Bulletin of the Seismological Society of America, Vol. 80, No. 2, pp. 354-371, April 1990

MOMENT-TENSOR INVARIANTS: SEARCHING FOR NON-DOUBLE- COUPLE EARTHQUAKES

BY D. W. VASCO*

ABSTRACT

A method for assessing the presence of a non-double-couple component in seismic sources is presented. The equation for the principal values of the moment tensor is a cubic equation with coefficients which are polynomial combinations of the tensor components. These coefficients are invariants of the tensor that relate to its symmetry and do not depend on the coordinate system used to describe the source. Each coefficient has a physical interpretation: The constant coefficient is determined by how well one or two double-couples describe the source, and the quadratic coefficient is the volume change associated with the source. If volume change is assumed to be absent, the constant term represents a measure of the double-couple nature of the source. A seismic waveform inversion method is presented which exactly explores the range of values of these properties. The method of extremal models is used to determine minimum and maximum values of the invariants subject to the constraint that the data must be satisfied within prescribed errors. For the invariant associated with the source volume change, this constraint results in a linear programming problem and a global solution may be found. For the other invariants a nonlinear programming problem results which may be solved by a reduced gradient algorithm. Two events were examined: A deep earthquake from the Bonin Islands and the Harzer nuclear explosion. A double-couple mechanism is essentially compatible with the Bonin Islands waveforms. Models with a positive volume change were required to fit the Harzer data. Some variation in the mechanism is possible due to gaps in the station distribution and possibly due to significant scattering from lateral heterogeneities. However, in spite of a range in the isotropic component, the event could not be fit with a double-couple mechanism.

INTRODUCTION

Since the inception of the moment-tensor concept (Gilbert, 1971), there has been conflicting evidence on the mechanism of some earthquakes. Early investigators noted evidence for a precursory isotropic (volume change) component in the source mechanism of a few deep earthquakes (Dziewonski and Gilbert, 1974; Gilbert and Dziewonski, 1975). The reports were subsequently questioned (Okal and Geller, 1979) on the basis of effects introduced by lateral heterogeneity and the relative insensitivity of waveforms to the isotropic component. Recently, Singh and Ben- Menahem (1988) reported that far-field displacements do not explicitly depend on the trace of the moment tensor, suggesting that the non-double-couple nature of an event may not manifest itself there. Even before this, it had been noted (Knopoff and Randall, 1970; Randall and Knopoff, 1970) that deviatoric moment tensors, those containing no isotropic component, need not be pure double-couples.

Later researchers have followed up this early work on deep events (Dziewonski and Woodhouse, 1983; Giardini, 1983; Giardini, 1984) with systematic analysis of deep seismicity. In particular, they searched for a deviation from a pure double-

* Formerly at: Earth Sciences Division Geophysics Laboratory Hanscom AFB, MA 0173.

354

MOMENT-TENSOR INVARIANTS 355

couple mechanism as measured by the parameter

X2

max(I X~ l, I X3 I) '

Xl, X2, and Xa being the eigenvalues of the moment tensor, Xl => X2 --> X3. In the latter study, it was found that c varies widely but in general decreases with increasing moment. As a function of depth there is again much scatter in e but two thirds of earthquakes deeper than 350 km have positive deviations corresponding to a predominance of the compression (Giardini, 1984). Silver and Jordan (1982) have developed a method for the estimation of the isotropic component of the moment tensor at long periods. For the events studied, one shallow and one deep, an isotropic component was detected at a 90 percent confidence level for the deep event, though more recent analysis (Riedesel and Jordan, 1989) finds a double-couple explanation consistent with the data.

Waveform inversion at shorter wavelengths have similarly produced indications, though ambigious, of non-double-couple events. A fairly recent example was the controversy over the mechanism of the May 1980 Mammoth Lakes, California, sequence (Ekstr6m, 1983; Geller et al., 1983; Julian and Sipkin, 1983; Wallace, 1985). Again, the waveform data could be satisfied by a number of alternative models: a double-couple with a complex rupture, lateral heterogeneity, and tensile failure. In other volcanic regions, such as Mount St. Helens (Weaver et aI., 1983) and Tori Shima, Japan (Kanamori et al., 1986), there is evidence for earthquakes produced by magma and magmatic gas transport.

The examples presented above illustrate that throughout the history of seismic source investigations there have been indications of non-double-couple sources. Almost inevitably, such observations have been followed by controversy and alter- native explanations. There seems to be a need to assess the significance of reported non-double-couple components. This paper presents a method for the exact mapping of error bounds on seismic waveforms into bounds on certain moment-tensor properties. The properties in question are the three invariants of the moment tensor: the trace, the determinant, and the sum of the determinants of the diagonal minors. These unique, coordinate-free properties of the equivalent source have physical interpretations. The trace is a measure of the volume change associated with the event, while both the trace and determinant vanish if, and only if, the source is a double-couple. It is not intended that this technique be used as a means of searching large earthquake catalogs for non-double-couple events. There are other methods, such as the eigenvalue decomposition, better suited for such rapid determination. Rather, it is more useful for detailed examinations of specific events which, in initial examination, do not have a pure double-couple mechanism. Finding upper and lower bounds on these invariants allows one to determine if significant volume change must be associated with the source or if a non-double-couple mechanism is required in order to satisfy the data. Furthermore, the range of models that adequately fit the data is an indication of how well-constrained the source properties are. Note that the calculation of bounds on source properties should be used in conjunction with the estimation of a seismic moment tensor and is not presented here as an alternative to moment-tensor determination.

