Seismological Research Letters, Volume 60, No. 2, Apnl _ June, 19g9

A Student's Guide to and Review of Moment Tensors

M. L. Jost and R. B. Herrmann

Department of Earth and, Atmospheric SciencesSaint Louis Uniuersity

P. O. Bos B09OSt. Louis, MO 69156

ABSTRACTA review of a moment tensor for describing a general seismic point source ispresented to show a second order moment tensor can be related to simpler seismicsource descriptions such as centers of expansion and double couples, A review ofliterature is followed by detailed algebraic expansions of the moment tensor intoisotropic and deviaroric components. Specific numerical examples are provided inthe appendices for use in testing algorithms for moment tensor decomposition.

INTRODUCTIONA major research interest in seismology is the

description of the physics of seismic sources. A commonapproach is the approximation of seismic sources bv amodel of equivalent forces that correspond to the linearwave equations neglecting non-linear effects in the nearsource region (Geller, 1976; Aki and Richarcls, 1gg0; Ken_nett, 1983; Bullen and Bolt, l98b). Equivalent forces aredefined as producing displacements at the earth's surfacethat are identical to those from the actual forces of thephysical process at the source. The equivalent forces aredetermined from observed seismograms that contain infor_mations about the source and path and distortions due tothe recording. Hence, the principle problem of source stu_dies is the isolation of the source effect by correcting forinstrument and path.

The classical method of describing seismic sources,having small dimensions compared to the wavelengths oiinterest (point source approximation) is by their strength(magnitudes, seismic moment) and tleir fault plane solu_tion (Flonda, 1962; Hirasarva and Stauder, 1g65;Herrmann, 1975). Recently, seismic moment tensors havebeen used routinely for describing seismic point sources(e.g. Kanamori and Given, lgg2; Dziewonski and Wood_lrouse, 1083b; Dziewonski et al. , Igg3a-c, lgg4a_c; Giar_dini, 1984; Ekstrcjm and Dziewonski, 19g5; Dziewonskiet al., l985ad, 1986a-c, 1982a-f; Ekstrcjm et al., IggT:Siphin, 1087; PDtr monthly listings published by NEIS).Gilbert (1970) introduced moment tensors for calculatingthe displacement at the free surface which can beexpressed as a sum of moment tensor elements times thecorresponding Green's function. An elastodynamicGreen's function is a displacement field due to an uni_directional unit impulse, i.e. the Green,s function is theimpulse response of the medium between source andreceiver. The response of the medium to any other timefunction is the convolution (Arfken, 1gg5) of that tirneflnction with the impulse response. The Grecn's functiondepends on source and receiver coordinates, the earthmodel, and is a tensor (Aki and Richards, 19g0). Theiinearity between the moment tensor and Green's functionelements was first used by Gilbert (tOZ:) tor calculatingmoment tensor elements flom obser.vations (moment ten_sor inversion). The concept of scismic momenL lensorc

was further extended by Backus and Mulcahy (1976), andBackus (I9ZZa, b). Moment tensors .an bL determinedfrom free oscillations of the earth (e.g. Gilbert andDziewonski, 1975), long-period surface waves (.,g.McCowan, 1976; Mendiguren, 1gZ7; patton and Aki,1979; Patton, 1980; Kanamori and Given, 19g1, 1982;Romanowicz, 1981; Lay et al., 79g2; Nakanishi andKanamori, 1982, lg84) or long-period body waves (e.g.Stump and Johnson, 1977; Strelitz, 1g7g, 1gg0; WarJ,1980a, b; F\Lch et a/., 1980; Fitch, 19g1; Langston, 19gl;Dziewonski et al., I}SI; Dziewonski and Woodhouse,1983a, b). Throughout this Student's Guide, we willfocus on second-rank, time independent moment tensors(Appendix I). We refer to Dziewonski and Gilbert (tg14),Gilbert and Dziewonski (1975), Backus and Mulcahy119?.9)^, P.r.\ls (toz7a), Srump and Johnson (r077), Strel_itz (1980), Sipkin (1081), and Vasco and Joinson {1g88}for a description of time dependent moment tensors.Higher order moment tensors are discussed by Backus andMulcahy (1976), Backus (I9zZz, b), and Dzlewonslii andWoodhouse (tssSa).

The reason that moment tensors are important isthat they completely describe in a first order approxima_tion the equivalent forces of general seismic point sources.The ecluivalent forces can be correlated to pirysical sourcemodels such as sudden relative displacement at a faultsurface (elastic rebound model by H. F,. Reicl, 1910).rapidly propagating metastable phase transitiorr, (Orriron,1963), sudden volume collapse due to phase transitions, orsudden volume increa^se due to explosions (Kennett, 1gg3;Vasco and Johnson, 1g8g). The equivalent forcesrepresenting a sudden displacement on a fault plane formthe familiar double couple. The equivalent forces of a sud_den change in shear modulus in presence of axial strainare represented by a linear vector dipole (Knopoff andRandall, 1970). In conclusion, a seismic moment tensor isa general concept, describing a variety of seismic sourcemodels, the shear dislocation (double couple source) beinrju>t one of t hem.

The equivalent forces can be determined from ananalysis of the eigenvalues and eigenvectors of themoment tensor (Appendix I). The sum of the eigenvaluesof the momeni tensor describes the volume change in thesource (isotropic componeni of the moment tensor). If the

JI

, ..,., . t . .; ' ,

Jost and Flerrmann

sum'is positive, the isotropic component is due to anexplosion. The source has an implosive component if thesum is negative. If the sum of the eigenvalues vanishes,then the moment tensor has only deviatoric components.

The deviatoric moment tensor represents a pure doublecouple source if one eigenvalue equals zero. If none of theeigenvalues vanishes and their sum still equals zero, themoment tensor can be decomposed into a major andminor double couple (Kanamori and Given, 1981), or adouble couple and a compensated linear vector dipole(CLVD) (Knopoff and Randall, 1970). A CLVD is adipole that is corrected for the effect of volume change,describing seismic sources which have no volume change,net force, or net moment. In general, a complete momenttensor can be the superposition of an isotropic componentand three vector dipoles (or three CL\lD's, or three doublecouples, Ben-Menahem and Singh, 1981).

This Student's Guide is an extension of "A student'sguide to the use of P- and S- wave data for focal mechan-ism determination" (Herrmann, 1975). Hence, emphasis isgiven illustrating the relations between classical faultplane solutions and seismic moment tensors. Addressinggeneral seismic point sources, we provide examples ofmoment tensor decompositions into basic equivalentsource representations, as contributions of dipoles or dou-ble couple sources. Clarification of terms such as majorand minor double couple or compensated linear vectordipole is provided. Moment tensor inversion schemes arebriefly summarized. In the appendices, examples of theuse of notation by different authors are given along withsome numerical results which are useful for testing com-puter programs. Furthermore, the formulation of t,hebasic Green's functions by Herrmann and Wang (1985) isconnected to a simple moment tensor inversion scheme.

GENERAL ELASTODYNAMIC SOURCEBy using Ihe representation lheorem for seismie

sources (Aki and Richards, 1980), the observed displace-ment do at an arbitrary position x at the time t due to a

distribution of equivalent body force densities, f 1,, \n a

source region is

€d,n(x,t) : I I Gn1,(x,t;r,4 /*(r,t) dv(fl dT, (1)

-m I/

where Gn6 are the components of the Gleen's functioncontaining ihe propagation effects, and V is the sourcevolume where /1 are non-zero. We assume the summationconvention for repeated indices (Arfken, 1985). The sub-script n indicates the component of the displacement.Throughout, we will use the following coordinate system

(Figure 1): The x-axis points towards north, the y-axistowards east, and the z-axis down (this system is righthanded). Then, er, er, and ez are the unit vectors

towards north, east, and vertically down, respectively.

By assuming that the Green's functions varysmoothly within the source volume in the range ofmoderate frequencies, the Green's functions can be

expanded into a Taylor series around a reference point tofacilitate the spatial integration in (1) (I(ennett,1983;Arfken, 1985). The expansion is usually done around thecentroid "

: €. The physical source region is character-ized by the existence of the equivalent forces. These forces

North

z

Fig. 1. Definition of the Cartesian coordinates (x,y,z).The origin is at the epicenter. Strike is measuredclockwise from north, dip from horizontal down,and slip counterclock'wise from horizontal. u and yare the slip vector and fault normal, respectively(modified after Aki and Richards, 1980).

arise due to differences between the model stress and theactual physical stress (stress glut, Backus and Mulcahy,1976). Outside the source region, the stress glut vanishesas do the equivalent forces. The centroid of the stressglut is then a weighted mean position of the physicalsource region (Backus, 1977a; Aki and Richards, 1980;Dziewonski and Woodhouse, 1983a). It seems that thecentroid of the stress glut gives a better position for theequivalent point source of an earthquake than the hypo-center which describes just the position of rupture initiali-zation. The Taylor series expansion of ihe components ofthe Green's function around this new reference Doint is

(2)

?i^-€i) Gnk,j,. . ;(x,t;€,t)

The comma between indices in (2) describes partial deriva-tives with respect to t,he coordinates after the comma.We define the components of the time dependent forcemoment lensor as :

,.:, | /It[u, j^(€,t):1 (r;,-€1,) ?i_-€i^)l tG,t) dv. (s)v

If conservation of linear momentum applies, such as for a

source in the interior of a body, then a term in M6 does

not exist in (3). With the Taylor expansion (2) and ihedefinition of the time dependent moment tensor (3), thedisplacement (t) can be written as a sum of terms whichresolve additional details of the source (multipole expan-

Gnlr(x,t;r,t):@12

-l vJr \Jrl

38

I '*"ryi

t-

A Student's Guide to and Review of Moment Tensots

sion, Backus and Mulcahy, 1976; Stump and Johnson,L977; Ak\ and Richards, 1980; Kennett, 1983; Dziewonskiand Woodhouse, 1983a; Vasco and Johnson, 1988):

d^(x,t) : (4)

@t-

P, ^ Gnk,i, i-(x,t,E,t) r Mrj,"'i^(€'t)

'

where * denotes the temporal convolution. By using aseismic signal that has much longer wavelengths than thedimensions of the source (point source approximation), weneed to consider only the first term in (4) (Backus andMulcahy, 1976; Stump and Johnson, 1977). Note, thatsingle forces will not be present in (a) if there are noexternally applied forces (indigenous source). The totalforce, linear and angular momentum must vanish for theequivalent forces of an indigenous source (Backus andMulcahy, 1976). The conservation of angular momentumfor the equivalent forces leads to the symmetry of theseismic moment tensor (Gilbert, 1970).

We assume t,hat all components of the time depen-dent seismic moment tensor in (a) have the same timedependence s(t) (synchronous source, Silver and Jordan,

1982). Neglecting higher order terms, we get (Stump andr

^EE\Jonnson. lv/ / |

d,n(x,t): M*i I c"r,j " s(t) I (5)

Mlri are constants representing the components of thesecond order seismic moment tensor M, usually termediie moment tensor. Note that the displacement d, is a

linear function of the moment tensor elements and theterms in the square brackets. If the source time functions(t) is a delta function, the only term left in the squarebrackets is Go1,1 describing nine generalized couples. Thederivative of a Green's function component with respectto the source coordinate €i is equivalent to a single couplewith arm in the {i direction. For & : 7, i.e. force in thesame direction as the arm, the generalized couples are vec-

tor dipoles (F igure 2; Maruyama, 1964). Thus, themoment tensor component Mp; gives the excitation of thegeneralized (k,j) couple.

