A geometric setting for moment tensors

Geophysical Journal International Geophys. J. Int. (2012) 190, 476–498

doi: 10.1111/j.1365-246X.2012.05491.x

A geometric setting for moment tensors Walter Tape1 and Carl Tape2 1 Department 2 Geophysical

of Mathematics, University of Alaska, Fairbanks, Alaska 99775, USA Institute and Department of Geology & Geophysics, University of Alaska, Fairbanks, Alaska 99775, USA . E-mail: [email protected]

Accepted 2012 March 30. Received 2012 March 16; in original form 2011 December 5

SUMMARY We describe a parametrization of moment tensors that is suitable for use in algorithms for moment tensor inversion. The parameters are conceptually natural and can be easily visualized. The ingredients of the parametrization are present in the literature; we have consolidated them into a concise statement in a geometric setting. We treat several familiar moment tensor topics in the same geometric setting as well. These topics include moment tensor decompositions, crack-plus-double-couple moment tensors, and the parameter that measures the difference between a deviatoric moment tensor and a double couple. The geometric approach clarifies concepts that are sometimes obscured by calculations. Key words: Theoretical seismology.

GJI Seismology

1 I N T RO D U C T I O N The seismic moment tensor is a mathematical description for earthquake sources. A moment tensor can represent a single earthquake, as in the case of a small-to-moderate event, or it can represent a subevent within a large, long-duration earthquake. Moment tensors range qualitatively from the classical double couple, which is associated with relative slip along a fault plane, to ‘non-deviatoric’ moment tensors that are associated with events other than conventional earthquakes. These events include those in volcanic and geothermal regions (e.g. Julian & Sipkin 1985; Foulger et al. 2004; Minson et al. 2007), as well as nuclear explosions, mine collapses and events associated with glaciers or hydraulic fracturing (Dreger & Woods 2002; Ford et al. 2009; Walter et al. 2009; Baig & Urbancic 2010). See also the reviews by Julian et al. (1998) and Miller et al. (1998). Computationally a moment tensor is a 3 × 3 symmetric matrix that describes an earthquake source as a sum of three force couples. Recorded seismic waveforms can be modelled as a linear combination of derivatives of Green’s functions, which describe the seismic wavefield at a station due to an impulsive source for a given structural model. The coefficients of the linear combination are the six independent entries of the moment tensor matrix. An inverse problem can be constructed in which measurements between recorded and simulated seismograms are used to estimate a moment tensor for a particular earthquake (e.g. Dziewonski et al. 1981). Non-deviatoric events have been recorded by seismometers. Their moment tensors have been inferred using ‘full’ moment tensor inversion algorithms that place no constraints on the six elements of the moment tensor (e.g. Dufumier & Rivera 1997; Minson & Dreger 2008). The unknown model vector describing the moment tensor has six elements, and therefore the model parameter space is six-dimensional. The parameters need not, however, be the matrix entries. In this paper we do two things:

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(i) Specifically: We define a parametrization of moment tensors that is conceptually natural. Although we perform no inversions in this study, we envision the parametrization as the basis for samplingbased, non-linear moment tensor inversions. Associated with the parametrization is a natural scheme for displaying the results of moment tensor inversions—both the moment tensors themselves and the uncertainties associated with full moment tensor inversions. The parametrization is given in Proposition 3 of Section 7. (ii) More generally: We present familiar concepts of moment tensors in a conceptual setting that complements calculations with pictures whenever feasible. Moment tensors are drawn as beachballs. Beachball patterns (moment tensor source types) are depicted on what is essentially the lune of Riedesel & Jordan (1989), and beachball orientations are depicted in a solid block. Any moment tensor is then determined by a pair of points—one in the lune and one in the block—together with a positive number that gives its norm. In this way we get a coherent picture of all moment tensors. We use several topics to illustrate this geometric approach. In Section 4 we consider moment tensor decompositions, and in Section 9 we consider crack-plus-double-couple moment tensors. In Section 8 we consider the parameter that measures how much an earthquake source departs from the classic model of slip along a fault, and we suggest the longitudinal coordinate γ on the lune as an alternative to . More generally, in Tape & Tape (2012) we favour the lune as an alternative to the Tk source-type plots of Hudson et al. (1989). In Section 10 we use the lune to plot source types from several recent studies. Many authors, as we do, have organized moment tensors according to eigenvalues (moment tensor source types), or eigenvectors (moment tensor orientations), or both (e.g. Hudson et al. 1989; Pearce & Rogers 1989; Riedesel & Jordan 1989; Frohlich 1992, 2001; Dufumier & Rivera 1997; Julian et al. 1998; Kagan 2005; C 2012 The Authors C 2012 RAS Geophysical Journal International

A geometric setting for moment tensors

Figure 1. Home for beachball patterns (moment tensor source types)—the fundamental lune L of the unit sphere. Each point on the lune represents a beachball pattern. The magenta equatorial arc is for deviatoric tensors, the red meridian is for sums of double couple and isotropic tensors, and the black arc is for crack + double couple tensors having Poisson ratio ν = 1/4 (Section 9). Above the upper blue arc all beachballs are only red, and below the lower blue arc all balls are only white. ISO, isotropic; DC, double couple; LVD, linear vector dipole; CLVD, compensated linear vector dipole; C, tensile crack with Poisson ratio ν = 1/4.

Minson & Dreger 2008; Sileny & Milev 2008; Walter et al. 2010; Chapman & Leaney 2012). For representing source types, Riedesel & Jordan (1989) proposed what is essentially our Fig. 1. Chapman & Leaney (2012) recently compared the source-type plots of Riedesel and Jordan with the Tk plots of Hudson et al. (1989). As we do, Chapman and Leaney prefer the plots of Riedesel and Jordan; that is, they prefer the representation on the sphere.

2 L I N E A R T R A N S F O R M AT I O N S , B E A C H B A L L S A N D λ- S PA C E A moment tensor M is a certain kind of linear transformation of three-space, and it can therefore be thought of as a vector field. That is, at each point e in space it attaches an arrow M(e). In particular, M attaches an arrow to each point of the unit sphere. Where the arrows point outwards, the sphere is coloured red (say), and where the arrows point inwards, the sphere is coloured white. Except for scale, this is the beachball associated with M. The moment tensor determines the beachball. Mathematically, moment tensors are exactly the self-adjoint linear transformations. They are the transformations that have three mutually perpendicular eigenvectors. The associated principal axes are in the directions of the eigenvectors, but they are lines rather than vectors. Each principal axis is associated with an eigenvalue λ, where, for any point e on the principal axis, M(e) = λ e. Along each principal axis the attached arrows are therefore radial, with the sign of λ determining whether the arrows point directly to or directly away from the origin. The principal axes and their corresponding eigenvalues determine the moment tensor. The beachball is a device for keeping track of the principal axes and their eigenvalues, and hence of the moment tensor itself. If the C

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Figure 2. Beachball determined by the point = (λ1 , λ2 , λ3 ) = (1, −1, 0) and by the frame vectors (eigenvectors) u1 (red), u2 (blue) and u3 (yellow). The numbers λ1 , λ2 , λ3 are the eigenvalues of the beachball’s moment tensor that correspond to u1 , u2 , u3 . Since λ1 > 0, the beachball is coloured red in the direction of u1 . Since λ2 < 0, the beachball is coloured white in the direction of u2 . The brown planes are the coordinate planes λ1 = 0, λ2 = 0, λ3 = 0.

eigenvalues do not all have the same sign and are distinct, then the symmetry of the beachball reveals the principal axes, and the colouring pattern determines the corresponding eigenvalues, except for a scale factor. The principal axes can be specified by giving a (right-handed) frame of orthonormal vectors u1 , u2 , u3 . For many purposes the frame and the principal axes are nearly interchangeable, but since the principal axes are considered to be only undirected lines, then each ordered triple of principal axes corresponds to four frames. At times the distinction between frames and principal axes will therefore need to be clear. So a moment tensor is determined by giving a frame u1 , u2 , u3 to serve as the eigenvectors, together with three numbers λ1 , λ2 , λ3 to serve as the corresponding eigenvalues. The eigenvectors give the orientation of the beachball, whereas the eigenvalues determine the pattern on the ball and its size. The triple = (λ1 , λ2 , λ3 ) can be thought of as a point in ‘λ-space’. Each point and each triple u1 , u2 , u3 together determine a moment tensor, that is, a beachball (Fig. 2). At a typical point there are infinitely many beachballs, all with the same pattern and size but with different orientations. A change in normally changes at least one of the pattern and size; the result is a new infinite collection of beachballs, all with the new pattern and size. All moment tensors appear this way. See Fig. 3. Fig. 4 shows some standard moment tensors in λ-space. The deviatoric beachballs are those with λ1 + λ2 + λ3 = 0. They are on the ‘deviatoric plane’ that contains the origin and that has normal vector in the 111 direction—the isotropic axis. The double couples are the deviatoric beachballs that have determinant zero, that is, λ1 λ2 λ3 = 0, so they lie on the lines in space that are the intersections of the λ-coordinate planes with the deviatoric plane. The compensated linear vector dipoles (CLVDs; Knopoff & Randall 1970) are the beachballs that are deviatoric and that have two equal eigenvalues.

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W. Tape and C. Tape 2.1 Conventions A given moment tensor corresponds to just one linear transformation, but it has many matrix representations. We will write moment tensors as matrices, but the matrices will always be with respect to the ‘standard basis’. For much of this paper, the standard basis can be chosen arbitrarily. However, if strike, slip, and dip in eqs (27) are to have their conventional meanings, then the standard basis must be north-west-up. When specifying points on the unit sphere, we will sometimes give their directions from the origin rather than the normalized points themselves. Thus the point referred to as (1, −2, 1) is really √1 (1, −2, 1). 6 Seismologists usually depict beachballs as viewed from above, and with the top hemispheres removed. We do not do that; we want to be able to show whole beachballs from various perspectives in three dimensions. Appendix D has a glossary of notation.

√ Figure 3. Beachballs at 1 = 31(1, −1, 0) and 2 = (1, −6, 5). The colouring patterns are determined as explained in Fig. 2. As before, the red, blue and yellow arrows on the beachballs indicate the frame vectors u1 , u2 , u3 , respectively. The patterns at 1 are all the same, with any two of the beachballs there differing by a rotation. The same is true at 2 .

3 A F I R S T PA R A M E T R I Z AT I O N O F MOMENT TENSORS A square matrix U is orthogonal if UU T = I. It is a rotation matrix if it is orthogonal and det U = 1. Orthogonal matrices preserve lengths and angles. Rotation matrices in addition preserve handedness. We use U to denote the group of rotation matrices. For = (λ1 , λ2 , λ3 ) and U a rotation matrix, we let ⎛ ⎞ λ1

⎜ [] = ⎝ 0 0

0

0

λ2

0 ⎠,

0

λ3

⎟

(1a)

[]U = U []U −1 .

(1b)

(More generally, if A is a moment tensor and U is a rotation matrix, then AU = U A U −1 .) As in Fig. 5, the matrix [] describes a beachball with eigenvalues λ1 , λ2 , λ3 and with corresponding eigenvectors in the x, y, z coordinate directions. The matrix []U describes the same beachball after being rotated by the matrix U. As a purely physical object, the beachball would be transformed by applying U to each of its points. But as a moment tensor—an operator—the beachball is transformed by conjugation by U, as in eq. (1b).

z z

y Figure 4. Some standard moment tensors—a double couple (DC), two CLVDs and a positive isotropic tensor (ISO). The double couple and the two CLVDs are in the deviatoric plane (purple, λ1 + λ2 + λ3 = 0). Double couples are at the intersections of the deviatoric plane with the coordinate planes. Isotropic tensors are on the isotropic axis (blue, normal to the deviatoric plane). Principal axes p1 , p2 , p3 are shown on the double couple.

