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Frequency- and Laplace-domain misfits

Five complementary misfits operating on the temporal Fourier transform \(D(\omega) = \sum_t d(t) e^{-i\omega t}\) are bundled here. All of them accept an inclusive frequency-index band freq_band=(k0, k1) so they double as multi-scale FWI building blocks (Bunks et al. 1995).

1. Frequency-domain L2 (Pratt-Shin-Hicks 1998)

\[ \mathcal J_\Omega = \tfrac12\sum_{\omega\in\Omega}|D_{\mathrm s}(\omega)-D_{\mathrm o}(\omega)|^2. \]

2. Phase-only frequency-domain (Bednar-Shin-Pyun 2007)

\[ \mathcal J_\phi = \tfrac12\sum_{\omega\in\Omega}|\Delta\Phi(\omega)|^2, \]

with \(\Delta\Phi = \arctan2(\sin(\Phi_{\mathrm s}-\Phi_{\mathrm o}), \cos(\Phi_{\mathrm s}-\Phi_{\mathrm o}))\) wrapped to \((-\pi,\pi]\). Amplitude weighting (\(w(\omega)=|D_{\mathrm o}(\omega)|/\max|D_{\mathrm o}|\)) suppresses bins where the amplitude is so small that the phase is meaningless; toggled with amplitude_weight=True/False.

3. Amplitude-only frequency-domain (Shin-Min 2006)

\[ \mathcal J_A = \tfrac12\sum_{\omega\in\Omega}(|D_{\mathrm s}|-|D_{\mathrm o}|)^2. \]

Because \(|D(\omega)|\) is invariant under a time shift \(t\to t+\tau\), this misfit is completely insensitive to traveltime errors — useful when you want to invert the amplitude separately from kinematics.

4. Shin–Min log misfit (Shin & Min 2006)

\[ \mathcal J_{\log} = \tfrac12\sum_{\omega\in\Omega}|\log D_{\mathrm s}(\omega)-\log D_{\mathrm o}(\omega)|^2. \]

Using \(\log D = \log|D|+i\,\arg D\), this decomposes cleanly as

\[ \mathcal J_{\log} = \underbrace{\tfrac12\sum(\log|D_{\mathrm s}|-\log|D_{\mathrm o}|)^2}_{\text{log-amplitude misfit}} \;+\; \underbrace{\tfrac12\sum|\Delta\Phi|^2}_{\text{phase misfit}}. \]

We wrap the phase difference to \((-\pi,\pi]\) to avoid the unwrapping pathologies of the naive complex logarithm (Choi & Alkhalifah 2013).

5. Laplace-domain L2 (Shin & Cha 2008)

The data are first damped by \(e^{-st}\) (\(s>0\) a real damping rate) and then compared in time-domain L2. Equivalent to a time-domain L2 on \(\hat d(t) = e^{-st} d(t)\). Strongly weighting early arrivals enables macro-velocity recovery from cycle-skipped data (Shin & Cha 2008).

The Laplace–Fourier variant (Shin & Cha 2009) uses complex damping \(s=\sigma+i\omega\); in this library that is recovered by combining LaplaceL2Loss(s=σ, dt=dt) with FrequencyDomainL2Loss(freq_band=...) on the damped data.

API

from sweep_loss import (
    FrequencyDomainL2Loss, FrequencyPhaseLoss, FrequencyAmplitudeLoss,
    LogarithmicShinMinLoss, LaplaceL2Loss,
)

FrequencyDomainL2Loss(freq_band=(5, 30))(syn, obs)
FrequencyPhaseLoss(amplitude_weight=True)(syn, obs)
FrequencyAmplitudeLoss()(syn, obs)
LogarithmicShinMinLoss()(syn, obs)
LaplaceL2Loss(s=2.0, dt=1e-3)(syn, obs)

Tests

tests/test_frequency.py checks:

  • FrequencyDomainL2Loss matches the explicit 0.5 * sum |rfft(s)-rfft(o)|^2,
  • band restriction \(\Omega \subsetneq\) all-frequencies gives a strictly smaller misfit,
  • all four loss classes give 0 for identical inputs,
  • LogarithmicShinMinLoss equals (log_amp)^2 + (wrap dphi)^2 numerically,
  • FrequencyAmplitudeLoss is essentially invariant to a small time shift,
  • phase-only misfit grows monotonically with shift magnitude,
  • LaplaceL2Loss reproduces the explicit damped-L2 formula,
  • gradients flow through all five misfits,
  • invalid arguments raise.

References

  • Pratt, R. G., Shin, C. & Hicks, G. J. (1998). Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion. Geophys. J. Int. 133 (2), 341-362. doi:10.1046/j.1365-246X.1998.00498.x
  • Shin, C. & Min, D.-J. (2006). Waveform inversion using a logarithmic wavefield. Geophysics 71 (3), R31-R42. doi:10.1190/1.2194523
  • Bednar, J. B., Shin, C. & Pyun, S. (2007). Comparison of waveform inversion, part 2: phase approach. Geophys. Prospect. 55 (4), 465-475. doi:10.1111/j.1365-2478.2007.00618.x
  • Shin, C. & Cha, Y. H. (2008). Waveform inversion in the Laplace domain. Geophys. J. Int. 173 (3), 922-931. doi:10.1111/j.1365-246X.2008.03768.x
  • Shin, C. & Cha, Y. H. (2009). Waveform inversion in the Laplace-Fourier domain. Geophys. J. Int. 177 (3), 1067-1079. doi:10.1111/j.1365-246X.2009.04102.x
  • Bunks, C., Saleck, F. M., Zaleski, S. & Chavent, G. (1995). Multiscale seismic waveform inversion. Geophysics 60 (5), 1457-1473. doi:10.1190/1.1443880
  • Choi, Y. & Alkhalifah, T. (2013). Frequency-domain waveform inversion using the phase derivative. Geophys. J. Int. 195 (3), 1904-1916. doi:10.1093/gji/ggt351