One moment-tensor invariant, the isotropic component (as measured by the trace), has been studied early on in the investigation of moment tensors (Dziewonski and Gilbert, 1974; Gilbert and Dziewonski, 1975; Okal and Geller, 1979; Silver and

356 D.W. VASCO

Jordan, 1982; Julian, 1986; Vasco and Johnson, 1989). All the methods, except for the latter two, use a least-squares procedure to determine the trace. There is an inherent nonuniqueness in least-squares solutions due to the trade-off between model resolution and model variance. There are indications (Singh and Ben- Menahem, 1988) that far-field waveforms are relatively insensitive to the isotropic moment-tensor component, and hence this property is poorly resolved. By estimat- ing bounds on the moment tensor, the sensitivity of the waveforms can be shown explicitly.

As mentioned previously, events that are not pure double-couples need not contain an isotropic component; the compensated linear vector dipole (CLVD) model for fluid intrusion is one such example (Julian and Sipkin, 1983). However, considera- tion of a general non-double-couple source is an inherently nonlinear problem due to the cubic relationship between the principal values and vectors and the compo- nents of the moment tensor. Recently, there have been graphical methods suggested for identifying non-double-couple moment-tensor components (Pearce et al., 1988; Hudsen et al., 1989; Riedesel and Jordan, 1989). These are methods for portraying a set of previously determined moment-tensor components and their associated variances such that the presence of a non-double-couple component is clear. As such they are useful for identifying potential non-double-couple events. The method of Riedesel and Jordan (1989) suffers from a difficulty in that perturbation theory is used to approximate the mapping between the moment-tensor variances and the variances of the principal components and principal vectors. This approximation, which first appeared in Strelitz (1980), is not valid in the presence of significant moment-tensor variance. Indeed, as demonstrated in Vasco and Johnson (1989) and as I hope to show here, waveform mismatches between predicted and observed seismograms can map into significant variations in mechanism. The computation of bounds presented here is an exact mapping from confidence intervals on the waveform data to bounds on the moment-tensor invariants. Furthermore, the bounds are unique for a given set of data confidence intervals (Parker, 1975). Clearly, the bounds depend on estimates of the errors in the seismic waveforms. It is most desirable to have a pr ior i error estimates but, because Green's function errors are difficult to calculate, I also present a way to use deviations from a least- squares model predictions to generate error estimates.

MOMENT-TENSOR INVARIANTS

Several ways have been suggested to decompose the symmetric moment tensor. For example, it may be represented as a sum of an isotropic component plus a major and minor double-couple (Gilbert, 1980) or as a combination of an isotropic tensor plus a double-couple and a CLVD (Knopoff and Randall, 1970). Unfortunately, such decompositions are not unique and depend nonlinearly, as a cubic, on the moment-tensor elements. This nonuniqueness led me to consider coordinate inde- pendent properties of the moment tensor, the moment-tensor invariants. One such invariant, the isotropic component, has been examined by many others (Dziewonski and Gilbert, 1974; Gilbert and Dziewonski, 1975; Okal and Geller, 1979; Silver and Jordan, 1982; Vasco and Johnson, 1989), but not explicitly as a moment-tensor invariant. In addition, Silver and Jordan (1982) considered the total scalar seismic moment, an invariant measure of the earthquake size and radiated energy. The isotropic component is a measure of only one possible non-double-couple mode. There are non-double-couple models, such as the CLVD, which are deviatoric. Indeed, excluding phase changes, there are few processes in nature that result in a

MOMENT-TENSOR INVARIANTS 357

net volume loss or creat ion resolvable at seismic wavelengths tha t are long relative to the source dimension. The total scalar seismic m o m e n t measures the radiated fault energy, not the depar ture of the source f rom a single double-couple. Thus, it is necessary to consider other invar iants of the m o m e n t tensor in order to test for the presence of a general non-double-couple component . In this section, I consider the m o m e n t tensor for a point source with the t ime dependence removed.

