamma CDF having parameters p and q, the compoundCDF of earthquake magnitudes (CAMPBELL, 1982) takes the form:

FM (m � mmax)=ÍÃ

Ã

Á

Ä

0Cb

�1−

� pp+m−mmin

�qn,

1,

for mBmmin,

for mmin5m5mmax,for m\mmax.

(47)

where Cb is a normalizing coefficient. It is not difficult to show that p and q can beexpressed through the mean and variance of the b value, where p=b( /(sb )2 andq= (b( /sb )2. The symbol b( denotes the mean value of the parameter b, sb is theknown standard deviation of b and describes its uncertainty, and Cb is equal to

Cb=�

1−� p

p+mmax−mmin

�qn−1

. (48)

Equation (48) is known as the Bayesian exponential-gamma CDF of earthquakemagnitude.

This way of dealing with the uncertainty of parameter b is far from unique. Forexample, for the same purpose MORTGAT and SHAH (1979) used a combination ofthe Bernoulli and the beta distributions. DONG et al. (1984) as well as STAVRAKA-

SIS and TSELENTIS (1987) used a combination of uniform and multinomial distribu-tions. An excellent study on how to handle varied uncertainties which are presentin the parameters, the model, and the data, can be found in the paper by RHOADES

et al. (1994).Knowledge of equation (47) makes it possible to construct the Bayesian version

of estimators T-P and K-S. Following (47), the PDF of Bayesian exponential-gamma distribution is equal to

fM (m � mmax)=

>bCb

� pp+m−mmin

�q+1

,

0,

for mmin5m5mmax

for mBmmin, m\mmax

. (49)

Thus, following (25) and (28), the Bayesian extension of the T-P estimator and itsapproximate variance become

m/ max=mobsmax+

1nCbpq

� pp+mobs

max−mmin

�−(q+1)

(50)

and

Var(m/ max)$s2M+

1(Cbpq)2

� pp+mobs

max−mmin

�−2(q+1)�n+1n3

n. (51)

Parametric-historic Hazard Analysis 433Vol. 152, 1998

The Bayesian version of the T-P estimator (equation 50) will be denoted as T-P-B.Unfortunately, the derivation of the Bayesian version of the K-S estimator is

not so straightforward. By definition (equation 33), it requires the calculation of theintegral

D=& mobs

max

m min

[FM (m � mobsmax)]n= (Cb )n & mobs

max

m min

�1−

� pp+m−mmin

�qnn

dm, (52)

which does not have a simple solution. It can be shown, under the condition thatn is a positive integer, that the integral (52) can be expressed simply as

D=(Cb )n

b

�−q ln(r)+ %

n

i=1

(−1)i

i�n

i�

(1−r i · q)n

, (53a)

where r=p/(p+mobsmax−mmin).

Alternatively, an approximate solution can be obtained through an applicationof Cramer’s procedure—used in the derivation of the asymptotic extreme distribu-tions. Following CRAMER (1948), for large n the value of [FM (m � mobs

max)]n isapproximately equal to exp{−n [1−FM (m � mobs

max)]}, and therefore integral (52)takes the form

D=c1& mobs

max

m min

exp�−nCb

� pp+m−mmin

�qndm,

where c1=exp[−n(1−Cb )]. Further simple calculations lead to an approximatesolution (which is the sum of an infinite number of terms):

D=pc1

q%�

i=0

(−1)i d i

i !(i−1/q)![1−r (q · i−1)], (53b)

or equivalently,

D=d1/q+2 exp[nrq/(1−rq)

b[G(−1/q, drq)−G(−1/q, d)], (53c)

where d=nCb and G( · , · ) is the incomplete gamma function. This leads to theBayesian version of the K-S estimator

m/ max=mobsmax+D, (54)

and its variance

Var(m/ max)$s2M+D2. (55)

We will denote the Bayesian version of the K-S estimator (equations 53a,b,c and54) as K-S-B.

