Source Depth and Mechanism Inversion at Teleseismic ...rses.anu.edu.au/~brian/PDF-reprints/2000/bssa.90.1369.pdf · Bulletin of the Seismological Society of America, 90, 6, pp. 1369–1383,

1369

Bulletin of the Seismological Society of America, 90, 6, pp. 1369–1383, December 2000

Source Depth and Mechanism Inversion at Teleseismic Distances

Using a Neighborhood Algorithm

by K. Marson-Pidgeon, B. L. N. Kennett, and M. Sambridge

Abstract We performed nonlinear waveform inversion for source depth, timefunction, and mechanism, by modeling direct P and S waves and correspondingsurface reflections at teleseismic distances. This technique was applied to moderatesize events, and so we make use of short period or broadband records, and utilizeSV waveforms in addition to P and SH. For the inversion we used a direct searchmethod called the neighborhood algorithm (NA), which requires just two controlparameters to guide the search in a conceptually simple manner, and is based on therank of a user-defined misfit measure. We use a simple generalized ray scheme tocalculate synthetic seismograms for comparison with observations, and show that theuse of a derivative-free method such as the NA allows us to easily substitute morecomplex synthetics if necessary. The source mechanism is represented in two dif-ferent ways; the superposition of a double-couple component with an isotropic com-ponent, and a general moment tensor with six independent components. Good resultsare obtained with both synthetic input data and real data. We achieve good depthresolution and obtain useful constraints on the source-time function and source mech-anism, including an isotropic component estimate. Such estimates provide importantdiscriminants between man-made events and earthquakes. We illustrate inversionwith real data using two earthquakes, and in both cases the source parameter estimatescompare well with the corresponding centroid moment tensor solutions. We alsoapply our technique to a known nuclear explosion and obtain a very shallow depthestimate and a large isotropic component.

Introduction

Accurate estimates of source depth and source mecha-nism for small to moderate size seismic events are importantfor a number of reasons, in particular for monitoring theComprehensive Nuclear Test Ban Treaty (CTBT). The chal-lenge for monitoring the CTBT is discrimination betweennumerous smaller earthquakes and nuclear explosions, in or-der to provide low-yield threshold monitoring. Accuratedepth estimates can provide an important discriminant be-tween man-made events and earthquakes. If the depth canbe accurately determined to lie outside the range of currentdrilling techniques, then the event is unlikely to be man-made. However, if the source depth is very shallow, furtherinvestigation may be required to determine whether it is anearthquake or a nuclear explosion. A further important dis-criminant for explosions is whether the source mechanismhas a significant isotropic component. A number of tech-niques have been developed that provide routine estimatesof source location and mechanism for medium to large seis-mic events using long-period body and surface waveforms(e.g., Dziewonski et al., 1981; Sipkin, 1994; Kawakatsu,

1995), as yet such automated techniques are not well devel-oped for smaller events.

The time separation of the direct body waves (P and S)and their corresponding surface reflections (pP, sP and pS,sS) increases rapidly with increasing source depth, and onlyslightly with epicentral distance. The relative amplitudes ofthe phases depend on the source mechanism, and the positionof the takeoff angles in relation to the source radiation pat-tern (e.g., Pearce, 1977, 1980). Thus significant constraintson source depth and mechanism can be obtained by mod-eling the composite effect of the direct and surface-reflectedphases. For large events recorded at long periods, it is im-possible to distinguish between the direct waves and theirsurface reflections for very shallow sources, which limits thedepth resolution (e.g., Dziewonski et al., 1981). However,for smaller events recorded at higher frequencies, it is pos-sible to identify a distinct interference pattern produced bythe different arrivals, even for source depths of only a fewkilometers. For deeper sources the depth phases separateclearly in time, and the source depth can be estimated di-

1370 K. Marson-Pidgeon, B. L. N. Kennett, and M. Sambridge

rectly. However, for shallow sources waveform modeling ofthe depth phases is needed.

The source parameters of moderate size earthquakeshave commonly been determined using least-squares inver-sion of long-period regional seismograms (e.g., Wallace andHelmberger, 1982). However, the use of regional waveformshas a number of limitations, such as poor depth resolution.The use of teleseismic data has a number of advantages, butuntil recently, most methods for determining the source pa-rameters of shallow teleseismic earthquakes have only beenapplied to large magnitude events (e.g., Langston, 1976; Na-belek, 1985). Such methods use a least-squares inversion oflong period P and SH waveforms to determine the sourcedepth, time function, and fault orientation. Recent work onobtaining depth and mechanism estimates at far regional andteleseismic distances for smaller events includes that ofGoldstein and Dodge (1999), who used P-waveform mod-eling.

In this article, we attempt to determine the source pa-rameters (depth, mechanism, and time function) of shallow,moderate size events using waveform inversion of teleseis-mic data. The use of broadband stations in a CTBT contextmeans that both P and S waves are likely to be recorded formoderate size events, even at teleseismic distances. Wemodel P and S and their corresponding surface reflectionsusing generalized ray theory (Langston and Helmberger,1975). We use short period or broadband velocity recordsof moderate size events, allowing better depth resolutionthan can be achieved using long-period waveforms. We donot make any correction for instrument response. Moderatesize events are not likely to be recorded teleseismically atmany stations, so it is important to exploit fully all the avail-able information. To this end, we make use of SV informa-tion in addition to P and SH information. A relative ampli-tude approach is used for a number of reasons: firstly, theuse of high-frequency data requires a detailed knowledge ofstructure in order to accurately model absolute amplitudes;secondly, array beams are more likely to preserve relativeamplitudes rather than absolute amplitudes.

The source mechanism is represented in two differentways. The first is in terms of the superposition of a double-couple with an isotropic (explosive) component. The double-couple is specified by strike angle, dip angle, and rake angle,and the seismograms depend nonlinearly on each of theseangles. The second representation is through a general mo-ment tensor with six independent components (e.g., Aki andRichards, 1980) that allows for a possible compensated lin-ear vector dipole component in addition to double-coupleand isotropic components. At fixed depth, the seismogramsdepend linearly on the moment tensor components, but non-linearity is introduced through variable source depth. Forboth mechanism representations we can test the significanceof any isotropic estimate by applying restrictions to theamount of isotropic component allowed in the inversion.

We use a fully nonlinear method of inversion rather thana linearized inversion to provide greater flexibility and avoid

possible dependence on the starting parameters. Since weare using fairly high-frequency data, we may need to makeallowances for a broad suite of crustal phases in order toaccurately model the observed seismograms. The use ofmore complex synthetics in a linearized inversion schemewould make the partial derivatives difficult to calculate.However, the use of a derivative-free method overcomes thisproblem. For the inversion we use a direct search methodcalled the neighborhood algorithm (NA) (Sambridge,1999a,b), which makes use of geometrical constructs knownas Voronoi cells to drive the search in parameter space. Thecells are used to construct an approximate misfit surface ateach iteration, and successive iterations concentrate sam-pling in the regions of parameter space that have low datamisfit. The algorithm is conceptually simple and its ease ofuse encourages us to apply it to the problem of waveforminversion for source parameters. We are able to easily testvarious assumptions and explore the use of different earthstructure models and seismogram calculation schemes ifnecessary, which would be more difficult if the inversionmethod relied on the calculation of partial derivatives.

