DERIVING SOURCE-TIME FUNCTIONS USING ... - OpenSeismo

Bulletin of the Seismological Society of America, Vol. 79, No. 3, pp. 711-730, June 1989

DERIVING SOURCE-TIME FUNCTIONS USING PRINCIPAL COMPONENT ANALYSIS BY D. W. VASCO ABSTRACT

Factors such as source complexity, microseismic noise, and lateral heterogeneity all introduce nonuniqueness into the source-time function. The technique of principal component analysis is used to factor the moment tensor into a set of orthogonal source-time functions. This is accomplished through the singular value decomposition of the time-varying moment tensor. The adequacy of assuming a single source-time function may then be examined through the singular values of the decomposition. The F test can also be used to assess the significance of the various principal component basis functions. The set of significant basis functions can be used to test models of the source-time functions, including multiple sources. Application of this technique to the Harzer nuclear explosion indicated that a single source-time function was found to adequately explain the moment tensor. It consists of a single pulse appearing on the diagonal elements of the moment-rate tensor. The decomposition of the moment tensor for a deep teleseism in the Bonin Islands revealed three basis functions associated with relatively large singular values. The F test indicated that only two of the principal components were significant. The principal component associated with the largest singular value consists of a large pulse followed 16-sec later by a diminished pulse. The second principal component, a long-period oscillation, appears to be a manifestation of the poor resolution of the moment-rate tensor at low frequencies. INTRODUCTION

A key factor in the study of earthquake sources is the time variation of the rupture process. In estimating scalar moments, this variation is often assumed to have some simple form such as a trapezoid or triangle. However, it is not necessary for an earthquake to have a simple displacement history. Multiple sources with different mechanisms are sometimes proposed to explain anomalous focal mechanisms. Many debates about moment-tensor solutions or focal mechanisms of seismic events have centered on the source-time function and its complexity. For example, complex sources may possibly be confused with non-double-couple earthquakes (Sipkin 1986). A recent example of this can be found in the debate over the 1980 sequence of earthquakes (ML > 6) in Long Valley caldera (EkstrSm, 1983; Julian and Sipkin, 1985; Wallace, 1985). Wallace and Ekstr6m attributed the complex waveforms to nearly simultaneous dip-slip and strike-slip on closely spaced faults. Julian and Sipkin (1985), on the other hand, explain the data with a non-doublecouple mechanism. This paper presents a method to empirically derive the source-time function of an event if such a function exists. If no single function exists, the best set of successive least-squares approximations are presented. This set of orthogonal functions may be used as a basis with which to construct the set of moment-tensor components. The number of such basis functions determines the degree of nonuniqueness possible in the source-time function. If more than one basis function is needed, then any linear combination of these functions may serve as a basis set for the moment tensor. 711

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D.W. VASCO

The method begins with an inversion of the seismic waveforms for the timevarying moment tensor. The importance of the time-varying formulation is that it has the degree of freedom necessary to describe closely spaced multiple sources and motion on curved faults. This inversion can be done by any of a number of methods (McGowan, 1976; Stump and Johnson, 1977; Sipkin, 1982). In what follows, I consider the moment-rate tensor, the time derivative of the moment tensor, rather than the moment tensor. I use the frequency domain method of Stump and Johnson (1977), which expands the source in a Taylor series about the source point ~, resulting in a system of convolutions in time, 1

u~(x, t) = Y~ ~. Gki,jl,..j~(x, t; ~, t') X Mijl...j~(]i, t' ), where x represents a temporal convolution, uk is the kth component of displacement observed at time t, Gki(x, t; r, t') are the Green's functions containing propagation effects for a source at r recorded by a station at x, and M~j~...jo(ti, t) are the time derivatives of the moments of the equivalent body forces (Gilbert and Dziewonski, 1975). The summation convention is assumed for repeated indices. If the source dimensions are small compared to the wavelengths of interest, then it is necessary to keep only the term for n -- 1 in the previous expansion and this results in uk(x, t) = Gki,j(x, t; 0, 0) x Mij(O, t)

(1)

for ~ = O. By Fourier transforming this convolution, a matrix equation is derived, u~(x, f) = G,:,.j(x, f; O, 0)Mij(0, f)

