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sweep_loss — formula report & bibliography

This document gathers, in one place, the mathematical definition and DOI-linked source paper of every misfit (loss) implemented in the sweep_loss package. It is also the canonical landing page for the mkdocs site (docs/report.md).

Tensor layout throughout:

\[ d_{\mathrm s}, d_{\mathrm o} \in \mathbb R^{n_s\times n_t\times n_r\times n_c} \quad(\text{shots} \times \text{time} \times \text{receivers} \times \text{channels}) \]

with residual \(r = d_{\mathrm s} - d_{\mathrm o}\).

Family Misfits implemented
A. Data-domain Lp / robust M-estimators L2, L1, Huber, pseudo-Huber, hybrid L1/L2, Cauchy, Tukey, Geman-McClure, Student-t
B. Correlation / amplitude-normalised trace-normalised L2, global correlation (NCC)
C. Travel-time / picking cross-correlation travel-time
D. Envelope & phase envelope (p=1,2 / log / squared), instantaneous-phase, exponentiated phase, envelope+phase combined, time-frequency phase/envelope
E. Frequency / Laplace domain freq-L2 (Pratt), phase-only, amplitude-only, Shin-Min log, Laplace L2
F. Convolution / matching filter Adaptive Waveform Inversion (Warner-Guasch), Luo-Sava deconvolution, OT of Matching Filter (OTMF)
G. Density-domain divergences NIM (Liu / Donno), Jensen-Shannon divergence (Yan 2024)
H. Optimal transport W1 (Métivier), W2 via CDF (Engquist-Froese / Yang), Sinkhorn divergence, GSOT (graph-space)
I. Dynamic warping Soft-DTW
J. Local attribute local similarity (Fomel)

A total of 34 PyTorch loss classes implementing ≈ 38 named misfits across families A-J (the variants of EnvelopeLoss, FrequencyXxxLoss, SinkhornLoss(debiased=...), OTMFLoss(order=1|2) count as separate misfits in the literature).

All references below carry a DOI link. Sections D-H reference the same "anti-cycle-skipping property" — these are misfits that remain monotone with respect to a time shift on a Ricker, well past the half-wavelength shift where the plain L2 misfit oscillates. This is exercised in the corresponding test files (test_<family>.py).

A. Data-domain Lp norms and robust M-estimators

A.1 L2 / least-squares — L2Loss

\[ \mathcal J_{L_2} = \tfrac12 \sum r^2. \]
  • Lailly, P. (1983). The seismic inverse problem as a sequence of before-stack migrations. In: Conference on Inverse Scattering, SIAM, 206-220. (no DOI)
  • Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics 49 (8), 1259-1266. doi:10.1190/1.1441754
  • Virieux, J. & Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics 74 (6), WCC1- WCC26. doi:10.1190/1.3238367

A.2 L1 — L1Loss

\[ \mathcal J_{L_1} = \sum |r|. \]
  • Crase, E., Pica, A., Noble, M., McDonald, J. & Tarantola, A. (1990). Robust elastic nonlinear waveform inversion: application to real data. Geophysics 55 (5), 527-538. doi:10.1190/1.1442864
  • Brossier, R., Operto, S. & Virieux, J. (2010). Which data residual norm for robust elastic frequency-domain full waveform inversion? Geophysics 75 (3), R37-R46. doi:10.1190/1.3379323

A.3 Huber — HuberLoss

\[ h_\delta(r) = \begin{cases}\tfrac12 r^2 & |r|\le\delta\\ \delta(|r|-\tfrac12\delta) & |r|>\delta\end{cases}. \]
  • Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Stat. 35 (1), 73-101. doi:10.1214/aoms/1177703732
  • Guitton, A. & Symes, W. W. (2003). Robust inversion of seismic data using the Huber norm. Geophysics 68 (4), 1310-1319. doi:10.1190/1.1598124