We verify the inversion scheme with a number of syn-thetic tests. The use of a simple generalized ray schemeproves to be effective for inversion when compared withmore accurate synthetic seismograms. Synthetic tests with aperturbed source location demonstrate the relative robust-ness of the inversion scheme. The NA approach also workswell with real data, and we illustrate the performance fortwo well recorded earthquakes with moderate magnitude, forwhich comparison can be made with the centroid momenttensor solutions produced by Harvard University. The firstillustration is for an event off the east coast of Honshu, Ja-pan, for which improved results can be produced with anadaptive inversion allowing for an oceanic layer above thesource. The second illustration is for an event in southernXinjiang, China, for which inversion using a standard ve-locity model works well. We also apply our technique to aknown nuclear explosion and obtain a very shallow depthestimate and large isotropic component.

Teleseismic Synthetic Seismogram Calculation

Accurate synthetic seismograms at teleseismic distancescan be calculated by using a modified reflectivity approachwith a slowness integral adapted to the epicentral distanceto the receiver (Marson-Pidgeon and Kennett, 2000). Thisapproach allows a full treatment of conversions and crustalreverberations but requires significant computational effort.

In order to evaluate seismograms for varying sourcedepth and mechanism rapidly so that a fully nonlinear in-version scheme can be employed, we use a simplified ap-proach for calculating synthetic seismograms at teleseismicdistances. Generalized ray contributions are generated forthe direct waves (P and S) and their surface reflected phases(pP, sP and pS, sS) following the method of Langston andHelmberger (1975).

Source Depth and Mechanism Inversion at Teleseismic Distances Using a Neighborhood Algorithm 1371

For the P-wave response, we first calculate the follow-ing expression for the near-source effects, including the free-surface reflections,

S ixT S ixTP pP[R] � [R ] e � [R ] [R ] eP D P U P F PPS ixTsP� [R ] [R ] e (1)U S F PS

where and are the upward and downward source ra-S SR RU D

diation terms, RF is the free-surface reflection matrix nearthe source, and TP, TpP and TsP are the travel times for P,pP, and sP, respectively.

For the S-wave response the following expressions arecalculated:

S ixT S ixTS pS[R] � [R ] e � [R ] [R ] eS D S U P F SPS ixTsS� [R ] [R ] e (2)U S F SS

S ixT S ixTS sS[R] � [R ] e � [R ] [R ] e , (3)H D H U H F HH

where TS, TpS, and TsS are the travel times for S, pS, and sS,respectively.

The displacement at the receiver corresponding to eitherthe P or S response is then calculated using the followingexpression:

�i p/2u(r, 0, x) � �W R(p, x)Q(x)ixM(x)e , (4)F

which allows for a phase shift of p/2 due to complete re-flection. The free-surface amplification factor at the receiveris given by WF, and the term �ixM(x) represents the far-field source-time function, which we specify to be a trape-zoid. At teleseismic distances and for shallow sources, a sin-gle ray parameter is used, where p is the geometric slownessfor the direct wave. The effect of attenuation is given by thefollowing,

1 xQ(x) � exp �ixt* lnm� � ��p 2p (5)

1�exp � |x|(t* � t* � t*)s m r2� �

where the effects of velocity dispersion are included in themantle. The loss factors for the source, mantle, and receiverstructures (given by respectively) are calculatedt*, t*, t*s m r

using an attenuation model. In this study we use the ak135velocity model of Kennett et al. (1995), with the correspond-ing attenuation profile of Montagner and Kennett (1996). Wemake an allowance for different source and receiver struc-tures with a common mantle structure beneath a separationlevel at around 200 km depth.

The Neighborhood Algorithm

The NA is a direct search method of inversion that pref-erentially samples those regions of a multidimensional pa-rameter space that have acceptable data fit. It has the abilityto search efficiently by sampling simultaneously in differentregions of parameter space. The NA shares some character-istics with other nonlinear methods such as simulated an-nealing and genetic algorithms. Such methods often have anumber of control parameters that have to be tuned for eachproblem, whereas the NA requires just two control parame-ters that guide the algorithm in a conceptually simple man-ner. A further advantage of the NA over other direct searchmethods is that only the rank of the misfit function is usedto compare models. This is of particular significance for seis-mic waveform inversion, as it avoids problems associatedwith scaling of the misfit function and allows any type ofuser-defined misfit measure to be employed. These pointsare discussed in more detail in Sambridge (1999a).

Unlike previous methods, the objective is to generatean ensemble of models with acceptable data fit rather thanseeking a single optimal model. The entire ensemble canthen be used to extract robust information about the modelparameters, such as resolution and trade-off. This is per-formed within a Bayesian framework and is discussed inmore detail in Sambridge (1999b). Even though global op-timization is not the primary objective of the NA, it has beenshown to work well in this respect for both receiver functioninversion (Sambridge, 1999a) and seismic event location(Sambridge and Kennett, 2001). Here we use the NA to seeka good match to seismograms recorded at teleseismic dis-tances, and so obtain estimates of the source depth, timefunction, and mechanism. We do not make use of the en-semble inference at this stage.

The NA is conceptually simple and makes use of simplegeometrical constructs. It exploits the self-adaptive behaviorof Voronoi cells, which are nearest neighbor regions definedby a suitable distance norm (Voronoi, 1908). Voronoi cellsare used to drive the search in parameter space. The size andshape of each Voronoi cell is uniquely determined by allprevious samples, and the cells are used to construct an ap-proximate misfit surface at each iteration. The behavior ofthe search algorithm is controlled by two parameters, ns andnr, and can be summarized by the following:

1. First, an initial set of ns models are generated randomly,and a misfit measure is calculated for each model.

2. Next, the nr models with the lowest misfit are determined,and a uniform random walk is performed inside their Vo-ronoi cells in order to generate a new set of ns models.

3. The above steps are then repeated by calculating the mis-fit function for the most recently generated ns models,and sampling inside the new Voronoi cells of the nr mod-els with lowest misfit. At each stage the size and shapeof the Voronoi cells automatically adapt to the previouslysampled models. This allows each successive iteration to


concentrate sampling in regions of parameter space thathave low data misfit.

The two control parameters, ns and nr, need to be tunedfor each specific problem. For small ns and nr, the algorithmis fairly localized in nature, whereas for larger ns and nr, thealgorithm is more exploratory in nature. Note by definition,nr � ns. For this application to waveform inversion, we havefound that broad sampling with a large nr value gives themost satisfactory results. In practice, we also specify a max-imum number of iterations and take the best fitting solutionfound as our preferred model.