(2)

for each frequency. This sytem of equations may be solved for the moment-rate tensor by generalized inversion (Stump and Johnson, 1977) or solved for bounds on moment-rate tensor properties by extremal inversion (Vasco and Johnson, 1989). After a time-varying moment-rate tensor has been derived, one is left with the task of interpreting the results. Can the data be explained by a single source-time function or is it complicated by a time-varying mechanism? What are the minimum number of mechanisms needed to explain the data? The method of principal components is suited to address such concerns. PRINCIPAL COMPONENT ANALYSIS

The method of principal component or factor analysis is directed toward determining the main factors contributing to a set of observations ( J6reskog et al., 1976; Menke, 1984; Savage, 1988). In the case of the moment-rate tensor, it is assumed that each component of the time-varying tensor is a weighted sum of canonical source-time functions plus an error term Mrr= ArrBT + E "r.

(3)

In the above equation, M is the n x 6 moment-tensor matrix, for n time points. The moment-rate tensor has been transformed into the time domain. It is factored into B, an n x 6 matrix of source-time basis functions, and A, a 6 x 6 matrix of basis function weights. The matrix E contains the errors due to microseismic noise, lateral heterogeneity and nonlinearity in the source, which map into the moment-

DERIVING SOURCE-TIME FUNCTIONS

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tensor solution. For the time being I will neglect the error term. Equation (3) states that the moment-tensor components, the rows of M T, can be written as a linear combination of a set of basis functions, the rows of B "r. The decomposition of equation (3) is nonunique; there are n x 6 knowns and (n + 6) x 6 unknowns, but a simple and useful one is in terms of the singular value decomposition of M v', M T = UAV T.

(4)

The matrix M T is represented as the product of U, a 6 x 6 column-orthogonal matrix, A, a 6 × 6 diagonal matrix of singular values, and the transpose of V, a n × 6 row-orthogonal matrix. Making the identification UA = A T vr=

Brr

results in the factorization in equation (3). This strategy offers a number of advantages over other decompositions. First, singular values are very useful in identifying the dimension of the basis set needed to explain the observations. The singular values act as weighting factors for the columns of B, the source-time basis functions. If one singular value is very large in relation to the other five, only one basis function is contributing significantly to the moment-rate-tensor components. Second, the set of basis vectors defined by the columns of B may be ordered in terms of their ability to explain the tensor. In particular, the column of B, bl, associated with the largest singular value, Xl, maximizes the sum of the projections onto the set of moment-rate-tensor components, 6

2 my.b1 j=l (Menke, 1984). The vector mj is the j t h column of the moment-rate-tensor matrix M. The column vector associated with the next largest singular value, b2, is orthogonal to this and acts to maximize the projections of the residuals left after the effect of bl is removed (J6reskog et al., 1976; Savage, 1988). Thus, the set of column vectors of B form an orthogonal set that are ordered by the ability to reduce the residuals of equation (3). Lastly, the singular value decomposition is very stable and efficient numerical algorithms to accomplish it are widely available (Lawson and Hanson, 1974; Press et al., 1986). Although one can arbitrarily recombine the basis functions by a linear transformation, the properties of the singular value decomposition make it desirable for initially factoring the time-varying moment tensor and assessing the basis set. Before proceeding with the singular value decomposition, the effect of errors on the estimates of the principal components needs to be considered. There are a number of sources of error in estimating the moment-rate tensor, and all of them map into the error term, E, in the decomposition of equation (4). First, there is the presence of microseismic noise due to environmental vibrations at the receiver site. Errors introduced by such noise can be estimated directly by measuring pre- or postevent ambient noise. There is also signal-generated noise caused by scattering from lateral heterogeneities at various scale lengths. Such scattering can occur anywhere along the ray path and can be accounted for by improvements in modeling

