Source Finiteness and Rupture Propagation Using Higher ... · Bulletin of the Seismological Society of America, Vol. 96, No. 4A, pp. 1241–1256, August 2006, doi: 10.1785/0120050164

1241

Bulletin of the Seismological Society of America, Vol. 96, No. 4A, pp. 1241–1256, August 2006, doi: 10.1785/0120050164

Source Finiteness and Rupture Propagation Using Higher-Degree

Moment Tensors

by Juan Li, Yu-Mei He, and Zhen-Xing Yao

Abstract Higher-degree moment tensor representation of seismic sources is ob-tained for a horizontally layered homogeneous medium. A Taylor series expansionof Green’s function around a reference source position and time is made, and thisenables us to approximate the seismic radiation in both the regional and teleseismicdistances through a sequence of terms representing increasingly detailed aspects ofthe source behavior. Source coefficients and orientation factors of the first- andsecond-degree moment tensors are obtained. The representation is applied to a uni-lateral rupture Haskell fault model, and the synthetic seismograms of different modelscalculated by the higher-degree moment tensors are compared with the theoreticalsolutions for a propagating source. Our results show that, the representation of higher-degree moment tensors up to degree 2 can describe the response of a moving sourcewell enough, and it’s possible to use the moment series as a tool for calculatingseismograms from finite and propagating faults in the forward sense. The computa-tion takes much less time than the method of summing point sources over the faultsurface. The information yielded by the higher-degree moments may solve problemssuch as the fault-plane ambiguity and the space–time evolution of the rupture prop-agation of an earthquake.

Introduction

The moment tensor representation has been applied ex-tensively to study the properties of seismic sources, and thezero-degree moment tensor representation for a point sourceis especially widely used, both for the purpose of modelingthe seismic wave field and for obtaining source parameters.It’s possible to approximate the effect of a moment tensordensity by a limited number of moment tensors of degree 0when the source region is relatively small compared with thewavelengths. The information, for example, moment cen-troid location and time provided by the inversion of fiveindependent moment tensor components, has become widelyavailable and the basic procedures have been standardized(e.g. Dziewonski et al., 1995). The zero-degree moment ten-sor, however, represents averages over the fault in whichonly the long-wavelength part of the seismograms can beused, and it does not contain the spatial information of thesource, and kinematic source parameters, like the geometryand rupture velocity of the fault.

By far one of the most difficult parts in computing thesynthetic seismogram is solving the propagation effectsthrough realistic velocity structures. Only for a few grosslysimplified models (e.g., whole space or half-space) are an-alytical solutions possible. Backus (1977a,b) attempted togive the source representation using a glut moment tensor

density by a limited number of moment tensors of degree 0and higher. Doornbos (1982) considered the cases in thelong-wave approximation and gave the formulation of seis-mic response represented in terms of 20 source parametersfor a homogeneous half-space model, and estimated that forsources with Ms � 6, the relative contribution of second-degree moments may be of the order of 10% or more, evenin long-period seismograms. Stump and Johnson (1982)studied the importance of higher-degree moments in mod-eling finite propagating faults in a half-space structure.McGuire et al. (2001) tried to get the space–time distributionfrom the information provided by the second-degree mo-ments for a teleseismic event. However, the propagation ef-fects are more prominent in the regional or near-field dis-tances, and no detailed investigation of the contribution ofhigher-degree moments have been made for a more realisticlayered earth model.

In this article, the higher-degree moment tensor repre-sentation of a propagating source for a horizontally layeredhomogeneous medium was given. The source coefficientsand orientation factors of the first- and second-degree mo-ment tensors for the calculation of Green’s functions wereobtained. We applied the representation to a simple unilat-eral rupture Haskell fault model. The synthetic seismograms

1242 J. Li, Y.-M. He, and Z.-X. Yao

of higher-degree moment tensors propagating in a homo-geneous or a horizontally layered medium were comparedwith the seismograms generated by using a summation ofpoint sources over the fault plane, and the effects of differentmodel parameters, for example, the characteristic length offault plane and the average rupture velocity, were investi-gated.

Representation of Higher-Degree Moment Tensors

Given the equivalent force system f(x, t), the responseof the Earth u(x, t) can be expressed as an integral over thesource region V (e.g., Aki and Richards, 1980):

i iu (x, t) � G (x, n; t, s) f (n, s)dV, (1)*j j�where (x, n; t, s) is the Green’s function, representing theiGj

ith component of displacement at position x and time t forsingle force excitations in the jth direction at point n andtime s, and the asterisk indicates temporal convolution.

To adapt to our present purposes, defining a momenttensor density distribution m(n, s), which is nonzero onlywithin the source volume V, and the displacement vectoru(x, t) can be given by:

i iu (x, t) � G (x, n; t, s) m dV, (2)*j,k jk�where means , denoting differentiation with re-i iG �G /�nj,k j k

spect to the kth Cartesian spatial coordinate nk, and dupli-cated indices indicate summation.

Expanding the Green’s function directly around aiGj,k

suitably chosen reference source position n0:

i i 0G (x, n, t, s) � G (x, n , t, s)j,k j,ki 0 0� G (x, n , t, s)(n � n )j,kl l l (3)i 0 0 0� G (x, n , t, s)(n � n )(n � n )j,klm l l m m

� …,

the later terms in the preceding equation involve higher-degree moment tensors, which contain information about thespatial distribution of failure in an earthquake and have greatpotential value for studying source finiteness and rupturepropagation (Stump and Johnson, 1982; Julian, 1998). Thedisplacement can now be written as:

i i 0u (x, t) � G (x, n , t, s) M (n, s)*j,k j,ki 0� G (x, n , t, s) M (n, s) �…, (4)j,kl jk,l*

where Mjk,l . . . are spatial moment tensors of the stress glutof order s (i.e., having s spatial indices k, l, . . .).

