Application of the EEPAS Model to Forecasting Earthquakes of Moderate Magnitude in Southern California

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1 Applcaton of the EEPAS Model to Forecastng Earthquakes of Moderate Magntude n Southern Calforna Davd A. Rhoades Davd A. Rhoades GNS Scence INTRODUCTION The EEPAS (Every Earthquake a Precursor Accordng to Scale) model s a method of long-range forecastng that uses the prevous mnor earthquakes n a catalog to forecast the major ones. It s based on the precursory scale ncrease (Ψ) phenomenon, whch nvolves an ncrease n the magntude and rate of occurrence of mnor earthquakes close to the source regon of a major event n preparaton, ncludng most recent major earthquakes n Calforna (Evson and Rhoades 2002, 2004). The perod of tme occuped by the ncrease scales wth magntude, but t s on the order of 15 years for an M 7 event, fve years for an M 6 event, and one or two years for an M 5 event. Wth a oneyear tme horzon as specfed for the Regonal Earthquake Lkelhood Models (RELM) testng n southern Calforna, t s therefore feasble to consder usng the model to forecast earthquakes of M 5 and above. The model has prevously been ftted to the New Zealand earthquake catalog usng earthquakes of magntudes exceedng 3.95 to forecast those exceedng magntude It was shown to explan the data much better than a baselne model that s n prncple tme-nvarant and has a locaton dstrbuton based on proxmty to the epcenters of past earthquakes. In the same form, and wth the same magntude thresholds, t was tested on Calforna over the perod and agan performed much better than the baselne model (Rhoades and Evson 2004). In the same form, but wth magntude thresholds one unt hgher, t was tested on Japan over the perod and produced a smlar result (Rhoades and Evson 2005), albet wth a smaller advantage over the baselne model. In order to ft well to lower magntudes down to M 6.25, some adjustment of parameters, and n partcular the magntude-scalng parameter, was found to be necessary. Recently, the model has been ftted at moderate magntudes down to M 4.75 to the hgh-qualty catalog of the Kanto regon, central Japan, prepared by the Natonal Research Insttute for Earth Scence and Dsaster Preventon (NIED). In the Kanto regon, earthquakes occur over a wde range of depths. The model was found to be nformatve down to 120 km. The results support the concluson that the Ψ phenomenon s a feature of the preparaton process not only of the largest earthquakes but also of most shallow earthquakes of moderate magntude (Rhoades and Evson 2006). In the less-complex regon of southern Calforna, nearly all of the earthquakes are shallow, wth more than 99% orgnatng at depths less than 20 km. The catalog of southern Calforna, lke that of the Kanto regon, s of hgh enough qualty to support the fttng of the EEPAS model at moderate magntudes. EEPAS MODEL The EEPAS model adopts predctve scalng relatons derved emprcally from 47 examples of the Ψ phenomenon by Evson and Rhoades (2004) n the earthquake catalogs of Calforna and northern Mexco, Japan, Greece, and New Zealand. These relatons, shown n fgure 1, are lnear regressons of manshock magntude M m, the logarthm of precursor tme T P, and the logarthm of precursor area A P, on precursor magntude M P. There M P s the average of the three largest magntudes n the precursory perod, T P s the tme between the onset of Ψ and the major earthquake, and A P s the area occuped by the precursory earthquakes, major earthquake, and aftershocks. In the EEPAS model, the queston of recognzng the Ψ phenomenon n advance of the major earthquake s set asde and every earthquake s regarded as a (possble) precursor of larger earthquakes to follow t n the long term. An earthquake s contrbuton to the earthquake-rate densty depends on ts magntude, whch s treated as an ndvdual nstance of M P. The rate densty λ(t,m,x,y) of earthquake occurrence s defned for any tme t, magntude m, and locaton (x,y), where m exceeds a threshold magntude m c and (x,y) s a pont n a regon of survellance R. Each earthquake (t,m,x,y ) contrbutes a transent ncrement λ (t,m,x,y) to the future rate densty n ts vcnty, gven by λ ( t, m, x, y) = w f ( t ) g ( m) h ( x, y) (1) where w s a weghtng factor that may depend on other earthquakes n the vcnty, and f 1, g 1 and h 1 are denstes of the probablty dstrbutons for tme, magntude, and locaton, respectvely. The magntude densty g 1 s assumed to take the form 110 Sesmologcal Research Letters Volume 78, Number 1 January/February 2007

