A geometric comparison of sourcetype plots for moment tensors

Geophysical Journal International Geophys. J. Int. (2012) 190, 499–510

doi: 10.1111/j.1365-246X.2012.05490.x

A geometric comparison of source-type plots for moment tensors Walter Tape1 and Carl Tape2 1

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

2 Geophysical

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

SUMMARY We describe a general geometric framework for thinking about source-type plots for moment tensors. We consider two fundamental examples, one where the source-type plot is on the unit sphere, and one where it is on the unit cube. The plot on the sphere is preferable to the plot on the cube: it is simpler, it embodies a more natural assumption about eigenvalue probabilities and it is more consistent with the conventional Euclidian definition of scalar seismic moment. We describe the source-type plots of Hudson, Pearce and Rogers in our geometric context, and we find that they are equivalent to a plot on the cube. We therefore suggest the plot on the sphere as an alternative. Key words: Theoretical seismology.

The equal area Tk source-type plots of Hudson et al. (1989) are a kind of map of source types of seismic moment tensors (Fig. 1). They have been used, for example, by Pearce & Rogers (1989), Julian et al. (1998), Bowers & Hudson (1999), Ford et al. (2009), Foulger et al. (2004), Ross et al. (1999), Walter et al. (2009) and Baig & Urbancic (2010). Those authors used the plots for theoretical purposes as well as to display inferred source types for various real events: earthquakes, glacier events, mine collapses, volcanic events, hydraulic fracturing and nuclear explosions. We describe a general geometric setting for thinking about source-type plots, and we consider two fundamental examples—one where the plot is on the unit cube and one where it is on the unit sphere. Each case is associated with an assumption about eigenvalue probabilities, and it turns out that the assumption for the sphere is more plausible than that for the cube. We consider the source-type plot of Hudson in our geometric context, and we find that it is essentially a plot on the cube. We therefore recommend the plot on the sphere as an alternative. The plot on the sphere was proposed and used by Riedesel & Jordan (1989). It is less well known than the Hudson Tk plot, though it has been used by, for example, W´eber (2006), Sileny & Milev (2008) and Sileny et al. (2009). In part of a larger paper, Chapman & Leaney (2012) recently compared the plot on the sphere with the Tk plot—the same problem that we consider here. As we do, they prefer the plot on the sphere. In fact, they dismiss the fundamental assumption of Hudson et al. (1989) regarding eigenvalue probabilities and therefore do not consider the so-called equal area aspect of the Tk plots. We agree that the Hudson assumption is strange, but we do not find it logically impossible. In general, the approach of Chapman & Leaney (2012) is complementary to ours, being more analytic and less geometric. C

2012 The Authors C 2012 RAS Geophysical Journal International

We stress that source-type plots are meant to accommodate ‘full’ moment tensors. That is, the only constraint on the moment tensors is that they be symmetric. However, source-type plots include no information about moment tensor orientation or magnitude.

2 S O U RC E T Y P E S A S B E AC H B A L L PAT T E R N S The geometric approach that we take is described in more detail in Tape & Tape (2012, sections 2 and 5). In that approach, we refer to source types as beachball patterns, and we think of moment tensors as beachballs in the usual way. That is, a beachball is determined by the eigenvalues and eigenvectors of the moment tensor. The eigenvalues determine the size and the pattern of the ball, and the eigenvectors determine the orientation. With the eigenvalue triple = (λ1 , λ2 , λ3 ) fixed, we get all beachballs having a fixed pattern and size but having varying orientations. We imagine all those beachballs to be located at the point in space, as suggested in Fig. 2.

3 O N A N A R B I T R A RY S U R FA C E We now describe how to depict source types (beachball patterns) on a fairly arbitrary surface. The surface can be chosen in various ways, with the choice determining the character of the source-type plot. In subsequent sections, we consider the special cases where the surface is the unit cube or unit sphere. If there is no order specified for eigenvalues, then a typical beachball will appear at six points in space, the six points being permutations of each other (Fig. 3). Since one of the six points will be in the 60◦ ‘fundamental wedge’ W that consists of points with λ1 ≥ λ2 ≥ λ3 , we can assume from the start that ∈ W.

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1 I N T RO D U C T I O N

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W. Tape and C. Tape v Explosion k = 1

k = 0.5

(-1, 5/9) Tensile crack

k=0 (0, 0) DC

T=

T=

-1

5 - 0.

(-1, -1/5)

(1, 1/5)

0.5

(-1, 0) CLVD

1

T=

T=0

(-1, 1/3) LVD

T=

.5

k = -0

(1, 0) CLVD (1, -1/3) LVD

u

(1, -5/9) Tensile crack

Implosion k = -1

Figure 1. Equal-area source-type plot, adapted from Hudson et al. (1989). DC, double couple; LVD, linear vector dipole; CLVD, compensated linear vector dipole. Although the diagram is in the uv-plane, points are identified with their Tk coordinates (Section 7).

