Skip to content

Local-similarity (Fomel) misfit

Definition

Fomel (2007) introduces the local similarity attribute: a windowed normalised cross-correlation that varies smoothly in time:

\[ \gamma_\sigma(\tau) \;=\; \frac{\sum_t w_\sigma(t-\tau)\,d_{\mathrm s}(t)\,d_{\mathrm o}(t)} {\sqrt{\sum_t w_\sigma(t-\tau)\,d_{\mathrm s}^2(t)\,\cdot\,\sum_t w_\sigma(t-\tau)\,d_{\mathrm o}^2(t)}}, \]

with \(w_\sigma\) a Gaussian window of half-width \(\sigma\) samples. The implementation evaluates the three sliding sums by 1-D convolution with the same Gaussian kernel (replicate-padded so the result has the input length).

The FWI misfit is

\[ \mathcal J_{\mathrm{LS}}(m) \;=\; \tfrac12 \sum_{\text{trace}}\sum_\tau w_\sigma^E(\tau)\,(1-\gamma_\sigma(\tau))^2, \]

with an energy weight \(w_\sigma^E(\tau) = \tfrac{(d_{\mathrm s}^2\!\ast\!w_\sigma)(\tau)\cdot(d_{\mathrm o}^2\!\ast\!w_\sigma)(\tau)}{\max_\tau\!(\cdot)}\) that down-weights silent regions of the trace (so the boundaries do not contribute spurious (1-0)² = 1 to the misfit).

Properties

  • Amplitude-invariant per window: scaling either signal by any positive constant leaves \(\gamma_\sigma(\tau)\) unchanged. Tested explicitly.
  • Time-localised: a brief flipped-polarity window contributes strongly to the misfit, whereas the global NCC averages it down to almost nothing.

When to use

  • Time-lapse FWI where you want to localise the residual in time.
  • As an attribute / monitoring quantity to visualise the misfit alongside the inversion.

API

from sweep_loss import LocalSimilarityLoss, local_similarity_loss
LocalSimilarityLoss(sigma_samples=8.0)(syn, obs)
local_similarity_loss(syn, obs, sigma_samples=8.0)

Tests

tests/test_local_similarity.py checks:

  • zero for identical inputs (with energy-weighted boundary handling),
  • invariance under per-trace amplitude scaling,
  • monotone growth with small shifts,
  • a brief polarity flip is more visible to local than to global NCC,
  • gradients flow,
  • parameter validation.

References

  • Fomel, S. (2007). Local seismic attributes. Geophysics 72 (3), A29-A33. doi:10.1190/1.2437573