Waveform Inversion Scheme

We perform waveform inversion using the NA for thefollowing model parameters: source depth, source-timefunction, and source mechanism. The source-time functionis specified as a trapezoidal function in time and is definedby the initial rise time, with the three time segments in theratio 1:3:1. This is suitable for small events, but more com-plex forms with more parameters can be used for largerevents. For the source mechanism we have experimentedwith two different styles of representation; a double-coupleplus isotropic component representation, and a general mo-ment tensor representation.

When we use the double-couple plus isotropic compo-nent mechanism, we have six parameters to be determined,and are therefore working in a six-dimensional parameterspace. For each of these parameters we have to specify upperand lower bounds to set up the search volume for the neigh-bourhood algorithm. For the strike angle we search between0 and 360�, for the dip angle we search between 0 and 90�,and for the rake angle we search between 0 and 360�. Theisotropic component is represented by an isotropic momenttensor, with a weighting factor between 0 and 5 to allow fora large range of values.

In contrast, the moment tensor representation requireseight parameters and the six moment tensor componentsneed to be able to take both positive and negative values.The search in this case will lie in an eight-dimensional pa-rameter space, and we anticipate that some combinations ofmoment tensor components will likely be better determinedthan others, which will impose some extra structure on thecharacter of the misfit function in the eight-dimensionalspace. These lineations have the potential to influence thenature of the search process for an optimal misfit.

For discrimination purposes, we need to provide accu-rate constraints on the significance of any isotropic compo-nent in the source mechanism estimate. The source inversioncan be performed with no restriction on the allowable iso-tropic component, or alternatively with a restriction im-posed. If the source mechanism is represented as the super-position of a double-couple and an isotropic component, itis simple to impose the restriction that the source mechanismis a pure double-couple by limiting the range of values the

isotropic component is allowed to take. When using a mo-ment tensor representation for the source mechanism we canimpose the restriction of zero trace (i.e., Mxx � Myy � Mzz

� 0), and consequently have a seven-dimensional parameterspace. The fit of the two types of inversion can then be com-pared in order to determine the significance of any estimatedisotropic component. For the case of an earthquake, if a largeisotropic estimate is obtained for an unconstrained inversion,then we would expect a lower misfit to be obtained whenthe isotropic component is restricted. On the other hand, foran explosion, we would expect to obtain a lower misfit whenno restriction is imposed on the isotropic component.

The forward modeling for each set of model parametersis performed using generalized ray theory as previously de-scribed. We need to quantify the misfit between the observedand synthetic seismograms for each model. The model issought that most closely matches the synthetics to the ob-served data, but we do not know the exact probability dis-tribution for the data errors, therefore we cannot define anabsolute measure of data misfit. Since we are using a deriv-ative-free inversion scheme, based on only the rank of themisfit function, we can investigate the use of different misfitmeasures. We are not restricted to the conventional squaredresidual measure (i.e., a L2 norm) which is commonly usedfor linear inversion. We consider the following misfit mea-sures: normalized cross-correlation and Lp norm measureswith p � 1, 1.5, 2. In each case, we apply a weighting factorto each station, depending on the signal to noise ratio for theparticular record.

The normalized cross-correlation measure we use isgiven by,

Ns1 obs synM � (1 � C(u , u ))S (6)� n n nn�1N �S�s

where Ns is the number of stations, is the nor-obs synC(u , u )n n

malized cross-correlation between the observed and pre-dicted seismograms at the nth station, calculated for an ap-propriate window, Sn is the signal-to-noise ratio at the nthstation, and �S� � is the averaged signal-to-noise�1N R Ss n n

ratio across all the stations.The Lp norm misfit measure we use is given by the

following expression,

t2N 1/ps1 obs syn pM � S dt|u (t) � u (t)| (7)p � n n n� � �n�1N �S� t1s

where p � 1, 1.5, 2.Tests with synthetic data show that the most stable re-

sults are obtained when using a L2 norm misfit measure. TheL2 norm measure converges to the correct result in a mod-erate number of iterations, even when multiple minima arepresent. In contrast, the L1 and L1.5 norms tend to lead tolocal minima instead, in this case the more robust statisticsare too tolerant of misfits in the seismograms. With a nor-


Table 1Station Locations Relative to the Western Iran Event

Station Distance (�) Azimuth (�)

KEV 34.97 347TOL 41.41 291SCP 88.72 323SLR 65.47 201COL 77.56 7

MAJO 68.41 60BJI 51.26 65

LZH 43.29 74KMI 46.70 89

malized cross-correlation measure, it is possible to fit thewaveforms at all but one station, for example, and therebyget a low misfit value. The normalized cross-correlationmeasure tends to favor those models that fit the data well,whereas the L2 norm measure tends to penalize those modelsthat do not fit the data. In the following examples we haveused an L2 norm misfit measure.

Since we are using a relative amplitude approach in thisstudy, both the observed and synthetic seismograms are nor-malized to have unit maximum amplitude prior to inversion.This means that factors such as the scalar moment do notneed to be included, and sensitivity to site amplification ef-fects is minimized. In order to overcome the problem oftravel-time variability due to factors such as lateral hetero-geneity and source location errors, the data and syntheticsare aligned prior to inversion to avoid spurious fits. Eachobserved and synthetic seismogram is aligned to an arbitraryreference time using a cross-correlation technique. The timewindow used in the inversion is 51.2 sec for both P and S,and the window starts 20 sec before the time of arrival forthe observed examples, and 5 sec before the time of arrivalfor the synthetic examples. A longer time window may beneeded for deeper sources, which have a greater time sepa-ration between the various phases. The synthetic seismo-grams are calculated for a broad range of frequencies, be-tween 0.01 and 2 Hz. No filtering was performed for thesynthetic tests, however, the real data used in the observedexamples were filtered using parameters discussed later.

The control parameters (ns and nr) used in this appli-cation of the neighborhood algorithm were obtained aftertrials over a limited range of values. Those parameters thatgive consistently stable results with good convergence werechosen. We are able to achieve a good match to the seismicwaveforms, with good correspondence between the best-fitmodel and the input model, even though exhaustive testingof the control parameters has not been undertaken. We havechosen a value of 16 for the parameter ns, with a value of 8for the parameter nr. The neighborhood algorithm is theninitiated by generating 16 random models in parameterspace, and the same number of models are generated at eachsubsequent iteration by resampling the 8 Voronoi cellswithin which the lowest current misfits are attained. Thealgorithm continues for 40 iterations, and tests show thatperforming more iterations does not significantly improvethe best-fit model.

The following steps are performed in the inversion forsource depth, mechanism, and time function:

1. First, we need an estimate of the source depth to initiatethe inversion, and this is taken from a source locationprocedure also using the NA for available arrival-timeinformation (Sambridge and Kennett, 2001) (or from bul-letin sources if these are available). We also need to takethe estimates of distance, azimuth, and expected arrivaltime at each station from the initial location estimate.

2. This hypocentral estimate is then linked to the waveform

inversion, which uses the neighborhood algorithm tosearch a range of source depths around the depth esti-mate, as well as searching a range of source-time functionparameters and source mechanisms.

3. Forward modeling is performed for each set of modelparameters using generalized ray theory, and the misfitbetween the observed and predicted seismograms is cal-culated.

