Earthquake source parameters for seismic hazard ... - CiteSeerX

10th European Conference on Earthquake Engineering , Duma (ed.) © 1995 Balkema, Rotterdam. ISBN 90 5410 528 3

Earthquake source parameters for seismic hazard assessment: How to obtain them from geologic data, historic seismicity and relative plate motions L.R.Jordanovski Seismological Observatory; University St. Cyril & Methodius, Faculty of Natural Sciences & Mathematics, Skopje, Macedonia

M. I.Todorovska Department of Civil Engineering, University of Southern California, Los Angeles, Calif., USA

ABSTRACT: The paper first reviews the elements of a probabilistic seismic hazard model, and the type of earthquake source information needed for seismic hazard calculations . The parameters that are used directly as input in hazard calculations are determined from earthquake catalogues and analysis of geologic explorations. Emphasis is put on procedures to obtain frequency of occurrence and magnitude distribution of the events. The paper also reviews selected probabilistic models of earthquake occurrences, and implications on ground motion estimates.

1 INTRODUCTION 1.1

The elements of a seismic hazard model

For assessment of hazard of strong ground motion at a site, it is necessary to gather information on the possible earthquakes, on the regional attenuation of ground motion with distance, and on the characteristics of possible realizations of ground motion at the site from a particular earthquake. The earthquake information includes spatial, temporal and magnitude distribution of earthquakes, and other detail such as length of rupture for a given magnitude, predominant fault mechanism, stress drop e.t.c. The attenuation of strong ground motion with distance is region dependent and is determined from strong motion recordings, when available and sufficient (frequency dependent attenuation, Trifunac and Lee, 1990), or isoseismal maps (Anderson, 1978; Trifunac et al., 1988). Realizations of ground motion at a site are usually described in terms of intensity of shaking, peak amplitude of ground motion, or spectral characteristics (Fourier or response spectra). This ground motion is then used to estimate response of structures or soils. The Fourier or response spectra are the most direct description of ground motion that has been used so far to define design loads for structures. These are evaluated via scaling equations obtained by regression analyses involving combinations of various parameters: earthquake magnitude, epicentral distance, depth of focus, geologic site condition, local soil condition. Scaling equations have also been developed in terms of local intensity of shaking instead of magnitude, to be used where regional attenuation only of intensity is available.

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Ground motions may be evaluated for deterministic scenarios of earthquake occurrence (e.g., assuming that the "maximum earthquake" has occurred, "the maximum earthquake" being the event that produces the largest motions at the site from all other possible events). The uncertainties in such estimates would be implied by dispersion in the scaling models (these would reduce with availability of larger amount of strong motion recordings). Another possibility is a probabilistic approach including effects from all possible magnitude earthquakes occurring at any of the earthquake sources. Such an option are uniform probability spectra, e.g., described in Todorovska 1994). The detail of the earthquake source information has to be consistent with the attenuation and scaling laws for strong ground motion to be used by the hazard analysis. This paper will focus on the earthquake source information. 1.2 Historic seismicity and geology

The essential information about the earthquake sources is where, when and of what size damaging earthquakes will occur during a specified exposure time (the lifetime or the service time of structures). Main sources of this information are earthquake catalogues and geology. Due to the incompleteness of and uncertainties in earthquake catalogues and geologic databases, these two sources are considered to be the main contributors to the uncertainties in seismic hazard assessment. The historic seismicity is an evidence of where and how earthquake occurred in the past. It helps identify active seismic regions, map surfaces of ac-

tive faults and subduction zones, determine the width of the seismogenic zone in a region, and determine the size of the ruptured area of a large earthquake (from loci of the aftershocks). It also helps determine the proportion of large relative to smaller earthquakes that are likely to occur (bvalue), the mean occurrence rates and the distribution of the return periods of particular types of events . (The term "interoccurrence interval" is often used instead of "return period".) However, this information is often not sufficient due incompleteness (or variable degree of completeness) in covering small and large events. This is caused by the short ( in geologic terms ) observational period compared to the return periods of large events, the nonstationary nature of earthquake occurrence , and the failure to report and document smaller events . Earthquakes may occur at locations where there was no evidence of prior occurrence according to the earthquake catalogue . Also, it is always possible that the largest earthquake (a rare event) has not occurred during the period covered by the catalogue . For larger earthquakes, observational period of at least several cycles is required to estimate reliably the return period . Some analyses of hazard are based only on occurrence rates from historic seismicity. Statistical methods are sometimes applied to improve the estimates resulting from incompleteness of the catalogues. Information from geology, typically going back 102 to 107 years, complements the information from the earthquake catalogues . Earthquakes leave permanent scars in the earth crust, by displacing the geologic structure along the fault surface. The geologic slip is determined by measuring offsets of geomorphic features and geological units and determining the age of each unit. This may involve large uncertainties . Also, some of the offset may be of other nature than seismic creep e.g.). From the amount of slip and the ( date, the periodicity and size of the largest events can be determined. By adding the slip that has occurred during a time interval , long term average slip rates , u, can be obtained. These rates are then used to estimate occurrence rates, as explained in what follows. 2 AVERAGE OCCURRENCE RATES AND MAGNITUDE DISTRIBUTION OF EARTHQUAKES 2.1 Seimic moment, Mo The quantity that can be related both to the geological deformation and to the magnitude is the seismic moment, Mo. By definition , the seismic moment released during an earthquake is

