EFFECTS OF MAGNITUDE UNCERTAINTIES ON SEISMIC HAZARD ESTIMATES

Transcription

1 79 EFFECTS OF MAGNITUDE UNCERTAINTIES ON SEISMIC HAZARD ESTIMATES Davd A RHOADES And Davd J DOWRICK SUMMARY Magntude uncertantes affect dfferent coponents of an estate of sesc hazard n a varety of ways, but ethods are avalable for counterng such effects. Uncertantes n easured agntudes cause bas n estates of the Gutenberg-Rchter actvtyrate paraeter and, f agntude uncertantes are correlated wth agntude, n the b-value also. The cobned effect can easly aount to a factor of two n estatng the frequency of occurrence of large earthquakes. The bases can be corrected f standard errors of agntude estates are known. In attenuaton odellng, gnorng agntude uncertantes can nflate the resdual varance and lead to spurous ters beng ncluded n the odel. Explct treatent of agntude uncertantes forally nvolves an extenson of the usual rando effects regresson odel. The axu agntude that s assued to be possble n a gven source regon s hghly nfluental on sesc hazard estates, but also subect to uch uncertanty. A substantal part of ths uncertanty can be quantfed fro relatons between earthquake agntude and source densons, and used to adust estates of the frequency-agntude relaton. The effects of agntude uncertantes on sesc hazard estates are potentally large. Allowance for such uncertantes should be a standard part of sesc hazard assessent. INTRODUCTION The standard probablty tree ethodology of probablstc sesc hazard assessent (PSHA) allows for the explct treatent of any knds of uncertanty. These nclude both epstec uncertanty (e.g., uncertanty n the paraeters of the frequency-agntude relaton for a gven source regon, and n the for and paraeters of relatons for attenuaton of strong shakng) and aleatory uncertanty (.e., natural varablty about the ftted relatons). However, the analyss of uncertanty seldo extends to a foral treatent of uncertantes n the data on whch the relatons are based. Such uncertantes are not readly captured by the probablty tree approach; rather they affect the qualty of the ftted relatons, and as shown below, ay be a cause of bas or of spurous varablty. Earthquake agntude estates play a crucal part n the estaton of both the frequency-agntude relaton and attenuaton relatons. The standard ethods for fttng both types of relaton gnore agntude uncertantes. The data sets used to estate these relatons are seldo hoogeneous, and nclude agntudes wth dfferent levels of uncertanty. Ths s because the qualty of agntude deternaton vares spatally for a gven sesograph network, and sesograph networks theselves are subect to frequent changes due to staton outages and less frequent changes due to aor upgradng. Modern catalogues usually contan an estate of the standard error of ndvdual agntude deternatons. For early agntude data, an average standard error applcable to the era of the catalogue can usually be adduced. Insttute of Geologcal and Nuclear Scences, P.O. Box , Lower Hutt, New Zealand. Eal: d.rhoades@gns.cr.nz Insttute of Geologcal and Nuclear Scences, P.O. Box , Lower Hutt, New Zealand. Eal: d.dowrck@gns.cr.nz

2 Sesc hazard analysts are often faced wth the need to select fro the avalable data to ensure that data of low qualty (.e., hgh uncertanty) wll not bas the results. Mantanng data qualty thus necesstates dscardng soe nforaton. An alternatve approach s to use all the avalable nforaton, and to accord to each data pont the weght that s due to t, gven ts uncertanty. Ths s the approach that s pursued here. It s ade possble by usng ethods whch allow for the uncertanty n ndvdual data values. MAGNITUDE UNCERTAINTIES IN THE GUTENBERG-RICHTER LAW Let us consder an earthquake catalogue wth observed agntudes ( x, =,, n) and correspondng standard errors (, =,, n). Suppose that the catalogue agntudes are free fro any systeatc bas. If they are deterned as the average over a nuber of statons, then the central lt theore assures the approxate noralty of the dstrbuton of each catalogued agntude,.e., x ~ N (, ) () where s the (unknown) true agntude. Rhoades (996) noted that, n lght of the Gutenberg-Rchter frequency-agntude relaton log N = a bm (Gutenberg and Rchter, 944), the pror dstrbuton of gven b, or, equvalently, gven β, where β = b log e 0, has densty f ( β ) exp( β ) () and showed that the posteror dstrbuton, gven x,, and β s x,, β ~ N ( x β, ) (3) Thus an observed agntude x has an assocated bas that depends on ts uncertanty; the larger the uncertanty, the larger the bas. Frequency-agntude relatons estated fro real catalogues wthout allowng for uncertantes are therefore based also. Tnt and Mularga (985) showed that f all the earthquakes have a coon standard devaton, then the bas n the a-value of the Gutenberg-Rchter relaton can be corrected by a = agr β log0 e (4) where a GR s the estate obtaned fro the observed agntudes. In ths case, observaton error does not cause bas n estates of β. Standard ethods approprate to exact agntude data ay be used, e.g., the axu lkelhood ethod of Ak (965) = β 0 (5) where s the average agntude exceedng soe threshold of copleteness 0, or the refneent of ths ethod whch allows for agntude roundng (Utsu, 966). In the case where the standard devatons dffer between earthquakes, the actvty rate for earthquakes exceedng a gven agntude s estated by λˆ( ) = T n = [ F ( )] (6) where T s the the te perod of the catalogue and F s the posteror cuulatve dstrbuton of, =,, n. Rhoades (996) showed that ths estate has varance 79

