Skip to content

Optimal Transport of the Matching Filter (OTMF)

Definition

Sun & Alkhalifah (2019) combine adaptive matching-filter and OT ideas:

  1. Compute the Wiener filter \(w(\tau)\) that maps \(d_{\mathrm s}\ast w\) to \(d_{\mathrm o}\) (same regularised Wiener filter as in AWILoss).
  2. Preprocess \(w\) to a non-negative density \(\hat w(\tau)\) with one of positive ∈ {"square", "abs", "linear", "exp"}.
  3. Measure Wasserstein distance between \(\hat w\) and the Dirac delta at zero lag:
\[ W_2^2(\hat w, \delta_0) = \int \tau^2\,\hat w(\tau)\,\mathrm d\tau,\qquad W_1(\hat w, \delta_0) = \int |\tau|\,\hat w(\tau)\,\mathrm d\tau. \]

order=2 returns the second moment (default); order=1 returns the first absolute moment.

Why it works

For a pure time shift \(\Delta t\) between \(d_{\mathrm s}\) and \(d_{\mathrm o}\), the Wiener filter is a Dirac at lag \(\Delta t\) convolved with a small low-pass kernel. Its second moment is

\[ W_2^2(\hat w, \delta_0) \approx (\Delta t)^2 + \mathrm{baseline}, \]

so the OTMF misfit is convex and monotone in \(|\Delta t|\), even when L2 oscillates due to cycle skipping. The matching-filter step also absorbs amplitude / waveform discrepancies — OTMF is amplitude-invariant under positive scaling of either input.

API

from sweep_loss import OTMFLoss, otmf_loss
OTMFLoss(dt=1e-3, epsilon=1e-4, positive="square", order=2)(syn, obs)

Tests

tests/test_otmf.py checks:

  • small but non-zero baseline for identical inputs (low-pass-kernel width from the Tikhonov stabiliser),
  • amplitude invariance under positive rescaling of either input,
  • monotone growth with shift,
  • monotone across an L2 cycle-skipping range,
  • order=2 baseline-subtracted growth matches \((\Delta t)^2\) to ~15%,
  • order=1 monotone growth,
  • gradients flow,
  • parameter validation.

References

  • Sun, B. & Alkhalifah, T. (2019). Adaptive traveltime inversion. Geophysics 84 (4), U13-U29. doi:10.1190/geo2018-0595.1
  • Sun, B. & Alkhalifah, T. (2019). The application of an optimal transport to a preconditioned data matching function for robust waveform inversion. Geophysics 84 (6), R923-R945. doi:10.1190/geo2018-0413.1
  • Sun, B. & Alkhalifah, T. (2019). Stereo optimal transport of the matching filter. SEG Tech. Progr. Expanded Abstracts, pp. 1505-1509. doi:10.1190/segam2019-3199662.1