Inversion of waveforms for extreme source models with an application to the isotropic moment tensor component

Transcription

1 Geophysial Journal (1989) 97, 1-18 nversion of waveforms for extreme soure models with an appliation to the isotropi moment tensor omponent D. W. Vaso and L. R. Johnson Center for Computational Seismology, Lawrene Berkeley Laboratory, and Department of Geology and Geophysis, University of California, Berkeley, CA 94120, USA Aepted 1988 July 15. Reeived 1988 June 9; in original form 1987 Marh 14 1 NTRODUCTON A major problem in seismology is the determination of the nature of seismi soures. The analysis of earthquake waveform data is the hief method of studying the faulting proess. Other methods suh as deep drilling and surfae strain measurements are more expensive and time onsuming. Furthermore, they do not offer a dynami piture of the faulting proess. Therefore, one must turn to the elasti wave radiation for details of the faulting proess. We hose to use bodywaves beause of the higher resolution they ontain and beause of the high quality digital waveforms whih are beoming available from global digital networks suh as the Global Digital Seismi Network (GDSN). Waveform inversion makes use of the whole seismograms, using all the information ontained in the time series. With few exeptions, most moment tensor inversions of waveforms have relied on some form of least squares to derive a best fitting solution (Gilbert & Dziewonski 1975; Gilbert & Buland 1976; MCowen 1976; Kanamori & Given 1981; Stump & Johnson 1977; Dziewonski, Chow & SUMMARY We have developed a new approah to the inversion of waveform data for the time-varying moment tensor. The method produes the soure model whih minimizes the modulus squared of any linear ombination of moment tensor omponents, subjet to the onstraint that the data are satisfied within speified onfidene intervals. This method allows the determination of possible soure models other than the least-squares solution, enabling one to determine the signifiane of ertain moment tensor properties; for example, the presene or absene of a volume hange (isotropi omponent) in the soure. Syntheti tests were used to examine the effet of miroseismi noise and lateral heterogeneity on the extreme models of the isotropi omponent. Lateral heterogeneity is found to have a strong effet on the estimation of the isotropi omponent of the moment tensor. The method was tested by using long-period waveforms from the Global Digital Seismi Network to estimate the isotropi part of the moment tensor of a deep Bonin slands earthquake. Modelling indiates that more than 10 per ent of the mehanism might have to be isotropi for detetion of volume hange in the presene of 10 per ent random noise and only 2 per ent lateral heterogeneity. The least-squares solution indiates that a relatively large hange in volume was involved in the soure mehanism. However, the minimum extreme solution shows that this volume hange is not atually required by the data and thus may not be signifiant. The method was also tested on near-soure data from the nulear explosion Harzer. n this ase, in spite of fairly large error bounds, it an be onluded that the soure has a lear explosive omponent. Key words: extremal inversion, inversion, moment tensor, volume hange Woodhouse 1981; Sipkin 1982). The notable exeptions to this approah are the L, minimization of Fith, MGowan & Shields (1980) and Tanimoto & Kanamori (1986) and the linear programming approah of Julian (1986). n an approah related to the latter paper we put forth an inversion method to examine the range of time-varying moment tensor models whih agree, within speified onfidene bounds, with observed body-waveform data. Using this tehnique it is possible to derive extreme models, that is, models whih make some property of the soure a minimum or maximum. As will be detailed later, the properties will be in terms of the modulus squared of linear ombinations of the moment tensor omponents. Many properties an be put in the form of a linear ombination of moment omponents. Six examples are given in Julian (1986), these inlude the explosiveness, thrust-like nature, horizontal extensiveness and vertial ompensated linear vetor dipole (CLVD) nature of the soure. These properties an be used to answer interesting geophysial questions, for example, to deide if an event is a nulear explosion rather than an earthquake. One new feature of our approah is the use of quadrati 1