Figure 3. The same beachball (i.e. moment tensor) appearing at six different points = (λ1 , λ2 , λ3 ), the six points being permutations of each other. The sphere is oriented so that we are looking down its 111-axis, with the deviatoric circle λ1 + λ2 + λ3 = 0 (purple) of the sphere in 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 . The red, blue and yellow lines emanating from each beachball are the first, second and third principal axes, respectively. Each ball is coloured red in the direction of the most positive eigenvalue, white in the direction of the most negative.

4 ON THE CUBE In our first example the surface L is the unit cube C, or rather it is the portion of C within the fundamental wedge W. So the pattern for an eigenvalue triple ∈ W is on C; it is the point πC () where the ray through meets C. For ∈ W, πC () =

= = max max(|λ1 |, |λ2 |, |λ3 |) max(|λ1 |, |λ3 |)

LC = πC (W) = C ∩ W. Figure 2. Some beachballs (moment tensors) at eigenvalue triples 1 = √ 31(1, −1, 0) and 2 = (1, −6, 5). The beachball patterns at 1 are all the same, with any two of the beachballs there differing by a rotation. The same is true at 2 . The pattern contains the moment tensor√information that is neither orientation nor size. The radius of the sphere is 62.

Two eigenvalue triples 1 and 2 give the same beachball pattern when they differ from each other by at most a permutation and a positive scalar factor (Fig. 4). Thus the triples (1, 2, 3), (2, 1, 3), (4, 2, 6) would all give the same pattern. Since is assumed to be in W, the possibility of permutations is eliminated. Triples that have the same pattern as must therefore lie on the ray through . ‘The’ pattern for will be the point π () that is the radial projection of to a surface L ⊂ W of our choosing. That surface will be our representation of beachball pattern space. The surface L should have the property that every ray in W meets it exactly once. Other than that, L can in principle be arbitrary, but a sensible choice of L gives a sensible representation of beachball pattern space.

(1)

Both C and LC are shown in Fig. 5. The set LC —the representation of beachball pattern space—consists of two half-faces of the cube. All beachball patterns can be depicted on LC itself. Alternatively, as in the lower diagram, the representation can be made planar by constructing reasonable coordinates u and v for LC and then using the uv-coordinate domain DC (the parallelogram, in this example) as the representation. For the uv-coordinates in the figure, the parametrizing function would be gC : DC → LC defined by (1, u, v − 1) v≥0 (2) gC (u, v) = (λ1 , λ2 , λ3 ) = (1 + v, u, −1) v ≤ 0. The function gC takes the two triangles of the parallelogram and puts them back on the λ1 = 1 and λ3 = −1 faces of the cube, where they came from. The representation of beachball pattern space on the unit cube is implicit in some literature. The parameter , which is supposed to measure the departure of a deviatoric pattern from being a pure double couple, is a signed distance on a cube (Tape & Tape 2012, section 8). Silver & Jordan (1982) mention a possible definition of scalar moment as M0 = max ; that would give a constant C 2012 The Authors, GJI, 190, 499–510 C 2012 RAS Geophysical Journal International

Plots of moment tensor source types

Figure 4. (Top panel) Eigenvalue triples all having the same pattern as = (4, −5, −7). They constitute the six red rays (two are hidden) and result from by permuting its entries and multiplying by a positive scalar. As in Fig. 3, the mirror planes λ1 = λ2 , λ2 = λ3 and λ3 = λ1 divide space into six congruent infinite wedges. The wedge facing us is the fundamental wedge W, for which λ1 ≥ λ2 ≥ λ3 . The six red rays are symmetric with respect to the mirror planes. (Bottom panel) The eigenvalue triple = (4, −5, −7) ∈ W and its pattern π (). The pattern π () is the radial projection of to the surface L. The surface L consists of all patterns and is thus a representation of beachball pattern space. But the representation depends on L—the representation is L. The choice of L is at our discretion, and the surface shown here is meant to look arbitrary. The 111-axis is vertical in both diagrams.

M 0 = 1 everywhere on the unit cube. And our parallelogram DC in the uv-plane in Fig. 5 is similar to the Tk source-type parallelogram of Hudson et al. (1989).

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Figure 5. Representation of beachball pattern space on the unit cube. The unit cube C in λ-space is oriented so that the 111-axis is vertical and so that the wedge W is facing us. On the cube the two gridded half-faces are the intersection LC of the cube with W. Those half-faces are the representation of beachball pattern space; each point on it corresponds to a beachball pattern. Alternatively, as at the right, the same half-faces—imagined to be hinged along their common cube edge—can be laid flat in the uv-plane to form a parallelogram DC that gives an equivalent representation of beachball pattern space. See also Fig. 14 (Section 9), where some familiar beachball patterns are identified on DC .