A.4 Pseudo-Huber / Hybrid L1/L2 — PseudoHuberLoss, HybridL1L2Loss

\[ \tilde h_\delta(r) = \delta^2\bigl(\sqrt{1+(r/\delta)^2}-1\bigr). \]
  • Bube, K. P. & Langan, R. T. (1997). Hybrid L1/L2 minimisation with applications to tomography. Geophysics 62 (4), 1183-1195. doi:10.1190/1.1444219
  • Ha, T., Chung, W. & Shin, C. (2009). Waveform inversion using a back-propagation algorithm and a Huber function norm. Geophysics 74 (3), R15-R24. doi:10.1190/1.3112572
  • Charbonnier, P., Blanc-Feraud, L., Aubert, G. & Barlaud, M. (1997). Deterministic edge-preserving regularisation in computed imaging. IEEE Trans. Image Process. 6 (2), 298-311. doi:10.1109/83.551699

A.5 Cauchy / Lorentzian — CauchyLoss

\[ \rho^{\mathrm C}_c(r) = \tfrac{c^2}{2}\log(1+(r/c)^2). \]
  • Black, M. J. & Anandan, P. (1996). The robust estimation of multiple motions. CVIU 63 (1), 75-104. doi:10.1006/cviu.1996.0006
  • Crase, E., Pica, A., Noble, M., McDonald, J. & Tarantola, A. (1990). Robust elastic nonlinear waveform inversion: application to real data. Geophysics 55 (5), 527-538. doi:10.1190/1.1442864

A.6 Tukey biweight — TukeyLoss

\[ \rho^{\mathrm T}_c(r) = \begin{cases}\tfrac{c^2}{6}\bigl(1-(1-(r/c)^2)^3\bigr) & |r|\le c\\ c^2/6 & |r|>c\end{cases}. \]
  • Beaton, A. E. & Tukey, J. W. (1974). The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16, 147-185. doi:10.1080/00401706.1974.10489171
  • Bube, K. P. & Nemeth, T. (2007). Fast line searches for the robust solution of linear systems in the hybrid l1/l2 and Huber norms. Geophysics 72 (2), A13-A17. doi:10.1190/1.2431639

A.7 Geman-McClure — GemanMcClureLoss

\[ \rho^{\mathrm{GM}}_c(r) = c^2 \frac{(r/c)^2}{1+(r/c)^2}. \]
  • Geman, S. & McClure, D. E. (1985). Bayesian image analysis: an application to single photon emission tomography. Proc. Stat. Comp. Sect., ASA, 12-18. (no DOI; see permanent record)

A.8 Student-t negative log-likelihood — StudentTLoss

\[ \rho^{\nu,\sigma}(r) = \tfrac{\nu+1}{2}\log\bigl(1+\tfrac{1}{\nu}(r/\sigma)^2\bigr). \]

Limits: \(\nu = 1\) is Cauchy; \(\nu\to\infty\) is Gaussian / L2.

  • Aravkin, A. Y., van Leeuwen, T. & Herrmann, F. J. (2011). Robust FWI using Student-t distribution. SEG Tech. Progr. Expanded Abstracts, pp. 2669-2673. doi:10.1190/1.3627747
  • Aravkin, A., Burke, J. V. & Friedlander, M. P. (2013). Variational properties of value functions. SIAM J. Optim. 23 (3), 1689-1717. doi:10.1137/120899157

B. Correlation / amplitude-normalised

B.1 Global correlation (NCC) — GlobalCorrelationLoss

Per trace with unit-norm \(\hat d = d/\|d\|_2\):

\[ \mathcal J_{\mathrm{NCC}} = \sum_{\mathrm{trace}}(1-\langle\hat d_{\mathrm s},\hat d_{\mathrm o}\rangle). \]
  • Choi, Y. & Alkhalifah, T. (2012). Application of multi-source waveform inversion to marine streamer data using the global correlation norm. Geophysical Prospecting 60 (4), 748-758. doi:10.1111/j.1365-2478.2012.01079.x
  • Routh, P., Krebs, J., Lazaratos, S., et al. (2011). SEG Tech. Progr. Expanded Abstracts, 2433-2438. doi:10.1190/1.3627697

B.2 Trace-normalised L2 — TraceNormalizedL2Loss

\[ \mathcal J_{\mathrm{nL2}} = \tfrac12\sum_{\mathrm{trace}}\sum_t (\hat d_{\mathrm s}-\hat d_{\mathrm o})^2 = \mathcal J_{\mathrm{NCC}}. \]

Same DOIs as B.1.

