CHARACTERISTIC EARTHQUAKE MAGNITUDE: MATHEMATICAL VERSUS EMPIRICAL MODELS

Transcription

1 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna CHAACTEISTIC EATHQUAKE MAGNITUDE: MATHEMATICAL VESUS EMPIICAL MODELS ABSTACT : G. Grandor, E Guagent 2 and L. Petrn 3 Emertus Professor, Dept. of Structural Engneerng, Poltecnco d Mlano, Mlano, Italy 2 Full Professor, Dept. of Structural Engneerng, Poltecnco d Mlano, Mlano, Italy 3 Assstant Professor, Dept. of Structural Engneerng, Poltecnco d Mlano, Mlano, Italy Emal: grandor@polm.t, guagent@polm.t, lpetrn@polm.t In te lterature concernng te caracterstc ypotess, one basc queston s wdely dscussed: s t possble to justfy (by statstcal tests) favourng te caracterstc magntude model for te nterpretaton of avalable catalogues? No generally accepted answer as been gven now a days. In a prevous paper (Grandor et al., 28) we analyzed a dfferent queston, peraps more useful from te engneerng pont of vew: s t possble to judge (on te bass of statstcal tests) wc one of two competng magntude models s more relable (all oter tngs beng equal) for te evaluaton of a specfc azard quantty at a gven ste? In tat paper we descrbed a metod wc can gve an answer to ts queston, and we studed te controversy surroundng te comparson between caracterstc-type magntude models and te classc doubly truncated exponental model. We found tat n many cases a caracterstc magntude model s more relable tan te exponental model. In te present paper we recall te man features of te metod and we apply t to te comparson between a matematcal model F M and an emprcal (non parametrc) dstrbuton F*. Te am s to fnd an emprcal F* wc s more relable tan F M, tanks to te substantal reducton of possble errors due to te use of a wrong model F M. We do not gve a general metod for te constructon of suc F*, nor we mantan tat t exsts n all cases. We smply sow ow, n a study case, we found te way to construct a very satsfactory F*. KEYWODS: Magntude dstrbuton, credblty of te model, comparson between competng models.. INTODUCTION In te frame of probablstc sesmc azard analyss, appled to a gven ste X, one problem s te coce of an approprate matematcal model of te magntude-frequency law for te events tat can strke ste X. Te comparson between te relablty of competng models can be based on te agreement wt te current fault segmentaton concepts, observatons and mecancs-based eartquake smulatons (Wu, Cornell and Wntersten, 995). However, as regards dscrmnant statstcal tests, t s generally recognzed tat te sesmc record n most sesmc zones s too sort for a meanngful statstcal comparson: all te relatons proposed n te lterature appear consstent wt te avalable sesmcty catalogs (Araya and Der Kuregan, 998), even f dfferent relatons may lead, all oter tngs beng equal, to mportant dfferences n te fnal results of te sesmc azard analyss. In a recent paper (Grandor et al., 28) we consdered te comparson between two competng magntude models on te bass of te followng crteron: nstead of askng wc one of te two models explans better te data of te avalable catalog, we ask wc one s more relable for te estmaton of a specfc target quantty, A, related to te sesmc azard at te gven ste X. In te same paper t s sown ow te crteron works wen te comparson s between two matematcal dstrbuton models. Te am of te present paper s to study, wt te same crteron, te competton between a matematcal model and a non parametrc emprcal dstrbuton F*, free (by ts nature) from modelng errors.