3 Var[ λˆ( )] = T n [ F = ( ) ] (7) In ths case there s a potental for bas n the estate of β. It s apparent that a bas wll occur f the standard error s correlated wth agntude, as llustrated n Fgure. Fgure. Scheatc plot of the bas n the estated Gutenberg-Rchter relaton n the presence of agntude uncertantes. Fro equaton (4), = β log e. 0 Rhoades (996) showed how to correct the bas n β by eployng an teratve backfttng procedure, based on the relaton β n = 0 = n = f ( ) d [ F ( )] 0 (8) For the dstrbuton of equaton (3), the ndvdual ters n the nuerator of (8) are gven by 0 f ( ) d = ( x β ) β + 0 x + 0 x + Φ φ β (9) where φ and Φ denote the standard noral densty and cuulatve dstrbuton functon, respectvely. A ore coplex, but nevertheless coputable, forula apples f agntude roundng s allowed for (Rhoades, 996). In ether case, the backfttng procedure s to use an ntal estate of β to get an ntal estate of f () by equaton (3), and then to apply equatons (8) and (3) alternately untl the estate of β converges, usually n ust a few teratons. Equaton (4) suggests an alternatve approxate procedure that stll nvolves teraton, but avods the need to evaluate noral ntegrals. Note that the corrected a-value n equaton (4) s the value that would be obtaned by 3 79

4 applyng the usual a-value estate to observed agntudes that have each been reduced by suggests that a sple correcton β. Ths = x β (0) to each observed agntude x, and the applcaton of the standard axu lkelhood procedures for estatng a- and b-values, should correct the bas n both paraeters, when the vary. In the case of b-value estaton, ths eans alternatng between equatons (5) and (0) nstead of equatons (3) and (4). Ths proposed approxate ethod nvolves uch less coputaton. Fgure. Estates of b-value fro sulated catalogues wth agntude uncertantes postvely correlated wth agntude usng: standard axu lkelhood (Ak, 965), axu lkelhood corrected for roundng (Utsu, 966), axu lkelhood corrected for roundng and agntude uncertantes (Rhoades, 996), and standard axu lkelhood wth agntude correcton (0) (Ak - MC). The true b-value s. The estates are presented as (a) box plots of 00 sulatons and (b) the ean and 95% confdence lts fro 00 sulatons. The calculaton of b-values usng the proposed approxate ethod of standard axu lkelhood after agntude correcton (Ak - MC) s copared wth the procedure of Rhoades (996) wth correcton for roundng and agntude uncertantes, and the procedures of Ak (965) and Utsu (966), n 00 sulated catalogues. In the sulated catalogues, agntude uncertantes are postvely correlated wth agntude, and, followng Rhoades (996), each catalogue has 5000 earthquakes conforng to the Gutenberg-Rchter relaton wth b-value,.7 and = 0.( + u ), where u s unforly dstrbuted on the nterval (0,). Observed agntudes are rounded to decal place. For calculaton of b-values the agntude threshold s taken as The sulaton results are gven n Fgure n the for of (a) boxplots whch show the edan, quartles and extrees of the dstrbuton of b-values and (b) approxate 95% confdence ntervals for the ean b-value. It can be seen that the Ak (965) procedure, whch gnores agntude uncertantes, and the Utsu (966) procedure, whch allows for roundng but otherwse gnores agntude uncertantes, gve estates that are sgnfcantly based. The Rhoades (996) ethod corrects the bas, and the Ak - MC ethod s only argnally based. Gven ts relatve splcty, t has uch to recoend t as a practcal ethod. 4 79