From Fig. 3 it should be clear that our λ-space need have no particular relation with ordinary physical space. This differs from Riedesel & Jordan (1989); their analogues of our λ1 , λ2 , λ3 axes are in the direction of the T, N and P axes of a moment tensor that they are depicting.

y

x

[Λ]

x

[Λ]U

Figure 5. A first parametrization []U of moment tensors. The matrix [] describes the beachball with eigenvalues = (λ1 , λ2 , λ3 ) and with corresponding eigenvectors in the x, y, z coordinate directions. The matrix []U describes the same beachball after being rotated by the matrix U. Here = (0, 1, −1) and U is a 20◦ rotation about the x-axis. C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


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Proposition 1. Let = (λ1 , λ2 , λ3 ) and let U be a rotation matrix. Then M = []U is the moment tensor whose eigenvalues are λ1 , λ2 , λ3 and whose corresponding eigenvectors are the columns of U. Proof . M(U ei ) = U []U −1 (U ei ) = λi U ei , where e1 , e2 , e3 is the standard basis for R3 . Thus U e1 , U e2 , U e3 —the columns of U—are eigenvectors of M with eigenvalues λ1 , λ2 , λ3 . Since U is a rotation then U e1 , U e2 , U e3 are orthonormal and hence M is a moment tensor. The notation []U emphasizes that the moment tensor is given by and U. The point determines the pattern and size of the beachball, and U gives the orientation. The rotation matrix U is the (eigen-) frame associated with the representation []U , and the columns of U—the eigenvectors—are the frame vectors.

4 MOMENT TENSOR DECOMPOSITIONS 4.1 Closest double couple If some moment tensors all happen to have the same principal axes, then their frames can be chosen to be the same as well. In that case the distances and angles between their eigenvalue triples are the same as distances and angles between the corresponding moment tensors. This section has some examples. Suppose we wish to find the closest double couple MDC to the given moment tensor M = [(2, 1, −3)]. ‘Close’ means close in R9 . That is, the space M 3×3 of 3 × 3 matrices is essentially R9 and hence has its own scalar product with associated distances and angles. So the distance between M and MDC makes sense. In λ-space the double couples occupy the six black spokes in Fig. 6. Although at each point on a spoke there are infinitely many moment tensors, all with different orientations, it turns out that in looking for MDC we need only consider one moment tensor at each point, namely, the one that has the same frame as M. And because the frames are the same, distances in λ-space are the same as distances in M 3×3 . The desired MDC can therefore be seen by inspection of the figure, and it can be calculated either by trigonometry or, better, using eq. (7). The result is MDC = [(5/2, 0, −5/2)]. What makes this simple is Proposition 8 of Appendix C. It guarantees that the closest double couple to M will have the same frame as M.

4.2 Lengths and angles for moment tensors To be more explicit about scalar products of matrices, define the function F : M 3×3 → R9 by ⎛ ⎞ a11

⎜ F ⎝ a21 a31

a12

a13

a22

a23 ⎠ = (a11 , a12 , a13 , a21 , a22 , a23 , a31 , a32 , a33 ).

a32

a33

⎟

(2)

Figure 6. Finding the closest double couple MDC to M = [(2, 1, −3)]. The candidates for MDC are the moment tensors that have the same frame as M and that are located on the six black spokes in λ-space. Thus on the spoke in the (0, 1, −1) direction, for example, the candidates would have the form s[(0, 1, −1)], s > 0. Among all spokes the only serious candidates would be at the three red dots, and, of those, the indicated MDC is clearly closest to M. Distances between points in λ-space give correct distances between the corresponding moment tensors because the moment tensors have the same frames. The λ-space is oriented so that we are looking √ down on the deviatoric plane (grey), which is shown as a disc of radius 14 = M. All six black spokes are in the deviatoric plane.

scalar product in M 3×3 . The M 3×3 scalar product has the expected properties. Also AU · BU = A · B,

U ∈ U,

A, B ∈ M 3×3 .

(4)

Therefore, in the notation of eq. (1b), [1 ]U · [2 ]U = 1 · 2

(5a)

∠ [1 ]U , [2 ]U = ∠(1 , 2 )

(5b)

[1 ]U − [2 ]U = 1 − 2

(5c)

[]U = .

(5d)

Thus if two moment tensors have the same frame U, then the angle and distance between them can be correctly depicted in λ-space.

1

Then the scalar product of matrices is defined by A · B = F(A) · F(B),

A, B ∈ M 3×3 ,

4.3 Decompositions (3)

where the dot on the right-hand side is the ordinary scalar product in R9 . Then lengths and angles in M 3×3 are defined in terms of the

1

In the continuum mechanics literature this scalar product is often written as A : B.

C


Treatments of moment tensors (e.g. Wallace 1985; Jost & Herrmann 1989) often ‘decompose’ moment tensors, that is, they write them as sums of other moment tensors. When the component tensors are required to have the same frame as the given tensor, the decompositions can be depicted in λ-space. Although the decompositions given below are well known, the accompanying figures may nevertheless be useful.

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W. Tape and C. Tape

If M = []U (eq. 1b), where = 1 + 2 , then M = [1 + 2 ]U = [1 ]U + [2 ]U = M1 + M2 .

(6)

Thus each decomposition = 1 + 2 of the vector gives rise automatically to a decomposition M = M 1 + M 2 . (See, however, Section 4.3.2.) But it is important to realize that M, M 1 and M 2 all have the same principal axes, since U is the same for all three. Thus although we can generate moment tensor decompositions with abandon, using eq. (6), the decompositions are very special. Section 9 has examples of moment tensor decompositions in which the principal axes for the components are not the same as for the given moment tensor. The remainder of this section (Section 4.3) has examples of moment tensor decompositions based on eq. (6). It is enough just to give the decomposition of the vector . We will loosely refer to as the moment tensor itself, whereas in fact it is only the triple of eigenvalues. Let pr be the vector projection of on . That is, pr =

· . 2

(7)

Then the decomposition of into its deviatoric and isotropic parts DEV and ISO is = pr(1,1,1) + ( − pr(1,1,1) ) =

tr tr (1, 1, 1) + − (1, 1, 1), 3 3 ISO

(8)

DEV

where tr (trace) is λ1 + λ2 + λ3 . The deviatoric part DEV can be further decomposed into its closest double couple and complement. Equivalently, the original can be decomposed into its components on the isotropic axis, the closest double couple axis, and the CLVD axis that is orthogonal to both: = pr(1, 0, −1) + pr(−1, 2, −1) + pr(1, 1, 1) , DC

CLVD

ISO

(λ1 > λ2 > λ3 ). (9)

As a vector identity, eq. (9) is correct for all . The restriction λ1 > λ2 > λ3 is added so that the double couple component pr(1, 0, −1) is the closest double couple to , and so that the corresponding moment tensor decomposition is well defined, as explained in Section 4.3.2. Fig. 7 illustrates both eqs (8) and (9).

For the decomposition using the near minor DC (Fig. 8b), the analogue of eqs (11) is similar. We omit it. DC and CLVD. For a decomposition into a double couple and CLVD, we take the axis for the double couple component 1 to be the closest DC axis to . Then, as with the decomposition into two double couples, 2 must lie on the line through and parallel to that axis. That leaves three choices for the CLVD component 2 : the ‘far’ CLVD, the ‘near’ CLVD and the ‘orthogonal’ CLVD, shown in Figs 8(c)–(e). If 2 is to be the orthogonal CLVD component (Fig. 8e), then =

λ1 − λ3 λ2 (1, 0, −1) + (−1, 2, −1), 2 2 DC = closest DC

The decompositions in this section are stated for = (λ1 , λ2 , λ3 ) in the fundamental sector D of the deviatoric plane, or in its interior D0 : D = { : λ1 + λ2 + λ3 = 0, λ1 ≥ λ2 ≥ λ3 }, (10)

Two double couples. Suppose (deviatoric) is to be decomposed into ‘major’ and ‘minor’ double couples 1 and 2 . There are three DC axes—lines through the origin and a double couple—with the ‘major’ DC axis being (in our terminology) the closest of the three to . If the point 1 is required to lie on the major DC axis, then 2 must lie on the line through and parallel to that axis, as in the diagrams of Fig. 8. That leaves two possibilities for the double couple 2 ; it can be the ‘near’ (to ) or the ‘far’ (from ) minor double couple.

( ∈ D0 )

(12)

orthogonal CLVD

In eq. (12), the magnitude M 0 = (λ1 − λ3 )/2 of DC is familiar from Dziewonski & Woodhouse (1983). Eq. (12) is the special case of eq. (9) in which is deviatoric. If 2 is the near CLVD component (Fig. 8d), then ⎧ (2λ1 + λ3 )(1, 0, −1) + λ2 (1, 1, −2) ( ∈ D, λ2 ≥ 0) ⎪ ⎪ ⎪ ⎨ =

DC

near CLVD

⎪ (λ1 + 2λ2 )(1, 0, −1) + (−λ2 )(2, −1, −1) ( ∈ D, λ2 < 0). ⎪ ⎪ ⎩ DC near CLVD (13)

The equations for the case where 2 is the far CLVD component (Fig. 8c) are similar, but the domain for is D0 rather than D. The equations are omitted. Three double couples. Ben-Menahem & Singh (1981) and Jost & Herrmann (1989) give a decomposition of a deviatoric into three double couples. That is, is written as a linear combination of (1, 0, −1), ( − 1, 1, 0) and (0, −1, 1). Since the three double couples are necessarily linearly dependent, there are infinitely many choices for the coefficients in the linear combination, and the decomposition is far from unique. Thus = t(1, 0, −1) + (t − λ1 )(−1, 1, 0) + (t + λ3 )(0, −1, 1), DC

4.3.1 Decomposition of deviatoric tensors

D0 = { : λ1 + λ2 + λ3 = 0, λ1 > λ2 > λ3 }.

In general, if = (λ1 , λ2 , λ3 ) is deviatoric and λ1 > λ2 > λ3 , then the decomposition using the far minor DC is (Fig. 8a) ⎧ (−λ3 )(1, 0, −1) + λ2 (−1, 1, 0), ( ∈ D0 , λ2 ≥ 0) ⎪ ⎪ ⎪ ⎪ ⎨ major DC far minor DC = ⎪ ( ∈ D0 , λ2 < 0). λ1 (1, 0, −1) + (−λ2 )(0, −1, 1), ⎪ ⎪ ⎪ ⎩ (11) major DC far minor DC

DC

(λ1 , λ2 , λ3 distinct)

DC

(14)

where t is arbitrary. For Jost & Herrmann, t = (λ1 − λ3 )/3. The decompositions into two double couples (Figs 8a and b) are special cases of eq. (14) where one of the coefficients is zero. From the examples above, it is clear that it is trivial to concoct decompositions. The substantive part comes in providing physical rationales for them (e.g. Frohlich 1994; Julian et al. 1998; Miller et al. 1998). And we emphasize again that the decompositions of this section are special in that the component moment tensors have the same principal axes as the given tensor. 4.3.2 A subtle point Eq. (6) contains more than meets the eye. For a given moment tensor M with eigenvalue triple = 1 + 2 , the equation appears to give a decomposition of M into moment tensors M 1 and M 2 having C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


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Figure 7. Decompositions = DEV + ISO (eq. 8) and = DC + CLVD + ISO (eq. 9). The grey rectangle is in the deviatoric plane, and its edges ¯ 1¯ (CLVD) directions. The volumes of the beachballs are proportional to their norms. We are referring to the moment are parallel to the 101¯ (DC) and 12 tensors in terms of their s, but note that all of the moment tensors have the same principal axes. = (3, 2, −2), DEV = (2, 1, −3), ISO = (1, 1, 1), DC = (5/2, 0, −5/2) and CLVD = (−1/2, 1, −1/2). The orientation of the sphere is the same as that of the decomposition diagram; the deviatoric plane (grey) is horizontal, and the isotropic axis—the 111 direction—is vertical.

