ORIGINAL ARTICLE

Moment tensor decompositions revisited

Václav Vavryčuk

# The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract The decomposition of moment tensors intoisotropic (ISO), double-couple (DC) and compensatedlinear vector dipole (CLVD) components is a tool forclassifying and physically interpreting seismic sources.Since an increasing quantity and quality of seismic dataallow inverting for accurate moment tensors andinterpreting details of the source process, an efficientand physically reasonable decomposition of momentand source tensors is necessary. In this paper, the mostcommon moment tensor decompositions are revisited,new equivalent formulas of the decompositions are de-rived, suitable norms of the moment tensors arediscussed and the properties of commonly usedsource-type plots are analysed. The Hudson skeweddiamond plot is introduced in a much simpler way thanoriginally proposed. It is shown that not only the Hud-son plot but also the diamond CLVD–ISO plot and theRiedesel–Jordan plot conserve the uniform distributionprobability of moment eigenvalues if the appropriatenorm of moment tensors is applied. When analysingmoment tensor uncertainties, no source-type plot isclearly preferable. Since the errors in the eigenvectorsand eigenvalues of the moment tensors cannot be easilyseparated, the moment tensor uncertainties project intothe source-type plots in a complicated way. As a conse-quence, the moment tensors with the same uncertaintiesproject into clusters of a different size. In case of ananisotropic focal area, the complexity of moment

tensors of earthquakes prevents their direct interpreta-tion, and the decomposition of moment tensors must besubstituted by that of the source tensors.

Keywords Dynamics andmechanics of faulting .

Earthquake source observations . Seismic anisotropy .

Theoretical seismology

1 Introduction

The moment tensor describes equivalent body forcesacting at a seismic point source (Burridge and Knopoff1964) and is a basic quantity evaluated for earthquakeson all scales from acoustic emissions to large devastat-ing earthquakes. The most common type of the momenttensor is the double-couple (DC) source which repre-sents the force equivalent of shear faulting on a planarfault in isotropic media. However, many studies revealthat seismic sources often display more general momenttensors with significant non-double-couple (non-DC)components (Julian et al. 1998; Miller et al. 1998). Anexplosion is an obvious example of a non-DC source,but non-DC components can also be produced by thecollapse of a cavity in mines (Rudajev and Šílený 1985;Šílený and Milev 2008), by shear faulting on a non-planar (curved or irregular) fault (Sipkin 1986), bytensile faulting induced by fluid injection in geothermalor volcanic areas (Ross et al. 1996; Julian et al. 1997)when the slip vector is inclined from the fault and causesits opening (Vavryčuk 2001, 2011) or by seismic anisot-ropy in the focal area (Kawasaki and Tanimoto 1981;

DOI 10.1007/s10950-014-9463-y

V. Vavryčuk (*)Institute of Geophysics, Academy of Sciences,Boční II/1401, 14100 Praha 4, Czech Republice-mail: vv@ig.cas.cz

Received: 19 January 2014 /Accepted: 23 September 2014 /Published online: 16 October 2014

J Seismol (2015) 19:231–252

Vavryčuk 2005; Roessler et al. 2004, 2007). Complica-tions also arise if the source is situated at a materialinterface (Vavryčuk 2013).

In order to identify which type of seismic source isphysically represented by the retrieved moment tensor,Knopoff and Randall (1970) proposed decomposing themoment tensors into three elementary parts: the isotro-pic (ISO), double-couple (DC) and compensated linearvector dipole (CLVD) components. Although the mo-ment tensor decomposition is not unique and manyother decompositions have been proposed, thedecomposition of Knopoff and Randall (1970) provedto be useful for physical interpretations and becamewidely accepted. This decomposition was further devel-oped and applied by Sipkin (1986), Jost and Herrmann(1989), Hudson et al. (1989), Kuge and Lay (1994),Vavryčuk (2001, 2005, 2011) and others. Furthermore,Hudson et al. (1989) and Riedesel and Jordan (1989)proposed graphical representations of the DC and non-DC components in order to identify visually the mostappropriate physical source corresponding to the re-trieved moment tensor (for a geometric comparison ofboth approaches, see Tape and Tape 2012b).

Since an increasing quantity and quality of seismicdata allow inverting for accurate moment tensors andinterpreting the details of the source process, an efficientand physically reasonable decomposition of moment ten-sors is necessary. This has recently motivated severalauthors to revisit the existing decompositions (Chapmanand Leaney 2012; Zhu and Ben-Zion 2013) and source-type plots (Chapman and Leaney 2012; Tape and Tape2012a, b) and to develop their modifications. In thispaper, I summarize the physical conditions imposed onthe moment tensor decompositions and present newequivalent formulas for themost commonmoment tensordecompositions. I introduce theHudson skewed diamondplot in a much simpler way than originally proposed. Icompare several alternative moment tensor decomposi-tions and source-type plots and discuss their advantagesand drawbacks. I show differences in moment and sourcetensors (also called the potency tensors) and point out thesignificance of the source tensor decomposition for earth-quake source interpretations in anisotropic media.

2 Orientation and type of source

The seismic moment tensor M is a symmetric tensorwhich can be decomposed using eigenvalues and an

orthonormal basis of eigenvectors in the following way:

M ¼ M 1e1e1 þM 2e2e2 þM 3e3e3; ð1Þwhere

M 1≥M 2≥M3 ð2Þand vectors e1, e2 and e3 define the T (tension), N(intermediate or neutral) and P (pressure) axes,respectively.

Using spectral decomposition (1), we separate twobasic properties of the moment tensor: the orientation ofthe source defined by three eigenvectors, and the typeand size of the source defined by three eigenvalues.Since the eigenvalues are independent, the type of thesource can be represented as a point in three-dimensional (3-D) space (Riedesel and Jordan 1989)

m ¼ M 1e1 þM 2e2 þM 3e3; ð3Þwhere vectors e1, e2 and e3 define a coordinate sys-tem in this space. In order to get a unique represen-tation, the eigenvalues must be ordered according toEq. (2). Consequently, the points representing thesource type cannot cover the whole 3-D space butonly its wedge called the ‘source-type space’. Thechoice of the coordinate system and the metric ofthe source-type space differ for individual momenttensor decompositions.

3 Standard moment tensor decomposition

3.1 Definition of ISO, DC and CLVD

In this decomposition, moment tensorM is diagonalizedand further restructured to form three basic types of asource: the isotropic (ISO) and double-couple (DC)sources, which have a clear physical interpretation (seeSection 3.3), and the compensated linear vector dipole(CLVD) source, which is needed for the decompositionto be mathematically complete (Knopoff and Randall1970; Dziewonski et al. 1981; Sipkin 1986; Jost andHerrmann 1989; Hudson et al. 1989):

M ¼ MISO þMDC þMCLVD

¼ M ISOEISO þMDCEDC þMCLVDECLVD; ð4ÞwhereEISO,EDC andECLVD are the ISO, DC and CLVDelementary tensors (also called the base tensors), and

232 J Seismol (2015) 19:231–252

MISO, MDC and MCLVD are the ISO, DC and CLVDcoordinates in the 3-D source-type space. The elemen-tary tensors read:

EISO ¼1 0 00 1 00 0 1

24

35;

EDC ¼1 0 00 0 00 0 −1

24

35;

EþCLVD ¼ 1

2

2 0 00 −1 00 0 −1

24

35;

E−CLVD ¼ 1

2

1 0 00 1 00 0 −2

24

35;

ð5Þ

where ECLVD is ECLVD+ or ECLVD

− depending on whetherM1+M3−2M2≥0 or M1+M3−2M2<0, respectively.Hence, the CLVD tensor is aligned along the axis withthe largest magnitude deviatoric eigenvalue. The basetensors have ordered eigenvalues according to Eq. (2).This condition is needed for the base tensors to lie in thesource-type space (i.e. in the space filled by momenttensors with ordered eigenvalues). The norm of all basetensors calculated as the largest magnitude eigenvalue(i.e. the maximum of |Mi|, i=1,2,3) is equal to 1. Thiscondition is called the unit ‘spectral norm’ and physi-cally means that the maximum dipole force of the basetensors is unity.

