~-2MT(o~)M(o~), MTgcc.asu.edu/advseismo/readings/1989_Bulletin of the... · Bulletin of the...

Bulletin of the Seismological Society of America, Vol. 79, No. 1, pp. 85-100, February 1989

DISPLAY AND ASSESSMENT OF SEISMIC MOMENT TENSORS

BY MARK A. RIEDESEL AND THOMAS H. JORDAN

ABSTRACT

Any moment-rate tensor can be expressed as M(o~) = ~-2MT(o~)M(o~), where MT is the total scalar moment and IVI is the source mechanism, a symmetric tensor

of unit Euclidean norm describing the geometry of the source. We present a

method for the graphical display of the second-order tensor I~1 and its uncertain-

ties, the latter specified by a fourth-order covariance tensor, var[IVI]. The display

allows the statistical significance of non-double-couple components to be tested

by inspection. It is also useful in evaluating the frequency dependence of the

source and assessing differences among moment-tensor estimates. We provide

a detailed description of the plotting method and some examples of its application

to source-mechanism spectra recovered from IDA data, including an earthquake

in the Kermadec Islands that has a large non-double-couple component.

INTRODUCTION

The Fourier transform of the moment-rate tensor M(w) for an event located at

ro is related to the spectrum of the far-field seismogram, u (r, w), by a linear equation

(Gilbert, 1971; Aki and Richards, 1980, p. 53):

u(r, ~) = G(r, ro, ~):M(w). (1)

The components of the tensor G are the transfer functions appropriate to the

particular displacement component, computable from a model of the earth and

knowledge of the instrument response. Methods for retrieving source structure

based on equation (1) are now standard for studying earthquakes and explosions

(e.g., Gilbert and Dziewonski, 1975; Buland and Gilbert, 1976; Mendiguren, 1977;

Strelitz, 1978; Aki and Patton, 1978; Masters and Gilbert, 1979; Fitch et al., 1980;

Ward, 1980; Dziewonski et al., 1981; Kanamori and Given, 1981; Langston, 1981;

Romanowicz, 1982; Sipkin, 1982; Nhb~lek, 1984).

Most of the seismic energy from earthquakes can be modeled as waves radiated

by a propagating shear dislocation, which at long wavelengths can be represented

by a localized double couple (Burridge and Knopoff, 1964). The double-couple model

has been widely applied, but it is becoming clear that deviations from this idealized

mechanism occur at even low frequencies (EkstrSm and Dziewonski, 1985; Kana-

mori et al., 1986; Sipkin, 1986a). An obvious advantage of the moment-tensor

formulation is that it incorporates terms describing non-double-couple behavior.

Formally speaking, the time-dependent seismic moment tensor is defined as the

spatial integral of the stress-glut field F(r, t) (Backus and Mulcahy, 1976). M thus

contains information related to the net compression or dilatation within a source

volume and the net variation in the orientation of a propagating dislocation. The

extraction of M from global digital networks should permit seismologists to place

routinely such integral constraints on the details of the rupture process. However,

for long-period waves in the far field, source complexities often (though not always)

manifest themselves by small, second-order deviations from a pure double couple,

and care must be taken in extracting any non-double-couple signal from the various

noise processes associated with real seismic data.

Inversion algorithms that recover the entire moment-rate tensor can also provide

formal measures of its estimation uncertainties, so that it should be straightforward

85

86 MARK A. RIEDESEL AND THOMAS H. JORDAN

to evaluate the statistical significance of any non-double-couple components, at

least in principle. Unfortunately, the manipulation of M, a second-order tensor,

and its estimation variance, var[M], a fourth-order tensor, can be rather cumber-

some. In practice, moment-tensor catalogs almost never include the complete

description of errors contained in var[M] but give only the marginal uncertainties

in the estimated parameters (e.g., Sipkin, 1986b; Dziewonski et al., 1987).

This paper discusses a graphical method for the display of seismic moment

tensors specifically constructed to test the significance of non-double-couple com-

ponents and to facilitate the comparison of different moment-tensor solutions. We

have employed this method in our previous work on low-frequency moment tensors

(Riedesel and Jordan, 1982; Riedesel et al., 1986), and our experience suggests that

its utility is sufficient to warrant broader use in source studies. We therefore provide

a detailed description of the method and some examples of its application.

THE SOURCE-MECHANISM TENSOR

The six independent components of M(~0) for an arbitrary seismic source can

vary independently with frequency; at a particular frequency, they are generally

complex numbers with differing phases. Here we restrict our attention to the special

case of "synchronous sources," whose components have the same Fourier phase

(Silver and Jordan, 1982). We can always adjust this phase to be zero, so that M(w )

is real. In the time domain, the six independent component functions of a synchron-

ous source have the same temporal centroid, as defined by Backus (1977, equation

5.5e), and our phase convention takes this centroid to be the temporal origin.

