''Parametric-historic'' Procedure for Probabilistic Seismic Hazard ...

Pure appl. geophys. 154 (1999) 1–22 0033–4553/99/010001–22 $ 1.50+0.20/0

‘‘Parametric-historic’’ Procedure for Probabilistic Seismic Hazard Analysis Part II: Assessment of Seismic Hazard at Specified Site ANDRZEJ KIJKO1 and GERHARD GRAHAM1

Abstract—A new methodology for probabilistic seismic hazard analysis is described. The approach combines the best features of the ‘‘deductive’’ (CORNELL, 1968) and ‘‘historic’’ (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 the authors’ work (KIJKO and GRAHAM, 1998) was dedicated to developing efficient statistical procedures that can be used for the evaluation of this parameter. In Part II the approach of a probabilistic seismic hazard assessment at a given site is described. The approach permits the utilization of incomplete earthquake catalogues. It is assumed that a typical catalogue contains two types of information: historical macroseismic events that occurred over a period of a few hundred years and recent, instrumental data. The historical part of the catalogue contains only the strongest events, whereas the complete part can be divided into several subcatalogues, each assumed complete above a specified threshold of magnitude. The author’s approach also takes into account uncertainty in the determination of the earthquake magnitude. The technique has been developed specifically for the estimation of seismic hazard at individual sites, without the subjective judgment involved in the definition of seismic source zones, in which specific active faults have not been mapped and identified, and where the causes of seismicity are not well understood. As an example of the application of the new technique, the results of a typical hazard analysis for a hypothetical engineering structure located in the territory of South Africa are presented. It was assumed that the only reliable information in the assessment of the seismic hazard parameters in the vicinity of the selected site comes from a knowledge of past seismicity. The procedure was applied to seismic data that were divided into an incomplete part, containing only the largest events, and two complete parts, containing information obtained from instruments. The simulation experiments described in Part I of our study have shown that the Bayesian estimator K-S-B tends to perform very well, especially in the presence of inevitable deviations from the simple Gutenberg–Richter model. In the light of this fact value m/ max =6.66 90.44, which was obtained from the K-S-B technique, was regarded as the best choice. At an exceedance probability of 10−3 per annum, the median value of peak ground acceleration on rock at the site is 0.31 g, and at an exceedance probability of 10−4 per annum, the median peak ground acceleration at the site is 0.39 g. The median value of the maximum possible acceleration at the site is 0.40 g, which was calculated from attenuation formulae by assuming the occurrence of the strongest possible earthquake, e.g., with magnitude m/ max = 6.66 at distance 10 km. Key words: Seismic hazard, incomplete catalogues, no seismic source zones.

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 8411424, E-mail: [email protected] [email protected]

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Andrzej Kijko and Gerhard Graham

Pure appl. geophys.,

1. Introduction In the procedure presented here, cognizance is taken of two potential difficulties which one faces in the process of probabilistic seismic hazard analysis (PSHA). They are: (i) the incompleteness of seismic catalogues, and (ii) the requirement of specifying seismic source zones. The available catalogues usually contain two types of information: macroseismic observations of the largest seismic events that occurred over a period of a few hundred years, and complete instrumental data for relatively short periods of time. The complete part can often be divided into several subcatalogues, each one complete above a given threshold of magnitude (Fig. 1). The second problem arises from the fact that the technique most often used for PSHA — a procedure developed by Cornell, requires specification of seismic source zones (CORNELL, 1968). Unfortunately, tectonic provinces or specific active faults have not often been identified and mapped, and the causes of seismicity are not well understood. Frequently different seismogenic zone specifications lead to significantly different assessments of hazard. In addition, the Cornell-based seismic

Figure 1 Illustration of data which can be used to obtain basic seismic hazard parameters for the area in the vicinity of the selected site by the procedure used. The approach permits the combination of the largest earthquake data and complete data which have variable threshold magnitudes. It allows the use of the largest known historical earthquake (m obs max ) which occurred before the catalogue began. It also accepts ‘‘gaps’’ (Tg ) when records were missing or the seismic networks were out of operation. Uncertainty in earthquake magnitude is also taken into account in that an assumption is made that the observed magnitude is the true magnitude subjected to a random error which follows a Gaussian distribution which has zero mean and a known standard deviation sM . (After KIJKO and SELLEVOLL, 1991)

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‘‘Parametric-historic’’ Hazard Analysis