eigenvalue triples 1 and 2 . But a frame U is involved, and if M can be written in the form []U in more ways than one, then M 1 and M 2 might not be well-defined functions of M. If M = []U = []V , with = 1 + 2 , and if the entries of are distinct, then V = UW for some W = X π , Y π , Z π or I, by Proposition 5 of Appendix A. And then [1 ]V = [1 ]U W = [1 ]U ; we get the same M 1 from eq. (6) whether using U or V . And similarly for M 2 . The moment tensors M 1 and M 2 are well-defined functions of M. By way of contrast, consider the decomposition of the CLVD = (1, 1, −2) into its DC and orthogonal CLVD components 1 and 2 (eq. 12). We attempt to mimic the argument of the previous paragraph, with M = []U = []V and = 1 + 2 . Because has repeated entries, this time we can conclude only that V = UW for some W in Cz , the group of rotational symmetries of a vertical cylinder (Proposition 6 of Appendix A). The rotation W , for example, might be Z π/6 , a thirty degree rotation about the z-axis. In that case, V = UZ π/6 = [UZ π/6 U −1 ]U, so that the frame V would be the result of rotating the frame U thirty degrees about U e3 = V e3 , the main symmetry axis of the beachball for M; see Fig. 9. Now [1 ]U = [1 ]V and [2 ]U = [2 ]V , and thus neither M 1 nor M 2 is a well-defined function of M. Or one can take the point of view that there are now infinitely many moment tensor decompositions of M determined by the vector decomposition = 1 + 2 . The complication in this example is that the beachballs [1 ] and [2 ] do not have all of the symmetries possessed by the beachball []. Equivalently, the triple is more degenerate than 1 and 2 . In general, a necessary and sufficient condition that eq. (6) determine a well-defined decomposition of M is that 1 and 2 each be at least as degenerate as . That is, if two entries of coincide, then the corresponding entries of 1 should coincide, and the corresponding entries of 2 should coincide. (If the condition fails, one still gets a decomposition, but the decomposition depends on C


which of the infinitely many possible frames U for M is chosen.) If has distinct entries—by far the most common case—then the condition is automatically satisfied and hence the moment tensor decomposition is well defined. But in the Fig. 9 example above, = (1, 1, −2) had two equal entries, while 1 and 2 did not. 5 B E A C H B A L L PAT T E R N S 5.1 Permuting the eigenvalues As in Fig. 10, a typical moment tensor will appear at six points in λ-space, the points being permutations of each other. Each transposition of = (λ1 , λ2 , λ3 ) corresponds to a reflection in one of the mirror planes λ1 = λ2 , λ2 = λ3 , λ3 = λ1 , and the six points are therefore symmetrically disposed with respect to the planes. The mirror planes divide λ-space into six congruent infinite wedges, with each wedge corresponding to an eigenvalue ordering. Our ‘fundamental wedge’ W is W = { ∈ R3 : λ1 ≥ λ2 ≥ λ3 }.

(15)

The technicalities of the permutation of eigenvalues and of the associated permutation of eigenvectors are deferred to Appendix A. 5.2 Beachball patterns Informally, the beachball pattern is the moment tensor information that is neither orientation nor size. Precisely, we define the pattern ˆ of a moment tensor M by ˆ =

(λ1 , λ2 , λ3 ) = (λ21 + λ22 + λ23 )1/2

λ1 ≥ λ2 ≥ λ3 ,

(16)

where λ1 , λ2 , λ3 are the eigenvalues of M, ordered as indicated. Beachball patterns lie on the ‘fundamental lune’ L (Fig. 11) defined

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W. Tape and C. Tape

Figure 8. (a,b) Decompositions of a deviatoric into two double couples. (c–e) Decompositions of into a double couple and CLVD. The bottom diagram is the same as the deviatoric portion of Fig. 7. In all diagrams = (2, 1, −3).

by L = { ∈ R : λ1 ≥ λ2 ≥ λ3 , = 1}. 3

(17)

Thus L is the set of beachball patterns (moment tensor source types). ˆ = (λˆ 1 , λˆ 2 , λˆ 3 ) is the pattern for M and if ρ = M [same If as , by eq. (5d)], then ρ λˆ 1 , ρ λˆ 2 , ρ λˆ 3 are the eigenvalues of ˆ = ρ[] ˆ also has eigenvalues ρ λˆ 1 , ρ λˆ 2 , ρ λˆ 3 , then M. Since [ρ ] ˆ M = ρ[]U for some rotation matrix U. Thus the function ˆ U, ρ) → ρ[] ˆ U (,

ˆ ∈ L, U ∈ U, ρ > 0

(18)

parametrizes all moment tensors. It expresses each moment tenˆ its orientation U and its norm ρ. sor in terms of its pattern , The parametrization is more efficient than the parametrization ˆ 1, ˆ 2 ∈ L, (, U ) → []U implicit in Proposition 1, since if then ˆ 1 ]U = ρ2 [ ˆ 2 ]U ⇒ ρ1 = ρ2 and ˆ1 = ˆ 2. ρ1 [ 1 2

(19)

Moreover, there is at least an intuitive feeling that, for fixed ρ ˆ is uniformly distributed on the lune L, then ρ[] ˆ U is and U, if itself uniformly distributed. But without some restrictions on U the parametrization is not yet truly efficient (Section 6). When specifying patterns, we will often give them as directions from the origin rather than as normalized points. Thus the pattern referred to as (1, −1, 0) is really √12 (1, −1, 0). 5.3 Coordinates for beachball patterns The (deviatoric) longitude γ and colatitude β are natural coordinates for the fundamental lune L and hence for beachball patterns. They are ordinary spherical coordinates but with pole at (1, 1, 1) rather than (0, 0, 1). The coordinate γ ranges from −π /6 to π /6, and β ranges from 0 to π . See Fig. 11. The moment tensors with β = π /2 are the deviatoric tensors, and the deviatoric moment tensors with γ = 0 are the double couples. C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


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Figure 9. Decomposition of a deviatoric vector into its DC component 1 and orthogonal CLVD component 2 , similar to Fig. 8(e). Here, however, is (1, 1, −2), a CLVD. Even with the moment tensor M fixed, the decomposition M = M 1 + M 2 determined by eq. (6) depends on the frame chosen for M: The vectors , 1 , 2 and the moment tensors M are the same in both diagrams, but the frames U and V for M are different, and as a result the moment tensors M 1 and M 2 in the right-hand diagram differ from their counterparts in the left. Thus neither M 1 nor M 2 is a well-defined function of M alone. Instead, there are infinitely many moment tensor decompositions of M, all corresponding to the same vector decomposition = 1 + 2 . No such problem arises when has distinct entries. Here 1 = (3/2, 0, −3/2) and 2 = (−1/2, 1, −1/2). The frames U and V differ by a 30◦ rotation about u3 = v3 .

Together with the usual radial coordinate ρ, the coordinates γ and β are spherical coordinates for the fundamental wedge W (eq. 15). The rotation matrix that takes the (1, 1, 1) direction to the (0, 0, 1) direction and that takes the (1, 0, −1) direction to the (1, 0, 0) direction is √ ⎞ ⎛√ 3

1 ⎜ U = √ ⎝ −1 6 √

2

0

− 3

2 √ 2

⎟

−1 ⎠. √ 2

Hence λ-coordinates are found from γ , β, ρ coordinates by ⎛ ⎞ ⎛ ⎞ λ1

ρ cos γ sin β

⎜ ⎟ ⎟ T ⎜ ⎝ λ2 ⎠ = U · ⎝ ρ sin γ sin β ⎠.

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ρ cos β

λ3

And in the other direction,

Figure 10. Multiple appearances of the same beachball. A typical beachball will appear at six different points in λ-space, the six points being permutations of each other. As a result, the totality of moment tensors at any one of the six points would coincide with the totality of moment tensors at any of the others. Here λ-space is oriented so that we are looking down the 111 axis, with the deviatoric plane coinciding with the plane of the paper. The green lines are the mirror planes λ1 = λ2 , λ2 = λ3 , λ3 = λ1 , seen edge-on. Reflection in any one of them corresponds to interchanging two of the entries in (λ1 , λ2 , λ3 ). The red, blue and yellow lines emanating from each beachball are the first, second and third √ principal axes, respectively. The radius of the sphere is (5, 1, −6) = 62.

Deviatoric moment tensors with γ > 0 have two positive eigenvalues; they are the beachballs with white caps and a red band. Those with γ < 0 have two negative eigenvalues; they are the beachballs with red caps and a white band. The moment tensors with β = 0 are the positive isotropic tensors, and the ones with β = π are the negative isotropic tensors. To get a uniform distribution of patterns on the lune, we can let b = cos β and then take a uniform distribution of points (γ , b) on the rectangle B defined by −π /6 ≤ γ ≤ π /6, −1 ≤ b ≤ 1. C


tan γ =

−λ1 + 2λ2 − λ3 , √ 3(λ1 − λ3 )

(21a)

cos β =

λ1 + λ2 + λ3 , √ 3

(21b)

ρ = . If M is deviatoric then eq. (21a) reduces to √ 3 λ2 tan γ = (λ1 + λ2 + λ3 = 0). λ1 − λ3

(21c)

(22)

An alternative coordinate to β is the latitude δ, for which β + δ = π/2.

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The coordinate δ more explicitly measures the departure from being deviatoric, and it puts the closest double couple at γ = δ = 0. Table 1 gives γ and δ for the special patterns (orange points) indicated on L in Fig. 1. 6 B E A C H B A L L O R I E N TAT I O N S In this section we will parametrize moment tensor orientations in terms of strike, slip (i.e. rake) and dip angles. We will speak as

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Figure 11. (a) The fundamental lune L [orange, eq. (17)], the same lune as in Fig. 1. Each point on the lune is a beachball pattern (moment tensor source type). The isotropic axis is vertical, hence the deviatoric plane (purple) is horizontal. The three green planes are the mirror planes—the planes of symmetry for the three permutations that are transpositions. Each of the six lunes delineated by the mirror planes corresponds to an ordering of λ1 , λ2 , λ3 , and each could serve as a fundamental lune. The upper blue arc is λ3 = 0; on and above it all beachballs are red. The lower blue arc, nearly out of sight, is λ1 = 0; on and below it all balls are white. Seven beachballs are shown, all on the same meridian γ = − 10◦ . Balls from top to bottom have colatitude β = 0◦ , 43◦ , 70◦ , 90◦ , 110◦ , 143◦ , 180◦ (latitude δ = 90◦ , 47◦ , 20◦ , 0◦ , −20◦ , −53◦ , −90◦ ). (b) Five deviatoric beachballs. The plane of the paper is the deviatoric plane. The parameter is discussed in Section 8. Table 1. Parameters for some special patterns (source types) as shown on the fundamental lune L in Fig. 1. The coordinates γ and δ are longitude and latitude on L, and λ1 , λ2 , λ3 are unnormalized eigenvalues. The parameter is λ2 /max (|λ1 |, |λ3 |) (Section 8). γ

δ

λ1

λ2

λ3

ISO (upper) ISO (lower) DC

– – 0

− π2 0

1 −1 1

1 −1 0

1 −1 −1

– – 0

CLVD (left)

− π6

0

2

−1

−1

− 12

π 2

0

1

1

−2

√1 3 −1 − sin √1 3

1

0

0

LVD (right)

π 6 − π6 π 6

Unnamed (left)

− π6

− sin−1

π 6 − π6 π 6

sin−1

CLVD (right) LVD (left)

Unnamed (right) C (left) C (right)

0

0

−1

1 2 − 12 1 2

2 3

0

−1

−1

− 12

2 3 5 −1 sin √ 33 − sin−1 √5 33

1

1

0

1 2 − 12 1 2

sin−1

3

1

1

−1

−1

−3

if the moment tensor is a double couple, so that the traditional interpretation of the moment tensor as expressing slip along a fault plane is clear. However, the parametrization that we arrive at (Proposition 2) will work regardless of whether the moment tensor is a double couple.

To express rotations, we define the matrices Rξ (v) = rotation through angle ξ about axis v; X ξ = Rξ (e1 ) = rotation through angle ξ about the x-axis; Yξ = Rξ (e2 ) = rotation through angle ξ about the y-axis;

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Z ξ = Rξ (e3 ) = rotation through angle ξ about the z-axis. If U is a rotation then Rξ (U v) = U Rξ (v)U −1 .

