Weighted Iterative Sigma-Space Inversion (WISSI)

Combines the four classic fault slip inversion algorithms (Michael, 1983; Angelier, 1990; Yamaji and Sato, 2006; and Hansen, 2013) into a single coherent framework operating in the Yamaji-Sato 5-sphere sigma-space.

Usage

slip_inversion_wissi(
  x,
  weights = NULL,
  sigma_alpha_deg = 10,
  gamma_max = 10,
  n_anneal = 8L,
  max_iter = 50L,
  tol_deg = 1e-04,
  run_stage4 = TRUE
)

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

sigma_alpha_deg

Estimated slip direction measurement error in degrees. Used in Stage 4 analytic uncertainty. Default 10.

gamma_max

Maximum sense annealing sharpness parameter. Higher values commit more strongly to the predicted slip sense. Default 10. Set to 0 to disable sense annealing (fully sense-agnostic, like Hansen 2013).

n_anneal

Number of annealing steps (outer loop of Stage 3). Default 8.

max_iter

Maximum inner iterations per annealing step. Default 50.

tol_deg

Convergence tolerance in angular stress distance (degrees). Default 1e-4.

run_stage4

Logical. Compute analytic uncertainty (Stage 4). Default TRUE.

Value

A named list with:

stress_tensor

3x3 reduced stress tensor (Cartesian frame)

y

6D unit y-vector on \(S^5\)

principal_axes

unit vectors of principal stress axes (max to min)

principal_vals

eigenvalues of stress_tensor (decreasing)

alpha

per-fault angular misfit (unsigned, 0-90°)

alpha_signed

per-fault signed misfit (0-180°)

mean_alpha

mean angular misfit across all faults (°)

suspected_flipped

row indices where alpha_signed > 90°

n_flipped_sense

number of faults whose sense was corrected in Stage 3

slips_corrected

sense-corrected slip matrix used in final inversion

mu

per-fault magnitude weights from Stage 2/3

phi_sense

per-fault tanh sense confidence from Stage 3

eigenvalue_gap

\(\lambda_2 - \lambda_1\) of \(M5\) (condition number proxy)

M5_eigvals

all 5 eigenvalues of final \(M5\)

unc

Stage 4 uncertainty list (if run_stage4 = TRUE): Cov5: 5x5 covariance matrix in sigma-space; Cov_y6: 6x6 covariance matrix (y-space); eigval_gap: eigenvalue gap (same as above); cov_eigvals: eigenvalues of Cov5; sigma1_unc: approx 1\(\sigma\) uncertainty on \(\sigma_1\) orientation Phi_unc: approx 1\(sigma\) uncertainty on \(\phi\)

n_iter_total

total number of inner iterations

References

Stephan (in prep.)

See also

Other stress-inversion: Fault_PT(), slip_inversion(), slip_inversion_angelier(), slip_inversion_hansen(), slip_inversion_hansen_boot(), slip_inversion_michael(), slip_inversion_simple(), slip_inversion_yamaji_sato()

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_wissi(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) * "%"))
}))