50 Physics ofthe Earth and Planetary Interiors, 47 (1987) 5061Elsevier Science Publishers By., Amsterdam Printed in The NetherlandsMaximum-likelihood estimation of hypocenter with origin timeeliminated using nonlinear inversion techniqueNaoshi Hirata and Mitsuhiro MatsuuraGeophysical Institute, Faculty of Science, University of Tokyo, Tokyo 113 (Japan)(Received February3, 1986; revisionaccepted July 2, 1986)Hirata, N. and Matsuura, M., 1987. Maximum-likelihood estimation of hypocenter withorigintimeeliminated usingnonlinear inversion technique. Phys. EarthPlanet. Inter., 47: 5061.A newalgorithm isappliedto inverting arrival timedata forhypocenterlocation. Thealgorithmincorporates bothobserved and prior data from a Bayesianpoint ofview. Wedefine marginal probabilitydensity function(pdDtoeliminate the origintime from the location problem; the posterior pdf of hypocenterparameters is integrated overthewhole range of the origintime.Thebest estimate of the hypocenter isdefined asaset of spatial coordinates whichmaximizes the marginal pdf. Assuming Gaussian errors in both observed and prior data,we obtain a simple algorithm.Estimation errors of parametersare evaluatedby an asymptotic covariance matrix, with which anasymptotic posteriorpdf iscomputed.Thealgorithm is appliedto observed data andis tested. An exampleof analysis is given for aftershocks of the 1969Gifuken-chubu earthquake (M= 6.6) reported by the Japan Meteorological Agency(JMA). The spatial distribution ofthe aftershocks is supposedto be Gaussian with standarddeviation of 15km.A center of the aftershock distribution,whichgives the priorestimates of hypocenters, is also estimated from observed data.Results of the nonlinear inversionof arrival timedataare examinedinterms of the asymptotic posteriorpdf. Wefound that relocated hypocenters of the aftershocks are concentrated in a narrow region of 23 km in width, while thehypocenters previously reported by JMA have awide distribution of 57 km.1. Introduction especially in depth. Forevents insideor on theborder of the network, theoretical estimates ofWidelyused computer programs for hypocenter error give areasonable measure ofactualerrors.location(e.g., Bolt, 1960; Herrinet al., 1962; Aki, However, theestimatessometimes fail for events1965) use aleast-squares iterative techniquede- outsidethe network; the theory leads tothe un-scribed by Jeffreys (1959) and originally attributed derestimation oferrors. Jameset al. (1969) dem-to Geiger (1910).Flinn (1965) discussed a method onstrated by numerical experiments that the errorfor evaluating errors involved in solutions and causedbyinadequate station distribution for adeveloped the theoryof confidenceregions ad- particular event located outside theseismic arrayvanced byGeiger (1910). Peters and Crosson is much larger thanthat caused by random errors(1972) applied Wolbergs (1976) method of predic- in dataor by inadequatemodeling of avelocitytionanalysistothe evaluationof estimationer- structure. Lilwall and Francis (1978) andrors. This method has beenused to investigate~the Duschenes et a!. (1983)investigatedtheproblemcapabilityof earthquake location byan actual foranoceanbottomseismographicarray withaseismic network (e.g.,Ishii and Takagi, 1978). smallnumber ofstations. They showedthat theTheanalysesshowthat theerror involved in resolvingpower of the small arrayis poor forhypocenter locationislarge outsidethenetwork, outsideeventsandthat thetheorybasedonthe0031-9201/87/$03.50 1987Elsevier Science Publishers B.V.51classical least-squaresmethod does not give the 2.1. Observation equations and prior datareal distributionof error.Theproblemof hypocenter location, whichis We consider both P andS arrivals at nstationsone oftheoldest inverseproblems in seismology, of aseismic network. Observational equations arehas been investigatedby many authorsfrom vari- Ito1 1f~(x)1 f 11 + l e p i (1)ous points of view: Buland (1976) applied the QR I P Ialgorithm; Bolt (1970)andJordanandSverdrup L t~j = Lfs(x)] +L1] L; i(1981) usedthe generalizedinversion algorithm, withUhrhammer (1980)discussedtheresolving powerof a small seismic network in termsof thesingular t0 = ~ Tp ~ p1 p2 ~ pfl]valuedecomposition ofacoefficient