The m o m e n t tensor m, is a symmetr ic , real valued 3 × 3 mat r ix (Aki and Richards, 1980)

f m l l m2~ FYt31

m=lm21 I. \ m 3 1 m32 m 3 3 /

(1)

The diagonalization of this a r ray results in a cubic equat ion for the eigenvalues,

7 3 - 3 iso(m)~ 2 + minor (m)~ - det (m) = 0 (2)

with iso(m), minor (m) , and det (m) defined below. The symmet ry of the mat r ix m insures tha t the roots of equat ion (2) are real. Because the eigenvalues of this mat r ix are unchanged by any nonsingular coordinate t ransformat ion , the coefficients of this polynomial are invariants , independent of any coordinate sys tem used to represent the m o m e n t tensor. In t e rms of the momen t - t enso r elements, the coeffi- cients are the de te rminan t of the m o m e n t tensor

det (m) = m l l m 2 2 m 3 3 + 2m21rn32m31 - m l l m ~ 2 - m 2 2 m ~ l - m 3 3 m ~ l , (3)

the sum of the diagonal minors of the de te rminan t of the tensor

minor (m) = rnl lrn22 + m22m33 + rn33rnl l - rn~l - rn~2 - m ~ l

minor (m) = ml~ m21 .~_ m22 FYt32 _~ T~t33 ]n31 m2~ m22 m32 m33 m3~ m~l

(4)

and, its isotropic componen t

iso(m) = rnll + rn22 + m33 3 (5)

These coefficients have physical significance and may be in terpre ted in t e rms of source properties. For example, iso(m), which is the sum of the diagonal e lements of the m o m e n t tensor divided by 3, is a measure of the volume change associated with the source. The quant i ty minor (m) vanishes when any two eigenvalues of the m o m e n t tensor vanish as may be easily seen by expressing it in principal coordinates. Similarly, det(m), which is the de te rminan t of m, can be in terpre ted as a measure of the depar ture of the mechan i sm from two-dimensional i ty . Consider the cubic equat ion when det (m) vanishes:

V(~2 _ 3 iso(m)~ + minor (m)) = 0 (6)

358 D.W. VASCO

which has at least one zero root. Because at least one root mus t remain nonzero for an observable seismic source, one or two eigenvalues can vanish. Physically, this corresponds to the requi rement tha t the source consists of fewer t han three couples: ei ther shear on a fault (two non-zero 7) or extension of a p lanar surface (one nonzero 7). The la t ter mechan i sm may be caused by fluid or gas injection into a region. I t may also appear if surface roughness, perhaps associated with asperities, forces the fault surfaces to move toward or away from each other. In this case, there would be a superposi t ion of shear on the fault and mot ion normal to the fault producing an apparen t volume change. I f the source is constra ined to have no net volume change, iso(m) = 0, the cubic in equat ion (6) becomes

.(V2 + minor (m)) = O.

Because only real roots are possible minor (m) < 0 and

= + ~ / - m i n o r ( m )

are the other two roots. We have el iminated mot ion perpendicular to the fault and are left with the famil iar double-couple mechanism. Conversely, if the source is a double-couple then the m o m e n t tensor is of the form

mij = Mo(, inj + ,jni) (7)

(Aki and Richards, 1980), where M0 is the seismic m o m e n t (a scalar quanti ty) , v is the normalized slip vector, and n is the normal to the fault surface. Because the slip direction lies in the fault plane, normal to n, ,knk = 0, and it follows tha t iso(m) = 2 ~Movknk ---- 0. Similarly, it can be easily shown tha t for the m o m e n t tensor of the double-couple source given above tha t de t (m) = 0. Therefore, the vanishing of de t (m) and iso(m) are necessary and sufficient conditions for a double-couple source. This means tha t the quant i t ies

[ iso(m) [ + ] de t (m) [

and

dc(m) = - i s o ( m ) 2 - de t (m) 2 (8)

can be used as d iscr iminants between double-couple and non-double-couple sources. The later quanti ty, denoted here by dc(m), has a m a x i m u m at the mos t double- coupleqike solution. Conversely, the least double-couple-like mechan i sm is the min imum of equat ion (8). Each t e rm in these equations can be scaled to account for the higher order of the cubic det (m) relative to the trace. As the two invar iants det(m) and iso(m) are sufficient to characterize the general non-double-couple behavior of a seismic event, the invar ian t minor (m) will not be discussed further.

Because the invar iants give coordinate-free informat ion about the source, it is not necessary to make assumpt ions about source or ienta t ion in order to determine whether a source is a non-double-couple. Thus, even though there are infinitely m a n y ways to decompose a source into componen t couples, the de te rminan t and trace are unique propert ies of the m o m e n t tensor. Because they contain impor tan t physical in terpre ta t ions tha t relate to the dimensional i ty of the source, they are valuable source proper t ies for s tudying deviations f rom a pure double-couple mech-

MOMENT-TENSOR INVARIANTS 359

anism. In the section that follows, I will present a method for determining bounds within which these quantities lie, given error bounds on the waveform data.