The set LS is a lune of the unit sphere. The lune, instead of the two cube half-faces, now represents beachball pattern space (see Fig. 6). Convenient coordinates on the lune—and indeed for points in the entire wedge W—are the longitude γ , measured eastwards from the = (1, 0, −1) direction, and either the colatitude β or the latitude δ = π /2 − β. (The poles of the lune are ± √13 (1, 1, 1) rather than (0, 0, ±1), and the equator is in the deviatoric plane λ1 + λ2 + λ3 = 0). As in section 5 of Tape & Tape (2012), the angles γ and β are given by −λ1 + 2λ2 − λ3 √ 3 (λ1 − λ3 ) λ1 + λ2 + λ3 . cos β = √ 3

tan γ = 5 ON THE SPHERE In our second example the unit sphere S replaces the unit cube. Now the pattern for an eigenvalue triple ∈ W is on S. Eq. (1) must be replaced by πS () =

= 1/2 2 2 λ1 + λ2 + λ23

LS = πS (W) = S ∩ W. C

2012 The Authors, GJI, 190, 499–510 C 2012 RAS Geophysical Journal International

(3)

(4)

On the lune the longitude γ ranges from −π /6 to π /6, and the latitude δ ranges from −π /2 to π /2. The latitude δ is zero for deviatoric patterns, and δ and γ are both zero for double-couple patterns. The representation of patterns on the lune can be made planar by using the γ δ-plane. Or δ can be replaced by b = cos β, in which case equal areas in the γ b plane correspond to equal areas

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W. Tape and C. Tape density is constant on the cube half-faces LC . The probability and density functions for sphere probability are denoted by PS and pS , respectively, and those for cube probability are denoted by PC and pC . Then the lune is an equal area representation with respect to PS , and the cube half-faces are an equal area representation with respect to PC . The difference between the two probabilities PC and PS is substantial, as shown in Fig. 7. The regions A and A are radial projections of each other, so they represent the same set of patterns and hence have the same cube probability PC (A) = PC (A ). (Their sphere probabilities are equal by the same reasoning, but not equal to their cube probabilities.) Similarly, B and B have the same cube probability. Since A and B are on the cube and have the same area, they too have the same cube probability, namely, (area of square)/(area of cube face) = 1/64. (Here, we are taking the domain to be the two cube half-faces LC or, equivalently, the single orange face.) Hence the regions A and B on the sphere also have cube probability PC (A ) = PC (B ) = 1/64. But their sphere probabilities are not the same: PC (A ) = (area of A )/(area of lune) = 0.0281 and, similarly, PC (B ) = 0.0074. To further quantify the difference between cube probability and sphere probability, let f = πS λ =1 be the radial mapping from the 1 λ1 = 1 cube face to the sphere. Then

Figure 6. Representation of beachball pattern space on the unit sphere S. The unit sphere in λ-space is oriented so that the 111-axis is vertical and so that the wedge W is facing us, as in Fig. 5. In contrast to Fig. 5, it is now the lune LS = S ∩ W (orange) that is the representation of beachball pattern space, rather than the two half-faces of the cube. ISO, isotropic; DC, double couple; LVD, linear vector dipole; CLVD, compensated linear vector dipole; C, tensile crack with Poisson ratio ν = 1/4. The purple arc consists of deviatoric patterns, and the black arc consists of crack plus double-couple patterns with ν = 1/4. Points are labelled without their normalizing factors; the double couple labelled (1, 0, −1), for example, is really √1 (1, 0, −1). 2

on the lune. We think, however, that the representation is more compelling and transparent if left on the lune itself.

6 P RO B A B I L I T I E S The two cube half-faces LC and the lune LS , both shown in Fig. 14 (Section 9), are topologically similar. There is nevertheless a significant difference between the two representations of beachball pattern space. Each representation brings its own suggestion of what constitutes reasonable probabilities for beachball patterns (i.e. source types). If a surface L—whether the half-faces, or the lune, or some exotic surface like the one in Fig. 4—is to be used to display clustering of beachball patterns, then equal areas on L should have the same probability. In fact, the probability P that a pattern occur in a region A of L should be equal to the fractional area of A P(A) =

area of A . area of L

(5)

Equivalently, the probability density for patterns should be constant on L. Following Hudson, we will refer to such representations as equal area representations. But whether a surface is an equal area representation depends on the operative probability model. We say that the probability is ‘sphere probability’ if the probability density is constant on the lune LS , and we say it is ‘cube probability’ if the