DOUBLtr COUPLE SOURCESThe moment tensor components in (5) in an isotropic

medium for a double couple of equivalent forces are given

by

(*'*) (*'v) (*,r)

v

zFig. 2. The nine generalized couples representing G,1,,i in (5) (modified

after Aki and Richards, 1980).

z

l-

39

Jost and Flerrmann

Mtj:ttA(%u1 *u1 u1) ,

where p is the shear modulus, A is the area of the faultplane, u denotes the slip vector on the fault surface, andru is the vector normal to the fauit plane (Aki and

Richards, 1980; Ben-Menahem and Singh, 1981). Notethat the contributions of the vector of t,he fault normal yand the slip vector u are symmetric in (6). From the sym-metry of M, we note that the roles of the vectors u and zcould be interchanged without affecting the displacementfield; i.e. the fault normal could equivalently be the slipvector and vice versa. This well known fault plane - auxi-liary plane ambiguity cannot be resolved from the seismicradiation of a point source. Hence studies of locations ofaftershocks, surface faulting, rupture directivity, or staticfinal displacements (Backus, 1977a) need to be done inorder to resolve this ambiguity.

The term up ui I ui u1, in (6) forms a tensor, D,describing a double couple. This tensor is real and sym-metric, giving real eigenvalues and orthogonal eigenvec-tors (Appendix I). The eigenvalues are proportional to (1,0, -1). Hence, the characteristic properties of a momenttensor represeniing a double couple are i) one eigenvalueof the moment tensor vanishes, and ii) the sum of theeigenvalues vanishes, i.e. the trace of the moment tensor iszero (the other two eigenvalues are constrained to equalmagnitude but opposite sign).

Let t, b, and p designate the orthogonal eigenvectorsto the above eigenvalues (Herrmann, 1975; Backus, 1977a;Dziewonski and Woodhouse, 1983a).

t:+(z*u) (7)v2'b:uXu (S)

p:+(r_,r) (o)-y2

The tensor D corresponding to the terms in the bracketsin (6) can be diagonalized (principal axis transformation,see Appendix I), where the eigenvectors give the directionsof the principal axes. The eigenvector b corresponding tothe eigenvalue zero gives the null-axis, the eigenvector tcorresponding to the positive eigenvalue gives the tensionaxis, T, and the eigenvector p corresponding to the nega-tive eigenvalue gives the pressure axis, P, of the tensor.These axes can be related to the corresponding axes of thefault plane solution, since we are focusing on pure doublecouple sources. The P-axis is in the direction of max-imum compressive motion on the fauli surface; the T-axisis the direction of maximum tensional motion. Note thatthe P- and T-axes inferred from the motion on the faultsurface are not necessarily identical to the axes ofmaximum tectonic stress, since the motion can be on apreexisting plane of weakness rather than on a newlyformed fault plane that would correspond to the max-imum tectonic stress (McKenzie, 1969). However, thisambiguity cannot be resolved from the seismic radiation.In order to determine the direction of rnaximum tectonicstress, additional geological data such as in stlt stressmeasurements and frictional forces is necessary. Lackingthis kind of information, it is generally assumed that theP- and T- axes found from the seismic wave radiation aresomewhat indicative of the direction of tectonic stress.

The double couple ur uj + ui tt1, czrn equivalently bedescribed by its eigenvectors (Gilbert, 1973).

us u1 *uj u*:ts t1 -p1 p* (10)

:0.5 [(4 * pr) Gi - n1) -t (tt - pt) (t1 + n1\Comparing the terms in (10), we find the relation betweentension and pressure axes and slip vector and fault normal(Appendix I):

1u:iT(t+p) (11)

":+( t-p) (12)y2The other nodal plane is defined by

Itr:iT(t-p) (13)

1u:Ar(t+p) (14)

If strike, O, dip, 6, and slip, \, of the faulting areknown, the slip vector u and the fault normal z are givenby (Aki and Richards, 1980)

u : u ( cos \ cos iD + cos 6sin \sin O ) e,

* d ( cos \sin O - cos 6sin \ cos O ) e, (15)

-z sin6sin\e, ,

where d is the mean displacement on the fault plane. Thefault normal z is

u : -sin|sinO e, *sin6cos0 eo - cos6e, . (f0)

The scalar product of u and u is zero. The strike of thefault plane, O, is measuled clockwise from north, with thefault plane dipping to the right when looking alcng thestrike direction. Equivalently, the hanging wall is then tothe right (Figure 1). The dip, d, is measured down fromthe horizontal. The slip, \, is the angle between the strikedirection and the direction the hanging wall moved rela-tive to the foot wall (the slip is positive when measuredcounterclockwise as viewed from the hanging wall side),The range of the fault orientation parameters are

0<o<2r, o,--5<+,1975; Aki and Riehards,moment is

Mo:ltAi

and -r <).<zr (Herrrnann,

1980). The sealar seismic

Equation (6) together with (15), (t6), and (17) lead tothe Cartesian components of the symmetric momenr len-sor in terms of strike, dip, and slip angles.

M," : -Mo (sin 6 cos \ sin 2O * sin 26 sin \ sin2 $)

Myy : M, (sin 6 cos \ sin 2O - sin 26 sin \ cos2 rD)

M.,: It[o(sin 26sin \) (18)

M"y : Mo(sin 6 cos \ cos 2iD * 0.5 sin 26sin \ sin 2O)

Mr, : -Mo(cos 6 cos \ cos Q * cos 26 sin \ sin Q)

Mo, : -Mo(cos 6 cos \ sin O - cos 26 sin \ cos O)

Different notation of the moment tensor elements are dis-cussed in Appendix IL In Appendix III, several simple

(6)

(17 )

40

r --:i--'

A Student's Guide to and Review of Moment Tensors

moment tensors are related to fault plane solutions.Body-wave and surface wave radiation patterns from a

source represented by a moment tensor are discussed byKennett (1988).

Since the seismic moment tensor is real and sym-metric, a principal axis transformation can be found,diagonalizing M (Appendix I). The diagonal elements are

the eigenvalues rn; of M. Then, the scalar seismic momentcan be determined from a given moment tensor by

M,:+( I-,1+l*,1) , (1e )

where m, and m2 are the largest eigenvalues (in the abso-

lute sense). The seismic moment can equivalently be

estimated by the relations (Silver and Jordan, 1982):

GENERAL SEISMIC POINT SOURCtrS

In this section, it is assumed that the seismic source

cannot be described by a pure double couple mechanism.

The moment tensor is represented as sum of an isotropicpart, which is a scalar times the identity matrix, and a

deviatoric part.

In order to derive a general formulation of themoment tensor decomposition, let's consider the eigen-

values and orthonormal eigenvectors of the moment ten-sor. Let rn; be the eigenvalue correspo_nding to the ortho-normal eigenvector a; : (a;rra;u,a;r)T. Using the ortho-normality of the eigenvectors (Appendix I, (A1.5))' we can

write the principal axis transformation of M in reverse

order as:

"t: ["rrr".]-

lt" a2r asz

: laty azy agy

lor, a2z a3,

From (2t), we findeigenvectors and mo

M"" : mt af, + m2 a], + m" a!,

trlyy : mt o?y I m2 a]o I ms a]u

M,": mL af, + m, o], + ^, o!,

M"y : ffir arz a1, * m2 a2t a2y I mg ag, ag, 122)

Mr, : ffit atz at, * mz ez, a2z + m3 a3, ag,

Mr, : mt ary ar, *.mz azy a2, * mg a3y a3,

The effect of the eigenvalue decomposition (21) is that a

new orthogonal coordinate system, given by the eigenvec-

tors, has been defined. In this new coordinate system, thesource excitaiion is completely described by a linear com-

bination of these orthogonal dipole sources.

m in (21) is the diagonalized moment tensor. The ele-

ments of m are the eigenvalues of M. We now define the

0

lr(M)0 ,,,:',]* ,t

-'

where tr(M) : m1* m2 * ms is the trace of themoment tensor and m; is a set of diagonal matriceswhose sum yields the second term in (23). The purelydeviatoric eigenvalues rnj of ihe moment tensor are

r ffit I *z +:t : ^, - ! rr(M) . (24)mi:mi - 3 B

The first term on the right hand side (RHS) of (Z:)describes the isotropic part of the moment tensor. Theeigenvalues of the isotropic part of the moment tensor are

important for quantifying a volume change in the source.The second term describes the deviatoric part of themoment tensor consisting of purely deviatoric eigenvalues,which are calculated by subtracting ll3 tr (M) from each

eigenvalue of M. This deviatoric part of the moment ten-sor can be further decomposed, where the number ofterms or the specific form of the decomposition will be

discussed in the next sections. Obviously, a multitude ofdifferent decompositions are possible. In Appendix fV, wegive some numerical examples illustrating several methodsof moment tensor decomposition.

Vector DipolesA moment, tensor can be decomposed into an isotre

pic part and three vector dipoles. In equation (23) let N :3 and

Applying (21) to m-1, we get for the {irst deviatoric term(i:1) in the decomposition

o?o aryat,

aryarz o?,

* ?^^\:mr arar , ('Jtl

where we identified the matrix as the dyadic arat (Appen-dix I). The dyadic a1a1 describes a dipole in the directionof the eigenvector a1. By applying (21) to m2 and ms in(25), we get similar expressions involving z"2a2 and asas,

describing the second and third deviatoric terms in the

decomposition. Finally, equation (21) can be written forthe decomposition into three linear vector dipoles along

the direciions of the eigenvectors of M as

nsor decomposition

"y' ,,,h,]

j,]

by rewriting m as

(23)

'.:[ry]':tryY (20)

l"':l

|;il (21)

lf.,o oll,,. at! o,,l

Ilr m2 o Il"r, a2y or,l.

I L. o '31 lo3, asy or.l

relations between components of thement tensor elements:

[-;oo] [o o ol [oo olm,:lo ool,*,-lo-;ol. *.:loo ol. e5)

[oool loool [oo*;j

4l

Jost and Herrmann

|Il*f o ol f,io o I-,:+lo -.iol, *:+l'o o.l[o o ol lo o-*i]f. o o l l-*; o ol

*":llr*; o I I l^ -ol (28)"'3-3 l" "'2 " l' -n:e lu m2 |

tt o -^;l Io o ol

f. o ol [--Jo ol*,:llo-.;ol rl ol' 3l - l'-u:slu u

-,tr o -Jl Lo o-il

where each m; is equivalent to a pure double couplesource (Appendix III). Notice that each double coupleconsists of two linear vector dipoles (c.f. (25), (26) and(2S)), e.g. @i lZ) ("r", - arz2) for m1. Each dipole con-sists of two forces of equal strength but opposiie direction(c.f. tr'igure 2). Then, the double couple can be seen thisway: The first couple is formed by one force of eachdipole, one force pointing in the positive a1, the other inthe negative a2 direction. The corresponding other coupleis constructed by the complementary force of each dipole,pointing toward ihe negative a1 and positive a2 direction.

Using (21) with (23) and (28), we get the result that amoment tensor can be decomposed into an isotropic partand three double couples.

IY: i(rn r*mr*ms)I (22)

* mi ararlml a2a2+m! asas,

which is identical ro (22) and equation (a.sS) in Ben-Menahem and Singh (1981).

Double CouplesNext, we decompose a moment tensor into an isotro-

pic part and three double couples. For the deviatoric partin(23)letN:6and

r _ L,M: ^ (m ,*m"*nr.)I* ^ (m,-m") (a,a,-ana") (29)3 3'11

+ l( m "-m.) (ana,,-a"a,1)1;(m,,-m, ) (a,1a"-arar) .3' 3'which is identical to equation (4.57) in Ben-Menahem andSingh (1981).