matrixforalinearsystem. These studiesshowthat theprob- t~= [tO t0 ~0 si s2~. . . snlemof earthquake location sometimes becomes ~ =1 (x), 1p2 (x), ..., f~(x)]under-determined. In suchcases it is important tomake a compromise betweenestimation error and i ,,, = [.f51(x), f52(x), ...,resolution. Inthe context of inversetheory, theunderestimation of estimation errors is simply due 1 = [1,1, ..., j ]Tto the fact that, in the classical least-squares e~= [e~1, e~2,..., epn]Tmethod,a covariancematrixofestimationerrorsinvolves only theerrorsdueto randomnoisein e5 = [e~i, e52, ..., esn]Tdata, not those due to poor resolution (e.g.,Here, thesuperscript T denotes thetranspose ofaMatsuura and Hirata,1982). vectoror a matrix, x =(x, y,z)Tis spatial coor-Jackson(1979) demonstrated that the explicit dinates of a quake focus, t is an origin time, t~ isuse ofprior information always reduces the ill- anobserved arrival timeof P-phaseat the i-thposed linear system to a well-posed linear system.station, t~, is anobservedarrival ofSphase, e~,Jacksons approach wasextended to thenonlinear and e51 are random errors in t~1and t~. f~and ~caseby Tarantola andValette(1982a) andJack- are theoretical travel times of P- and S-waves fromson andMatsuura (1985). Their approaches to the focus tothei-th station. Weassume both e~nonlinear inversion were applied to the earth- and e~are Gaussianwith zeromean and covari-quake location problem by Tarantola andValette ance and E~(1982b) and also by Matsuura (1984).Matsuura (1984) showed a simple algorithm to e~ N(O, Er), e~N(O, E5) (2)find a maximum likelihood solution, whileand they are statistically independentTarantolaandValette(1982b)claimedthedirectuseofa posterior probability density functionof C(e~,c) = 0 (3)hypocenter parameters. Inthe present study, we Thenalikelihoodfunctionof observed dataisapplyMatsuuras method toactuallyobservedgiven bydata to demonstrate that the Bayesian approach iseffective in the routineworkofearthquakeloca- p(t~,t~:x, t)=c exp(s1(x, t)/2) (4)tion. withs1(x, t)2. Bayesianapproach tothe location problem =(t~ f~(x) lt)TE~1(t~ f~(x)it)+(t~ f5(x)lt)TE~1(t~ f5(x) it)Here we derive a maximum likelihood solution (5)by following Matsuura (1984) and introduce somenotations, wherec isanormalizationfactor.If we useonly52observeddata anddefineanoptimal solutionby The integrationin(12)canbe analyticallyper-the hypocentral parameters (x, 1) which maximize formed,becausetheargument of theexponentialthe likelihoodfunction, the solution is identical to function in (10)has a quadratic formwith respectthe classical least-squares solution which mini- to t. Thus we have the marginal pdf ofthe spatialmizes the weighted sum of residuals s1(x,t) in (5). coordinatesWeuse the prior estimate xof x as data, p(x: t~, t~)=cexp[ q(x)/2] (13)subject to unknown errors dx=x+d (6) withAs fortheorigin time t, we havenoinformation q(x) ItP f~(~) ]T [to f~(~) 1I Fl ~prior to observation, and assume a uniform distri- Lt~ f5(~)j Lt~butionfor t. The errors d are supposedtobeGaussian, with zero mean and covariance D +(xo x)TD_1(xo x) (14)d N(0, D) (7) F[I 1 1E~I 0 ]11E~ 0Thenthe joint probability density function(pd~ a E~ 0 E~1] (15) - U ] L Iof the hypocentral parameters prior to observationis given bya=lT(E~1+E~1)l (16)p(x, t)=cexp[ (x0 x)TD1(x0 x)] (8) where Idenotesa 2n X 2nidentity matrixand Uwhere c is a normalization factor. A square root of is a 2n x 2n matrix all of whose elements areeach diagonal element a~of Dis referred to unity.hereafter as a standard error inpriorestimation.We can use as a measure of uncertainty in the 2.3.Maximum likelihood estimatei-th component of prior estimate x.The maximumlikelihood estimate ~, which2.2. Posterior probability maximizes the marginal pdf of (13), minimizesq(x) in(14). For ~ the variation of q(x)withWith the use of Bayes theorem, the joint pdf of respect tox must vanish.Thus we get aset ofhypocentral parameters posterior to observed data equations which ~ satisfiesis expressed as It~ATFI PfP(x)l+D_1(Xo~)o (17)p(x, t: t;, t~)=cp(t~, t~:x, t)p(x, t) (9)Here, p(t~,t~:x, t) is alikelihood functionof whereobserveddata, p(x, 1) is thepriorpdfof hypo-A=grad1~~ (18) central parameters, and cisanormalization