Equations (4) and (5) uniquely define values MISO,MDC and MCLVD expressed as follows:

M ISO ¼ 1

3M 1 þM 2 þM 3ð Þ; ð6Þ

MCLVD ¼ 2

3M 1 þM 3−2M 2ð Þ; ð7Þ

MDC ¼ 1

2M 1−M 3− M 1 þM 3−2M 2j jð Þ; ð8Þ

where MCLVD also includes the sign of the elementaryCLVD tensor. If the elementary CLVD tensor is consid-ered with its sign as in Eq. (4), then MCLVD should becalculated as the absolute value of Eq. (7).

In seismological practice, however, we do not evalu-ate values MISO, MDC and MCLVD in Eqs. (6–8) butrather scalar seismic moment M and relative scale fac-tors CISO, CDC and CCLVD defined as:

CISO

CCLVD

CDC

24

35 ¼ 1

M

M ISO

MCLVD

MDC

24

35; ð9Þ

where M reads

M ¼ M ISOj j þ MCLVDj j þMDC; ð10Þand scale factors CISO, CDC and CCLVD satisfy the fol-lowing equation:

CISOj j þ CCLVDj j þ CDC ¼ 1: ð11ÞEquations (6–10) imply that CDC is always positive

and in the range from 0 to 1; CCLVD and CISO are in therange from −1 to 1. Consequently, the decomposition ofM can be expressed as:

M ¼ M CISOEISO þ CDCEDC þ CCLVDj jECLVDð Þ; ð12Þwhere M is the norm of M calculated using Eq. (10)and represents a scalar seismic moment for a generalseismic source. The absolute value of the CLVD termin Eq. (12) is used because the sign of CLVD isincluded in the base tensor ECLVD. Equation (12)can be used also for composing M from known scalefactors CISO, CDC and CCLVD and scalar moment M(see Appendix A).

Note that Eqs. (6–8) look different from those pub-lished by Jost and Herrmann (1989) but are equivalent.Also, Hudson et al. (1989) use different formulas andnotation, but the resultant τ–k decomposition is the sameas the CLVD–ISO decomposition except for the sign ofthe CLVD axis (for details, see Section 6.1).

3.2 Definition of the scalar seismic moment

Although the decomposition intoMISO,MDC andMCLVD

using base tensors (5) is quite common and is used bymany authors, the definition of scale factors CISO, CDC

and CCLVD in Eq. (9) is more ambiguous because of thevariability of the definitions of scalar seismic momentM.In the above approach, M is defined by Eq. (10) as thesum of spectral norms of the individual components ofmoment tensor M. The same value of M is produced bythe norm proposed by Bowers and Hudson (1999)

M ¼ MISOj jj j þ MDEVj jj j; ð13Þwhere ||MISO|| and ||MDEV|| are the spectral norms of theisotropic and deviatoric parts of moment tensor M,respectively.

J Seismol (2015) 19:231–252 233

Note that the simplest option whenM is calculated asthe spectral norm of complete moment tensor M:

M ¼ max M 1j j; M 2j j; M 3j jð Þ ð14Þcannot be applied in Eq. (9) because it leads to theviolation of Eq. (11) if CISO and CCLVD are ofopposite signs. For example, Vavryčuk (2001,2005) uses the spectral norm of M defined inEq. (14) for calculating CISO, but factors CCLVD

and CDC must then be scaled by another constantto satisfy Eq. (11):

CISO ¼ M ISO

M;CCLVD ¼ MCLVD

M;CDC ¼ MDC

M; ð15Þ

where

M ¼ MCLVDj j þMDC

1− M ISOj j.M

: ð16Þ

If CISO and CCLVD are of the same signs,Eqs. (15–16) yield the same scale factors CISO,CDC and CCLVD as when applied norms defined inEq. (10) or Eq. (13).

Nevertheless, a consistent scaling of the ISO, DC andCLVD components by the spectral norm of M can stillfind applications, for example, when constructing theHudson skewed diamond source-type plot (seeSection 6.1).

On the contrary, Silver and Jordan (1982) preferdefining M in a quite different way—using the Euclid-ean (Frobenius) norm:

M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2M 2

1 þM 22 þM 2

3

� �r; ð17Þ

where factor 1/2 is used for consistency with thestandard definition of the scalar seismic moment(Aki and Richards 2002, their Eq. 3.16). For thepure DC source, norms (10) and (17) yield thesame moment M; hence, they apparently seem tobe equally suitable for being used in the momenttensor decomposition. However, using the Euclideannorm for M in the above decomposition is mathe-matically inconvenient and introduces difficultiesbecause it is inconsistent with the definition ofthe norms of the base tensors (5). A mathemati-cally consistent decomposition based on exclusive-ly using the Euclidean norm will be described inSection 5.2.

3.3 Physical properties of the decomposition

The decomposition of the moment tensor is performed inorder to physically interpret a set of nine dipole forcesrepresenting a general point seismic source and to easilyidentify some basic types of the source in isotropicmedia:

& The explosion/implosion is an isotropic source, andthus it is characterized by CISO=±1 and by zeroCCLVD and CDC.

& Shear faulting is represented by the double-coupleforce and characterized by CDC=1 and by zero CISO

and CCLVD.& Pure tensile or compressive faulting is free of shear-

ing and thus characterized by zeroCDC. However, thenon-DC components contain both ISO and CLVD.The ISO and CLVD components are of the samesign: they are positive for tensile faulting but negativefor compressive faulting (Vavryčuk 2001, 2011).

& The shear-tensile (dislocation) source defined as thesource, which combines both shear and tensilefaulting (Vavryčuk 2001, 2011), is characterized bynon-zero ISO, DC and CLVD components. Thepositive values of CISO and CCLVD correspond totensile mechanisms when the fault is opening duringrupturing. The negative values of CISO and CCLVD

correspond to compressive mechanisms when thefault is closing during rupturing. The ratio betweenthe non-DC and DC components defines the anglebetween the slip and the fault.

& Shear faulting on a non-planar fault is characterizedgenerally by a non-zeroCDC andCCLVD. TheCISO iszero because no volumetric changes are associatedwith this type of source.

3.4 Graphical representation of the decomposition

The moment tensor decomposition can be displayed andeasily interpreted using a diamond source-type plot(Fig. 1). The plot shows the position of the source inthe CLVD–ISO coordinate system in which the DCcomponent is represented by the colour intensity. Asource with pure or predominant shear faulting is locat-ed at the origin of coordinates or close to it. Explosionsand implosions are located at the top and bottom vertexof the diamond, respectively. Pure tensile and compres-sive cracks are plotted at the margins of the diamond.Points along the CLVD axis correspond to faulting on

234 J Seismol (2015) 19:231–252

non-planar faults and points in the first and third quad-rants of the diamond correspond to shear-tensile sources.