CLVDAlternatively, a moment tensor can be decomposed

into an isotropic part and three compensated linear vectordipoles. Adding terms like m1 and m, in (Za) gives a

CL\fD, 2a1a1 - a,2z"2 - a,sas. This CL\D represents a

dipole of strength 2 in the direction of the eigenvector a1,

and two dipoles of unit strength in the directions of theeigenvectors z, znd as, respectively. The decompositioncan then be expressed as:

IlM: ^ (m r*m"*mil+ ^-m 1(2a1a1-aoa,2-aeae) (30)3' "', 3

ll* - m "(2a"ao-ar E r -& raq )* ):m.(2a.aa-a, 1a 1-a,2a,2),33"'

which is identical to equation (a.56) in Ben-Menahem andSinsh (1s81).

Major and Minor CoupleNext, we will decompose a moment tensor into an

isotropic component, a major and minor double couple.The major couple seems to be the best approximation of ageneral seismic source by a double couple (Appendix fV)'since the direciions of the principal axes of the momenttensor remain unchanged. The major double couple is con-

structed in the following .way (Kanamori and Given, 1981;

Wallace, 1985): The eigenvector of the smallest eigenvalue(in the absolute sense) is taken as the null-axis. Let'sassume that l-J l>l*il> l-i I i" 1ze;. In (z:), letN:2 and use the deviato'ric-conaition -i+- i+*i': o

io obtain

Dl :

0

-m10

term indouble

(32)

Ittl: 5(m 1*mr*m")I (34)

* m[ (a"a"-arar) +mi (apy-a2a2) .

Double Couple - CLVDFollowing Knopoff and Randall (1970) and Fitch el

ol. (1980), we can decompose a moment tensor into an iso'tropic part, a double couple and a compensated linear vec-tor dipole. T,et's assume again that l-J l> l*i l>l*i I

in (23). We can write the deviatoric part in (Z:) as (N :1)

lroolm,:mj lo (r-r) ol (35)

Lo o rlwhere F : - mi I m[ ard (F-1) : *i I *i. Note that0<f <0,5. This constraint on F arises from the deviatoric.ooaition *i+*i+*[ : o. We can decompose (35)

--"r *YY.m'

fnl

0

0I r'j,] -:l (31)

ol

Applying (21) to m-1, we get the first deviatoricthe decomposition which corresponds to a purecouple termed major couple.

f. o otl"''lplMAr : | ", ". ", I lo -.i o I l""t IL ',"ll " ll'_lt o -Jl 1".'lInstead of Lhe major double couple, a besl double couplecan be constructed similarly by replacing mj in (:Z) lythe average of the largest two eigenvalues (in the absolutesense, Giardini, 1984). Applying (21) to m2 gives thesecond deviatoric term in the decomposition which also

corresponds to a pure double couple termed minor couple.

L.'l--,,'^r I l- l ''lMMt!\ - L"t .r "r l*, l"l | (33)

L".tlThe complete decomposition is then:

42

I

A Student's Guide to and Review of Moment Tensors

F,:*r(1 -2r) [3; l]*-," l-t'

into two parts representing a double couple and a CL\E

*;1 ""where we assumed that the same principal stresses

produce the double couple as well as the CL\D radiation.The complel,e decomposition (21) is then:

'|

tvt: *(* 1*m2*ms)I + rn;(l-zF) (zsas-a2a2) (37)o

+ m! r (2arar-arar-arar) .

To estimate the deviation of the seismic source fromthe model of a pure double couple, Dziewonski ec ai.(1981) used the parameter

(38)

where mfi,n is the smallest eigenvalue (in the absolutesense) and -lu* ir the largest (in the absolute sense),given by (24). From (35), we see that e : F. For a puredouble couple source, mi,,n :0 and e :0; for a pure

CL\aD, e : 0.5. Alternatively, € can be expressed in per-centages of CL\D (multiply e by 200. The percentage ofdouble couple is (f-2.; * 100). Dziewonski and Wood-house (1983b, see also Giardini, 1984) investigated thevariation o{ e versus seismic moment and earthquake spa-tial distribution on the surface of the earth.

MOMENT TENSOR INVtrRSIONThere are various methods of inversion for moment

tensor elements. The inversion can be done in the time orfrequency domain. Different data (e.g. free oscillations,surface- and body waves; different seismogram com-ponents) can be used separately or combined. In addition,certain a priori constraints such as tr (M) : O, or Mr, :My, : 0 can be imposed to stabilize the inversion, result-ing in a decrease in number of resolved moment tensorelements. In this Student's Guide, we briefly outline cer-

tain approaches and refer to the original papers lorlurther reference.

Gilbert f1970) introduced the seismic moment tensor

for calculating the excitation of normal modes (Saito,1967) of free oscillations of the earth. Gilbert (1973) sug-gested an inversion scheme for moment tensor elements inthe frequency domain. Gilbert and Dziewonski (1975)

used free oscillation data for their moment tensor inver-sion. Gilbert and Buland (1976) investigated on the smal-

Iest number of stations necessary for a successful inversion(see also Stump and Johnson, lS77). McCowan (1976)'

Mendiguren (1977), Patton and Aki (1979), Patton (1980),

Romanowicz (1981), I{anamori and Given (1981, 1982),

Lay et ol. (19E2), Nakanishi and I(anamori (1982, 1984),

and Scott and l{anamori (1985) used long-period surface

waves (typically low pass filtered at 135 sec). Stump andJohnson (1977), Strelitz (1978, 1980), Ward (r980a, b),FiLch et al. (1980), Langston (1981), Dziewonski ef ol.

(19s1), and Dziewonski and Woodhouse (1983a, b), used

moment tensor inversion for body wave data (typicallyIow pass filtered at 45 sec). A comparison between

moment tensors from surface waves and body waves was

done by Fitch et al. (1981). Dziewonski et al. (1981) sug-gested an iterative inversion method, solving for themoment tensor elements and the centroid location (Backus

and Mulcahy, 1976; Backus, 1977a; see Dziewonski and

Woodhouse, 1983a for a review). The reason for thatapproach is that moment tensor elements trade off withthe location of the earthquake. The lateral heterogeneityof the earth was considered in inversion methods by Pat-ton (1980), Romanowicz (1981), Nakanishi and Kanamori(1982), and Dziewonski et al. (1084c).

The moment tensor inversion in the time domain can

use the formulation in (5) (e.g. Gilbert, 1970; McCowan,1976; Stump and Johnson, 1977; Strelitz, 1978; Fitchet al.,l98O Ward, 1980b1 Langston, 1981). If the source

time function is not known or the assumption of a syn-chronous source is dropped (Sipkin, 1986), the frequency

domain approach is chosen (e.g. Gilbert, 1973; Dziewonskiand Gilbert. 1974; Gilbert and Dziewonski, 1975; Gilbertand Buland, 1976; Mendiguren, 1977; Stump and John-son, 1977; Patton and Aki, 1979; Patton, 1980; Ward,1980a, Kanamori and Given, 1981; Romanowicz, 1981):

dn6,f):MriU)c*,iU) (3s )

Both approaches, (s) and (39) lead to linear invelsions inthe time or frequency domain, respectively' The advan-

tage of linear inversions is that a large number of fastcomputational algorithms are available (e.g. Lawson and

Hanson, 1974; Press et al. , 1987)' We can write either (5)

or (39) in matrix form:

d:Gm (40)

In the time domain, the vector d consists of n sampled

values of the observed ground displacement at various

arrival times, stations, and azimuihs. G is a n X 6

matrix containing the Green's functions calculated using

an appropriate algorithm and earth model, and rn is a

vector containing the 6 moment tensor elements to be

determined (Stump and Johnson, 1977). In the frequencydomain, (aO) can be written separately for each frequency.d consists of real and imaginary parNs of the displacementspectra. Weighting can be introduced which actuallysmoothes the observed spectra subjectively (Mendiguren,1977; see also Ward, 1980b for weighting of body-wavedata in the time domain). In the same way, G and mcontain real and imaginary parts. m contains also thetransform of the source time function of each momenttensor element. If constraints are applied to the inversion,then m can contain a smaller number of moment tensor

elements. In such a case, G has to be changed accordingly'We refer to Aki and Richards (1980) for the details ofsolving (+0) for m (Note that (40) is identical to their(12.88)).

The following presents an outline of the processing

st,eps in a moment tensor inversion. The first step is thedata acquisition and the preprocessing, We need datawith good signal to noise ratio that are unclipped andthat have a good covefage of the focal sphere (Satake,

1985). Glitches (non-seismic high amplitude spikes due tonon-linearity of instruments e.g. Dziewonski ef a/., 1981)

have to be identified and possibly removed. Analog datahave to be digitized. The effect of non-orthogonality ofthe analog recorder must be corrected. The digitized

tffimax

43

Jost and Herrmann

record has to be interpolated and resampled with a con-stant sampling rate. At this point, a comparison of thesampled waveform with the original one can help to iden-tify digitization errors. The horizontal components willbe rotated into radial and transverse components. Lineartrends have to be identified and removed. The instrumenteffect is considered next (for WWSSN data see Hagiwara,1958; for SRO data see McCowan and Lacoss, 1978; forIDA data see Agnew et al., 1976). We can use either oneof the two approaches: i) we can remove the instrumenteflect from the observed data and compare with theory orii) we can apply the instrument response to the syntheticGreen's functions and compare with observed data. Thenominal instrument response can be used or the calibra-tion of the instrument can be checked by using f.e. thecalibration pulse on the record. In addition, the polarityof the instruments should be velified, e.g. from records ofl<nown nuclear explosions. High frequency noise in thedata is removed by low-pass filtering. Amplitudes are

corrected for geometrical spreading and reflections at thefree surface of the earth (Bullen and Bolt, 1985). For sur-face waves, the moving window analysis (Landisman ec

o/., 1969) is applied in order to determine the group velo-city dispersion. From this analysis, we can identify thefundamental mode Rayleigh and Love waves which canthen be isolated.

Second, synthetic Green's functions are calculated.Notice that the Green's functions are dependent on theearth-model, the location of the point source (centroid ofthe stress glut, or epicenter and focal depth), and thereceiver position.

The third step is the proper inversion, i,e. the solu-tion of (40) (Aki and Richards, 1980). Usually, the inver-sion is formulated as least squares problem (Gilbert, 1973;Gilbert and Buland, 1976; Mendiguren, 1977; Stump andJohnson, 1977). However, using other norms can haveadvantages in situations where less sensitivity to gross

errors like polarity reversions is required (Claerbout andMuir, 1973; Fitch el al., 1980; Patton, 1980).

The source time funclion in (5) is often assumed tobe a step function (Gilbert, 1970, 1973; McCowan, 1976;Stump and Johnson, 1977;Patton and Aki, 1979; Patton,1980; Ward, 1980b; Dziewonski el a/., 1981; Kanamoriand Given, 1981). Aiming at the recovery of source timefunctions, Burdick and Mellman (1976) used a powerfuliterative waveform inversion method based on optimizingthe cross-correlation between observed, long-period body-wave trains and synthetics. The same approach was used

by Wallace et al. , (198f) in order to invert for fault planesolutions. Other methods were employed by Strelitz (r980)and I{ikuchi and Kanamori (1982) for large earthquakes(see also Lundgren et a/. , 1988). Christensen and Ruff(1985) reported on a trade-off between source time func-tion and source depth for shallow events.

If the focal depth is not known, then a linear inver-sion can be done for each depth oui of a number of trialdepths. The most probable depth will minimize the qua-

dratic error between observed and theoretical waveforms(Mendiguren, 1977; Patton and Aki, 1979; Patton' 1980;

Romanowicz, 1981). The influence of source depth on theresults of the moment tensor inversion was investigatedby Sipkin (1982; Dziewonski el al. , 1987b). Differences insource depth influence the relative excitation of normal

modes, causing systematic errors in the inversion.