M(~) contains information on both the magnitude and radiation pattern of a

seismic event. These two aspects of the source are conveniently separated by writing

M(~0) as the product of the total scalar moment and the source-mechanism tensor

(Silver and Jordan, 1982):

M(w) = ~f2MT(w)l~I(w). (2)

The total moment is defined to be one-half the squared Euclidean length of M,

MT(~) = [!M(w):M(w)] 1/2, (3)

which reduces to the ordinary seismic moment M0 in the special case of a double

couple, l~I is a symmetric tensor of unit Euclidean norm, whose eigenvector expan-

sion is written

3

l~I = Y~ hi ei 6i (4) i = l

where

3

Y~ ~,i 2 = 1. i=1

The roll-off of MT as a function of frequency can be used to determine the

characteristic source duration ¢c (Silver and Jordan, 1983), whereas variation of

with frequency gives information on the temporal changes in the radiation pattern.

DISPLAY AND ASSESSMENT OF SEISMIC MOMENT TENSORS 87

However, the methods optimized for the recovery of the source-mechanism spectra

~I(w) are not, in general, optimal for the estimation of the total-moment spectra

MT(W). We have therefore developed a "two-pass" algorithm for estimating the

moment-rate tensor at low frequencies, the first pass providing l~I and the second

MT. Our aims in formulating this algorithm were twofold: (1) to enable the

estimation l~I(w) and MT(w) over a wide frequency band with good frequency

resolution, thereby constructing the time-dependence of the source; and (2) to

provide good estimates of the uncertainties in M(~0) for use in evaluating departures

from the double-couple model. Silver and Jordan (1982, 1983), Riedesel (1985), and

Riedesel et al. (1986) discuss the details of the computational procedures.

In analyzing the uncertainties in M, it is convenient to make use of an isomorph-

ism between symmetric, three-dimensional, second-order tensors and six-dimen-

sional vectors. Following Silver and Jordan (1982), we define the Cartesian com-

ponents of a vector m in terms of those of M by

m = (Mrr, Moo, M~,, ,J-2Mro, "q/-2Mr4,, ~/-2Mo¢). (5)

This isomorphism, written m ¢=> M, preserves Euclidean norm; i.e., m • m = M:M.

Moreover, it allows us to represent the fourth-order eovarianee tensor var[M] by a

second-order covariance matrix Vm --- var[m]. We suppose m is a sample from a

Gaussian distribution with a mean mo ¢=~ Mo and a covariance matrix Vm ¢=~ var[M] 1

whose eigenvalues are small compared to M y 2 = 7 mo • too. Then, to first order in

these small quantities, rh ¢~ l~I is a sample from a Gaussian distribution with mean

liao ¢* l~Io and a eovariance matrix (Silver and Jordan, 1982),

V~a = (2MT)-2(I -- moriao) • Vm • (I - lhorho). (6)

In our analysis, we assume an estimate of V~a is available. An algorithm for obtaining

V~a from data in narrow, low-frequency bands is in Riedesel (1985).

GRAPHICAL DISPLAY OF THE SOURCE MECHANISM

Source mechanisms more general than a double couple have been displayed by

plotting either the far-field P-wave radiation patterns or by partitioning the mech-

anism into elementary source mechanisms. In the first method, the P-wave radiation

pattern of the entire moment-rate tensor is plotted on a projection of the focal

sphere. Knopoff and Randall (1970) expand the radiation pattern into a sum of

zeroth-plus second-order spherical harmonics, but these six numbers do not display

the size of any deviatoric, non-double-couple components in the source mechanism

in an obvious way. In the second technique, 1~I is decomposed into an isotropic

component plus either a major plus a minor double couple (Gilbert, 1980) or a

double couple plus a compensated linear-vector dipole (Knopoff and Randall, 1970),

and the component mechanisms plotted separately. Using either of these techniques,

it is difficult to assess visually the size of any non-double-couple components, at

present there is no method of visually displaying the uncertainties in ]~I with either

of these plotting methods. Therefore, the significance of the non-double-couple

parts of M cannot be evaluated using these plots. Given the difficulties in construct-

ing a quantitative display of a general second-order mechanism tensor and its

associated fourth-order covariance tensor using these standard representations, we

adopt a different approach.

8 8 M A R K A. R I E D E S E L AND T H O M A S H. J O R D A N

We order the eigenvalues of l~I such that hi ==- k2 --- k3. With this convention, the

eigenvalue pattern for a pure double couple is (1, 0, -1)/~/2, and the eigenvectors

{6i: i = 1, 2, 3} correspond to the usual definitions of the tension (T), neutral (N),

and compression (P) axes, respectively. In this paper, the principal axes (61, 62, 63)

will be plotted using the symbols T, N, P), even when M is not a pure double couple.