M (s) � m (n, s)dVjk jk� (5)

0 0M (n , s) � (n � n )m (n, s)dV.jk,l l l jk�By expanding about a suitably chosen referenceiGj

source point and replacing the convolutions by summationswith the aid of temporal moments, Doornbos (1982) andDahm and Kruger (1999) obtained an equation with 90source parameters (moment tensors up to degree 2) for theasymptotic wave functions in the far field. With the “smooth-ing” assumption that the six components of the moment ten-sor density have a common space and time history, the num-ber of source parameters will be greatly reduced to 20. Herethe same basic procedure is adapted and the smoothingsource time is assumed through this article. However, notethat we do not apply the asymptotic wave functions in thefar field as used in Doornbos (1982) to evaluate the spatialderivatives in equation (4). Instead, we use the exact wavefunction, which can be used in the calculation of a finitepropagating source for both teleseismic and local distances.In the time domain, the complete representation of higher-degree moment tensor can then be obtained as the following:

1i i i 2 iˆ ˙ ¨u (x, t) � M G � D(s)G � D(s )G*jk j,k j,k j,k� 21 i i˙� D(n )G � D(sn )G (6)l j,kl l j,kl2

11 i� D(n n )G � Ol m j,klm � 2��2 R

with

M � m dVjk jk�0M � (n � n ) m dVjkl l l jk�

0 0M � (n � n )(n � n )m dVjklm l l m m jk�ˆ ˙M � M dsjk jk�

0ˆ ˙M D(s) � (s � s ) M ds (7)jk jk�ˆ ˙M D(n ) � M dsjk l jkl�

0ˆ ˙M D(sn ) � (s � s ) M dsjk l jkl�ˆ ˙M D(n n ) � M dsjk l m jklm�

2 0 2ˆ ˙M D(s ) � (s � s ) M ds .jk jk�The preceding complete expansion is made around an ap-

Source Finiteness and Rupture Propagation Using Higher-Degree Moment Tensors 1243

Figure 1. Coordinate system for the dislocationformulation. z is positive downward.

propriate source position n0 and a reference source time s0

both in spatial and temporal spaces, and it enables us toapproximate the seismic radiation u through a sequence ofterms that represent increasingly detailed aspects of thesource behavior (Kennett, 2001).

As shown in the following sections, this transform ofthe representation of displacement enables us to calculatethe Green’s function, its first and second differentiation withrespect to the Catesian spatial coordinate nk, and the differentintegration terms in equation (7), which are related to M fora simple propagating model, for example, uni- or bilateralpropagating Haskell model analytically.

For a simplified point source, only the first term is re-tained, and the displacement for an arbitrary fault with strike�, dip d, and rake angle k or equivalently moment tensorsolution Mij can be calculated using algorithms like the re-flectivity method and the generalized rays theory. Here ageneralized reflection-transmission coefficient matrix withintegration of wavenumber k is applied to calculate theGreen’s function, and Bouchon’s discreet wavenumber sum-mation scheme is adapted to calculate the wavenumber in-tegration in the frequency domain (Yao and Harkrider, 1983;He et al., 2003).

Calculation of Green’s Functions

We give a brief description of how to derive sourcecoefficients and orientation factors for a given zero-degreemoment tensor following the methods used in wave-motionproblems (Yao and Harkrider, 1983). The same procedurewill be applied to derive the source coefficients of the dif-ferentiation of Green’s functions with respect to the spatialcoordinate nk, which will be used in the summation ofhigher-degree moment tensor displacement calculation.

When a single concentrated force in direction n withtime function f(t) acts on the source point, in the frequencydomain the solution of elastodynamic equation has the fol-lowing expression (e.g., Ben-Menahem and Singh, 1981):

�ik RbF(x) e2u � d kij ij b2 �4pqx R2 �ik R �ik Rb �� e � e

� , (8)� ��x �x Ri j

where the subscript i denotes the displacement componentwith respect to some convenient reference system, j is thedirection of the single body force, q is density, F(x) is theFourier transform of the force f(t), dij is the Kronecker func-tion, k� � x/� and kb � x/b; and R is the distance betweenthe source and receiver.

Physically, a seismic source can be expressed by theP-, SV-, and SH-wave motion radiated from the source.

Adapting the concepts of displacement potentials for a rec-tangular point-shear dislocation in an infinite elastic me-dium, the seismic function, or the displacement potentials ofthe source can be written in terms of the wave-number k,and circular frequency x for an arbitrarily oriented dislo-cation within the coordinate system shown in Figure 1:

� �M D(x) �a|z�z |0 sP e J (kr)kdku � A m ms � m� 2� �4pq �x m�0 0

� �M D(x) �b|z�z |0 sSV e J (kr)kdkw � A m m (9)s � m� 2� �4pq �x m�0 0

� �� M D(x) �b|z�z |0 sSH e J (kr)kdk,v � B m ms � m� 2� �4pq �x m�0 0

where, us, ws, and vs are the P-, SV-, and SH-displacementpotentials in frequency domain. Pm, SVm, and SHm are calledsource coefficients, Am and Bm are orientation factors, whichare functions of the fault geometry, for example, k, d, h fora simple shear dislocation model; zs is the z coordinate ofsource; , ; Jm(kr) is the2 2 1/2 2 2 1/2a � (k � k ) b � (k � k )� b

ordinary Bessel function of order m; M0 is the seismic mo-ment and D(x) is the Fourier transform of the source timehistory. For a shear dislocation point source, the ordersm � 0, 1, 2 and only 5 items of orientation factors withcorresponding source coefficients are retained, which will beshown in the following section.

The introduction of the basic solution of elastodynamicequation (equation 8) and the displacement potentials (equa-tion 9) enables us to follow the same deduction procedureto obtain the source coefficients and orientation factors forthe calculation of higher-degree moment tensors andiGj,kl

by completing the partial derivatives with respect toiGj,klm

coordinate nl and nm. As shown in the later sections, no for-mal discrepancy of the displacement field exists for differentorders of moment tensors. The radiation patterns, however,


Figure 2. The layered half-space consists of n lay-ers over a half-space at the bottom. The source de-noted as a star is located at a depth of zs; and� �z zj j

denote points just below and above the interface zj.

become much more complex as more lobes are introducedas the degree of the moment tensor increases.

Calculation of for a Layered MediumiGj,k

Considering a model with n � 1 horizontal, homoge-neous, and isotropic layers overlaying a semi-infinite me-dium (Fig. 2), the vertical, radial, and tangential displace-ment fields on the free surface for a buried point source canbe expressed as (He et al., 2003):

21 d ˙w (t) � D(t) A W (t)*z � m m� �4pq dt m�0

21 d ˙q (t) � D(t) A Q (t) (10)*r � m m� �4pq dt m�0� 21 d ˙v (t) � D(t) B V (t)*h � m m� �4pq dt m�1

with

(1) (1) (2) (2)A W � A W � A W0 0 0 0 0 0(11)