2 2 D. W. Vaso and L. R. Johnson programming for minimizing the modulus squared of the linear ombinations of moment tensor omponents rather than using linear programming for minimizing a linear ombination of the omponents. Quadrati programming is the minimization or maximization of a quadrati funtional subjet to linear equality and inequality onstraints (Dantzig 1963). This approah was taken beause of problems in extending the linear programming approah to waveform inversion. Furthermore, it is diffiult to interpret linear ombinations of moment tensor omponents in the omplex frequeny domain and to transform these properties into the time domain. As will be shown, through the use of Parseval's theorem, properties derived through the quadrati programming approah may be onveniently interpreted in both the frequeny domain and the time domain. n addition, the quadrati programming method takes the same order of time as the generalized least-squares approah and has proven to be very effiient for waveform inversion. We expliitly solve for the time-varying moment tensor, not assuming a soure-time funtion. The use of the time-varying moment tensor is a key feature of the method beause it allows for motion over urved and ompliated fault surfaes. Multiple rupture events also pose no diffiulty in the time-varying formulation. An appliation of extreme models, the one prinipally addressed in this paper, is to determine the signifiane of the isotropi omponent in the moment tensor. This has importane in at least two areas: nulear explosion seismology and the determination of deep and intermediate earthquake mehanisms. Nulear explosions are known to be mainly dilatational soures. However, beause of soure omplexity and sattering due to lateral heterogeneities, the moment tensor ontains non-dilatational omponents. t is neessary to determine the signifiane of these other omponents. Furthermore, extreme models allow the determination of the smallest and the largest explosive solutions. This is useful for the estimation of bounds on yields. n general, earthquakes are thought to have doubleouple mehanisms. Oasionally, however, investigators desribe events whih ontain non-double-ouple omponents suh as an isotropi trae (Dziewonski & Gilbert 1974; Silver & Jordan 1982). n partiular, deep and intermediate earthquakes in subdution zones may be assoiated with volume derease. The very high pressures and the possibility of phase hanges make the fault mehanis very diffiult to model. Therefore, the nature of the faulting deep within the earth is still an open question. The role of volume hange an be addressed by the alulation of extreme models. For example, it is possible to ompute the model with the minimum total squared trae amplitude and to ompare the trae of this model with the deviatori omponent. This gives an indiation of the relative volume hange assoiated with the event. The omputation of suh a model results in a quadrati programming problem as will be shown below. 2 METHOD OF NVERSON Using the representation theorem for seismi soures, a onnetion an be made between soure parameters and observed displaements in tenns of equivalent body fores. This is a fore system whih would produe displaements of the earth's surfae equivalent to that from the true physial situation, i.e. a heterogeneous fault system with a non-linear rheology. A general indigenous soure an be expressed in terms of equivalent body fores (Bakus & Mulahy 1976), resulting in the relationship where uk is the k-th omponent of displaement, Gki(x,t; r, t') are the Green's funtions ontaining propagation effets, h(r, t') are the sums of equivalent body fores, and V is the soure region where the h(r, t') are non-zero. The summation onvention is assumed for repeated indies. The Gki(x, t; r, t') may be expanded in a Taylor series (Stump & Johnson 1977) about the point r = E, " 1 Gki(x? t; t') = 7 (rl - El) ' ' ' (rn - En),=on. Gki,j,...jn(x, t; E> t'). Define the fore moment tensor and the displaement beomes r " 1 uk(x, f) = 2 - Gki,jl...jn(x7 t; E, 0) X Mij1...jn(E, where X represents a temporal onvolution. f the soure dimensions are small ompared to the wavelengths of interest it is neessary to keep only the term for n = 1 in the previous expansion, resulting in for 5 = 0. By Fourier transforming this onvolution a matrix equation is derived, uk(x>f)=gki.j(x,f;o, O)Mij(O,f), (2) for eah frequeny. t is possible to solve either equation (1) or equation (2) for the moment tensor in the time or frequeny domain respetively. n order to solve the time domain equation (1) one may assume a soure time funtion whih is the same for all moment tensor omponents, onverting the onvolution into a multipliation. Alternatively, the onvolution equation may be written as a matrix equation and the resulting large blok Toplitz system of equations may be solved by a least squares error algorithm (Sipkin 1982). Another approah, the one taken in this paper, is to work in the frequeny domain, solving equation (2). Eah omponent may have an arbitrary soure-time funtion and it is only neessary to solve a muh smaller system of equations suessively at eah frequeny of interest. This system of equations may be solved by least squares, minimizing the 1' norm of the residuals, where is the index vetor (i, j). The total number of station omponents is denoted by K. The index 1 indiates north t),