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Suppose now that a moment tensor M is written []U , so that the matrix U is a frame of eigenvectors for M. Suppose also that the frame vectors u1 , u2 , u3 (columns of U) are arranged in order of decreasing size of their eigenvalues (λ1 ≥ λ2 ≥ λ3 ). If the frame is rotated 45◦ about its own second vector u2 , then we get a frame V whose vectors v1 , v2 , v3 are the slip vector S, the nodal vector N × S and the fault plane normal vector N associated with one of the two fault planes of M (Figs 12 and 13). Here we are temporarily abandoning convention and not insisting that N be upward. Since V = Rπ/4 (u2 ) · U = Rπ/4 (U e2 ) · U = U Yπ/4 U −1 U = U Yπ/4 , then U = V Y−π/4 , M = []V Y−π/4 .

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C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


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Figure 12. Frame U of eigenvectors u1 , u2 , u3 of a double couple, and frame V of slip, nodal and normal vectors v1 = S, v2 = N × S and v3 = N for one of the two fault planes of the double couple. If the frame vectors of U are ordered according to decreasing size of their eigenvalues, as here, then the frames U and V differ by a 45◦ rotation about their common second vector u2 = v2 .

Of the two frames U and V , the frame U is the more natural mathematically, but V is more directly tied to the strike, slip and dip angles. The slip and normal vectors S and N, and hence M = []V Y−π/4 , can be expressed in terms of the strike, slip and dip angles κ, σ and θ. As in Fig. 14, K = Rφ (e3 ) · e1 = Z φ · e1

(strike vector)

(27a)

N = Rθ (K) · e3

(normal vector)

(27b)

S = Rσ (N) · K

(slip vector),

(27c)

where the standard basis is e1 (north), e2 (west), e3 (zenith), and where φ = −κ, so that positive κ will be clockwise. Since we are allowing the normal N to be downwards, then the dip angle θ varies from 0 to π rather than the conventional 0 to π /2. Since the matrix V is a function of S and N, namely, V (S, N) = (S, N × S, N),

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Figure 14. Diagram for eqs (27) and (40), showing strike vector K, slip vector S and normal vector N. Everything green is horizontal.

then the moment tensor M = []V Y−π/4 is a function of κ, σ and θ (and ). (The vectors S, N × S and N must be regarded as column vectors, and in fact they are the columns of V .) If N is not vertical then eqs (27) define κ (mod 2π ), σ (mod 2π ) and θ (0 < θ < π ) as functions of (S, N), using the fact that K=

e3 × N e3 × N = . e3 × N sin θ

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6.1 An efficient parametrization If is fixed and the angles φ, σ , θ are allowed to vary, with −2π ≤ φ ≤ 0, −π ≤ σ ≤ π and 0 ≤ θ ≤ π , then the parametrization

Figure 13. Strike vector K, slip vector S and fault normal vector N, together with strike angle κ, slip angle σ and dip angle θ . The block on the N side of the fault slips in the direction S relative to the other block. The quadrant of the double couple beachball between S and N should therefore be red. Our slip vector S is a unit vector, so S gives only the direction, not the magnitude, of slip. C


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Figure 15. The same double couple moment tensor, with four different triples of strike, slip and dip coordinates (κ, σ , θ ). The four triples correspond to slip and normal vector pairs (S, N), (−S, −N), (N, S) and (−N, −S). Black arrows are the normal vectors, and orange arrows are the slip vectors. Green arrows are the strike vectors. The discs are the fault planes for the normals. The pairs (S, N) and (−S, −N) correspond to one fault plane and motion, and the pairs (N, S) and (−N, −S) correspond to the other. One of the four coordinate triples (κ, σ , θ ), namely the one with θ ≤ π /2 and |σ | ≤ π /2, must be in the set P of Proposition 2.

(φ, σ, θ ) → []V Y−π/4 generates all moment tensors having eigenvalues , but it does so with duplication, since more than one pair (S, N) give the same moment tensor, as illustrated in Fig. 15. To confirm the duplication, let Px = Y−π/4 · X π · Yπ/4 , Py = Y−π/4 · Yπ · Yπ/4 = Yπ ,

⎛

0

0

⎜ Px = ⎝ 0

1

−1

1

0

(30)

⎞ ⎟

0 ⎠,

⎛

0

⎜ Pz = ⎝ 0 −1

0

⎞

0

−1

−1

0 ⎠,

0

0

⎟

(31)

V (S, N) · Px = V (N, S), V (S, N) · Py = V (−S, −N),

(32)

V (S, N) · Pz = V (−N, −S). Since X π , Y π , Z π all commute with [], then Px , Py , Pz all commute with []Y−π/4 , and []V (S, N) Y−π/4 = []V (−S, −N) Y−π/4 = []V (N, S) Y−π/4 = []V (−N, −S) Y−π/4 .

(33)

In other words, the four slip and normal vector pairs (S, N), (−S, −N), (N, S) and (−N, −S) all give the same moment tensor. See Fig. 15. For the parametrizing function, we want to choose just one of the four pairs. Let us assume for the moment that neither S nor N is vertical. If κ , σ , θ are coordinates for the sign-changed pair (S , N ) = (−S, −N), then θ = π − θ

σ = −σ

(for sign change)

(34a)

(for sign change).

(34b)

Thus 0 ≤ θ ≤ π /2 or 0 ≤ θ ≤ π /2. If instead κ , σ , θ are coordinates for the swapped pair (S , N ) = (N, S), then cos σ = K · S =

(e3 × N) · S , sin θ

(e3 × N ) · S sin θ (e3 × S) · N = . sin θ =

(36)

Since (e3 × S) · N = −(e3 × N) · S, then

Pz = Y−π/4 · Z π · Yπ/4 . Then

cos σ = K · S

(35)

cos σ sin θ = − cos σ sin θ

(for swap).

(37)

Thus cos σ and cos σ have opposite signs, and so |σ | ≤ π /2 or |σ | ≤ π /2. The same conclusion holds for a combined swap and sign change, since the sign change of the slip and normal pair only changes the sign of σ , not the magnitude. Thus one of the four pairs (S, N), (−S, −N), (N, S) and (−N, −S) associated with eq. (33) will have its σ and θ coordinates satisfying |σ | ≤ π /2 and 0 ≤ θ ≤ π /2; again see Fig. 15. And if one of S or N happens to be vertical, then the other is horizontal, and then one of the four slip and normal pairs will have σ = ±π /2 and θ = π /2, and so the conclusion still holds: there is no loss in generality in assuming |σ | ≤ π /2 and 0 ≤ θ ≤ π /2. For fixed we now have an efficient parametrization of moment tensors that have eigenvalues . Proposition 2, next, is a summary. In it we use the parameter h = cos θ instead of θ , to make the frame V of slip, nodal and normal vectors uniformly distributed (Section 6.2). The form for the slip and normal vectors in the proposition is consistent with eq. (4.88) of Aki & Richards (2002). Proposition 2. Let = (λ1 , λ2 , λ3 ) and let P (Fig. 16) consist of points (κ, σ , h) such that 0 ≤ κ ≤ 2π,

−

π π ≤σ ≤ , 2 2

0 ≤ h ≤ 1.

Let V = V (κ, σ, h) = (S, N × S, N) be (from eqs 27 and 28) ⎛ ⎞ √ cos κ cos σ h sin κ cos σ − 1 − h 2 sin κ ⎜ ⎟ − cos κ sin σ ⎜ +h sin κ sin σ ⎟ ⎜ ⎟ ⎜ ⎟ √ ⎜ ⎟ 2 ⎜ − sin κ cos σ h cos κ cos σ − 1 − h cos κ ⎟. ⎜ ⎟ ⎜ +h cos κ sin σ ⎟ + sin κ sin σ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ √ √ 2 2 1 − h sin σ 1 − h cos σ h C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


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Figure 16. Physical map of the solid block P that is the orientation parameter space of Proposition 2. Except on the very top (h = 1), each line segment of P parallel to the σ -axis corresponds to a distinct fault plane. Moving along such a segment corresponds to varying the slip angle σ in a diagram like Fig. 13 while leaving the fault plane unchanged. Also see Fig. 17.

Then the function (κ, σ, h) → []V Y−π/4 ,

(κ, σ, h) ∈ P

(38)

parametrizes all moment tensors having eigenvalues (in any order). If λ1 , λ2 , λ3 are distinct then the parametrization is one-to-one (but not quite onto) on the set P0 ⊂ P consisting of points with 0 ≤ κ < 2π , −π /2 < σ < π /2, 0 < h < 1. Proof . It only remains to verify the claim about one-to-one. Suppose that []V1 Y−π/4 = []V2 Y−π/4 , where Vi = V (κi , σi , h i ) = (Si , Ni × Si , Ni ) and (κi , σi , h i ) ∈ P0 . [We need to show (κ 1 , σ 1 , h1 ) = (κ 2 , σ 2 , h2 ).] Letting U i = V i Y −π/4 , we find from Proposition 5 in Appendix A that U 2 = U 1 W , where W is one of I, X π , Y π or Z π . That in turn means (eqs 30 and 32) that (S2 , N2 ) is one of (S1 , N1 ), (−S1 , −N1 ), (N1 , S1 ) or (−N1 , −S1 ). But due to eqs (34b) and (37), only one of (S1 , N1 ), (−S1 , −N1 ), (N1 , S1 ) and (−N1 , −S1 ) can have its coordinates in P0 . Therefore (S1 , N1 ) = (S2 , N2 ) and (κ 1 , σ 1 , h1 ) = (κ 2 , σ 2 , h2 ). If λ1 > λ2 > λ3 , then the angles κ, σ and θ = cos −1 h are the strike, slip and dip angles for one of the two fault planes associated with the closest double couple to []V Y−π/4 . Fig. 16 shows the locations in P of parameters (κ, σ , h) for some familiar types of faults and slips. See also Fig. 17, which shows several double couple moment tensors at their parameter points in P. Although P is big enough to accommodate all double couple moment tensors (or any other moment tensors with fixed ), it only directly depicts about a half of the associated faults and slips, namely, those with |σ | ≤ π /2. The other half have |σ | ≥ π /2 and are the second fault planes of the double couples. They are found from (S, N) (first and third columns of V ) by swapping S and N and then, if necessary, changing the signs of both, so that the new normal is upwards.

6.2 Uniformity of the frames, not the moment tensors If we define V = Rσ (N) · Rθ (K) · Z φ , then V is a rotation matrix that takes e1 and e3 to S and N, respectively. In detail, from Fig. 14, Zφ

Rθ (K)

Rσ (N)

Zφ

Rθ (K)

Rσ (N)

V : e1 −→ K −→ K −→ S, V : e3 −→ e3 −→ N −→ N. C


(39)

Since V (Proposition 2) also takes e1 and e3 to S and N, then V = V , that is, V = Rσ (N) · Rθ (K) · Z φ .

(40)

The coordinates φ and θ determine N = V e3 and K, and then σ tells how much to rotate K about N to get S = V e1 . Since φ − π /2 and θ are the ordinary spherical coordinates (longitude and colatitude, respectively) for N = (cos(φ − π/2) sin θ, sin(φ − π/2) sin θ, cos θ) ,

(41)

then N will be uniformly distributed on the unit sphere if φ (i.e. −κ) is uniformly distributed and if h = cos θ is uniformly distributed on the interval [−1, 1]. If also σ is uniformly distributed then the frame V should be uniformly distributed. This is the reason for using the parameter h instead of θ in Proposition 2; a uniform distribution of (κ, σ , h) gives a uniform distribution of frames. A uniform distribution of frames does not, however, ensure a uniform distribution of moment tensors. This is easy to see when γ ≈ ±π /6, since in that case the beachball nearly has cylindrical symmetry. A uniform distribution of frames will waste frames that differ from each other approximately by a rotation whose axis is close to the cylinder axis. The same point has been made by Hudson et al. (1989). For a more rigorous discussion of uniformly distributed frames, see Miles (1965) and Robbin (2006).