area A. The area A is estimated from the length and the depth of the aftershock zone, and µ is estimated from the density, p, and the shear wave velocity, /3 (µ = p/32 ). The seismic moment is also related to the low frequency radiation from the source and to the earthquake magnitude. The following relationship has been determined statistically between the seismic moment and the magnitude Mo = 10c+dM (2) When M is the moment magnitude , c Pe 16. and d 1.5 (Mo is in dyne-cm). From Eqns (1) and (2), it follows that the seismic slip is an exponential function of the earthquake magnitude . An earthquake with magnitude one unit larger produces uA that is about 30 times larger . For fixed A , this implies that the largest earthquakes dominate in the contribution to the total cumulative slip. Eqn (1 ) is applicable to a fault with shear deformation . That formulation is appropriate for fault segments along the plate boundaries (California, e.g .). There are alternative equations for the cases where the deformation is within a volume rather than along a surface (when numerous faults dominate the deformation of the volume). In this case , the seismic moment is treated as a tensor quantity rather than as a scalar , and the deformation is represented by the strain tensor. 2.2 Magnitude distribution of earthquakes It is usually assumed that the distribution by magnitudes of the number of earthquakes in a fixed time interval is exponential , of the form (Gutenberg- Richter relationship) N(M) = 10a-bM)1(Mmax - M) (3) where N(M) is the average number of earthquakes per year with magnitude larger than M, 3f is the Haviside step function , Mmax is the maximum possible earthquake in the zone and a and b are constants . The b- value determines the proportion of large relative to small earthquakes and is region dependent (ranges between 0.5 and 1.2). The value of a (productivity ) depends on the specific fault or zone , and is a measure of the activity (Trifunac , 1993 ). The maximum magnitude depends on the size of the fault and on the width of the seismogenic zones. ( Mmax is smaller for a shallow seismogenic zone and shorter faults. Another form of the Gutenberg Richter relationship is n(M) = 10a1-6ME(Mmax - M) (4) in terms of the density

Mo=aAu (1) n(M) = -dN/dM

where A is the ruptured area, µ is the shear modulus and u is the average seismic slip over the

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(5)

or the incremental form

2.3 Relationship between seismicity and long term average of geologic slip rates

nAM (M) = 10a-bM j (Mmax - M) (6)

where noM(M) = N (M -

- N (M + AM

J

(7) is the number of earthquakes with magnitude within an interval AM centered at M. For small AM nLM(M) = n(M)AM. (8) It should be noted that Eqns (3) and (4), although being alternatives in describing the magnitude distribution of earthquakes , are not equivalent. Differentiating Eqn (3 ) with respect to M results in a Dirac 's Delta-function , b(x), in the expression for the density n(M). This consequence is considered by some investigators non-physical, and Eqns (4) and (5) are preferred . In engineering applications , the effects of earthquakes with size smaller than a threshold magnitude, Mmin, ere usually neglected , resulting in truncation from below of the Gutenberg -Richter relationship ( Eqn (3 ), ( 4) and (6)). Engineers do not always agree on the value of Mmin . Lee and Trifunac ( 1985 ), for example , use Mmin = 2.75, while others have suggested Mmin as high as 5. The choice of Mmin is important in probabilistic seismic hazard exposure evaluation (Todorovska 1994). While the effect of small earthquakes may not be significant unless these occur very close to the site , due to the high likelihood of occurrence, their contribution to the hazard may not be negligible. The mechanism of release of seismic energy during smaller and large earthquakes differs because of different boundary conditions . ( Small earthquakes are considered to be those that rupture entirely within the seismogenic zone and large ones are those that rupture the whole width of this zone). Therefore, changes in the slope of the recurrence curve are expected at the transition magnitude MT, which depends on the width of the seismogenic zone and on the fault geometry Trifunac, 1993 ). For fault segments along the p ate boundaries (e.g. the Parkfield segment of the San Andreas fault in California) observations have suggested that the largest earthquakes (that rupture the whole fault segment ) exhibit stronger regularity than the smaller earthquakes (aftershocks ) and have mean occurrence rate that does may not obey the exponential recurrence relationship for the other events ( Anderson and Luco , 1983 ; Singh et al ., 1983). The characteristic earthquake model has been proposed to model such behavior. The recurrence curve for such segments may be represented as a superposition of the distribution in Eqn ( 4) with some maximum magnitude Ma and a b-function or a distribution at Mmax•