4. Once the algorithm has completed the specified numberof iterations, the best fitting source depth, time function,and mechanism is taken as our preferred model.

Synthetic Tests

Synthetic tests have been performed to investigate theaccuracy of the depth and source mechanism inversionscheme. We have simulated an event for which a sourcemechanism determined from long-period recordings was re-ported in the Bulletin of the International SeismologicalCenter (ISC). The sample event occurred in western Iran, on20 June 1990. We use the National Earthquake InformationCenter (NEIC) moment tensor solution to calculate the syn-thetics, with a source depth of 17 km. A trapezoidal source-time function is used with a rise time of 1.5 sec. The ob-served noise-free seismograms are calculated using bothgeneralized ray theory, and an adaptation of the reflectivitymethod, including the effect of shallow reverberations andconversions (Marson-Pidgeon and Kennett, 2000). We alsoperform tests where errors are introduced into the sourcelocation estimate, to determine how robust the inversion isin the presence of noise. In each case the best-fit model iscompared to the input model to determine the accuracy ofthe inversion scheme. The inversions are performed usingnine stations in the teleseismic distance range, whose loca-tions are given in Table 1.

Simple Tests

We undertake a varied set of tests with synthetic datato verify the effective operation of the NA. We first performtests of the inversion scheme by calculating observed seis-mograms using the simple generalized ray scheme previ-ously described. We generate seismograms for each station


Table 2Comparison of Inversion Results with the Input Model, Using a

Double-Couple Plus Isotropic Term Representation of theSource Mechanism

TypeDepth(km)

SourceFunction

(sec)Strike

(�)Dip(�)

Rake(�)

IsotropicWeighting Misfit

Model (faultplane 1)

17.0 1.5 202 38 156 0.0

Model (faultplane 2)

311 76 54

P inversion 16.7 1.6 120 88 254 0.8 0.091Constrained P

inversion17.1 1.5 197 37 155 0.0 0.006

S inversion (T) 17.2 1.5 113 79 302 2.0 0.003S inversion

(R and T)16.7 1.6 197 30 155 0.1 0.008

P and Sinversion*

17.8 1.5 119 87 301 0.5 0.17

Constrained P andS inversion*

17.7 1.6 312 73 61 0.0 0.12

*The observed seismograms have been calculated using the full repre-sentation.

using the epicentral distances and azimuths shown in Table1 with the NEIC mechanism and the ak135 reference model.We then attempt to match these observed seismograms tothe predicted seismograms with the aid of the NA.

First, we use a representation of the source mechanismin terms of a double-couple plus an isotropic component,and use only the P waveforms as recorded on the verticalcomponent. For an unconstrained inversion, quite a largeisotropic component is obtained (Table 2), due to the limitedsampling of the source radiation pattern with the availablestation distribution. The fault plane parameters are poorlyconstrained due to the dominance of the isotropic compo-nent, however, the source depth and time function are wellconstrained. If we impose the restriction of zero isotropiccomponent and perform the inversion again, the resultingsource mechanism is a much better representation of theNEIC mechanism, with fault plane parameters which areclose to the input values (Table 2). A somewhat lower misfitvalue is obtained, which indicates that a pure slip dislocation(an “earthquake”) is compatible with the data.

The S-wave group provides a somewhat different sam-pling of the source radiation, and it is therefore useful toexamine how an inversion based solely on S can perform.Commonly only the transverse component of S is used insource inversion. However, the use of SH waveforms alonecannot provide any constraints on the isotropic component(Table 2). When we use both the radial and transverse com-ponents of the S-wave data, and so sample both SV and SHinformation, we obtain a much better result from the inver-sion, since now we have much more information on the ra-diation pattern. The isotropic component is negligible, andthe source mechanism is well constrained, along with thesource depth and time function (Table 2). This shows thatthe inclusion of SV-wave data can provide the extra infor-mation needed to constrain the source mechanism.

We have shown that good results are obtained usingsimple synthetics, therefore, we proceed with a more real-istic set of tests where we calculate the observed seismo-grams using the full teleseismic representation described inMarson-Pidgeon and Kennett (2000). The use of the morecomplete calculation scheme introduces some noticeablechanges in the character of some of the waveform segments,as we now include shallow reverberations and conversions.We continue to use the simplified approximations in the in-version, because of the speed of computation, but now theobservations and predictions are generated using very dif-ferent schemes, which provides a more realistic test of theinversion scheme.

We perform joint inversions of both P and S waveformsfor the new dataset. We use the vertical component of P andboth the radial and transverse components of S. The com-bined S-wave data were given half the weighting of the P-wave data when calculating the misfit function. We haveused both of the representations of the seismic source in theinversions, with differing dimensionality of the parameterspace.

The first two tests use the representation of the sourcemechanism in terms of a double-couple plus an isotropiccomponent. First, an unconstrained joint inversion is per-formed, and a significant isotropic component is introducedwith a substantial departure from the NEIC solution (Table2); however, the source depth and time function are wellconstrained. We can make a check as to whether the data iscompatible with a pure slip dislocation source (an “earth-quake”) represented by just a double-couple. We thereforeundertake a second inversion with the isotropic componentconstrained to zero (effectively a five-parameter space). Thestrike, dip, and rake angles are well constrained, with valuesthat are very close to the input values. The results for thesource depth and time function agree well with the originalinput. A lower misfit is obtained, and the waveform matchis improved when the isotropic component is constrained tozero and thus the pure double-couple model is to be favored.

For the same dataset as the last two tests, we carry outa joint P- and S-waveform inversion using an eight-param-eter space where the mechanism is represented in terms ofsix independent moment tensor components. The sourcedepth and time function fit well when compared with theinput model (Table 3). A comparison between the observedand predicted seismograms is shown in Figure 1, and anexcellent waveform match is achieved. It is difficult to judgeimmediately the comparison of the moment tensor compo-nents, however, plotting the radiation patterns reveals thatthe recovered source mechanism is very similar to the ob-served mechanism except for a small isotropic component.In order to compare the isotropic components, we calculatethe normalized trace of the moment tensor. This gives avalue of 0.4 for the isotropic component, which is smallerthan that obtained using the alternative source mechanism


Table 3Comparison of Inversion Results with the Input Model Using a

Moment Tensor Representation of the Source Mechanism

TypeDepth(km)

Sourcefunction

(sec) Mxx Myy Mzz Mxy Mxz Myz Misfit

Model 17.0 1.5 0.34 �0.73 0.39 �0.27 �0.63 �0.36P and S

inversion16.8 1.7 0.61 �0.62 0.67 �0.24 �0.85 �0.51 0.14

The observed seismograms have been calculated using the full represen-tation.

representation (which gives a value of 1.0). This is probablybecause the moment tensor representation allows for a com-pensated linear vector dipole component, whereas the alter-native representation does not, therefore requiring a largerisotropic component in order to fit the data. We note that themisfit level achieved is slightly better than for the uncon-strained P and S inversion with the double-couple and iso-tropic component.

