The distribution of random moment tensors’ eigenvectors underpins some of the assumptions for source decompositions such as the Hudson decomposition. Hudson et al. (1989) assume a uniform prior distribution of eigenvalues.
However, there are several possible methods for generating the random moment tensors with different eigenvalue distributions, discussed in detail in Appendix B.
Fig. 2.24 shows that randomly generated normalised three dimensional matrices sampled using algorithm B.1 have an unordered eigenvalue distribution that is not uniform, and the ordered eigenvalues (Eq. 2.34) are strongly peaked distributions. Although the normalisation of the eigenvalues does affect the distributions, it does not affect the transformation to the Hudson or lune source parameters.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 Λ −1 0 1 0 Λ1 −1 0 1 0 Λ2 −1 0 1 0 Λ3
Figure 2.24: Distribution of unordered and ordered eigenvalues for a normalised random
distribution of moment tensors (Sampled using algorithm B.1).
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 Λ −1 0 1 0 Λ1 −1 0 1 0 Λ2 −1 0 1 0 Λ3
Figure 2.25: Distribution of unordered and ordered eigenvalues, following the assumptions of
Hudson et al.(1989). The red lines correspond to the normalised probability distributions which are given by Eqs 2.98 - 2.100 for the ordered eigenvalues(Hudson et al., 1989).
Hudson et al. show that for uniformly distributed eigenvalues (Fig. 2.25), the distributions
of the ordered eigenvalues are:
p (Λ1 = Λ) =
3
8(1 + Λ)
2
p (Λ2 = Λ) = 3 4 1 − Λ2, (2.99) p (Λ3 = Λ) = 3 8(1 − Λ) 2 . (2.100)
There is a factor of two difference between Eq. 2.99 and the PDF given in Hudson et al. Eq. 42, accounting for normalisation of the PDFs.
The assumptions2 to arrive at Hudson et al. Eqs 41 and 42 are valid for random moment tensors sampled using algorithm B.3 from a uniform eigenvalue distribution. Consequently, a plot of u, v has a uniform prior probability density for this sampling. This is easily illustrated by Fig. 2.26, which show the τ, k and u, v distribution for three different approaches to generating random moment tensors. Those generated from algorithm B.1 have a higher density just above and below the origin, and moment tensors sampled from a uniform distribution on the fundamental eigenvalue lune again have a non-uniform distribution in the τ, k and u, v plots. The distribution of moment tensors distributed according to Eqs. 2.98-2.100 is, as expected, uniform on the corresponding u, v plot, but not the τ, k plot.
Figure 2.26: τ, kand u, v distribution of moment tensors. (a) and (d) are random moment tensors
sampled from the multi-dimensional normal distribution (Section B.1 and Fig. 2.24), (b) and (e) are moment tensors with uniform random eigenvalues (Fig. 2.25) and (c) and (f) are moment tensors with eigenvalues distributed uniformly on the fundamental eigenvalue lune (Fig. 2.27).
2The description of Hudson et al. Eq. 30 as the “combined probability distribution for Mx, My, and Mz taking values X, Y , and Z respectively, (X > Z > Y )” reflects the sampling from the uniform box described by the uniform a priori distribution assumed in Hudson et al. Eq. 41. However, it is important to note that the triple can always be ordered appropriately, and that the probability reflects the probability of obtaining an eigenvalue triple (a, b, c) which is then ordered largest to smallest, and not that a always corresponds to X.
Eigenvalues generated from uniform sampling on the lune (Fig. 2.27) are uniform when unordered, due to the dimensionality of the inner product (Appendix A), but do not match the ordered Hudson et al. eigenvalue distribution (Fig. 2.25).
The distribution of these sources on the fundamental eigenvalue lune (Fig. 2.28) shows that the different approaches again have different patterns. The Hudson et al. eigenvalue distribution has a maximum at two points and strong sampling at the isotropic limits, but poor sampling at the double-couple points.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 Λ −1 0 1 0 Λ1 −1 0 1 0 Λ2 −1 0 1 0 Λ3
Figure 2.27: Distribution of unordered and ordered eigenvalues, following the assumptions of Tape
and Tape(2012b) to generate a uniform distribution of eigenvalues on the fundamental eigenvalue lune.
The distribution of the moment tensors sampled using algorithm B.1 has a maximum at the double-couple source and reduces outwards. It is also important to note that the moment tensors distributed according to 2.24 have very few if any samples at the implosive and explosive extrema, unlike the other case when the eigenvalues are uniformly distributed.
Figure 2.28: Lune type plot of the distribution of moment tensors. (a) is the distribution of random
moment tensors sampled from the six-dimensional normal distribution (Section B.1 and Fig. 2.24), (b) is the distribution of moment tensors with uniform random eigenvalues (Fig. 2.25) and (c) is the distribution of moment tensors with eigenvalues distributed uniformly on the fundamental eigenvalue lune (Fig. 2.27).
Tape and Tape(2012b) observe that when sampling the eigenvalues according to Hudson et al. “from the point of view of the sphere dweller - and of us - the distribution of Λs would
retain some very peculiar directional dependencies”, which concurs strongly with Fig. 2.28. This also shows that the fundamental eigenvalue lune is not an equal area plot with respect to random moment tensors sampled using algorithm B.1.
Using the probabilities derived in Appendix B (Eqs 8.15 and 8.16), it is possible to construct a parameterisation for which the source-type distribution is uniform. The trans- formation can be derived from the cumulative distribution functions (CDFs) for the parameter probability distribution functions (PDFs) (Eqs B.10 and B.11) to give
η = 1 2(sin 3γ + 1) , (2.101) ξ = B (x; 5.7479, 5.7479) B (5.7479, 5.7479) , (2.102) where B (x; α, β) = ´x 0 t α−1(1 − t)β−1
dt is the incomplete Beta function, and
CLVD CLVD Explosion Implosion DC TC + TC −
Figure 2.29: η − ξtype plot. The red lines
show constant Poisson’s ratios, clockwise from the deviatoric axis
ν = −1, −0.5, 0, 0.1, 0.25, 0.4, 0.5, with the dashed line corresponding to ν = 0.5.
B (α, β) = ´0∞ tα−1
(1+t)α+βdt is the complete Beta
function, or Euler Integral of the first kind. The source-type end members are at the edges of the plot (Fig. 2.29), and the distribution of the random moment tensors from algorithm B.1 is now uniform in these parameters, as shown in Fig. 2.30. Unfortunately, while this plot has the desirable equal area property for this distribution, it does not have the useful properties of the elastic parameters of the fundamental eigenvalue lune. Fig. 2.29 shows the contours of constant Pois- son’s ratio, which are great circles on the fundamental eigenvalue lune, and correspond to curved lines in this plot.
Figure 2.30: η, ξplot of the distribution of moment tensors. (a) is the distribution of random
moment tensors sampled from the six-dimensional normal distribution (Section B.1 and Fig. 2.24), (b) is the distribution of moment tensors with uniform random eigenvalues (Fig. 2.25) and (c) is the distribution of moment tensors with eigenvalues distributed uniformly on the fundamental eigenvalue lune (Fig. 2.27). The white lines correspond to grid lines for the η and ξ coordinates.