The earthquake occurrence rates should be consistent with the rate of deformation of the earth crust and with the permanent deformations caused by earthquakes . From geological observations, average slip rates on faults can be estimated over long periods of time (102 to 107 year. Besides the large uncertainty associated wi th geologic slip rates , and the possibility of aseismic deformation (creep), another problem that occurs for faults with small rates ( less that 0.01 mm/year ) is that geological slip rates are an average over very long time interval , during which there may have been fluctuations . Slip rates averaged over Holocene time (- 104 years ) are preferred to these averaged over Cenozoic time (6.6 x 107 years). From the average geologic slip rates, u, the average moment rate Mo can be estimated. Mo = pAug.

(9)

The moment rate, Mo, can be related to the occurrence rate, n(M). Anderson (1979) and Molnar (1979) suggested independently how this can be done. Regardless of the shape of N(M), the following must hold Mm a:

Mo = Mo(M)n(M)dM

(10)

where Mo is the total moment rate, Mo(M) is the moment released by an earthquake with magnitude M (see Eqn (2)) and n(M)dM is the number of earthquakes per year with magnitude within the interval (M - dM/2, M + dM/2). Bounding the magnitudes through which seismic moment may be released, and assuming a shape for n(M) (e.g., given the b-value, region dependent) the productivity, a, can be estimated. The b-value is usually determined from seismicity data, and Mmax from the characteristics of the seismogenic zone. Thus, Eqn (10) is used as a constraint to evaluate a. If the magnitudes of possible events are discretized, the integral in Eqn (10) transforms into a sum, from some minimum magnitude, Mm;n, to the maximum magnitude, Mmax. Due to the shorter observational period and occurrence of creep, the seismic slip rate is usually smaller than the geologic slip rate. For the seismic slip rate to be close to the geologic slip rate, the observational period should include several cycles of the largest earthquake on the fault. The slip rates (from seismicity or geologic slip) should be consistent with geodetic measurements of strain accumulation (by a global positioning system). For southern California, for example, the global rates for the region are consistent with the relative plate motion. This is not the case,

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however, for the individual faults in the same region.

3 EARTHQUAKE OCCURRENCE MODELS There is an intrinsic uncertainty associated with the earthquake occurrence, and, therefore, it is appropriate to represent the return period (interoccurrence interval) of events of a particular size by a random variable rather than by a deterministic constant. This uncertainty differs from the uncertainty in determining the average occurrence rates (parametric uncertainty). Since it is a part of the nature of the process, it cannot be eliminated, while the parametric uncertainty can be reduced (and theoretically eliminated) with better quality of data. The earthquake occurrence models in seismic hazard assessment can be grouped in three major groups: probabilistic (with or without memory, the earthquake occurrence is a statistical process described by probability distribution functions), deterministic ("literal model", no intrinsic uncertainty, the earthquakes occur exactly as specified by the mean rates, and predictions (can be associated with a subjective likelihood of happening). Anderson and Trifunac (1977) and Lee and Trifunac, 1985 show how these different models can be combined and used in a single estimate, and the program NEQRISK has the capability to deal with such situations. The simplest probabilistic model for earthquake occurrence is the Poissonian model. The assumptions for this model are that the earthquakes occur independently from each other, that the chance for one earthquake occurring during small time interval At is proportional to At, and the chance for more than one is of order (At)2. The return period, TM, of events with magnitude between M - Ai and M + A2 has exponential distribution, with mean µTi,r = ,.(M) and standard deviation ATM = uTM (one parameter distribution). The number of events of the same size during time interval t, Ni(t), is Poissonian. The probability that a spectral amplitude exceeds a given level during exposure time, say Y, from any possible earthquake of size M at a source depends on the expected number of occurrences of such events during the exposure time. For the Poissonian model, that number is independent on the history of past occurrences, and is a function only of the average rate of occurrence, and of the length of the exposure period (memoryless model) E[NM(Y)] = HAM(M)Y.