C. Travel-time / picking

C.1 Cross-correlation travel-time — CrossCorrelationTraveltimeLoss

Per trace, with \(\tau^\star = \arg\max_\tau \int d_{\mathrm s}(t) d_{\mathrm o}(t+\tau)\,dt\):

\[ \mathcal J_{\mathrm{CCT}} = \tfrac12 \sum_{\text{trace}} (\tau^\star)^2. \]

The smooth, differentiable surrogate replaces \(\arg\max\) with a correlation-weighted centroid (van Leeuwen & Mulder 2010 form).

  • Luo, Y. & Schuster, G. T. (1991). Wave-equation travel-time inversion. Geophysics 56 (5), 645-653. doi:10.1190/1.1443081
  • van Leeuwen, T. & Mulder, W. A. (2010). A correlation-based misfit criterion for wave-equation traveltime tomography. Geophys. J. Int. 182 (3), 1383-1394. doi:10.1111/j.1365-246X.2010.04681.x
  • Marquering, H., Dahlen, F. A. & Nolet, G. (1999). Three-dimensional sensitivity kernels for finite-frequency traveltimes. Geophys. J. Int. 137 (3), 805-815. doi:10.1046/j.1365-246x.1999.00837.x
  • Wang, S., Song, P., Tan, J., Xia, D., Zhao, B. & Mao, S. (2024). Differentiable Traveltime Misfit for Wave-Equation Tomography. 85th EAGE Annual Conference & Exhibition, Oslo, Norway, Expanded Abstracts. doi:10.3997/2214-4609.202410170. Replaces the power-of-cross-correlation centroid (van Leeuwen-Mulder 2010, our default) with a softmax-of-cross-correlation centroid; same family of smooth-argmax surrogates.

D. Envelope and instantaneous-phase misfits

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

D.1 Envelope misfits — EnvelopeLoss

  • \(\mathcal J_E^{(p)} = \tfrac{1}{p}\sum |E_{\mathrm s}-E_{\mathrm o}|^p\) for \(p\in\{1,2\}\) (Wu 2014)
  • \(\mathcal J_E^{\log} = \tfrac12\sum(\log E_{\mathrm s}-\log E_{\mathrm o})^2\) (Bozdağ 2011)

  • \(\mathcal J_{E^2} = \tfrac12\sum(E_{\mathrm s}^2-E_{\mathrm o}^2)^2\) (Chi 2014)

  • 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

D.2 Instantaneous-phase misfit — InstantaneousPhaseLoss

\[ \mathcal J_\phi = \tfrac12 \sum [w_\phi(t)\Delta\phi(t)]^2, \qquad \Delta\phi = \arctan2(\sin(\phi_{\mathrm s}-\phi_{\mathrm o}),\cos(\phi_{\mathrm s}-\phi_{\mathrm o})). \]
  • Bozdağ et al. (2011) — same DOI as D.1.
  • 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

D.3 Exponentiated phase — ExponentiatedPhaseLoss

Normalised analytic signal \(\tilde s(t) = a(t)/E_s(t) = e^{i\phi_s(t)}\):

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

D.4 Envelope + phase combined — EnvelopePhaseLoss

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

D.5 Time-Frequency phase / envelope — TimeFrequencyPhaseLoss

Gabor STFT \(G_s, G_o\) with magnitude \(A\) and phase \(\phi\). Envelope term \(\tfrac{1}{2}\sum_{t,\omega}(|G_s|-|G_o|)^2\); phase term uses the unit-circle residual \(|e^{i\phi_s}-e^{i\phi_o}|^2\) weighted by \(|G_o|\). Mixed by \(\alpha\).