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hazard assessment procedure requires, for each zone, a knowledge of the model parameters (in the simplest case the Gutenberg–Richter parameter b, the level of completeness of seismic data mmin , the mean seismic activity rate l, and the upper limit of earthquake magnitude mmax ), which cannot be determined reliably for areas that are small or have a very incomplete seismic history. The problem of insufficient data for small areas is often overcome by selecting larger seismic source zones, but such a procedure is in conflict with the whole concept of selection of independent seismogenic provinces. In addition, a purely formal enlargement of the potential seismic source zones results in underestimation of the seismic hazard (XU and GAO, 1997). Therefore, a procedure that accepts the varying quality of different parts of the catalogue and does not require specification of seismic source zones would be an ideal tool for analyzing and assessing seismic hazard. Bearing in mind the above requirements, the authors propose a procedure which permits incompleteness of earthquake catalogues and is free from the subjective judgment involved in the determination of seismic source zones. The new approach combines the best of the deductive (CORNELL, 1968) and non-parametric historic (VENEZIANO et al., 1984) procedures and, in many cases, is free from the basic disadvantages characteristic of each of them. (For a detailed analysis of the deductive (CORNELL, 1968) and non-parametric historic (VENEZIANO et al., 1984) procedures the reader may consult Part I of our study). Since the new proposed approach is parametric, following the scheme of MCGUIRE (1993), it could be classified as a parametric-historic procedure. Application of the new procedure consists essentially of two steps. The first step is applicable to the area in the vicinity of the site for which knowledge of the seismic hazard is required. In this respect the procedure is similar to Cornell’s deductive approach and requires an estimation of area-specific parameters. The parameters depend on the PSHA model selected; in this case they are the area-specific mean seismic activity rate l, the Gutenberg–Richter parameter b and the maximum regional magnitude mmax . The approach is open to any alternative parameterization. The second step is applicable to a specified site, and consists of an assessment of the parameters of amplitude distribution of the selected ground motion parameter. Since in each step parameters are estimated through the maximum likelihood procedure, by applying the Bayesian formalism, any additional geological or geophysical information (as well as various uncertainties) can easily be incorporated. The new procedure is consequently capable of giving a realistic assessment of seismic hazard in areas of both low and high seismicity, including cases where the catalogues are incomplete. In the present form, the procedure allows assessment of the PSHA in terms of peak ground acceleration, peak ground velocity or peak ground displacement. An extension of the procedure and assessment of the entire spectrum of ground motion is straightforward.

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If the procedure is applied to all grid points of, and around, a region, then a map of seismic hazard for the entire region can be obtained. To the best of the authors’ knowledge, the approach closest in conception to theirs is that of FRANKEL (1995), in which seismic hazard is mapped in the central and eastern United States. The Frankel approach has been adopted by LAPAJNE et al. (1997) in modeling seismic hazard in Slovenia. We also present results of a typical PSHA for a site of a hypothetical engineering structure (HES).

2. Assessment of Area-specific Parameters for the Case of Incomplete Data Sets Since the technique applied for the assessment of area-specific seismic hazard parameters is similar to the procedure described by KIJKO and SELLEVOLL (1989, 1992), only the main points of the procedure are presented, namely, those that are required for understanding the next step of this approach, which is estimating seismic hazard parameters that are characteristic of a specific site. In the sequel, all the formulae are derived for the model most often used in engineering seismology, viz. the one assuming the doubly truncated Gutenberg– Richter frequency-magnitude relationship and a Poisson distribution of earthquake occurrence. The procedure is also open to any alternative parameterization.

2.1. Extreme Magnitude Distribution as Applied to the Largest Earthquake Data Let us assume that in the vicinity of the specified site of the HES (i) the occurrence of the main seismic events in time can be described by a Poissonian process with an area-specific mean activity rate l and (ii) earthquake magnitudes m are distributed according to the doubly truncated Gutenberg–Richter-based cumulative distribution function (CDF) FM (m mmin , mmax ). It should be noted that, henceforth, notations like fM (m mmax ), and FM (m mmax ), (Part I, Equations (6) and (7)), will be replaced by fM (m mmin , mmax ) and FM (m mmin , mmax ), respectively. Hence, in all distribution functions, both the lower and the upper limits of the magnitude range will be shown. The CDF of the largest magnitudes occurring during the time interval t, given by KIJKO and SELLEVOLL (1989, 1992), is F max M (m m0 , mmax , t) =

exp{−l0 t[1− FM (m m0 , mmax )]}−exp(−l0 t) , 1− exp(−l0 t)

(1)

where l0 l(m0 ) = l[1 − FM (m0 mmin , mmax )] is the mean activity rate of earthquake occurrence within the specified area with magnitude m0 and above, m0 is the lower earthquake magnitude in the extreme part of the catalogue, and m0 ] mmin . The parameter l l(mmin ) is the mean activity rate of earthquakes with magnitude

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mmin and above. Magnitude mmin is the minimum threshold magnitude for the entire catalogue. The condition in the choice of its value is that mmin cannot exceed m0 , or any level of completeness of any part of the complete subcatalogue (see Fig. 1). In most cases we are dealing with a high enough activity rate l0 and long enough time intervals t. Hence, the term exp(−l0 t) can be ignored.

2.2. Treatment of Uncertainty in Earthquake Magnitude Determination If the error in the determination of magnitude is assumed to be normally distributed with standard deviation sM (TINTI and MULARGIA, 1985a,b), the probability density function (PDF) and cumulative distribution function CDF of the apparent magnitude become (KIJKO and SELLEVOLL, 1992; GIBOWICZ and KIJKO, 1994) fM (m mmin , mmax , sM )= fM (m mmin , mmax )C(m, sM ),

(2)

FM (m mmin , mmax , sM )= FM (m mmin , mmax )D(m, sM )

(3)

respectively, where

! ex 2

2

C(m, sM ) =

erf

D(m, sM ) = A1 erf

n "

mmax − m

2sM

m − mmin

2sM

+ x + erf

m− mmin

2sM

+ 1 + A2 erf

−2C(m, sM )A(m) 2[A1(m)],

n n

−x

mmax − m

2sM

,

(4)

−1

(5)

sM denotes the standard error of earthquake magnitude determination, A(m)= exp(−bm), A1 =exp(−bmmin ), A2 = exp(−bmmax ), erf(·) is the error function, x= bsM / 2, and the magnitude m is unbounded from both ends. Further application of the distribution functions (2) and (3) requires renormalizations. If mmin is the cut-off value of apparent magnitude, at and above which the earthquakes are complete, then its normalized PDF f0 M (m mmin , mmax , sM ) is zero up to mmin , and is equal to fM (m mmin , mmax , sM )[1 − FM (mmin mmin , mmax , sM )] for m ] mmin . Similarly, the normalized CDF of apparent magnitude is F0 M (m mmax , sM ) =

FM (m mmin , mmax , sM )− FM (mmin mmin , mmax , sM ) . (6) 1− FM (mmin mmin , mmax , sM )

In fact, f0 M (m mmin , mmax , sM ) and F0 M (m mmin , mmax , sM ) are conditional distributions of m given that m ]mmin . Finally, from the assumed model of apparent magnitude it follows that the ‘‘true’’ mean activity rate l(m) must be replaced by its ‘‘apparent’’ counterpart, l0 (m), according to approximate relation (TINTI and MULARGIA, 1985a,b)

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l0 (m)= l(m) exp(x 2).