because it takes time to accumulate sufficient amount of strain energy. Evidence for this hypothesis is the regular occurrence of large earthquakes on fault segments along the plate boundaries (e.g., segments of the Mexican subduction zone, Alaska and Chile). Models with time dependent hazard function and with smaller scatter are needed to predict such behavior (Sykes, 1971; Thacher, 1984; Sykes and Nishenko, 1984). One such model is the characteristic earthquake model. The characteristic earthquake is the largest earthquakes on the segment that ruptures the entire segment , after which the process is completely renewed (Nishenko and Buland, 1987). The characteristic earthquakes are an independent statistical population beside the population of smaller earthquakes (including also aftershocks of the large earthquakes) whose occurrence is Poissonian. Nishenko and Buland (1987) concluded that the observed distribution of the return period of characteristic earthquakes on segments along the circum Pacific plate is best represented by the Lognormal distribution. Jordanovski et. al (1989) used the same distribution for characteristic earthquakes of the Guerrero, the Michoacan and the Oaxaca segments of the Mexican subduction zone, in seimic hazard evaluation for Mexico City. Other models with memory are timepredictable and slip-predicable models or SemiMarkov models. The problem with application of time dependent models is the uncertainty in (i) fault segmentation and (ii) the average value and variance of the return period, to be determined from limited data, which weakens the justification for replacing the Poissonian model. It is meaningful to apply such models only for sites for which the hazard is governed by the characteristic earthquakes. It is often the case that the hazard at the site is governed by closer faults which are not strongly periodic, than by a distant strongly periodic sources. The hazard rate hLN (t ) for a process with lognormally distributed return period, TLN, depends on time as V rInt-al h LN (t) = St

L s J Int -al - Std L J

where cp and b are the density and the cumulative standard normal distribution functions, and A and S are the mean and the variance of In TLN. Known the hazard rate, the conditional expected number of earthquakes during the next Y years can be calculated by

(11)

Therefore, this model is not consistent with the seismic gap hypothesis, because once a large earthquake has happened on a fault segment, the chance for another one shortly after is small

(12)

Y+t

E ]NM(te

+Y)I te]

= f c h LN (r)dr

(13)

t

where to is the elapsed time since the previous earthquake, and where time t is set to zero each

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time a characteristic event occurs . The shape of hLN(t) is such that it is equal to zero at t = 0, and has a broad maximum at time about and exceeding the mean return period. Todorovska ( 1994 ) studied possible implications of the assumption of time dependent hazard rate , h(t), on uniform probability response spectra. A hypothetical segment of the San Andreas fault in southern California (about 350 km long) was chosen with return perio d of events with magnitude M > 7 equal to 160 years ( Mmax = 8 was assumed ) which exhibits strongly periodic behavior . This segment was modeled as a vertical surface 15 km deep . Other 29 faults and a diffused region of seismicity also contribute to hazard of ground motion (these were represented as lines and as a surface , respectively ). PSV spectral amplitudes PSVLN and PSVEX were evaluated for models with exponentially (TEX) and lognormally (TLN) distributed return period of the characteristic events. It was assumed that median of TLN is equal to the average of TEX. The difference of the predicted PSV levels by the two models was measured by the factors, f and fl, defined as PSVLN f

(14)

PSVEX '

and PSVLN - PSVEX fi =

a*

same value of p; for p = 0.1 or smaller , all the differences are within one o*. The study concluded that, according to the described measure of significance, the Poissonian model is sufficiently good for engineering hazard assessments in the Los Angeles metropolitan area.

4 SUMMARY AND CONCLUSIONS This paper discussed the main elements of assessment of hazard of strong ground motion at a site. These parameters can be obtained by analysis of data from earthquake catalogues or from geologic explorations. Also, information from both of these two sources can be combined, through the seismic moment. The paper also reviewed probabilistic models for earthquake occurrence. The main elements of a procedure for seismic hazard assessment are: a model for the earthquake sources, attenuation law for the region and scaling laws for the quantity to be estimated. Trifunac (1994) concentrates on identifying and analyzing various properties of the source used in hazard assessment. Lee and Manic discussed the attenuation with distance. Todorovska et al., show examples of hazard assessment for Bulgaria. Todorovska (1994) presents an analysis of implications on ground motion from variation of the earthquake source parameters.

(15) REFERENCES

where a* = IPSVLN(P = 0.1)-PSVLN

(P = 0.5)1, (16)

is a measure of the standard deviation for the estimate of PSVLN. For engineering purposes, the factor f is of interest, and f > 2, for instance, is considered to be significant. The factor f, indicates how large the difference is in terms of the standard deviation for the estimates. Values If, I > 1, can be considered as significant. The study showed that, for a given exposure period, the difference is the largest immediately after the occurrence of a characteristic event (the Poissonian model overestimates the ground motion). Both the conservative and nonconservative estimates by the Poissonian model differ by a factor less than 2 relative to those by the time dependent model (S = 0.1, 0.2, 0.5 an 0.85 were considered). The difference is larger if the time dependent model has a narrow distribution (S is small). Nishenko and Buland, 1987, suggested a generic value S 0.2 for the Circum Pacific earthquakes. The difference also depends on the confidence level, and is the largest for very small probability of exceedance. For the generic value of S is small, the nonconservative estimates by the Poissonian model are within one a* even for probability of exceedance p = 0.01, while the conservative estimates may be above one a* for the

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