Systematic errors in the inversion are also due todeviations of the earth-model from the actual propertiesof the earth, aflecting the synthetic Green's functions.This is a fundamental problem in the sense that we areable to separate the source effect from the observedseismogram only to a limited accuracy (Mendiguren, 1977;Langston, 1981; Silver and Jordan, 1982; O'Connell andJohnson, 1988). A major problem is the effect of lateralheterogeneity of the earth (Engdahl and I(anamori, 1980;Romanowicz, 1981; Gomberg and Masters, 1988; Sniederand Romanowicz, 1988). For example, a relative changeof 0.5 Vo due to lateral heterogeneity can cause a misloca-tion in the order of of 50 km at epicentral distances ofabout 90 degrees (Dziewonski and Woodhouse, 1983b).Giardini (1984) and trkstrdm and Dziewonski (1985)reported on regional shifts in centroid positions due tolateral heterogeneity. In the inversion, lateral hetero-geneity is often neglected, i.e. the calculation of theGreen's functions is usually based on parallel layers oflateral homogeneity (Harkrider, 1964, 1970; Langston andHelmberger, 1975; Harkrider, 1976). Nakanishi andKanamori (1082) included the effect of lateral hetero-geneity into the moment tensor inversion. Anotherapproach was developed for earthquakes within a smallsource area: a calibration event is declared (mechanismknown); the spectral ratio of any earthquake in thatregion and the calibration event will result in isolating thedifference in source effects - the influence of the path iseliminated (Patton, 1980). It seems that the errors due tolateral heterogeneity are usually large enough to make a

statistical significant detection of an isotropic componentof t,he moment tensor difficult (Okal and Geller, 1979;Silver and Jordan, 1982; Vasco and Johnson, 1988).

Patton and Aki (1979) investigated the influence ofnoise on the inversion of long-period surface wave data.They found thai additive noise such as backgroundrecording noise does not severely affect the results of a

linear inversion. However, multiplicative noise (signalgenerated noise) caused by focusing, defocusing, mul-tipathing, higher mode or body wave interference, andscattering distorts the inversion results significantly(overestimation or underestimation of moment tensor ele-

ments, deviation from the source mechanisml Patton,1980; Ward, 19E0b). Finally, body waves of events withmoments larger than 1027 dyne-cm are severely affected byfiniteness of the source and directivity. If not correctedfor, an inversion can lead to severe errors in the momenttensor elements (Dziewonski et al., 1981; Kanamori andGiven, 1981; Patton and Aki, 1979; Lay et aL., 1982;Giardini, 1984).

The inversion has only a limited resolution ofmoment tensor elemenls for certain data. If we have spec-tra of fundamental mode Rayleigh waves only. the con-slraint that the trace of the moment tensor vanishes (novolume change) must be applied (Mendiguren, 1977; Pat-ton and Aki, 1979). This constraint is linear. Anotherconstraint which is often applied in addition is that one

eigenvalue vanishes (approximating the source by a dou-ble couple). This constraint is not linear (Strelitz, 7978;Ward, 1980b). In stich a case, the inversion is iterative,using a Iinearized version of the constraints (Ward,1980b). For earthquakes at shallow depths (less than

M

I

A Student's Guide to and

about 30 km), the moment tensor elemenLs M* and M*corresponding to vertical dip slip faulting are not wellconstrained from long-period surface wave data since therelated excitation functions assume very small values nearthe surface of the earth (Fitch el o/., 1981; Dziewonski ef

al., 1981; Kanamori and Given, 1981, 1982; Dziewonskiand Woodhouse, 1983a). In order to overcome this prob-lem, additional independent data, such as fault strike(observed surface breakage) can be introduced into theinversion. Another approach is to constrain these

moment tensor elements to be zero. Thus, possible faultmechanisms are restricted to vertical strike slip or 45

degree dip slip (Kanamori and Given, 1981, 1982).

In Appendix V, we relate the Green's functions in theformulation of Herrmann and Wang (1985) to a simplemoment tensor inversion scheme. This inversion exampleis aimed at testing computer programs.

CONCLUSIONA seismic moment tensor describes the equivalent

forces of a seismic point source. The eigenvectors are theprincipal axes of the seismic moment tensor. For puredouble couple sources, the principal axis corresponding tothe negat,ive eigenvalue is the pressure axis, the principalaxis corresponding to the positive eigenvalue is the tensionaxis, and the principal axis corresponding to the eigen-

value zero gives the null axis. The pressure, terxion, and

null axes can be displayed in the familiar focal mechanismplot (fault plane solution). For general seismic sources' we

can decompose the seismic moment tensor. Fimt, we can

separate out the isotropic component which describes thevolume change in the source. The leftover part of themoment tensor has, in general, three nonvanishingeigenvalues. This deviatoric part of the moment tensorcan be decomposed into a number of simple combinationsof equivalent forces. Obviously, there is no uniquemoment tensor decomposition, i.e. unique model ofequivalent forces. We outlined methods of determiningmoment tensor elements from observations, indicatingthat recording noise as well as systematic errors due to an

insufficient knowledge of the Green's functions can intreduce errors into the moment tensor elements. This sug-gests caution when apparent non-double couple sources

result from the inversion

Randall and Knopoff (1970), Gilbert and Dziewonski(1975), Dziewonski el a/. (1981), Kanamori and Given(1981, 1982), Dziewonski and Woodhouse (t983b), Giar-dini (1984), and Scott and Kanamori (1985) reported thatsome seismic sources cannot be described by a pure doublecouple. One explanation is that some fault planes show a

complex geometry (Dziewonski and Woodhouse, 1983b).Another explanation can be that some sources deviatefrom the model of a sudden shear dislocation; they can be

due to a rapidly propagating phase transition (Knopoffand Randall, 1970; Dziewonski and Woodhouse, 1983b).

However, the simple inversion experiment in Appendix Vpointed out that the deviation from a pure double couplecan also be due to the presence of noise in the data(Stump and Johnson, 1977; Patton and Aki, 1979; Pat-ton, 1980; Ward, 1980b; Wallace, 1985; O'Connell and

Johnson.1988).

Review of Moment Tensors

ACKNOWLEDGMENTSWe thank two anonymous reviewers for their helpful

criticism of the manuscript. Critical remarks by OznurMindevalli are appreciated. Funds for this research were

provided by the Defense Advanced Research ProjectsAgency under contract F49628-87-K-0047 monitored bythe Air Force Geophysics Laboratory.

APPENDD( IIn the following, we give some mathematical

definitions of tensors, the eigenvalue problem and dyadicsfollowing Arfken (1985).

Let M be a moment tensor of second rank (order).Then, M is represented as a 3X3 matrix in a given refer-ence frame. Let a4 be the cosine of the angle between thep axis of another coordinate system and the k axis. Thenthe components of M, M17, transform into the new refer-ence frame by the relation

Mpq : atp aiq M*i , (A1.1)

where we need to sum over repeated indices (summationconvention).

Given a moment tensor M, let's assume that there isa vector a and a scalar m such that

Ma:m a (A1.2)

a is called eigenvector of M and rz is the correspondingeigenvalue. For calculating the eigenvalues and eigenvec-tors of a given moment tensor (solving the eigenvalueproblem), we transform (A1.2)

(M-*I) a:0 , (A1.3)

where I is the identity matrix. (A1.3) is a system of 3

simultaneous homogeneous linear equations in a; . Non-trivial solutions are found by solving the secular equation(characteristic polynomial)

det(M-rnI) :g , (A1.4)

where "det" means the determinant. (A1.4) is a polynemial of bhird degree. It has three real roots, i.e. eigen-values, since the moment tensor is real and symmetric(Faddeeva, 1959), Substituting each eigenvalue m; into(A1.3) gives the corresponding eigenvector a;. The eigen-

vectors are orthogonal. Multiplying each eigenvector byits inverse norm, we get the orthonormal eigenvectors,renaming them as a;:

a; a1 :6;1 (A1.5)

I(nor,ving the eigenvectors, we can diagonalize M (princi-pal axis transformation). Let A be the matrix whosecolumns are the orthonormal eigenvectors of M' From(A1.5), we see that A is orthogonal : A? : A-1. Then,Ar M A : m, where m is diagonal, consisting of theeigenvalues of M.

We represent a dyadic by writing two vectors a and

b together as ab (see Appendix A in Ben-Menahem andSingh, 1981). These two vectom forming the dyadic are

not operating on each other, but define a 3X3 matrix. Leti, j, and k be unit vectors along a right handed Cartesiancoordinate system. The dyadic ab is defined as

45

a. b,, a. b,

o, bo o, b,

arby arb,

For a : b, we get (26). The multiplication of a vector cfrom the left is

c ij : [(ic" * jc, * kc")'i]j : c,j (A1.7)

If the dyadic is symmetric, the multiplication of any vec-tor with the dyadic is commutative, i.e. ab : ba. Ingeneral, we can understand a dyadic as a tensor of secondrank. By a proper choice of the coordinate system, a sym-metric dyadic can always be transformed into diagonalform (principal axis transformation). As an example, wecan rewrite (10) using dyadics (Gilbert, 1973):

rru + utr: tt - pp

:0.5 [(t+pxt-p) + (t-p)(t+n)l .

APPENDIX IIThe PDE monthly listings (NEIS) routinely publish

centroid moment tensor solutions in the notation of thenormal mode theory following Dziewonski et ai. (r9sr).For reference, the spherical moment tensor elements, /;.in the notation of Gilbert and Dziewonski (1975),Dziewonski et al. (1981), and Dziewonski and Woodhouse(1983a) are compared to the moment tensor elements as

given in (18) following Aki and Richards (19s0).

ft:Mu:Mut _ itr _ lnJ 2 - tvt oe - rvrrr

fs:M6o:Myy

| 4: M,6 : M", (A2.1)

ls:M,d:-Mu,I a: Meo: - Mz!

'where (r,@,/) are the geographical coordinates at thesource. O is the colatitude (O : 0 at the north pole) and

/ is the longitude of the point source. The sign of the off-diagonal moment tensor elements depend on the orienta-tion of the coordinate system (Fitch el a/., 1981). But theeigenvalues and the eigenvectors of the moment tensor inthe formulation of (1S) or (A2.1) are identical, which canbe shown by comparing the solutions to the secular equa-tion (Appendix I). This result is expected since physicallaws should not depend on the choice of the referencefreme The slin wecf61 s and fault normal z are(Dziewonski and Woodhouse, 1983a)

u : t (- cos \ cos O - cos 6sin \ sin O ) e6

Az (cos\sinO-cosrsin\cosQ)e6 (L2.2)

+uslnAsln0eiland

Jost and Flerrmann

z: sin 6sin O es * sin 6cos O e6 * cos 6e, (A2.3)

These two equations are identical Lo (a.I22) in Ben-Menahem and Singh (1981). The differences in sign com-pared to (15) and (16) can be fully explained by noting

(,{1.6) that e, : - Q, ed: ey, and es : - ex; in other words,ee, ed, and er are unit vectors towards south, east, andup, respectively (defining a right handed system).

APPENDD( IIIIn order to gain some experience in the relationships

between a moment tensor and a fault plane soluiion, threesimple focal mechanisms are discussed in detail. Thesewill be vertical strike slip, 45 degree dip slip, and verticaldip slip faults. These three fault plane solutions form acomplete set : The seismic radiation from a dislocation ona plane dipping an arbitrary angle (but striking north-south) can be expressed as a linear combination of thesethree solutions (Burridge et al., 1964; Ben-Menahem andSingh, 1968).

(A1.8)

The eigenvalues and eigenvectors of this tensor(trxample 4.6.1 in Arfken, 1985, see also Appendix I) areshown in Table A.1 (The components of the eigenvectorsare north, easL, and down).