(1, λ2 , λ3 ) f (λ2 , λ3 ) = (1, λ2 , λ3 ) ∂f ∂ f 2 2 −3/2 . ∂λ × ∂λ = 1 + λ2 + λ3 2 3

(6)

The quantity ∂f /∂λ2 × ∂f /∂λ3 is the Jacobian factor for f ; it measures the ratio of transformed area to original area. Its reciprocal, evaluated at f −1 () = (1, λ2 /λ1 , λ3 /λ1 ), is proportional to the density for cube probability on the portion of the sphere associated with the λ1 = 1 cube face. Normalizing and adding the analogous expression for the λ3 = −1 cube face, we find the cube probability density pC () on the lune LS to be ⎧ 1 ⎪ ⎪ ∈ LS , λ1 + λ3 ≥ 0 ⎪ ⎨ 4λ3 , 1 (7) pC () = ⎪ −1 ⎪ ⎪ ∈ LS , λ1 + λ3 ≤ 0. ⎩ 3, 4λ3 3/2 On the lune the density pC varies by √ a factor of 3 ≈ 5 between = (1, 0, 0) and = (1, 1, 1)/ 3. From the point of view of the sphere (surface) dweller, the cube dweller overestimates points near the cube vertices, and he underestimates them near the face centres. Which representation for patterns is to be preferred—the lune or the two cube half-faces? The choice is in part between sphere probability and cube probability. A representation for beachball pattern space is a baseline reference on which real data can be displayed. To us, it is more natural to use sphere probability as the underlying principle for constructing the representation. That is, it is more natural to have the baseline assumption be that all directions are equally likely for a pattern, as opposed to having them favour certain directions. Thus any non-uniformities in real data plotted on the lune are easy to interpret; we understand what the standard is that the data are deviating from. The lune is also simpler. With the half-faces there is always a dichotomy, since the two faces have to be treated differently. The dichotomy can be baffling if one sees only the parameter domain— the parallelogram—without realizing that it is associated with the cube (e.g. Fig. 1). C 2012 The Authors, GJI, 190, 499–510 C 2012 RAS Geophysical Journal International


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Figure 7. Comparison of cube probability PC with sphere probability PS . The squares A and B on the cube have cube probability PC (A) = PC (B) = 1/64. The corresponding regions A and B on the sphere have sphere probability PS (A ) = 0.028 and PS (B ) = 0.007, differing by a factor of 4. But A and A represent the same set of patterns, and B and B represent the same set of patterns. So from the point of view of the sphere dweller, the cube dweller overestimates points in the direction of cube vertices and underestimates them in the direction of cube face centres. The lower diagrams show contours of cube probability density pC on the sphere. In the lower right diagram, the lune LS from Fig. 14 has been added; this is the we are concerned with. Lighter √portion of the sphere that √ √ shades of orange and green have higher pC ; maxima of pC are at the isotropic points ±(1, 1, 1)/ 3 and at Q1 = (1, 1, −1)/ 3 and Q2 = (1, −1, −1)/ 3, and minima are at the LVDs (1, 0, 0) and (0, 0, − 1).

To us, the advantages of the lune are self-evident, but either the lune or the cube half-faces can be made to work as a representation of patterns. Moreover, in spite of our loose talk about sphere dwellers, etc. the space of eigenvalue triples should not be confused with ordinary physical space. Once the two spaces are properly recognized as distinct, sphere probability feels less inevitable. So our preference for the lune is at least in part personal; we doubt that we can justify it upon purely logical grounds.

with the wedge W). That implies that the probability for patterns is cube probability PC , that is, that the probability density for patterns is constant on the cube half-faces LC . To see why, note that the cube max = L is just a concentric scaled version of the unit cube C . Under the Hudson assumption, the probability of a pattern being in a given disk of area B on a face of C is therefore proportional to the volume Bh/3 of the cone whose base is the disk and whose vertex is the origin (Fig. 8). Since cone height h ≡ 1, the probability per unit area on the cube face is constant.

7 T K S O U RC E - T Y P E P L O T S Next we describe the Tk source-type plots of Hudson et al. (1989) in our geometric context. Due to the geometric slant and to notational changes, our description only approximates the original.