2 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna 2. THE METHOD We call F te unknown true magntude dstrbuton F M (m; ϑ) = P(M m) and we assume tat: ) t s ndependent of te space and tme coordnates of te events; 2) te avalable catalog s a random sample S drawn from F. We compare te competng models all oter tngs beng equal. Precsely, we assume tat te models are appled to a test-ste for wc all oter elements tat contrbute to te estmaton of A are known and ndependent of te magntude dstrbuton F M. As a consequence, f a magntude dstrbuton F j s gven (bot form and parameters), ten a known procedure Z, appled to F, yeld te quantty A j : j A j = Z(F j). (2.) In partcular, obvously, A = Z(F ) s te true value of A. Te fundamental tool for te acevement of te comparson s te evaluaton, for eac model F, of te foreseeable errors n estmatng A, under a gven ypotetcal true dstrbuton F. Te dstrbuton of suc errors wll be descrbed by: ) te mean value  m of ndependent estmatons obtaned from random samples S drawn from F wt te same sze as te avalable catalog; 2) te standard devaton σ of te estmates of A ; 3) an ndcator Δ tat we call te credblty of te model F wt respect to F : {  } A ka Δ = P <, (2.2) were  s te estmator of A wt te model F, and te parameter k defnes a conventonal lmt. Te selecton of te form F s affected wt te epstemc uncertanty, wle te statstcal uncertanty, due to te randomness of te sample S, concerns te estmaton of te parameters. Δ s a syntetc ndex tat accounts for bot tese uncertantes and s connected, troug te parameter k, wt a level of error wc s consdered meanngful n te estmaton of A. 3. THE COMPAISON BETWEEN MATHEMATICAL MODELS Let F and F 2 be te matematcal forms of two competng models. A prelmnary basc approac proceeds troug te followng four steps. Te frst step s te analyss of te errors of F under te ypotess tat te true magntude dstrbuton as te same matematcal form as F (.e. F s te rgt model). Te results of ts analyss gve a measure of te statstcal uncertanty connected wt te use of te model F. A second step regards agan te errors of F, but under te alternatve ypotess tat te model F s wrong (n partcular because te trut as te matematcal form of te competng model F 2 ). Ts second experment s representatve of te robustness of te model F. Te trd and fourt steps, wt analogous procedure, gve an dea of te statstcal uncertanty and te robustness of te model F 2. Te results of te basc approac open nterestng statstcal perspectves, as sown for nstance by te applcaton to te followng case. Te test-ste X as te features tat are plausble for a ste located n a sesmc zone of Soutern Italy. Te events are unformly dstrbuted over te zone and follow a Posson process. Te rate of occurrence s.3 events per year. Te number of events n te catalog s n=4, and te lower cutoff of te catalog s m =4. Te target quantty A s te peak ground acceleraton (PGA) wt 5-year return perod at te ste X: A=a(5). We assume tat for te estmaton of a(5) an error larger tan 2% s meanngful from te engneerng pont of vew;.e. we assume k=.2. 2

3 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna Te frst model, F E, s te classc doubly truncated exponental dstrbuton derved from te Gutenberg and cter relaton. Te second model, F C, s a mxture between te exponental and a lnear dstrbuton defned n te two fxed non overlappng ranges [m m E ] and [m E m ], a ybrd model n wc te relatve frequency of strong caracterstc eartquakes s gven by te parameter p, tat s te wegt of te lnear dstrbuton component. By ypotess m -m E =.5. Te correspondng probablty denstes f E and f C are sown n te Fgures and 2. Fgure Exponental model m =4, m =7, b=.9 Fgure 2 Hybrd model m =4, m =7, b=.9,p=.5 p = area between m E and m Te numercal computatons ave been carred out by a systematc use of te Montecarlo metod for te producton of te random samples S, and by te maxmum lkelood (ML) metod for te estmaton of parameters. Frst step of te basc approac: te ypotetcal trut F s a truncated exponental dstrbuton wt te parameters m = 7, b =. 9, sortly ndcated exp(7,.9). Te selected model F s exactly a truncated exponental model exp(m,b), wose parameters ave to be estmated from eac one of te random samples S. However, te ML estmator of m s based (Psarenko et al., 996), on te oter and, at present tere s no generally accepted metod for estmatng m (Kjko, 24). At ts pont, we ntroduce a smplfyng ypotess, te nfluence of wc wll be dscussed later n detal: we assume tat m as been correctly estmated (e.g. on te bass of geologcal elements);.e. te estmate mˆ concdes wt te true value m. Tus te model becomes exp(7,b) and from eac sample S only te b-value as to be estmated; gven m te ML estmator of b s unbased. Te results of te numercal computatons are sown n Table 3., frst row. Te mean value  m concdes wt A ; te dsperson of te estmates leads to a credblty Δ=.68. Second step. Let us see wat appens wt te same model f te trut s dfferent; for nstance a ybrd dstrbuton wt te same parameters m = 7 and b =. 9 as n te prevous case, but wt a 5% of te events concentrated between magntude 6.5 and 7: sortly, ybr(7,.9,.5). Stll keepng te ypotess tat te true m s known, te results (Table 3., second row) sow ow great may be te nfluence of te epstemc uncertanty on te relablty of te truncated exponental model: te value of A s remarkably underestmated and te credblty becomes very small. Trd and fourt steps. Te beavor of te ybrd model (Table 3.2) s qute dfferent: t gves good estmate of A, wt credblty.6. f t s te rgt model; and beaves very well even f te trut s a truncated exponental dstrbuton. Te g effectveness of te ybrd model derves from te fact tat, on te one and, te parameter p s strctly connected wt te relatve frequency of strong eartquakes and, on te oter and, te quantty a(5) s manly governed by suc events. 3