Perturbed Source Location

In this set of synthetic tests we have perturbed thesource location, thus providing a more realistic test of theinversion scheme in the presence of errors and noise. Weuse three test cases, where the latitude and longitude ofthe source location are perturbed by random values in therange �0.05�, and the source depth is perturbed by randomvalues in the range �5 km. These values were chosen toreflect the errors in the source location for this well con-strained event, as reported in the Bulletin of the ISC.

The perturbed source location is used to calculate theobserved seismograms, while the unperturbed source loca-tion (Table 1) is used to calculate the predicted seismograms.In this way we can simulate the effect of errors in the sourcelocation, where the actual hypocenter differs from the esti-mated hypocenter that is used in the waveform inversion.The observed seismograms are calculated using the full tele-seismic representation. The inversion is performed in twoways: firstly with the waveforms shifted in time in order toalign the direct arrivals on both the observed and predictedseismograms, and secondly without correcting for the timeshifts introduced by the source location perturbation.

In the first set of tests we represent the source mecha-nism in terms of a double-couple plus an isotropic compo-nent, and perform inversions using P-wave data as recordedon the vertical component. Inversions are performed withthree different source perturbations, for both aligned and un-aligned data. The source depths and time functions recov-ered from the various inversions match the input values well.For the aligned data the recovered source depths are within�0.9 km of the input values, and the source-time functionsare within �0.06 s. For the unaligned data we match thesource depths to within �2.3 km, and the source time func-tions to within �0.06 s. In most cases a large isotropic com-

ponent is obtained due to the lack of constraint on this pa-rameter from the use of P-wave data alone, with the givenstation distribution. Therefore the strike, dip, and rake anglesare poorly constrained due to the dominance of the isotropiccomponent in these cases. One test case with unaligned dataobtained a fairly small isotropic component, and so thestrike, dip, and rake angles are well constrained. Two out ofthe three test cases obtained lower misfit values with un-aligned data than with aligned data.

We use one of our test cases to provide an illustrationof inversion with a perturbed source location. The sourcelocation has been perturbed by �0.05� in latitude, 0.02� inlongitude, and 1.8 km in source depth (giving a source depthof 18.8 km). We use a moment tensor representation of thesource mechanism and perform a joint inversion of both P-and S-wave data. We do not correct for the time shifts intro-duced by the perturbation so that timing errors of up to0.5 sec are present.

We first perform an inversion using P, SV, and SH-wavedata with a moment tensor representation and an uncon-strained isotropic component. The resulting source depth andtime function (19.4 km and 1.5 s respectively) match theinput values well. Plotting the radiation patterns reveals avery close match to the input source mechanism (Fig. 2),with a negligible isotropic component. This is an ideal case,assuming we have P, SV, and SH information available,however this may not always be so. We perform anotherinversion using only P and SH information to simulate amore realistic case. The main difference now is that we haveless constraint on the isotropic component due to the loss ofSV information. In fact, we now obtain a significant isotropiccomponent, with a normalized trace of 0.2. The recoveredsource mechanism is a fairly good representation of the inputmechanism (Fig. 2), even though it contains a significantisotropic component. The resulting source depth of 19.2 kmmatches the model well, as does the recovered source-timefunction of 1.6 sec. A comparison between the observedseismograms and the predicted seismograms calculated fromthe best fit model is shown in Figure 3. A good match isobtained for both the vertical component of P and the trans-verse component of S, as would be expected due to the closematch between the recovered model parameters and the in-put parameters.

In order to determine whether the estimated isotropiccomponent is real, or just an artifact of the inversion, weperform the same inversion again but with the constraint ofzero trace imposed. Thus we have a seven-dimensional pa-rameter space to search, as we now have only five indepen-dent moment tensor components to determine, rather thansix. As can be seen in Figure 2, the recovered source mech-anism provides a better match to the input mechanism thanthe previous inversion with no constraint on the isotropiccomponent. Again the source depth and time function arewell constrained with recovered values of 19.0 km and 1.6sec, respectively. A slightly better seismogram fit is achieved(Fig. 3), and consequently a lower misfit value is obtained


Figure 1. Comparison between the com-plex observed (black traces) and predicted(grey traces) seismograms for the western Iranevent, using a moment tensor representation ofthe source mechanism. The vertical componentof P (left), and the radial (middle) and trans-verse (right) components of S are displayed.

(0.179 compared to 0.194). Thus it seems that the estimatedisotropic component was an artifact of the inversion, and isnot significant.

Illustrations Using Observed Data

Honshu Event

For our first illustration of source mechanism and depthinversion with observed data, we use an event that occurredoff the east coast of Honshu, Japan (Fig. 4) on 14 May 1998.This event has an estimated mb of 5.8, so is large enough tohave a centroid moment tensor (CMT) solution (Dziewonskiet al., 1999b), providing a model with which we can com-pare our inversion results. The estimated depth from theCMT solution is 19 km. The waveforms have been obtainedfrom the Incorporated Research Institutions for SeismologyData Management System (IRIS DMS). Ten stations are usedin the inversion, however, only five of the stations have suit-able S waveforms due to low signal-to-noise ratios. Thewaveforms are bandpass filtered in the range 0.01 Hz to 0.6Hz before inversion.

Since this earthquake occurred beneath the ocean, it isimportant to allow for the correct structure above the source.This can easily be achieved, since we are using a derivative-free inversion scheme with an allowance for different struc-tures at the source and receiver ends. As expected, we arebarely able to achieve a satisfactory seismogram fit using thestandard ak135 velocity model, however, the addition of awater layer above the source results in a significant improve-ment to the fit. Additional phases such as pwP, swP, swS

and pwS need to be calculated to take into account reflectionsoff the water interface as well as off the seabed interface.Water depths in the range 1 to 2 km were tested, and a waterdepth of 1.25 km was found to give the best fit. A P-wavevelocity of 1.5 km/sec was used, with a density of 1.03 g/cm3, for the water layer. The inversion is performed usingP, SV, and SH waveforms, and a moment tensor represen-tation of the source mechanism is used.

The fit between the observed and synthetic seismogramsis quite good when a water layer above the source is included(Fig. 5). Note that the P waveforms at the stations WRABand CCM (which have small P arrivals) are fit very poorlyusing the standard ak135 model, however, with the inclusionof a water layer above the source the waveform fit at thesetwo stations is greatly improved. The relative amplitudes atstation WRAB are still not fit very well, but the relative tim-ing and overall pulse shape are fit. Station PMG is fit poorly,even when taking into account the suboceanic source, andthe likely explanation is the presence of sedimentary layersnear the receiver, causing large amplitude reverberations af-ter the initial P arrival. Such effects are not modeled in ourinversion.