2 g m a b m M M ( m) = exp 2 2 M M σ π σ where a M, b M, and σ M are parameters (fgure 1A). The tme densty f 1 s assumed to take the form 2 H ( t t ) t t 1 log( ) a b m T T f ( t ) = exp 1 ( t t ) σ ln( 10) 2π 2 T T (3) σ where H ( s) = 1 f s > 0 and 0 otherwse, and a T, b T, and σ T are parameters (fgure 1B). The locaton densty h 1 s assumed to take the form ( x x ) + ( y y ) h ( x, y) = exp 1 2 bam (4) 2πσ 10 2σ 2 10 b A m A A where σ A and b A are parameters (fgure 1C). The total rate densty s obtaned by summng over all past occurrences, ncludng earthquakes outsde R, whch could affect the rate densty wthn R: (2) Fgure 1. Predctve relatons, derved from 47 examples of the Ψ phenomenon by Evson and Rhoades (2004). (A) Manshock magntude M m on precursor magntude M P ; (B) Precursor tme T P on M P ; (C) Precursor area A P on M P. λ ( t, m, x, y) = µλ ( t, m, x, y) + η( m ) λ 0 ( t, m, x, y ) (5) t t0 ; m m0 where μ s a parameter, λ 0 s a baselne rate densty, t 0 s the tme of the begnnng of the catalog, and η s a normalzng functon. The overall form of the rate densty (equaton 5) s smlar to that for epdemc-type models (Ogata 1989, 1998; Console and Murru 2001; Console et al. 2003; Jackson and Kagan 2000). Lke the epdemc-type models, the structure s formally that of a branchng process. However, conceptually there s a bg dfference; the dea here s not that a small earthquake trggers a larger one but that t provdes evdence that a larger earthquake may be n preparaton (Evson and Rhoades 2001). Also, the component dstrbutons here do not follow power laws n tme, locaton, or earthquake moment. The normal, lognormal, and bvarate normal dstrbutons adopted n equatons (2), (3), and (4), respectvely, are chosen to be consstent wth normally dstrbuted errors n the predctve relatons (fgure 1). In the case of magntudes, the standard frequency-magntude relaton s preserved overall by the normalzng functon η, whch makes the long-run average rate densty under the EEPAS model, as a functon of magntude,.e., averaged over tme and locaton, conform as well to the Gutenberg-Rchter frequency-magntude relaton (Gutenberg and Rchter 1944) as the catalog tself does. When a lower magntude threshold m 0 s appled for precursors, an adjustment must be made to compensate for the mssng contrbuton from earthquakes wth magntudes below m 0. See Rhoades and Evson (2004) for detals. As n prevous applcatons of the EEPAS model, the baselne model adopted here s fashoned from a model proposed by Jackson and Kagan (1999) n whch the rate densty λ 0 depends on proxmty to past earthquakes (PPE) n the catalog wth M > m c. For detals of the PPE model, see Rhoades and Evson (2004, 2005). Sesmologcal Research Letters Volume 78, Number 1 January/February

3 Fgure 2. Map of southern Calforna showng (A) regon of survellance R (enclosed by dashed polygon) for the EEPAS model and epcenters of earthquakes wth M > 4.95 durng the perod ; (B) epcenters of earthquakes wth M > 2.45 durng the perod APPLICATION TO SOUTHERN CALIFORNIA The EEPAS model was ftted to the Advanced Natonal Sesmc System (ANSS) Worldwde Earthquake Catalog, whch s contrbuted to by members of the U.S. Councl of the Natonal Sesmc System and avalable from the Northern Calforna Earthquake Data Center, to maxmze the lkelhood of forecastng earthquakes over the perod wthn the regon of survellance R n southern Calforna (fgure 2A). To assess the sutablty of the catalog, we examned the catalog of Fgure 3. (A) Cumulatve numbers of earthquakes wthn the whole regon of fgure 2 exceedng certan magntude levels as a functon of tme; (B) Rato of numbers of earthquakes N(M > m + 0.5) / N(M > m) n three-year ntervals and smooth local regresson fts. The sold lne shows the expected value of the rato under catalog completeness and a Gutenberg-Rchter b- value of 1. all earthquakes snce the begnnng of 1932 wth M > 2.45 and wth epcenters n the lattude range N and longtude range W. The dstrbuton of epcenters s shown n fgure 2(B). The cumulatve number of earthquakes exceedng certan magntude thresholds s shown n fgure 3(A). The approxmately constant slope of the curve for M > 2.45 snce about 1980 ndcates homogenety of the catalog n ths range. In order to judge the varaton of magntude completeness wth tme, ratos of the number of earthquakes exceedng certan magntude thresholds have been plotted n fgure 3(B). Let N(M > m) be the number of earthquakes exceedng magntude 112 Sesmologcal Research Letters Volume 78, Number 1 January/February 2007

