Carl Tape's website - beachball gallery

A geometric perspective on seismic moment tensors

Below are some example figures related to the papers listed here.

I have posted online some matlab scripts, GMT scripts, and mathematica scripts relevant for interpreting and plotting moment tensors. These files are available within the compearth repository on github.

They can be downloaded from github as a zipped file or using git from the command line.

Please email me with corrections or suggestions.

Beachball representing slip on a fault plane (Tape and Tape, 2012, Figure 13).

(a) The fundamental lune L [orange], 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 latitude δ = 90, 47, 20, 0, −20, −53, −90. (b) Five deviatoric beachballs. The plane of the screen is the deviatoric plane. [Tape and Tape, 2012a, Fig. 11]

Home for beachball patterns (moment tensor source types)—the fundamental lune 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. 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. [Tape and Tape, 2012a]

Five published compilations of full moment tensors, represented on the fundamental lune. The dashed line is the crack + double couple arc for ν = 1/4. [Tape and Tape, 2012a]

2D version of the fundamental lune and example data sets. See 3D version above.

[above] Beachballs plotted on the 2D version of the fundamental lune.The double couple is at the center; its orientation changes for each example.

Four regimes for moment tensors. Above and to the right of the arc λ2 = 0 (blue), moment tensor beachballs have red bands and white caps (λ2 > 0 > λ3). Below and to the left of it they have white bands and red caps (λ2 > 0 > λ1). Above the arc λ3 = 0 (red), beachballs are all red (λ3 > 0). Below the arc λ1 = 0 (white), they are all white (λ1 < 0). [Tape and Tape, 2013, Fig. S1]

Some moment tensors with middle eigenvalue = 0. The angle alpha (plotted as orange contours) is the angle between the fault normal vector and the slip vector. For moment tensors with λ2 = 0, alpha gives the angular extent of either of the white lunes of the beachball.

Some crack tensors, which lie on the boundary of the lune. [Tape and Tape, 2013]

phi = 15 (zeta = 90)

phi = 45 (zeta = 90)

phi = 75 (zeta = 90)

phi = 105 (zeta = 90)

[above] Alternative depiction of four of the beachballs represented in Tape and Tape (2013), Figure 6. (These figures were plotted using a Mathematica notebook available in the compearth github repository.)

Any moment tensor (M) can be expressed a linear combination of two orthogonal tensors: a crack tensor (K) and a double couple (D) such that the crack plane of K is a shear plane of D (Minson et al., 2007). A key point is that K and D do not have the same basis. The angle zeta measures the amount of K (crack) within M; it ranges from zeta = 0 at the center (D) to zeta = 90 at the lune boundary. note: zeta is not the same as the arc distance to the lune center point. [Tape and Tape, 2013]

Previously published moment tensors of real events, here plotted on the lune. [Tape and Tape, 2013]

Previously published moment tensors for induced events, here plotted on the lune. The line between (1,0,0) and (0,0,-1) marks the subset of moment tensors with middle eigenvalue = 0. [Tape and Tape, 2013, Fig. S14]

The uniform parameterization of moment tensors. The parameterization handles moment tensor eigenvalues (top row) separately from orientation parameters (bottom row). [Tape and Tape, 2015, Fig. 1]

Lune with contour plot of probability density for eigenvalue triples (λ1, λ2, λ3) of moment tensors. The density is maximum for double couples (center of lune) and falls off to zero at the lune boundary, where λ2 = λ1 or λ2 = λ3. [Tape and Tape, 2015, Fig. 2]

Geometric representation of the probability distribution for ξ₀, the minimum rotation angle between two sets of principal axes. The green surface is the portion of the sphere that is on or inside the cube. As t increases, the sphere grows and the surrounding cube shrinks. The probability density (above) is proportional to the area of the green surface. The letters in the plot above correspond to the letters (or t values) at the left. The fifth percentile, tau05 = 35.7^o, is labeled. The angle varies from 0^o to 120^o (t3 = 2π/3). [Tape and Tape, 2012c]

Figures motivating the use of ω_dc instead of ξ₀ for quantifying the difference between two double couple moment tensors (Figures 14 and 15, Tape and Tape, 2012, "Angle between principal axis triples").

Left (Fig. 14). The moment tensor M0 is at the origin. The measure ξ₀considers the difference in orientation between M0 and M1 to be the same as that between M0 and M2, namely ξ₀ = π/2 in both cases. Yet, intuitively, the beachball M1 resembles M0 more closely than does M2. In fact, since their colors are reversed, M2 and M0 are as different as moment tensors of the same size can be, whereas M0 and M1 are not so different from each other, especially in the x direction, where both are red. Since all three beachballs have the same eigenvalue triple, any differences in appearance ought to be attributable to differences in orientation. Since the parameter ξ₀does not capture these differences, ξ₀ may not always be the best measure of separation for moment tensor orientations.

Right (Fig. 15). Comparison of ξ₀and ω for 90^o rotations. The angle ξ₀ is constant, ξ₀ = π/2, whereas ω_dc varies from π/3 at U1 to π at U2. The orientations U1 and U2 are the 90^o rotations about the x- and y-axes, the same as in Fig. 14. The ω_dc values for U1 and U2 seem to capture our intuition from Fig. 14 better than do the ξ₀ values.

Probability density of random moment tensors, displayed on three different source-type plots. The distribution is uniform on the v-w rectangle. It is elevated on the center of the eigenvalue lune (the center is a double couple). It is peculiar on the T-k plot.