3 while indies 2 and 3 represent east and down, respetively. /ii\ The vetor M is given by the real and imaginary omponents of the moment tensor: im M,, im M33 This results in the generalized inverse solution whih may be written in terms of the singular-valued deomposition. n addition to being a well-known, effiient proedure, this method allows one to form the data and model resolution matries and the unit ovariane matrix (Menke 1984). t is important to find a best fitting model whih the least squares solution provides, but it is also useful to use the data to answer more speifi geophysial questions. For example, is a pure double-ouple soure suffiient to satisfy the data, or is some volume hange present in the soure? To answer suh a question it is neessary to find the extremum of a linear ombination of the moment tensor omponents, the sum of the diagonal elements of the moment. Alternatively, one ould minimize the modulus squared of this quantity, the approah we advoate. The searh for the extrema is subjet to the onstraint that the data uk are satisfied within some onfidene intervals k. Thus we seek or subjet to uk - Ek GkMl Uk + Ek, k = 1,2,..., K. (5) The A,[ matrix is ruial beause it defines the properties whih will be bounded. By hanging the elements of A,!, a variety of different soure harateristis an be onsidered: the largest and smallest vertial strike-slip fault soure, the most and least thrust-like soure, et. Table 1 presents the oeffiients of the quadrati form (Aml in equation 4) for a number of soure properties. By hanging the sign of the elements the problem is hanged from a minimization to a maximization. The system of equalities and inequalities given above results from the loal imposition of error bounds for eah onstraint equation, that is, the onfidene interval for eah datum is a strit interval whih no datum may exeed. f the data set is large it is unlikely that some intervals will not be exeeded (Oldenburg 1983). As an alternative to this imposition of absolute onfidene bounds on the data it is (3) (4) nversion of waveforms for extreme soure models 3 Table 1. Coeffiients for the quadrati form in equation (4) for a variety of moment tensor properties. The real and imaginary omponents have idential oeffiients. QP oeffiients Vertial Vertial Soure Explosive Thrust-like CLVD strike-slip possible to impose a statistial bound. Using this approah, one would require that the i-th datum is satisfied within some unspeified error bound ei. Now the onstraint will be that the sum of the errors for eah onstraint should not exeed some pre-determined value E. Algebraially, instead of the onstraints in equation (5) we have, uk 5 GklM + ek K 2 eise i-1 ekro k=l,2,..., K. k = 1, 2,..., K The additional onstraint has been imposed that all the e, s are positive. f the errors ei were independent, normal, random variables, the value of E might be alulated using a priori estimates of their mean and standard deviations a, (Parker & MNutt 1980). Unfortunately, the statistial parameters of errors assoiated with waveforms are not known a priori. n addition to miroseismi noise, whih may be estimated by pre-event noise samples, there is signal generated noise due to errors in the estimation of the Green s funtion. This noise, in both phase and amplitude, is often more signifiant than miroseismi noise and must be aounted for. We will estimate these bounds by taking the differene between the observed data and data predited by a model derived through an inversion proedure suh as least squares. Another method to estimate errors on the waveforms is through Monte Carlo simulation of data sets. This involves using the least squares solution to generate a data set whih is then perturbed by an a priori distribution of errors. The statistis of this data set may be omputed and used to ompute onfidene intervals with whih to derive extreme models. Unfortunately, present-day models of lateral heterogeneity are not adequate to fully represent the Green s funtions errors. Three-dimensional models of attenuation are still preliminary (Durek et al. 1988) and the statistial parameters of any of the models have not yet been determined. The algorithm used to minimize equation (3) subjet to the onstraints of equation (5) or (6) is the well known

