Direct inversion For Fault slip in $S^5$ — slip_inversion_yamaji

Stress tensor inversion via the Yamaji and Sato (2006) eigenvector method.

Usage

slip_inversion_yamaji_sato(x, weights = NULL, flip = FALSE)

Arguments

x: object of class "Pair" or "Fault" with at least 4 rows.
weights: numeric. Weightings for the faults. Must have the same length as x
flip: logical. Flip if you want to have the negative stress tensor, i.e. sigma 1 and 3 will be flipped.

Value

Same output as slip_inversion() plus

y: 6D unit y-vector on $S^5$ representing the tensor
alpha: per-fault angular misfit (unsigned, 0-90°)
mean_alpha: mean angular misfit across all faults

Details

A reduced stress tensor is represented as a unit 5-vector $x$ on the 5-sphere $S^5$, embedded in the 6D space of symmetric tensors via the $y$-vector (Eq. 6):

$$y = (1/\sqrt{2}) \times (s_{11}, s_{22}, s_{33}, \sqrt{2} \times s_{23}, \sqrt{2} \times s_{31}, \sqrt{2} \times s_{12})$$

with normalisation constraints (Eqs. 11-12):

$\sigma_I = s_{11} + s_{22} + s_{33} = 0$ (deviator)

$\sigma_{II} = (s_{11}^2+s_{22}^2+s_{33}^2)/2 + s_{23}^2+s_{31}^2+s_{12}^2 = 1$ (unit norm)

These constraints confine $y$ to the intersection of a hyperplane and the 6-sphere, which defines the 5-sphere $S^5$.

For each fault datum $\{n, \text{slip}\}$, define the 6-vector $\epsilon'$ (Eq. 8, derived from Eq. 4) such that the Wallace-Bott condition becomes:

$\epsilon'_i \cdot y = 0$ (shear stress perpendicular to $b_i = n_i \times \text{slip}_i$)

The optimal $y$ minimises $\sum_i (\epsilon'_i \cdot y)^2$, i.e. the smallest-eigenvalue eigenvector of $M = \sum_i \epsilon'_i \otimes \epsilon'_i$.

The computation is carried out in the 5D deviator subspace via an explicit orthonormal basis $B$ ($6 \times 5$), constructed as the complement of the deviator null direction $h = (1,1,1,0,0,0)/\sqrt{3}$, to avoid degeneracy in the $6\times 6$ projected matrix.

The sense condition (Eq. 5 / Eq. 10) selects between the two antipodal solutions via a majority vote on $\tau_i \cdot \text{slip}_i > 0$ (positive work).

Distance metrics

Angular stress distance (or Lode angle) $\Theta$ (Eq. 19): great-circle distance on $S^5$, $\Theta = \arccos{|y_1 \dot y_2|})$, range $[0, 90]$ deg for reduced tensors.

Orife-Lisle distance $D$ (Eq. 26): Euclidean chord on $S^5$, $D = 2 \sin(\Theta/2)$, range $[0, 2]$.

Michael distance $D_M$ (Eq. 34): $D_M = 1 - \cos(\Theta)$, range $[0, 2]$.

Relationship to misfit angles (Eq. 37): $\bar{d}$ approx $\Theta$, where $\bar{d}$ is the mean angular misfit between predicted and observed slip directions.

References

Yamaji, A., & Sato, K. (2006). Distances for the solutions of stress tensor inversion in relation to misfit angles that accompany the solutions. Geophysical Journal International, 167(2), 933–942. doi:10.1111/j.1365-246X.2006.03188.x

Examples

set.seed(20250411)

nx <- length(angelier1990)
par(mfrow = c(2, nx / 2))

invisible(lapply(seq_len(nx), function(i) {
  # inversion
  x <- angelier1990[[i]]
  res <- slip_inversion_yamaji_sato(x)

  # some stress shape
  phi_val <- round(res$stress_shape$phi, 2)

  # misfit
  rup_val <- round(res$misfit$rup_mean, 2)

  # Plot the faults (color-coded by RUP%) and show the principal stress axes
  stereoplot(title = names(angelier1990)[i], guides = FALSE)
  stereo_shmax(res$SHmax)
  fault_plot(x, col = assign_col(res$misfit$rup))
  points(res$principal_axes, col = 1:3, pch = 16, cex = 1.5)
  text(res$principal_axes,
    label = rownames(res$principal_axes),
    col = 1:3, adj = -.25
  )
  legend("topleft", col = 2:4, legend = rownames(res$principal_axes), pch = 16)
  title(sub = bquote(Phi == .(phi_val) ~ "|" ~ bar("RUP") == .(rup_val) * "%"))
}))

dev.off()
#> null device 
#>           1

Direct inversion For Fault slip in \(S^5\)