For pure tensile and shear-tensile sources, the ISO/CLVD ratio depends on the elastic properties of themedium surrounding the source (Vavryčuk 2001,2011). In isotropic media, this ratio is:

CISO

CCLVD¼ 3

4

vPvS

� �2

−1: ð18Þ

Hence, the point representing pure tensile faulting inFig. 1 (black dot) can be close toCISO=1 (correspondingto an explosion) for high values of vP/vS but also close toCCLVD=1 for low values of vP/vS. The limiting cases are:

vPvS

→∞ andvPvS

¼ 2ffiffiffi3

p ; ð19Þ

describing fluids and the lower limit of stable solids (λ=−2/3μ), respectively. Similar conclusions can be drawnfor pure compressive faulting.

Note that the above-mentioned basic types of sourcescannot be located in the second or fourth quadrants of

the diamond source-type plot. Moment tensors in thisarea may indicate errors of the moment tensor inversiondue to noise in data, limited data coverage, an inaccuratevelocity model, a more complicated source model orfaulting in anisotropic media.

As mentioned earlier, in isotropic media, CISO=±1and CDC=1 correspond to an explosion/implosion andto shear faulting, respectively. Their physical meaning isthus straightforward. However, the moment tensor withCCLVD=±1 does not correspond to any simple physicalseismic source, and the presence of CLVD in momenttensors often causes confusions and poses questions asto whether it is necessary to introduce the CLVD. Thedecomposition described earlier indicates that theCLVD component is required to render the decomposi-tion mathematically complete, and the CLVD cannotthus be avoided. Although it has no simple physicalmeaning itself, it can be interpreted physically in com-bination with the ISO component as a product of tensilefaulting. In the case of a pure tensile crack, the majordipole of the CLVD component is aligned with thenormal to the crack surface, and the volume change

DC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CLVD

ISO

o

o

oexplosion

implosion

shearcrack

tensile

crack.

compressive

crack

.

Fig. 1 The diamond CLVD–ISOplot with the positions of the basictypes of seismic sources. Thearrows indicate the range of thepossible positions of momenttensors for pure tensile orcompressive cracks

J Seismol (2015) 19:231–252 235

associated with the opening crack is described by theISO component. However, in the case of shear-tensilefaulting, the CLVD major dipole need not be normal tothe fault, and also the orientation of DC component isnot simply related to the fault plane.

4 Source tensor decomposition

A simple classification of sources based on the mo-ment tensor decomposition is possible in isotropicmedia only. In anisotropic media, the problem ismore complicated. The moment tensor of anexplosive/implosive source is characterized byCISO=±1 in isotropic as well as anisotropic media,but the moment tensor of an earthquake source isaffected not only by the geometry of faulting but alsoby the elastic properties of the material in the focalzone. Depending on these properties, the momenttensors can take a general form with non-zero ISO,DC and CLVD components even for simple shearfaulting on a planar fault (Vavryčuk 2005). For thisreason, physical interpretations of earthquake sources(i.e. shear-tensile dislocation sources) in anisotropicmedia should be based on the decomposition of thesource tensor, which is directly related to the geom-etry of faulting.

The source tensor D of a dislocation source (alsocalled the potency tensor, see Ben-Zion 2001,2003; Ampuero and Dahlen 2005) is a symmetricdyadic tensor defined as (Ben-Zion 2003; Vavryčuk2005):

Dkl ¼ uS

2sknl þ slnkð Þ; ð20Þ

where unit vectors n and s denote the fault normal andthe direction of the slip vector, respectively, u is the slipand S is the fault size. In anisotropic media, the relationbetween the source and moment tensors reads(Vavryčuk 2005, his Eq. 4):

Mij ¼ cijklDkl; ð21Þand in isotropic media

Mij ¼ λDkkδij þ 2μDij; ð22Þwhere cijkl is the tensor of elastic parameters, and λ andμ are the Lamé’s parameters. While the moment andsource tensors diagonalize in anisotropic media in

different systems of eigenvectors and thus their relationis complicated, the eigenvectors of the moment andsource tensors are the same in isotropic media and theirdecomposition according to the formulas in Section 3.1yields similar results.

The properties of the moment and source tensordecompositions for shear and tensile sources in iso-tropic and anisotropic media are illustrated in Figs. 2and 3. Figure 2 shows the source-type plots for shear-tensile sources with a variable slope angle (i.e. thedeviation of the slip vector from the fault, seeVavryčuk 2011) situated in an isotropic medium.The plot shows that the scale factors of the ISO andCLVD components are linearly dependent for bothmoment and source tensors. For the moment tensors,the line direction depends on the vP/vS ratio (Fig. 2a).For the source tensors, the line is independent of theproperties of the elastic medium and the CISO/CCLVD

ratio is always 1/2 (see Fig. 2b and Appendix B).This property is preserved even for anisotropic me-dia. On the contrary, the moment tensors can behavein a more complicated way in anisotropic media.Figure 3 shows the ISO and CLVD components ofthe moment tensors of shear (Fig. 3a) or shear-tensile(Fig. 3b) faulting in the Bazhenov shale (Vernik andLiu 1997). This complicated behaviour prevents astraightforward interpretation of moment tensors interms of the physical faulting parameters. Therefore,first the source tensors must be calculated from mo-ment tensors and then interpreted. If the elastic prop-erties of the medium in the focal zone needed forcalculating the source tensors are not known, theycan be inverted from the non-DC components of themoment tensors (Vavryčuk 2004, 2011; Vavryčuket al. 2008). Note that the retrieved medium parame-ters do not refer to local material properties of thefault but to the medium surrounding the fault anddepend on the wavelength of the analysed radiatedwaves.

5 Alternative decompositions

Obviously, moment and source tensor decompositionscan be proposed in many alternative ways. They differin the definition and scaling of the base tensors. Here welist several alternative approaches and mention theirpros and cons.

236 J Seismol (2015) 19:231–252

5.1 A new simplified moment tensor decomposition

The properties of the decomposition described byEqs. (6–10) depend on the scaling of base tensors

which control the fractional amounts of the ISO, DCand CLVD components present in a general momenttensor. If the normalization is properly modified, thefinal moment decomposition formulas can be

ISO

CLVD

ISO

3

2

3

1.5

1.4

1.3

1.2

CLVD

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DC

)b)a

Fig. 2 The source-type plots for the moment (a) and source (b)tensors for shear-tensile faulting in an isotropic medium charac-terized by various values of the vP/vS ratio (the values are indicatedin the plot). Red dots source tensors, black dots moment tensors.