7.1 Hudson’s probability is cube probability Hudson assumed that eigenvalue triples = (λ1 , λ2 , λ3 ) are uniformly distributed in a cubical box max ≤ L (or in its intersection C


7.2 On the four triangles LH Thus, from our perspective, Hudson was looking for a representation of patterns that was equal area with respect to cube probability PC . The cube half-faces LC themselves would be fine, but he took a less direct route. He constructed a set LH consisting of the four triangles shown in Fig. 9. The upper and lower vertices ±(1, 1, 1) of LH are on the axis of the wedge W, and the left and right vertices (1, − 12 , − 12 ) and ( 12 , 12 , −1) are on the wedge faces proper, so the

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W. Tape and C. Tape

Figure 8. The insight showing that uniform distribution of eigenvalue triples (λ1 , λ2 , λ3 ) in a cube max ≤ L gives constant probability density for patterns on the unit cube. An observer who looks at two equal disks on a wall must look through the same volume of space for each (see Section 7.1).

surface LH has the necessary property that every ray in W meets it exactly once. Hence, like the cube half-faces and the lune, the set LH is a representation of beachball pattern space. But LH is not an equal-area representation with respect to PC . For it to be so, its triangles would each have to be parallel to a face of the unit cube. Two of them are, but two are not. Hudson had already parametrized LH with essentially the following function h, illustrated in Fig. 9. For |T| ≤ 1 and |k| ≤ 1, ⎧ 1 1 ⎪ ⎪ (1 − T ) (1, 0, −1) , , −1 +T ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ DC ⎨ CLVD h(T, 0) = ⎪ 1 1 ⎪ ⎪ , − (1 + T ) (1, 0, −1) −T 1, − ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ DC CLVD

h(T, k) = (1 − |k|) h(T, 0) + k(1, 1, 1) deviatoric

|k| ≤ 1.

T ≥0

T ≤ 0, (8a)

(8b)

where, for (u, v) ∈ DH , ⎧ 2u 2v ⎪ ⎪ , 0 ≤ u/4 ≤ v (Region A) ⎪ ⎪ 2 + u − 2v 2 + u ⎪ ⎪ ⎪ ⎪ ⎪ u ⎪ ⎨ ,v u ≤ 0, v ≥ 0 1−v (T, k) = ⎪ ⎪ ⎪ u v ⎪ ⎪ , u/4 ≤ v ≤ 0 (Region B ) ⎪ ⎪ ⎪ 1 − v 1 − 2v ⎪ ⎪ ⎩ −(T (−u, −v), k(−u, −v)) otherwise. (10)

isotropic

He then wrote T and k in terms of new coordinates u and v, thus giving a new parametrization gH : DH → LH of the four triangles. The domain DH is the parallelogram in the uv plane shown in Fig. 10, and the function gH is defined by gH (u, v) = h(T (u, v), k(u, v)),

Figure 9. The Hudson et al. (1989) representation LH of beachball pattern space. It consists of four triangles in space that together are the analogue of the two cube half-faces LC in Fig. 5 or the lune LS in Fig. 6. Each triangle is itself planar, but no two of the triangles are coplanar. Vertices are at the double couple (1, 0, −1), the isotropic points ±(1, 1, 1), and the CLVDs (1, − 21 , − 12 ) and ( 12 , 12 , −1). The red and blue lines are coordinate contours for Hudson’s coordinates T and k, respectively. The larger diagram illustrates the parametrization in eq. (8); for T and k positive, the point h(T, 0) is on the deviatoric line segment between the double couple (1, 0, −1) and the CLVD ( 12 , 12 , −1), and then = h(T, k) is on the segment between h(T, 0) and the isotropic point (1, 1, 1).

(9)

Region A (Hudson’s terminology) is the orange portion of the first quadrant of the uv-plane in Fig. 10, and Region B is the orange portion of the third quadrant. In Fig. 10 the parallelogram DH has been subdivided into uv grid squares of area 1/6 × 1/6, and the corresponding ‘squares’ are shown on LH and LC . The squares E, E and E in the figure all represent the same set of patterns and so they all have the same cube probability PC . Likewise F, F and F have the same cube

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Figure 10. The Tk source-type plot as an equal area parametrization of the two cube half-faces LC . The mapping gH parametrizes points on the four triangles LH in terms of u and v. When gH is followed by the radial mapping πC to the cube, the resulting composite g is an equal area parametrization; it transforms areas in the parallelogram DH to areas in the cube by a constant factor of 3/2. Equal areas in DH therefore represent equal cube probabilities, which is what Hudson wanted. (Left) Hudson’s representation LH of beachball pattern space, the same as in Fig. 9 but with u and v-coordinate contours instead of T and k. The brain wants to bend the triangles along the two short red line segments, but the four triangles LH are unchanged from Fig. 9. (Middle) Parameter domain DH for the uv-coordinates. (Right) The radial projection of LH to the unit cube. Colours correspond in all diagrams, and the colours have the same meanings as in Fig. 14.