(7)

2.3. The Likelihood Function of Area-specific Seismicity Parameters From the definition of PDF and from relations (1) and (6) it follows that the PDF of the strongest earthquake within a period t with apparent magnitude m]m0 and standard deviation sM , is f0 max M (m m0 , mmax , tM ) = l0 0 tf0 M (m m0 , mmax , t, sM ) · exp[−l0 0 t[1− F0 M (m m0 , mmax , t, sM )] 1− exp(−l0 0 t) .

(8)

After introducing the PDF (8), one can construct the likelihood function of the strongest earthquake magnitudes from the extreme part of the catalogue. Such a function depends on the unknown area-characteristic parameters (l, b), and becomes n0

L0 (l, b) = const 5 f0 max M (m0j m0 , mmax , t0j , sM0j ).

(9)

j=1

In relation (9), m0j is the apparent magnitude of the strongest earthquake occurring during the time interval tj , sM0j is the value of its standard deviation, j= 1, . . . , n0 , and n0 is the number of earthquakes in the extreme part of the catalogue. The time intervals tj are calculated according to the same formula as described in KIJKO and SELLEVOLL (1989). Const is a normalization factor independent of parameters l and b. It is assumed that the second, complete part of the catalogue can be divided into ns subcatalogues (Fig. 1). Each of them has a time span Ti and is complete, starting (i) from the known magnitude m min . For each subcatalogue i, mij is the apparent (i) magnitude, mij ]m min , and sMij is its standard deviation, j =1, . . . , ni , where ni denotes the number of earthquakes in each complete subcatalogue and i = 1, . . . , ns . If the size of seismic events is independent of their number, the likelihood function of earthquake magnitudes present in each complete subcatalogue i, is equal to Li (l, b)=Li (b) · Li (l),

(10)

which is the product of the function of b, Li (b), and the function of l, Li (l). Following the definition of the likelihood function of a set of independent observations, the function Li (b) is of the following form: ni

(i) Li (b) = const 5 f0 M (mij m min , mmax , sMij ). j=1

(11)

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The assumption that the number of earthquakes per unit time is a Poisson random variable gives a form of Li (l) equal to Li (l) = const(l0 i ti )ni exp(−l0 i ti ),

(12)

where const is a normalizing factor and l0 i is the apparent, mean activity rate for the complete subcatalogue i. The relation between the apparent activity rate and the ‘‘true’’ one is defined by relation (7). For the ith complete subcatalogue the true mean activity rate is equal to (i) (i) ) =l[1−FM (m min mmin , mmax )]. li l(m min

Relations (10) – (12), for i=1, . . . , ns , define the likelihood functions for each complete subcatalogue. Finally, the joint likelihood function of all data in the catalogue, extreme and complete, is given by: ns

L(l, b)= 5 Li (l, b).

(13)

i=0

The maximum likelihood estimates l and b are the values of l. and b. that maximize the likelihood function (13). From a formal point of view, the maximum likelihood estimate of mmax is simply the largest observed earthquake magnitude m obs max . This follows from the fact that the likelihood function (13) decreases monotonically for mmax . Therefore, by including one of the formulae for mmax [the T-P estimator (Part I, equation (39)) or its Bayesian version (Part I, equation (50)) or the K-S estimator (Part I, equation (45)) or its Bayesian version (Part I, equation (54))], and by putting ( ln L(l, b)/(l =0 and ( ln L(l, b)/(b = 0, we obtain a set of equations determining the maximum likelihood estimate of the area-specific parameters l. , b. and m/ max . Such a set of equations is given by KIJKO and SELLEVOLL (1989), and can be solved by an iterative scheme.

3. Assessment of the Seismic Hazard for a Gi6en Site 3.1. Distribution of Peak Ground Accelerations Recorded at the Site To express seismic hazard in terms of peak ground acceleration (PGA) the aim would be to calculate the conditional probability that a single earthquake of random magnitude M at a random distance R will cause a PGA equal to, or greater than, an acceleration of engineering interest amin . For this purpose the assumption (e.g., BOORE and JOYNER, 1982; AMBRASEYS, 1995) is used that in the range of interest the attenuation curve of the PGA a, has the following form: ln(a) = c1 + c2 · M+f(R)+ o,

(14)

where c1 and c2 are empirical constants, M is the earthquake magnitude, and f(R) is a function of earthquake distance R. The term o is a random error which has been

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observed (MCGUIRE, 1978; CAMPBELL, 1981; JOYNER and BOORE, 1981) to have a normal (Gaussian) distribution. Usually f(r) is of the form f(R)= c3 · R+ c4 · ln(R),