6.3 From moment tensor to strike, slip and dip We are given a moment tensor M with distinct eigenvalues. Where does M get located in the set P of Proposition 2? The answer is largely contained in Fig. 12. We find a frame U of eigenvectors of M that are ordered according to decreasing size of their eigenvalues. From U we get the frame V = U Y π/4 . Then the four candidates for slip and normal vector pairs (S, N) are (v1 , v3 ), (−v1 , −v3 ), (v3 , v1 ) and (−v3 , −v1 ). We calculate σ and θ for each, using eqs (27) and (29). The pair (S, N) that is wanted is the one with |σ | ≤ π /2 and 0 ≤ θ ≤ π /2. Some qualification is needed to accommodate exceptional cases such as horizontal faults; see Appendix B.

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Figure 17. Some double couple moment tensors, each located at its parameter point (κ, σ , h) in P. Black arrows are fault normal vectors, green are strike vectors and orange are slip vectors. The discs are the fault planes for the normals. At the right are normal dip slips, σ = −π /2, and at the left are reverse dip slips, σ = +π /2. At the centre are strike-slips, σ = 0. For σ = 0 the slip vector coincides with the strike vector and is not seen. All strike angles here are κ = π /2. In this diagram the κ and σ axes are aligned with the spatial x (north) and y (west) axes as shown. The dimensions of P are not drawn to scale, and the vertical axis is sin −1 h = π /2 − θ (elevation angle of N) rather than h, both done in order to display the beachballs better.

7 PA R A M E T R I Z I N G A L L M O M E N T TENSORS Combining the results of Proposition 2 and eq. (18), we have an efficient and conceptually reasonable parametrization of the space M of all moment tensors (Proposition 3, next). Recall that γ and β = cos −1 b are longitude and colatitude for the lune L, and that κ, σ and θ = cos −1 h are the strike, slip and dip angles for the closest double couple. Also recall that π π (42a) B = (γ , b) : − ≤ γ ≤ , −1 ≤ b ≤ 1 , 6 6 π π P = (κ, σ, h) : 0 ≤ κ ≤ 2π, − ≤ σ ≤ , 0 ≤ h ≤ 1 . (42b) 2 2

Define M : B × P × R+ → M by ˆ VY , M(γ , b, κ, σ, h, ρ) = ρ[]

Proposition 3.

−π/4

where ˆ = (γ , b) V = V (κ, σ, h)

[from eq. 20 with ρ = 1], [from Proposition 2],

ρ > 0. Then the function M parametrizes all moment tensors. It is one-toone (but not quite onto M) if its domain is restricted to B0 ×P0 ×R+ , where π π B0 = {(γ , b) : − < γ < , −1 < b < 1} ⊂ B, 6 6 π π P0 = {(κ, σ, h) : 0 ≤ κ < 2π, − < σ < , 0 < h < 1} ⊂ P. 2 2 (B0 consists of coordinates for patterns that have distinct entries.)

Proof . It only remains to verify the claim about one-to-one. So ˆ 1 ]U = ρ2 [ ˆ 2 ]U , where ˆ i = (bi ), bi ∈ B0 , suppose that ρ1 [ 1 2 Ui = V (pi ) Y−π/4 , pi ∈ P0 , ρ i > 0. Then ˆ 1 ]U = ρ2 [ ˆ 2 ]U = |ρ2 |, |ρ1 | = ρ1 [ 1 2 ˆ 1 ]U and [ ˆ 2 ]U are equal and hence have and so ρ 1 = ρ 2 . Then [ 1 2 ˆ 2 is a permutation of ˆ 1 . Since the same eigenvalues, so that ˆ1 = ˆ 2 = . ˆ Then [] ˆ V (p ) Y ˆ V (p ) Y . ˆ i ∈ L, then = [] 1 −π/4 2 −π/4 ˆ are distinct and hence p1 = p2 , Since bi ∈ B0 then the entries of ˆ1 = ˆ 2 then by Proposition 2, since pi ∈ P0 . Since bi ∈ B0 and b 1 = b2 . Fig. 18 illustrates the making of the moment tensor M = ˆ VY in Proposition 3. The coordinates γ and b determine ρ[] −π/4 ˆ = (λˆ 1 , λˆ 2 , λˆ 3 ) ∈ L, and the coordinates κ, σ , h deterthe pattern ˆ has eigenvalues ˆ mine the rotation matrix V . The beachball [] and eigenvectors e1 , e2 , e3 , the standard basis vectors. That beachball is then rotated 45◦ clockwise about the y-axis, resulting in the ˆ Y ; the vectors e1 and e3 are slip and normal vectors beachball [] −π/4 for one of the two slip motions associated with it. The vectors V e1 and V e3 —the first and third columns of V —must therefore be slip and normal vectors for the beachball []V Y−π/4 , which is the result of an additional rotation by means of V . Lastly the ball would be scaled by the factor ρ, not shown in the figure. In Appendix B we construct a set P1 , with P0 ⊂ P1 ⊂ P, such that, when restricted to B0 ×P1 ×R+ , the function M in Proposition 3 is one-to-one and parametrizes all moment tensors that have distinct eigenvalues. For seismological applications the fine distinction between P0 , P1 and P is apt to be irrelevant, since minor duplication of moment tensors is usually acceptable within grid search algorithms, but the fact that duplication occurs—on the boundary of P—does matter. The duplication shows dramatically that distances C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


489

ˆ VY ˆ Figure 18. Creating the moment tensor [] −π/4 of Proposition 3, with pattern and slip–nodal–normal frame V = {S, N × S, N}. For purposes of ˆ The standard basis vectors e1 , e2 , e3 are its eigenvectors. ˆ is taken to be a double couple, namely, ˆ = √1 (1, 0, −1). (a) The double couple []. illustration, 2

◦ ˆ Y (b) The double couple [] −π/4 . The clockwise 45 rotation about the y-axis has made the slip and normal vectors S0 and N0 of the new tensor coincide with e1 ˆ VY ˆ ˆ ˆ and e3 . (This depends on the fact that λ1 ≥ λ2 ≥ λ3 .) (c) The double couple [] −π/4 , the result of applying the rotation matrix V to the previous beachball. The first and third columns of V are necessarily slip and normal vectors for the new tensor. Here V is a (minus) 25◦ rotation about the x-axis.

ˆ and norm ρ, the moment tensors at p0 = (κ0 , π/2, cos θ0 ) Figure 19. (Top) Duplication of moment tensors on the back face σ = π /2 of P. For fixed pattern and p1 = (κ0 + π, π/2, sin θ0 ) will be the same. (The points p0 and p1 are at the centres of the balls, on the back face.) The motions are reverse dip slips that differ from each other by a swap of their slip and normal vectors. As a result of the duplication, half of the back face of P can be dispensed with in ˆ = √1 (1, 0, −1). (Bottom) Contour plot of the distance function parametrizing moment tensors, as indicated in Fig. B1. Here κ 0 = θ 0 = π /3, ρ = 1 and 2 ˆ V (p)Y d(p) = M(p) − M(p0 on the back face. (M(p) is the moment tensor [] −π/4 .) That is, d(p) is the distance from the ball at p = (κ, π/2, h) to the ball at p0 , but with the distance being in moment tensor space and hence measured as explained in Section 4.2. Distances between moment tensors can be quite different from distances in κσ h-space.

in moment tensor space are not as they appear in P. Fig. 19 is an illustration. The moment tensors at p0 and p1 are the same, and hence the distance between them is zero, and this is reflected in the contour plot of distance. So although moment tensors with fixed C


pattern and norm can be plotted in P, the interpretation of the results can be tricky. By way of contrast, moment tensors with fixed orientation can be plotted in λ-space, and distances between moment tensors are then the same as distances in λ-space (eq. 5c).

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We intend to incorporate the parametrization in Proposition 3 into an algorithm for moment tensor inversion. That is, the parametrization will be used to construct a grid for use in a grid search. The radial coordinate ρ should be truncated, both above and below, at some reasonable values. Then, rather than search over equal increments in ρ, we will let ρ = eq and then search over equal increments in q. Since scalar seismic moment is usually defined as M0 = √12 ρ, and since moment magnitude M w ∝ ln M 0 , searching over equal increments of q corresponds to searching over equal increments of M w . For the remaining five variables, equal increments are probably acceptable, though the resulting moment tensors will not be as equally distributed as we would like. 8 T H E PA R A M E T E R Seismologists have used a certain parameter to measure the extent to which a deviatoric eigenvalue triple = (λ1 , λ2 , λ3 ) departs from being a double couple. We follow Giardini (1984) and define () =

λ2 , max(|λ1 |, |λ3 |)

(λ1 ≥ λ2 ≥ λ3 ).

(43)

However, most authors reverse the sign, giving = −λ2 /max (|λ1 |, |λ3 |) (e.g. Giardini 1983; Kuge & Kawakatsu 1990). We use the formulation in eq. (43) because it makes , λ2 and the deviatoric longitude γ all have the same sign. The parameter γ serves the same purpose as , and we think that γ is more transparent than . Fig. 11(b) shows some sample values of γ and for various deviatoric . The angle γ varies from −π /6 to π /6, with γ = 0 at the double couple. The parameter varies from −0.5 to 0.5, again with = 0 at the double couple. To relate and γ analytically: Without loss of generality, can be assumed to lie on the unit cube (with λ1 + λ2 + λ3 = 0 and λ1 ≥ λ2 ≥ λ3 ), in which case is λ2 . If λ2 ≥ 0 then = (1 − λ2 , λ2 , −1) and then, from eq. (22), √ √ 3 λ2 3 = (44) tan γ = , (λ2 ≥ 0). λ1 − λ3 2− The case λ2 ≤ 0 is similar. Combined with eq. (44), it gives √ 3 . tan γ = 2 − ||

(45)

Fig. 20 shows how and γ are related geometrically. √ Since the (signed) distance from the double couple D to is 2, the parameter is essentially measuring separation between D and as a straight line distance on a cube, whereas γ , being an angle, measures separation as arc length on a sphere. To us γ is more natural, but either γ or will give a workable measure. In this section (Section 8) the triple has up to now been assumed to be deviatoric (and in W). But γ is defined for all in W (eq. 21a), and γ () = γ (DEV ). The definition of can be extended to all of W as well, by defining () = (DEV ),

∈ W.

(46)

Then eq. (45) is immediately seen to be correct not just for deviatoric but for all in the wedge W. From eq. (45) one can check that and tan γ are numerically close. The T coordinate of Hudson et al. (1989) is 2; that is, T () = 2() for all ∈ W. This follows from fig. 9 of Tape & Tape (2012) and from the fact that T () = T (DEV ), () = (DEV ), T () = T (t) and () = (t), (t > 0); we can therefore assume = h(T, 0) as in fig. 9, in which case T () = T and () = T /2.

Figure 20. Relation between γ and . The cut-away portion of the unit cube is bounded by the two wedge planes and the deviatoric plane. The point D is the double couple (1, 0, −1), and the points C and C are the CLVDs (1/2, 1/2, −1) and (1, −1/2, −1/2). With the deviatoric point = (λ1 , λ2 , λ3 ) on the unit cube and in the fundamental wedge W, the parameter is λ2 , √ and the signed distance D is 2. Both and γ measure the amount of CLVD in the deviatoric part of , but to us γ is simpler.

9 CRACK + DC MOMENT TENSORS This section is based in part on Minson et al. (2007). We treat CDC moment tensors—‘crack-plus-double-couple’ tensors. Such a moment tensor is supposed to describe a source that is a crack together with a shear, where the plane of the crack coincides with the fault plane of the shear, and where the crack opens in the direction normal to that plane. If the plane of the crack is horizontal, then according to Minson et al. (2007), or Aki & Richards (2002), the tensile moment tensor C for the crack has the form [in our notation (eq. 1a)] C = m c [(1, 1,

1 ν

− 1)],

(47)

where ν is the Poisson ratio of the medium and where mc is a scalar. If, in the shear, the upper half-space is moving in the positive x-direction, then the moment tensor DC for the shear has the form (Fig. 18b) DC = m 0 [(1, 0, −1)]Y−π/4 .

(48)

The moment tensor M for the CDC is then ⎞ ⎛ mc

⎜ M = DC + C = ⎝ 0

m0

0

m0

mc

0

0

m c ( ν1

⎟ ⎠.