To examine the non-double-couple properties of a moment-rate tensor we use

equation (4) to construct on the surface of the focal sphere unit vectors whose

components in the principal axis coordinate system are the eigenvalues of different

types of source mechanisms. The vector which describes a general source mechanism

is:

3

= 2 ~i6~. (7) i=1

A double-couple source mechanism has the vector representation

a = (61 - 63)145 . (8)

Two possible compensated linear-vector dipole vectors (CLVD) (Knopoff and

Randall, 1970) are

1A 3' = ±6 ~e3)/~/~ (9) i ( 6 1 - 2 2 -

and

3 i ' 1~ -~6 - e3)/4~-~ ~- , (10) = (7el + 2 2

and the vector corresponding to a dilatational, purely isotropic source is

= (61 + 62 + 63)/V~. (11)

The vector ] is orthogonal to the plane containing d, i, and ] ' , so that a . ] = ] .

= i , ] = o. The vector ~ characterizes the source mechanism which, in general,

can be expressed as a linear combination of a pure double couple, a CLVD,.and a

purely isotropic component (Knopoff and Randall, 1970). The principal axes specify

the orientation of the mechanism with respect to the geographic coordinate system;

a knowledge of ~, and the 6~ 's is sufficient to reconstruct ~vI. The vector ¢i corresponds

to a pure double-couple mechanism, and if M is a pure double couple, ~ will be

colinear with d, i.e., ~ • d = _1 (Fig. la). Similarly, 1 and ] ' correspond to pure

compensated linear-vector dipoles (Fig.Alb) , and ] represents a purely isotropic

source. The great circle which connects d, 1, and i ' on the unit sphere defines the

subspace on which ~ must lie if I~ is a purely deviatoric (i.e., tr M = 0) source (Fig.

lc).

After ordering the principal axes, we arrange them to form a right-handed

coordinate system. For plotting purposes, any eigenvector which lies in the upper

half of the focal sphere is reflected through the origin on to the bottom half of the

focal sphere and is plotted with an upward pointing triangle. Eigenvectors lying on

the lower half of the focal sphere are plotted withdownward pointing triangles. The

same operations are performed on the vectors h, d, l, l ' , and ]. The deviatoric great

DISPLAY AND ASSESSMENT OF SEISMIC MOMENT TENSORS 8 9

i)c

CLVD

FIG. 1. The source mechanism plot and initial body-wave radiation patterns for four different l~I's plotted using an equal area projection. On each plot, the three principal axes ~1, e2, and ~a, corresponding to eigenvalues )~1 ~ X2 -- Xs, are plotted as T, N, and P on the lower focal hemisphere. The mechanism ks @araeteri~ed by the unit vector ~, = • X,e ~, which can be compared with the canonical unit vectors d, I, 1', and i, representing a pure double couple, two different pure eompensated linear-vector dipoles, and a pure dilatation, respectively. The dashed line shows the locus of all deviatoric mechanisms. Vectors are plotted as • if they are on the lower focal hemisphere and as a • if they are projected from the upper hemisphere. For a pure double couple the axes T, N, and P are the conventionally defined tension, neutral, and eompression axes, respectively. The contours of the P-wave first motion are shown in the circles on the right. The region containing the solid contour lines is the eompressional quadrant, the dashed contour lines are in the dilatational quadrant, and the thick eontours are the nodal lines. (a) ~ pure double-couple M with the eigenvalue pattern (1, 0, -~1), for which the source meehanism veetor X lies direetly on top of the double-couple vector d. (b) An M which is a pure CLVD with the eigenvalue pattern (1, -0.5, -045). For this type of mechanism the souree meehanism vector X lies"direetly on top of the CLVD veetor 1. (c) A purely deviatorie but non-double-eouple source mechanism M is shown. The s"ouree mechanism vector X lies on the deviatoric great circle between the veetors I and d, indicating that M has both a^non-zero double-couple component and a non-zero CLVD component. (d) A general source meehanism M. This example is a linear epmbination of a double-couple, a CLVD, and an isotropic eomponent. The source mechanism vector X lies off the deviatoric great circle, showing the presence of the isotropie portion of the source. The shaded portion shows that area in which any possible source mechanism vector X must lie given that the eigenveetors el, e2, and e3 are ordered to correspond to the eigenvalues },1 _-> h2 --> ~3.

circle is p lo t t ed as a do t t ed line. T h e r e su l t i ng lower hemisphe re is pro jec ted onto a

p l ane u s ing a n equal area p ro jec t ion for p lo t t ing , j u s t as is done for P -wave f i rs t

m o t i o n plots .