(15)

where c3 , and c4 are empirical constants and ln(·) denotes the natural logarithm. It can be seen that an application of a straightforward approach, like CORNELL’s (1968) or the elegant technique developed by BENDER (1984), implies that the probability that a random earthquake magnitude M in the magnitude range mmin 5M 5 mmax will cause a PGA at the site exceeding a, is: Pr[PGA ] a]=

exp(−bmmax )− y · exp[−b(x− c1 )/c2 ] , exp(−bmmax )− exp(−bmmin )

(16)

where x = ln(a), y(·) is a function of the form

&

rmax

exp[b · f(r)/c2 ] · fR (r) dr,

(17)

r min

and mmin is the minimum value of an earthquake magnitude capable of generating a PGA a of engineering interest, i.e., PGA] amin . The value of function y(·) is characteristic for each site. According to formula (17) its value depends on the spatial distribution of seismicity fR (r), on the functional form of PGA vs. distance [function f(r) in formula (15)], and on the value of b=b ln (10). For each site, the value of function y(·) is constant. Finally, from relation (16) it follows that at any given site the logarithm of the PGA is distributed according to the same type of distribution as earthquake magnitude, i.e., negative exponential—the form of the familiar Gutenberg– Richter distribution. The two distributions differ only in the value of their parameters. If the parameter of magnitude distribution is equal to b, the parameter of the distribution of ln(PGA) is equal to b/c2 . Indeed, from (16), after normalization, the CDF of the logarithm of PGA is: FX (x xmin , xmax )=

exp(−gxmin )− exp(−gx) , exp(−gxmin )− exp(−gxmax )

(18)

where g= b/c2 , xmin =ln(amin ), xmax = ln(amax ), and amax is the maximum possible PGA at the site. It must be noted that the functional form of the CDF (18) depends on the functional form of the attenuation relation formula (14). Choosing a different functional form of the attenuation relation may result in a different functional form of the CDF (18).

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3.2. Distribution of the Largest PGA Recorded at the Site and an Estimation of its Parameters For most engineering applications, a function of particular interest is the CDF of the maximum PGA expected at the site during a specified time interval t. In order to estimate this function one has to make certain assumptions regarding the temporal distribution of the earthquakes recorded at the site. Once again, the common assumption made in engineering seismology is accepted, viz. that the occurrence of events with PGA a, (where a] amin ) at the site follows a Poisson distribution, with mean activity rate l l(x= xmin ). This assumption implies that earthquakes which cause a PGA at the site equal to, or exceeding a certain value a, can be represented by a Poisson process with average occurrence rate l(x)=l[1− FX (x xmin , xmax )], where x =ln(a). Hence the CDF of the logarithm of the largest PGA recorded at the site, within a period of time t, is given by a formula which is analogous to (1), i.e., F max X (x xmin , xmax , t) =

exp{−lt[1− FX (x xmin , xmax )]}−exp(−lt) . 1−exp(−lt)

(19)

The difference between the meaning of the mean activity rate l here, and that introduced in Section 2.1, should be noted. In Sections 2.1–2.2, the parameter l characterizes the mean rate of seismicity for the whole selection area in the vicinity of the site for which the PSHA is performed. Here, the value of l describes the mean activity rate of the selected ground motion parameter experienced at the site. The CDF (19) of the logarithm of the largest PGA observed at the site is doubly truncated. The first truncation, from the bottom, xmin = ln(amin ), represents the chosen threshold for acceleration of engineering interest. The second truncation, xmax =ln(amax ), is an unknown parameter representing the logarithm of the maximum possible PGA at the site. Therefore, for a given amin , the seismic hazard at the site is determined by three parameters: l, g, and amax . In order to estimate these parameters, the largest PGAs a1 , . . . , an felt at the site of interest are selected from n consecutive time intervals t1 , . . . , tn , and the maximum likelihood procedure is used. For a specified amax , the likelihood function of the sample x1 , . . . , xn , where xi = ln(ai ) (i= 1, . . . , n), is a function of the unknown parameters l and g, and is given by n

L(l, g) = 5 f max (xi xmin , xmax , ti ), X

(20)

i=1

where f max X (xi xmin , xmax , ti ) is the PDF of the logarithm of the largest PGA occurring at the site during a specified time interval t. From the definition of the PDF, and from formula (19), the PDF f max X (x xmin , xmax , t) is equal to max f max X (x xmin , xmax , t) = ltF X (x xmin , xmax , t)fX (x xmin , xmax ),

(21)

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where fX (x xmin , xmax ) is the PDF of x and is equal to fX (x xmin , xmax )=

g exp(−gx) . exp(−gxmin )− exp(−gxmax )

(22)