The dots correspond to the sources with a specific value of theslope angle (i.e. the deviation of the slip vector from the fault). Theslope angle ranges from −90° (pure compressive crack) to 90°(pure tensile crack) in steps of 3°

ISOa)

CLVD

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DC

b) ISO

CLVD

Fig. 3 The source-type plots for pure shear (a) and shear-tensile(b) faulting in an anisotropic medium. The black dots in a corre-spond to 500 moment tensors of shear sources with randomlyoriented faults and slips. The black dots in b correspond to themoment tensors of shear-tensile sources with strike=0°, dip=20°and rake=−90° (normal faulting). The slope angle ranges from−90° (pure compressive crack) to 90 ° (pure tensile crack) in steps

of 3°. The red dots in a and b show the corresponding sourcetensors. The medium is transversely isotropic with the followingelastic parameters (in 109 kgm−1 s−2): c11=58.81, c33=27.23, c44=13.23, c66=23.54 and c13=23.64. The medium density is2,500 kg/m3. The parameters are taken from Vernik and Liu(1997) and describe the Bazhenov shale (depth of 12.507 ft)

J Seismol (2015) 19:231 252 237–

simplified. For example, if we adopt base tensors asfollows:

EISO ¼ 2

3

1 0 00 1 00 0 1

24

35;

EDC ¼1 0 00 0 00 0 −1

24

35;

EþCLVD ¼ 2

3

2 0 00 −1 00 0 −1

24

35;

E−CLVD ¼ 2

3

1 0 00 1 00 0 −2

24

35;

ð23Þ

the relative ISO, DC and CLVD scale factors read

CISO

CCLVD

CDC

24

35 ¼ 1

2M

M 1 þM 2 þM 3

M 1 þM 3−2M 2

M 1−M 3− M 1 þM 3−2M 2j j

24

35;ð24Þ

where

M ¼ 1

2M 1 þM 2 þM 3j j þM1−M 3ð Þ; ð25Þ

and the ISO/CLVD ratio for shear-tensile faulting inisotropic media reads

CISO

CCLVD¼ 3

8

vPvS

� �2

−1

2: ð26Þ

The other properties of the decomposition:explosion/implosion producing CISO=±1, shear faultingbeing characterized by CDC=1, and pure tensile faultingproducing no DC, remain unchanged (see Fig. 4). Theformula for M is identical with Eq. (12,) but the scalarseismic momentM is now defined by Eq. (25). Note thatif we adopt Eq. (25) as the formula for the tensor norm,the norms of the base tensors in Eq. (23) are equal to 1similarly as for the standard moment tensor decomposi-tion. Also, the scalar seismic moment for the pure DCsource coincides with the standard definition. The dis-advantage of this decomposition, however, is that the

magnitude of dipole forces is different for differentelementary source tensors, so that the physical insightinto the decomposition is not as straightforward as forthe standard decomposition.

5.2 Euclidean moment tensor decomposition

The decomposition into the ISO, DC and CLVD com-ponents can also be performed by normalizing basetensors EISO, EDC and ECLVD using the Euclidean(Frobenius) norm:

Ek k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2EklEkl

r¼ 1; ð27Þ

where the summation over repeated indices is ap-plied. The coefficient 1/2 is used in Eq. (27) toproduce the scalar seismic moment consistent withthe standard definition for the pure DC source. Ac-cording to Chapman and Leaney (2012), the basetensors are defined as:

E�ISO ¼

ffiffiffi2

3

r 1 0 00 1 00 0 1

24

35;

E�DC ¼

1 0 00 0 00 0 −1

24

35;

E�CLVD ¼ 1ffiffiffi

3p

1 0 00 −2 00 0 1

24

35;

ð28Þ

where the asterisk is used to distinguish between thebase tensors defined in Eq. (28) and in Eq. (5) used inthe standard decomposition. Both sets of base tensorsdiffer in two basic aspects: in the normalization and inthe definition of the CLVD tensor. The CLVD tensor inEq. (28) is rotated to lie along the N axis but not alongthe P or T axes as in Eq. (5). The modification of theCLVD is needed for the base tensors to form an orthog-onal system:

E�ISO

� �kl E

�DC

� �kl ¼ E�

ISO

� �kl E

�CLVD

� �kl

¼ E�DC

� �kl

E�CLVD

� �kl¼ 0; ð29Þ

238 J Seismol (2015) 19:231–252

and thus applying the Euclidean norm to be meaningful.The Euclidean norm producesMISO

∗ ,MCLVD∗ andMDC

∗ asfollows:

M *ISO ¼ 1ffiffiffi

6p M1 þM 2 þM 3ð Þ; ð30Þ

M *CLVD ¼ 1

2ffiffiffi3

p M 1 þM 3−2M 2ð Þ; ð31Þ

M *DC ¼ 1

2M 1−M 3ð Þ: ð32Þ

Scalar seismic moment M* reads:

M * ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2M 2

1 þM22 þM 2

3

� �r

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2M 2

ISO þM 2CLVD þM 2

DC

� �r: ð33Þ

The scale factors CISO∗ , CCLVD

∗ and CDC∗ are

defined as:

CLVD

3

2

3

1.5

1.4

1.3

)b)a ISOISO

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

CLVD

1.2

ISO

CLVD

c)

DC

ISO

CLVD

d)

Fig. 4 The source-type plots for shear-tensile faulting in isotropicmedia (a, b) and for pure shear (c) and shear-tensile (d) faulting inanisotropic media for the simplified decomposition (seeSection 5.1). The dots in a and b correspond to the shear-tensilesources with the slope angle ranging from −90° (pure compressivecrack) to 90° (pure tensile crack) in steps of 3°. The black dots in acorrespond to the moment tensors; the red dots in b correspond to

the source tensors. The black dots in c correspond to 500 momenttensors of shear sources with randomly oriented faults and slips.The black dots in d correspond to the moment tensors of shear-tensile sources with strike=0°, dip=20° and rake=−90° (normalfaulting). The red dots in c and d show the corresponding sourcetensors. The medium in c and d is transversely isotropic (see thecaption of Fig. 3)

J Seismol (2015) 19:231–252 239

C*ISO

C*CLVDC*

DC

24

35 ¼ 1

M*� �2

sign M*ISO

� �M*

ISO

� �2sign M*

CLVD

� �M*

CLVD

� �2M *

DC

� �2

264

375;

ð34Þ

in order to satisfy

C*ISO

�� ��þ C*CLVD

�� ��þ C*DC ¼ 1; ð35Þ

similarly as for the standard decomposition (11).Let us briefly discuss the mathematical and physical

consequences of the decomposition based on using theEuclidean norm. Despite the attractive mathematicalproperties of the Euclidean norm, applying this normto the moment tensor decomposition leads to conse-quences which are rather undesirable and might disqual-ify this decomposition from being routinely employedin seismological practice. The reasons are as follows:

1. As mentioned earlier, the major dipole of the CLVDbase tensor lies along the N axis in order to satisfy theorthogonality condition (29). However, this conditionapplies to the full 3-D space but not to the source-typespace. As a consequence, the newly defined CLVD basetensor does not satisfy the ordering of eigenvalues pre-scribed by Eq. (2) and does not lie in the source-typespace. If we decompose any general moment tensorusing base tensors EISO

∗ , EDC∗ and ECLVD

∗ :

M ¼ M*ISO þM*

DC þM*CLVD

¼ M *ISO E*

ISO þM *DC E*

DC þM *CLVD E*

CLVD; ð36Þ

we find that the last term on the rhsMCLVD∗ is ‘unstable’

because it lies outside the source-type space. If MCLVD∗

is further decomposed as an individual tensor, its eigen-values must be reordered according to Eq. (2), andMCLVD

∗ subsequently splits into another non-zero MDC∗

and MCLVD∗ . This property violates the basic condition

imposed on the base vectors (tensors) in linear vectorspaces.2. As a consequence of the previous point, the relativescale factor CCLVD

∗ can never attain the value of ±1 evenfor any of the following pure CLVD moment tensors:

M1 ¼ 1

2

2 0 00 −1 00 0 −1

24

35;

M2 ¼ 1

2

−1 0 00 2 00 0 −1

24

35;

M3 ¼ 1

2

−1 0 00 −1 00 0 2

24

35;

ð37Þ

because, first, their eigenvalues have to be ordered, andafter that they can be decomposed. Their scale factorsare CCLVD

∗ =1/4 and CDC∗ =3/4.