fac-tor. Substituting (4) and (8) into (9), we obtain I ~(x) IL S Jx=t~p(x, I: t, t~)= c exp(s2(x, t)/2) (10) and F isdefined by (15). We adopt thefollowingwith iterative algorithmto solve (17)=s1(x, t) +(x0 x)TD~(x0 x) (11) Xk+l=Xk +akCkrk 0 < < 1 (19)withwheres1(x, t) isdefinedby(5). Because wearemostinterestedinthespatial coordinates of the Ck =(A~FA k + D 1) 1 (20)quake focus, we eliminate the origin time byandcalculating the marginal pdf [to f (xk)lp(x: t;, t~)f~p(x, t: t~,t~)dt (12) rk=A~F I + D~(x0xk) (21)t~fS(xk).i53Here, Ak is given by(18) by replacing ~ withXk, is used as a measure of the estimationerror in theand the factor ctk is an adjustable parameter, i-th element of parameters ~. We will referwhichis chosen so that the posterior pdf (13) hereafter to s, as astandarderror inthe i-thbecomes larger thanthat of onestepearlier. The element. We define an asymptotic posterior pdf byiterations start from x =xand proceed until r o\ I / \ \ Ik p~x:t ,t )=cexp~s~x),/2) ~29becomes acceptably small. Theongintime corre- psponding to thehypocenter ~ is defined as whereE1 0 tf (~) s(x)= (x~)TC~(x ~) (30)t=(1/a)[1T1T] _________ p p ( 2 2 )0 E_i t~f( ~) and c is a no rm al iz atio nfactor. To plotcontoursS o fc o ns tantp ro b ab il ity de ns ity fo re ac h parameterwh ic h m axim iz e s ( 10) unde rth e c o ns trainto f x = pair z =(z1, z2 )T, we wil l us e marginal covariancei C2 o fz . Sup p o s e z is a linear combinationof xWe m ay de fine analternative solution (x*, t* ) ~= Bx 3 1which maximizes the posterior joint pdfp(x, t: t~,t~)in (10). However,under the assump- thentio n we m ade , it c an b e p ro ve d th at the latter c =BCBT (32)s o l utio n ( x*, t* ) is ide ntic al to th e former solu-tion (~,1) : we h ave as s um e d th e total ignorance Theasymptotic marginal pdffor zb e c o m e sofprior estimate on tin(8). ~(z: t~,t~) c exp(s(z)/2) (33)2.4. Asymptotic statistics withs ( z ) = (z ~)TC;1(z ~) (34)Under the assumption of linearity for f(x) andaro und x= we canderivea relationbetweenestimates ~andatrue solutionx (Jacksonand ~=B~ (35)Matsuura,1985) where c is a normalization factor. Although HA is= HAx +Xix +He + Kd (23) commonly called the resolution matrix (Jackson,th 1972; Wiggins, 1972), following Jackson and Mat-wi suura (1985), we will call HAand Xi partialH=M1ATF (24) resolution matrices. The importance ofeach dataK=M1D1 (25) type is evaluatedby the trace of each matrixproduct for observational dataand 3M = ATFA +D1 (26) h=~(HA)1~/3 (36)where e arerandom errorsin observeddata withzeromeanandcovariance E, anddarethosein and for prior data,prior data with zero mean and covariance D.From(23) theasymptotic covariance matrix C for k =~ (KI)~~/ 3 (37)estimation error is obtainedas 1=1where3 is thetotal number ofunknown parame-C=HEHT+KDKT=M1 (27~ / ters and thesum of hand kis always unity.where M is defined by (26). Note that Cis identi-cal to Ck in (19) if we put ~=Xk. A square root ofthe i-th diagonal element of the covariancematrix 3. Prior dataC Toapply themethoddeveloped here to actuala1 = (28) data, weneedappropriate priorestimate xofa54hypocenter. Any data independent of arrival time 3.2.Aftershock distribution with unknown~tdata can provide prior information about thehypocenter: amplitude data or, almost equiv- The assumption of (41) maynot be adequate inalently, aspatial distributionofseismicintensity actual cases, where aftershocks may occur only inmaygive the prior information. Aphysical or a narrow region of the fault zone, or the center ofstatistical considerationon asequence of earth- theaftershock distribution may not be located inquake occurrence also gives us priorinformation the closevicinityof themainshock. Then itisonthe hypocenter. For example, thespatial distri- morerealisticto estimatethecenter pfromob-bution of aftershocks is highly correlated with the serveddata. Supposethat we haveobservedmfault zoneof amain shock. Herewe present a aftershocks whosehypocenters xk (k =1,..., m)statistical model on the spatial distribution of will be estimated from msetsofobserved arrivalaftershocks. times t~(k =1,..., m) at nkstations. Each arrivalis related to xkby an observational equation with3.1. A model for occurrence of aftershocks