4 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna Table 3.. esults obtaned wen te truncated exponental s te rgt model (frst row) and wen t s a wrong model (second row). model trut A Â m σ Δ exp(7,.9) exp(7,b) ybr.(7,.9,.5) Table 3.2. esults obtaned wen te ybrd s te rgt model (frst row) and wen t s a wrong model (second row). model trut A Â m σ Δ ybr.(7,.9,.5) ybr.(7,b,p) exp(7,.9) In concluson, te results of te above descrbed basc approac (wc s one of te examples contaned n Grandor et al., 28) would suggest te ypotess tat F C s more relable tan F E for te estmaton of a(5) at ste X. However, before acceptng ts ypotess, two furter exploratons are needed. Frst, te results of te comparson must be n favour of F C for a plausble range of te true parameters m, b. Second, f anoter model F D s consdered to be plausble, te two models F C and F E must be compared also under te ypotess tat te trut as te form F D. If all te above mentoned tests are n favour of F C, t can be concluded tat a purely statstcal scenaro strongly supports te ypotess tat F C s more relable tan F E for te estmaton of a(5) at ste X. Note tat, beng te comparson based on random samples S ( possble catalogs) drawn from eac ypotetcal true dstrbuton, te result of te competton does not depend on te data contaned n te really avalable catalog, te role of wc deserves a few comments. Te catalog cooperates wt geologcal and geopyscal knowledge by suggestng plausble models. Moreover, once a model as been cosen, te catalog s essental for te estmaton of te parameters. Te pont s precsely te coce between competng models, gven tat all te relatons proposed n te lterature appear consstent wt te avalable sesmcty catalogs. However, t sould be noted tat te result of te competton between two models depends on te azard quantty tat one wants to nfer from te magntude dstrbuton. For nstance t may appen tat F s more relable tan F 2 for te estmaton of a(5), wle F 2 s more relable tan F for te estmaton of a(5). Ts s te reason wy t as been suggested (Grandor et al. 23) to compare te competng models lookng at ter credbltes Δ and Δ 2 n te estmaton of te target quantty A. It s true tat te real dstrbuton F s not known, but n tat paper t as been sown ow, startng from te avalable catalog, t s possble to obtan, under some reasonable ypoteses, te probablty tat Δ > Δ 2. In oter words, tanks to te ntroducton of te credblty ndex, te catalog can lead to a dscrmnatng symptom;.e. t s not always true tat te proposed models are equally consstent wt te avalable sesmc catalogs: t depends on te comparson crteron. Two ways are ten open to te am of obtanng more strngent results. One way s te metod tat as been summarzed as yet, wc s only based on te matematcal structure of te two models and on te features of te ste, wle t s ndependent of te avalable catalog. In a second way, on te contrary, te catalog becomes te man tool leadng, troug a non parametrc procedure, to a substantal reducton of possble errors due to wrong matematcal modellng. Ts second way s llustrated n wat follows. 4. A NON PAAMETIC POCEDUE As we observed before, te robustness of te ybrd model, as far as te estmaton of a(5) s concerned, depends manly on te fact tat te parameter p derved from eac sample S s strctly connected wt te relatve frequency of strong eartquakes n te sample. If we abandon matematcal models and from eac sample we derve an emprcal dstrbuton F*, for nstance te cumulatve frequency polygon (CFP), we could expect to obtan a procedure wt robustness smlar to tat of te ybrd model. Actually, te CFP follows by ts nature te tal of te sample. Let us try wt a very smple constructon of te emprcal dstrbuton: te values of F* are derved from eac sample S for magntude less tan 5,6,6.5,7,7.5 (an example n Fgure 3). Te Table 4. sows te results obtaned wt ts non parametrc procedure, compared wt tose obtaned from te matematcal models. 4