TABLE A.1

trIGtrNVAIT]E EIGtrN\ECTOR

"g : (a,i *: iia"b,

I iiao b"

I kiarb,II o""

: I orb,

lo,u,

arj * a,k) (b,i + bli + 6zk)

I iia,bu I ika,b,

* ijarbo I ikaob,

* kiarby + kkarbz

Vertical strike slip faultThe following focal mechanism is assumed: (strike) O

: 0" (dip) d : 90', and (slip) \ : 0'. From (15) and(16), the slip vector on the fault plane is q: (1,0,0) andthe vector normal to the fault plane is z: (0,1,0). Themoment tensor can be deiermined from (f 8).

lr',olM:lMo o ol (A3.1)tlL0 0 0l

0Mo

( 0.0000, o.oooo,-1.0000)( 0.7071, 0.707l, o.oooo)

-Mn Go.zoz r,0.7071,0.0000)

The eigenvector b corresponding to the eigenvaluezero is the null-axis, the eigenvector t corresponding tothe positive eigenvalue gives the tension axis, T, and theeigenvector p corresponding to the negative eigenvaluegives the pressure axis, P, of a focal mechanism.

The focal mechanism is obtained by using (7)-(14)(Herrmann, 1975). For the irend and plunge (in degrees)of the X-, Y-, null-, T-, and P-axes, we get (90, 0), (180,o), (270,90), (45, 0), and (135,0), respectively. The trendof both the P and T axes can be shifted by 180' (FigureA.la); i.e. the P-axis can also be described by (3150,00)and the T-axis by (2250, o0 ). This ambiguiiy can be fol-lowed through to the moment tensor: The sign of an

eigenvector is not constrained by the solution of the eigen-value problem (Arfken, 1985). However, any choice ofsign leads to the same focal .mechanism.

'J -'*-ry1l:r"''

I

46

A Student's Guide to and Review of Moment Tensors

z.Nb

Fig. A.1. Focal mechanisms of a vertical strike slip fault (strike : 00, dip

- 90', slip : 0'), (u), a 45 degree dip slip fault (strike : 00, dip: 450 , slip : 900 ), (b), and a vertical dip slip fault (strike : 00,dip : ggo , slip : 90'), (c) (Appendix III).

TABLtr A.3

EIGENVALT]E EIGEN\ECTOR

45 degree dip slip faultThe following focal mechanism is assumed: (strike) O

: o', (dip) 6 : 450, and (slip) \ : 90'. From (15) and

(16), u : (0,-0.7071,-o.707L) and u : (0,0.7071,-0.7071).The moment tensor is calculated from (18).

rllo o o I

M:lo-Mo o | (A3.2)

lo o toJThe corresponding eigenvalues and eigenvectors are

shown in Table A.2.

-Mn ( 0, 1, 0)

The fault plane solution is obtained from (7)-(14)(Herrmann, 1975). For the trend and plunge (in degrees)

of the X-, Y-, null-, T-, and P-axes, we get (90, 45), (270,

45), (360, 0), (180, 90), and (270, o), respectively. Thetrend of the P and null axes can be shifted by 180' (trig-ure A.lb) to (900 , 00 ) and (180', 0o ), respectively'

The corresponding eigenvalues and eigenvectors are

shown in Table A.3.

0Mo

(-1.0000, 0.0000, 0.0000)( 0.0000, o.7o7 t,-0.707 1)

Ma: 1

(A4.1)

1 4.2)

(A4.3)

tatt\

The first monent tensor represents an explosion, theothers are the familiar ones from Appendix III, represent-ing a vertical stlike-slip, a 45 degree dip-slip, and a verti-cal dip-slip fault, respectively. All four moment tensorsare superimposed in order to describe a complex source

that is dominated by a vertical strike slip mechanism.

TABLtr A.2

EIGtrNVALT]E EIGEN\ECTOR

-Mn ( 0.0000, 0.7071.0.7071)

The fault plane solution is obtained from (7)-(1a)(llerrmann, 1975). For the trend and plunge (in degrees)

of the X-, Y-, null-, T-, and P-axes, we get (0, 90), (90, 0),(180, 0), (27o, 45), and (90, 45), respectively. The trendof ihe null axis can be shifted by 1800 (Figure A.1c) to(360" o').

APPENDD( IVIn the following, examples of the five methods of

moment tensor decomposition are presented.

In order to construct a moment tensor that does notlead to a simple dou

0Mo

(-1, o, o)( o, o,-1)

Mr:1

Mz:6

Me :3

ble couple meehanism. le!

?31011

a 3l001o ol-rol011

1

00

0I0

0

0

0

[o o o

l: j,?

Vertical dip slip faultThe following focal mechanism is assumed: (strike) O

: o', (dip) 6 : 900 , and (slip) \ : 900. From (15) and

(fo), u : (0,0,-1) and u: (0,1,0). The moment tensor is

calculated from (18).

lo o o I' -tn | (A3.3)M:10 o

I

lo-to 0 J

Aa

Jost and Herrmann

N

Fig. A.2. Focal mechanisms of the double couples from the moment tensordecomposition (Appendix rv). (u) major couple of the moment ten-sor in (A4.5), elementary moment tensor EMTS in (.44.6), andsecond term on the RHS of (A4.9) (strike:3550, dip :80', slip: 16'), (b) elementary moment tensor EMT2 in (.44.6) (strike :1250, dip : 630, slip : -950), (c) elementary moment tensorEMT4 in (Aa.6) (strike:1990, dip:440, slip:63').

The result is

':[a 5j, I[o -r 4 ](A4.5)

Table A.4 shows the eigenvalues of (Aa.5) and thecorresponding eigenvectors, which are the principal axes ofM.

TABLtr A.4

EIGtrNVAI,L]E EIGEN\GCTOR

the eigenvector (-0.2938, -0.1397, -0.9456) with eigenvalue2.9 as the null-axis. The fault plane solution of the majordouble couple gives for the X-, Y-, null-, T-, and P-axes(in degrees): (172, 16), (265, 10), (25, 71), (219, 18), and(128, 4), respectively (Figure A.2a). The major doublecouple gives a good estimate of the major contribution tothe faulting which is predominantly strike slip (compareFigures A.1a and A.2a).

Next, the moment tensor in (Aa.s) is decomposedinto an isotropic part and three double couples following(29) which is evaluated by using (26) together with thedata in Table A.4. The numbering of the eigenvalues andeigenvectors in (29) follows the columns of Table A.4, butthat is noi relevant to the solution. The calculation gives

[rool Io.+s+z o.3ee5 --o.broelM: l l0 1 0l+ 0.67e4 | 0.3995 0.3396 -032201

[o o rl [--o.sroo 4l22o -o.7s361

I o.roza o.e22r -0.1882.|+ 4.2rr0

| o.s22r -0.2623 4.2477 I (,4.4.6)

[-0.1882 -0.2477 0.0951 J

3.85235.8904

-6.7427

(-0.2938, -0.1397, -0.9456)( o.z\s2, 0.b992, -0.3170)( 0.6109. -0.7883. -0.0734)

The sum of the eigenvalues is equal to 3, which is theexpected value for the sum of the eigenvalues of (A4.1),describing an explosion.

In order to calculate the deviatoric part of the givenmoment tensor, the isotropic part is removed by subtract-ing one third of the trace of (A4.5) from each diagonal ele-

ment. The solution to the corresponding eigenvalue prob-lem leads to the same eigenvectors as above. Thisindicates that the principal axes of the complete momenttensor are the same as the principal axes of thecorresponding deviatoric tensor. The deviatoric eigen-

values are 2.8523, 4.8904, and -7.7427 in the order ofTable A.4 (see (2a)). From (38), e : o.37, i.e. the givenmoment tensor has a double couple component of 26 7o

and a CL\ID componeni of 74 %o.

For the determination of the major couple from (32),

we identify the eigenvecior (0.6109, -0.7883, -0.0734)

corresponding to the deviatoric eigenvalue of -7'7 as theP-axis, the eigenvector (0.7352, 0.5992, -0.3170)

corresponding to the eigenvalue of 4.9 as the T-axis, and

[-o.zsos 0.5226 o.Bz27I

+ 3.5316 | 0.5226 -0.6019 0 0743

L o.z22z 0.0743 0.8882

This equation is identical to (Aa.s). The first ele-

mentary moment tensor (EMTI) on the RHS of (Aa.6)describes the explosion (isotropic component of (Aa.5))and is identical to (Aa.1). The last three elementarymoment tensors on the RHS (EMT2, EMT3, trMT4,respectively) represent pure double couple sources sincethe eigenvalues of each tensor is 0 and * 1. The three ele-

mentary moment tensors have identical eigenvectorswhich are the same vectors as shown in Table A.4.However, the correlation between eigenvector and eigen-value (i.e. null-, P-, and T-axes) varies. Note that replac-

48

_J *"T*_ w

A Student's Guide to and Review of Moment Tensots

ing M14 by -Mr,i switches the sign of the eigenvalues

(leavin! the eigenvectors untouched), which is identical tointerchanging the P- and T-axes.

From the eigenvalues and the eigenvectors, the faultplane solution for each elementary moment tensor is

determined and shown in Table A.5.

TABLE A.5

EMT2 EMT2 EMTs EMT3 EMT4 EMT4A)ilS TRD PLG TRD PLG TRD PLG

(des.) (deg.) (dee.) (deg') (dee.) (des.)

x 36 26 r72 16 324 38

Y 226 63 265 10 109 46

NLILL r28 4 25 71 2L9 18

T 2r9 18 2r9 18 25 7lP257rr2841284

The focal mechanisms corresponding to EMT2 -EMT4 are shown in Figures A.2b, A'2a, and A.2c, respec-

tively. Note that the positions of the axes remain fixed inthese figures, where only the correlation to the eigenvalues

changes. The fault plane solution representing the thirdelementary moment tensor EMT3 in (A+.0) is identical tothe fault plane solution of the major couple (Figure A.2a).Notice that this solution has also the largest coefficient in(,44.6). This solution is an approximation to the majorcontributor of the moment tensor (Figure A.1a and

(A4.2)). However, the other fault plane solutions (Figure

A.2b and A.2c) do not show similarities to the input faultmechanisms (Figure A.1b and A.1c).

The seismic moments of the elementary moment ten-

sors are given by the coeflfrcients in (Aa.6). The sum of the

seismic moments of the elementary moment tensors is 1'4

times larger than the seismic moment of the composite

moment tensor in (A4.5).

Next, the moment tensor in equation (Aa'5) isdecomposed into an isotropic part and three vector dipoles

following (27) which is evaluated by using (26) together

with Table A.4.rf

11 0 0l

M: I lo 1-o l+ 2.8523

[o o tl

0.0863 0.0110 0.277S

0.0410 0.0195 0.1321

0.2779 0.1321. 0.8941

moments of the elementary moment tensors are given bythe coefficients in (Aa.7) which are identical to the devia-toric eigenvalues of (A4.5). This exercise demonstratedthat vector dipoles are related to the eigenvectors scaled

by the corresponding eigenvalue of a given moment ten-sor, which makes an evaluation of (A4.7) obsolete.

Alternatively, the moment tensor in equaiion (A4.5)can be decomposed into an isoiropic part and three com-pensated linear vector dipoles using (30).

[r o ol [--o.zarr o.r23r

M:1lo l ol+1.2841 lo.l23l --o.e415

L0 o rl 10.8336 0.3e63

I o.ozrs r.32lo --o.6eel

I+ 1.e635 | 1.3216 0.0773 -0.56e7

|

L-0.69e1 -0.5697 -0.69851

I o.rtso -r.4447 -0.1345.|

-2.2476 l:.t++z 0.8642 0.1734 l.[-0.1345 O.1734 -0.9838 J

0.8336.l

0.3e63 |

1.6823 l

(A4.8)

This equation is identical to (Aa.5). The seismic

moments of the elementary moment tensors are given bythe product of the respective coefficient and V3 Theeigenvalues and eigenvectors for (A4.8) are shown inTable .4.6, using the same notation as above. Note thatthe eieenvectors are identical to those in Table A.4.