4 Fgure 4. (A) Frequency magntude relaton for earthquakes n R n the learnng perod The lne shown has slope b = (B) Maxmum lkelhood estmate of the Gutenberg-Rchter b- value n R for , and ts uncertanty (+ 2 standard errors), as a functon of mnmum magntude threshold. m n a tme nterval. Under the assumpton of catalog completeness and a Gutenberg-Rchter b-value of 1, the expected value of the rato N(M > m + 0.5) / N(M > m) s 0.316, shown by the sold horzontal lne n fgure 3(B). If the catalog s adjudged to be suffcently complete when ths rato drops below 0.4, then t s suffcently complete for M > 3.95 from 1932, for M > 3.45 and M > 2.95 from about 1970, and for M > 2.45 from about On ths bass, wth a learnng perod of , the magntude threshold for all earthquakes s set at m 0 = Ths s adequately low, gven that the typcal dfference n magntude between precursory and major earthquakes s a lttle more than one unt (fgure 1A) and that the magntude threshold for targeted earthquakes s m c = There are 79 earthquakes exceedng m c n the learnng perod, wth epcenters nsde R. The data from the earler perod s needed for estmatng parameters of the baselne model as well as provdng data on precursory earthquakes. The Gutenberg-Rchter b-value s an mportant parameter for normalzaton of the EEPAS model. A frequency-magntude plot of earthquakes wthn the regon of survellance over the perod s shown n fgure 4(A), together wth the maxmum lkelhood Gutenberg-Rchter relaton for M > The graph s farly lnear for M < 5.0, wth mnor devatons above the lne at hgher magntudes ndcatng a decrease n b-value. The maxmum lkelhood estmate of the b-value (Ak 1965) vares slghtly wth the choce of lower magntude threshold (fgure 4B) and takes values rangng from 1.01 to 0.87 wth thresholds rangng from 2.45 to For the EEPAS model, earthquakes above magntude 4.0 contrbute most to the rate densty for magntudes 5.0 and above. Hence the value b = 0.96, whch s the estmate for a lower magntude threshold of 4.0, s adopted. The magntude scalng parameter b M was set to 1.0, whch corresponds to perfect scalng of magntudes and was also adopted for fttng of the model to the whole of Japan and the Kanto catalogs (Rhoades and Evson 2005, 2006). The parameters a M, σ M, a T, b T, σ T, b A, σ A, and μ were then all ftted to maxmze the lkelhood of the catalog usng the downhll smplex method (Nelder and Mead 1965). As n prevous applcatons of the model, two weghtng strateges were tred: the equalweghtng strategy w = 1 for all, and a strategy to down-weght aftershocks. The latter strategy has proved superor n most prevous applcatons, but here, as for the Kanto regon wth m c = 4.75, the equal-weghtng strategy s better. In ths case t has a bg advantage, whch amounts to a gan of about 20 n the log lkelhood. Therefore, only the results for the equal-weghtng strategy are reported here. As n most prevous applcatons, the optmal value of μ s 0. Wth b M = 1, μ = 0, and w = 1, the normalzaton functon reduces to the constant 2 σ M η = exp β + a M (6) 2 where β = bln(10). The optmal parameter values are lsted n table 1. These values dffer from those n prevous applcatons n several respects. In partcular, σ M, b T, b A, and σ T are relatvely large, and a M and σ A are relatvely small, compared wth the correspondng values n prevous applcatons. Thus the contrbuton that an earthquake of gven magntude makes to the future rate densty has a magntude dstrbuton wth a relatvely low mean and large spread, a tme dstrbuton wth a relatvely hgh mean and large spread, and a locaton dstrbuton wth a relatvely small spread. A tentatve explanaton for these dfferences s offered, as follows. The magntude threshold m c = 4.95 s well below the largest earthquake n the catalog durng the learnng perod (.e., the M 7.3 Landers manshock of 28 June 1992). In such crcumstances, a szeable proporton of earthquakes exceedng m c can occur as aftershocks of much larger events. For example, n the sx months followng the Landers manshock there were 19 Sesmologcal Research Letters Volume 78, Number 1 January/February