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D.W. ¥ASCO

the propagation. Without detailed three-dimensional velocity and attenuation models, it is difficult to estimate signal-generated errors. Finally, there are errors associated with the estimation of M, the moment-rate tensor. For example, in the frequency-by-frequency construction of the moment-rate tensor presented in equation (2), there may be problems in determining the moment-rate-tensor components at certain frequencies. The condition number of the moment-tensor inversion, the ratio of the largest to smallest eigenvalue, can be used as a measure of the numerical stability of the inversion. Condition numbers up to 27 were measured for the Harzer nuclear explosion, discussed below, at the high (greater than 5 Hz) and low (less than 1 Hz) frequencies. These moment-rate errors at particular frequencies are transformed to errors at all points in the time domain. Errors in the assumed depth of an event also result in deviations of the moment tensor. Finally, there are the errors that may occur from the assumption that the moment-rate tensor may be represented as a factorization into source and time factors. The separation into time and source factors works best for a source with a time invariant geometry. That is, the source-time function may vary over time but the proportion of the moment-tensor components activated does not change. If other moment components are activated with time and begin to interfere, the basis functions will represent the total moment-rate tensor. Therefore, the individual principal components may not reflect the individual physical mechanisms. This is a consequence of nonuniqueness introduced by multiple interfering sources. In the extreme case, a continuously changing mechanism, all the singular values will be significant. This is because all the basis functions are needed to describe such a moment-rate tensor. As will be seen in an example below, if the mechanisms are separated in time, the results can be quite close to the underlying physical situation. Given the possible sources of error presented above, it would be useful to assess their effect on the principal component decomposition. The decomposition of equations (3) and (4) is not linear and not amenable to conventional linear analysis. The most direct method for obtaining confidence limits on decomposition parameters, such as the singular values, is Monte Carlo simulation (Press e t al., 1986; Savage, 1988). The Monte Carlo method requires knowledge of the probability distribution of the errors on the data. As mentioned earlier, errors caused by signalgenerated noise are not well determined. One way to estimate the data errors is to compute the difference between the observed data and the data predicted by the least-squares estimate of the moment-rate tensor. If errors contributing to the waveform data are regarded as normally distributed, the significance of each principal component can be estimated through the F-test (Menke, 1984; Press et al., 1986). This is a statistical test of the similarity of variances that can be used to test the significance of the principal component expansion. That is, if there are two models with different numbers of parameters, does one model fit the data significantly better? The number of degrees of freedom for the p t h order principal component representation is given by ~p = m - p ( n + 7). The number of knowns is given by m and the number of unknowns corresponds to the number of parameters for the singular value decomposition when only p principal components are considered. The F ratio used is the ratio of the chi square (x 2) of the principal component expansion using only the first principal component to the x 2 of the p t h order expansion, 2

F = X,,_2 ..~"

DERIVING SOURCE-TIME FUNCTIONS

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The distribution of F is well known for normally distributed variables (Menke, 1984). Using F, one can test the proposition that the ~(2 for a single component model is smaller than the x ~ for a p component model. If we can reject this hypothesis with great confidence, then we can be fairly certain that the extra principal components reduce the variance. This will be illustrated in the following examples and applications. DATA AND MODEL ERRORS

The worldwide distribution of broadband, digital, three-component seismometers, such as Geoscope (Romanowicz et al., 1984) and the planned IRIS network (IRIS, 1984), will provide an unprecedented data set for the examination of seismic sources. In addition, models of lateral heterogeneity of seismic velocities (Dziewonski, 1984; Woodhouse and Dziewonski, 1984) and attenuation (Durek et al., 1988) have been presented recently. These developments should make it possible to study teleseismic events in greater detail than ever before. Deep (>300 kin) teleseismic events are especially interesting because their physical mechanism has not been demonstrated with certainty. When analyzing such events, it is important to estimate the complexity of the source-time function because of possible tradeoffs between the sourcetime function and the source geometry. The methods presented above can be used to quantitatively estimate the complexity of the source-time function. This will be illustrated by the decomposition of the moment-rate tensor for a Bonin Islands earthquake, of magnitude 5.6 and at a depth of 467.7 km, that occurred on 4 October 1985. The specifics of the event and the results of the inversion for the moment tensor are given in Vasco and Johnson (1989). Before this event is analyzed, a set of moment-rate tensor factorizations for a number of synthetic data sets will be considered. As mentioned previously, it is difficult to assess the significance of principal component analysis without estimates of the errors in the moment-tensor components. The question of the significance of these results can be addressed qualitatively through synthetic modeling. I considered a situation identical to the above Bonin Islands event but with a source mechanism shown in Figure 1, a compensated linear vector dipole (CLVD: MlI = -2.0 x 101~dyne-cm/sec, M22 = 4.0 x 10 ~6dyne-cm/sec, M:~3 = -2.0 x 101~ dyne-cm/sec). The source-time function is a step with a thirdorder polynomial rise, a rise time of 10 sec, and beginning at 8 sec. Note that the moment is of the form M(t) = Mo • f (t), a constant geometry and a varying sourcetime function. When I applied principal component analysis to the moment tensor, only one singular value was nonzero. The principal component associated with the largest singular value, PC 1, recovered the polynomial source-time function used to generate the data. For the most part, the other basis functions were oscillatory functions with the frequency of oscillation increasing with the component number. The exception was the third component, PC3, a low-frequency oscillation that is probably a symptom of the poor resolution at the lower frequencies. The averaging functions for the basis set assigned the correct weights to each principal component recovering the CLVD of Figure 1 when only the first principal component is used. Thus, for a perfect Earth, the results are quite good. If Gaussian microseismic noise is added to the seismograms, with an amplitude of 40 per cent of the maximum value of the displacement, the structure of the singular values changes (Fig. 2a). Now six singular values can be seen, though the maximum value is over five times larger than the next one. The basis functions associated with the small singular values consist primarily of higher-frequency