3. The spectral norm (10) used in the standard decom-position keeps the magnitude of the largest eigenvalueequal to 1. This physically means that the predominantlinear vector dipoles in all base tensors have the samemagnitude. Hence, the scale factors CISO, CCLVD andCDC measure directly the relative magnitudes of thedipole forces in the source. For example, if the momenttensor is composed of the following MISO and MDC

tensors:

MISO ¼1 0 00 1 00 0 1

24

35 ;

MDC ¼ 21 0 00 0 00 0 −1

24

35;

ð38Þ

then CDC is obviously twice larger than CISO, CDC=2/3and CISO=1/3. However, applying the Euclidean normleads toCDC

∗ larger than CISO∗ by a factor of 8/3: CDC

∗ =8/11 and CSO

∗ =3/11. This might complicate a physicalinsight into the decomposition.4. Defining the major dipole of the CLVD along the Naxis causes the moment tensor for pure tensile (or com-pressive) faulting to contain a non-zero DC component(see Chapman and Leaney 2012, their Eq. B12a). If thevP/vS ratio is 1.73, the DC percentage is 18 %. Thisproperty is awkward because it violates the basic re-quirement imposed on moment tensor decompositionsthat the presence of shear faulting is measured by theamount of DC. Consequently, we expect pure tensilefaulting to generate no DC since it is free of shearing.Obviously, a decomposition which does not respect

240 J Seismol (2015) 19:231–252

such conditions complicates the physical interpretationsof a seismic source.

5.3 Crack-plus-double-couple model

Minson et al. (2007) analysed non-double-couple mech-anisms of volcanic earthquakes on Miyakejima, Japan,and introduced a model which combines shear faultingwith crack opening, calling this model the ‘crack-plus-double-couple’ (CDC) model. The authors further de-veloped a non-linear scheme for directly inverting forthe parameters of this source model from seismic obser-vations. Since the model does not describe all sourcetypes, only two eigenvalues of moment tensor M areindependent. For this reason, the authors do not distin-guish between ISO and CLVD components and calcu-late just the percentages of the DC and non-DC compo-nents using the following formula:

Cnon−DC ¼ MC

M 0 þMC100%; ð39Þ

where M0 and MC are the scalar moments of the shearand tensile cracks defined as

MDC ¼ M 0

1 0 00 0 00 0 −1

24

35;

Mcrack ¼ MC

1 0 00 1 0

0 0λþ 2μ

λ

264

375;

ð40Þ

where λ and μ are the Lamé coefficients of the mediumin the focal zone.

If we compare the CDC model with the other previ-ously developed models, we find that the CDCmodel is,in fact, similar with the tectonic model of Dufumier andRivera (1997) or with the shear-tensile model ofVavryčuk (2001, 2011). Hence, the CDC, tectonic andshear-tensile models can all be treated in a standard wayusing the decomposition of the moment and sourcetensors into the ISO, DC and CLVD components asdescribed in Sections 3 and 4. The standard decompo-sition has particularly two basic advantages compared tothe CDC model. Firstly, it is not necessary to assume apriori that the source is described by the shear-tensilemodel. This type of a source is recognized by the lineardependence between the ISO and CLVD scale factors

when analysing moment tensors of the shear-tensilesources in the same (isotropic) focal area. Secondly,the CDC decomposition cannot be applied withoutknowledge of the vP/vS ratio (or equivalently of thePoison ratio) in the focal area. The standard decompo-sition does not require knowing this value. It is evenpossible to determine this value from the ISO/CLVDratio (see Eq. (18)).

However, the idea of inverting for the parameters ofthe shear-tensile source model directly from seismicobservations in order to stabilize the inversion is originaland valuable since it was applied later also by otherauthors (Vavryčuk 2011; Stierle et al. 2014a, b).

5.4 Other decompositions

When seismologists started to analyse general momenttensors, many alternative approaches of decomposingthe moment tensors into ‘elementary’ force systemswere proposed. Ambiguities mostly appeared indecomposing the deviatoric part of the moment tensor.It was decomposed, for example, into three DCs, intomajor and minor DC, into two DCs with the same Taxesor into the DC and variously oriented CLVDs (for areview, see Julian et al. 1998). Over time, these decom-positions were mostly abandoned because they provedunsuitable for mathematical or physical reasons. Let ussummarize a few of the main reasons:

1. The source-type space is a wedge in full 3-D space.Therefore, decomposing the moment tensor into ISOand three DCs is an unnecessary overparameterizationwith no clear advantage.2. Except for the moment tensor decompositions de-scribed in Sections 3.1 and 5.1, no other approach offersdecomposing the moment tensors into elementary forcesystems which all lie in the source-type space. If any ofthe elementary force systems is outside the source-typespace, the decomposition introduces mathematical dif-ficulties because these force systems cannot be consid-ered as the base vectors in this space. This means thatthe individual components of the moment tensor violatethe ordering condition (2), and when further analysed,they split into other components. Mathematically, thismeans that the decomposition is not a linear transforma-tion within the source-type space.3. The decompositions fail to be sufficiently general forphysical interpretations. For example, the decomposi-tion of the deviatoric part of the moment tensor into a

J Seismol (2015) 19:231–252 241

major DC and some minor residual component is justi-fied and appropriate if the analysed moment tensordescribes shear faulting on a planar fault in isotropicmedia. The major DC would correspond to the trueshear faulting, and the residual component would rep-resent errors due to the inversion or due to data limita-tions. However, if the source represents shear-tensilefaulting or pure tensile faulting, this decomposition isnot helping in the interpretations because it cannot beused for identifying this type of a source. This appliesalso to the other mentioned decompositions.

6 Alternative source-type plots

All moment tensors fill a source-type space which is awedge in the full 3-D space. The magnitude of thevector in this space is the scalar moment, and its direc-tion defines the type of the source. In order to identifythe type of the source visually, it is convenient to plot allunit vectors of the source-type space in a 2-D figureusing some projection. We distinguish three basic kindsof the source-type plots depending on the norm appliedto the source-type space (see Fig. 5):

1. Diamond CLVD-ISO plot. If the norm is defined asthe sum of the spectral norms of the individual tensorcomponents using Eq. (10), we map points which coverpartly the surface of two pyramids with a commonhexagonal base (see Section 3.4).2. Hudson’s skewed diamond plot. If the spectral norm(Eq. 14) is used, we map points which cover partly thesurface of a cube (see Section 6.1).3. Riedesel–Jordan lune plot. If the Euclidean norm(Eq. 17) is used, we map points which cover partly thesurface of a sphere (see Section 6.2).

If we assume a random distribution of moment ten-sors in the source-type space, the three above-mentionedsurfaces of unit distance are covered by the momenttensors uniformly. The projections of these surfaces into2-D plots are chosen to conserve this property.