5 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna Fgure 3 CFP of a sample S drawn from F =exp(7,.9) Table 4. Comparson between te emprcal F* and te matematcal models. trut A model A m Δ Δ w Δ* Δ*/Δ exp(7,b).9.68 exp(7,.9).9 ybr.(7,b,p) F*.2.58 ybr.(7,b,p).36.6 br.(7,.9,.5).38 exp.(7,b) F* Wt smplfed symbols, Δ s te credblty of te rgt matematcal model (MM), Δ W s te credblty of te wrong matematcal model and Δ*s te credblty of te emprcal F*. As expected, te beavor of te non parametrc procedure s smlar to tat of te ybrd model. Te exponental model sows a slgtly larger credblty wen t s te rgt model, but t s by far loosng te competton f te trut s te ybrd dstrbuton. Te results of Table 4. are partally a remake and partally an extenson of tose publsed n a prevous paper (Grandor et al., 26). Te fact tat a smple emprcal procedure may ave a credblty Δ* not far from te credblty Δ of te MM s nterestng; and t s wort examnng closely a few aspects of ts comparson. Let us consder n detal te comparson between te frst and te trd row of Table 4.,.e. between te MM and te emprcal F* (wen te trut s a truncated exponental dstrbuton). Te comparson s affected wt two man approxmatons. Frst, we dd not take nto account te uncertanty n te estmaton of m : te ntroducton of ts uncertanty would decrease te credblty Δ. On te oter and, te very smple tecnque adopted for te constructon of F* s open to mprovements, tat would lead to an ncrease of te credblty Δ*. It s mportant to analyze te quanttatve nfluence of tese two possble correctons on te rato Δ*/Δ. It s true tat tere s no generally accepted metod for estmatng m. However, te applcatons of varous metods descrbed n te lterature gve some nformaton about te uncertanty of suc estmaton. For nstance, Psarenko et al. (996) n te case of Soutern Italy, wt 44 events M 5 and max observed M=7., fnd for te estmate mˆ a standard devaton (SD) of te order of.5. Kjko (24) fnds for Soutern Calforna wt a non parametrc procedure mˆ = 8.54 ±. 45. In te same paper Kjko uses also syntetc data generated accordng to a G- relaton wt m =6, m =8 and b=. He fnds tat te bas of te estmate mˆ s low: t does not exceed. unt of magntude f te number of eartquakes n te catalogue s N=5. However te bas s larger f (m -m )>2: t may rc.3 unts of magntude f (m -m )=3, as n our ste X. In order to take nto account n a syntetc approxmate way te avalable nformaton, gven te condtons of ste X, we do not assgn now to mˆ te true value m, but we assume te estmate mˆ to be a random 5

6 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna varable: normally dstrbuted wt mean m =7 and SD=.7 n te range 5.65 mˆ ; beng te resdual probabltes unformly dstrbuted n te two extreme classes 5.55 mˆ and 8.35 mˆ Wt reference to te ypotetcal trut F = exp(7,.9 ), te MM wll be now ndcated exp( mˆ, b ) and, n order to remember ts knd of estmaton for m, te relatve credblty wll be ˆΔ. We consdered 27 classes of mˆ, wdt. unt, eac class beng dentfed by ts center ( mˆ = 7 means 6.95 mˆ 7.5 ). Te credblty of te model exp ( mˆ, b ) wt respect to te trut F = exp(7,.9) s gven by 5 Δˆ = p Δˆ (4.) 4 were p s te probablty tat mˆ s n te class, and exp ( mˆ, b). ˆΔ s te correspondng credblty of te model Te values of ˆΔ n te feld > (tat s mˆ 7 ) are smply obtaned by assumng for eac class gven m = mˆ (nstead of gven m = 7 as n Table 3.). As sown n Table 4.2, te credblty ˆΔ decreases regularly wen m ncreases, even f te varatons are not dramatc. Table 4.2 Hypotetcal trut F = exp(7,.9). p =prob. tat mˆ s n class. ˆΔ = credblty of te MM f mˆ s n class. class p ˆΔ p ˆΔ =.3329 In te feld < (tat s mˆ 7 ), t s dffcult to catc te credbltes ˆΔ, due to te fact tat many samples drawn from F wll ave maxmum observed magntude larger tan mˆ. In order to overcome ts dffculty, we accept te approxmate ypotess tat te dstrbuton of te credbltes ˆΔ n te feld mˆ 7 s 6