It is interesting to note the strong similarities between the site-characteristic likelihood function (20) and the area-characteristic likelihood function (9). Both functions define the likelihood of a set of largest observations: function (9) is based on observations of the strongest earthquake magnitudes, while function (20) is based on the largest logarithms of the ground motion acceleration. Both likelihood functions are described by the same type of distribution functions, i.e., negative exponential, truncated from both ends. The only difference is that the area-characteristic likelihood function (9) is constructed from a PDF that explicitly takes into account uncertainty in the determination of earthquake magnitude, while the site-characteristic likelihood function (20) is based on values of ln(PGA), which are calculated from the attenuation formula, and their uncertainties, therefore are determined only implicitly from the uncertainty of the attenuation formula itself. These similarities between area-characteristic and site-characteristic distribution functions have important consequences. From the fact that the CDF of earthquake magnitudes (Part I, Equation (7)) and the logarithm of PGA recorded at the site (this paper, Equation (18)) are of the same type, it follows that the value of the maximum possible PGA at the site, amax , can be estimated according to the same procedure as mmax , where in the formulae (6) and (7) of Part I, the parameter b is replaced by g, the mean seismic activity rate for the area, l, is replaced by the mean seismic activity rate at the site, l, and the maximum observed magnitude m obs max is obs replaced by the logarithm of the maximum observed PGA, ln(a max ). It follows that the maximum likelihood estimators of parameters l and g (say l. and g/ ) can be obtained by making the likelihood function (20) as large as possible under the condition that one of the formulae from Part I for mmax (i.e., Equations (39), (45), (50), or (54)) is included, where the parameter b is replaced by g, the activity rate for the area, l, is replaced by the activity rate at the site, l, and the maximum observed magnitude m obs max is replaced by the logarithm of the maximum observed PGA, ln(a obs ). max It is not difficult to show that for a given value of xmax , maximization of the likelihood function (20) is equivalent to the solution of two simultaneous equations: ( ln L(l, g)/(l = 0 and ( ln L(l, g)/(g = 0, which take the form Á Ã Í Ã Ä

#

$

1 tA 2 −tA t exp(−lt) = − , l A2 −A1 1 −exp(−lt) 1 B −B1 (tA2 − tA)(B2 − B1 ) tB2 − tB = x− 2 +l − , A2 −A1 (A2 − A1 )2 A2 − A1 g

n

where A1 =exp(−gxmin),

A2 = exp(−gxmax ),

(23)

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B1 =xmin exp(−gxmin ),

#

$

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B2 = xmax exp(−gxmax ),

1 n t exp(−lti ) t exp(−lt) = % , n i = 1 1− exp(−lti ) 1 −exp(−lt) n

t= % ti /n, i=1 n

tA = % ti A(xi )/n, i=1

n

x= % xi /n, i=1 n

tB= % ti B(xi )/n, i=1

A(x) = exp(−gx) and B(x) =x exp(−gx). It should be noted that when the expression exp(−lt) in the first of equations (23) is small enough, the second term may be neglected. This will be the case when the time intervals t are long enough and/or the site-characteristic, mean activity rate l is high enough. Equations (23) then take the simpler form (KIJKO and DESSOKEY, 1987): Á 1 = tA 2 − tA , Ã A2 − A1 l Í (24) Ã 1 =x− tAx− tA 2 xmax . Ä g tA− tA2

3.3. Inclusion of Additional Information One of the strongest points of this procedure for PSHA is the ability to take into account any additional information on its parameters. This could come from a knowledge of seismogenic zones or from any independent geophysical or geological sources. This ability is especially important when seismic event catalogues are highly incomplete (there is not enough historical information to assess model parameters with the required accuracy), or the area for which the PSHA is required has a low level of seismicity. Also, the role of independent information is especially important when assessing hazard for the low probabilities of exceedance (e.g., less than 0.001/year). For purposes of the demonstration of the procedure, it is assumed that in order to obtain more reliable estimates of PSHA, independent information regarding the Gutenberg – Richter parameter b for an area in the vicinity of the given site should be included. It is assumed that the approximate value of bprior and its standard deviation sprior are available, since, for example, they were known from historic, independent data, or were taken to be the same as for another area of similar seismogenic features. Alternatively, as bprior and sprior , the values of b. /ln (10) and its standard deviation s/ b /ln(10), which were obtained from the maximization of likelihood function (13), can be used. The most efficient way to include a priori information is to apply the formalism of the Bayesian estimation, which is described very comprehensively e.g., by BOX and TIAO (1973), or by TARANTOLA (1987). Following the Bayesian approach, the

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value of bprior is understood as a priori information about an unknown b value and can be written in the form bprior = b+db,

(25)

where b is the unknown, ‘‘true’’ value of parameter b, which is subject to the unknown error db. Often, if no information leading to the choice of the particular type of distribution of db is available, an assumption is made that the errors db are Gaussian, with mean equal to zero and known standard deviation sprior . Then the a priori PDF of the parameter b is given by p(b) =

1 sprior 2p

exp −

1 2s 2prior

n

(b−bprior )2 .

(26)

Following a Bayesian formalism, the joint PDF of the model parameters, a posteriori to observed data, is proportional to the likelihood function (20), multiplied by the a priori PDF p(b). Taking into account that parameter g =b/c2 , where c2 is a known constant from the attenuation formula (14), the a posteriori PDF of parameters l and b is given by p(l, b)= const L(l, b/c2 )p(b).

(27)

The const is the normalizing factor that ensures that the integral p(l, b), over the whole domain of the parameters l and b, is equal to one. According to Bayesian formalism, the best estimates of l and b are those that maximize the a posteriori PDF (27). The same approach can be applied in order to incorporate a priori information about the activity rate l or, in general, about any unknown model parameter.