6.1 Hudson’s skewed diamond plot

Hudson et al. (1989) introduced two source-type plots: adiamond τ–k plot and a skewed diamond u–v plot. Theτ–k plot is the diamond CLVD–ISO plot described in

Section 3.4 but with the opposite direction of the CLVDaxis. The skewed diamond plot is introduced in order toconserve the uniform probability of moment tensor ei-genvalues normalized by spectral norm (14). If we as-sume that the eigenvalues can attain values between −1and +1 and satisfy the ordering condition (2), all pointsthen fill a volume forming a wedge inside a cube (seeTape and Tape 2012b, their Fig. 5). The unit vectors filltwo half-faces on this cube (see Fig. 6a, b). In the upper

half-face (yellow colour), the normalized eigenvalueM 1

equals 1, while in the lower half-face (red colour), the

normalized eigenvalue M 3 equals −1. The dashed linein Fig. 6b shows the deviatoric sources:

M 1 þM 2 þM 3 ¼ 0: ð41ÞThe two half-faces in Fig. 6b are further skewed, the

dashed line defining deviatoric sources (axis u) and thedotted line connecting the explosive and implosivesources (axis v) to be mutually perpendicular and of unitlength. Themoment tensor with eigenvaluesM1,M2 andM3 is projected into the u–v plot using the followingsimple equations:

u ¼ −2

3M 1 þM 3−2M 2

� ;

v ¼ 1

3M 1 þM 2 þM 3

� ;

ð42Þ

where

Mi ¼ Mi

max M 1j j; M2j j; M 3j jð Þ; i ¼ 1; 2; 3: ð43Þ

Equation (42) is similar to Eqs. (6–7) for the standarddecomposition into the CLVD and ISO except for scal-ing. It is emphasized that the eigenvalues in Eq. (42) arenormalized to their maximum absolute magnitude, hence

M 1 ¼ 1 for the upper half-face (in yellow colour) and

M 3 ¼ −1 for the lower half-face (in red colour). Equa-tion (42) ensures that the skewed diamond u–v plotsatisfies the requirement of the uniform probability dis-tribution function (PDF) for sources with random eigen-values because the coordinates u and v depend linearly on

M 1 , M 2 and M 3 . Note that in contrast to Hudson’soriginal derivation which is rather complicated and treatsthe individual quadrants of the plot separately, the pre-sented formula is simple and valid for all quadrants.

Figure 7 shows the mapping of a regular grid in CISO

and CCLVD calculated using Eqs. (6–10) into the

242 J Seismol (2015) 19:231–252

diamond τ–k plot and into the skewed diamond u–v plot.Figure 7a proves that the τ–k plot is the CLVD–ISO plotwith the reversed CLVD axis. Figure 7b shows that theCLVD–ISO grid is deformed in the first and third

quadrants of the u–v plot. This is the reason why thesources satisfying the spectral norm (14) display a uni-form PDF in Fig. 7d but a non-uniform PDF in Fig. 7c.In contrast, the sources satisfying the norm defined by

a) b)

c)

Fig. 5 Surfaces of unit vectors in the source-type space. The normof moment tensor is defined as: a the sum of spectral norms of thebase tensors (see Eq. (10)), b the spectral norm of moment tensor

M (see Eq. (14)), and c the Euclidean norm of moment tensor M(see Eq. (17)). The colours are used for improving the 3-D visu-alization and have no physical meaning

J Seismol (2015) 19:231–252 243

Eq. (10) display a uniform PDF in Fig. 7e but a non-uniform PDF in Fig. 7f. Hence, the norms must beapplied consistently when generating and decomposingthe moment tensors as well as when plotting the sourcetypes.

6.2 Riedesel–Jordan lune plot

A different approach is suggested by Riedesel andJordan (1989) who introduced a compact plot displayingboth the orientation and type of source on the focalsphere. The moment tensor is represented by a vectorof the eigenvalues of M in a coordinate system formedby the eigenvectors ofM (P, T and N axes):

m ¼ M 1e1 þM 2e2 þM 3e3: ð44Þ

The vector is normalized using the Euclidean normand thus projected on a part of the sphere called the‘lune’ (Tape and Tape 2012a) which covers one sixth ofthe whole sphere (see Fig. 5c). This surface is projectedusing a lower-hemisphere equal-area projection into a 2-D plot (see Fig. 8a). However, as pointed out by Chap-man and Leaney (2012), this representation is not opti-mum for several reasons. Firstly, vectors characterizingpositive and negative isotropic sources (explosion andimplosion) are physically quite different, but they aredisplayed in the same area on the focal sphere in thisprojection. Secondly, the analysis of uncertainties of a

[-1,-1,-1]

[ 1, 1, 1]

[ 1, 1,-1][ 1,-1,-1]

[-1,-1,-1]

[ 1, 1, 1]

[ 1, -0.5, -0.5]

[ 0.5, 0.5, -1]

[ 1, 0 ]

[ 4/3, 1/3]

[ -4/3, -1/3]

[ 0, 1 ]

[ 0, -1 ]

M2

[ -1, 0 ]

a) b)

c)

3M

1M

2M

3M

1M

2M

3M

1M

Fig. 6 A geometric interpretation of the Hudson’s skewed dia-mond source-type plot. a The cube representing the volume of themoment tensor eigenvalues. b Half-faces corresponding to M1=1(in yellow colour) and M3=−1 (in red colour). c The skewed

diamond u–v plot. The dashed line defines the positions of thedeviatoric sources; the dotted line is the ISO axis. The arrows in band c show the orientation of axes of the individual eigenvalues

244 J Seismol (2015) 19:231–252

moment tensor solution by plotting a cluster of vectorsm includes both effects—uncertainties in the orientationas well as in the source type. This is fine if the momenttensor is non-degenerate. However, difficulties arisewhen the moment tensor is degenerate or nearly

degenerate because small perturbations cause significantchanges of eigenvectors.

The mentioned difficulties can be avoided by fixingthe coordinate system. If we fix the coordinate axes inthe form:

)b)a

)d)cISO

CLVD-

ISO

CLVD-

ISO

CLVD-

ISO

CLVD-

)f)eISOISO

CLVD-

CLVD-

Fig. 7 The Hudson’s diamond τ–k plot (a, c, e) and the skeweddiamond u–v plot (b, d, f). The CLVD−means the reversed CLVDaxis. The dots in a and b show a regular grid in CISO and CCLVD

from −1 to 1 with step of 0.1. The dots in c and d show 3,000sources with randomly generated moment tensors satisfying thespectral norm (14). The dots in e and f show 3,000 sources with

randomly generated moment tensors satisfying the norm (10).Plots d and e indicate that the distribution of sources is uniformif the moment tensors are generated and decomposed using thesame norm. On the contrary, if the norms are mixed, the distribu-tion of sources is non-uniform (see plots c and f)

J Seismol (2015) 19:231–252 245

e1 ¼ 1ffiffiffi3

p ;1ffiffiffi6

p ;1ffiffiffi2

p� �T

;

e2 ¼ 1ffiffiffi3

p ;−2ffiffiffi6

p ; 0

� �T

;

e3 ¼ 1ffiffiffi3

p ;1ffiffiffi6

p ;−1ffiffiffi2

p� �T

;