5 UNCERTAINTIES IN MAXIMUM MAGNITUDE Estates of axu regonal agntude ax are subect to uch uncertanty. Kko and Sellevoll (989) and Kko and Graha (998) have gven and copared ethods for estatng ax by extrapolaton of earthquake frequency-agntude data. However, snce agntude deternatons are usually adequate over only the last few decades and are not necessarly a good gude to axu possble regonal agntudes, ust ax n practce often be estated ndependently of hstorcal sescty data. For ths purpose, statstcal relatonshps between source densons and earthquake agntudes (e.g., Wells and Coppersth, 994) are useful. A recent exaple of such a relatonshp s shown n Fgure 3. Fgure 3. Regresson of earthquake agntude aganst rupture area usng worldwde data, n whch rupture area s the product of the surface rupture length and the downdp wdth of the rupture. After Strlng et al. (998). Suppose that on the bass of such a relatonshp, ax s estated to be norally dstrbuted wth ean µ and standard devaton. Let us consder the estaton of the whole frequency-agntude relaton, ncludng the tal-off at the hgh agntude end, usng both a sescty catalogue and perfect nforaton on the axu agntude. Suppose that equaton (0) has already been appled to adust for ndvdual observed agntude uncertantes. The frequency-agntude relaton s assued to be (negatve) exponental between the threshold of copleteness 0 and the unknown axu agntude ax. Let x denote the largest agntude n the catalogue. Then we fnd that the condtonal densty of agntudes exceedng 0 s c exp[ β ( 0 )] x f ( β, µ, ) = () µ µ Φ c exp[ β ( 0 )] > x Φ x 5 79

6 Integratng over ( 0, ), the constant c can be shown to satsfy c = β Φ µ x β Φ exp [ β ( µ ) β ] x µ 0. () The log lkelhood of the earthquake catalogue s log L = n = log f ( β, µ, ) (3) whch can be optsed nuercally to estate β and hence c. The effect of uncertanty n ax on the agntude dstrbuton so obtaned s llustrated n Fgure 4, n whch the densty of equaton () has been estated fro a sulated catalogue of 00 earthquake agntudes exceedng 0 = 5.0 wth µ=7.5 and a range of values of. Fgure 4. Effect of uncertanty n axu agntude ax on estaton of frequency-agntude relaton. Probablty densty ftted to 00 earthquakes of agntude 5.0 and above, wth ~ N ( µ, ), where µ=7.5 and =0, 0., 0. and 0.3. ax MAGNITUDE UNCERTAINTIES IN ATTENUATION RELATIONS Estaton of attenuaton relatons for strong-oton data requres careful treatent of uncertantes because of the structure of the data. Strong-oton data sets typcally consst of a large nuber of observatons generated by a uch saller nuber of earthquakes. The between-earthquake and wthn-earthquake coponents of varaton have to be treated separately. The parttonng of the error varance nto the two coponents s accoplshed n the rando effects regresson odel (e.g. Abrahason and Youngs, 99). Earthquake agntude uncertanty s one factor that contrbutes to the apparent rando earthquake effect, and hence to the between-earthquake coponent of varance. If uncertan agntudes are treated as exact n the rando effects odel, then the between-earthquake coponent of varance s nflated by the agntude 6 79

7 uncertanty. Rhoades (997) ntroduced explct allowance for agntude uncertantes nto the rando effects attenuaton odel. He proposed the followng odel: y = α + βm + f ( r, θ) + η + ε (4) for =, k; =, n, where the, y are observatons of soe strong oton paraeter, the M are uncertan earthquake agntudes, the r are dstances of the observatons fro the earthquake source, and α, β and the vector θ represent unknown paraeters. The between-earthquake varatonsη and wthn-earthquake varatons ε are assued to be ndependently and norally dstrbuted wth zero ean and unknown varances τ and varances, respectvely. The M are assued to to be norally dstrbuted wth known eans s. Equaton (4) can then be recast as and known y = α + β + f ( r, θ) + ξ + ε (5) where now ξ ~ N (0, βs + τ ). Ths odel can be ftted by an extenson to the procedure proposed by Abrahason and Youngs (99) for the rando effects odel. Rhoades (996) showed that for the Joyner and Boore (98) peak horzontal acceleraton attenuaton data, 57% of the rando effects coponent could be explaned by agntude uncertantes alone, and n partcular by the large uncertanty assocated wth usng local agntude M L as a surrogate for oent agntude M w. The odel (5) has been appled by Dowrck and Rhoades (999) to estatng attenuaton relatons for Modfed Mercall ntenstes n New Zealand earthquakes. In that study, the odellng of agntude uncertanty allowed earthquake agntudes of four dfferent types wth wdely varyng uncertantes to be ncluded n the study, wthout fear of contanatng the between-earthquake coponent of varance. The reoval of the agntude uncertanty fro the rando effects, as s accoplshed by odel (5), proves estaton of the between-earthquake coponent of varance and facltates further odellng to explan ths coponent of varance by fttng other physcally eanngful varables such as tectonc settng and focal echans. EFFECT ON HAZARD ESTIMATES Modellng of agntude uncertantes wll not necessarly ake a bg dfference to the assessed hazard n every case. However, the effects are not always trval ether. It does offer both quanttatve and qualtatve proveents n earthquake hazard estaton and soetes the effects ay be substantal. For exaple, usng the New Zealand catalogue of local agntudes and assocated standard errors for the perod , Rhoades (996) showed that allowng for agntude uncertantes gave consstently hgher b-value estates than the standard axu lkelhood ethod. The ncreases were as hgh as for soe subsets, whch aounts to a 40% reducton n the estated rate of occurrence of earthquakes of agntude 7 and above, when the lower agntude threshold s 0 =4.0. Ths reducton s ncreased to 60% f the bas n a-value deternaton s also allowed for. In the case of axu agntude uncertantes, t s clear fro Fgure 4 that the uncertanty on an assued axu agntude has the potental to arkedly affect estates of the rate of occurrence of earthquakes at the hgh end of the agntude scale. Snce t s the large earthquakes that are the ost portant fro a hazard pont of vew, realstc estaton of axu agntude uncertanty s a atter of great portance n ost sesc hazard assessents. The attenuaton uncertanty often akes a sgnfcant contrbuton to the overall uncertanty n sesc hazard studes. Allowng for agntude uncertantes here, by reducng reducng the between-earthquake coponent of varance, can be expected to brng about a oderate reducton n the overall attenuaton uncertanty. 7 79