MOMENT-INVARIANT DETERMINATION

An examination of the general non-double-couple nature of a source requires more than the determination of the moment-tensor components. The cubic rela- tionship between the moment-tensor components and the principal components results in an inherently nonlinear problem. However, as will be seen below, the moment-tensor components and their accompanying variances can be used to construct approximate upper and lower bounds on the moment-tensor invariants through a nonlinear mapping. Alternatively, there is a method for mapping data confidence bounds exactly into bounds on the moment-tensor invariants: The method of extreme models (Parker, 1972, 1974, 1975; Sabatier, 1977a, b; Vasco and Johnson, 1989). Before these methods are discussed, I present a summary of results from linear inverse theory that bear upon the problem at hand.

Methods for solving linear inverse problems, Hm = u, such as the determination of the moment-tensor components, are well-documented (Aki and Richards, 1980; Menke, 1984; Tarantola, 1987). Of particular interest here is the commonly used maximum likelihood inverse (Aki and Richards, 1980) Hm~,

HT.~ = CaaHT(HCaaH T + Cuu) -1,

where H represents the Green's function, u the waveform data, and m the moment tensor components. The data covariance matrix is denoted by Cu~ while the a priori model covariance is Caa. The a posteriori covariance of the model parameters is given by

C m m --1 --1 T = HmlCuu(Hml) (9)

(Aki and Richards, 1980). There is an important trade-off between the model parameter covariance and the model parameter resolution, R, which is a measure of the averaging of the "true model," and is written in terms of the maximum likelihood inverse

R = Hm~H

(Menke, 1984). A very direct relationship exists between the two quantities (Nowack and Lutter, 1988)

Cmm = R ( I - R ) C . a .

Another formulation, put forward by Tarantola (1987), estimates a mean model

-I -1 T -i rh = (HTCm~H + C~ ) (H C ~ u + C21mp)

and an a posteriori covariance matrix,

Cram = (HTCS2H + C; J ) -~ (10)

where mp is an a priori model estimate. In this approach the a posteriori co- variance matrix and the model parameter resolution are related by (Nowack

360

and Lutter, 1988)

D. W. VASCO

Cram = (I - R)C.~.

In the case of poor resolution, the a posteriori covariances are given by the a priori covariances.

Using the equations just presented, it is possible to derive confidence intervals on the moment tensor, m~j _+ 5mij, where I have returned to the notation of the previous section on moment-tensor invariants. These intervals can be used in conjuction with the equations of the moment-tensor invariants, equations (3), (4), and (5), to arrive at signed upper and lower bounds of the invariants. For the isotropic component, which is linear in the moment-tensor components, the limits are

(mll "l- m22 + m33) + (6mH + (~m22 -{- ~m33).

For the other invariant, det(m), the relationship is cubic in the confidence intervals. It is a combinatorial problem to find an upper and lower bound of this quantity. Intervals derived in this way are not statistically rigorous, however, and can only be used to roughly evaluate the significance of a given non-double-couple compo- nent. A more direct approach is to map confidence intervals on the data into upper and lower bounds on certain model properties, without using the linear inverse solution as an intermediate step. This procedure, known as the method of extremal bounds or extremal models, has been developed for model properties which are linear functionals of the parameters (Parker, 1972, 1974, 1975; Sabatier 1977a, b). As such it can be applied directly to the computation of bounds on the isotropic moment-tensor invariant (Vasco and Johnson, 1989). The basic idea is to constrain the model by the data and some confidence interval on the data

u - a _ - < H m _ - < u + a . (11)

The waveform data is the vector u, and ¢ are the fixed confidence bounds on the data. These are point confidence intervals; it is also possible to constrain the data using a global confidence bound,

- ~ =< H m =< u + e

u - e _ _ < H m _ _ > ~

E el <=E i =1

e > O

Here, el are non-negative error variables and E is a bound on the total misfit. Constraining the model by either system of equations, one can then find the models which make some linear property, cTm, a maximum or minimum. Again, the isotropic component of the moment tensor is one simple example of a linear property. Methods of solving such problems, particularly the simplex method (Dantzig, 1963), are widely available (Land and Powell, 1973; Menke, 1984; Press et al., 1986).

MOMENT-TENSOR INVARIANTS 361

For the study of the other moment-tensor invariant, the method of extreme models must be generalized. The constraints on the model remain linear but the model property, the moment-tensor invariant det(m), is now a polynomial of the moment-tensor components, possibly as high as sixth degree (in the case of the non-double-couple measure dc(m)). Such constrained minimization or maximiza- tion problems present no fundamental difficulty and solutions may be found by readily available techniques (Gill et al., 1981; Beale, 1988). The primary difficulty introduced by a nonlinear Objective functional is the presence of local extrema. Therefore, to ensure that the solution is not merely a local extrema, it is necessary to use a number of starting solutions. Fortunately, there are a number of such models available in the solution of the linear problem given above and in the minimum and maximum trace solutions. Perhaps a method-like simulated annealing (Kirkpatrick et al., 1983) can be used to better explore the model space.