7 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna symmetrcal to tat of te feld mˆ 7. So te credblty of te MM becomes: Δ ˆ = =.63 (4.2).e. te uncertanty n te estmaton of m reduces te credblty of te MM from.68 (Table 3.) to.63: a reducton of te order of 7%. As to te emprcal F*, ts constructon may be canged n many ways (see for nstance Grandor et al., 24) Here we propose a correcton tat fulfls two man condtons: frst, to keep a g smplcty and, second, to ncrease as muc as possble te credblty of te emprcal procedure. Startng from te smple F* of Fgure 3, te corrected dstrbuton F s gven by: F = F m m + ( F ) m. (4.2) Te constructon of F s very smple and, wat s more mportant, te factor can be adjusted n order to obtan a better performance of te emprcal procedure. For nstance, n te case of ypotetcal trut F = exp(7,.9), te Fgure 4 sows te nfluence of te factor on te results obtaned wt te emprcal F. Wt =.25 te expected estmate of A ( A m )concdes wt A and te credblty of te emprcal F s even larger tan te credblty of te MM ( Δ / ˆ Δ =.4). Te factor as been adjusted lookng at te ypotetcal trut exp(7,.9). However, wt =.25, te emprcal procedure F s very robust; Table 4.3 sows te results concernng dfferent ypotetcal truts. In all te consdered cases, keepng =.25, we obtaned Δ ˆΔ. Te applcaton of te non parametrc procedure tat we just descrbed refers to te evaluaton of a(5) at te ste X; te results obtaned cannot be smply extrapolated to oter cases. Tey only sow tat, once a specfc matematcal model F as been selected, t s wortwle to explore te possble exstence of a non parametrc F* (our F s just an example) avng credblty Δ* practcally equal to te credblty ˆΔ of F (remember tat ˆΔ s evaluated under te optmstc ypotess tat F s te rgt model). If suc a F* exsts, ts adopton nstead of te matematcal F reduces substantally possble errors due to epstemc uncertanty. 5. CONCLUSIONS Te comparson between two plausble competng magntude models s often dffcult because te avalable catalog s too sort. Ts dffculty can be overtaken by comparng te foreseeable errors made by te two models (n te estmaton of te target quantty A) under approprate ypotetcal true magntude dstrbutons. Followng ts metod t s possble to obtan a statstcal scenaro wc suggests ratonal decsons wen facng te coce between two matematcal magntude models. In partcular, we consdered te controversy surroundng te comparson between te classc exponental model F E and a caracterstc type model F C, appled to te estmaton of a(5) at a gven test ste X. Te results of a prelmnary basc approac are clearly n favour of F C. Furter results are contaned n our prevous paper (Grandor et al., 28). Once a matematcal model F as been selected (watever metod as been used for te selecton) te non parametrc procedure may lead to a furter reducton of possble errors due to wrong matematcal modelng. Ts appens f an emprcal F* succeeds n reacng a credblty Δ* practcally equal to te credblty ˆΔ of te model F. In ts case let us call A* te value of a(5) obtaned from te emprcal F* appled to te avalable catalog, and A te value correspondng to F. In spte of te fact 7

8 Te 4 t World Conference on Eartquake Engneerng October 2-7, 28, Bejng, Cna Δ ˆΔ t may be tat A* and A dffer notably from one anoter; ts would be a symptom suggestng tat F s a wrong model and tat A* s more relable tan A. Fgure 4 Performance of * F as functon of n te case F = exp(7,.9) trut * Table 4.3 Performance of Δ / ˆΔ * F (=.25) versus MM trut * Δ / exp. (7,.9).4 ybr. (7,.9,.5). exp. (7,.3).4 ybr. (7,.3,.5).27 exp. (7.2,.9).2 ybr. (7.2,.9,.5).7 exp. (7.2,.3).5 ybr. (7.2,.3,.5).9 ACKNOWLEDGEMENT Part of ts work as been carred out under te fnancal auspces of te Convenzone INGV-DPC troug te Project No. 2 (Development of a dynamcal model for sesmc azard assessment at natonal scale). Suc support s gratefully acknowledged by te autors. EFEENCES Araya,. and Der Kuregan, A. (988). Sesmc Hazard Analyss: mproved models, uncertantes and senstvtes. UCB/EEC-9/ Grandor, G., Guagent, E. and Taglan, A. (23). Magntude Dstrbuton versus Local Sesmc Hazard. Bulletn of te Sesmologcal Socety of Amerca 93, Grandor, G., Guagent, E. and Petrn, L. (24). About te statstcal valdaton of probablty generators. Bollettno d Geofsca Teorca e Applcata 45(4): Grandor, G., Guagent, E. and Petrn, L. (26). Eartquake,catalogues and modellng strateges. A new testng procedure for te comparson between competng models. Journal of Sesmology : 3, Grandor, G., Guagent, E. and Petrn, L. (28). Statstcal grounds for favourng te caracterstc magntude model. a case study. Bulletn of te Sesmologcal Socety of Amerca, n press. Kjko, A. (24). Estmaton of te Maxmum Eartquake Magntude, m max. Pure and Appled Geopyscs 6, Psarenko, V.F., Lyubusn, A.A., Lysenko, V.B. and Golubeva, T.V. (996). Statstcal Estmaton of Sesmc Hazard Parameters: Maxmum Possble Magntude and elated Parameters. Bulettn of. te Sesmologcal Socety of Amerca 86, Wu, S.-C., Cornell, C.A. and Wntersten, S.T. (995). A Hybrd ecurrence model and ts mplcaton on sesmc azard results. Bulettn of. te Sesmologcal Socety of Amerca 85, -6. ˆΔ 8