probability. But E and F have the same area on the cube and hence have the same cube probability. Hence E and F, back in DH , have the same cube probability. Thus equal areas in DH have the same cube probability; DH is an equal area representation of patterns with respect to cube probability. The congruency of the uv grid squares (parallelograms) on the cube makes it clear at a glance. In Fig. 10 the function that maps DH to the cube is g = πC gH . That is, g takes a point (u, v) ∈ DH to the four triangles and then projects it radially to the cube. The function g is much simpler than its component function gH (eqs 8–10). For (u, v) ∈ DH , ⎧ u u ⎪ v ≥ u/4 ⎨ 1, + v, −1 − + 2v 2 2 (11) g(u, v) = ⎪ ⎩ 1 − u + 2v, u + v, −1 v ≤ u/4. 2 2 The first case, v ≥ u/4, is orange in Fig. 10 and follows from eqs (1), (8) and (10). That case pertains to the λ1 = 1 face of the cube. The second case, v ≤ u/4, is green; it pertains to the λ3 = −1 face. From eq. (11), ⎫ ∂g = (0, 1/2, −1/2) ⎪ ⎪ ⎪ ⎪ ∂u ⎪ ⎪ ⎪ ⎬ ∂g = (0, 1, 2) v ≥ u/4. ∂v ⎪ ⎪ ⎪ ⎪ ⎪ ∂g ⎪ ∂g ⎪ ⎭ ∂u × ∂v = 3/2

(12)

Thus, all areas in the orange region of the parallelogram DH are being transformed to the cube with a constant Jacobian factor of 3/2. The same is true in the green. The function g is therefore an equal area parametrization of the cube half-faces LC . Again we see C


that Hudson’s source-type plot is an equal area representation with respect to cube probability. 7.3 Tk plot probability versus sphere probability Since Tk plot probability is cube probability, the comparison between Tk plot probability and sphere probability is already implicit in Fig. 7. Nevertheless, Fig. 11(c) makes the comparison explicit by showing contours of sphere probability density pS directly on the Tk source-type plot. As expected, the density pS is found to vary by a factor of 33/2 ≈ 5 over the plot (eq. 15). Thus if sphere probability is the ‘true’ probability, then there is considerable distortion in the Tk source-type plot. Patterns that appear uniformly distributed on the Tk source-type plot would in fact be about five times as dense near the linear vector dipoles as near the isotropic points. The two points ±(4/3, 1/3), like the two isotropic points, have minimum pS , and so the patterns would be as sparse there as at the isotropic points. To see how Fig. 11(c) is made: From eq. (6) the sphere probability density on the λ1 = 1 cube half-face is −3/2 1 + λ22 + λ23 , (13) pS (1, λ2 , λ3 ) = 2π/3 the normalizing factor 2π /3 being one-sixth of the surface area of the sphere. The contours of pS on the cube half-faces LC are then circles (Fig. 11a). They map to circles on the flattened faces DC using the inverse of gC of eq. (2). Then those circles map to ellipses on the Tk plot DH , using the matrix R=

4/3 −2/3 1/3 1/3

.

(14)

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W. Tape and C. Tape Since both gC and R are equal area mappings, the ellipses are indeed contours of sphere probability density on the Tk plot. And that density is, from eqs. (13), (2) and (14), ⎧ 2 −3/2 ⎪ ⎨ 9 u − uv + u + 5v 2 − 4v + 2 , v ≥ u/4 pS (u, v) = 4π 2 ⎪ ⎩ pS (−u, −v), v ≤ u/4. (15) 7.4 Tk coordinates Fig. 9 suggests the meaning of T and k: The coordinate k is a measure of the isotropic part of a pattern, and T is a measure of the extent to which the deviatoric part departs from being a pure double couple. In Fig. 12 the contours of T and k are plotted on three different surfaces: the two half-faces LC of the cube, Hudson’s four triangles LH , and the lune LS . The radial projection functions πC , πS and πH take one surface to another. On the lune the contours are not so different from the contours of γ and δ—the lines of longitude and latitude. On LH a contour of k consists of two contiguous line

Figure 11. Contours of probability density pS for sphere probability. Lighter shades have higher pS . (a) and (b) Contours of pS on the two cube half-faces LC and on the flattened cube faces DC . The contours are circles, with pS varying by a factor of 33/2 between the cube vertices (blue points) and the face centres (red points). (c) Contours of pS on the Tk source-type plot DH . This diagram is the result of distorting Fig. 11(b) using the linear mapping of eq. (14); the circles in Fig. 11(b) become ellipses, and the red squares A and B become parallelograms A and B . The red regions A, A , B, B all have cube probability PC = 1/64, but their sphere probabilities are PS (A) = PS (A ) = 0.028 and PS (B) = PS (B ) = 0.007, as indicated in Fig. 7.

