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Exponentiated-phase misfit (Yuan et al. 2020)

Definition

Normalise the analytic signal by its envelope:

\[ \tilde s(t) = \frac{s(t) + i\,\mathcal H[s](t)}{E_s(t)} = e^{i\phi_s(t)},\qquad E_s(t) = \sqrt{s^2(t)+\mathcal H^2[s](t)}. \]

The exponentiated-phase misfit (Yuan et al. 2020, eq. 7) is the L2 distance between the two unit-modulus complex signals:

\[ \chi_{\mathrm{EP}}(m) = \tfrac{1}{2}\sum \int_0^T |\Re\tilde s - \Re\tilde d|^2 + |\Im\tilde s - \Im\tilde d|^2 \,\mathrm dt. \]

Why "better than instantaneous phase"

InstantaneousPhaseLoss (Bozdağ et al. 2011) takes \(\phi = \arctan(\mathcal H[d]/d)\) and subtracts; this has a branch cut at \(\phi = \pm\pi\) that must be wrapped (we use atan2(sin, cos)). The exponentiated phase replaces this with a smooth division and never crosses the branch cut: it is everywhere \(C^\infty\).

Properties

  • Amplitude invariant (exactly when eps=0): \(\tilde{(\alpha s)} = \tilde s\) for any \(\alpha > 0\). With eps > 0 the invariance is broken by \(O(\epsilon/E_s^2)\) — set eps=0 if you can guarantee no zero-amplitude samples enter the loss.
  • Bounded: \(|\Delta\Re|^2 + |\Delta\Im|^2 \le 4\) per sample, so the total misfit is at most \(2N_{\mathrm{samples}}\).
  • Polarity sensitive: flipping \(s\to -s\) rotates \(\tilde s\) by \(\pi\), so the loss is large (this distinguishes it from the envelope misfit).

API

from sweep_loss import ExponentiatedPhaseLoss
ExponentiatedPhaseLoss(eps=1e-8)(syn, obs)

Tests

tests/test_exponentiated_phase.py checks:

  • zero for identical signals,
  • amplitude invariance with eps=0,
  • per-sample upper bound of 2,
  • monotone growth with shift,
  • polarity sensitivity,
  • no jump across the \(\pm\pi\) branch cut (this is the main motivation for the loss — verified on a sinusoid swept through \(\pi\)),
  • gradients flow,
  • parameter validation.

References

  • Yuan, Y. O., Bozdağ, E., Ciardelli, C., Gao, F. & Simons, F. J. (2020). The exponentiated phase measurement, and objective-function hybridisation for adjoint waveform tomography. Geophys. J. Int. 221 (2), 1145-1164. doi:10.1093/gji/ggaa063
  • Gao, F., Yuan, Y. O., Ciardelli, C., Simons, F. J., Bozdağ, E. & Tromp, J. (2023). Review of misfit functions for adjoint full waveform inversion in seismology. Geophys. J. Int. 235 (3), 2794-2820. doi:10.1093/gji/ggad372