Some examples of the p l o t t i n g m e t h o d are shown in F igure 1. I n F igure l a we

p lo t a pure double-couple M, for which the source m e c h a n i s m vector ~, lies d i rec t ly

on top of the double-couple vec tor d. F igure l b is a p lo t for a pure CLVD source

m e c h a n i s m wi th p r inc ipa l va lues (1, - 0 . 5 , -0.5)/~ff~, for which ~ is iden t ica l to the

CLVD vector 1. A type of CLVD co r r e spond ing to the p r inc ipa l va lues

(0.5, 0.5, - 1 )L~ /~ is a t an equal d i s t ance a long the devia tor ic great circle on the

o the r side of d a n d is p lo t t ed as 1 ' . A n ]~I which is pure ly devia tor ic b u t n o n - d o u b l e -

couple is shown in F igure lc . ~ for th i s m e c h a n i s m lies on the devia tor ic great circle,

9 0 MARK A. RIEDESEL AND THOMAS H. JORDAN

but has both a pure double-couple and a CLVD component and so plots between (1

and 1. It is also possible for a purely deviatoric mechanism to lie between a and l ' ,

with all deviatoric ~'s somewhere on the deviatoric great circle between 1 and ] ' ,

inclusive. Figure ld is a plot of a general source mechanism which, in this example,

is a linear combination of equal parts double-couple, compensated linear vector-

dipole, and purely isotropic sources. Here ~ is not colinear with any of d, l, ] ' , or

]. The fact that it plots off of the deviatoric great circle demonstrates the existence

of the isotropic component; the fact that ), does not lie on the great circle which

connects a and ] is due to the presence of a deviatoric, non-double-couple compo-

nent. Using the above conventions for plotting, ~ may lie only on a restricted portion

of the surface of the focal sphere. The region in which any mechanism will fall is

defined by adding increasing portions of an isotropic component to a deviatoric

mechanism. This means that, for any ~I, ~ will lie in the orange-slice shaped sectors

defined by the great circles that pass through 1 and ] and through l ' and ~. This

region is shaded in Figure ld.

ANALYSIS OF UNCERTAINTIES

Our error analysis is based on a simple application of first-order perturbation

theory. The notation is simplified by the use of the isomorphism between m and

M. For example, the perturbation in an eigenvalue due to a perturbation ~ l~I to an

initial source mechanism estimate l~Io results in an g/[ which has the vector

representation

rh = Zho + ~m, (12)

where ~ Iia • rho = 0. To first order, the perturbation to the eigenvalues of l~Io are

~ki = ~ m • fiii. (13)

where fiii ¢=> lqii - e~i. The eigenvalues ~,~ are stationary with respect to first order

perturbations in the eigenvectors ~. This implies that, to first order, the error

ellipse constructed for k will contain only information on the uncertainty in the

principal moments and not on the uncertainty in the orientation of the principal

axes. The relationship between ~ and the vectors d, i, i ' , or i is determined only by

the eigenvalues of l~I, since the same 6~'s are used to construct them all. Therefore,

the error ellipse for h will be a true representation of the uncertainty in, and the

significance of, any non-double-couple components of l~I.

To display the uncertainty in rh on the plot we need the covariance matrix for

),, V~,. The covariance matrix for the vector (~1, h2, )~3) has components

[V~]~ s = ( ~ X ~ : ) = l]i~. V~ • fiss. (14)

To obtain V~, we transform V~ from the principal axis system to the geographic

coordinate system. The confidence ellipse for ~ is computed from the eigenvector

expansion of V~, and projected onto the surface of the focal sphere for display.

To construct confidence ellipses for the ~ we calculate the first order perturbation,

~ , to an eigenvector ~i (Mathews and Walker, 1970, Chap. 10, pp. 286-288),

5e , - - X e~ (15)

DISPLAY A N D A S S E S S M E N T OF SEISMIC M O M E N T T E N S O R S 91

The covariance matrix for en is

V a = (~en~en) = E Y~ • V~ • ~ t i m , (16)

Confidence ellipses for each of the en are constructed from the eigenvector expansion

of its corresponding Va. The matrices Va are only the marginal uncertainties of the

principal axes, rather than the complete covariance, which involves terms ( ~ei~ej ).

Thus we can calculate the trade-off among the uncertainties in different eigenvec-

tors, but these uncertainties cannot be easily displayed on the plot. For solutions

with nearly degenerate principal axes, the uncertainty in the direction of these axes

will, in general, be even greater than is indicated by their error ellipses on the plot.

The 95 per cent confidence ellipse around ~ allows various hypotheses about the

source mechanism to be tested by inspection. For example, the hypothesis that l~I

is a pure double couple can be rejected at the 95 per cent confidence level if (1 lies

outside of ~'s confidence ellipse. The presence of a significant isotropic component

can be tested by observing whether or not any portion of the deviatoric great circle

lies within ),'s confidence ellipse. If it does, then the hypothesis that the mechanism

is purely deviatoric cannot be rejected, whereas if the ellipse includes no part of the

deviatoric great circle, the mechanism has a significant isotropic part at the 95 per

cent confidence level. The use of the plotting method with errors is illustrated in

the following applications to global seismic data.

EXAMPLES

Since 1977 the IDA network of long-period accelerometers has provided data

from a sufficient number of well-distributed stations to allow the routine estimation

of moment-rate tensors for most earthquakes with seismic moments larger than

about 10 ~9 N-m (Agnew et al., 1976). Riedesel and Jordan (1985) and Riedesel et al.