4 4 D. W. Vaso and L. R. Johnson simplex algorithm for linear programming. To minimize the quadrati funtional in equation (4), subjet to the onstraints, requires quadrati programming (Dantzig 1963). By finding the minimum and maximum of the funtional subjet to the onstraints it is possible to derive bounds on soure properties from bounds on the data. This is one method to explore the range of possible models rather than derive a single solution. As mentioned previously, formulating the problem in terms of the square of the modulus of a linear sum of moment tensor omponents allows the transition between moment tensor properties in the time and frequeny domains. This follows from Parseval s theorem whih states that the integral of the square of the modulus of a funtion is equal to the integral of the square of the modulus of the Fourier transform of the funtion (Braewell 1978). f one onsiders the linear funtional F(w,) = C,M, as a finite Fourier transform of the time series f(t), then the square of the modulus of F(wj) may be represented as a quadrati form involving a vetor M omposed of the real and imaginary parts of the moment tensor omponents, F(WJF(WJ* = M~AM, (7) where F(wi)* is the omplex onjugate of F(ui). Beause of Parseval s theorem, the sum of the funtion in equation (7) over all frequenies has the same value as the sum of the amplitude squared of the time series f(t) over all time. Fortunately, our time series are finite and our seismi spetra are band-limited, due to instrument transfer funtions. Hene, the extremum of this sum is the same in both the time and frequeny domain. Therefore, it is possible to ompute the minimum or maximum value of the square of the modulus of any linear ombination of moment tensor omponents in the frequeny domain, sum the extreme values over a finite number of frequenies, and interpret this diretly in the time domain. n what follows we will onsider the moment rate tensor rather than the moment tensor. The rate tensor has no stati offset and hene returns to zero over time. 3 DEEP EARTHQUAKES Debate has ontinued about the presene of an isotropi moment tensor omponent in deep and intermediate subdution zone earthquakes (Dziewonski & Gilbert 1974; Okal & Geller 1979; Silver & Jordan 1982; Hodder 1984; Ftiedesel & Jordan 1985). t seems reasonable to expet subduting regions to be areas of ompation aompanied by dewatering, high fluid pressure, rak losure and phase hanges. The essential question is if any of these phase hanges are meta-stable. Only then an suh reations produe rapid failure whih exites high-frequeny waves in the earth. Reently, high pressure experiments (Kirby 1987; Meade & Jeanloz 1988) have deteted shear instabilities assoiated with phase hanges. The most reent work reorded sudden failure and aousti emissions when simulated oean lithosphere was subjeted to pressures of up to 20 GPa at temperatures below 900 K. f suh physial proesses are oumng they should be seismially detetible though the assoiated volume hange may be small. Reently, there have been indiations that silent or slow earthquakes may be oumng in the earth (Beroza & Jordan 1988). These events, whih are not aompanied by high-frequeny seismi waves, may ontain an isotropi omponent. When inverting for the moment tensor, errors may be introdued due to a number of effets. Energy from miroseisms an mask small arrivals and ontaminate the waveforms. Some investigators have emphasized the possible trade-off between lateral heterogeneities and a possible isotropi omponent (Okal & Geller 1979). n addition, soure omplexity suh as urved faults and multiple rupture an give rise to non double-ouple moment tensor solutions (Sipkin 1986). Given miroseismi noise, omplex soures, and lateral heterogeneities, the derivation of extreme models seems well suited to assess the presene or absene of an isotropi moment omponent. With this in mind we onsider a 467.7km deep, magnitude 5.6 earthquake near the Bonin slands reorded by 15 GDSN long period instruments (Fig. 1 shows the station distribution). The 15 stations span a distane range from 38.5 to 91.7 but have a gap in overage in the southeast quadrant. Shown in Fig. 2 are the 15 long period, vertial GDSN seismograms. The first 256s of the reords were used in the inversions. The signal-to-noise ratio is quite high (onsider the pre-event miroseismi noise) and the arrivals oherent aross the array of stations. 4 TESTS WTH SYNTHETC DATA To illustrate the method and also explore its apabilities, we first onsider some tests with syntheti data. We adopt the same number and distribution of stations as for the Bonin slands earthquake (Fig. l), assuming the event is at a depth of km with the soure speified by the time-varying moment rate tensor shown in Fig. 3. This is a dip-slip event with a superimposed isotropi omponent. The body waves from the event (P, pp and sp) were alulated by the WKBJ method (Chapman 1978) for a PREM earth model (Dziewonski & Anderson 1981) and then low-pass filtered to obtain the vertial omponent syntheti seismograms shown in Fig. 4. Random noise with an amplitude 10 per ent that of the maximum signal has been added to eah seismogram to simulate miroseismi noise with a signal-to-noise ratio of 10: 1. This example is only intended to simulate miroseismi noise whih should be unorrelated for the global station distribution onsidered. This is not the ase with signal-generated noise arising from lateral heterogeneities whih may be orrelated when reeiver rustal strutures are similar. We now ask the question, what an be said about the possibility of an isotropi omponent in the soure given the noisy, band-limited seismograms? One way to answer this question is to ompute the least-squares solution at eah frequeny and then Fourier transform the results into the time domain where the moment tensor trae an be omputed. Suh results are shown in Fig. 5 where it an be seen that they do a good job of reovering the known solution of Fig. 3. But the question remains, beause of the miroseismi noise in the seismogram, what is the range of possible moment tensor traes? One way to quantify this is to find the models with the maximum and minimum trae amplitudes squared and still satisfying the data within the errors introdued by the noise. That is, minimize the