From the above relations it follows that the assessment of the maximumregional magnitude mmax requires knowledge of the area-specific mean seismicactivity rate l and the Gutenberg-Richter parameter b. The maximum likelihoodprocedure for the assessment of these two parameters is presented in Part II.

Andrzej Kijko and Gerhard Graham434 Pure appl. geophys.,

8. Some Comparisons of Estimators I and II

One way to compare the performance of estimators based on Procedures I andII is to use an empirical approach in a Monte-Carlo simulation study.

We chose a value of mmax=6.5, b=2, (or equivalently b$0.87), and mmin=3.0,and then simulated earthquake magnitudes from a Gutenberg-Richter-based CDF(7). We then compared the two estimated magnitudes m/ (I)

max and m/ (II)max, computed

according to Procedure I and Procedure II respectively, with the true value ofmmax=6.5, to ascertain which estimator performed better. This operation wasrepeated 1000 times in order to discern a general pattern with respect to the relativeperformance of the estimators.

Two properties of the estimators were studied: bias and mean error. It should berealized that when an estimator is biased (i.e., it has a systematic error), it is oftenpossible to find some simple correction that removes the bias altogether. Even whenthis cannot be done, a biased estimate can be used, provided only that we are ableto demonstrate that the amount of the bias is small, which is often the case in largesamples.

On the other hand, even when an estimator is unbiased it is of little use if wedo not know the extent of its dispersion about the true value. A natural measure ofdispersion about the given point is provided by the second moment. Accordingly,the efficiency of our two estimators for the maximum regional magnitude mmax, viz.m/ (I)

max and m/ (II)max, can be compared by means of the corresponding mean errors

E(m/ (I)max−mmax)2 and E(m/ (II)

max−mmax)2. Clearly the estimate with the smallermean error is the more efficient one.

Figure 1 illustrates the performance of the two ‘‘standard’’ estimators, viz. T-Pand K-S, in the case of significant uncertainty in the Gutenberg-Richter parameterb. During the data generation procedure, the b value was subjected to a random,normally distributed error with mean equal to zero and standard deviation, sb, equalto 25% of the b value. We have repeated our simulations 1000 times for differentnumbers of earthquakes, ranging from 50 to 250.

Figure 1a displays the average of 1000 solutions computed by estimators T-P andK-S, respectively. The K-S solutions are significantly closer to the ‘‘true’’ value ofmmin=6.5 than the corresponding solutions provided by estimator T-P.

Figure 1b illustrates respective mean errors. The figure shows that in terms ofmean error, the estimator K-S is more efficient than T-P. The superiority of estimatorK-S is seen especially clearly when the number of earthquakes is small (say 50–100);in this case the mean error in the estimation of mmax by T-P is several times largerthan for estimator K-S. The large errors (up to 2 units of earthquake magnitude),practically disqualify the use of the T-P estimator when there is significant uncertaintyin the b value and when the number of observations is small.

The next three figures show the performance of estimators T-P and K-S and theirBayesian counterparts (viz. T-P-B and K-S-B) in the presence of uncertainties in theGutenberg-Richter parameter b.

Parametric-historic Hazard Analysis 435Vol. 152, 1998

Figures 2 and 4 show the application of these estimators in two situations: whenthe presence of uncertainty in the Gutenberg-Richter parameter b is taken intoaccount (mmax is estimated through the respective Bayesian distributions), and whenthe uncertainty in the b value is ignored.

Figure 1Comparison of performance of estimators Tate-Pisarenko (T-P) and Kijko-Sellevoll (K-S) based on 1000synthetic catalogues where the ‘‘true’’ value of mmax=6.5 and the b value was subjected to a random,normally distributed error with mean equal to zero and standard deviation equal to 25% of b value. Thesuperiority of estimator K-S is clearly seen, especially for a small number of earthquakes. (a) Mean value

of mmax estimation. (b) Mean error of mmax estimation.