(1) (1) (2) (2)�A Q � A Q � A Q0 0 0 0 0 0

1(1)A � � (M � M )0 11 2221(2)A � � M0 332

A � M cos A � M sin A1 13 z 23 z(12)1

A � (M � M ) cos 2A � M sin 2A2 11 22 z 12 z2�B � �M sin A � M cos A1 13 z 23 z

1B � � (M � M ) sin 2A � M cos 2A ,2 11 22 z 12 z2

where Mij is the six independent components of momenttensor in the geographic coordinate; D(t) is the time historyand it should be emphasized here that the smoothing as-sumption—the moment tensor has the same time history—is applied. Am is the orientation factor indicating the radiationpattern of the seismic energy. Wm(t), Qm(t), and Vm(t) arestep responses that correspond to the vertical, radial, andtangential displacements of three fundamental shear dislo-cations: 45 dip-slip fault (m � 0, 45� DS), dip-slip fault (m� 1, DS) and strike-slip fault (m � 2, SS), respectively. Inthe frequency domain, they have the following expressions:

�w J (kr)kdkW (x) � m mm �0

m�Q (x) � [q J� (kr) � v J (kr)]kdk (13)m m m m m�

0 krm��

V (x) � [q J (kr) � v J� (kr)]kdk ,m m m m m�0 kr

where

dJmJ� (x) � (14)m dx

qm 1 RS �1 RS˜� Rev(z ) (I � R R) T1 D U4x� �wm (15)

� �P Pm mSL FS �1 SL(I � R R ) R �D U D � �� SV SVm m

1 RS �1 RS˜v � Rev(z ) (I � R R) Tm 1 D,SH U,SH4x (16)SL FS �1 SL � �(I � R R ) (R SH �SH ).D,SH U,SH D,SH m m

Pm, SVm, and SHm are source coefficients for P, SV, and SHwaves, and the plus and minus superscripts denote the down-or up-going waves, respectively. I is a unit matrix, and thesubscript SH refers to the SH problem. Rev(z1) is the receiverfunction on the free surface, and all the other items are thegeneralized reflection or transmission matrix coefficient, forexamples and are the reflection coefficient matrix orRS RSR TD U

transmission matrix for the down-going or up-going wavesbetween and . The source coefficients for a zero-degree� �z zR S

moment tensor representation have been used widely, whichhave the following forms:


(1) 2 (1) (1)m � 0 P � �k /a SV � �ek SH � 00 0 0(2) (2) (2)P � �2a SV � �2ek SH � 00 0 0

2 2 2m � 1 P � �2ek SV � (k �2k )/b SH � ek /k1 1 b 1 b

2 2m � 2 P � �k /a SV � �ek SH � k /b2 2 2 b

where e � 1 for the plus superscript and �1 for the minussuperscript.

Calculation of and for a Layered Mediumi iG Gj,kl j,klm

Following the basic deduction procedure described inthe preceding section, we can obtain the source coefficientsand orientation factors for the higher-degree moment tensors

and . It can be expected that with the increasingi iG Gj,kl j,klm

of the degree of moment tensors, more coefficients will beintroduced into the equation. In the frequency domain,

and , the displacement potentials contributed by1 1 1u , w vs s s

the first-degree moment tensor can finally be described as:

3 �M D (x)01 �a z�z⎢ ⎢su � A P e J (kr)kdks � m m m� 2� �4pq �x 0m�0

3 �M D (x)01 �b z�z⎢ ⎢sw � A SV e J (kr)kdks � m m m� 2� �4pq �x 0m�0

3 �� M D (x)01 �b z�z⎢ ⎢sv � B SH e J (kr)kdks � m m m� 2� �4pq �x 0m�0

(17)

where,

(1) (1) (2) (2) (3) (3)A P � A P � A P � A P0 0 0 0 0 0 0 0

(1) (1) (2) (2) (3) (3)A SV � A SV � A SV � A SV (18)0 0 0 0 0 0 0 0�(1) (1) (2) (2) (3) (3)B SH � B SH � B SH � B SH0 0 0 0 0 0 0 0

(1) (1) (2) (2) (3) (3)A P � A P � A P � A P1 1 1 1 1 1 1 1

(1) (1) (2) (2) (3) (3)A SV � A SV � A SV � A SV (19)1 1 1 1 1 1 1 1�(1) (1) (2) (2) (4) (4)B SH � B SH � B SH � B SH1 1 1 1 1 1 1 1

(1) (1) (2) (2)A P � A P � A P2 2 2 2 2 2

(1) (1) (2) (2)A SV � A SV � A SV (20)2 2 2 2 2 2�(1) (1) (2) (2)B SH � B SH � B SH2 2 2 2 2 2

the superscript 1 on the left term of equation (17) denotesspecially the contribution by the first-degree moment tensor,and the source coefficients and orientation factors of the first-degree moment tensors are as follows:iGj,kl

(1) 2 (1) 2 2 (1) 2m � 0 P � ek SV � k(2k � k )/2b SH � ek /20 0 b 0 b

(2) 2 (2) (2)P � ek /2 SV � kb/2 SH � 00 0 0

(3) 2 (3) (3)P � �ea SV � �kb SH � 00 0 0

1 1(1) 3 (1) 2 (1) 2m � 1 P � k /a SV � ek /4 SH � � k k/b1 1 1 b4 4

(2) (2) 2 2 (2) 2P � �2ka SV � e(2k � k ) SH � bk /k1 1 b 1 b

(3) (3) 2 (3)P � �ka SV � �ek SH � 01 1 1

1(4) (4) (4) 2P � 0 SV � 0 SH � � k k/b1 1 1 b4

1(1) 2 (1) (1) 2m � 2 P � � ek SV � �kb/2 SH � ek /22 2 2 b2

(2) 2 (2) 2 2 (2) 2P � �ek SV � �k(2k �k )/2b SH � ek /22 2 b 2 b

1 13 2 2m � 3 P � � k /a SV � �ek /4 SH � k k/b3 3 3 b4 4


(1)A � M D(n ) � M D(n )0 13 1 23 2

(2)A � (M � M )D(n )0 11 22 3

(3)A � M D(n )0 33 3

(1)A � [(3M � M ) D(n ) � 2M D(n )] cos A � [(M � 3M ) D(n ) � 2M D(n )] sin A1 11 22 1 12 2 z 11 22 2 12 1 z

(2)A � (M cos A � M sin A ) D(n )1 13 z 23 z 3

(3)A � M [D(n ) cos A � D(n ) sin A ]1 33 1 z 2 z

(4)A � �2(M � M ) [D(n ) cos A � D(n ) sin A ]1 11 22 1 z 2 z

(1)A � [(M � M ) cos 2A � 2M sin 2A ] D(n )2 11 22 z 12 z 3

(2)A � [M D(n ) � M D(n )] cos 2A � [M D(n ) � M D(n )] sin 2A2 13 1 23 2 z 13 2 23 1 z