(49)

− 1)

The eigenvalues λ1 ≥ λ2 ≥ λ3 of M are mc mc + α, λ2 = m c , λ3 = − α, λ1 = 2ν 2ν

(50)

C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International

A geometric setting for moment tensors where α=

m 2c

1 − 2ν 2ν

2 + m 20 .

(51)

Hence λ2 = ν. λ1 + λ3

(52)

Eq. (52) holds for any CDC moment tensor, since eq. (49) can be conjugated by any rotation matrix U, and since a suitable choice of U can achieve any desired configuration of crack plane and slip direction. Proposition 4 is a kind of converse to eq. (52). It tells how to find the double couple and tensile components for a moment tensor whose eigenvalues satisfy eq. (52). Proposition 4. If λ1 ≥ λ2 ≥ λ3 and λ2 /(λ1 + λ3 ) = ν, then the diagonal matrix [] = [(λ1 , λ2 , λ3 )] is a CDC moment tensor with Poisson parameter ν. Two decompositions of [] into a double couple and a tensile crack are (53a) [] = m 0 [(1, 0, −1)]Yχ− π + m c (1, 1, ν1 − 1) Yχ ,

491

Comparison with eqs (47) and (48) shows that the two terms on the right-hand side of eq. (55) are double couple and tensile tensors with a horizontal crack plane. Conjugating by Y χ gives eq. (53a) and rotates the crack plane through angle χ about the y-axis. Then conjugating eq. (53a) by X π gives eq. (53b), since X π Y χ = Y −χ X π . See Fig. 21. The second equality in eq. (54c) follows from eq. (21a) and √ sin(π/6 − γ ) 1 − 3 tan γ . (56) = √ sin(π/6 + γ ) 1 + 3 tan γ A consequence of eq. (54c) is √ cos 2χ = 3 tan γ .

(57)

Proposition 4 generalizes to an arbitrary M = []U , since eqs (53a) and (53b) can be conjugated by U. The beachballs, the crack planes and the slip directions all get rotated by U. Proposition 4 makes no assumptions about the Poisson ratio ν. Theoretical bounds for ν are −1 ≤ ν ≤ 0.5, and most crustal materials have ν in the vicinity of ν = 0.25 (Christensen 1996).

4

[] = m 0 [(1, 0, −1)]Y−χ+ π + m c (1, 1, ν1 − 1) Y−χ ,

(53b)

4

where Y χ is rotation through angle χ about the y-axis and where (54a) m 0 = λ1 − λ2 λ2 − λ3 , m c = λ2 , tan2 χ =

(54b) λ1 − λ2 sin(π/6 − γ ) = λ2 − λ3 sin(π/6 + γ )

0≤χ ≤

π . 2

(54c)

The normal vectors to the crack planes for the two decompositions are N1 = Yχ · (0, 0, 1) = (sin χ , 0, cos χ ), N2 = Y−χ · (0, 0, 1) = (− sin χ , 0, cos χ ), and the angle between the planes is 2χ . Proof . From eq. (54c) we find ⎞ ⎛ λ2 − λ3 λ1 − λ2 0 ⎜ λ1 − λ3 λ1 − λ3 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 0 Yχ = ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ λ1 − λ2 λ2 − λ3 ⎠ ⎝ − 0 λ1 − λ3 λ1 − λ3

⎜ =⎝

λ2

√

0 √ λ1 − λ2 λ2 − λ3

⎛ =

√ √ ⎞ λ1 − λ2 λ2 − λ3

0

0

0

λ1 − λ2 + λ3

⎛1

0

0

0

⎜0 ⎟ 0 ⎠ +λ2 ⎜ ⎝

1

0

0

0

0

λ1 + λ3 −1 λ2

0

⎜ λ1 − λ2 λ2 − λ3 ⎝ 0 1

1

⎞

= m 0 [(1, 0, −1)]Y−π/4 + m c (1, 1, C

⎟ ⎠

λ2

0

0 1 ν

− 1) .


Eq. (52) and Proposition 4 show that for any moment tensor M, M is a CDC for ν

⇐⇒

λ2 = ν, λ1 + λ3

(58)

where λ1 ≥ λ2 ≥ λ3 are the eigenvalues of M. Since the condition λ2 = ν(λ1 + λ3 ) describes a plane through the points (0, 0, 0) and (1, 0, −1), then on the fundamental lune the locus of CDC moment tensor patterns for ν is an arc of a great circle through the double couple (1, 0, −1). See Fig. 22. Fig. 23 shows CDC loci for several values of ν. If ν = −1 then (1, 1, −2) is a CDC pattern for ν, by eq. (58), and so the CDC locus is the deviatoric arc β = π /2. And if ν = 1/2 then (1, 1, 1) is a CDC pattern for ν, and the CDC locus is the meridian γ = 0 consisting of sums of double couple and pure isotropic patterns. Fig. 23 also shows two CDC beachballs on the meridian γ = −10◦ of the lune. According to eq. (54c) or eq. (57), meridians are contours for the function 2χ that gives the angle between crack planes in the CDC decompositions, so here the angles are the same for the two beachballs.

1 0 M O M E N T T E N S O R PAT T E R N S F O R REAL EVENTS

and then []Y−χ = Y−χ · [] · Yχ ⎛

9.1 CDC patterns on the lune

⎞ ⎟ ⎟ ⎠

(55)

According to Minson et al. (2007), the Miyakejima volcanic earthquake swarm in the year 2000 contained many events whose sources were far from being double couples and in fact were far from being deviatoric. For each of 18 of the Miyakejima earthquakes, Minson computed five moment tensors: one was unconstrained, and the other four were constrained so that one was a double couple, one was deviatoric, one was double couple plus isotropic, and one was a CDC with ν = 0.25. In Fig. 24 we have plotted the patterns (source types) of their moment tensors on our fundamental lune L. Fig. 25 shows patterns from five published compilations of full moment tensor inversions, including those of Minson above. The moment tensors of Walter et al. (2009, 2010) are derived from events that were within glacial ice or at the base of the glacier. Most of these are extremely isotropic, with β ≤ 10◦ , but one cluster of

492

W. Tape and C. Tape

Figure 21. Illustrating the decompositions in eqs (53a) and (53b), which are here abbreviated to [] = D1 + C1 and [] = D2 + C2 , respectively. The plane through each beachball Di and C i is the plane of the crack. The original [] is shown again on the right-hand side, with both of the possible crack planes together. The beachballs are drawn so that their volumes are proportional to their norms. The arrows are in the x, y, z coordinate directions. Those directions constitute a principal axis system for the original [], but the principal axis systems for Di and C i are different from each other and different from the principal axis system for []. These decompositions are therefore more subtle than those of Section 4, in which all principal axis systems were the same. Here = (2, 1, −4).

Figure 22. CDC moment tensors for ν = −0.5. Their patterns lie on a great circle arc through the double couple (1, 0, −1). The planes through the beachballs are the two possible crack planes. The orientations of the balls depend on the choice of principal axes (red, blue and yellow arrows). The plot shows the angle 2χ between fault planes as a function of the longitude γ on the lune (eq. 57, independent of ν). The points are given without their normalizing factors. C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


493

Figure 23. CDC moment tensor loci (blue arcs) for ν = −1, −0.5, 0, 0.2, 0.5. The special case ν = −1 gives the deviatoric tensors, and the special case ν = 0.5 gives the moment tensors that are double couple plus pure isotropic. The contours of 2χ , which is the angle between the two possible CDC crack planes, are meridians on the lune, and so the two CDC beachballs shown here, both on the meridian γ = −10◦ , have the same angle between crack planes. For Earth’s crust, only values of ν somewhere near ν = 0.25 are realistic.

events from Walter et al. (2009), ‘surface cluster B’, has a moderate negative isotropic component. The 18 Miyakejima volcanic events from Minson et al. (2007) are predominantly positive isotropic with negative γ values. The 26 geothermal events from Foulger et al. (2004) have moderate isotropic components, either positive or negative. The 32 Nevada events of Ford et al. (2009) comprise 17 nuclear explosions, 12 earthquakes and three collapses (two mines, one cavity). The three different event types fall into three clusters, as presented in Ford et al. (2009) and evident in Fig. 25. The moment tensors derived from nuclear explosions are strongly isotropic, though not as isotropic as most of the glacier events. The GCMT catalogue contains deviatoric moment tensors from most global events, 1976–2011, with magnitudes M w > 5.4 (Dziewonski et al. 1981). Because the GCMT algorithm constrains the output tensors to be deviatoric (with λ1 ≥ λ2 ≥ λ3 ), their patterns are confined to the deviatoric arc β = π /2 of the lune. Moment tensor orientations can be similarly displayed on the solid block P, but they are less informative unless the viewer has the option of rotating the block within a digital animation.

1 1 S U M M A RY A N D D I S C U S S I O N In Proposition 1 moment tensors are written in the form []U . The triple = (λ1 , λ2 , λ3 ) determines the pattern and size of the associated beachball, and the rotation matrix U gives the orientation. No assumption is made about the order of the eigenvalues λ1 , λ2 , λ3 or about U. As a result, this parametrization, that is, (, U ) → []U , is inefficient, in the sense that it is far from being one-toone; many pairs (, U ) can give the same moment tensor []U . Nevertheless, this parametrization is sometimes useful. It is the sensible way to express moment tensors all of which have the same principal axes, since the moment tensors can then be depicted in C


Figure 24. (Top) Moment tensor patterns (source types) for the Miyakejima earthquake events EVT1 (orange), EVT3 (green) and EVT5 (blue) as calculated by Minson et al. (2007). They calculated five moment tensors for each event: a double couple, a deviatoric tensor, a CDC tensor with ν = 0.25, a double couple plus isotropic tensor, and a full (unconstrained) moment tensor. The double couple patterns are omitted here, since they all coincide. (Bottom) Full moment tensor patterns for all 18 of the Miyakejima earthquakes in Minson.

λ-space, with distances in λ-space giving true distances between moment tensors. In Proposition 3, moment tensors are written in the form ˆ is restricted to the fundamental lune L (Fig. 1), ˆ V Y , where ρ[] −π/4 and where the matrix V (or rather its coordinate triple) is in the block ˆ gives the pattern, the P (Fig. 16), and where ρ > 0. The triple matrix V gives the slip–nodal–normal frame for the closest double couple to the moment tensor, and ρ gives the scalar seismic moment via M0 = √12 ρ. A moment tensor is therefore depicted as a point in L, a point in P and a point on the real half-line. This parametrization, ˆ V, ρ) → ρ[] ˆ V Y , is nearly one-to-one and should that is, (, −π/4 be suitable for use in moment tensor inversions, though it is not as uniform as we would like. The lune L represents moment tensor patterns (source types) in a straightforward way. The block P represents moment tensor orientations based on strike, slip and dip angles. The block representation is therefore natural from a traditional geology point of view (Fig. 16), but distances in P are not proportional to distances in moment tensor space (Fig. 19).

494

W. Tape and C. Tape

Figure 25. (Left-hand side) Some non-deviatoric moment tensor patterns (i.e. source types) on the fundamental lune. Five data sets are shown: Walter et al. (2010), purple; Walter et al. (2009), blue; Minson et al. (2007), green; Foulger et al. (2004), orange; Ford et al. (2009), red. The points of Walter et al. (2010) are near the north pole and are barely visible. The dashed line is the theoretical locus of CDC patterns (Section 9) with ν = 0.25. Also see Section 10. (Right-hand side) Distribution of angle γ for 35 070 moment tensors from the GCMT catalogue. Since GCMT tensors are constrained to be deviatoric, their patterns would plot on the equatorial arc of the lune, and γ is therefore the only parameter needed to specify the patterns.