ð45Þ

in the north-east-down coordinate system, we obtain theplot (Fig. 8b) suggested by Chapman and Leaney (2012)which resembles theCLVD–ISOdiamond source-type plot(Fig. 1) but adapted to a spherical metric. The basic sourcetypes are characterized by the following unit vectors:

eISO ¼ 1ffiffiffi3

p e1 þ e2 þ e3ð Þ ¼ 1; 0; 0ð ÞT ; ð46Þ

eDC ¼ 1ffiffiffi2

p e1−e3ð Þ ¼ 0; 0; 1ð ÞT ; ð47Þ

eCLVD ¼ffiffiffi2

3

r1

2e1−e2 þ 1

2e3

� �¼ 0; 1; 0ð ÞT ; ð48Þ

eþCLVD ¼ffiffiffi2

3

re1−

1

2e2−

1

2e3

� �¼ 0;

1

2;

ffiffiffi3

p

2

� �T

; ð49Þ

)b)a

)d)c

ISO

DC

CLVD

CLVD

P

T

N

deviatoric

solutions

+

-

ISOISO

ISO

CLVD CLVD

CLVD

Fig. 8 The Riedesel–Jordan source-type plot. a The originalcompact plot proposed by Riedesel and Jordan (1989) displayingboth the orientation of the moment tensor eigenvectors (P, T andNaxes), basic source types (ISO, CLVD, DC) and the position of thestudiedmoment tensor (red dot). b–dAmodified Riedesel–Jordanplot proposed by Chapman and Leaney (2012). The dashed area

in b shows the area of admissible positions of sources. The dots inc show a regular grid in CISO and CCLVD from −1 to 1 with step of0.1. The dots in d show 3,000 sources defined by the momenttensors with randomly generated eigenvalues normalized by theEuclidean norm of M. Plot d indicates that the distribution ofsources is uniform

246 J Seismol (2015) 19:231–252

e−CLVD ¼ffiffiffi2

3

r1

2e1 þ 1

2e2−e3

� �¼ 0; −

1

2;

ffiffiffi3

p

2

� �T

:

ð50Þ

The basic properties of this projection are exempli-fied in Fig. 8c, d: Fig. 8c shows the mapping of a regulargrid in CISO and CCLVD calculated using Eqs. (6–10),and Fig. 8d indicates that the PDF of sources witheigenvalues M1, M2 and M3 randomly distributed overthe lune is uniform. Obviously, if the moment tensorsare normalized using other than the Euclidean norm, thePDF of sources in the Riedesel–Jordan plot will be non-uniform, for example, when points uniformly distribut-ed on the cube are projected on the sphere.

As discussed in Section 5.2, the coordinate sys-tem in the decomposition with the Euclidean normmeets difficulties. The ISO and DC axes lie in the

source-type space, but the CLVD axis (with theCLVD’s major dipole oriented along the N axis) liesoutside this space. As a consequence, the CLVDdoes not lie in the source-type plot and the CLVDscale factor can never attain a value of ±1 but onlyof ±1/4 at the most. Another difficulty with thisdecomposition mentioned in Section 5.2 is that thetensile crack is characterized by a non-zero DCcomponent.

The above-mentioned problems with the CLVDscaling can be removed if the CLVD axis in theRiedesel–Jordan plot is rescaled by the multiplicationfactor of 4. Consequently, base tensors ECLVD

+ or ECLVD−

defined in Eq. (5) will attain values of CCLVD=±1,similarly as in the standard decomposition. In addition,it might appear to be more practical to stretch theCLVD axis and to transform the lune plot into acircular plot.

)b)a

)d)c ISO

CLVD-

ISO

CLVD

ISO

CLVD

ISO

CLVD

Fig. 9 Distribution of random sources in four different source-type plots. a The diamond CLVD–ISO plot using the standarddecomposition (see Section 3.1), b the diamond CLVD–ISO plotusing the simplified decomposition (see Section 5.1), c theHudson’s skewed diamond plot (see Section 6.1), and d the

Riedesel–Jordan plot (see Section 6.2). The dots show 3,000sources defined by the moment tensors with randomly generatedcomponents in the interval from −1 to 1. The distribution ofsources is non-uniform for all source-type plots

J Seismol (2015) 19:231–252 247

6.3 Analysis of moment tensor uncertainties usingsource-type plots

Source-type plots are often used for assessing uncer-tainties of the ISO, DC and CLVD components ofmoment tensors. In this section, we compare severalsource-type plots and discuss how the errors areprojected into them.

The idea behind using the source-type plots forassessing errors is simple. The moment tensor is notplotted just as one point corresponding to an optimumsolution but as a cluster of acceptable solutions. The sizeof the cluster measures uncertainties of the solution.Obviously, this approach is reasonable if the same sizeof a cluster corresponds to the same uncertainties inde-pendently of the position of the cluster in the source-type plot. Most seismologists assume that this propertyis uniquely satisfied in the skewed diamond plot because

randomly generated eigenvalues cover uniformly thisplot (Fig. 7d). However, this is not quite true for tworeasons. Firstly, if the appropriate norm of eigenvalues isapplied, the other source-type plots also display a uni-form PDF for randomly generated eigenvalues (seeSections 6.1 and 6.2). Secondly, when the uncertaintiesof the individual moment tensor components areanalysed, the moment tensor is not in the diagonal form.After diagonalizing the moment tensor, the errors areprojected into the errors of eigenvalues in a rather com-plicated way. This is demonstrated in Fig. 9, whichshows sources with moment tensors having all six inde-pendent components randomly distributed with a uni-form probability in the interval from −1 to 1. The figureindicates that some source types are quite rare, in par-ticular, sources with high explosive or implosive com-ponents. This observation is common for all source-typeplots and confirms the well-known fact that the

)b)a

)d)c ISO

CLVD-

ISO

CLVD

ISO

CLVD

ISO

CLVD

Fig. 10 Distribution of pure DC sources contaminated by randomnoise in four different source-type plots. a The diamond CLVD–ISO plot using the standard decomposition, b the diamond CLVD–ISO plot using the simplified decomposition, c the Hudson’sskewed diamond plot, and d the Riedesel–Jordan plot. The dots

show 1,000 DC sources defined by the elementary tensorEDC (seeEq. 4) and contaminated by noise with a uniform distributionbetween −0.25 and 0.25. The noise is superimposed to all tensorcomponents

248 J Seismol (2015) 19:231–252

distribution of eigenvalues for random matrices is notuniform (Mehta 2004).

More realistic sources are modelled in Figs. 10 and11: the pure DC and ISO sources defined by tensorsEDC

and EISO using Eq. (5) are contaminated by randomnoise with a uniform distribution in the interval from−0.25 to 0.25. The noise is superimposed to all sixtensor components, and 1,000 random moment tensorsare generated. As expected, the randomly generatedmoment tensors form clusters, but their shape is differ-ent for different projections, and their size depends alsoon the type of source. For the DC source (Fig. 10), themaximum PDF is in the centre of the cluster whichcoincides with the position of the uncontaminatedsource. In the diamond CLVD–ISO plot (Fig. 10a) andHudson’s skewed diamond plot (Fig. 10c), the cluster isasymmetric, being stretched along the CLVD axis. Amore symmetric cluster is produced in the diamondCLVD–ISO plot (Fig. 10b) when the simplified decom-position is applied (see Section 5.1) and in the Riedesel–Jordan plot (Fig. 10d). However, the symmetry of the

cluster is apparent in the Riedesel–Jordan plot becausethe CLVD and ISO axes are of different lengths. Asignificantly higher scatter of the CLVD componentscompared to the ISO components in the moment tensorinversions has been observed and discussed also inVavryčuk (2011). For the pure ISO source (Fig. 11),the clusters are smaller than for the DC source, and themaximum PDF is outside the position of the uncontam-inated source. This means that the ISO percentage issystematically underestimated due to the errors of theinversion for highly explosive or implosive sources.This is interesting and should be taken into accountwhen interpreting real observations.