The recovered source mechanism is a good represen-tation of the mechanism obtained from the CMT inversion(Fig. 6). This is encouraging given the limited sampling ofthe source radiation pattern due to the small number of sta-tions and rather poor azimuthal coverage. When the oceaniclayer above the source is taken into account the recoveredsource depth is 18 km, which is very close to the CMT depthof 19 km. The source depth obtained when performing the


NEIC

N

E

PN

E

SHN

E

SV

P, SV and SHunconstrained

N

E

PN

E

SHN

E

SV

P and SHunconstrained

N

E

PN

E

SHN

E

SV

P and SHconstrained

N

E

PN

E

SHN

E

SV

Figure 2. (top) Source radiation patterns for P, SV, and SH waves for the westernIran event using the NEIC moment tensor solution. Lower three sets of radiation patternsare the result of various inversions (described in the text). The plotted symbols representthe average take-off angle and azimuth for various phases; solid circles for P, solidtriangles for pP, solid (grey) diamonds for pS; open circles for S, open triangles for sS,and open diamonds for sP.

inversion with the standard ak135 model is 22 km. Thus thetwo depth estimates differ by only 4 km, which illustratesthat a reasonably well-constrained source depth can be ob-tained, even when the use of a standard velocity model suchas ak135 is not a good representation of the true structure.

There are other factors that are not modeled, whichcould explain discrepancies in the seismogram fit. Firstly,the earthquake occurred within a subduction zone, thereforethe effect of dipping structure could be important. Our syn-thetics are calculated assuming a horizontally stratified me-dium. Near source structure is likely to have influenced thewaveforms at WRAB, PMG, and CCM, since the propagation

paths to these stations start out along the strike of subductionzones; the Honshu and Izu-Bonin zones for WRAB and PMG,and the Kuril zone for CCM. There is also a 1- to 2-km thicksedimentary layer beneath the ocean off the Pacific coast ofnorthern Honshu, as revealed by a seismic refraction study(Asano et al., 1981). In a study of earthquakes along thenorthern Honshu arc, Seno and Kroeger (1983) found that itwas necessary to include this sedimentary layer, because thereflection from the bottom of the sediment is very large.Perhaps this can explain some of the arrivals seen betweenP and pP, which are not being fit. There also seems to beevidence for significant differences in S-wave attenuation


10. 20. 30. 40.

Time [s]

Z:P34.97

∆

41.41

88.72

65.47

77.56

68.41

51.26

43.29

46.70

10. 20. 30. 40.

Time [s]

T:SHKEV

TOL

SCP

SLR

COL

MAJO

BJI

LZH

KMI

Figure 3. Comparison between the complex ob-served (black traces) and predicted (grey traces) seis-mograms for the western Iran event with a perturbedsource location, and a moment tensor representationof the source mechanism. The results from two in-versions are displayed: unconstrained isotropic com-ponent (dark grey traces) and constrained isotropiccomponent (light grey traces). The vertical compo-nent of P (left) and the transverse component of S(right) are displayed.

Figure 4. Location of the Honshu event and sta-tion distribution used in the inversion.

between stations, which could be contributing to the fairlypoor fit between the observed and synthetic S waveforms.

Southern Xinjiang Event

The event used as our second illustration occurred inthe continental region of southern Xinjiang, China (Fig. 7)on 6 April 1997. The event has an estimated mb of 5.6, witha CMT depth estimate of 15 km. Thus we can again compareour inversion results with the CMT solution (Dziewonski etal., 1999a). A depth estimate of less than 15 km is accept-able, since it is well known that the CMT method cannotresolve depths shallower than 15 km (Dziewonski et al.,1987). The waveforms used in the inversion have been ob-tained from the IRIS DMS. Again, ten stations are used inthe inversion, but only five of the stations have suitable SVand SH waveforms. The waveforms are bandpass filtered inthe range 0.01 Hz to 0.7 Hz before inversion.

Inversion with the standard ak135 velocity model workswell in this case, since the earthquake occurred in a conti-nental region. We first perform a joint P and S inversion,with a suitable weighting factor applied to the S-wave datadue to differing amounts of P and S information available.A moment tensor representation of the source mechanism isused. A significant isotropic (implosive) component is ob-tained, with a normalized moment tensor trace of �1.1. Thisisotropic estimate is not really surprising given the samplingdistribution of the radiation pattern (see Fig. 9). Even thougha significant isotropic component is recovered, the seismo-gram fit is good. The isotropic component is then con-strained to be zero, and the inversion is performed again. Agood fit is achieved for the P- and SV-wave seismograms,but not for the SH seismograms. It seems that an introducedisotropic component may be compensating for other factors,such as the use of the wrong velocity model. Thus, when aconstrained inversion is performed, it may be best to applya nonzero constraint and allow for a small, but not significantisotropic component. We therefore perform the inversionagain, with the isotropic component constrained to have thevalue �0.1. An excellent fit is achieved for the P-wave seis-mograms, and a reasonable fit is achieved for the S-waveseismograms (Fig. 8). The recovered source depth is 15 km,which is the same as the CMT estimate. We obtain goodcorrespondence between the source mechanism estimateobtained from the NA inversion and the CMT inversion(Fig. 9).

As an illustration of the flexibility of the inversionscheme, we have also performed inversion with more com-plex synthetic seismograms using the method of Marson-Pidgeon and Kennett (2000), which allows for a full treat-


10. 20. 30. 40.

Time [s]

Z:P

DAV

36.62

KMI

36.95

LSA

43.42

CHTO

43.52

PMG

49.54

NIL

55.00

WRAB

60.45

OBN

66.39

KIV

69.89

CCM

87.46

R:SV

DAV

36.62

PMG

49.54

WRAB

60.45

KIV

69.89

CCM

87.46

10. 20. 30. 40.

Time [s]

T:SH

DAV

36.62

PMG

49.54

WRAB

60.45

KIV

69.89

CCM

87.46

Figure 5. Comparison between the observed(black traces) and predicted (grey traces) seismo-grams for the Honshu event with the inclusion of awater layer above the source. Shown are the verticalcomponent of P and the radial and transverse com-ponents of S. Each set of traces is annotated with thestation name and epicentral distance.

ment of crustal conversions and reverberations. Since we areusing a derivative-free inversion scheme, this is easy toachieve, however, it does require significant computationtime. A comparison between the observed and predictedseismograms for the two different synthetic computationschemes is shown in Figure 8. It can be seen that both cal-culation schemes produce similar looking predicted seis-mograms, though there are extra arrivals evident on the com-plex traces. The use of the more complex synthetics in theinversion gives a depth estimate of 15 km, which is the sameas that obtained using generalized ray theory. Similar esti-mates of the source mechanism are also obtained. This jus-tifies the use of a simple generalized ray scheme in this case,however, it may be necessary to use more accurate syntheticsfor problematic events.

Indian Nuclear Test

As a test of the discrimination capabilities of ourmethod, we perform inversion for a known nuclear explo-sion: the Indian nuclear test of 11 May 1998. The U.S. Geo-logical Survey (USGS) reported a mb of 5.2 for this explo-sion, which implies an approximate yield of 10–15 kt(Wallace, 1998). The waveforms used in the inversion havebeen obtained from the IRIS DMS. We found only six sta-tions in the teleseismic distance range with good P-wavesignals (Table 4). No detectable S-wave energy was ob-served at any of the stations, thus providing evidence to sus-pect this event of being a nuclear explosion. In the followingwe have performed inversion using just the P waveforms,which were bandpass filtered in the range 0.01 Hz to 1.5 Hzbefore inversion.