thesame formas (1). Individual hypocenter x~isa randomvariable defined by (38) and (39).Therefore we obtain a set of equations asFirst, we introduce astochastic model fromwhichwecalculateaprobability density p(x) ofx=p+~ ~k....N(9 CE) 1spatial coordinates ofhypocenter. Thestochasticmodel for occurrence of aftershocks is that the t~=fk(x) + lt +e e N(0,hypocenters are Gaussian,with meanpandco- (k =1,..., m) (42)variance CE: for thek-th events= p +~ (38) wherefk(xk)are theoretical travel times for thek-th aftershock, t an origin time and ekrandomwhere ~ is Gaussian with zeromean and covari- noise in t~.We suppose that the errors~ and~,ance CE ek and eh, and~ and eare statistically indepen-dentN(O, CE) (39)Then the joint pdfof x is expressed as C(E~,~h) = 01~fork h (43)p(xk: p)=cexp[ (xk )TC1(k p)/2] C(ek, eh) =0)(40) andwherec isa normalizationfactor.If theparame- C(~,e)=0for any pair of k and h (44)ters pand CE aregiven, the mostprobableesti- It shouldbe noted that since (42) has a newmatex1 of the hypocenter is obviously pand, the unknown parameter p, we have to solve (42)uncertaintyin xt is evaluatedby CE simultaneouslywith respect top and x~(k=x= p} 1,..., m). It will be shown, however, that the= (41) problem canbeseparatelysolvedusing an itera-D CE tive scheme.We know that aftershocksare, in general, distrib- Since ~ ande~arestatisticallyindependent,uted in the vicinityof the main shock. The dimen- the posterior joint pdfof p and x1 ( =1 m)sions of the aftershock area are nearly the same as given t~(k =1,..., m) becomesthefault dimensionsofthemain shock, which isroughly estimatedfroma magnitudeofthemain p(p,x, t~:t~k=1,...,m)shock. Thus wemay takenthehypocenter of themain shock as p, and the square of its fault =cexp[ q1(p, xe, t~ k=1,..., m)/2]dimensionas the diagonal elements of CE. (45)55with withqi(p,xk, tk; k=1,...,m) (53)=~ (p_xk)TC~1(p_xk) , k=1 . .k =1 Since the maximumlikelihoodestimate x kfor them k-th events is coupled with those for all others~h+ ~ (t~fk(x)11k)T (h ~ k), we shouldsolve (52) for alltheXksimul-k=1 taneously. The non-linear eq. 52, however, has thexE~(t~ fk(x~) ltk) (46) sameformasthat of(17)if ~isgiven. Further-more, x can beinterpreted as an averageof hypo-In thesame way as in section 2.2, we eliminate the centers. Thus,wewill use aniterativescheme toorigin timestk.The marginal pdf is given by solve (52): first, weset 5~to the hypocenter of thek main shock. Second, (52) is solved with respect tox: t~k= 1 m) each~k with given ~using thescheme procedure=~ ~ dti dtm as discussed in section 2.3. Third, ~ is corrected byJ L (53) with the estimated*k (k =1,..., m). Then aI k k new set of ~k (k =1,..., m)is determined for theXpIp,x t :t k=1 ... m) . . . . / x, and the iteration will proceed until x converges.= cexp[ q2(p, xk; k = 1 m)/2] (47) The iterative procedure suggests that if we havealarge numberof aftershoks whosehypocenterswith are determined, thenwe canuse anaverage ofq( xk.k = 1 m ~ those hypocenters as the prior estimate of the2. P a.., / hypocenter of another aftershock because the=~ (t~~k (xk) )TFk (t~~k (xk)) average is scarcely affected by the new aftershock.k 1 That is,inpractice, it issufficienttoestimatepfromapart of the data set using the scheme+~ (p x1o)TC~1(p xk) (48) discussed above.Once we have obtaineda reliablek=1 estimate of p, thenwe canuse it forthe priorwhere Fkhas the same form as F in (15). Thus, the estimate of x for other events.maximumlikelihood estimates ~ and ~i mustsatisfy the following equations4. Nwnerical example~C1(1i~)=0 (49)k=1 E 4.1. HYPNLIand We present an example with actual data toA~Fk(t~ fk(~)) + C~(1i ~)=0 show howour method of hypocenter location1k=1 m~ so~ works. All the following calculations are per-/ . / formed by a computer program for asingle eventwhereAk hasthe sameformas Ain(6). From location algorithm, whichis referred toas HY-(49)~iis expressedin terms of ~! as PNLI (Hypocenter determinationby anonlinearm inverse method) hereafter. Input data for HY-p =~ xk/m (Si) PNLI are as follows: (1) control parameters whichk 1 are independent of individual earthquakes. TheSubstituting (51) into (50), we obtain parameters specifycoordinates (latitude, longitudeA~Fk(t~ fk(~.