5 nversion of waveforms for extreme soure models 5 F@re 1. Station distribution of GDSN instruments reording the Bonin slands earthquake of 1985 Otober 4. The earthquake epienter is denoted by a ross at the enter of the map and the stations by triangles. The magnitude 5.6 event was at a depth of km. 3 funtional in equation (4) with the elements of the A,,,, matrix given by the explosive terms in Table 1. To use the quadrati programming methods desribed above it is neessary to first estimate bounds on errors in the data. n this example, absolute onfidene bounds will be used, i.e. onstraints (5) will be onsidered. The bounds were omputed by taking the least-squares solution in Fig. 5 and prediting the displaement observed at eah station. The differene between the vetor of observed omponents, u, and the vetor of predited omponents, up, is used as an estimate of the errors for the data, E = u -up. n analogy with the 95 per ent onfidene interval of normal distributions, twie E was used as a onfidene interval on the data. These bounds on the real and imaginary omponents of the data as a funtion of frequeny are shown in Fig. 6 for the third station. Using these estimates of the error bounds it is possible to ompute the model with the minimum squared moment tensor trae modulus and the results are shown in Fig. 7. Sine this is an extreme solution, it exhibits onsiderably more deviation from the known solution than does the least-squares solution of Fig. 5. A similar omputation an be performed for the maximum trae modulus, and the moment rate tensor traes for the two extreme models, the maximum and minimum sum of squared traes, are ompared with the least-squares moment rate tensor trae in Fig. 8. As expeted, the least-squares solution lies between the two extreme solutions. What the extreme solutions provide is a measure of the unertainty in the least-squares solution due to random noise in the data. The effet of inreasing the

6 E+00 O.llE+Ol 0.43E E E+00 34: $: Q : 48E:QQ ,34E+00 O.l9E+00 8: 3bE%' Figure 2. GDSN long period seismograms from the Bonin slands event. The early P arrivals (P, pp, and PP) are present in this setion. M11 M2 1 M E E E+19 M E+15 M E.20 M E Qur 3. Moment tensor soure used for the syntheti modelling. The soure onsists of a dip-slip event with a superimposed isotropi omponent. The isotropi omponent onstituted 60 per ent of the soure.

7 nversion of waveforms for extreme soure mod& E E E E E-09?$: %&El Q: 34F E E-08 Figure 4. Syntheti seismograms with 10 per ent noise, from the soure model in Fig. 3 and reorded at the stations in Fig. 1. The WKBJ method was used to ompute the seismograms using the PREM veloity struture. noise in the data is shown in Fig. 9. Here, the minimum extreme traes are shown for varying signal-to-noise ratios. The bounds get progressively wider as the additive noise inreases, the lower bound tending to zero with greater noise. This shows how widening onfidene bounds on the data, aused by a dereasing signal-to-noise ratio, leads to wider bounds on the isotropi moment tensor omponent. Lateral heterogeneity in both attenuation and veloity struture also affets moment tensor estimation beause it introdues phase and amplitude errors into the data. f the Green s funtion used differs from the real earth, differenes between the observed data and the predited data will our. The errors in the Green s funtion enter the above data onfidene intervals through the assumed data resolution matrix R,. This is a matrix whih relates the observed data to the data predited assuming a partiular veloity struture (Green s funtion) suh as PREM. For an over-determined problem it an be written diretly in terms of the assumed Green s funtion G, (Aki & Rihards 1980; Menke 1984), Contained in R, are the effets of the experimental setup (station distribution, miroseismi noise, et.) as well as the effets of lateral heterogeneity. To make this learer R, an be written in terms of the true Green s funtion G,. We make the assumption that the true Green s funtion is a perturbation of the assumed Green s funtion, G, = G, + y. (9) We assume that llyll<< ~ ~GJ, the matrix norm of the Green s funtion perturbation is muh less than the matrix norm of assumed Green s funtion. Substituting equation (9) into equation (8) results in the following expression for R,, R, = (G,- y)(g:gt - G:y - y G, - yty)- x (G: - Y3. (10) Negleting terms higher than first order in y, fatoring out GTG, and expanding the inverse results in, R, = G,(G:G,)- G: + G,(G:G,)-~G:YG: + G,(G:G,)-~~ G,G: - Y(G:G,)- G: - G,(G:G,)- ~~. The first term on the right is the resolution matrix in terms of the true earth model and Green s funtion, G,.