(21)A � [(M � M )D(n ) � 2M D(n )] cos 3A � [(M � M )D(n ) � 2M D(n )] sin 3A3 11 22 1 12 2 z 11 22 2 12 1 z

(1)B � M D(n ) � M D(n )0 13 2 23 1

(1)B � �[(3M � M )D(n ) � 2M D(n )] sin A � [(M � 3M ) D(n ) � 2M D(n )] cos A1 11 22 1 12 2 z 11 22 2 12 1 z

(2)B � (�M sin A � M cos A )D(n )1 13 z 23 z 3

(3)B � M [�D(n ) sin A � D(n ) cos A ]1 33 1 z 2 z�(4)B � 2(M � M ) [D(n ) sin A � D(n ) cos A ]1 11 22 1 z 2 z

(1)B � [�(M � M ) sin 2A � 2M cos 2A ]D(n )2 11 22 2 12 z 3

(2)B � �[M D(n ) � M D(n )] sin 2A � [M D(n ) � M D(n )] cos 2A2 13 1 23 2 z 13 2 23 1 z

B � �[(M � M )D(n ) � 2M D(n )] sin 3A � [(M � M ) D(n ) � 2M D(n )] cos 3A3 11 22 1 12 2 z 11 22 2 12 1 z

We can see that for the first-degree moment tensor, the orderm varies from 0 to 3, and the orientation factors appear morecomplex, which breaks the normal symmetry radiation pat-tern of P-SV waves generated by a point-source model, andthis property may be further applied to distinguish the faultplane from the auxiliary plane.

Appendix gives source coefficients and orientation fac-tors of the second-degree moment tensors , in whichiGj,klm

the order m reaches to 4, meaning that Bessel functions withorder 4 should be involved in the integration to get the sur-face displacement contributed by the higher-degree momenttensors.

Application to Haskell Fault Model

Assuming a unilateral rupture Haskell fault model (e.g.,Lay and Wallace, 1995) with fault length L, width W, rupturevelocity v (Fig. 3); the final slip DD is uniform, then therupture duration Trup � L/v. If the rupture plane is in the(x, y) plane, the stress glut rate is defined by:

m � lDDd(t � x/v) [H(x) � H(x � L)]jk (22)[H(y � W/2) � H(y � W/2)]d(z),

where H is the Heaviside step function, d is Dirac’s deltafunction, and l is the shear modulus; the stress glut rate is

nonzero for �W/2 � y � W/2, 0 � x � L, and 0 � t �L/v.

For a unilateral rupture Haskell model, we can get theintegration in equation (7) analytically:

2Lˆ ˆ ˆM � m LW, M D(s) � M � s (23)jk jk jk jk 0� �v

with

ˆm � lDDf , M � lDDLW, M � M f . (24)jk jk 0 jk 0 jk

A vector with rake angle k on a fault plane (Fig. 4) canbe expressed in the geophysical coordinate by its three com-ponents , where:(1) (1) (1)r � r • ( f , f , f )1 2 3

(1)f (k) � cos k cos A � sin k cos d sin A1 z z

(1)f (k) � cos k sin A � sin k cos d cos A (25)2 z z

(1)f (k) � �sin k sin d .3

If the rupture propagates along the x direction, and the anglebetween x� and rupture direction is kL, then we can use kL

to express vector r:


Radiation Pattern

The plot at the top of Figure 5 illustrates the radiationpatterns of various moment tensor terms for the radial com-ponents in the (x, z) plane, and the bottom one is for thetangential components. To give an example, only radiationpatterns of terms Mxz , MxzD(nx) and MxzD(nxnx) are shownin the figure. The response of zero-degree moment tensor isrepresented by a typical quadrupole radiation pattern for apoint-source shear fault. With the increasing of the degreeof moment tensors, directivity effects break the symmetrybetween the rupture and auxiliary plane. Complexity of theradiation patterns increases significantly and more lobes areintroduced into the radiation patterns. It is expected that therelatively small propagation velocity makes this effect moreobvious for S waves. It is possible that, if observational datafrom a variety of azimuths were available, the informationin the higher-degree moment radiation patterns could beused to stack the seismograms, and with the enhancing ofthe effects of the higher-degree moments, the fault finitenessand the real fault plane could be resolved.

Seismograms Generated by Higher-Degree MomentTensor Representation

First, seismograms from finite propagating sources inhomogeneous media will be generated using the precedingequations of higher-degree moment tensor representation.The seismograms are compared with the theoretical ones fora propagating source. The seismograms for a propagatingsource are calculated by the commonly used approach:superposing seismograms from several hundred denselyspaced point sources, which approximates a rupturingsources with finite size (Anderson, 1976; Hartzell et al.,1978). While summing the synthetic seismograms for nu-merous points on the fault surface, the spacing of the pointsources is selected to be close enough to avoid spatial alias-ing, and this process is especially computationally expen-sive.

To study properties of the seismograms generated byhigher-degree moment tensor, several different fault modelcases (Table 1) in the simi-infinite homogeneous mediumwith parameters vp � 6.2 km/sec, vs � 3.5 km/sec, q � 2.7g/cm3, k � 90�, kL � 90�, d � 90�, and Az � 90� areselected. The characteristic length of the fault plane, the av-erage rupture velocity vrup, the epicentral distance betweenthe source and receiver are allowed to vary in these cases.

The propagating source model is shown in Figure 3. Forsimplicity, we assume that the width W is much shorter thanthe length of the fault L, and thus the effect of the width canbe neglected. The width of the fault plane is chosen as0.1 km. A simple ramp source function with a finite rise timeT � 1 sec (Fig. 6) for each propagating point source on thefault plane is chosen.

The vertical, radial, and tangential components ofground displacement for case 1 at azimuths of 30�, 210�, and

Figure 3. Geometry of the unilateral rupture Has-kell fault model with fault length L, width W, andrupture velocity v.

Figure 4. Transform of the coordination system.k is the rake angle of a vector r in the (x�, y�) plane;the rupture propagates along the x direction; kL andk0 are angles between x and x�, x and the vector r,respectively.

(1) (1) (1)r • f (k) � x f (k ) � y f (k � 90�). (26)i i L i L

Using s0 � 2L/v, (kL)L/2 and T � L/v, we can0 (1)n � fl l

get the other integration terms in equation (7):

M D(s) � 0jk

2T2ˆ ˆM D(s ) � Mjk jk� �12L(1) 0ˆ ˆM D(n ) � M f (k ) � njk l jk l L l� �2 (27)L(1)ˆ ˆM D(sn ) � M f (k )jk l jk l L 12

2L(1) (1)ˆ ˆM D(n n ) � M f (k ) f (k )jk l m jk l L m L� 122W(1) (1)� f (k � 90�) f (k � 90�) .l L m L �12

The preceding coefficients are in accordance with thosegiven by Dahm and Kruger (1999).