In what follows, I consider the linear inversion for the scalar moment tensor resulting from a spatial point source assumption. The inverse problem for this tensor of rank 2 is fundamentally unique (Backus and Mulcahy, 1976) if the waveform errors are Gaussian and not large and the model is well-resolved. I then use the maximum of the discrepencies between the observed waveforms and those predicted using the generalized inverse moment tensor and a base noise level, ¢,, to compute waveform errors,

~(t) = max(z~, l u - Hml ) . (12)

The question is then, given these discrepencies, what is the range of a particular moment-tensor invariant? These then are local bounds of the invariants about the least-squares solution. If a priori estimates of z(t) are available, they could be readily incorporated. It might be possible for example, to include the mean absolute amplitude of scattered energy occuring between predicted arrivals as the base noise level for subsequent arrivals. Given the initial least-squares solution and the waveform bounds, the nonlinear programming problem can be solved locally, using a reduced-gradient algorithm (Beale, 1988) as interpreted in Murtagh and Saunders (1978).

A DEEP EVENT FROM THE BONIN ISLANDS

On 4 October 1985, a deep (467.7 km) earthquake of body-wave magnitude 5.6 occurred under the Bonin Islands. The area, south of Japan, is one of active subduction, with many deep and intermediate events. The vertical seismograms for the event are shown in Figure 1. The event in question has already been studied using time-varying moment-tensor inversion (Vasco and Johnson, 1989). Moment- tensor models with largest and smallest total volume changes were computed in addition to calculating a generalized inverse for the source. No significant volume change was observed for this event, though the difference between the largest and smallest traces allowed by the data confidence bounds was large. The principal axes of the generalize inverse were (Vasco and Johnson, 1989) (plunge,azimuth): tension (10,42), intermediate (23,146), and compression (40,320), in agreement with the Harvard solution (Bull. ISC, 1985). The source-time function from this event consisted of a pulse followed 19 sec later by a similar, smaller pulse (Vasco, 1989). The generalized inverse was slightly contaminated by a low-frequency oscillation due to poor conditioning of the inverse at these frequencies. Because of this, it was necessary to convolve the Green's function with the source-time function and to

362 D . W . VASCO

Z 0 I===4

<

U/

{JHTO

~HIO

dTAO

NWAO

KEV

R~NT

ODH

KON

R~ON

dRFO

ANMO

v v ~ "

I i i I I i i i I i I , I I I

38.5

4 2 . 8

47.8

64.0

71.8 l:~

72.7 t l~

> 80.5

82.9 0

83.9

88.7

90.5

91.7

MAX. AMP.=0.559"~

0.0 64.0 128.0 192.0 258.0 PRED . . . . OBS

TIME (S) FIG. 1. Vertical GDSN waveforms for the Bonin Islands earthquake. Also presented are the predicted

arrivals generated by a least-squares solution for the point source moment tensor. The maximum amplitude is given in microns.

TABLE 1

MOMENT-TENSOR SOLUTIONS FOR THE BONIN ISLANDS EARTHQUAKE

Bonin Islands Mechanisms (×1026 dyne-cm/sec)

ro l l 1Tt21 D~t22 m31 17t32 D~t33 tr(m) det(m) 1/~

E I m~j/61 Z I mJ61

Least squares 0.278 0.317 -0.324 -0.063 0.338 -0.239 -1.097 0.007 Minimum iso(m) 0.403 0.356 -0.226 -0.057 0.328 -0.215 -0.144 -0.034 Maximum iso(m) 0.109 0.180 -0.405 -0.056 0.336 -0.286 -2.543 0.018 Maximum dc(m) 0 .416 0.399 -0.230 -0.065 0.323 -0.230 -0.160 -0.001 Minimum dc(m) 0 .109 0.180 -0.405 -0.056 0.336 -0.286 -2.543 0.018

f i l te r t he d a t a be fore a t t e m p t i n g an inve r s ion . T h e G r e e n ' s f u n c t i o n was c o n s t r u c t e d us ing the W K B J t e c h n i q u e ( C h a p m a n , 1978) a n d the P r e l i m i n a r y Re fe r ence E a r t h M o d e l (Dz i ewonsk i a n d A n d e r s o n , 1981).

A f t e r t h e c o n v o l u t i o n a n d f i l te r ing , t h e va r i ous e x t r e m e source m o d e l s s a t i s fy ing the d a t a w i t h i n p r e s c r i b e d e r ro r b o u n d s were f o u n d us ing t h e m e t h o d f rom the p r e v i o u s sec t ion for p o i n t con f idence i n t e r v a l s ( equa t ion 11). T h e r e su l t i ng m o m e n t - t e n s o r mode l s a re p r e s e n t e d in T a b l e 1, a n d t h e f i r s t m o t i o n s f rom t h e mode l s a re p l o t t e d in F igu re 2. F i r s t , a g e n e r a l i z e d inve r se was c o n s t r u c t e d for t he p o i n t source so lu t ion , wh ich is d i s p l a y e d in t h e figure. C o m p r e s s i o n a l f i r s t m o t i o n s a re f o u n d in t h e d a r k e n e d reg ions whi le d i l a t a t i o n a l f i r s t m o t i o n s occur in t he wh i t e regions . No t i ce t h a t i t is n o t a pu re doub le -coup le ; i t d e v i a t e s s o m e w h a t f rom sl ip on a faul t .