TABLE ,4'.6

EMT EIGENVAI-IIE trIGENVECTORo1

-I

q

( 0.6109,-0.7883,-0.0734)( o.7 352, 0.5992,-0.3170)(-0.2938.-0.1397.-0.9456)

-1-12

(-0.2938,-0.1397,-0.9456)( 0.6109,-0.7883,-0.0734)( o.7352, 0.5992,-0.3170)

-l-t2

( 0.7352, 0.5992,-0.3170)(-0.2938,-0.1397,-0.9456)( 0.6 109,-0.7883,-0.0734)

I o.s+os o.44ob -0.2330.|+ 4.8e04 | o.++os 0.35e1 -0.18ee

I

L-o.2330 -0.1899 0.1005 J

I o.sztz -o.48lo -0.0448.l-7.7427 l-o.nt'u o6214 0.0578

I.L-0.0148 0.0578 o.oo54 I

Next, the moment tensor in (Aa.s) is decomposed

into an isotropic part, a double couple and CL\D follow-ing (37), whbre e : F : 0.3684.

-0.1882

0.0951

(A4.e)

[rool Io.rozr oe22rM:r l0 1ol+2037e I0.e221 4.2623

L0 0 ll L--o.1882 4.2477

I o.rroo -r.4447 -o.rs4sl- 2.8523 l_r.+++z 0.8642 0.1734 l.

l-o.ta+s 0.1734 -0.98881

144.7 )

This equation is identical to (Aa.s). In the notationused above, each of the elementary moment tensors

trMT2, EMT3, and trMT4 have two eigenvalues equal tozero, the third one equals one. EMT2 is represented by theeigenvector (0.2938, 0.1397, 0.9456)' EMT3 by l-O.7352'-0.5992, 0.3170), and EMT4 by (0.6109, -0.7883, -0.0734).

These vector dipoles are mutually orthonormal. Noticethat these vector dipoles are identical to the eigenvectors

of equation (A4.5), which are the principal axes of the ten-

sor (Table A.4). EMT2 represents the null-, EMT3 thetension-, and EMT4 the pressure axis. The seismic

This equation is identical to (Aa.5). Notice that thesecond term on the RHS corresponds to EMT3 in (ea.6)and to the major double couple. These three tensors allhave the same fault plane solution (Figure A.2a). Thethird term in (A+.0) corresponds to EMT4 in (e+.s),representing a CL\D (see Table A.6).

As final remark, Iet's consider the decompositionequations (27), (25), and (30) for a simple double couplesource (e : 0 ), e.g. let ffit: - ffi2: I, and rn. : g'

Then, M : ar&r - a2a2 for all three equations. That is,

we get one pure double couple out of the decomposition.For a CL\|D (e :0.5), let's assume that m1: tn2:-7t

49

and ma : 2. Then all three formulas give M : 2 asag -arat - a2a,2' representing one CL\ID.

APPENDIX VIn this section, we relate the Green's functions in the

formulation of Herrmann and Wang (1985) to a momenttensor inversion scheme. Following the theory given byHerrmann and Wang (1985), the Fourier transformed dis-placements at the free surface at the distance r from theorigin due to an arbitrarily oriented double couple

without moment is

d,,(r,z4,a): ZSS Ar + ZDS A2 + ZDD As

d,(r,z4,w): r?SS Ar + RDS 42 + RDD As (A5.1)

d6(r,z4,u): ?SS A4+ TDS As ,

where d,, is the vertical displacement (positive upward), d,

is the radial displacement, and d6 is the tangential dis-placement (positive in a direction clockwise from north).The functions ZSS, ZDS, ZDD, RSS, RDS, RDD, TSS,and TDS together with ZEP and REP are the ten Green'sfunctions required to calculate a wave fieid due to an arbi-trary point dislocation source or point explosion buried ina plane layered medium (Wang and Herrmann, 1980;

Herrmann and Wang, 1985). As before, let' u:(u",u,u")and u--(u,uo,ur) be the dislocation vector and vectornormal to the fault plane, respectively. Note that (15) and(16) are identical to the formulation used by Herrmannand Wang (19s5), where our u equals their f and our zequals their n (f : x-axis, 2 - y-axis, 3 : z-axis). Then

A r:(u, u " -

u, u o ) cos(2 a z ) *(u, u, *u o u ")

sin(2 a z )

A 2:(u, u, *u, u,) cos(az )*(u o u, *u, u o) sin(az )

A g:11', u, (A5.2)

A n:(u, u, -u o u o ) sin(2 a z )-(u, u

o *u o u,) cos(2 a z )

A ,:(uru,Iu rur) sin(az)-(uo u rlu, uu) cos(az ) ,

where az is the azimuth of observation. Equivalently,

,q ,: !(u,,-Mur) cos(2az)lM,o sin(2az), 2' tt'

Az: M* cos(az)*Mo, sin(az)

ar: -f,$r,,+Myy)1

A n: |(M,,-Mou) sin(2az)-M,u cos(2az)

As: -My, cos(az)*M," sin(az) .

These equations are identical to (A5.2) which can be

proven by using (18) together with (15) and (r0). Notethat the coefficients given in (A5.3) agree with themoment tensor elements as defined by Aki and Richards(1980; (A5.3) differs in sign with the coefficients of Langs-

ton (1981) due to conventions on displacements and

Green's functions).

Note that either definition of the coefficients of the

Green's functions can be used for the calculation of the

displacement at the free surface, depending on whether

the focal mechanism or the moment tensor is given. Here,

equations (A5.3) and (A5.1) are rised in order to developan inversion scheme for the moment tensor elements. We

regroup and assume the presence of an isotropic com-ponent (ZtrP + 0, RtrP l0):

1,,d,(r,z+,,a): M," lf cos(zaz) - ry. +l

t-**,,1#cos(2cz) -ry.+].*,,1+]+ u*lzss sin(zaz)]

+ u",lzos.or(or)]

+ uu,fzos rir'(or)]

d,(r,z 4,u) : r-\ry cos(2az) - ry. #]* r,,1# cos(2az)- ry..f.]-rrrl+f* *"olass sin(zcz) ]

* r,,lnDs cos(az) ]

+ Mo"lBDs sin(az)]

d, 6(r,z :o,w) : * -1ry,i'(2,, ) ]

**,,1#,i,l2,,)]

* M* lttt "o"(zol]

Jost and flerrmann

(A5.3)

(A5.4)

(A5.5)

(A5.6)

TI+ M",ITDS sin(az)l

tl+ M!,I-TDS cos(az)1.

Equations (A5.4), (A5.5), and (.{5.6) each set up a

moment tensor inversion scheme. Equations (As.a) and(A5.5) are formulated for the general case where the inver-sion expects a moment tensor that is a composition of an

isotropic part and a deviatoric part. An inversion based

on transverse data, (.45.6), cannot resolve Mrr.ln such a

case, we assume that the moment tensor is purely devia-toric and constrain M", -- - (M* t M*). The same con-straint can be applied to (A5.4) and (A5.5) in the case of

- .. -f ---qFFTa".I

I

50

an inversion that looks for a pure deviatoric moment ten-sor (set formally ZEP : RtrP : 0 in (A5.4) and (A5.5)'Dziewonski et cl. . 1981).

From the last three equations we see that theobserved displacement at the free surface is a linear com-

bination of the station specific Green's functions, withinthe square brackets, with the moment tensor elements as

scalar multipliers. We also note that if the source timefunction is known and a point source approximation is

acceptable, the moment tensor elements are independentof frequency (linear inversion) and similar equations arise

relating observed time histories to temporal Green's func-tions within the square brackets.

Next, we performed a simple moment tensor inver-sion using the vertical component of synthetic teleseismic

P-wave first motion peak amplitudes as suggested byStump and Johnson (1977). We assumed a pure devia-

toric source (ZEP :0 in (A5.4)).

LeL azl, '.., azn be azimuths of n different stations'

Then the expressions in the square brackets of (A5.4)

define components of a matrix as a;1(az;) , ..., a;s(az;) forthe i-th azimuth. A system of linea"r equation arises:

d,(az ) a 1(az 1) a 6(az ) lvf rx

Muu

Mz!

M",

Mu"

A Student's Guide to and Review of Moment Tensors

(A5.7)

Fig. A.4. Assumed focalseismograms: st rike1100 (Appendix V).

zss

2.388E-Ot

ZDI)

7.323E-09

ZDS

6, rr02E-09

Fig. A.3. Synthetic Green's functions ZSS, ZDD, and ZDS(Herrmann and Wang, 1985) for a half-space model( V" : 8 km/sec, Vs : 4.6 km/sec, P : 3.3

gl tm3, h:30 km, t" :0.7) calculated by usingthe Haskell formalism. The time window ranges

from 4.0 to 55.1 sec (dt : 0.05 sec). Maximumampli[udes are in cm (Appendix V).

dr(azn)

For observations at more than 5 distinct azimuths,lhe system (A5.7) is overdetermined. The solution can be

reached by the classical least squares approach. The five

moment tensor elements can be determined by using thenumerical stable singular value decomposition. We

imposed the deviatoric constraint Mr, : - (M* + Mw).Hence the inversion gives a purely deviatoric source.

However, we were not constraining one eigenvalue as zero

(double couple), letting the inversion tell us about doublecouple and CL\ID components' The eigenvalues and

eigenvectors can be calculated using the Householder

transformation with further QL decomposition. Theimplementation of these numerical concepts was done

using code by Press et al. (1987).

In the following, some results of inveriing syntheticdata are presented. First, Green's functions were calcu-

lated using a Haskell formalism for a simple half-space

model ( Vp : 8km/sec, V, : 4.6 km/sec and p : 3'3

gl cm3, h : 30 km ). Figure A.3 shows the three basic

Green's functions ZSS, ZDD, and ZDS. The assumed

focal mechanism (Figure A.4: strike : 1800, dip : 46',slip: 11go)is the same as used by Herrmann (1975, Fig-ure 2). Teleseismic P-wave first motions were synthesized

at 12 equidistant azimuths (epicentral distance : 500 ).Not,e that an instrument response was not included in thesynthetics. Due to the simple model and the fact thai allstations are equidistant from the source, a correction foranelastic attenuation ( t- :0.7 ) or geometrical spread-

ing is not required. A correction for an extended source is

,rot .rr..r.u.y since the moment used is 1020 dynecm and

the duration of the source time function is 0.2 sec. We

used (A5.a) for time domain measurements.

aor(azn) ans(azn)

mechanism for the synthetic: 1800, dip : 40'. slip :

5l

illl | "

]ost and Herrmann

TABLE A.7: RESULTS OF THE MOMENTTENSOR TNVERSION (MAJOR COUPLE)

Cme 0 Cme I Cme II Cme III Cme IV Cme V

if,

Mru

M""

M4

M""

Mo"

0

-0.925

0.925

-0.220

-o.262

-0.163

-0.037

-0.902

0.939

-0.199

-o.262

-0.168

-0.050

-0.951

1.002

-0.200

-0.260

-0.162

-0.109

-0.966

1.075

-0. r94

-0.264

-0.168

-0.202

-1.023

1.225

-0.176

-o.257

-0.156

0.301

- 1.091

0.791

0.257

-0.t72

-0.324

EV(NUrL)

EV(T)

EV(P)

%ofDQ

% of CL\D

Mo

0.00

1.00

- 1.00

100

0

r.00

-0.04

1.01

-0.97

s2

8

0.99

-0.05

|.o7

- 1.02

90

10

1.04

-0.1I

1.14

-1.03

8l

19

1.09

-0.20

1.28

-1.08

69

al

1 .19

0.26

o.92

-1.18

56

44

1.07

STRII(E

DIP

SLIP

180.0

40.0

110.0

l/v.c

39.7

109.0

r7s.2

40.1

107.8

r77.8

39.9

106.1

176.4

40.5

103.2

2l1.8

40.8

123.1

STRtr(E

DIP

SLIP

334.6

52.8

7 4.O

335.4

52.9

7 4.9

330.5

52.2

75.0

51.9

77.O

339.3

50.8

7S.0

351.1

56.8

64.8

r (TRD)

r (PLG)