Some basic characteristics of the higher-degree momentsummation can be deduced from Figure 7. In this case, thelength of the fault is 1 km, and the average rupture velocityon the fault plane is 3 km/sec, generating a rupture time Trup

of 0.33 sec, which is much smaller than the rise time 1 sec.We can see from the comparison at azimuths of 30� and 210�that, with the degree of the moment tensor increasing, thefrequency content of the contribution of the higher momenttensor also increases, and the total contribution including the

90� are given in Figure 7. Here no instrument or attenuationoperator have been convolved with the responses. The dis-placements are plotted to the same scale for those at azi-muths of 30� and 210� but to a different scale for radial andvertical responses at the azimuth of 90�. The lowest dottedline represents responses of the zero-degree moment ten-sor—the commonly used point-source model. The seismo-gram of the middle solid line is the total contribution of thezero-, first-, and second-degree moment tensors; and thethick solid line above is the theoretical seismograms calcu-lated by the summing of densely propagating point sourceson the fault.

Figure 5. Angular radiation patterns for the radial (panels labeled with R) andtangential (panels labeled with T) displacement components for excitations of variousdegree moment tensors: Mxz, Mxz D(nx) and Mxz D(nxnx). The fault and auxiliary planesare indicated by solid and dashed line, respectively.

Table 1Finite-Fault Rupture Models

Model Parameters Case 1 Case 2 Case 3 Case 4

R (epicentral distance) (km) 15 15 15 160H (depth) (km) 8 8 8 8L (km) 1.0 5.0 5.0 5.0W (km) 0.1 0.1 0.1 0.1vrup (rupture velocity) (km/sec) 3.0 3.0 1.5 1.5T (rise time) (sec) 1 1 1 1Trup (rupture time) (sec) 0.33 1.67 3.33 3.33

Figure 6. A ramp source time function for eachmoving point on the fault.


is still acceptable. This is especially true for the tangentialcomponent response. With the increasing of the degree ofmoment tensors, the frequency content of the seismogramsof the summed moments also increases. Comparing the seis-mograms at azimuths of 30� and 210�, we can see that theresponses of higher-degree moment tensors converge to thetheoretical seismograms of finite fault well, whereas the con-tribution of zero-degree moment tensor is quite differentfrom the theoretical one. At an azimuth of 30�, the fault isrupturing toward the observation point, and the frequencycontent of the propagating fault response is clearly higherthan that of a static point source model and the peak of thedisplacement amplitude moves forward clearly. It has an op-posite effect for the case at 210�. The propagation effectbecomes evident when the rupture duration is larger than therise time.

Synthetic seismograms for case 3 are shown in Figure 9,in which the rupture velocity is changed to 1.5 km/sec com-pared with 3.0 km/sec of case 2. With the decreasing ofrupture velocity, the rupture time increases a lot, and thepropagation effect is expected to be much more obvious, andthis can be clearly seen from the dotted line and thick linein Figures 8 and 9. The little fluctuation in the seismogramsis contribution of the higher-degree moment tensors.

Case 4 is almost the same as case 3, except that theobservation is made at a regional distance of 160 km. Thethree components of displacement for this model at azimuths

higher-degree moment tensor converges more to the theo-retical summation of the propagating models, although thepoint source approximation also works to some degree.

At an azimuth of 90�, the first half of the fault is rup-turing toward the observation point, whereas the second halfruptures away, which makes the constructive interferenceeffects smaller. However, the point-source approximationpredicts no P-SV motion at all. That means the zero-degreeterm is zero at this azimuth. After expanding the momentsaround the center of the fault, the contribution of the first-and second-degree terms become the major one. The solu-tions converge and match the summed seismograms of thepropagating source points very well. The observed maxi-mum amplitude of the vertical and radial components is,respectively 1.3% and 0.8% of that at an azimuth of 30�because of the limited effects of constructive interference.In these seismograms, the effects of both ends of the faultare observed, and the P and SV phases can be clearly iden-tified.

Figure 8 is the displacement responses for case 2, whichis almost the same as case 1, except that the fault length isextended to 5 km, leading to a rupture duration Trup � 1.67sec. The effect of this little change is illustrated next. Withthe increasing of the fault length, the responses of the higher-degree moment could not be ignored, and the propagationeffect becomes much more evident than case 1, in which thepropagation effect is minor and point-source approximation

Figure 7. Comparisons of synthetic seismograms generated by the higher-degreemoment tensor representation and those calculated by the summation of densely prop-agating points on the fault. Comparisons are shown for seismograms recorded at vary-ing azimuths at 30� (a), 210� (b), and 90� (c), respectively. The bottom dotted line, themiddle thin line, and the top thick line represent the contribution of zero-degree momenttensor, the higher-moment summation, and the theoretical one. The maximum mag-nitude of vertical and radial displacements at an azimuth of 90� is 1.3% and 0.8% ofthat at an azimuth of 30�, respectively. The seismograms are calculated for case 1(Table 1).


Figure 8. Comparisons of synthetic seismograms generated by the higher-degreemoment tensor representation and those calculated by the summation of densely prop-agating points on the Fault. Comparisons are shown for seismograms recorded at vary-ing azimuths at 30� (a) and 210� (b), respectively. The bottom dotted line, the middlethin line, and the top thick line represent the contribution of zero-degree moment tensor,the higher-moment summation, and the theoretical one. The seismograms are calculatedfor case 2.

Figure 9. Same as in Figure 8, except it’s for case 3 and recordings at Az � 30�(a) and Az � 210� (b).


quency property of different synthetic records begin to di-verge. The synthetic propagating seismograms and the re-sponses generated by moment summation shown as the solidand dashed lines bifurcate at about 0.4 Hz, which is a goodexample of the propagating effect at different azimuths. Thisalso means that at 0.1�0.4 Hz, the first- and second-degreemoment tensor components become comparable in size tothe zero-degree term, and if one wishes to do some inversionproblems in studying the source finiteness using the higher-degree moment terms, the appropriate frequency windowsfor study should be chosen with care. The spectra for theseismogram at 30� holds much more energy than that at 210�in this frequency band, yielding an apparent higher cornerfrequency at 30� than at 210�. The spectra amplitude of thezero-degree moment tensor, however, lies between the twosolid lines at azimuths of 30� and 210�, showing no effectof the propagation.