(1986) have developed a method to recover the seismic moment-rate tensor from

low-frequency digital seismic data. In this technique, m is estimated using a system

of equations isomorphic to (1),

g • m = u ( 1 7 )

where g <=> G. The data functionals u are integrals of the complex acceleration

spectra over narrow frequency bands (0.1 mHz in this study) centered on the

fundamental spheroidal modes. The frequency-domain averaging reduces the sen-

sitivity of the estimates to attenuation and splitting (Gilbert, 1973; Jordan, 1978).

The algorithm involves a nonlinear phase-equalization procedure to compensate for

centroid time shifts, the effects of unmodeled earth structure, and station timing

errors. In this study, estimates m ( ~ ) are obtained over the frequency interval 1 to

11 mHz by inverting the integrals in discrete 1-mHz bands centered on the

frequencies COn = (n + !) mHz; n = 1, 2, . . . 10. This interval contains 101

fundamental modes, about 10/mHz, so that for earthquakes with 10 IDA stations

available, approximately 100 complex numbers are used to derive the moment-rate

tensor in each 1-mHz band. Riedesel (1985) demonstrated that this procedure yields

reliable narrow-band estimates of the source-mechanism spectrum to frequencies

as high as 11 mHz and that the errors derived for the estimates adequately model

the uncertainties induced by noise in the data, including signal-generated noise.


It is common when inverting seismic data to find the moment-rate tensor to

modify the system of equations so as to restrict the class of possible solutions.

Constraints on the solutions to these equations can then be applied in an ad hoc

manner or through the use of projection operators. The most common constraint,

which is used in the great majority of inversion schemes, is to constrain the trace

of the moment-rate tensor solution to be zero, thus eliminating any possible isotropic

part from the estimate. A simple linear constraint to make the trace of M zero can

be applied when solving equation (17), or projection operators can be employed by

solving the modified system of equations,

PI" g" m = P r . u. (18)

The projection operator PI annihilates those parts of g and u which contain all of

the information on any possible isotropic part of M, thus constraining the solution

to be purely deviatoric. This method has the advantage over the linear constraint

because it allows a self-consistent error analysis to be done on an estimate obtained

using equation (18). In either case, the source mechanism vector ), can then lie only

upon the deviatoric great circle somewhere between 1 and 1', in the region which

includes d. Figures la, lb, and lc are all examples of source mechanism plots for

purely deviatoric ]~I's.

Another restriction is sometimes employed when inverting data from a shallow

earthquake using long-period records. In this case the Mro and Mr, elements of the

moment-rate tensor cannot be well determined from the data. Some investigators

(Kanamori and Given, 1981) simply set these two elements of M equal to zero, and

fit for the remaining elements. However, in order to perform an error analysis on

an estimate of M, it is also necessary to project out the information contained in

Mro and Mr, from the least squares moment-rate tensor inversion equations:

Ps • g " m = P s • u. (19)

The projection operator Ps annihilates any possible information on Mro and Mr,

contained in the transfer functions g or in the data u. Using this projection, the

Mr0 and Mr, components of the resulting estimate of M will be zero, but the resulting

covariance matrix will reflect the constraints used in obtaining the solution. Any

such constrained M can be written as a sum of an isotropic part plus two double

couples, each of which can be either a vertical strike-slip or a 45 ° dip-slip mechanism.

M then has one principal axis colinear with the r (vertical) axis, with the other two

axes lying on the equatorial plane of the focal sphere. A more complete description

of the operators PI and Ps is found in Riedesel (1985).

The moment-rate tensor inversion method has been applied to 60 earthquakes

recorded on the IDA network between 1977 and 1985 (Riedesel, 1985; Riedesel and

Jordan, 1985). It was found that most are consistent with a double-couple model of

the source, with none having a significant isotropic component. Some events,

however, had a significant deviatoric, non-double-couple component, as determined

using the error analysis and plotting method presented above. Here we give four

representative examples from this data set, which illustrate the application of the

plotting technique and error analysis to earthquakes of different depths and source-

mechanism types.

DISPLAY AND A S S E S S M E N T OF SEISMIC M O M E N T TENS ORS 93

Michoacan

The Michoacan earthquake of 19 September 1985 was a large, shallow event (h

= 22 km; MT = 1.2 X 1021 N-m) that occurred in the Middle America trench, and

the event of 21 September 1985 was the largest aftershock (MT = 2.6 X 102o N-m).

The results from a multiband inversion of IDA data for these events have been

previously presented using this plotting method (Riedesel et al., 1986). The source-

mechanism spectrum l~I (w) was estimated by the moment-tensor inversion method

of Riedesel and Jordan (1985) (hereafter referred to as the Riedesel-Jordan method).

