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2-Wasserstein (Engquist–Froese / Yang) misfit

Definition

For two 1-D densities \(f, g\) with CDFs \(F, G\),

\[ W_2^2(f, g) \;=\; \int_0^1 \bigl(F^{-1}(z) - G^{-1}(z)\bigr)^2\,\mathrm dz. \]

We sample \(z\) uniformly on \((0,1)\) (n_quantiles knots, default 256), linearly interpolate the inverse CDFs from the cumulative sums of the two trace densities, and Monte-Carlo the integral. The loss returned is \(\tfrac12 W_2^2\) following the FWI convention of Yang, Engquist, Sun & Hamfeldt (2018).

Signed traces are first mapped to non-negative densities via the same positive argument as the NIM and W1 losses.

Key property

For a translated Gaussian density of shift \(\Delta t\),

\[ W_2^2 \;=\; (\Delta t)^2, \]

i.e. \(\tfrac12 W_2^2 = \tfrac12 (\Delta t)^2\) which is quadratic and convex in the shift. This is the cycle-skipping cure proposed by Engquist & Froese (2014). Tested in test_w2_squared_equals_shift_squared_for_translated_gaussian.

API

from sweep_loss import Wasserstein2Loss, w2_loss
Wasserstein2Loss(positive="square", dt=1e-3, n_quantiles=256)(syn, obs)

Tests

tests/test_w2.py checks:

  • zero for identical signals across positive transforms,
  • analytic identity \(\tfrac12 W_2^2 = \tfrac12(\Delta t)^2\) to relative 5% on translated Gaussians,
  • monotone growth with shift,
  • monotone across a range where L2 cycle-skips,
  • gradients flow,
  • parameter validation.

References

  • Engquist, B. & Froese, B. D. (2014). Application of the Wasserstein metric to seismic signals. Commun. Math. Sci. 12 (5), 979-988. doi:10.4310/CMS.2014.v12.n5.a7
  • Engquist, B., Froese, B. D. & Yang, Y. (2016). Optimal transport for seismic full waveform inversion. Commun. Math. Sci. 14 (8), 2309-2330. doi:10.4310/CMS.2016.v14.n8.a9
  • Yang, Y., Engquist, B., Sun, J. & Hamfeldt, B. F. (2018). Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion. Geophysics 83 (1), R43-R62. doi:10.1190/geo2016-0663.1