P (TRD)

P (PLG)

r92.7

/ D.O

75.9

6.6

194.9

70.1

6.7

rs4.7

77 .l

/ o.l

6.2

196.8

78.1

76.4

6.1

198.2

80.0

5.2

20s.7

67.3

98.8

5.5

For Cme 0, moment tensor elements are calculated from (18) msuming a double couple source

(strike - 180", dip :40', slip : 110'). The eigenvalues of the moment tensor coresponding to

the null-, T-, and P-axes are shown as EV(NULL), EV(T), and EV(P), respectively. Equation (38)

is used to determine the percentage of double couple or CLVD from the eigenvalues of the moment

tensor. The seismic moment is calculated using (20), The orientation of the fault plane and auxiliary plane is given together wit,h the trend and plunge of the T- and P-axes (Henmann, 1975).

cmes I - IV are for additive pseudo-random noise ( 0 %, 14 %, 28 vo, and.56 %, respectively) in

the synthetic seismograms at 12 diflerent azimuths. Case V msumes that one of the 12 seismo-

grams h* a r"versed polarirl (0 % p'"udo random noi.e)

Table A.7 displays the inversion results for the majorcouple. The moment tensor elemenis, the percentage ofdouble couple and CL\D, the seismic moment, and thefocal mechanism parameters are shown. For Case 0, themoment tensor elements were calculated from the givenfault plane solution and (18). Next, three experimentswere performed: 1.) synthetic seismograms were calculatedusing.the Flaskell method (Case I). Figure A.Sa shows thevertical component of a synthetic seismogram at an

azimuth of 0 degrees. 2.) Different amounts of pseudo-

random noise were added to the synthetic seismogramscalculated in Case I with amplitudes of * 0'25 X 10-" cm

(Case ll. Irigure A.5b). + 0.5 X lO-e em (Case lll. FigureA.Sc), and + 1.0 X 10-e cm (Case tV, Figure A.5d).

Averaged over the 12 azimuths' these noise levels

correspond to 14.%o, 28 %o, and 56 %o pseudo-random

adclitive noise, respectively. 3.) The final experiment(Case V) relates to possible polarity errors of seismo-

graphs. Hence it was assumed that one of the 12 seismo-grams of Case I had a wrong polarity.

The theoretical focal parameters (Case 0) agree

within the measurement errors with the observed ones(Case I). This justifies the technique. The effect of noise is

to severely distort the moment iensor elements. The iso-tropic moment tensor components seem to be more sensi-

tive to noise than the deviatoric ones. Notice that themoment tensor gains a contribution of a CL\rD due to thenoise. The percentage of CL\D versus double coupleincreases with increasing noise. The effect of randomnoise on the fault plane solution that is derived from themoment tensor elements is minor; i.e. the fault planesolution for the major double couple is very close to theoriginal focal mechanism. However, with increasing noise,

the fault plane solution deteriorates. 8 %o polar\zationerrors in otherwise perfect data lead to worse results than56 7o additive random noise (Case fV). A doubling of the

" -- -l

52

A Student's Guide to and Review of Moment Tensors

2.267E-O?

2" +73E-09

2.60rrE-09

2.98rrE-09

Fig. A.5. Vertical components of synthetic teleseismic

seismograms at 50 degrees and azimuth of 00. Thetime window ranges from 4.O Lo 29'6 sec (dt :0.05 sec). Maximum ampliiudes are in cm. (a) Nopseudo-random noise added; (b) - (d) pseudo'

random noise is added with amplitudes of' *.4.25 X 10-e cm, 40.50 X 10-e cm, *1'0 X 10-e

cm, respectively (APPendix V).

polarization errors gives meaningless results. Due to thesetup of the experiment, a minor couple would be a pureartifact of the noise.

REFERENCES

Agnew, D., J. Berger, R. Buland, W. Farrell, and F. Gil-bert (1S76). International deployment ofaccelerometers: a network for very long-periodseismology, EOS, Trans. Am. Geophys. Union 57,180-188.

Aki, K. and P. G. Richards (1980). Quantitatiue Seismol-ogy: Theory anil Methods, W. H. Freeman and Co.,New York, San Francisco, 932 pp.

Arfken, G. (1985). Mathematical Method,s for Physicists,3rd ed., Academic Press Inc., Orlando, Florida' 985

pp.

Backus, G. tr. (1977a). Interpreting the seismic glutmoments of total degree two or less, Geophys. J. R.Astr. Soc. 5L, l-25.

Backus, G. E. (1977b). Seismic sources with observableglut moments of spatial degree two, Geophys. J. R.Astr. Soc. 5L,27-45.

Backus, G. and M. Mulcahy (1976). Moment tensors andother phenomenological descriptions of seismicsources - I. continuous displacemenls, Geophys. f.R. Astr. Soc. 46,341-361.

Ben-Menahem, A. and S. J. Singh (1968). Eigenvectorexpansions of Green's dyads wiih applications togeophysical theory, Geophys. J. R. Astr. Soc. 16'4L7-452.

Ben-Menahem, A. and S. J. Singh (1981). Seismic Waaes

and, Sources, Springer Verlag, New York, 1108 pp.

Bullen, I{. tr. and B. A. Bolt (1985). An Introd,uction tothe Theory ol Seismology, 4Lh ed., CambridgeUniversity Press, Cambridge, 499 pp.

Burdick, L. J. and G. R. Mellman (1976). Inversion of thebody waves from the Borrego Mountain earthquaketo the source mechanism, Bull. Seism. Soc. Am66, 1485-1499.

Burridge, R., E. R. Lapwood, and L. Knopofl (1964). Firstmotions from seismic sources near a free surface,Bull. Seism. Soc. Am. 54, 1889-1913.

Christensen, D. H. and L. J. Ruff (1985). Analysis of thetrade-ofl beiween hypocentral depth and sourcetime function, Bull. Seism. Soc. Am. 75, 1637-1656.

Claerbout, J. F. and F. Muir (1973). Robust modelingwith erratic dzla, Geophysfcs 38, 826-844.

Dziewonski, A. M. and F. Gilbert (I974). Temporal varia-tion of the seismic moment tensor and the evidenceof precursive compression for two deep earth-quakes, Nature 247, 185-188.

a

53

Jost and Herrmann

Dziewonski, A. M. and J. H. Woodhouse (1983a). Studiesof ihe seismic source using normal-mode theory, inEarthquakes: Obseruation, Theory and, Interpreta-tion, H. Kanamori and E. Boschi, Editors, North-Holland, Amsterdam, New York, Oxford, 608 pp.

Dziewonski, A. M. and J. H. Woodhouse (fOSSb). Anexperiment in systematic study of global seismicity:centroid-moment tensor solutions for 201 moderateand large earthquakes of 1981, J. Geophys. Res.88,3247-3271.

Dziewonski, A. M., T.-A. Chou, and J. H. Woodhouse(1981). Determination of earthquake source param-eters from waveform data for studies of global andregional seismicity, J. Geophys. Res. 86, 2825-2852.

Dziewonski, A. M., A. Friedman, D. Giardini, and J. H.Woodhouse (1983a). Global seismicity of 1982:centroid-moment tensor solutions for 308 earth-quakes, Phys. Earth Planet. Interiors 33, 76-90.

Dziewonski, A. M., A. Friedman, and J. H. Woodhouse(1983b). Centroid-moment tensor solutions forJanuary-March 1983, Phys. Earth Planet. Interiorss3,71-75.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1983c). Centroid-moment tensor solutions forApril-June 1983, Pi.ys. Earth Planet. Interiors 33,243-219.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1984a). Centroid-moment tensor solutions forJuly-September 1983, Phys. Earth Planet. Interiors34, 1-8.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1984b). Centroid-moment tensor solutions forOctober-December 1983. Phus. Earth Planet. Interi-ors 34, 129-136.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(198ac). Centroid-moment tensor solutions forJanuary-March 1984, Phys. Earth Planet. Interiors34,209-219.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1985a). Centroid-moment tensor solutions forApril-June 1984, Phys. Earth Planet. Interiors 37,87-S6.

Dziewonski, A. M., J. E. Franzen, and J. H, Woodhouse(1985b). Centroid-moment tensor solutions forJuly-September 1984, Phys. Earth Planet. Interiors38,203-213.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1985c). Centroid-moment tensor solutions forOctober-December 1984. Phus. Earth Planet. Interi-ors 39,1,47-156.

Dziewonski, A. M., J, E. Franzen, and J. H. Woodhouse(1985d). Centroid-moment tensor solutions forJanuary-March 1985, Phys. Earth Planet. Interiors40,249-258.

Dziewonski, A. M., J, E. Franzen, and J. H. Woodhouse(1986a). Centroid-moment tensor solutions forApril-June 1985, P[Es. Earth Planet. Interiors 41,215-224.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1S86b). Centroid-moment tensor solutions forJuly-September 1985, Phys. Earth Planet. Interiors42,205-214.

Dziewonski, A. M., J. E. Franzen, and J. H. Woodhouse(1986c). Centroid-moment tensor solutions forOctober-December 1985, Phys. Earth Planet. Interi-ors 43, 185-195.

Dziewonski, A. M., G. Ekstrijm, J. E. Franzen, and J. H.Woodhouse (1987a). Centroid-moment tensor solu-tions for January-March 1986, Phys. Earth Planet.Interiors 45. 1-10.

Dziewonski, A. M., G. Ekstrijm, J. E. Franzen, and J. H.Woodhouse (1987b). Global seismicity of 1972:centroid-moment tensor solutions for 471 earth-quakes, Phys. Earth Planet. Interiors 45, 11-36.

Dziewonski, A.M., G. Ekstriim, J. E. Franzen, and J. H.Woodhouse (1987c). Centroid-moment tensor solu-tions for July-September 1986, Phys. Earth Planet.Interiors 46, 305-315.

Dziewonski, A. M., G. Ekstrijm, J. E. Franzen, and J. H.Woodhouse (1S87d). Global seismicity of 1978:centroid-moment tensor solutions for 512 earth-quakes Phys. Earth Planet. Interiors 48,376-342.

Dziewonski, A.M., G. Ekstriim, J. H. Woodhouse and G.Zwart (1987e), Centroid-moment tensor solutionsfor October-December 1986, Prlys. Earth Planet.Interiors 48, 5-17.

Dziewonski, A. M., G. Ekstrijm, J. E. Franzen, and J. H.Woodhouse (1987f). Global seismicity of 1979:centroid-moment tensor solutions for 524 earth-quakes Phys. Earth Planet. Interiors 48, 18-46.

Ekstrijm, G. and A. M. Dziewonski (1985). Centroid-moment tensor solutions for 35 earthquakes inWestern North America (1977-1983), Bull. Seism.Soc. Am. 75,23-39.

Ekstriim, G., A. M. Dziewonski, and J. H. Woodhouse(1987). Centroid-moment tensor solutions for the51 IASPEI selected earthquakes, 198G1984, P[gs.Earth Planet. Interiors 47.62-66.

trngdahl, E. R. and H. Kanamori (1980). Determination ofearthquake parameters, EOS, Trans. Am. Geophys.Union 6L.6G65.

54

Evison, F. F. (1963). Earthquakes and faults, Bull. Seism.

Soc. Am. 53' 873-891.

Faddeeva (1959). Computational Methods of Linear Alge-brc, Dover, New York, 252 PP.

Fitch, T. J. (1981). Correction and addition io 'Estima-tion of the seismic moment tensor from teleseismicbody wave data with applications to intraplate and

mantle earthquakes' by T. J. Fitch, D. W.McCowan, and M. W. Shields, J. Geophys. Res.