Figure 12. Contours of T (red) and k (blue) on the lune LS , on the two cube half-faces LC , and on the four triangles LH of Hudson. Radial projection takes one diagram to another. The viewpoints are the same for all three diagrams, with the 111 direction vertical. The triangles of LH in the NW and SE quadrants coincide with cube faces. Fig. 1 shows some T and k contours in the uv-plane. While T and k are natural coordinates on the four triangles LH , their contours become awkward on the cube and in the uv-plane. C 2012 The Authors, GJI, 190, 499–510 C 2012 RAS Geophysical Journal International

Plots of moment tensor source types segments, and so the corresponding contour on LS must consist of two contiguous great circle arcs. The contour has a peculiar look, but it is still close to being a line of latitude. And a contour of T is exactly a line of longitude. Implicit in the above is the understanding that the Tk coordinates (and Hudson’s uv-coordinates) have been extended from LH to all of W by requiring that they be constant on rays. Explicitly, T, k (and m) coordinates are defined on W by the function (T, k, m) → m h(T, k),

m > 0 (h, T, k from eq. 8).

(16)

Viewed in hindsight, the Tk coordinates and the closely related four triangles LH are irrelevant to the equal area feature of Hudson’s Tk plots. In Fig. 10, for example, the set LH is dispensable; what matters is the equal area mapping g, going directly from the parallelogram DH to the cube. Eq. (8) defines the Tk coordinates and eq. (11) defines g; neither equation requires the other.

7.5 The distribution of the eigenvalue triples Hudson assumed that the eigenvalue triples were uniformly distributed within a cube max ≤ L. As explained in Section 7.1, it followed that the patterns were uniformly distributed on the unit cube (or the two half-faces). The initial assumption is suspect, since it implies that, up to a point, the number of events between scalar seismic moment M 0 and M 0 + M 0 should increase with increasing M 0 —in fact, that it should be proportional to M02 —whereas real earthquakes decrease in frequency with increasing M 0 (Gutenberg & Richter 1944). But the initial assumption could be replaced with the more general assumption that the (volume) probability density for an eigenvalue triple have the form p() = f (max ) for some function f . Patterns would still be uniformly distributed on the cube, and, with the proper choice of f , the number of events between M 0 and M 0 + M 0 could be made to tail off reasonably with increasing M 0 . But from the point of view of the sphere dweller— and of us—the distribution of s would retain some very peculiar directional dependencies.

8 NORMS AND SCALAR SEISMIC MOMENT Intuition suggests that the definition of scalar seismic moment M 0 should be tied to the choice of baseline probability for patterns: Patterns on a level surface for M 0 should all be equally likely with respect to the baseline probability. The moment M 0 is given by a norm defined for eigenvalue triples = (λ1 , λ2 , λ3 ). At least four norms have been considered. For ∈ W, 1/2 (17a) = λ21 + λ22 + λ23 max = max(|λ1 |, |λ2 |, |λ3 |)

(17b)

H = ISO max + DEV max

(17c)

GCMT =

|λ1 | + |λ3 | , 2

(17d)

where in eq. (17c) λ1 + λ2 + λ3 (1, 1, 1) 3 = − ISO .

ISO = DEV C


(18)

507

The conventional definition of seismic moment is M0 = √12 (e.g. Silver & Jordan 1982; Dahlen & Tromp 1998; Stein & Wysession 2003; Shearer 2009). That is, the definition is in terms of the standard Euclidian norm (eq. 17a). As an alternative, Silver & Jordan (1982) considered but did not adopt the definition M0 = max . A second alternative definition is that of Bowers & Hudson (1999), namely, M0 = H ; their definition has been used by Ford et al. (2009) and Walter et al. (2009). A third alternative definition is M0 = GCMT . That is the definition used in the GCMT catalogue (Dziewonski et al. 1981), but there the context is deviatoric tensors. Jost & Herrmann (1989) listed both the GCMT definition and the Euclidian definition, but their context is not completely clear. Bowers & Hudson (1999) apparently meant the GCMT definition to apply to all moment tensors, and as an illustration they calculated it for an isotropic tensor and a tensile crack. In W the level surfaces of the four norms in eqs (17a–d) are scaled versions of, respectively, the lune LS , the two cube half-faces LC , the four triangles LH of Hudson, and a surface that we will call LGCMT . Thus the definitions M0 = √12 and M0 = max are naturally associated with sphere probability and cube probability (respectively). The definitions M0 = H and M0 = GCMT are associated with probabilities whose densities are constant on LH and LGCMT . Fig. 13 shows level surfaces M 0 = 1 for the three alternate definitions of M 0 . Since the Euclidian moment M0 = √12 is proportional to the ordinary distance of from the origin, the values of the Euclidian moment are evident on the various surfaces. All four definitions agree at the double couple (1, 0, −1). On the surface in Fig. 13(a), where the max moment √ max is identically 1, the Euclidian moment varies by a factor of 3 between the face centre and the vertices. On the deviatoric segments (purple), it varies by a √ factor of 4/3. On the surface in Fig. 13(b), it is the Bowers and Hudson moment H that is identically 1. On that surface the variation of the Euclidian moment is the same as in Fig. 13(a), and indeed the extremes occur at the same points on the two surfaces. Incidentally, the norms max and H agree in the NW and SE quadrants, but not in the NE and SW. The definition M0 = H goes naturally with the four triangles LH , whereas the definition M0 = max goes naturally with the planar Tk source-type plot (Fig. 1), since the probability there is cube probability. On the surface in Fig. 13(c) the GCMT moment GCMT is identically 1. The Euclidian moment varies by a factor of 2 between the double couple (1, 0, √−1) and the point (2, 2, 0). On the deviatoric segments it varies by 4/3. The level surface GCMT = 1 (Fig. 13c) is the set LGCMT alluded to above. It is the portion of the square cylinder |λ1 | + |λ3 | = 2 that lies within the wedge W. The parallelogram on LGCMT is bounded by the wedge planes and by the two planes λ1 = 0 and λ3 = 0. On the parallelogram, since λ1 ≥ 0 ≥ λ3 , then GCMT = 12 (λ1 −λ3 ), which is the ‘double-couple moment’. That is, it is the Euclidian moment of the closest double couple to . But GCMT = 12 (λ1 − λ3 ) elsewhere on LGCMT . What if the double-couple moment were used everywhere? That is, what if moment were defined to be M0 = 12 (λ1 − λ3 ) (not a true norm), not just for λ1 ≥ 0 ≥ λ3 but everywhere on the wedge? Then the two triangles in Fig. 13(c) would be eliminated and the parallelogram would be extended to an infinite vertical strip on which Euclidian distances d would be unbounded. To sum up: We have four definitions of scalar seismic moment M 0 . One consideration in choosing among them is: On what sort of