Twelve stations were employed for each earthquake, with 11 common to both. The

data were edited to eliminate nonlinearities in the initial body waves, the R1 surface-

wave packets, and, in some cases, R2 surface waves. All stations recorded R3 and

A B A B 1 to 2 1 to 2 6 to 7 6 to 7

2 to 3 2 i~o 3 7 to 8 7 to 8

3 to 4 3 to 4 8 to 9 8 to 9

4 to 5 4 to 5 9 to 10 9 to 10

5 ~o 6 5 to 6 10 to 11 lO to 1i

@0 @@ 1985 ;862

FIG. 2. Source-mechanism spectra 1VI(~) for the Michoacan earthquake of 19 September 1985 over the frequency interval of 1 to 11 mHz. Each circle is a focal-sphere plot of the normalized moment tensor M ( w ) in a 1-mHz band derived from 12 IDA records by the normal-mode method of Riedesel and Jordan (1985). Cglumn A shows unprojected mechanisms, whereas columns B show projected mechanisms for which tr M, Mre, and Mr¢ have been annihilated by the orthogonalization procedure described in Riedesel (1985). The solutions are consistent with the hypothesis that the mechanism is a pure double couple, at the 95 per cent confidence level. This is shown by the fact that, in colunn A, the 95 p e r c e n t confidence ellipse for the source-mechanism vector k (thick line) contains the double-couple vector d in all 10 bands. The 95 per cent confidence ellipses on the princil~al axes T, N, and P are done in thin lines. Similarly, the 95 per cent confidence limit error bars for k in column B also always contain the vector k. No frequency dependence is seen in the mechanism.


later wave groups without any apparent nonlinear distortion. Five hours of record

following the first good time point were used in the analysis. Transfer functions

relating the moment-rate tensor to acceleration were generated from model 1066A

(Gilbert and Dziewonski, 1975) assuming the centroid latitudes, longitudes, and

depths reported by Harvard (EkstrSm and Dziewonski, 1986): 17.97°N, 102.07°W,

22 km for the main shock and 17.61°N, 101.48°W, 22 km for the aftershock.

The results of the inversions for the main event are reproduced in Figure 2. There

is no frequency dependence of the source mechanism, and all of the solutions are

consistent with the hypothesis that the source is a double couple. The second set of

solutions (column B) demonstrates the effect of constraining the trace of M to be

zero, as well as projecting out all information on the Mre and Mr¢ components of

the moment-rate tensor (Riedesel, 1985). The 10-band average mechanism for the

19 September 1985 Michoacan event is displayed in Figure 3a. For comparison, the

Harvard CMT solution for this event (EkstrSm and Dziewonski, 1986) is shown in

Figure 3b. The 10-band average mechanism plot for the 19 September 1985 with

both the trace constrained to be zero and with Mro and Mr¢ projected out is shown

in Figure 3c. The effect of projection on the source mechanism is evident in the

restriction of the principal axes and in the elimination of the isotropic component.

The 10-band average mechanism for the 21 September 1985 aftershock is plotted

in Figure 3d. The graphical method clearly shows the similarity between the

mechanisms for the two events.

Harvard

Aftershock

Fro. 3. (a) The average of the 10 bands in Figure 2, column A for the Michoacan event of 19 September 1985. (b) The Harvard CMT solution for the Michoacan event obtained from GDSN data by EkstrSm and Dzienowski (1986). (c) The average mechanism for the Michoacan event with the eonstraints M~0 = Mr, = 0 and tr[M] -- 0 obtained by averaging the 10 solutions in Figure 2, column B. (d) The average mechanism for the aftershock of 21 September 1985, which has a source mechanism very similar to that of the main shock of 19 September 1985.


Honshu

The deep focus (439 km) Honshu earthquake of 7 March 1978 occurred 250 km

west of the Bonin Islands at 31.90°N latitude, 137.44°E longitude (CMT location

from Giardini, 1984). The Riedesel-Jordan method was used to find moment-rate

tensor solutions for this event using data from nine stations of the IDA network in

ten 1 mHz wide bands from I to 11 mHz. The results are plotted in Figure 4, which

includes a contour plot of the P-wave radiation pattern. The solutions are consistent

with the hypothesis that the event is a pure double couple in all but one band, since

is contained within the error ellipse of k for 9 of the 10 solutions. The solutions

are consistent with the hyi)othesis that the mechanism is purely deviatoric in every

band. The estimates of M indicate no significant frequency dependence of the

source. These results fail to confirm the results of Silver and Jordan (1982), who

found a significant isotropic part at the 95 per cent confidence level in the bands 7

to 8, 8 to 9, 9 to 10, and 10 to 11 mHz. This does not necessarily mean that the

results of Silver and Jordan are wrong, since their technique could have tighter

error bounds on the isotropic part. The average source mechanism for the event

shown in Figure 4 is obtained by weighing each individual solution by its covariance

matrix. The Harvard CMT solution (Giardini, 1984) is also plotted but does not

have errors included, since the full covariance matrices for the CMT solutions are