5 TABLE 1 EEPAS Model Parameter Values, Optmzed for the Regon of Survellance, , Except as Noted. * fxed ** see text Parameter Value m * m c 4.95* b 0.96** a M 1.00 b M 1.00* σ M 0.58 a T 1.49 b T 0.48 σ T 0.81 b M 0.61 σ M 0.66 μ 0.00 earthquakes of magntude 5.0 and greater wthn 50 km of the epcenter. The EEPAS model was not desgned to forecast aftershocks, but when t s formally optmzed the parameter values wll adjust to forecast both manshock and aftershocks as well as possble. Ths explans the relatvely low mean and large spread of the magntude dstrbuton. Now, the precursory earthquakes n the Ψ phenomenon usually consttute a Gutenberg-Rchter set of magntudes. The smaller magntudes n ths set would often be aftershocks of larger precursors, and f aftershocks were down-weghted ther contrbuton would be mnmzed. But when many aftershocks have magntudes exceedng m c, the precursory aftershocks can be used to help forecast the aftershocks, f the tme dstrbuton for a gven magntude of precursor s lengthened. Ths explans the relatvely hgh mean and large spread of the tme dstrbuton, as well the superorty of the equal-weghtng strategy. The relatvely small spread of the locaton dstrbuton s attrbuted to the tght clusterng of earthquake locatons close to major faults n southern Calforna (fgure 2). In vew of the above consderatons, the optmzaton of the parameters for m c = 4.95 has probably compromsed the ablty of the model to forecast larger earthquakes wth, say, M > 6, n order to mprove the forecastng of the majorty of the earthquakes n the target range,.e., those between M 5 and M 6. Nevertheless, the parameter values are the best to use for future RELM testng gven the magntude threshold, because the test perod s also lkely to nclude a smlar proporton of aftershocks above the threshold. There were N = 79 earthquakes wth M > m c n the learnng perod. The optmal value of the log lkelhood for the EEPAS model s The ncrease n log lkelhood ( ln L ) s 64.6 compared to the PPE baselne model and compared to the model of least nformaton,.e., a statonary and spatally unform Posson (SUP) model wth Gutenberg-Rchter magntude dstrbuton. Ths gves an nformaton rate per earthquake ( ln L / N ) of 0.82 compared to PPE and 2.04 compared to SUP. The former rate s hgher than n the applcatons to New Zealand and Japan but smlar to that for the whole of Calforna wth m c = The latter rate s hgher than that for New Zealand, the whole of Calforna, and the whole of Japan but slghtly less than that for the Kanto regon wth m c = CONCLUSION The applcaton of the EEPAS model to the catalog of southern Calforna ndcates that the model may be useful for longrange forecastng at moderate as well as large magntudes n ths regon. The nformaton rate per earthquake compared wth tme-nvarant sesmcty models s greater than or smlar to that obtaned n prevous studes. The EEPAS model, as optmzed for the perod , wll be submtted to the RELM testng exercse and used to make one-year forecasts, whch wll be updated annually. Ths study also gves mpetus to future research amed at usng the EEPAS model to further mprove long- and shortterm forecasts of moderate-to-large earthquakes n southern Calforna, and perhaps to provde new models, or new varants of the present model, for RELM testng. Frst, there s the queston of whether the model could be enhanced by adoptng a modfed magntude dstrbuton to explctly allow for aftershocks of forecasted events. By ths means, the present compromse n the parameter values of the tme and magntude dstrbutons to accommodate aftershocks mght be avoded. Secondly, although μ = 0 n the present model, and hence no tme-nvarant baselne model s ncorporated nto the forecasts, t may be advantageous to EEPAS to ncorporate some other tme-nvarant model more nformatve than PPE, f, as seems lkely, such a model should become avalable durng the RELM testng exercse. Fnally, EEPAS could be used n leu of a tmenvarant baselne model for ndependent sesmcty n conjuncton wth an epdemc-type short-term forecastng model to mprove daly forecasts of earthquake probabltes. In such a combnaton, the EEPAS model would be expected to mprove the forecasts of most manshocks, except for those that have mmedate foreshocks. ACKNOWLEDGMENTS Ths research was supported by the New Zealand Foundaton for Research, Scence and Technology under Contract No. C05X0402. Helpful revews of the manuscrpt were provded by Russell Robnson, Warwck Smth, and Bob Smpson. The author s grateful to Matt Gerstenberger for provdng nformaton on the requrements for RELM testng. REFERENCES Ak, K. (1965). Maxmum lkelhood estmaton of b n the formula log N = a bm and ts confdence lmts. Bulletn of the Earthquake Research Insttute, Tokyo Unversty 43, Sesmologcal Research Letters Volume 78, Number 1 January/February 2007

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