Andrzej Kijko and Gerhard Graham436 Pure appl. geophys.,

Figure 2Comparison of performance of estimator Tate-Pisarenko (T-P) and its Bayesian counterpart (viz. T-P-B)based on 1000 synthetic catalogues where the ‘‘true’’ value of mmax=6.5 and the b value was subjectedto a random, normally distributed error with mean equal to zero and standard deviation equal to 25%of b value. Application of Bayesian estimator T-P-B significantly reduces the bias as well as mean error.

(a) Mean value of mmax estimation. (b) Mean error of mmax estimation.

Figure 2a illustrates the results of the estimation of mmax by 2 estimators: T-Pand T-P-B. This figure illustrates that an application of the ‘‘standard’’ procedure,(viz. estimator T-P), which does not provide for the presence of errors in the bvalue, leads to significantly biased (overestimated) values of mmax. Figure 2b shows

Parametric-historic Hazard Analysis 437Vol. 152, 1998

the mean errors. Application of the respective Bayesian procedure significantlyreduces the bias as well as the mean error.

Figure 3 is analogous to Figure 2, however here the estimators of Procedure Ihave been replaced by the respective estimators of Procedure II, viz. K-S, andK-S-B. Figure 4 shows the performance of estimators T-P-B and K-S-B.

Figure 3Comparison of performance of estimator Kijko-Sellevoll (K-S) and its Bayesian counterpart (viz. K-S-B)based on 1000 synthetic catalogues where the ‘‘true’’ value of mmax=6.5 and the b value was subjectedto a random, normally distributed error with mean equal to zero and standard deviation equal to 25%of b value. Application of Bayesian estimator K-S-B significantly reduces the bias as well as mean error.

(a) Mean value of mmax estimation. (b) Mean errors of mmax estimation.

Andrzej Kijko and Gerhard Graham438 Pure appl. geophys.,

Figure 4Comparison of performance of estimators Tate-Pisarenko-Bayes (T-P-B) and Kijko-Sellevoll-Bayes(K-S-B) based on 1000 synthetic catalogues where the ‘‘true’’ value of mmax=6.5 and the b value wassubjected to a random, normally distributed error with mean equal to zero and standard deviation equalto 25% of b value. The superiority of estimator K-S-B is clearly seen, especially for a small number of

earthquakes. (a) Mean value of mmax estimation. (b) Mean error of mmax estimation.

The above numerical experiments have demonstrated that both the Bayesianestimators, particularly K-S-B, tend to perform well in the presence of significantuncertainty in the b value.

Parametric-historic Hazard Analysis 439Vol. 152, 1998

9. Remarks and Conclusions

This study is dedicated to the problem of the determination of the maximumregional earthquake magnitude mmax.

Two types of statistical procedures were developed and analyzed. Broadlyspeaking, the first type of procedure is more ‘‘straightforward,’’ while the secondone is more ‘‘advanced’’ and requires more complex calculations. It is assumed thatfor each of the procedures both the analytical form and the parameters of thedistribution function of earthquake magnitude are known.

Special formulae have been derived which take into account uncertainties in theparameters of the distribution function of earthquake magnitude.

The manner of applying derived general formulae to the Gutenberg-Richterfrequency-magnitude relationship is demonstrated.

Some comparison between all the procedures has been performed on theMonte-Carlo generated data sets, having a pre-set maximum regional magnitudemmax. Numerical tests show that in terms of bias and mean errors, the ‘‘advanced’’estimators (viz. K-S and K-S-B) are more efficient than the respective ‘‘straightfor-ward’’ estimators.

Acknowledgments

We are grateful to K. Aki for his careful reading of the draft version of themanuscript, critical review, suggestions, and very helpful comments. We also thankour colleague C. Randall for his kind help.

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(Received March 2, 1998, accepted April 21, 1998).