4. Example of Application: The PSHA for a Site of a Hypothetical Engineering Structure 4.1. Assessment of Area-characteristic Parameters As an illustration, the procedure as described above was used to estimate the seismic hazard parameters at the site of a hypothetical engineering structure (HES) with latitude 33.4°S and longitude 19.24°E, located in the territory of South Africa (Figure 2). The location of the HES is indicated by a dot on the map. The PSHA is based on the seismic events, recorded by the South African National Seismological Network of the Council for Geoscience (formerly the Geological Survey of South Africa), originating in the area of interest from the beginning of 1801. It is assumed that the only reliable information in the assessment of the seismic hazard parameters in the vicinity of the HES comes from a knowledge of past seismicity. No data were available which could provide us a firm basis for identification and grading of seismic source zones. No map of heterogeneity of the lithosphere and fault pattern, inferred from geological evidence of large earth-

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Figure 2 Seismicity map within a distance of 450 km from the site of the hypothetical engineering structure.

quakes in the past, and no coda wave attenuation were available. The natural choice for PSHA was therefore the set of statistical tools as derived above and in Part I of our work. The procedures were applied to seismic data limited to an arbitrarily chosen distance of 450 km from the site (Fig. 2). The compiled catalogue is incomplete in terms of historic events, but it is natural to expect that information on the largest earthquakes would be more complete and therefore their distribution could be used. Hence the catalogue is divided into an incomplete part (historic) and two complete parts, for which information from instruments was obtained (see Fig. 1). The first part of the catalogue contains the largest seismic events (with local magnitudes equal to or exceeding m0 = 4.0) that occurred during the period 1 January, 1801 to 31 December, 1970. It was assumed that, for all of these events, the standard deviation of the determination of magnitude was 0.4.

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The second part (from 1 January, 1971 to 31 December, 1990) includes the complete catalogue of seismic events, from magnitude level m (1) min = 3.8 upwards. The uncertainty of magnitude was assumed to be 0.2. The third part (from 1 January, 1991 to 31 December, 1995) includes the complete data from magnitude level m (2) min = 3.5 upwards. The uncertainty of magnitude was assumed to be 0.1. The first part of the catalogue contains 32 of the largest seismic events. The second part contains 13 events, with local magnitude 3.8 or larger, and the third contains 8 events, with local magnitude 3.5 or larger. The strongest known earthquake that occurred within the area was an event of local magnitude mL $6.3 of September 29, 1969. Since this event was very well investigated, it is assumed that the magnitude of this event was determined with a relatively small standard error sM , equal to 0.25. Maximization of the likelihood function (13) and application of the T-P estimator (Part I, Equation (39)) for the area surrounding the site of HES gives m/ max =7.28 91.26, b. =2.20 90.26, or equivalently b. = 0·939 0.11, and l. = 10.05 93.86. By replacing equation T-P by its Bayesian counterpart, i.e., the estimator T-P-B (Part I, Equation (50)), the maximization of the likelihood function (13) gives m/ max =6.94 90.69, b. =2.18 90.26, (b. = 0.939 0.11), and l. = 9.879 3.80. Repetition of the same procedure for the K-S estimator (Part I, Equation (45)) for the same area leads to m/ max = 6.859 0.60, b. = 2.189 0.26, (b. = 0.929 0.11), and l. =9.81 93.77. Finally, maximization of the likelihood function (13) and application of the Bayesian extension of the K-S estimator (i.e., K-S-B, Part I, Equation (54)) for the area surrounding the site of HES results in m max = 6.669 0.44, (b. = 0.929 0.11), and l. =9.65 93.73. All four estimators of the area-specific mean activity rate l. are calculated for the local magnitude mmin equal to 2.5. The standard deviations of the parameters l. and b. are calculated as positive square-roots of the diagonal elements of the approximate variance-covariance matrix D, where D−1 is symmetric, and is composed of the 2nd partial derivatives of the logarithm of the likelihood function (13) with respect to l and b Æ ( 2 ln(L) ( 2 ln(L) Ã (l 2 (l (b D−1 =−Ã 2 2 ( ln(L) ( ln(L) Ã (b 2 È (b (l

Ç Ã Ã . Ã l= l. É b= b.

)

(28)

A summary of the reuslts of the application of the four procedures for assessment of the maximum regional magnitude mmax is given in Table 1. Taking into account that simulation experiments described in Part I of our study have shown that the Bayesian estimator K-S-B tends to perform very well (especially in the presence of

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inevitable deviations from the simple Gutenberg–Richter model of distribution of earthquake magnitudes), the value m/ max = 6.669 0.44, obtained from applying the K-S-B technique, was used in all further computations.

4.2. Assessment of the Site-characteristic Parameters and Maximum PGA at the Site of HES The horizontal component of the PGA at the site of the HES was calculated using the attenuation curve obtained from averaging the PGA from earthquakes, worldwide, of different magnitudes and at different distances, where acceleration is measured in units of gravity, [g], and the coefficients in the attenuation Equations (14) – (15) are c1 =−3.13, c2 =0.617, c3 = −0.00675 and c4 = −0.79 (AMBRASEYS and SRBULOV, 1994). This attenuation equation predicts the mean value of ln(PGA), with a standard deviation sln(PGA) equal to 0.62. The PGA values refer to sites of ordinary, competent rock. This limitation is due to the fact that the data were obtained at seismological stations founded on hard rock. Despite formal similarities between the determination of the maximum regional magnitude and the maximum PGA at the site, reliable assessment of amax is even more difficult than for mmax . At least two different approaches to the assessment of this important parameter can be applied. The first approach is simple and is based on the concept of the ‘‘design’’ or ‘‘floating’’ earthquake. This approach can be seen as a special case of technique known as ‘‘scenario earthquake’’ (ISHIKAWA and KAMEDA, 1993) or ‘‘beta earthquake’’ procedure (MCGUIRE, 1995), which originally is applied to the entire spectrum of ground acceleration. According to this procedure, amax is the value of the PGA calculated from ground-motion attenuation formulae by assuming the occurrence of the strongest possible earthquake (e.g., with magnitude m/ max ) at a very short distance (say r0 =10 km). Unfortunately, there is an overwhelming number of observations worldwide that show that the PGA values at short distances from the earthquake hypocenter are extremely scattered. For example, the PGA observed at several stations located at a distance of about 10 km from the Northridge earthquake, which registered a magnitude of about 6.7, ranged from 0.3 Table 1 Results of the application of four procedures for estimation of the maximum regional magnitude mmax for the area surrounding the site of the hypothetical engineering structure m/ max estimation procedure