MOMENT-TENSOR INVARIANTS

MINIMUM TRACE MODEL MAXIMUM TRACE MODEL N N

N

MINIMUM DOUBLE-COUPLE MVO~V'AXIMUM DOUBLE-COUPLE MODEL N ' N

363

FIG. 2. Different source mechanism models for the Bonin Islands earthquake (least squares, minimum and maximum trace (iso (m)), and minimum and maximum double-couple (dc(m))). The p-wave polarity is projected onto the lower hemisphere, the compressional first motion denoted by the darkened portion.

Its mechanism is in accordance with the Harvard solution (tension axis = (16,49); intermediate axis = (23,146); compression axis = (61,287)) as well as with the frequency-domain result of Vasco and Johnson (1989). Having found this least- squares solution, the predicted displacements at the stations were computed and are shown in Figure 1. The first arrivals were matched fairly well but the later arrivals were not as well modeled. These mismatches were used in equation (12) to compute waveform errors. The average amplitude of the microseismic noise pre- ceeding the event was used to compute the base noise level (~ ) is this equation. Given these errors, what are the ranges of the moment-tensor invariants? By minimum and maximum of iso(m) I mean the minimum and maximum of its square, because it is not the signed magnitude but the absolute value that is of interest. The model which minimizes this measure of source volume change is shown in Figure 2 and does not resemble a double-couple. It is also very different from the maximum iso(m) solution. In fact, the ratio of one third of the moment trace to the average absolute value of the moment components has a value of -0.144, almost an order of magnitude less than that of the generalized inverse, -1.097 (see Table 1). The wide variation may be caused by the insensitivity of the waveforms to the trace of the source as well as by the errors in fitting the data. The minimum dc(m) (minimum double-couple) model satisfying the data within the confidence bounds is given in Table i and Figure 2. It deviates markedly from a double-couple mechanism and from the generalized inverse. In contrast, the model which most resembles a double-couple has a focal mechanism which is almost identical to a double-couple, with a determinant which is very small. The ratio of the cube root of the determinant to the average absolute value of the moment-tensor elements is -0.001. The corresponding value for the generalized inverse is 0.007, seven times as large. Thus, the waveform data can be satisfied, within the misfit of the generalized inverse solution, by a double-couple mechanism.

364 D . W . VASCO

For this event it seems that the extreme models differ greatly. This may be due to many factors, such as lateral heterogeneity or the temporal point source assump- tion. The effects of lateral heterogeneity can be addressed through the use of recently derived models (Woodhouse and Dziewonski, 1984; Durek et al., 1988). The temporal point source assumption need not be a restriction because the method may be formulated for a time-varying moment tensor.

DISCRIMINATES FOR EXPLOSIONS: HARZER NUCLEAR BLAST

In order to discern how well waveforms can be used to determine if an event was an explosion or an earthquake I examined seismograms from the Harzer nuclear explosion. The displacement field from this event was recorded by a near-field array of seven three-component accelerometers (Fig. 3) as described in Johnson (1988). The two components used in the inversion, the transverse and vertical, are shown in Figure 4, as are the predicted motions based upon a generalized inversion for the source moment tensor. There are gaps in the station coverage of this array to the east and west of the explosion due to the malfunction of a station. The explosion, in Pahute Mesa at the Nevada Test Site, was situated under 600 m of volcanic sediment that is very heterogeneous (Leonard and Johnson, 1987). The averaged, one-dimensional velocity structure of these authors was used in the inversion of the Harzer waveforms. The Green's function was computed using the reflectivity method as presented in Kind (1978), and the first 2 sec of the waveforms were inverted for the point moment tensor. A best-fitting source-time function for the event was derived using principal component analysis (Vasco, 1989). This function was convolved with the point source Green's function before the inversion was begun.

The generalized inverse solution and four extreme models were derived (Fig. 5 and Table 2) for Harzer. The generalized inverse (Fig. 5) consists totally of

. J "%

Test S i te B o u n d a r y . , ~ " ' ' : / - ' S ' ' - - ' ' ' " - - " \

' \ ~

l ".. / I \ .,.-~"

\ ". I ... \ ..~ .. ~ N

FIG. 3. Station (filled circles) distribution for the Harzer experiment. Shown here is the boundary of the Nevada Test Site (dashed-dot line), the Silent Canyon caldera boundary (dashed line), and the shot point (open circle).