508

W. Tape and C. Tape

Figure 13. Level surfaces M 0 ≡ 1 for three alternate definitions of seismic moment M 0 . The conventional definition M0 = 2−1/2 , in terms of the ordinary Euclidian distance d = , can be compared at a glance with the three alternate definitions; the circles on the surfaces are contours of d. (a) M0 = max (eq. 17b). The level surface is LC (Fig. 5). (b) M0 = H (eq. 17c). The level surface is LH (Fig. 9). (c) M0 = GCMT (eq. 17d). The level surface is LGCMT . It is the intersection of the square cylinder and the wedge W, and consists of a parallelogram and two triangles, the bottom one hidden. On the top triangle all eigenvalues are positive, and on the bottom all are negative. The GCMT catalogue uses this definition (i.e. M0 = GCMT ), but only for deviatoric tensors (purple line segment).

set of eigenvalue triples is it reasonable for M 0 to be constant? The Euclidian moment M0 = √12 is constant on the lune LS . The max moment M0 = max , the Bowers and Hudson moment M0 = H , and the GCMT moment M0 = GCMT are constant on, respectively, the two cube half-faces LC , the four triangles LH , and the parallelogram and two triangles LGCMT . Closely related to the four definitions of seismic moment are four choices for the baseline probability for beachball patterns (source types). The probability density can be chosen to be constant either on LS , LC , LH or LGCMT . To us the lune LS is the natural choice. Anything else should require compelling seismological justification.

LC (Fig. 10). A simpler equal-area parametrization of LC would be gC (Fig. 5), and there are many more. But the choice is not between different equal-area parametrizations of the two cube half-faces. The choice is between the cube and the sphere or, rather, between the cube half-faces LC and the lune LS . Which is the better representation for source-type space? The lune embodies what to us is a more natural and more useful hypothesis about eigenvalue probabilities. The lune is also consistent with the conventional definition of scalar seismic moment. And the lune is simple. We hope that Fig. 14 speaks for itself in this regard, but we also mention one example: The longitude γ and latitude δ on the lune are transparent measures that serve the same purpose as the more elusive T and k.

9 C O N C LU S I O N From our geometric point of view, the Hudson et al. (1989) sourcetype plot is an equal-area parametrization of the two cube half-faces

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



509

Figure 14. Five versions of beachball pattern space. At the left are the two cube half-faces LC and their parameter domain DC from Fig. 5. In the middle are Hudson’s four triangles LH and their parameter domain DH from Fig. 10. At right is the lune LS from Fig. 6. Familiar patterns (i.e. source types) are identified in the lower left diagram, and then the same colouring scheme is used throughout. ISO, isotropic; DC, double couple; LVD, linear vector dipole; CLVD, compensated linear vector dipole; C, tensile crack with Poisson ratio ν = 1/4. The purple segments (and arc on the lune) are the loci of deviatoric patterns, black are loci of crack plus double-couple patterns with ν = 1/4. Red segments correspond to cube edges, and yellow to diagonals of cube faces. The viewpoint is the same in the top three diagrams. Each of the three gets taken to another by radial projection.

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