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TIME (S) FIO. 1. Time-varying moment-rate tensor to be used for the synthetic modeling that follows. The mechanism is a compensated linear vector dipole (CLVD: MI~ = -2.0 x 10~c~dyne-em/sec, M~ = 4.0 x 10 j~ dyne-cm/see, M:~:~= -2.0 x 10~ dyne-cm/sec). oscillations due to the added noise. T h e d o m i n a n t principal c o m p o n e n t is a noisy version of the source-time function used to generate the m o m e n t tensor. In order to evaluate the significance of the c o m p o n e n t s associated with the second and third singular values, an F - t e s t was applied (Fig. 2b). T h e hypothesis we are examining is t h a t the residuals, when modeling the m o m e n t tensor with one principal component, are significantly lower t h a n if more principal c o m p o n e n t s are used. For significance values n e a r 1.0, we can decisively reject the hypothesis t h a t the × ~ for the single principal c o m p o n e n t case is less t h a n the x 2 when using more t h a n one principal component. Hence, for similar values, the extra p a r a m e t e r s in the higher principal c o m p o n e n t s do not significantly reduce the residuals of the fits to the data. T h e result in Figure 2b indicates t h a t the second a n d higher principal c o m p o n e n t s do not significantly lower the variance considering the extra p a r a m e t e r s they introduce. Therefore, a single source-time function, P C 1, adequately explains the data. N o t e t h a t the F test returns a probability of 0.5 when the F ratio is 1.0, w h e n the residuals are identical. T h i s is the case for the first point in Figure 2b, where the ratio is the x ~ of one c o m p o n e n t model to itself. T h i s is not an indication t h a t the one c o m p o n e n t model is inadequate. In addition, lateral variations were modeled using the spherical h a r m o n i c representations of W o o d h o u s e and Dziewonski (1984) and Dziewonski (1984). Using the m o m e n t - r a t e model in Figure I as a source, w a v e f o r m s were c o m p u t e d for each source-station pair by assuming an average velocity structure identical to the radial average of the models between the source a n d receiver. Thus, each source-station pair has a different velocity structure. T h e W K B J m e t h o d ( C h a p m a n , 1978) was

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Examining the other moment-rate-tensor components, PC3 does not seem to contribute to any of them except M21. Its presence in M21 may be because of the overlap between this source-time function and PC1. The F t e s t in Figure 8b indicates that this component does not significantly alter the fit to the data. Because of the low frequencies used and the high signal-to-noise ratio, I believe that lateral heterogeneity and microseismic noise are not chiefly responsible for the distribution of singular values. Much of this nonuniqueness is likely due to poor conditioning at the lower frequencies evidenced in PC2. In addition, source complexity may also be partly responsible for the high dimension of the source-time basis set but our confidence of this, as determined by the F test, is not high. THE HARZER NUCLEAR EXPLOSION

Nuclear explosions are another seismic source that is not completely understood. Seismic waveforms produced by nuclear explosions are much more complicated than expected for an isotropic explosion. Much of this complication can be explained by propagation in inhomogeneous media, but there may also be source effects. Release of regional tectonic stress, surficial spall, and nonlinear source processes have all been invoked to account for nonisotropic energy. In this section, principal component analysis will be used to examine the source-time behavior of a nuclear explosion. The Harzer explosion, detonated 6 June 1981, was recorded by an array of broadband accelerometers (Johnson, 1988). The near-field waveforms from distances of 1 to 11 km have been carefully analyzed and a stable moment-tensor

HARZER

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