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Deconvolution-based misfit (Luo–Sava)

Definition

For each trace, deconvolve the observation by the synthetic:

\[ \Psi(\tau) \;=\; \mathcal F^{-1}\!\Bigl(\tfrac{D_{\mathrm o}(\omega)}{D_{\mathrm s}(\omega)+\epsilon}\Bigr). \]

If the model is perfect, \(\Psi\) is a unit impulse at zero lag. The Luo–Sava misfit penalises departures from that with \(P(\tau)=\tau^2\):

\[ \mathcal J_{\mathrm{Decon}}(m) \;=\; \tfrac{1}{2}\sum_{\text{trace}}\sum_\tau \tau^2\,\Psi(\tau)^2. \]

The optional normalize=True mode divides by \(\sum_\tau \Psi^2\) to make the misfit amplitude-invariant, recovering the "AWI-style" variant of Choi & Alkhalifah (2018).

Relation to AWI

  • AWI convolves \(d_{\mathrm s}\) with a Wiener filter to match \(d_{\mathrm o}\), then penalises the lag spread of that filter normalised by its own L2-norm.
  • Luo–Sava deconvolves the observation by the synthetic, then penalises the lag spread of that deconvolution. Without normalisation it is amplitude-sensitive; with normalisation it is amplitude-invariant and closer to AWI in spirit.

API

from sweep_loss import DeconvolutionLoss
DeconvolutionLoss(dt=1e-3, epsilon=1e-4, normalize=True)(syn, obs)

Tests

tests/test_deconvolution.py checks:

  • normalize=True is invariant under positive amplitude scaling,
  • normalize=False is not (sanity check that the option matters),
  • monotone growth with small shifts,
  • monotone across a range where L2 cycle-skips,
  • gradients flow,
  • parameter validation.

References

  • Luo, S. & Sava, P. (2011). A deconvolution-based objective function for wave-equation inversion. SEG Tech. Progr. Expanded Abstracts, pp. 2788-2792. doi:10.1190/1.3627773
  • Choi, Y. & Alkhalifah, T. (2018). Time-domain full-waveform inversion of exponentially damped wavefield using the deconvolution-based objective function. Geophysics 83 (2), R77-R88. doi:10.1190/geo2017-0057.1
  • Zhu, H. & Fomel, S. (2016). Building good starting models for FWI using adaptive matching filtering misfit. Geophysics 81 (5), U61-U72. doi:10.1190/geo2015-0596.1