At frequencies above 1 Hz, however, all the momenttensor series for three components begins to diverge. It isexpected that if more accurate seismograms in higher-frequency bands are desired, then the moment tensors ofdegree 3 and greater should be included. Nevertheless, withthe increasing of the degree of moment tensors, the morespatial derivatives of Green’s function should be considered,leading to greatly increased number of necessary source co-efficients and orientation factors, and the introduction of nu-merical noise will become a serious problem in this case.

All the previous synthetic seismograms can be made fora new horizontally layered homogeneous media, because the

of 30� and 210� are shown in Figure 10. Clearly, the effectsof the first- and second-degree moments are now importanton both the amplitude and frequency content. For the verticaland radial components, the contribution of the higher-degreemoment tensors can be seen mainly in the shear and Ray-leigh waves. At an azimuth of 30�, the fault ruptures towardthe receiver, and the responses by the higher-degree momenttensors add constructively and yield an amplitude of theshear or Rayleigh wave much higher than the amplitude ofthe zero-degree term alone. For the cases at an azimuth of210�, the responses generated by the higher-degree termsadd destructively, leading to a more decreased amplitudethan the point-source model.

Viewing the moments in the frequency domain andcomparing them with the point-source model, we can gainfurther information provided by the moment tensor series.Figure 11 compares the three different displacement com-ponents at azimuths of 30� and 210� of the point-sourcemodel, the higher-degree moment tensor, and the propagat-ing point summation for case 4 in the frequency domain,respectively. At very long periods, we can see that all therecords generated by different models display almost thesame frequency content, which means that the zero-degreemoment tensor dominates the record and the use of point-source approximation in studying the source is adequate.Because the zero-degree moment tensor dominates the rec-ord at long periods, the scalar moment determined frompoint-source spectra will not be affected by propagation.

With the increasing of frequency, however, the fre-

Figure 10. Same as in Figure 8, except it’s for case 4 and recordings at Az � 30�(a) and Az � 210� (b).


Table 2Crustal Velocity Model Used for Synthetic Seismogram

Generation in Horizontally Layered Homogeneous Media

�

(km/sec)b

(km/sec)q

(g/cm3)Layer Thickness

(km)

6.2 3.5 2.7 328.2 4.5 3.4 /

sumed to be very small compared with the length of the fault,and thus the characteristic size of the fault could be repre-sented by the length L alone. In this problem, two charac-teristic times are involved, and they are the rise time T andthe rupture time Trup � L/vrup.

Comparing the effect of the two characteristic times,two important cases should be considered. (1) The rupturetime is shorter than the rise time: Trup � T. In this case, therupture front expands quickly, and covers the entire faultplane before the healing of the fault surface, and thus theentire fault is moving during an interval equal to T � Trup.In case 1, we have T � 1 sec and Trup � 0.33 sec. Thedisplacements at azimuths of 30� and 210� (Fig. 7) show thatthe responses of the first- and second-degree moment tensors

contribution of higher-degree moment tensors can be cal-culated without further difficulties. The response of the lay-ered media can be obtained using the generalized reflectivityand transmission coefficient methods (Yao and Harkrider,1983). Just to give a simple example, Table 2 shows thelayered velocity model used in another calculation. Theother parameters are all the same as case 4, except thatthe half-space problem is changed into a layered medium.Synthetic seismograms for this layered model are shown inFigure 12. By comparison of the theoretical, the zero-degree,and the higher-degree terms, the same conclusion could bedrawn as Figure 10. The variation of waveform is especiallyobvious for shear, Rayleigh, and SH waves. The directionof rupture propagation is clearly embodied in the higherterms, which could not be neglected when the much higherfrequency content of the seismograms is considered.

Discussions and Conclusions

While studying the source finiteness and rupture prop-agation using models in Table 1, the important parametersare the fault size L and W, the rise time T at each point onthe fault, and the rupture velocity of the propagating rupturefront vrup. For simplicity, the width of the fault plane is as-

Figure 11. Comparisons of the displacement spectra for case 4. The solid, dashed,and dotted lines represent the spectra of displacements of propagating source model,higher moment summation, and point-source model, respectively. The upper panel isfor model at an azimuth of 30� and the lower for 210�. (a) Vertical component. (b)Radial component. (c) Tangential component.


ponents add constructively and yield shear-wave amplitude,which is almost one and a half times that generated by apoint-source model. At an azimuth of 210�, the responsesadd destructively, and the squeezed amplitude is only 60–70% of that generated by a point source. The large variationof azimuthal amplitude changes can be observed in manyearthquakes and may help to explain the role of propagationeffect in the azimuthal magnitude changes as well as in pre-dicting ground motions from finite propagating faults.

At an azimuth of 90�, the point-source model alone pre-dicts no P-SV motions at all, whereas the match between thedisplacements generated by the higher-degree moment ten-sors and the summed propagating point-source model isgood enough (Fig. 7c). It could be deduced that in the syn-thetic data, one can observe the effects of finite faults andrupture propagation quite clearly near the nodal plane, inwhich no motion will be generated for a traditional pointsource. The little constructive interference makes the dis-placement amplitudes observed very small—almost a factorof 100 below the observations at 30� and 210�, but with theimproved signal-to-noise ratio of observed broadband wave-form data, or if the azimuthal coverage is good enough, orif some stacking could be done the use of nodal seismogramsmay be helpful in solving the fault-plane ambiguity for seis-mic data.

For some earthquakes, unilateral rupture is a sufficientmodel of the faulting process, but there are lots of earth-

contribute mostly to the higher-frequency contents, and noobvious propagation effect is observed. Considering thelong-period components, the overall shape of the responsesof a commonly used point-source model still works to somedegree, and the approximation of the point source to the realfinite source can be used to generate a satisfying displace-ment curve in this case.

(2) The rupture time is longer than the rise time: Trup �T. In this case, the rupture front expands slowly. The faulttears at one end of the fault and begins to move as the tearpropagates along the fault surface in the rupture direction.The healing of the fault surface begins at the initiation pointof rupture prior to the rupture front reaching to the end ofthe fault. Many earthquakes hold this relationship becauseof the relative small rupture velocity and large size of rupturearea. In case 2, we have T � 1 sec and Trup � 1.67 sec. Thedisplacements in Figure 8 show that, in this case, the prop-agation effect becomes more evident, and the point-sourcemodel fails to predict the displacement responses even forthe long-period contents. The contribution of the higher-degree moment tensors plays an important role in generatingthe propagation effects. In cases 3 and 4, the reduced rupturevelocity leads to a larger value of Trup and as expected, thepropagation effect becomes more obvious than case 2 does.The contribution of higher-degree moment tensors can beseen mainly in the shear and Rayleigh waves in case 4. Atan azimuth of 30�, the first- and second-degree moment com-

Figure 12. Comparison of displacement responses for case 4 in a horizontally lay-ered homogeneous medium (Table 2). The recordings are at Az � 30� (a) and Az �210� (b).