ˆ in Proposition 3 is expressed in terms of the The pattern longitude γ and colatitude β on the lune. The frame V is expressed in terms of strike κ, slip σ and dip θ . Then to make the parametrization more uniform, the parameters β and θ are replaced by b = cos β and h = cos θ . (Hence b is the coordinate on the isotropic axis, and h is the height of the unit fault normal N.) Thus a moment tensor is a function of the six parameters γ , b, κ, σ , h and ρ. The coordinates γ and b determine the moment tensor pattern, while κ, σ and h determine the orientation, and ρ determines the seismic moment. We anticipate using the parameters γ , b, κ, σ , h and ρ of Proposition 3 for sampling-based moment tensor inversions that either perform direct grid searches or use more sophisticated approaches such as simulated annealing, the Metropolis algorithm, or the neighbourhood algorithm. In these inversion algorithms, a misfit function quantifies the differences between a set of recorded seismograms and a set of synthetic seismograms produced from a moment tensor and a structural model. Parameters other than γ , b, κ, σ , h and ρ can be used as well, such as the six independent entries mij of the moment tensor. Regardless of the parameters chosen for the inversion, the lune L and the block P can be used to depict the set of plausible moment tensors, that is, the moment tensors that generate synthetic seismograms with reasonably low misfit values. Sections 4, 8 and 9 illustrate our geometric approach—and the power of pictures. Figures 7 and 8 in Section 4 show familiar decompositions. Though the figures are trivial, they seem to be missing from the literature and may be conceptually helpful. Fig. 20 in Section 8 graphically relates the parameters γ and , both of which are supposed to measure the departure of a deviatoric moment tensor pattern from being a double couple. Proposition 4 in Section 9 is a succinct description of crack-plus-double-couple decompositions. One such decomposition is shown in Fig. 21.

AC K N OW L E D G M E N T S We thank two anonymous reviewers for helpful suggestions.

REFERENCES Aki, K. & Richards, P.G., 2002. Quantitative Seismology, 2nd edn, University Science Books, San Francisco, CA. [2009 Corrected printing.] Baig, A. & Urbancic, T., 2010. Microseismic moment tensors: a path to understanding frac growth, Leading Edge, 29, 320–324. Ben-Menahem, A. & Singh, S.J., 1981. Seismic Waves and Sources, Springer-Verlag, New York, NY. Chapman, C.H. & Leaney, W.S., 2012. A new moment-tensor decomposition for seismic events in anisotropic media, Geophys. J. Int., 188, 343–370. Christensen, N.I., 1996. Poisson’s ratio and crustal seismology, J. geophys. Res., 101(B2), 3139–3156. Dreger, D. & Woods, B., 2002. Regional distance seismic moment tensors of nuclear explosions, Tectonophysics, 356, 139–156. Dufumier, H. & Rivera, L., 1997. On the resolution of the isotropic component in moment tensor inversion, Geophys. J. Int., 131, 595–606. Dziewonski, A. & Woodhouse, J.H., 1983. An experiment in the systematic study of global seismicity: centroid-moment tensor solutions for 201 moderate and large earthquakes of 1981, J. geophys. Res., 88(B4), 3247–3271. Dziewonski, A., Chou, T.-A. & Woodhouse, J.H., 1981. Determination of earthquake source parameters from waveform data for studies of global and regional seismicity, J. geophys. Res., 86(B4), 2825–2852. Ford, S.R., Dreger, D.S. & Walter, W.R., 2009. Identifying isotropic events using a regional moment tensor inversion, J. geophys. Res., 114, B01306, doi:10.1029/2008JB005743. Foulger, G.R., Julian, B.R., Hill, D.P., Pitt, A.M., Malin, P.E. & Shalev, E., 2004. Non-double-couple microearthquakes at Long Valley Caldera, California, provide evidence for hydraulic fracturing, J. Volc. Geotherm. Res., 132, 45–71. Frohlich, C., 1992. Triangle diagrams: ternary graphs to display similarity and diversity of earthquake focal mechanisms, Phys. Earth planet. Inter., 75, 193–198. Frohlich, C., 1994. Earthquakes with non-double-couple mechanisms, Science, 264, 804–809. Frohlich, C., 2001. Display and quantitative assessment of distributions of earthquake focal mechanisms, Geophys. J. Int., 144, 300–308. Giardini, D., 1983. Regional deviation of earthquake source mechanisms from the ‘double-couple’ model, in Earthquakes: Observation, Theory and Interpretation: Notes from the International School of Physics C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International

A geometric setting for moment tensors “Enrico Fermi” (1982: Varenna, Italy), Vol. LXXXV, pp. 345–353, eds Kanamori, H. & Boschi, E., North-Holland Pub., Amsterdam. Giardini, D., 1984. Systematic analysis of deep seismicity: 200 centroidmoment tensor solutions for earthquakes between 1977 and 1980, Geophys. J. R. astr. Soc., 77, 883–914. Hudson, J.A., Pearce, R.G. & Rogers, R.M., 1989. Source time plot for inversion of the moment tensor, J. geophys. Res., 94(B1), 765–774. Jost, M.L. & Herrmann, R.B., 1989. A student’s guide to and review of moment tensors, Seism. Res. Lett., 60(2), 37–57. Julian, B.R. & Sipkin, S.A., 1985. Earthquake processes in the Long Valley Caldera area, California, J. geophys. Res., 90(B13), 11 155–11 169. Julian, B.R., Miller, A.D. & Foulger, G.R., 1998. Non-double-couple earthquakes: 1. Theory, Rev. Geophys., 36(4), 525–549. Kagan, Y.Y., 2005. Double-couple earthquake focal mechanism: random rotation and display, Geophys. J. Int., 163, 1065–1072. Knopoff, L. & Randall, M.J., 1970. The compensated linear-vector dipole: a possible mechanism for deep earthquakes, J. geophys. Res., 75(26), 4957–4963. Kuge, K. & Kawakatsu, H., 1990. Analysis of a deep “non double couple” earthquake using very broadband data, Geophys. Res. Lett., 17(3), 227–230. Miles, R.E., 1965. On random rotations in R3 , Biometrika, 52, 636–639. Miller, A.D., Foulger, G.R. & Julian, B.R., 1998. Non-double-couple earthquakes: 2. Observations, Rev. Geophys., 36(4), 551–568. Minson, S.E. & Dreger, D.S., 2008. Stable inversions for complete moment tensors, Geophys. J. Int., 174, 585–592. Minson, S.E., Dreger, D.S., B¨urgmann, R., Kanamori, H. & Larson, K.M., 2007. Seismically and geodetically determined nondouble-couple source mechanisms from the 2000 Miyakejima volcanic earthquake swarm, J. geophys. Res., 112, B10308, doi:10.1029/2006JB004847. Mochizuki, E., 1989. On double couple solutions derived from moment tensors, Phys. Earth planet. Inter., 54, 1–3. Pearce, R.G. & Rogers, R.M., 1989. Determination of earthquake moment tensors from teleseismic relative amplitude observations, J. geophys. Res., 94(B1), 775–786. Riedesel, M.A. & Jordan, T.H., 1989. Display and assessment of seismic moment tensors, Bull. seism. Soc. Am., 79(1), 85–100. Robbin, J.W., 2006. The probability distribution of the angle of rotation, http://www.math.wisc.edu/∼robbin/angelic/rotationprobability.pdf (last accessed 2012 March). Sileny, J. & Milev, A., 2008. Source mechanism of mining induced seismic events: resolution of double couple and non double couple models, Tectonophysics, 456, 3–15. Strelitz, R.A., 1989. Choosing the ‘best’ double couple from a momenttensor inversion, Geophys. J. Int., 99, 811–815. Tape, W. & Tape, C., 2012. A geometric comparison of source-type plots for moment tensors, Geophys. J. Int., in press, doi:10.1111/j.1365246X.2012.05490.x (this issue). Wallace, T.C., 1985. A reexamination of the moment tensor solutions of the 1980 Mammoth Lakes earthquakes, J. geophys. Res., 90(B13), 11 171–11 176. Walter, F., Clinton, J.F., Deichmann, N., Dreger, D.S., Minson, S.E. & Funk, M., 2009. Moment tensor inversions of icequakes on Gornergletscher, Switzerland, Bull. seism. Soc. Am., 99(2A), 852–870. Walter, F., Dreger, D.S., Clinton, J.F., Deichmann, N. & Funk, M., 2010. Evidence for near-horizontal tensile faulting at the base of Gornergletscher, a Swiss Alpine Glacier, Bull. seism. Soc. Am., 100(2), 458– 472. Willemann, R.J., 1993. Cluster analysis of seismic moment tensor orientations, Geophys. J. Int., 115, 617–634.

APPENDIX A: PERMUTING THE E I G E N VA L U E S Redundancy in the parametrization M = []U (Proposition 1) arises from the fact that and U give more information than is needed to specify the moment tensor. The matrix U is a frame of C


495

eigenvectors of M, and is the corresponding eigenvalue triple. Although each eigenvector is associated with a specific eigenvalue, there is no pre-assigned order to either the eigenvalues or the eigenvectors. Thus the eigenvectors can be permuted in any of six ways, so long as the eigenvalues are suitably permuted to compensate. The directions of any two of the eigenvectors can also be reversed. In all, that makes for 6 × 4 = 24 possible frames U. So there are (at least) 24 pairs (, U ) that give the same moment tensor. This appendix makes precise the above informal description. Proposition 5, later, is used in the proofs of Proposition 2 and Proposition 7 and in Section 4.3.2. Proposition 6 is used in Section 4.3.2. Eq. (A4) is used in the proof of Proposition 8. Let V be the group of 24 rotational symmetries of the cube. Those are the 3 × 3 matrices that have determinant +1 and that have exactly one non-zero entry in each row and column, with the non-zero entry being ±1. An example is the matrix ⎞ ⎛ −1

0

⎜ V = ⎝0

0

⎟

−1 ⎠.

0

1

0

(A1)

0

For any matrix V = (vi j ) ∈ V, let PV be the permutation matrix PV = (|vi j |).

(A2)

Then for V, V1 , V2 ∈ V, PV1 PV2 = PV1 V2 ,

(A3a)

PW = I

(A3b)

(W = I, X π , Yπ , Z π ),

[]V = [PV ].

(A3c)

(In eq. A3c and elsewhere, must be treated as a column vector, in order to make the matrix multiplication work.) For example, if V is as in eq. (A1), then PV permutes the three entries of cyclically: ⎞⎛ ⎞ ⎛ ⎞ ⎛ 0

λ1

λ2

λ3

λ1

1

0

⎜ PV = ⎝ 0

0

⎟⎜ ⎟ ⎜ ⎟ 1 ⎠⎝ λ2 ⎠ = ⎝ λ3 ⎠,

1

0

0

and eq. (A3c) becomes [(λ1 , λ2 , λ3 )]V = [(λ2 , λ3 , λ1 )]. That is, we can conjugate by V or we can permute by PV ; the result is the same. Returning to the general case, we find from eq. (A3c) that, for U ∈ U and V ∈ V, []U V = [PV ]U [PV−1 ]U V = []U .

(A4)

According to eq. (A4), there are 24 pairs (PV−1 , U V )—one pair for each V in V—that all give the same moment tensor as (, U ). Proposition 5. Let = (λ1 , λ2 , λ3 ) have distinct entries. Then []U1 = []U2 iff U 2 = U 1 W for W = X π , Y π , Z π or I. Proof . Suppose []U1 = []U2 . Then W [] = []W , where W = U2−1 U1 . Then []W e j = W []e j = W λ j e j = λ j W e j , so that W e j is an eigenvector of [] with eigenvalue λj . Since λ1 , λ2 , λ3 are distinct, then the only eigenvectors of [] with eigenvalue

496

W. Tape and C. Tape

λj are multiples of e j , and so W e j must itself be a multiple of e j . And since W is orthogonal then in fact W e j = ±e j , So

⎛

±1

⎜ W = ⎝0 0

j = 1, 2, 3. ⎞

0

0

±1

0 ⎠.