8 8 D. W. Vaso and L. R. Johnson ROFJHN SYNTHETCS (0.1 NOSE) L M L / \ / \ \ / M g Figure 5. Least-squares solution for the moment rate tensor using the noisy seismograms. The soure moment rate tensor from Fig. 3 has essentially been reovered. The diagonal elements M,, M,? and M,, are superimposed on one another. Therefore, the onfidene bounds, E, in equation (5) are given by, E = ( - R, - G,(GfG,)- GTyGf - G,(G;G,)-*Y~G,G: + Y(G;G,)-~G~ + G,(G;G,)-~Y~). (11) R, does not depend on the perturbations of the Green s funtion, only on the struture of the experiment. t is now lear that, even if the experiment were strutured suh that the data were perfetly resolved, the presene of lateral heterogeneity, non-zero y, ould still produe non-zero onfidene bounds on the data. Therefore, if the differenes between the observed and predited data are used as onfidene bounds on the data, then the effet of model errors will be inorporated in the extreme model estimates. n the presene of lateral heterogeneity the extreme bounds on the model properties will be wider. A syntheti test was also performed to explore the effet of the lateral heterogeneity. The test was similar to the previous one with the earthquake at km depth and the station distribution as shown in Fig. 1. The mehanism is the same as before (Fig. 3) but now the earth model is not homogeneous. nstead, eah path onsists of a perturbed version of PREM. Speifially, a 2 per ent random perturbation in veloity was added at eah depth in the model with independent perturbations for eah ray path. The lateral heterogeneity results in the syntheti seismograms shown in Fig. 10 with errors in the phase and amplitude present. When the extremal solutions, minimum and maximum trae squared, are omputed and presented with the least-squares solution, they differ substantially (Fig. 11). By omparing these models with similar models derived with 10 per ent random noise present (Fig. 8) it is seen that the effet of lateral heterogeneity an be quite strong. These examples demonstrate some strengths and weaknesses of the approah taken so far. First, the error bounds derived depend diretly on the knowledge of the veloity struture, the point soure assumption, and the miroseismi noise. This dependene is through the least-squares solution whih is used to onstrut the predited data. For an over-determined problem, as our knowledge improves, for example through the modeling of lateral heterogeneities, the bounds on the data will narrow. However, with the absolute bounds used, the strit inequalities of equation (5) an be either too restritive or too wide. This is beause the

9 SPECTRA ERRORS (XlO-8 CM) 0.50 t + l l t z i REAL Fire 6. Error bounds on the real and imaginary omponents of the spetra for the reord from station 3. Eah point in the figure is at a different frequeny. BONN MNMUM (0. riose) o.20 ' 0.10 t ' \ ' \ ' M11 M2 1 M22 M3 1 M _ - -u. 10 t Fire 7. Model having the minimum squared modulus for the moment rate tensor trae. Essential elements of the original soure model are reovered but the diagonal elements are no longer idential. Furthermore, the M,, and the M31 elements are no longer zero.

10 BONN TRACES (0.1 NOSE) 1 a 00 t f F'l. 00 Figure 8. Comparison of the moment rate tensor traes of the three models: least squares, maximum total squared modulus, minimum total squared modulus. BONN MNMUM TRACES W t g - 1. O O ' ' ' ' ' ' ' ' 1 ' 1 F i i 9. Effet on the minimum total trae squared solution of varying the signal-to-noise ratio of the seismograms. t is seen here that as the perentage noise inreases the minimum dereases.

11 E E E E E H E-08 Figure 10. Seismograms with perturbed arrivals due to lateral veloity inhomogeneities. Two per ent random perturbations of the PREM model were applied to eah soure-reeiver path \ MN t Figure 11. Moment rate tensor traes of the least squares and maximum and minimum extreme trae solutions. These solutions are the result of inverting the perturbed seismograms in Fig. 10 while assuming a laterally homogeneous veloity struture (PREM).

12 12 D. W. Vaso and L. R. Johnson onfidene bounds must be satisfied exatly, no single point may exeed the bounds. Therefore, the bounds must be made wide in order for the probability that any data exeeds these bounds to be small (Oldenburg 1983). This an be improved by the use of the statistial bounds given in equation (6). 5 RESULTS FOR THE BONN SLANDS EARTHQUAKE Given the results of these syntheti tests whih illustrate the effets of random noise and earth heterogeneity, we an now return to the inversion of the data shown in Fig. 2 for the Bonin slands earthquake. Reall that the basi question to be answered is whether the soure of this earthquake had a signifiant isotropi omponent. As a prelude to the inversion of the atual data, one further set of syntheti tests was performed. These tests address the problem of determining if a omputed minimum trae is signifiantly different from zero. Using the station distribution and event loation idential to the Bonin slands earthquake and assuming 10 per ent random noise and 2 per ent lateral veloity perturbations, the minimum trae squared solution was obtained for a series of soures having different relative sizes of the isotropi omponent. We onsider an isotropi omponent to be identifiable if it an be distinguished from other features of the solution whih are due to the mapping of miroseismi noise and Green's funtion errors into the solution. These errors produe a noise threshold above whih a oherent isotropi omponent must protrude for it to be detetible. The results, whih are shown in Fig. 12, allow one to ask the question: What proportion of isotropi omponent must be present in the soure in order for it to be unambiguously identified in the inversion results? The answer is fairly high: more than 10 per ent of the soure has to be due to volume hange alone for its detetion. When the same exerise was onduted using perturbations of 5 per ent, at least 20 per ent of the soure had to be isotropi for detetion. Thus it may not be possible to disern a small isotropi moment tensor without more and better data and without better modelling of the veloity struture. Turning now to the data shown in Fig. 2, the least-squares solution is shown in Fig. 13. The main part of the soure is an initial pulse of about 20s duration, whih reflets the bandwidth of the instrument. For this initial pulse, the BONN SLANDS SMULATON (MN) 0 W m \ s 0. uu t.3u ! r+ z w ? : Fire U. Simulation of the moment rate traes in the minimum total trae squared solution for varying proportions of the isotropi omponent. The veloity struture was again assumed to be PREM while the struture used to generate the seismograms ontained 2 per ent perturbations of PREM. The proportion of isotropi omponent is given by the ratio of the isotropi omponent to the sum of the absolute values of the moment omponents.