m/ max 9SD

T-P T-P-B K-S K-S-B

7.289 1.26 6.9490.69 6.8590.60 6.6690.44

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to more than 1.0 g (SOMERVILLE et al., 1996). Similar, well-documented examples of high variations in the PGA at short distances can be found in a special issue of the Seismological Research Letters (Seism. Res. Lett. 68, No. 1, 1997), which summarizes the current state of attenuation relations for use in PSHA. Consequently, in order to apply this simple, deterministic-based procedure, a proper, probabilistic-based analysis of the uncertainty in the estimated PGA has also been incorporated. Following the original assumption made by AMBRASEYS and SRBULOV (1994), it is assumed that accelerations from earthquakes of magnitude m at distance r are lognormally distributed, with mean value ln(PGA) = c1 + c2 · m+c3 · r+c4 · ln(r),

(29)

and standard deviation sln(PGA) , where sln(PGA) is constant—independent of magnitude and distance. It must be noted that the common assumption about the Gaussian nature of distribution of ln(PGA) was recently questioned by GUSEV (1996). His investigation of ground-motion accelerations recorded in Mexico indicate the presence of a significant component of scattered value of ln(PGA) which is non-Gaussian. In this simple model of account of uncertainty in PGA estimation the latest results by GUSEV (1994) will not be incorporated and, following AMBRASEYS and SRBULOV (1994), it will be assumed that the random scatter of the logarithm of acceleration is indeed normal with standard deviation equal to sln(PGA) . Clearly, in addition to the uncertainty of the attenuation formula itself, there is another source of uncertainty which derives from the possibility of an erroneous determination of its parameters, viz. earthquake magnitude m and earthquake distance r. There are several possible treatments of the uncertainty of the model parameters and each of them will lead to different formulae for describing these uncertainties. An excellent discussion concerning how to handle all potential sources of uncertainty in the ground-motion attenuation relation can be found in papers by BENDER (1984), and BENDER and PERKINS (1993). More recently ATKINSON and BOORE (1997), and TORO et al. (1997), analyzed these uncertainties in terms of aleatory and epistemic uncertainties. ATKINSON and BOORE (1997) pointed out that classification of uncertainties as aleatory or epistemic is not always clear; there is often significant overlap between them. Fortunately, the treatment of uncertainties used here in estimation of PGA relations is very straightforward and its classification under either of the above categories has no influence on the final results. Following TINTI and MULARGIA (1995a,b), it is assumed that the earthquake magnitude m is determined with an error which is normally distributed and has mean zero and standard deviation sM . The same assumption is made concerning the error in the determination of earthquake distance, where the standard deviation of r is denoted as sR . After applying the simple procedure for the determination of approximate distributions of functions (e.g., BENJAMIN and CORNELL, 1970), it can

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be seen that for an earthquake with an apparent magnitude m, and located at a distance r, the value of ln(PGA) is approximately normally distributed with the mean given by the RHS of Equation (29), and the standard deviation given by sTOTAL = s 2ln(PAG) + c 22 s 2M + s 2R (c3 + c4 /r)2.

(30)

Therefore, if an earthquake of apparent magnitude m is located at a distance r from the site, the probability that this earthquake will cause a PGA equal to or greater than a is given by

Pr[PGA ]a]= 1− F

ln(a)−ln(a) ) , sTOTAL

(31)

where F(·) is the normal probability integral F(z)=(2p)−1/2 z− exp(−0.5t 2) dt, (ABRAMOWITZ and STEGUN, 1970), and a) is the median value of acceleration, with a) a(0.5) =exp[c1 +c2 m +c3 r + c4 ln(r)]. It should be recalled that the median is such that it divides the range of the random variable into intervals which have equal probability. Thus the probability that a random quantity is less than the median, coincides with the probability that it is greater than the median, and this probability is equal to 0.5. Therefore, half of the observations should have a value less than the median, and half should have a value greater than the median. Equation (31) makes it possible to assess the required value of maximum acceleration at the site together with its uncertainty. Following this approach, the value of amax is calculated as a median value of distribution of acceleration (31), calculated at the critical distance r0 and generated by the largest possible earthquake magnitude mmax : a/ max a/ max (0.5) = exp[c1 + c2 m/ max + c3 r0 + c4 ln(r0 )].

(32)

Following relation (31), the upper confidence limit of the a/ max , a/ max (0.84) is calculated as a solution of equation F

ln[(a/ max (0.84)]−ln(a/ max ) = 0.84. sTOTAL

(33)

The confidence level 0.84 was chosen by analogy with the normal distribution, for which the upper site confidence limit, with the confidence level 0.84, corresponds to the mean value increased by one standard deviation. It can be shown that, for a lognormal distribution of acceleration, the upper limit amax (0.84) takes the simple form a/ max (0.84)=a/ max · exp(sTOTAL ).

(34)

For the selected model of the PGA attenuation, where sln(PGA) = 0.62, for m/ max = 6.66, s/ M =0.44, r0 =10 km and sR = 5 km, the resulting median value of maximum PGA a/ max , expected at the site of the HES, is equal to 0.40 g, and its 84% upper limit, a/ max (0.84) = 0.82 g.