0

±1

⎟

Since det W = 1, then W = X π , Y π , Z π or I. The converse is eq. (A4) with V = W . If the entries of are distinct, then the beachball [] has only four rotational symmetries, namely, the four rotations I, X π , Y π , Z π mentioned in Proposition 5. (They are the same as the rotational symmetries of the unit cube when coloured with three colours, opposite faces being coloured the same.) If, say, λ1 = λ2 = λ3 , then the rotational symmetries of [] are the same as those of a vertical cylinder. The analogue of Proposition 5 is then as follows: Proposition 6. Let λ1 = λ2 = λ3 . Let Cz be the group of rotational symmetries of a vertical cylinder. That is, Cz consists of all rotations about the z-axis, together with all 180◦ rotations about axes in the xy-plane. Then []U1 = []U2 iff U 2 = U 1 W for some W ∈ Cz . The proof of Proposition 6 is nearly identical to that of Proposition 5 and is omitted. Proposition 6 has obvious variants for the cases λ2 = λ3 = λ1 and λ1 = λ3 = λ2 ; those, too, are omitted.

Figure B1. The set P1 . If has distinct entries, then on P1 the parametrization (κ, σ, h) → []V Y−π/4 in Proposition 2 is one-to-one and yet still produces all moment tensors with eigenvalues . The set P1 consists of the interior of P together with certain subsets of the boundary of P, namely, the green lines and rectangles shown here. The red lines are excluded.

set P1 is shown in Fig. B1. Following is an outline of the deleting procedure. The top face h = 1 of P is peculiar, but the peculiarity only stems from the familiar singularity of spherical coordinates at the poles. From eqs (40), (27) and (25), V = Rσ (N) · Rθ (K) · Z φ = Rθ (K) · Z σ · Rθ (K)−1 · Rθ (K) · Z φ = Z φ · X θ · Z −φ · Z σ · Z φ

Proposition 7. Let 1 and 2 have distinct entries. Then [1 ]U1 = [2 ]U2 iff there is V ∈ V such that U2 = U1 V and 2 = PV−1 1 . Proof . If [1 ]U1 = [2 ]U2 , then 1 and 2 are permutations of each other. Hence there is a permutation matrix P such that, with 1 and 2 regarded as column vectors, 1 = P2 . Choose V ∈ V so that P = PV . Then [1 ] = [P2 ] = [PV 2 ] = [2 ]V , [2 ]U2 = [1 ]U1 = [2 ]U1 V . Proposition 5 gives U 2 = U 1 V W for some W = X π , Y π , Z π or I. Letting V = V W , we have U 2 = U 1 V and PV−1 1 = PV−1 1 = 2 . The converse is eq. (A4). A P P E N D I X B : A S M A L L E R B L O C K P1 ⊂ P Throughout this appendix the triple = (λ1 , λ2 , λ3 ) is assumed to be fixed, with distinct entries. According to Proposition 2, the function (κ, σ, h) → []V Y−π/4 , with domain P, parametrizes all moment tensors having eigenvalues (in any order). The function is not quite one-to-one, due to the duplication of moment tensors that occurs on the boundary of P. According to the proposition, the function can be made one-to-one by restricting it to P0 ⊂ P. The function then fails, however, to produce all of the same moment tensors as before. In this appendix we construct an intermediate set P1 , with P0 ⊂ P1 ⊂ P, on which the function (κ, σ, h) → []V Y−π/4 is one-to-one and produces all moment tensors with eigenvalues . The idea is to delete the boundary portions of P where duplication occurs. The resulting

= Zφ · Xθ · Zσ = Z −κ · X θ · Z σ .

(B1)

Thus if h = 1 (i.e. θ = 0), then V = Z σ −κ , and so []V Y−π/4 is constant on each line σ − κ = constant (blue, Fig. B2) of the top face. That means that the entire top face of P can be replaced by, say, its centre line σ = 0. This amounts to agreeing that for a horizontal fault plane the strike vector—which is otherwise undefined—will coincide with the slip vector. Once we have agreed to keep the centre line of the top face, we can delete the four edges of P that are parallel to the κ-axis. The reasoning is illustrated in Fig. B2. Half of the back face σ = π /2 can also be deleted, due to the duplication illustrated in Fig. 19. The situation on the front face is similar. Half of the bottom face can be deleted as well, due to similar reasoning. The end face κ = 2π can obviously be deleted, due to the periodicity of κ. This leaves us with the set P1 in Fig. B1. The preceding reasoning, if fleshed out, shows that P1 is big enough so that the function (κ, σ, h) → []V Y−π/4 , when restricted to P1 , still produces all moment tensors having eigenvalues . The proof that the function is one-to-one on P1 is like the proof of oneto-one in Proposition 2. One has to argue that P1 is small enough so that only one of (S, N), (−S, −N), (N, S) and (−N, −S) can have its κ, σ , h coordinates in P1 . APPENDIX C: CLOSEST BEACHBALL Proposition 8 was used in Section 4.1 to find the closest double couple by drawing a picture. Proposition 8. Let , ∈ R3 , each with its entries distinct. Let M = []U0 and let M( ) = {[ ]U : U ∈ U}, C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International


497

Figure B2. Duplication of moment tensors on the top face of P and on the four edges of P that are parallel to the κ-axis. The moment tensors []V Y−π/4 are the same at the points (κ, −π /2, 1) (upper left), (κ + π , π /2, 1) (upper right), (κ + π , −π /2, 0) and (κ, π /2, 0), as well as everywhere on the diagonal blue line joining the upper two points. The four edges parallel to the κ-axis, as well as the entire top face, can therefore be replaced by the green centre line σ = 0 of the top face. Black arrows are normal vectors, green are strike vectors, orange are slip vectors. Here κ = π /3.

where U is the group of rotation matrices. [The set M( ) consists of all beachballs obtained by rotating the beachball [ ] arbitrarily.] If M is the closest moment tensor in M( ) to M, then M = [P ]U0 for some permutation P. Thus the eigenvector frame U 0 for M is also an eigenvector frame for M . Mochizuki (1989) states a special case of Proposition 8 with only a partial proof, saying that the remainder is easy. This makes us think that our proof can be simplified, but we do not see it. Strelitz (1989) also discusses a related problem. The proof of our lemma is due in part to Ed Bueler. Lemma 1. ⎛

Let U = (uij ) be orthogonal and let ⎞

u 21 u 31

⎜ U = ⎝ u 31 u 11 u 11 u 21

u 22 u 32

u 23 u 33

u 32 u 12

u 33 u 13 ⎠.

u 12 u 22

u 13 u 23

⎟

If rank U = 1 and if U = 0, then has the form (x, x, z), (x, y, y) or (x, y, x). Proof . Since rank U = 1, then some row of U , say the first, is not all zeros. Suppose all of the first row entries are non-zero. Since, again, rank U = 1, the second row is a multiple of the first: (u 31 u 11 , u 32 u 12 , u 33 u 13 ) = t (u 21 u 31 , u 22 u 32 , u 23 u 33 ) u 31 u 11 u 32 u 12 u 33 u 13 = = =t u 21 u 31 u 22 u 32 u 23 u 33

which contradicts the orthogonality of U. So the first row of U has the form (x, y, 0) [or (x, 0, z) or (0, y, z)]. The sum of its entries must be zero, again by orthogonality, and so the first row becomes (x, −x, 0). The matrix U is then row equivalent to ⎛

⎞

−1

0

⎜ ⎝0

0

0 ⎠.

0

0

0

1

⎟

With obvious modifications for the other cases, this proves the lemma. Proof of Proposition 8. It is enough to consider the case U 0 = I. We have U ı

c

−→ M( ) onto

↓

ı

↓

c

f

M3×3 −→ M3×3 −→ R1 , where M 3×3 consists of all 3 × 3 matrices, where ι is inclusion, where c (or just c) is conjugation, and where f (or just f ) is the distance function (squared). c (U ) = [ ]U , f (M ) = M − []2 .

u 11 u 12 u 13 = = =t u 21 u 22 u 23 (u 11 , u 12 , u 13 ) = t (u 21 u 22 , u 23 ), C


Now suppose M = c(U ) = U [ ]U T is the closest point of M( ) to M = []. Then M is a critical point of f on M( ), that is, the directional derivative of f at M is zero in all directions tangent to

498

W. Tape and C. Tape

M( ). With X t , Y t , Z t as in eq. (24), let ⎞ ⎛ 0 0 0 d Xt ⎟ ⎜ Kx = = ⎝ 0 0 −1 ⎠, dt t=0

0

⎛

dYt Ky = dt

⎜ =⎝ 0

t=0

d Z t Kz = dt

0

⎛

0 1

0

⎟ 0 ⎠,

0

0

−1

0

⎞

0

−1

⎜ = ⎝1 t=0

1

0 0

0

0

where = (λ1 , λ2 , λ3 ) and = (λ1 , λ2 , λ3 ). Since λ1 , λ2 , λ3 are distinct, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ u 21 u 31

u 11 u 21

u 23 u 33

λ1

u 32 u 12

⎟ ⎜ ⎟ ⎜ ⎟ u 33 u 13 ⎠ · ⎝ λ2 ⎠ = ⎝ 0 ⎠.

u 12 u 22

u 13 u 23

λ3

u 22 u 32

U

⎟

0 ⎠.

(u 11 u 21 λ1 + u 12 u 22 λ2 + u 13 u 23 λ3 )(λ1 − λ2 ) = 0,

⎜ ⎝ u 31 u 11

⎞

0

(u 31 u 11 λ1 + u 32 u 12 λ2 + u 33 u 13 λ3 )(λ1 − λ3 ) = 0

0

(C1)

0

Since U is orthogonal, the sum of each row of U must be zero. That gives ⎛ ⎞ ⎛ ⎞ 1

The vectors (matrices) d c(X t U ) |t=0 = K x M − M K x , dt d c(Yt U ) |t=0 = K y M − M K y , T2 = dt d c(Z t U ) |t=0 = K z M − M K z T3 = dt

T1 =

are tangent to M( ) at M . The condition that the gradient of f be orthogonal to T 1 , T 2 , T 3 at M reduces to (u 21 u 31 λ1 + u 22 u 32 λ2 + u 23 u 33 λ3 )(λ2 − λ3 ) = 0

0

⎜ ⎟ ⎜ ⎟ U · ⎝ 1 ⎠ = ⎝ 0 ⎠. 1

0

Hence rank U ≤ 2. Since in eq. (C1) is not a multiple of (1, 1, 1), then rank U ≤ 1. And since the entries of are distinct, then rank U = 1, according to Lemma 1, and so U must be the zero matrix. It follows that each row and column of U has exactly one nonzero entry, and that that entry must be ±1. Then since det U = 1 the possibilities for U are exactly the 24 rotational symmetries of the cube. Then from eq. (A3c), M = [ ]U = [PU ], as desired.

A P P E N D I X D : G L O S S A RY O F N O TAT I O N This glossary is not an index; third column entries are not comprehensive. b B, B0 D, D0 e1 , e2 , e3 h I K, S, N L M 3×3 M ⊂ M 3×3 M() P, P0 , P1 Rξ (v) R1 , R+ , R3 U V W X ξ , Y ξ , Zξ [], []U DEV ˆ β, γ δ φ θ κ, σ , θ ν ρ χ

b = cos β Closed and open γ b rectangles Closed and open deviatoric sectors Standard basis for R3 h = cos θ Identity matrix Strike, slip and normal vectors Fundamental lune Set of 3 × 3 matrices Set of moment tensors

Rotation matrix with axis v and angle ξ Reals, positive reals, three-space Group of rotation matrices Group of rotational symmetries of the cube Fundamental wedge Rotations through angle ξ about the x, y, z-axes Eigenvalue triple Deviatoric part of Moment tensor pattern (i.e. source type) Colatitude and longitude with pole (1, 1, 1) Latitude with pole (1, 1, 1) Ordinary spherical coordinate for longitude Ordinary spherical coordinate for colatitude Strike, slip and dip angles Poisson ratio (λ21 + λ22 + λ23 )1/2 Half-angle between possible crack planes in CDC

Section 6 Eq. (42a), Proposition 3 Eq. (10) Section 6 Section 6 Eq. (17) Section 4.1 Appendix C Proposition 3, Fig. B1, eq. (42b) Eq. (24) Section 3 Appendix A Eq. (15) Eq. (24) Eq. (1) Eq. (8) Eq. (16) Section 5.3 Section 5.3

Section 8 Section 6 Section 9 Eq. (21c) Fig. 22 C 2012 The Authors, GJI, 190, 476–498 C 2012 RAS Geophysical Journal International