13 M E+25 M2 1 M22 M3 1 M32 M Figure W. The least-squares moment rate trae solution for the Bonin slands event. This is the result of inverting the seismograms in Fig. 2 assuming a PREM earth. BONN SLAND TRACES !+ z w Figure 14. Comparison of the traes for the three models: least squares, maximum moment rate trae squared, and minimum moment rate trae squared. The minimum trae is essentially zero but the wide range in the solutions suggests that the isotropi omponent is not well onstrained.

14 14 D. W. Vaso and L. R. Johnson prinipal axes of the deviatori part of the soure have the approximate orientations (plunge, azimuth): tension axis = (10,42); intermediate axis = (45, 144); ompression axis = (40,320). This is similar to the Harvard solution (SC 1985): tension axis = (16,49); intermediate axis = (23, 146); ompression axis = (61,287). t is also generally onsistent with the fault plane solutions of other earthquakes in this region (Burbah & Frolih 1986). However, the primary interest in this study is the trae of the moment tensor and it appears that this is different from zero for the least-squares solution, beause in Fig. 13 the M33 omponent is larger than the M,, and MZ2 omponents. This trae from the least-squares solution is shown in Fig. 14 along with maximum and minimum trae solutions obtained with the quadrati programming approah. The error bound assumed in the extremal solutions was twie the root-mean-square error of the least-squares solution. The least-squares solution has a prominent negative isotropi omponent in the first 20 s whih ould be interpreted as a volume hange at the soure. The minimum extreme solution shows that, given the unertainty in the data and the Green s funtions used in the inversion, this volume hange suggested by the least-squares solution is probably not signifiant. On the basis of the syntheti tests and the fairly large signal-tonoise ratio in the data (Fig. 2), it seems likely that lateral heterogeneity is largely responsible for the width of the bounds and therefore the weakness of any onlusion whih may be drawn from these results. n order to test if the minimum volume hange is signifiantly different from zero, the best fitting solution with the trae onstrained to be zero ould be found. Then a Monte Carlo simulation, onsidering all error soures, ould be used to ompile a population of waveform data sets. An inversion of these data sets would give statistis on the zero trae solutions. This would allow one to statistially test if the minimum squared trae solution is signifiantly different from zero. We feel that, at present, models of veloity and attenuation lateral heterogeneity are not yet adequate for this. nstead, we rely on a omparison of the isotropi omponent with the other moment tensor omponents to estimate signifiane. 6 NUCLEAR EXPLOSON SOURCES One seismi soure whih is known to have a large isotropi omponent is a nulear explosion. t is an ideal ase for the omputation of extremal moment tensor models. Nearsoure data from a nulear explosion an be used to ompute upper and lower bounds on squared moment tensor trae, whih is a measure of the volume hange assoiated with the soure. Thus the solutions are, respetively, the most and least explosion-like solutions. This has important appliations in the verifiation of nulear explosions beause these solutions provide best and worst ases by whih to deide if an event was a nulear explosion. The Harzer experiment of 1981 June 6, a nulear explosion of equivalent magnitude 5.5 at a depth of 637m was reorded by eight, three-omponent, broadband, digital aelerometers. The details of the olletion and interpretation of the data an be found in Johnson (1988) Figure 15. Harzer experimental setup. Station distribution within the Silent Canyon aldera. The stations were three-omponent broadband aelerometers.