8 CONCLUSION The proper handlng of uncertantes of all knds s now an essental part of best practce n probablstc sesc hazard analyss. Uncertanty n agntude deternaton s one of the contrbutng factors that needs to be consdered. The ethods that are now avalable for dealng wth ths factor, n frequency-agntude relatons, axu agntudes and attenuaton relatons, elnate potental and actual bases and are not dffcult to pleent. They should becoe a standard part of practcal sesc hazard assessent. ACKNOWLEDGEMENTS Ths work was funded under FRST contract CO5804. In-house revews of ths paper were ade by J. Cousns and G. McVerry. REFERENCES Abrahason, N.A. and Youngs, R.R., 99. A stable algorth for regresson analyss usng the rando effects odel. Bull. Sesol. Soc. A. 8: Ak, K., 965. Maxu lkelhood estaton of b n the forula log N=a-bM and ts confdence lts. Bull. Earthquake Research Insttute, Tokyo Unversty, 43: Dowrck, D.J. and Rhoades, D.A., 999. Attenuaton of Modfed Mercall ntensty n New Zealand earthquakes. Bull. N.Z. Soc. Earthqu. Engnrg, 3: Gutenberg, B. and Rchter, C.F., 944. Frequency of earthquakes n Calforna. Bull. Sesol. Soc. A., 34: Joyner, W.B. and Boore, B.M. (98). Peak horzontal acceleraton and velocty fro strong-oton records ncludng records fro the 979 Iperal Valley, Calforna, earthquake. Bull. Ses. Soc. A., 7: Kko, A. and Graha, G., 998. Paraetrc-hstorc procedure for probablstc sesc hazard analyss. Part I: Estaton of Maxu regonal agntude ax. Pure Appl. Geophys., 5: Kko, A. and Sellevoll, M.A., 989. Estaton of earthquake hazard paraeters ffro ncoplet data fles, Part I, Utlzaton of extree and ncoplete catalogs wth dfferent threshold agntudes. Bull. Sesol. Soc. A., 79: Rhoades, D.A., 996. Estaton of the Gutenberg-Rchter relaton allowng for ndvdual earthquake agntude uncertantes. Tectonophyscs, 58: Rhoades, D.A., 997. Estaton of attenuaton relatons for strong-oton data allowng for ndvdual earthquake agntude uncertantes. Bull. Sesol. Soc. A., 87: Strlng, M., Rhoades, D. and Berryan, K., 998. Evaluaton of Wells and Coppersth (994) earthquake and fault relatonshps n the New Zealand context. EQC proect 97/49, Earthquake Cosson Research Foundaton. Tnt, S. and Mularga, F., 985. Effects of agntude uncertantes on estatng the paraeters n the Gutenberg-Rchter frequency-agntude law. Bull. Sesol. Soc. A., 75: Utsu, T., 966. A statstcal sgnfcance test of the dfference n b-value between two earthquake groups. J. Phys. Earth, 4: Wells, D. and Coppersth, K., 994. New eprcal relatonshps aong agntude, rupture length, rupture wdth, rupture area, and surface dsplaceent. Bull. Sesol. Soc. A., 84: