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Envelope misfits

Let \(E(t)=|a(t)|=\sqrt{d(t)^2+\mathcal H[d](t)^2}\) be the instantaneous envelope (modulus of the analytic signal). sweep_loss provides four envelope-based variants, all triggered through a single class EnvelopeLoss with boolean / integer switches.

Definitions

  • p = 2 (default), Wu, Luo & Wu (2014): \(\mathcal J_E^{(2)} = \tfrac12 \sum (E_{\mathrm s}-E_{\mathrm o})^2\).
  • p = 1: \(\mathcal J_E^{(1)} = \sum |E_{\mathrm s}-E_{\mathrm o}|\).
  • log (Bozdağ, Trampert & Tromp 2011, eq. 14): \(\mathcal J_E^{\log} = \tfrac12 \sum (\log E_{\mathrm s} - \log E_{\mathrm o})^2\).
  • squared (Chi, Dong & Liu 2014): \(\mathcal J_{E^2} = \tfrac12 \sum (E_{\mathrm s}^2 - E_{\mathrm o}^2)^2\).

All four operate sample-wise after Hilbert-transforming along the time axis.

Why it helps FWI

The envelope is non-oscillatory — it carries the low-frequency "modulation" of the signal even when the carrier waveform itself is heavily cycle-skipped. Inverting the envelope first gives a kinematic update that bypasses cycle-skipping; the subsequent waveform inversion can then converge in a much narrower basin (Wu, Luo & Wu 2014).

Sign invariance

\(E(t)=|a(t)|\) is invariant under \(d \mapsto -d\), so the envelope misfit cannot tell two polarity-flipped waveforms apart. This is a feature for amplitude-only inversion but a foot-gun for polarity-sensitive problems — add a polarity-aware term (or use the instantaneous-phase misfit) if you need it.

API

from sweep_loss import EnvelopeLoss
EnvelopeLoss(p=2)(syn, obs)                  # Wu 2014
EnvelopeLoss(log=True)(syn, obs)             # Bozdağ 2011 eq. 14
EnvelopeLoss(squared=True)(syn, obs)         # Chi 2014

Tests

tests/test_envelope.py checks:

  • envelope() matches scipy.signal.hilbert to float64 precision (with eps=0),
  • perfect match → 0 for all four variants,
  • numerical equality with the explicit per-sample formulas,
  • polarity invariance,
  • analytic-signal orthogonality property <Re(a), Im(a)> ≈ 0,
  • gradients flow,
  • invalid parameter combinations raise.

References

  • Bozdağ, E., Trampert, J. & Tromp, J. (2011). Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements. Geophys. J. Int. 185 (2), 845-870. doi:10.1111/j.1365-246X.2011.04970.x
  • Wu, R.-S., Luo, J. & Wu, B. (2014). Seismic envelope inversion and modulation signal model. Geophysics 79 (3), WA13-WA24. doi:10.1190/geo2013-0294.1
  • Chi, B., Dong, L. & Liu, Y. (2014). Full waveform inversion method using envelope objective function without low frequency data. J. Appl. Geophys. 109, 36-46. doi:10.1016/j.jappgeo.2014.07.010