M O M E N T - T E N S O R I N V A R I A N T S

HARZER VERTICAL COMPONENTS

H9

H4

H2

Z O HI

H8

H5

__ -- ~ ~ - - / ~ ~ ~ ~ ~

2.4

3.5

3.5

4.8

5.8

5.8

I I I I I I I I I I I I I I I

0,0 0,5 1.0 1.5

TIME (S)

8.8

tD V/I

Z C'I

Y~

V

MAX. AMP.=0.2738

2.0 PRED .... o~

365

HARZER TRANSVERSE COMPONENTS

H9

H4

H2

O Hi

H6

H8

H5

0 . 0 I I I f I I I I I I I I I I I

0.5 1.0 1.5

z.4

3,5

35 V/I

> 4.e 2 :

Q .

5.8

5.8

8.8

TIME (~)

MAX. AMP.=0.3210

20 PRED ..... o~

FIG. 4. Vertical and t ransverse velocity records of the Harzer event used in the inversion. The m a x i m u m velocities are given in uni t s of 100 cm/sec.

366 D. W. VASCO

MINIMUM TRACE MODEL MhXIMUM TRACE MODEL N N

0:O N

MINIMUM DOUBLE-COUPLE MODEL MAXIMUM DOUBLE-COUPLE MODEL

FIG. 5. Source mechanisms from the Harzer explosion (similar to Fig. 2).

TABLE 2 MOMENT-TENSOR SOLUTIONS FOR THE HARZER NUCLEAR EXPLOSION

Harzer Mechanisms (Xl022 dyne-cm/sec)

t~Zl 1 m21 /7222 r/231 /7532 ln33 tr(m) det(m) 1/3

[ mlj/6 [ ~ I miJ61

Least squares 0.407 -0.038 0.341 0.003 0.086 0.286 5.345 0.187 Minimum/so(m) 0.195 -0.043 0.000 0.003 0.065 0.121 4.437 -0.015 Maximum iso(m) 0.577 0.042 0.621 -0.012 0.068 0.410 5.576 0.497 Maximum dc(m) 0.195 -0.043 0.000 0.003 0.065 0.121 4.437 -0.015 Minimum dc(m) 0.577 0.042 0.621 -0.012 0.068 0.410 5.576 0.497

compressional radiation, as expected for an explosion. It was used as an initial solution with which to compute data misfits for bounds on the waveforms which are shown in Figure 4. Because of lateral heterogeneity, which was not accounted for in the one-dimensional velocity model, there were significant differences between the predicted and the observed waveforms. This is very apparent on the transverse component which should not contain any arrivals, given that the source was an explosion. These differences mapped into confidence bounds on the data. These bounds were used with the pre-event noise, which was quite small, to derive the a(t)s in equation (12). Within these point bounds, equation (11), it was possible to find the extreme models for the square of the moment- tensor invariant iso(m) and the combination of squared invariants dc(m). First, iso(m) was examined by the minimization procedure mentioned above. The most explosion-like source, has an identical radiation pat tern to the generalized inverse solution. Its normalized trace (ratio of trace to average absolute moment component) was slightly larger than the least-squares result, with a value of 5.576 (see Table 2). Also of interest was the minimum trace source, which bears some resemblence to an explosive mechanism,

MOMENT-TENSOR INVARIANTS

HARZER VERTICAL COMPONI NTS

H9 -- __ _ _.~ . . . . . ~ 2.4

Z O H1 5--,

H6

H8

H5

0.0 0.5 1.0 1.5

TIME (S)

367

3.5 tD

4.e Q

5.6

v 5.8

6.6

MAX. AMP.=0.2738

2.0 PRED .... o~

HARZER TRANSVERSE COMPONENTS

H9 ~ 2.4

H4

H2

Z O HI

F-, H6

H8

H5

3.5

3.5 Vll

4.e :Z

P~

5.6

v 5.8

6.6

MAX. AMP.=O.321Q I I I ) I I I I I I I I I I I

0.0 0.5 1.0 1.5 2.0 PRED . . . .

o~ TIME (~)

FIO. 6. Observed waveforms from the Harzer explosion and synthetic seismograms generated using the maximum trace model (maximum iso (m)).

368 D.w. VASCO

with a normalized trace (4.437) significantly different from zero. The area of dilatational first motion is exactly the azimuthal gap in station coverage to the west. It seems that as a result of the clustering of stations, the normalized trace can vary though it is required to be nonzero.

In addition, because this was high-frequency data, scattering caused widened confidence bounds. The synthetics, generated by the maximum trace solution, are shown in Figure 6. As required, they fit the observed waveforms within two standard errors.

When the sum of squares of the invariants det(m) and iso(m) are examined the results are similar to that for iso(m). The minimum double-couple solution is a purely compressional source (Fig. 5) with the same large isotropic component as the maximum explosion model (5.576, see Table 2). The ratio of the cube root of the determinant to the average absolute moment tensor value (0.497) is over twice as large as that of the least-squares result (0.187). However, the maximum double- couple model also differs somewhat from an explosive mechanism. The normalized iso(m) of this model, 4.437, is significant. The normalized cube of the determinant (-0.015) for the maximum double-couple is less than one tenth of the value of the generalized inverse, 0.187, but still significant. Thus, in spite of large waveform mismatches, the waveforms still cannot be satisfied by a double-couple mechanism.