References

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Anderson, J. G. (1976). Motions near a shallow rupturing fault: evaluatingof effects due to the free surface, Geophys. J. R. Astr. Soc. 46, 575–593.

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Dahm, T., and F. Kruger (1999). Higher-degree moment tensor inversionusing far-field broad-band recordings: theory and evaluation of themethod with application to the 1994 Bolivia deep earthquake, Geo-phys. J. Int. 137, 35–50.

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Dziewonski, A., G. Ekstrom, and M. Salganik (1995). Centroid-momenttensor solutions for April–June 1994, Phys. Earth Planet. Interiors88, 69–78.

Hartzell, S. H., G. A. Frazier, and J. N. Brune (1978). Earthquake modelingin a homogeneous half-space, Bull. Seism. Soc. Am. 68, 301–316.

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Julian, B. R. (1998). Non-double-couple earthquakes: theory, Rev. Geo-phys. 36, no. 4, 525–549.

Kennett, B. L. N. (2001). The Seismic Wavefield, Vol. 1: Introduction andTheoretical Development, Cambridge University Press, New York,189–207.

Lay, T., and T. C. Wallace (1995). Modern Global Seismology, AcademicPress. New York, 364–376.

McGuire, J. J., L. Zhao, and T. H. Jordan (2001). Teleseismic inversion forthe second-degree moments of earthquake space-time distributions,Geophys. J. Int. 145, 661–678.

Stump, B. W., and L. R. Johnson (1982). Higher-degree moment tensors—the importance of source finiteness and rupture propagation on seis-mograms, Geophys. J. R. Astr. Soc. 69, 721–743.

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Appendix on following two pages.

quakes nucleating in the center of the fault and spreading inboth directions. The source finiteness and rupture propaga-tion effect of this bilateral rupture can also be modeled byusing higher-degree moment tensor representation. The in-tegration terms in equation (7) can be obtained analyticallyas done in the Application to Haskell Fault Model sectionand then the coefficients can be applied to calculate the con-tribution of different-degree moment tensors. The propaga-tion effect of the bilateral rupture model, however, is muchless than that of a unilateral model, and the source time func-tion varies much less with azimuth. It is often impossible todistinguish a bilateral rupture from a point-source model(Lay and Wallace, 1995). So in this article, the representa-tion of higher-degree moment tensors is only applied to theunilateral rupture Haskell model as an illustration.

In summary, the moment tensor representation can beuseful in modeling source finiteness and rupture propagationon seismograms. Higher-degree moment tensor representa-tion of seismic sources is obtained for a horizontally layeredhomogeneous medium in this study. The representation isapplied to a simplified unilateral-rupture Haskell model, andregional synthetic seismograms contributed by the higher-degree moment tensors are compared with the theoreticalsolutions for a propagating source. Our results show that it’spossible to use the moment series as a tool for calculatingseismograms from finite and propagating faults in the for-ward sense, which will save more computation time than thetraditional summation of densely spaced points along thefault. The convergence of series summation to the theoreticalpropagation responses enables one to generate the syntheticseismograms at relatively higher frequencies. The brake inthe symmetry of the radiation patterns of the higher termsmay help us to solve the ambiguity problem between therupture and auxiliary plane. The limit of source parametersand the efficiency in calculating the synthetic seismogramsmake it possible to solve the worthy inverse problem of thespace–time evolution of the rupture propagation for a finite-fault model.

Acknowledgments

We thank L-X Wen for his comments and discussions of this work.Two anonymous reviewers provided helpful reviews for the revision of thisarticle. This work was supported by the National Natural Science Foun-dation of China (Grant Nos. 40404004 and 40404003).


Appendix

Source coefficients and orientation factors of thesecond-degree moment tensors can be obtained as iniGj,klm

the following, and the order m reaches to 4.

1 1 1(1) 4 (1) 3 (1) 2 2m � 0 P � � k /a SV � � ek SH � k k /b0 0 0 b8 8 4

(2) 2 (2) 2 2 (2) 2P � 2k a SV � ek(2k �k ) SH � k b0 0 b 0 b

1 1(3) 2 (3) 2 (3)P � k a SV � ekb SH � 00 0 02 2

1 1(4) 2 (4) (4)P � k a SV � ek SH � 00 0 02 2

(5) 3 (5) 2 (5)P � �a SV � �ekb SH � 00 0 0

1 1 1(1) 3 (1) 2 (1) 2m � 1 P � ek SV � k b SH � � ek k1 1 1 b2 2 2

1 1 1(2) 3 (2) 2 2 2 (2) 2P � ek SV � k (2k �k )/b SH � ek k1 1 b 1 b2 4 4

(3) 2 (3) 2 2 (3) 2 2P � �2eka SV � �b(2k �k ) SH � ek b /k1 1 b 1 b

(4) 2 (4) 2 (4)P � �2eka SV � �2k b SH � 01 1 1

1(5) (5) (5) 2P � 0 SV � 0 SH � � ek k1 1 1 b2

1(6) (6) (6) 2P � 0 SV � 0 SH � ek k1 1 1 b4

1 1 1(1) 2 (1) 2 (1) 2m � 2 P � k a SV � ekb SH � � k b2 2 2 b2 2 2

(2) 2 (2) 2 2 (2) 2P � 2k a SV � ek(2k �k ) SH � � k b2 2 b 2 b

1 1 1(3) 4 (3) 3 (3) 2 2P � k /a SV � ek SH � � k k /b2 2 2 b2 2 4

1 1(4) 2 (4) 3 (4)P � k a SV � ek SH � 02 2 22 2

1(5) (5) (5) 2P � 0 SV � 0 SH � � k k/b2 2 2 b4

1 1 1(1) 3 (1) 2 (1) 2m � 3 P � ek SV � k b SH � � ek k3 3 3 b2 2 2

1 1 1(2) 3 (2) 2 2 2 (2) 2P � ek SV � k (2k �k )/b SH � � ek k3 3 b 3 b2 4 4

1 1 14 3 2 2m � 4 P � k /a SV � ek SH � � k k /b4 4 4 b8 8 8

(1)A � (3M � M )D(n n ) � (M � 3M )D(n n ) � 4M D(n n )0 11 22 1 1 11 22 2 2 12 1 2