A wide range of source depths, from 0 to 35 km, aresearched in the inversion. We obtain a best-fitting sourcedepth of 0.6 km, which provides a good fit to the observedP-wave seismograms. Based on the depth estimate alone, wehave grounds to suspect this event of being man-made. Weuse a moment tensor representation of the source mecha-nism, with no restriction on the allowable isotropic compo-nent. The source mechanism estimate we obtain is notwholly explosive, as it predicts the presence of a smallamount of S-wave radiation. We do, however, obtain a largeisotropic component, and we estimate that the isotropic mo-ment is at least 50% of the total moment, using the methodof Bowers and Hudson (1999). This result is not really sur-prising, given the poor azimuthal coverage, and limited num-ber of stations (Table 4). The departure from an isotropicsource mechanism is in the area where the P-wave radiationpattern is not sampled due to lack of stations. If we had afew more stations, at different azimuths, we would be ableto obtain better constraints on the source mechanism.

We cannot directly compare our depth estimate with theactual depth, as the Indian government has not released thesedetails. However, we expect the depth of burial to be on theorder of a few hundreds of meters. Given the very shallowdepth estimate, and large isotropic component estimate, wehave provided strong evidence to suspect this event of being

a nuclear test. If this test had not been announced, our resultswith only a few stations would have provided grounds forfurther investigation.

Discussion and Conclusions

We have demonstrated the success of nonlinear wave-form inversion for moderate-sized events at teleseismic dis-tances using the neighborhood algorithm. We have extendedmethods for determining the source parameters of large tele-seismic earthquakes to smaller events. Unlike most previousmethods, we make use of SV information in addition to Pand SH information, and use short period or broadband re-cords instead of long-period waveforms. A relative ampli-tude approach is used in this study, which means that factorssuch as the scalar moment do not need to be included. Wehave been able to achieve good depth resolution and obtainuseful information on the source-time function and source


N

E

P

CMT SolutionN

E

SHN

E

SV

N

E

P

PredictedN

E

SHN

E

SV

Figure 6. Source radiation patterns for P, SV, and SH waves for the Honshu event,using the centroid moment tensor solution (top), and the predicted source mechanismobtained from the inversion (bottom). The plotted symbols represent the average take-off angle and azimuth for various phases; solid circles for P, solid triangles for pP,solid (grey) diamonds for pS; open circles for S, open triangles for sS, and open dia-monds for sP.

Figure 7. Location of the southern Xinjiang eventand station distribution used in the inversion.

mechanism. We have deliberately considered examples withjust a few stations, and so need to take advantage of both Pand S data to exploit differences in the source radiation pat-terns. Even a limited amount of S information can help toresolve ambiguities present in the P data alone. In particular,the inclusion of just a few SV observations can provide sig-nificant constraints on the isotropic component. Both of thesource representations work well with the NA. We havefound that the use of a full moment tensor allows for a moregeneral source mechanism than the use of a double-coupleplus isotropic component representation, and therefore givesbetter results.

We have performed a number of synthetic inversiontests, which demonstrate that our inversion scheme using theNA is able to achieve good matches to both simple noise-free seismograms and more accurate noise-free seismo-grams. In order to test the robustness of the inversion schemein the presence of noise, we have performed a number ofsynthetic tests with a perturbed source location. In all ourtests the source-time function and source depth have beenwell recovered, but limited sampling of the source radiationpattern with just a few stations means that it was possible toobtain source mechanisms that give a good representation ofthe seismograms but that differ significantly from the truemodel. Often a large isotropic component estimate is ob-tained, which results in a poor correspondence between the


10. 20. 30. 40.

Time [s]

Z:P

NRIL

30.64

HIA

31.67

SSE

36.72

YAK

38.72

DPC

43.10

KBS

46.76

MAJO 47.48

KMBO 54.40

BGCA

62.72

DBIC

79.51

R:SV

NRIL 30.64

HIA 31.67

SSE 36.72

YAK 38.72

MAJO

47.48

10. 20. 30. 40.

Time [s]

T:SH

NRIL

30.64

HIA

31.67

SSE

36.72

YAK

38.72

MAJO

47.48

Figure 8. Comparison between seismograms forthe southern Xinjiang event: observed seismograms(black traces) along with predicted seismograms us-ing simple synthetics (grey traces) and more complexsynthetics (light grey traces). Shown are the verticalcomponent of P, and the radial and transverse com-ponents of S. Each set of traces is annotated with thestation name and epicentral distance.

input and predicted source mechanisms. In this case it isdesirable to also perform an inversion constrained to just adouble-couple to see whether the isotropic estimate is realor just an artifact of the inversion. In particular for all thesynthetic earthquake tests we have performed, the misfit islower when the isotropic component is restricted, indicatingthat the data are compatible with a double-couple model.

Three illustrations using high-quality observed datahave been provided. For the two earthquake examples, theresulting source parameters compare well with the CMT so-lution in both cases. For the Honshu event it is necessary toperform an adaptive inversion and allow for an oceanic layerabove the source. This is easily achieved since we use aderivative-free inversion scheme with an allowance for dif-

ferent structures at the source and receiver ends. Eventhough we have used the ak135 reference model throughoutthis study, there is little computational cost in using custom-ized source and receiver models by modifying the near-surface velocities, thus changing the apparent amplification.Variations in S-wave attenuation away from the referencemodel can be accommodated by modifying the mantle at-tenuation operator. A reasonably well constrained sourcedepth is obtained for the Honshu event, even when the oce-anic layer is not included, indicating that the depth estimateis relatively insensitive to the velocity model used. However,more accurate depth and mechanism estimates are obtainedwhen a water layer above the source is incorporated. Thisillustrates the need to identify whether events are suboceanicwhen an accurate depth and mechanism estimate is required,such as when monitoring the CTBT.

Inversion for the southern Xinjiang event worked wellusing the standard ak135 reference model, as this providesa good representation of continental regions. Quite a largeimplosive component was recovered from the inversion dueto the sampling distribution of the radiation pattern. It ispossible that an introduced isotropic component may becompensating for other factors, such as the use of the wrongvelocity model, resulting in an improved seismogram fit withthe introduction of a small isotropic component. Thus whentesting the significance of any isotropic estimate, it may bebest to allow for a small, but not significant isotropic com-ponent in the inversion.

Our method also works well when applied to nuclearexplosions, as illustrated with the Indian nuclear test. A veryshallow source depth was obtained, along with a large iso-tropic component estimate. If this test had not been an-nounced, our results with only a few stations would haveprovided grounds for further investigation. This examplealso illustrates that our method works well using only P-wave information.