These solutions demonstrate the range of models which may fit the data within the errors of the least-squares solution. In the case of the maximum and minimum trace solutions, it is possible to compare the results to those of Vasco and Johnson's (1989) time-varying extremal moment models. In that work, we found measurable trace energy in the minimum trace solution though there were wide variations between the models. This was also the case in this study with the focal mechanisms (Fig. 5) of the minimum trace and maximum double-couple models differ somewhat from an explosion.

DISCUSSION AND CONCLUSIONS

I have presented a method to explore the range of certain moment-tensor properties compatible with a given data set. Two quantities in particular, the moment-tensor determinant and trace, have important physical significance and may be used to distinguish between purely double-couple events and non-double- couple events. One of these quantities, the moment-tensor determinant, is a cubic polynomial in the moment-tensor components. This is due to the fundamental nonlinearity in the mapping of the moment-tensor components into their principal components and cannot be avoided in the study of general non-double-couple sources. Methods to linearize this relationship or use perturbation theory must take into account that the actual variances on the moment-tensor components can be quite large. This was clearly demonstrated here by the wide range of solutions that fit the waveform data within the confidence bounds.

When few constraints are available, the solution of the inverse problem will be nonunique. In this case, it is best to invert the data for bounds on model properties. In particular, one may derive extreme models which have the least and greatest trace and determinant. These bounds can be used to assess the significance of observed non-double-couple mechanisms.

I have considered two very different data sets: GDSN teleseismic waveforms from a deep Bonin Island earthquake and near-field ground motion from the Harzer nuclear explosion. It was found that the earthquake, while having a slightly non- double-couple generalized inverse solution may be modeled by a double-couple. On

MOMENT-TENSOR INVARIANTS 369

the other hand, the waveforms from the Harzer nuclear explosion may not be modelled by such a mechanism. In spite of these conclusions, it seems that there are a range of models possible for the data sets given. Given this ambiguity, it is tempting to accept the generalized inverse result as a best estimate. However, this minimum norm inverse may be biased by such things as shallow depths and lateral heterogeneity. Furthermore, it is well-known that the weighting of the generalized inverse heavily favors large outliers in the data. Thus, it is important to consider how the moment-tensor properties may vary for all possible models.

It is difficult to obtain aprior i bounds on seismic waveforms, and I have discussed a technique to use data misfit to estimate waveform errors. Because this depends on the generalized inverse, some bias may be introduced into the estimate of the bounds. When no a priori bounds are available, it is important to assess the least- squares estimate on which the method is based. In particular, any sign of an underdetermined solution invalidates the approach for estimating confidence bounds on the data. Hence, it is important to examine the singular values of the generalized inverse for ill-conditioning. Over parameterized problems can have very small residuals and still be quite incorrect.

Incorporating a priori waveform bounds is the ideal approach but, at present, it is often not possible. If statistical models of heterogeneity are available for an area, then it is possible through Monte-Carlo modeling to estimate waveform errors. These may then be incorporated into the calculation of extreme models. For example, one could average recently derived velocity and attenuation models (Woodhouse and Dziewonski, 1984; Durek et al., 1988) over spherical shells to compute a mean radial velocity model and compute standard deviations for this mean model. Then, waveform statistics may be gathered by propagating energy through an ensemble of models with the mean and standard error derived from the lateral heterogeneity models. Work is currently proceeding in the analysis of teleseismic and regional codas (Dainty, 1989). One hope is to use arrays of seismo- meters to calculate the statistics of the scattering medium and waveform errors. Then, using these errors as a priori data bounds, one could determine bounds on the moment-tensor invariants.

The method of extreme models is suited to addressing a key question in seismic source observation: How universal is the double-couple source? Given advances in the determination of lateral heterogeneity on all scales, the haze shrouding the source may be lifted somewhat. Broadband data, such as currently collected by the GDSN, Geoscope, and IRIS worldwide arrays, will allow examination of the source at all periods with high signal-to-noise ratios. With these improvements we can test for the deviation from a pure double-couple source. The calculation of extreme models is one way to accomplish this.

ACKNOWLEDGMENTS

Much of this work was done while I was at the Geophysics Laboratory, United States Air Force, under Contract F19628-86-C-0224. This work was also partially supported by the Defense Advanced Research Projects Agency and was monitored by the Air Force Geophysics Laboratory under contract F19628-89- K-0017. Center for Computational Seismology Contribution no. 86.

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CENTER FOR COMPUTATIONAL SEISMOLOGY LAWRENCE BERKELEY LABORATORY AND

SEISMOGRAPHIC STATION UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720

Manuscript received 12 June 1989