15 nversion of waveforms for extreme soure models 15 and will only be briefly reviewed here. The near-soure network was azimuthally distributed around the epienter with stations from 2.4 to 6.6 km from the event (Fig. 15). The veloity reords are shown in Fig. 16. The network was loated in the Silent Canyon Caldera, a heterogeneous veloity struture. The averaged one-dimensional veloity struture in this aldera has been studied by Leonard & Johnson (1987) and their model was used in moment tensor inversions. A modified refletivity method (Kind 1978) was used to ompute the Green's funtions. Beause of diffiulties at one of the sites, seven stations were used in the inversion, giving a total of 21 omponents. The least-squares solution for the moment rate tensor of this overdetermined problem is shown in Fig. 17. The most important part of the moment rate tensor is the initial short-period pulse whih is followed by longer-period osillations whih are less oherent and more poorly resolved. This initial pulse is most pronouned on the trae omponents M,, MZ2 and M33, although there is still energy on the off-diagonal elements, suh as Mz3. The M,, and MZ2 elements are fairly similar in their time dependene, but the M,, element is somewhat different, ontaining a large seondary pulse at about 2 s. When this solution is ombined with the Green's funtions, predited seismograms are obtained whih do a reasonably good job of explaining the 19 R major features of the observed data in Fig. 16, with average orrelation oeffiients of about 0.5. However, there still remain signifiant differenes between the predited and observed seismograms, partiularly on the transverse omponents. Muh of this differene an probably be attributed to effets whih were not taken into aount in the modelling, suh as lateral heterogeneities and sattering in the wave propagation and seondary soure effets suh as spall. Given that the soure has been less than perfetly resolved by the least-squares solution for the moment tensor, what an be said about the unertainty in the explosive part of the soure? This question is answered in Fig. 18 whih shows the least-squares solution for the moment rate tensor trae along with the estimates for the minimum and maximum moment rate tensor traes. t is lear that an initial ompressional pulse is present on all three solutions. Thus, even in the presene of fairly large error bounds due to both random noise and defiienies in the modelling proess, it an be onluded that this soure has a lear explosive omponent. The extremal solutions are also useful in assoiating an unertainty with the first pulse on the trae, whih is diretly related to the yield of the explosion. Up to this point all of the alulations have employed the t 1 J 16. Rotated veloity reords from the Harzer explosion. Note the signifiant energy on the transverse omponents. Distane from the soure is dereasing upward.

16 M11 M2 1 MZZ M3 M32 M33, U Least-squares solution for the Harzer explosion s moment rate tensor. - u W rn \ u W z > 3 0 N X v W b 6 P= E-. z w F L HARZER TRACES Figure 18. Moment rate traes for the least squares, maximum and minimum trae squared solutions. There is a wide range in the models for the early pulse but a definite isotropi omponent is present in all models.

17 nversion of waveforms for extreme soure models 17 HARZER r MNMUM BOUNDS GLO - LOC..._ t Figre 19. Minimum moment rate trae squared solutions omputed using the strit data bounds of equation (5) and the global data bounds of equation (6). Again, both models have an obvious isotropi omponent. loal error bounds of equation (5), but there are situations where the global error bounds of equation (6) might be preferred. The results of using these two types of error bounds are ompared in Fig. 19 for the minimum trae solution. n this partiular ase the hoie of bound does not ause enough differene in the results to affet any of the onlusions based upon them. 7 CONCLUSONS A method has been developed by whih extreme models of the time-varying moment tensor may be onstruted. The method has the potential to answer many interesting geophysial questions. n partiular, it is now possible to assess the presene or absene of volume hange (isotropi omponent) in seismi soures. t has been ommonly assumed that the isotropi omponent of the moment tensor vanishes. We believe that this assumption has not been adequately examined and that this method an be used for this purpose. The two appliations desribed above illustrate two areas in whih extreme bounds on the isotropi omponent are partiularly useful: deep earthquakes and nulear explosions. Surely, many others are possible. One exiting aspet of the tehnique is that it makes full use of the waveforms. With the advent of new networks of wide dynami range, broadband digital seismometers suh as GEOSCOPE and RS greater resolution will be possible. As an be seen from the Harzer inversions, even high frequeny data an give exellent inversion results. The method desribed in this paper provides an extreme estimate of some aspet of the moment tensor, given the reorded seismograms and an estimate of their unertainty in either the time domain or frequeny domain. This unertainty should inlude all possible soures of error, inluding random noise, the soure loation, the earth model used in alulating the Green's funtions, and the method used to alulate the Green's funtions. n some situations it may be possible to make a priori estimates of all these errors, but in general this will not be possible. n this paper we have proposed the alternative proedure of using the residual of the least-squares solution as an estimate of the total error. While this proedure has the undesirable feature that the error estimate depends upon the data, it does provide a proedure that an be used in all instanes and it seems to have given reasonable results in the two ases onsidered. The method is flexible in the sense that the error onstraints an be applied in either a loal or global sense. ACKNOWLEDGMENTS We thank Shimon Coen for several enlightening onversations and for foring us to tell him what was happening. This researh was partially supported by the Defense Advaned

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