Finally, we conclude that the comparison of varioussource-type plots did not prove a clear preference for theHudson skewed diamond plot or for any other plot inerror interpretations. A uniform probability of momenteigenvalues in the source-type plots does not provideany specific advantage in the error analysis. Momenttensors displaying the same uncertainties can projectinto clusters of a different size.

)b)a

)d)c ISO

ISO

CLVD

ISO

CLVD

ISO

CLVD-

DCDC

CDCD

CLVD

Fig. 11 The same as Fig. 10 but for the pure ISO source contaminated by random noise

J Seismol (2015) 19:231–252 249

7 Discussion

The standard decomposition into ISO, DC and CLVDcomponents described in Section 3 and its modificationproposed in Section 5.1 are the only decompositionswhich define the base vectors (tensors) within the source-type space. This property implies that the individual partsof the decomposed moment tensor are uniquely definedand can exist as individual source types. Consequently,the base vectors are mapped inside the source-type plot.Since the source-type space is a wedge in 3-D space, thebase vectors of the source-type space cannot form anorthogonal basis.

The base vectors of other decompositions such as thedecomposition into the ISO and three DCs, the ISO andthe major and minor DCs or the ISO and two DCs withthe same T axes do not form a vector basis inside thesource-type space. This is somewhat inconvenient be-cause some of the decomposed parts of the momenttensors are not defined in the source-type space and thuscannot exist independently as individual types of source.This also applies to the approach of Chapman andLeaney (2012), who proposed the ISO, DC and CLVDdecomposition using the Euclidean norm and the CLVDwith the major dipole oriented along the N axis. As aconsequence, relative scale factor CCLVD

∗ can neverattain the value of ±1 for any moment tensor but only±1/4. This decomposition has also some other unde-sirable properties, for example, pure tensile faulting inisotropic media yields a non-zero DC component. Thisis against physical intuition since the DC is usuallyconsidered to be a measure of the amount of shearfaulting in seismic sources.

In analysing the general moment tensors, we facethe following key problems: how to define the scalarseismic moment and what is the most appropriatenorm of the moment tensor in the moment tensordecompositions. The standard decomposition de-fines the scalar moment as the sum of spectral normsof the individual moment tensor components. Inprinciple, the scalar moment could be defined alsoas the spectral norm of complete moment tensor M.However, this definition introduces difficulties in themoment tensor decomposition because it can pro-duce the sum of the ISO, DC and CLVD relativescale factors higher than 1 (provided the ISO andCLVD parts are of opposite signs). Another possi-bility is using the Euclidean norm for defining thescalar moment and consequently for decomposing

the moment tensors. However, applying the Euclid-ean norm is also tricky. This norm is appropriate fororthogonal spaces, for example, when working inthe 6-D space of full moment tensors. But thesource-type space of ordered eigenvalues of momenttensors is a wedge of the full 3-D space, and thus itis not orthogonal.

Different approaches to the moment tensor decom-position project into constructing various source-typeplots used in interpretations. We show that the Hudsonskewed diamond plot can be introduced in a muchsimpler way than originally proposed by Hudson et al.(1989). The plot can readily be constructed using theISO and CLVD components scaled to the spectral normof moment tensor M. The plot conserves the uniformdistribution probability of moment eigenvalues, but theanalysis indicates that it does not provide any essentialadvantage for error interpretations. Interestingly, alsoother source-type plots such as the diamond plot or theRiedesel–Jordan lune plot conserve the uniform distri-bution probability of moment eigenvalues if the appro-priate norm is adopted. But again, this provides no clearbenefit in the error analysis because the errors of theeigenvectors and eigenvalues of the moment tensorscannot be easily separated. Hence, the source-type plotsshould be viewed as a rather simple tool for visualizingthe basic classification of sources. A detailed and accu-rate analysis of seismic sources should always be per-formed using the complete moment tensors. For thesereasons, the most straightforward and comprehensibleapproach for the first source-type interpretations is prob-ably the simple diamond CLVD–ISO plot.

Acknowledgments I wish to thank Jan Šílený and an anony-mous reviewer for their detailed and constructive comments whichhelped to improve the paper significantly. The study was support-ed by the Grant Agency of the Czech Republic, Grant No.P210/12/1491.

Appendix A. Calculation of moment tensor Mfrom the scalar moment M and the ISO, DCand CLVD scale factors

The eigenvalues of moment tensor M are calculatedfrom scale factors CISO, CDC, CCLVD and scalar seismicmoment M defined in Eqs. (6–10) using the followingsimple formulas:

250 J Seismol (2015) 19:231–252

for CCLVD ≥0

M 1 ¼ M CISO þ CDC þ CCLVDð Þ;

M 2 ¼ M CISO−1

2CCLVD

� �;

M 3 ¼ M CISO−CDC−1

2CCLVD

� �;

ðA1Þ

and for CCLVD <0

M 1 ¼ M CISO þ CDC−1

2CCLVD

� �;

M 2 ¼ M CISO−1

2CCLVD

� �;

M 3 ¼ M CISO−CDC þ CCLVDð Þ:

ðA2Þ

Appendix B. The ISO, DC and CLVDdecompositionof the source tensor

Source tensor D of the shear-tensile model

D ¼ uS

2nsþ snð Þ

¼ uS

2

2n1s1 n1s2 þ n2s1 n1s3 þ n3s1n1s2 þ n2s1 2n2s2 n2s3 þ n3s2n1s3 þ n3s1 n2s3 þ n3s2 2n3s3

24

35ðB1Þ

has the following diagonal form (see Vavryčuk 2005, hisAppendix B):

Ddiag ¼ uS

2

n⋅sþ 1 0 00 0 00 0 n⋅s−1

24

35; ðB2Þ

where n ·s is the scalar product of two unit vectors nand s. Vector n is the fault normal, and s is thedirection of the slip vector. The maximum eigenvalueD1 ¼ uS

2 n⋅sþ 1ð Þ is positive or zero, the minimum

eigenvalue D3 ¼ uS2 n⋅s−1ð Þ is negative or zero.

Using Eqs. (6–10) we obtain

DISO ¼ 1

3D1 þ D2 þ D3ð Þ ¼ 1

3uS n⋅sð Þ; ðB3Þ

DCLVD ¼ 2

3D1 þ D3−2D2ð Þ ¼ 2

3uS n⋅sð Þ; ðB4Þ

DDC ¼ 1

2D1−D3− D1 þ D3−2D2j jð Þ

¼ 1

2uS 1− n⋅sj jð Þ; ðB5Þ

D ¼ 1

2DISOj j þ DCLVDj j þ DDCð Þ

¼ 1

2uS 1þ n⋅sj jð Þ; ðB6Þ

so the scale factors CISO, CDC and CCLVD read

CISO

CCLVD

CDC

24

35 ¼ 1

3 1þ n⋅sj jð Þ2n⋅s4n⋅s

3 1− n⋅sj jð Þ

24

35: ðB7Þ

It follows from Eqs. (B3-B4) that the ISO/CLVDratio is 1/2 independently of the slope angle α, calculat-ed as sinα=n⋅s (see Vavryčuk 2011).

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