  • 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
  • Kristeková, M., Kristek, J. & Moczo, P. (2009). Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals. Geophys. J. Int. 178 (2), 813-825. doi:10.1111/j.1365-246X.2009.04177.x
  • Kristeková, M., Kristek, J., Moczo, P. & Day, S. M. (2006). Misfit criteria for quantitative comparison of seismograms. Bull. Seismol. Soc. Am. 96 (5), 1836-1850. doi:10.1785/0120060012

E. Frequency / Laplace domain

E.1 Frequency-domain L2 — FrequencyDomainL2Loss

\[ \mathcal J_\Omega = \tfrac12 \sum_{\omega\in\Omega} |D_{\mathrm s}-D_{\mathrm o}|^2. \]
  • 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
  • Bunks, C., Saleck, F. M., Zaleski, S. & Chavent, G. (1995). Multiscale seismic waveform inversion. Geophysics 60 (5), 1457-1473. doi:10.1190/1.1443880

E.2 Phase-only — FrequencyPhaseLoss

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

E.3 Amplitude-only — FrequencyAmplitudeLoss

\[ \mathcal J_A = \tfrac12 \sum_{\omega\in\Omega} (|D_{\mathrm s}|-|D_{\mathrm o}|)^2. \]
  • Shin, C. & Min, D.-J. (2006). Waveform inversion using a logarithmic wavefield. Geophysics 71 (3), R31-R42. doi:10.1190/1.2194523

E.4 Shin-Min log misfit — LogarithmicShinMinLoss

\[ \mathcal J_{\log} = \tfrac12 \sum_{\omega\in\Omega} |\log D_{\mathrm s}-\log D_{\mathrm o}|^2 = \tfrac12 \sum (\log|D_{\mathrm s}|-\log|D_{\mathrm o}|)^2 + \tfrac12\sum|\Delta\Phi|^2. \]
  • Shin & Min (2006) — same DOI as E.3.
  • 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

E.5 Laplace L2 — LaplaceL2Loss

\[ \mathcal J_{\mathrm{Laplace}} = \tfrac12 \sum_t (e^{-s t}(d_{\mathrm s} - d_{\mathrm o}))^2. \]

F. Convolution / matching filter

F.1 Adaptive Waveform Inversion (AWI) — AWILoss

For each trace, with Wiener filter \(w = \arg\min \|d_{\mathrm s}\ast w - d_{\mathrm o}\|^2 + \epsilon\|w\|^2\):

\[ \mathcal J_{\mathrm{AWI}} = \tfrac12 \sum_{\mathrm{trace}} \frac{\sum_\tau (T(\tau) w(\tau))^2}{\sum_\tau w(\tau)^2},\quad T(\tau) = \tau\cdot\Delta t. \]
  • Warner, M. & Guasch, L. (2016). Adaptive waveform inversion: theory. Geophysics 81 (6), R429-R445. doi:10.1190/geo2015-0387.1
  • Warner, M. & Guasch, L. (2014). SEG Tech. Progr. Expanded Abstracts, 1089-1093. doi:10.1190/segam2014-0371.1
  • Guasch, L., Warner, M. & Ravaut, C. (2019). Adaptive waveform inversion: practice. Geophysics 84 (3), R447-R461. doi:10.1190/geo2018-0377.1

F.2 Optimal Transport of Matching Filter — OTMFLoss

Compute the Wiener matching filter \(w(\tau)\) as in AWI, preprocess to a density \(\hat w\), then measure Wasserstein distance to the Dirac at zero lag:

\[ W_2^2(\hat w, \delta_0) = \int \tau^2\,\hat w(\tau)\,d\tau, \qquad W_1(\hat w, \delta_0) = \int |\tau|\,\hat w(\tau)\,d\tau. \]
  • Sun, B. & Alkhalifah, T. (2019). Adaptive traveltime inversion. Geophysics 84 (4), U13-U29. doi:10.1190/geo2018-0595.1
  • Sun, B. & Alkhalifah, T. (2019). The application of an optimal transport to a preconditioned data matching function for robust waveform inversion. Geophysics 84 (6), R923-R945. doi:10.1190/geo2018-0413.1
  • Sun, B. & Alkhalifah, T. (2019). Stereo optimal transport of the matching filter. SEG Tech. Progr. Expanded Abstracts, pp. 1505-1509. doi:10.1190/segam2019-3199662.1