86, 9375-9376.

Fitch, T. J., D. W. McCowan, and M. W. Shields (1980).

Estimation of the seismic moment tensor fromteleseismic body wave data with applications tointraplate and mantle earthquakes, .I. Geophys

-Res. 85, 3817-3828.

Fitch, T. J., R. G. North, and M. W. Shields (1981).

Focal depths and moment tensor representations ofshallow earthquakes associated with the great

Sumba earthquake, J. Geophys. Res- 86' 9357-

9374.

Geller, R. J. (1976). Body force equivalents for stress-drop

seismic sources, Bull. Seism. Soc. Am. 661 1801-

1804.

Giardini, D. (1984). Sysiematic analysis of deep seismicity:

200 centroid-moment tensor solulions for earth-quakes betrveen 1977 and 1980, Geophys. J. R'

Astr. Soc. 77' 883-914.

Gilbert, F. (1970). Excitation of lhe normal modes of the

earth by earthquake sources' Geophys. J' R' Astr'Soc. 22,223-226.

Gilbert, F. (1973). Derivation of source parameters from

low-frequency spectra, Phil. Trans. R Soc. A 274,,tou-J / l.

Gilbert, F. and R. Buland (1976). An enhanced deconvolu-

tion procedure for retrieving the seismic momenttensor from a sparse network, Geophys. .I- R. Astr.Soc. 47,25L-255.

Gilbert, F. and A. M. Dziewonski (1075). An applicationof normal mode theory to the retrieval of struc-tural parameters and source mechanisms from

. seismic spectra, Phil. Trans. R. Soc A 278, 187-

269.

Gomberg, J. S. and T. G. Masters (1938). Waveformmodeling using locked-mode synihetic and

differential seismograms: application to determina-tion of the struclure of Mexico, Geophys. J. 94,1 I 3-21 8.

Hagiwara, T. (1958). A note on the theory of the elec-

tromagnetic seismograph, BulL. Earthquake Res.

Inst. Tokyo [Jniu. 36, 139-164.

A Student's Guide to and Review of Moment Tensors

Harkrider, D. G. (1964). Surface waves in multilayeredelastic media, 1. Rayleigh and Love waves fromburied sources in a multilayered elastic half-space,Bull. Seism. Soc. Am. 54,627-679.

Harkrider, D. G. (1970). Surface waves in multilayeredelastic media, 2. Higher mode spectra and spectralratios from point sources in plane layered earthmodels, Bull. Seism. Soc. Am. 60' 1037-1987.

Harkrider, D. G. (1976). Potentials and displacements fortwo theoretical seismic sources' Geoph,ys. J. R

Astr. Soc. 47,97-133.

Herrmann, R. B. (1975). A student's guide to the use of Pand S wave data for focal mechanism determina-tion, Earthquake Notes 46' 29-39.

Herrmann, R. B. and C. Y. Wang (1985). A comparison ofsynthetic seismograms, Bull. Seism. Soc Am. 75,41-56.

Hirasarva, T. and W. Stauder, S.J. (1965). On the seismicbody waves from a finite moving source, Bzl/.Seism. Soc. Am. 55,237-262.

Honda, H. (1962). Earthquake mechanism and seismicwaves, Geophysical Notes, Geophys. Inst., Fac. ofScience, Tokyo Uniu. 15,1-97.

Kanamori, H. and J. W. Given (1981). Use of long-periodsurface waves for rapid determination ofearthquake-source parameters, Phys. Earth Planet.Interiors 27, 8-3I.

Kanamori, H. and J. W. Given (1982). Use of long-periodsurface waves for rapid determination of earth-quake source perameters: 2. Preliminary determina-tion of source mechanisms of large earthquakes (Ms> 6.5) in 1080, Phys. Earth Planet. Interiors 3O,260-268.

I(ennett, B. L. N. (1983). Seismic Waae Propagation inStratified Media, Cambridge University Press,Cambridge, 342 pp.

I{ennett, B. L. N. (1988). Radiation from a moment-tensor source, \n Seism,oLogical Algorithm,s - Com-putational Methods and Computer Programs, D. J.Doornbos, Editor, Academic Press, San Diego, Cal-ifornia, 427-447.

I{iliuchi, M. and II. Iianamori (1982). Inversion of com-plex body u'aves, Bul/. Seisnt. Soc. Am. 72, 491-

506.

Iinopoff, L. and N{. J. Randall (1970). The compensatedlinear-vector dipole: a. possible mechrnism for deep

earthquakes, 1 Ceophys. Res. 75,4957-{963.

Landisman, M., A. Dzie'rvonski, and Y. Sato (1969).Recent improvements in the analysis of surfacewave observations, Geophys. J. R. Aslr. Soc. 17,

55 360-'103'

Jost and Herrmann

Langston, C. A. (1981). Source inversion of seismicwaveforms: The I(oyna, India, earthquakes of 13September 1967, Bull. Seism. Soc. Am. 71, l-24.

Langston, C. A. and D. V. Helmberger (tSZ5). A pro-cedure for modeling shallow dislocation sources,Geophys. J. R. Astr. Soc. 42,117-130.

Lawson, C. H. and R. J. Hanson (tSZ+). Soluing LeastSquares Problems, Prentice-Hall, Englewood Clifls,New Jersey.

Lay, T., J. W. Given, and H. Kanamori (1932). Long-period mechanism of the 8 November 1980 Eureka,California, earthquake, Bull. Seism. Soc. Am. 22,439-456.

Lundgren, P. R., E. A. Okal, and S. Stein (1988). Body-wave deconvolution for variable source parameterslapplication to the 1978 December 6 Kuriles earth-qnake, Geophys. J. 84,171-180.

Maruyama, T. (1964). Statical elastic dislocations in aninfinite and semi-infinite medium, Bull. EarthquakeRes. Inst. 42, 289-368.

McCowan, D. W. (1976). Moment tensor representation ofsurface wave sources, Geophys. J. R. Astr. Soc.44, 595-599.

McCowan, D. W. and R. T. Lacoss (1979). Transfer func-tions for the seismic research observatory seismo,graph system, Bull. Seism. Soc. Am. 68, b0l-b12. .

Mcl(enzie, D. P. (1969). The relation between fault planesolutions for earthquakes and the directions of theprincipal stresses, Bull. Seisrn. Soc. Am. bg, 59l-601.

Mendiguren, J. A. (1977). Inversion of surface wave datain source mechanism studies, J. Geophys. Res. 82,889-894.

Nakanishi, L and H. Kanamori (1982). Effects of lateralheterogeneity and source process time on the linearmoment tensor inversion of long-period Rayleighwaves, Bull. Seism. Soc. Am. 72,2063-2080.

Nakanishi, L and H. I(anamori (1984). Source mechanismsof twenty-six large, shallow earthquakes (MS >6.5) during 1980 from P-wave first motion andlong-period Rayleigh wave data, Bull. Seism. Soc.Am. 74,805-818.

O'Connell, D. R. H. and L. R. Johnson (lggg). Secondorder moment tensors of microearthquakes at TheGeysers geothermal field, California, Bull. Seism.Soc. Am. 78, 1674-1692.

Ohal, E. A. and R. J. Geller (1979). On the observabilityof isotropic seismic sources: the July 31, 1970Colombian earthquake, Phys. Earth planet. Interi-ors 18, 176-196.

Patton, H. (1980). Reference point equalization methodfor determining the source and path effects of sur-face waves, I. Geophys. Res. 85,821-848.

Patton, H. and K. Aki (1979). Bias in the estimate ofseismic moment tensor by the linear inversionmethod, Geophys. J. R. Astr. Soc. 59,479-498.

Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T.Vetterling (1987). Numerical Recipes: The Art ofScientific Computing, Cambridge University Press,Cambridge, 818 pp.

Randall, M. J. and L. Knopoff (1920). The mechanism atthe focus of deep earthquakes, J. Geophys. Res.75, 4965-4976.

Reid, H. F. (1910). Elastic rebound theory, Uniu. Calif.Publ,., Bull. Dept. Geol. Sci. 6,413-489.

Romanowicz, B. (1081). Depth resolution of earthquakesin central Asia by moment tensor inversion oflong-period Rayleigh waves: Effects of phase velo-ciiy variations across Eurasia and their calibration,J. Geophys. .Bes. 88, 5963-5984.

Saito, M. (1967). Excitation of free oscillations and surfacewaves by a point source in a vertically heterogene-ous earth, J. Geophys. Res. 72,3689-3699.

Satake, K. (1985). Effects of station coverage on momenttensor inversion, BulL. Seism. Soc. Am. 75, 165T-roo/.

Scott, D. R. and H. I{anamori (1985). On the consistencyof moment tensor source mechanisms with first-motion data, Phys. Earth Planet. Interiors 57, gZ-

ro7.

Silver, P. G. and T. H. Jordan (1982). Optimal estimationof scalar seismic moment, Geophys. J. R. Astr. Soc.70, 755-787.

Sipkin, S. A. (1982). Estimation of earthquake sourceparameters by the inversion of waveform data:synthetic waveforms, Phys. Earth Planet. Interiors30, 242-255.

Sipkin, S. A. (1986). Interpretation of non-double-coupleearthquake mechanisms derived from rnoment ten-sor inversion, J. Geophys. Res. gl

r SBl-542.

Siphin, S. A. (1987). Moment tensor solutions estimatedusing optimal filter theory for 51 selecied earth-quakes, 1980-1984, Phys. Earth PLanet. Interiors47, 67-79.

Snieder, R. and B. Romanowicz (1gSS). A new formalismfor the eflect of lateral heterogeneity on normalmodes and surface waves - I: isotropic perturba-tions, perturbaiions of interfaces and gravitationalperturbations, Geophys. J. 52, 207-222.

56

A Student's Guide to and Review of Monsrt Tcnss

Strelitz, R. A. (1978). Momentsource models, Geophys. J.364.

tensor inversions andR. Astr. Soc. 52. 859-

Wallace, T. C., D. V. Helnberger, and G. R. Mellman(1981). A trschnique for the inversion of regionaldata in source paramet€r studies, J. Geoph,ys. Res.86, 16791685.

Wang C. Y. and R. B. Herrmann (1980). A numericalstudy of P-, SV-, and SH-wave generation in aplane layered medium, Bull. Seism. Soc. Am. 7O,101t1036.

Ward, S. N. (1980a). Body wave calculations usingmoment tensor sources in spherically symmetric,inhomogeneous media, Geophys. J. R. Astr- Soc.60, 53-66.

Ward, S. N. (1980b). A technique for the recovery of theseismic moment tensor applied to the Oaxaca, Mex-ico earthquake of November 1978, Bull. Seism. Soc.Am. 7O,717-734.

Received June 23, 1988Revised October 4. 1988Accepted November 1, 1988

Strelitz, R. A. {1930). The fate of the downgoing slab: Astudy of the moment tensors from bJdy waves ofcomplex deep-focus earthquakes, piyr. EarthPlanet. Interiors 21, g3_96.

Stump, B. W. and L. R. Johnson (1977). The determina_tion of source properties by the linear inversion ofseismograms, Bull. Seism. Soc. Am. 02. 14gg_1502.

Vasco, D. W. and L. R. Johnson (19g8). Inversion ofwaveforms for extreme ,oor.a _od"ls with anapplication to the isotropic moment tensor com_ponent, in Regional Stutlies with Broadband Data,T. V. McEvilly and L. R. Johnson, Editors, ReportNo. 1, Air Force Geophysics Laioratory AFGLTR_88_0131.

Wallace, T. C. (i985). A reexamination of the momenr.tensor solutions of the 19g0 Mammoth Lakesearthquakes, J. Geophys. Res. gOr 11,1Z1_11,176.

57