Even though we have used a rather simple syntheticseismogram calculation scheme, which has its limitations,we have demonstrated that good results are obtainable formoderate size events using relatively high-quality data. Theuse of a derivative-free inversion method such as the NA hasa number of advantages over linearized inversion; there isno dependence on the starting parameters, and various mod-eling complexities are easily included. The interface be-tween the inversion and the waveform modeling is simple;the only information input into the forward modeling routineis the model parameters. In this way it is easy to substitutedifferent modeling routines and even combine differentschemes. We have demonstrated this flexibility for the eventin southern Xinjiang by substituting more complex syntheticseismograms into the inversion scheme. In this case the useof ray theory is justified as similar results were obtained,however, the ability to use more accurate synthetics in theinversion may be useful for problematic events. In suchcases the extra computational cost of calculating the syn-thetic seismograms would have to be weighed against the


N

E

P

CMT SolutionN

E

SHN

E

SV

N

E

P

PredictedN

E

SHN

E

SV

Figure 9. Source radiation patterns for P, SV, and SH waves for the southern Xin-jiang event, using the centroid moment tensor solution (top) and the predicted sourcemechanism obtained from the inversion (bottom). The plotted symbols represent theaverage take-off angle and azimuth for various phases; solid circles for P, solid trianglesfor pP, solid (grey) diamonds for pS; open circles for S, open triangles for sS, and opendiamonds for sP.

Table 4Station Locations Relative to Indian Nuclear Test Site

Station Distance (�) Azimuth (�)

XAN 32.57 68.59ULN 34.39 43.48OBN 37.88 327.33SSE 43.05 72.52YAK 51.46 30.70

COLA 83.29 15.92

possible benefit. One limitation of this method is that it doesnot directly provide any error bounds for the estimatedmodel parameters, although the distribution of well-fittingmodels in parameter space can provide a useful guide. Wehave been using the NA purely for parameter estimation,however, the entire ensemble of models generated can beused to extract measures of resolution and trade-off in themodel parameters, within a Bayesian framework (Sam-bridge, 1999b). Systematic nonlinear error analysis, whichrequires knowledge of the statistics of noise sources, willform the subject of further studies.

Acknowledgments

The comments of the reviewers and Associate Editor were very help-ful. This work was supported in part by Grant DSWA01-97-1-0023.

References

Aki, K., and P. G. Richards (1980). Quantitative Seismology: Theory andMethods, W. H. Freeman, San Francisco.

Asano, S., T. Yamada, K. Suyehiro, T. Yoshii, Y. Misawa, and S. Iizuka(1981). Crustal structure in a profile off the Pacific coast of north-eastern Japan by refraction method using ocean bottom seismometers,J. Phys. Earth. 29, 267–282.

Bowers, D., and J. A. Hudson (1999). Defining the scalar moment of aseismic source with a general moment tensor, Bull. Seism. Soc. Am.89, 1390–1394.

Dziewonski, A., T. A. Chou, and J. H. Woodhouse (1981). Determinationof earthquake source parameters from waveform data for studies ofglobal and regional seismicity, J. Geophys. Res. 86, 2825–2852.

Dziewonski, A. M., G. Ekstrom, J. E. Franzen, and J. H. Woodhouse(1987). Centroid-moment tensor solutions for January–March 1986,Phys. Earth Planet. Int. 45, 1–10.

Dziewonski, A. M., G. Ekstrom, and N. N. Maternovskaya (1999a). Cen-troid-moment tensor solutions for April–June, 1997, Phys. EarthPlanet. Int. 112, 1–9.

Dziewonski, A. M., G. Ekstrom, and N. N. Maternovskaya (1999b). Cen-troid-moment tensor solutions for April–June, 1998, Phys. EarthPlanet. Int. 112, 11–19.

Goldstein, P., and D. Dodge (1999). Fast and accurate depth and sourcemechanism estimation using P-waveform modeling: a tool for specialevent analysis, event screening, and regional calibration, Geophys.Res. Lett. 26, 2569–2572.

Kawakatsu, H. (1995). Automated near-realtime CMT inversion, Geophys.Res. Lett. 22, 2569–2572.

Kennett, B. L. N., E. R. Engdahl, and R. Buland (1995). Constraints onseismic velocities in the earth from travel times, Geophys. J. Int. 122,108–124.


Langston, C. A. (1976). A body wave inversion of the Koyna, India, earth-quake of December 10, 1967, and some implications for body wavefocal mechanisms, J. Geophys. Res. 81, 2517–2529.

Langston, C. A. and D. V. Helmberger (1975). A procedure for modellingshallow dislocation sources, Geophys. J. R. Astr. Soc. 42, 117–130.

Marson-Pidgeon, K., and B. L. N. Kennett (2000). Flexible computation ofteleseismic synthetics for source and structural studies, Geophys. J.Int. 143 (in press).

Montagner, J.-P., and B. L. N. Kennett (1996). How to reconcile body-wave and normal-mode reference earth models, Geophys. J. Int. 125,229–248.

Nabelek, J. (1985). Geometry and mechanism of faulting of the 1980 ElAsnam, Algeria, earthquake from inversion of teleseismic body wavesand comparison with field observations, J. Geophys. Res. 90, 12,713–12,728.

Pearce, R. G. (1977). Fault plane solutions using relative amplitudes of Pand pP, Geophys. J. R. Astr. Soc. 50, 381–394.

Pearce, R. G. (1980). Fault plane solutions using relative amplitudes of Pand surface reflections: further studies, Geophys. J. R. Astr. Soc. 60,459–487.

Sambridge, M. S. (1999a). Geophysical inversion with a neighbourhoodalgorithm. I. Searching a parameter space, Geophys. J. Int. 138, 479–494.

Sambridge, M. S. (1999b). Geophysical inversion with a neighbourhood

algorithm. II. Appraising the ensemble, Geophys. J. Int. 138, 727–746.

Sambridge, M. S., and B. L. N. Kennett (2001). Seismic event location:nonlinear inversion using a neighbourhood algorithm, Pageoph. (inpress).

Seno, T., and C. Kroeger (1983). A reexamination of earthquakes previ-ously thought to have occurred within the slab between the trenchaxis and double seismic zone, northern Honshu arc, J. Phys. Earth31, 195–216.

Sipkin, S. (1994). Rapid determination of global moment-tensor solutions,Geophys. Res. Lett. 21, 1667–1670.

Voronoi, M. G. (1908). Nouvelles applications des parametres continus ala theorie des formes quadratiques, J. Reine Angew. Math. 134, 198–287.

Wallace, T. (1998). The May 1998 India and Pakistan nuclear tests, Seism.Res. Lett. 69, 386–393.

Wallace, T. C., and D. V. Helmberger (1982). Determining source param-eters of moderate-size earthquakes from regional waveforms, Phys.Earth Planet. Int. 30, 185–196.

Research School of Earth SciencesThe Australian National UniversityCanberra ACT 0200, Australia

Manuscript received 31 January 2000.

Documents

Source Depth and Mechanism Inversion at Teleseismic ...rses.anu.edu.au/~brian/PDF-reprints/2000/bssa.90.1369.pdf · Bulletin of the Seismological Society of America, 90, 6, pp. 1369–1383,