F.3 Deconvolution-based — DeconvolutionLoss

\[ \Psi(\tau) = \mathcal F^{-1}\!\Bigl(\tfrac{D_{\mathrm o}}{D_{\mathrm s}+\epsilon}\Bigr),\quad \mathcal J_{\mathrm{Decon}} = \tfrac12 \sum \tau^2 \Psi(\tau)^2. \]
  • Luo, S. & Sava, P. (2011). A deconvolution-based objective function for wave-equation inversion. SEG Tech. Progr. Expanded Abstracts, 2788-2792. doi:10.1190/1.3627773
  • Choi, Y. & Alkhalifah, T. (2018). Time-domain full-waveform inversion of exponentially damped wavefield using the deconvolution-based objective function. Geophysics 83 (2), R77-R88. doi:10.1190/geo2017-0057.1

G. Density-domain divergences

G.1 NIM (CDF distance) — NIMLoss

\[ f = \sigma(d),\quad F(t) = \tfrac{\int_0^t f}{\int_0^T f},\quad \mathcal J_{\mathrm{NIM}} = \tfrac12 \sum \int_0^T (F_{\mathrm s}-F_{\mathrm o})^2\, dt. \]

G.2 Jensen-Shannon divergence — JensenShannonLoss

\[ \mathrm{JSD}(p, q) = \tfrac12\mathrm{KL}(p\|m) + \tfrac12\mathrm{KL}(q\|m),\quad m = \tfrac12(p+q). \]

Symmetric, bounded by \(\log 2\).

  • Yan, Y., Chen, X., Li, J., et al. (2024). Multiparameter shallow-seismic waveform inversion based on the Jensen-Shannon divergence. Geophys. J. Int. 238 (1), 132-155. doi:10.1093/gji/ggae143
  • Endres, D. M. & Schindelin, J. E. (2003). A new metric for probability distributions. IEEE T. Inf. Theory 49 (7), 1858-1860. doi:10.1109/TIT.2003.813506
  • Lin, J. (1991). Divergence measures based on the Shannon entropy. IEEE T. Inf. Theory 37 (1), 145-151. doi:10.1109/18.61115

H. Optimal transport

H.1 1-Wasserstein — Wasserstein1Loss

\[ W_1(f, g) = \int_0^T |F(t)-G(t)|\, dt. \]
  • Métivier, L., Brossier, R., Mérigot, Q., Oudet, E. & Virieux, J. (2016). Measuring the misfit between seismograms using an optimal transport distance: application to full waveform inversion. Geophys. J. Int. 205 (1), 345-377. doi:10.1093/gji/ggw014
  • Engquist, B. & Froese, B. D. (2014). Application of the Wasserstein metric to seismic signals. Commun. Math. Sci. 12 (5), 979-988. doi:10.4310/CMS.2014.v12.n5.a7

H.2 2-Wasserstein via CDF transport — Wasserstein2Loss

\[ W_2^2(f, g) = \int_0^1 (F^{-1}(z)-G^{-1}(z))^2\, dz. \]
  • Engquist, Froese & Yang (2016). Optimal transport for seismic full waveform inversion. Commun. Math. Sci. 14 (8), 2309-2330. doi:10.4310/CMS.2016.v14.n8.a9
  • Yang, Y., Engquist, B., Sun, J. & Hamfeldt, B. F. (2018). Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion. Geophysics 83 (1), R43-R62. doi:10.1190/geo2016-0663.1

H.3 Sinkhorn / entropic OT — SinkhornLoss

\[ \mathrm{OT}_\varepsilon(p, q) = \min_{\Pi \in \Pi(p,q)}\sum_{ij} C_{ij}\Pi_{ij} + \varepsilon \sum_{ij} \Pi_{ij}(\log\Pi_{ij}-1), $$ $$ S_\varepsilon(p, q) = \mathrm{OT}_\varepsilon(p,q) - \tfrac12 \mathrm{OT}_\varepsilon(p,p) - \tfrac12 \mathrm{OT}_\varepsilon(q,q). \]
  • Cuturi, M. (2013). Sinkhorn distances: lightspeed computation of optimal transport. NeurIPS 26. arXiv:1306.0895
  • Feydy, J., Séjourné, T., Vialard, F.-X., Amari, S., Trouvé, A. & Peyré, G. (2019). Interpolating between optimal transport and MMD using Sinkhorn divergences. AISTATS 22, 2681-2690. arXiv:1810.08278
  • Schmitzer, B. (2019). Stabilised sparse scaling algorithms for entropy regularised transport problems. SIAM J. Sci. Comput. 41 (3), A1443-A1481. doi:10.1137/16M1106018