not routinely published. The plots show that there is excellent agreement between

our solutions and the CMT result, particularly for the 10-band average and the

i to 2 5 to 6 9 to i 0

2 to 3 6 to 7 I0 to I i

3 to 4 ? to 8 Average

4 to 5 8 to 9 Harvard

Honshu

1978 66

FIG. 4. Source mechanism spectrum i~I(~) for the 7 March 1978 Honshu earthquake derived from nine IDA records using the Riedesel-Jordan method. The solutions are consistent with the hypothesis that the mechanism is a pure double couple, at the 95 per cent confidence level. This is shown by the fact that the 95 per cent confidence ellipse for the source mechanism vector h (thick line) contains the double-couple vector d in all but the 4- to 5-mHz band. The contours of the P-wave first motion are shown for each band.


bands 3 to 4, 4 to 5, and 5 to 6 mHz, where our solutions best fit the data as

indicated by the small confidence ellipses.

Kermadec Islands

A moderately large (Mo = 3.3 ! 1019 n-m) intermediate depth (224-km) earthquake

that had an unusual moment-tensor spectrum occurred on 26 January 1983 at

30.52°S latitude, 179.21°W longitude in the Kermadec trench. The source mecha-

nism spectrum computed using the Riedesel-Jordan method from 14 IDA records

and is shown in Figure 5. This event is not consistent with a double-couple solution

in 10 out of the 10 bands, and is consistent with a pure compensated linear-vector

dipole in 8 of the 10 bands.

For a pure CLVD el is completely determined, but e2 and e3 are degenerate, and

these axes can be chosen to be any two mutually perpendicular axes which are in

the plane orthogonal to el. For an ~I derived from real data the axes will probably

never be exactly degenerate. However, if estimating ~I at several different frequen-

cies, the ordering of the principal axes for the individual l~I's by the magnitude of

their eigenvalues can result in solutions which are very close to one another but

have quite different looking source mechanism plots.

For example, if 1~I1 has the eigenvalue pattern (1.2, -0.5, -0.7)/(1.48) with

corresponding eigenvectors (ea, eb, ec), then el = e,, e2 = eb, and e3 = e~, and the

1 to 2 5 to 6 9 to 10

2 to 3 6 to 7 10 to 11

3 to 4 7 to 8 Average

4 to 5 8 to 9 H a r v a r d

K e r m a d e e Is lands

1983 26

FIG. 5. The intermediate depth (224-km) Kermadec Islands event of 26 January 1983 shows significantly non-double-couple M's in all ten 1-mHz bands. The Riedesel-Jordan method was used to estimate the solutions from 14 stations of the IDA network. The non-double-couple nature of the source is indicate.d by the fact that none of the 95 percen t confidence level error ellipses on the source mechanism vectors k contain the double-couple vector d. The event is consistent with the hypothesis that the event is a pure CLVD in 8 of the 10 bands, since the error ellipses on the source mechanism vectors X contain the CLVD vector 1' in 8 of the 10 bands. The 10-band average and the Harvard CMT solution also have large non-double-couple components. This result is probably due to complex faulting but could conceiv- ably be caused by an unusual earthquake source process, such as a rapid phase transition in the

subducting slab.


source mechanism and body-wave first motion plots are those in Figure 6a. For

comparison, 1~I2 has eigenvalues (1.2, -0.7, -0.5)/(1.48) with the same corresponding

eigenvectors (6a, 6b, 6c). The principal axes are then reordered so that 62 = 6c, and

63 = 6b. This, in turn, changes the vectors X, d, 1, and 1', since they depend on the

values of 62 and 63 (equations 7 to 11). As a result, the mechanism plot c h a n t s

significantly (Figure 6b), even though the P-wave radiation patterns for M1 and M2

are virtually identical. The rather sudden shift in the appearance of the plot with a

relatively small change in g/I is unavoidable, since some convention for the ordering

of the eigenvectors must be chosen. However, this apparent instability in the plot

occurs only for mechanisms very close to a CLVD. For the Kermadec event, this

phenomenon can be seen in the solutions in the 4 to 5 mHz and 5 to 6 mHz bands

in Figure 5. These two bands have T and N axes which are very nearly the opposite

of each other because of the reordering of the principal axes due to small changes

in the values of the principal moments. For estimates of M obtained from real data

which are close to a CLVD, the two nearly degenerate axes have large error ellipses

which clearly show the large uncertainties in the orientation of the degenerate axes

within the plane perpendicular to the remaining principal axis. This can be seen in

all 10 bands of the Kermadec event plotted in Figure 5.

