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Instantaneous-phase and envelope+phase misfits

Let the analytic signal be \(a(t) = d(t) + i\,\mathcal H[d](t) = E(t)\,e^{i\phi(t)}\) with envelope \(E\) and instantaneous phase \(\phi\).

Definitions

Instantaneous-phase misfit (Bozdağ, Trampert & Tromp 2011, eq. 22):

\[ \mathcal J_\phi(m) = \tfrac{1}{2}\sum \bigl[w_\phi(t)\,\Delta\phi(t)\bigr]^2, \]

where \(\Delta\phi = \operatorname{wrap}(\phi_{\mathrm s}-\phi_{\mathrm o})\) and \(\operatorname{wrap}(x)=\arctan2(\sin x,\cos x)\in(-\pi,\pi]\) is the differentiable wrapped phase difference. The optional envelope weight \(w_\phi(t) = E_{\mathrm o}(t)/\max_t E_{\mathrm o}\) damps samples where the envelope is tiny (and the phase is meaningless).

Envelope + phase combined (Fichtner 2008; Yuan, Simons & Tromp 2016):

\[ \mathcal J_{E+\phi}(m) = (1-\alpha)\,\mathcal J_E + \alpha\,\mathcal J_\phi. \]

With \(\alpha=0\) → pure envelope misfit (Wu 2014); with \(\alpha=1\) → pure phase misfit; both endpoints reproduce the corresponding stand-alone losses exactly (asserted in the tests).

Properties

  • Amplitude scaling of syn: phase is invariant under a positive scalar multiplication; the envelope-weighted variant inherits this invariance (up to round-off).
  • Polarity sensitive: flipping the sign of syn shifts the phase by \(\pi\), so \(\mathcal J_\phi\) grows by a non-trivial amount — unlike the envelope misfit.
  • Bounded residual: \(|\Delta\phi| \le \pi\) ⇒ \(\mathcal J_\phi \le \tfrac12 \pi^2 \cdot N_{\mathrm{samples}}\).

When to use

  • Mitigating cycle-skipping while still using waveform information.
  • Combined envelope+phase inversion: \(\alpha\) controls a "low-frequency first, then high-frequency" hierarchy.

API

from sweep_loss import InstantaneousPhaseLoss, EnvelopePhaseLoss
InstantaneousPhaseLoss(envelope_weight=True)(syn, obs)
EnvelopePhaseLoss(alpha=0.5, envelope_log=False)(syn, obs)

Tests

tests/test_inst_phase.py checks:

  • perfect match → 0,
  • amplitude scaling of syn leaves the envelope-weighted misfit invariant (round-off tolerance only),
  • monotone growth with small time shifts,
  • polarity sensitivity (vs. the envelope misfit, which is not polarity sensitive),
  • wrapped-phase residual is always in \((-\pi,\pi]\),
  • \(\alpha=0\) matches the pure envelope loss,
  • \(\alpha=1\) matches the pure phase loss,
  • linear combination at intermediate \(\alpha\),
  • gradients flow,
  • module == functional aliases.

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
  • Fichtner, A., Kennett, B. L. N., Igel, H. & Bunge, H.-P. (2008). Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain. Geophys. J. Int. 175 (2), 665-685. doi:10.1111/j.1365-246X.2008.03923.x
  • Yuan, Y. O., Simons, F. J. & Tromp, J. (2016). Double-difference adjoint seismic tomography. Geophys. J. Int. 206 (3), 1599-1618. doi:10.1093/gji/ggw233