(2)A � M D(n n ) � M D(n n )0 13 1 3 23 2 3

(3)A � (M � M )D(n n )0 11 22 3 3

(4)A � M (D(n n ) � D(n n ))0 33 1 1 2 2

(5)A � M D(n n )0 33 3 3

(1)A � [(3M � M )D(n n ) � 2M D(n n )] cos A � [(M � 3M )D(n n ) � 2M D(n n )] sin A1 11 22 1 3 12 2 3 z 11 22 2 3 12 1 3 z

(2)A � [M (3D(n n ) � D(n n )) � 2M D(n n )] cos A � [M (D(n n ) � 3D(n n )) � 2M D(n n )] sin A1 13 1 1 2 2 23 1 2 z 23 1 1 2 2 13 1 2 z


(2)B � �[M (3D(n n ) � D(n n )) � 2M D(n n )] sin A1 13 1 1 2 2 23 1 2 z

� [M (D(n n ) � 3D(n n )) � 2M D(n n )] cos A23 1 1 2 2 13 1 2 z

(3)B � (�M sin A � M cos A )D(n n )1 13 z 23 z 3 3

(4)B � M [�D(n n ) sin A � D(n n ) cos A ]1 33 1 3 z 2 3 z

(5)B � 2(M � M )[D(n n ) sin A � D(n n ) cos A ]1 11 22 1 3 z 2 3 z

(6)B � 4[D(n n ) � D(n n )](M sin A � M cos A )1 1 1 2 2 13 z 23 z

(1)B � �[(M �M ) sin 2A � 2M cos 2A ]D(n n )2 22 11 z 12 z 3 3

(2)B � �[M D(n n ) � M D(n n )] sin 2A2 23 2 3 13 1 3 z

� [M D(n n ) � M D(n n )] cos 2A13 2 3 23 1 3 z

(3)B � �[M D(n n ) � M D(n n )] sin 2A � [(M2 11 1 1 22 2 2 z 11

� M )D(n n ) � M (D(n n )� D(n n ))] cos 2A22 1 2 12 1 1 2 2 z

(4)B � �M [(D(n n ) � D(n n )) sin 2A2 33 2 2 1 1 z

� 2D(n n ) cos 2A ]1 2 z

(5)B � [�M D(n n ) � M D(n n )] sin 2A � [(M2 11 2 2 22 1 1 z 11

� M )D(n n ) � M (D(n n )� D(n n ))] cos 2A22 1 2 12 1 1 2 2 z

(1)B � �[(M � M )D(n n ) � 2M D(n n )] sin 3A3 22 11 1 3 12 2 3 z

� [(M � M )D(n n ) � 2M D(n n )] sin 3A22 11 2 3 12 1 3 z

(2)B � �[M (D(n n ) � D(n n )) � 2M D(n n )] sin 3A3 13 2 2 1 1 23 1 2 z

� [M (D(n n ) � D(n n )) � 2M (D(n n )] cos 3A23 2 2 1 1 13 1 2 z

B � �[M (D(n n ) � D(n n )) � M (D(n n ) � D(n n ))4 11 2 2 1 1 22 1 1 2 2

� 4M D(n n )] sin 4A � [2(M � M )D(n n )12 1 2 z 22 11 1 2

� 2M (D(n n ) � D(n n ))] cos 4A12 2 2 1 1 z

Institute of Geology and GeophysicsChinese Academy of Sciences100029, Beijing, [email protected]

Manuscript received 2 August 2005.

(3)A � (M cos A � M sin A ) D(n n )1 13 z 23 z 3 3

(4)A � M [D(n n ) cos A � D(n n ) sin A ]1 33 1 3 z 2 3 z

(5)A � �2(M �M )[D(n n ) cos A � D(n n ) sin A ]1 11 22 1 3 z 2 3 z

(6)A � �4[D(n n ) � D(n n )] (M cos A � M sin A )1 1 1 2 2 13 z 23 z

(1)A � [(M � M ) cos 2A � 2M sin 2A ]D(n n )2 22 11 z 12 z 3 3

(2)A � [M D(n n ) � M D(n n )] cos 2A2 23 2 3 13 1 3 z

� [M D(n n ) � M D(n n )] sin 2A13 2 3 23 1 3 z

(3)A � [M D(n n ) � M D(n n )] cos 2A � [(M2 11 1 1 22 2 2 z 11

� M )D(n n ) � M (D(n n ) � D(n n ))] sin 2A22 1 2 12 1 1 2 2 z

(4)A � M [(D(n n ) � D(n n )) cos 2A2 33 2 2 1 1 z

�2D(n n ) sin 2A ]1 2 z

(5)A � [M D(n n ) � M D(n n )] cos 2A � [(M2 11 2 2 22 1 1 z 11

� M )D(n n ) � M (D(n n ) � D(n n ))] sin 2A22 1 2 12 1 1 2 2 z

(1)A � [(M � M )D(n n ) � 2M D(n n )] cos 3A3 22 11 1 3 12 2 3 z

� [(M � M )D(n n )� 2M D(n n )] sin 3A22 11 2 3 12 1 3 z

(2)A � [M (D(n n ) � D(n n )) � 2M D(n n )] cos 3A3 13 2 2 1 1 23 1 2 z

� [M (D(n n ) � D(n n )) � 2M D(n n )] sin 3A23 2 2 1 1 13 1 2 z

A � [M (D(n n ) � D(n n )) � M (D(n n ) � D(n n ))4 11 2 2 1 1 22 1 1 2 2

� 4M D(n n )] cos 4A � [2(M � M )D(n n )12 1 2 z 22 11 1 2

� 2M (D(n n ) � D(n n ))] sin 4A12 2 2 1 1 z

(1)B � (M � M )D(n n ) � M (D(n n ) � D(n n ))0 22 11 1 2 12 1 1 2 2

(2)B � M D(n n ) � M D(n n )0 13 2 3 23 1 3

(3) (4) (5)B � B � B � 00 0 0

(1)B � �[(3M � M )D(n n ) � 2M D(n n )] sin A1 11 22 1 3 12 2 3 z

� [(M � 3M )D(n n ) � 2M D(n n )] cos A11 22 2 3 12 1 3 z

Documents

Source Finiteness and Rupture Propagation Using Higher ... · Bulletin of the Seismological Society of America, Vol. 96, No. 4A, pp. 1241–1256, August 2006, doi: 10.1785/0120050164