Riedesel (1985) found that several intermediate depth earthquakes such as the

Kermadec event showed large significant non-double-couple components. This is

most likely due to complex faulting. In some instances, however, it might be

attributable to an intrinsically non-double-couple source process, such as a sudden

phase transition in a subducting slab. The Harvard CMT solution for this event

agrees well with our solutions and is also highly non-double-couple (Dziewonski et

al., 1983). The CMT solutions are computed using data which include the 50- to 70-

second body-waves, and the consistency of their solution with ours indicates that

the non-double-couple nature of this event may extend over the frequency band

from I to 20 mHz. Sipkin (1986a) has shown that for shallow earthquakes, apparent

non-double-couple source mechanisms can be caused by either source complexity

or by an inherently non-double-couple process, such as magma injection. An

examination of the Kermadec event at higher frequencies might help discriminate

among the possible explanations for the observations, if the processes producing

the high-frequency elastic radiation can be assumed to be the same as that producing

the low-frequency seismic waves.

F1G. 6. (a) A plot of the meel:~anism 1~I1 with eigenvalues ̂ (X~, Xb, Xc) = (1.2, -0.5, -0.7)/(1.48), a deviatoric mechanism close to 1. (b) A source mechanism M2 with eigenvalues (),~, Xb, he) = (1.2, --0.7~ --0.5)/(1.4:8). The redefinition of which eigenvector is,6a and whi@ is 63 for M2 compared to those for M1 results in changes in the positions of the vectors X, d, 1, 1', and i, since they are defined in terms of the eigenvectors 61, 62, and e3 acording to equations 7 to 11.


Harvard

Chagos Archipelago

FIO. 7. This plot of l~I estimated by the Riedesel-Jordan method for the 7 to 8 mHz band of the 30 November 1983 Chagos Islands earthquake illustrates the utility of the confidence ellipses in evaluating source-mechanism solutions. The solution has a very large apparent isotropie part, but it is consistent with a pure double-~couple mechanism, since the 95 per cent confidence level error ellipses on the source mechanism vector X contains the double-couple vector el.

This event demonstrates the utility of the plotting method and error analysis in

the evaluation of unusual events. The plot gives a clear display of the non-double-

couple nature of the solutions and provides an immediate statistical evaluation of

the significance of the non-double-couple components as a function of frequency.

Also, the instability in the nearly degenerate principal axes for solutions close to a

CLVD is clearly demonstrated by the error ellipses.

Chagos Archipelago

On 30 November 1983 a shallow earthquake with a moment of 4.1 x 1020 n-m

occurred at latitude 6.35°S, longitude 71.75°E in the Indian Ocean, at a nominal

depth of 10 km (Dziewonski et al., 1984). The earthquake has a normal fault

mechanism which may be associated with a "diffuse" plate boundary between an

Australian plate and an Indo-Arabian plate (Wiens et al., 1985). The l~I computed

for this event from 11 IDA records using the Riedesel-Jordan method in the band

7 to 8 mHz has the Mr0 and Mr~ components projected out and is shown in Figure 7. This solution has a very large apparent isotropic part, as indicated by the fact

that the source mechanism vector X lies far off the deviatoric circle. Also, the P-

wave radiation pattern for this mechanism is positive everywhere on the focal

sphere. However, the fact that the double-couple vector d lies within the error

ellipse of X indicates that the solution is consistent with the hypothesis that, at the

95 per cent confidence level, this event is a pure double couple in the 7 to 8 mHz

frequency band. The source mechanism plot clearly shows the large isotropic

component in the solution but also provides vital information on the uncertainty

in the determined l~I.

CONCLUSIONS

The method presented provides a visual display of all of the information on the

geometry of a point source moment-rate tensor estimate, M(~), along with the

uncertainties in the estimate, var[M (~) ]. The plot facilitates the rapid identification

of non-double-couple components and provides the information needed to quickly

evaluate their significance, allows the assessment of the frequency dependence of a

source mechanism, M(~0), and simplifies the comparison of different events with

each other. The plotting method and error analysis are valuable tools in the


evaluation of moment-rate tensor estimates which provide information not available

on other methods of visually displaying seismic sources. Applying this method to

estimates of moment-rate tensor inversions for 60 events using the method of

Riedesel and Jordan (1985), we found that most events were well described by a

pure double couple, with no events having a significant isotropic component.

However, a small number of events did have large, significant deviatoric non-

double-couple components.

ACKNOWLEDGMENTS

We would like to thank Duncan Agnew and the IDA group at the University of California, San Diego

for supplying us with the IDA data. Ken Creager supplied helpful contour plotting software. Cliff Frohlich

provided useful comments on the manuscript. This work was supported by NSF grants EAR-8519095

(UTA) and EAR-8518394 (MIT).

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INSTITUTE FOR GEOPHYSICS DEPARTMENT OF EARTH, ATMOSPHERIC AND PLANETARY

UNIVERSITY OF TEXAS SCIENCES

AUSTIN, TEXAS 78759 MASSACHUSETTS INSTITUTE OF TECHNOLOGY

(M.A.R.) CAMBRIDGE, MASSACHUSETTS 02139

(T.H.J.)

Manuscript received 7 May 1988

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