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Figure 3 The probability that an earthquake of magnitude m/ max =6.66 90.44 at a hypocentral distance of 10 9 5 km, will produce a PGA exceeding a given value at the site of the hypothetical engineering structure.

The probability that an earthquake of magnitude m/ max = 6.669 0.44, occurring at a hypocentral distance of 10 9 5 km, will produce a PGA exceeding a given value at the site of the HES is shown in Figure 3. Estimated a/ max (0.50) and a/ max (0.84) values of the maximum PGA a/ max are given in Table 2. Solution of the set of Equations (23) for the selected site of the HES, for amin =0.05 g, and a/ max =0.40 g, gives l. = 0.1239 0.014 [YEAR−1], and g/ = 3.499 0.35.

Table 2 Medians a/ max (0.5), and an upper 84% confidence limit a/ max (0.84), of maximum, horizontal PGA at the site of a hypothetical engineering structure obtained by the application of the ‘‘design’’ earthquake and four procedures analogous to the estimation of the maximum magnitude mmax a/ max estimation procedure ‘‘design’’ earthquake T-P T-P-B K-S K-S-B

a/ max (0.5)

a/ max (0.84)

0.40 1.10 0.97 0.32 0.32

0.82 6.68 3.95 0.72 0.69

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The second approach to the assessment of the maximum PGA at the site is purely probabilistic and is based on the fact that the earthquake magnitudes and the logarithms of PGA follow the same type of distribution function. Thus the value of maximum possible PGA, amax , at the site of HES can be estimated according to the same equations as the maximum regional magnitude mmax , where in all formulas the parameter b is replaced by g=b/c2 , the area-specific mean activity rate of seismicity, l, is replaced by its counterpart, characteristic for the site, and the maximum observed magnitude m obs max is replaced by the logarithm of the maximum observed PGA, a obs . Values of a/ max (0.50) and a/ max (0.84) obtained max from application of the four procedures based on equations derived for estimation of mmax (viz., T-P, T-P-B, K-S, and K-S-B) are given in Table 2. The estimated values of maximum possible acceleration differ significantly. For example, the application of K-S and K-S-B procedures gives coincidentally the same value of median acceleration a/ max , equal to 0.32 g, while the application of T-P and T-P-B formulas, gives the values equal to 1.10 and 0.97 g. It is interesting to note that the ‘‘design’’ earthquake approach, which describes the worst scenario and should therefore provide more conservative results, gives a/ max = 0.40 g, which is only slightly higher than the a/ max obtained from application of the K-S and K-S-B procedures. The suspicion that T-P and T-P-B procedures discussed above provide significantly overestimated median values of maximum accelerations can be confirmed from a comparison of the values of accelerations at the level of 84% of confidence. Such values are unacceptably large, 6.67 and 3.95 g, respectively (see Table 2). This confirms conclusions regarding the performance of different mmax estimators and also the fact that the procedures based on the ’‘straightforward’’ approach (viz., T-P and T-P-B) often provide results with large mean errors. Figure 4 depicts the probabilities that the given value of the horizontal component of PGA at the site of the HES will be exceeded in one, 50 and 100 years.

5. Remarks and Conclusions The technique used above for seismic hazard assessment has been developed specifically for the estimation of seismic hazard at individual sites without the subjective judgment involved in the definition of seismic source zones, inasmuch as specific active faults have not been identified and mapped and the causes of the seismicity are not well understood. This technique combines the best features of the ‘‘deductive’’ (CORNELL, 1968) and ‘‘historic’’ (VENEZIANO et al., 1984) procedures. Since the seismic hazard parameters are estimated by the maximum likelihood procedure, by applying the Bayesian formalism, any additional geological or geophysical information, both subjective and objective (as well as various kinds of uncertainties), can be easily

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Figure 4 Probabilities that a given peak ground acceleration will be exceeded in 1, 50 and 100 years for the site of the hypothetical engineering structure.

incorporated. The procedure described is consequently capable of giving a realistic assessment of seismic hazard in the areas of both low and high seismicity. Since the maximum regional magnitude, mmax , and the maximum site-characteristic PGA, amax , are of principal importance in any PSHA, some of the statistical techniques which can be used for evaluation of these important parameters are presented. A simple model for the assessment of uncertainty in estimating mmax and amax is also developed. This approach is particularly useful for the mapping of seismic hazard in areas where both large, historical observations and complete, recent instrumental observations are available. The procedure takes into account the incompleteness of the seismic catalogues. The method accepts mixed data of two types, one containing only the largest earthquakes and the other containing data sets complete from different thresholds of magnitude upwards. The procedure also incorporates uncertainty in earthquake magnitude determination. Based on the maximum likelihood principle, a system of equations has been derived for parameters of the models most often used in engineering seismology, viz., a doubly truncated Gutenberg–Richter frequency-magnitude, and a Poisson distribution of earthquake occurrence in time. It must be noted that the procedure is open to any alternative parameterization. Generalization for the case of an alternative distribution of earthquake size, as e.g., Pareto or Gamma distributions

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(KAGAN, 1993), which takes into account the presence of ‘‘characteristic earthquakes’’, is straightforward. If applied to a grid it will result in a map of seismic hazard, in which the contours of the expected largest PGA during specified time periods can be drawn. As an example of an application of the proposed methodology, the results of a typical PSHA for a site of a hypothetical engineering structure are presented.

Acknowledgments The authors wish to express their gratitude to K. Aki for his critical review, suggestions and very helpful comments. They also thank their colleague, C. Randall, for his kind support.

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