H.4 Graph-space OT — GSOTLoss

Per-trace assignment on \((t, d)\) with ground cost \(c(\cdot, \cdot) = \eta(t-t')^2 + (d-d')^2\) solved by Hungarian.

  • Métivier, L., Allain, A., Brossier, R., Mérigot, Q., Oudet, E. & Virieux, J. (2018). Optimal transport for mitigating cycle skipping in FWI: a graph-space transform approach. Geophysics 83 (5), R515-R540. doi:10.1190/geo2017-0807.1
  • Jonker, R. & Volgenant, A. (1987). A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38 (4), 325-340. doi:10.1007/BF02278710

I. Dynamic warping

I.1 Soft-DTW — SoftDTWLoss

\[ \mathrm{sDTW}_\gamma(d_{\mathrm s}, d_{\mathrm o}) = -\gamma\log\sum_{A\in\mathcal A} \exp\!\bigl(-\tfrac{1}{\gamma}\langle A, \Delta\rangle\bigr),\quad \Delta_{ij} = (d_{\mathrm s,i}-d_{\mathrm o,j})^2. \]
  • Cuturi, M. & Blondel, M. (2017). Soft-DTW: a differentiable loss function for time-series. ICML 70, 894-903. arXiv:1703.01541
  • Hale, D. (2013). Dynamic warping of seismic images. Geophysics 78 (2), S105-S115. doi:10.1190/geo2012-0327.1
  • Ma, Y. & Hale, D. (2013). Wave-equation reflection traveltime inversion with dynamic warping and full-waveform inversion. Geophysics 78 (6), R223-R233. doi:10.1190/geo2013-0004.1

J. Local-similarity attribute

J.1 Local similarity — LocalSimilarityLoss

\[ \gamma_\sigma(\tau) = \frac{\sum_t w_\sigma(t-\tau) d_{\mathrm s}(t) d_{\mathrm o}(t)}{\sqrt{\sum_t w_\sigma(t-\tau) d_{\mathrm s}^2 \cdot \sum_t w_\sigma(t-\tau) d_{\mathrm o}^2}}, \quad \mathcal J_{\mathrm{LS}} = \tfrac12 \sum w_\sigma^E(\tau)\,(1-\gamma_\sigma(\tau))^2. \]
  • Fomel, S. (2007). Local seismic attributes. Geophysics 72 (3), A29-A33. doi:10.1190/1.2437573

Misfits NOT implemented (and why)

These FWI objectives couple the misfit with the forward problem or require additional inputs (extended wavefields, model perturbations, …) that don't fit into a pure loss(syn, obs) interface. They are listed here for completeness:

  • Symes, W. W. & Carazzone, J. J. (1991). Velocity inversion by differential semblance optimisation. Geophysics 56 (5), 654-663. doi:10.1190/1.1443082
  • van Leeuwen, T. & Herrmann, F. J. (2013). Mitigating local minima in full-waveform inversion by expanding the search space. Geophys. J. Int. 195 (1), 661-667. doi:10.1093/gji/ggt258
  • Chauris, H. & Plessix, R.-E. (2012). Investigating the differential waveform inversion. 74th EAGE Conf. doi:10.3997/2214-4609.20149790

These can be added later by exposing the additional inputs (extended gather offsets, the wavefield, the model gradient, …) as constructor arguments — but the consensus is to treat them as separate inversion frameworks rather than loss(syn, obs) losses.

Plugging sweep_loss into the sweep propagator

The misfit API matches the one already used in geophyai/examples/FWI/2d/acoustic/torch/_fwi_marmousi_common.py, which currently computes the L2 misfit inline as

loss_sum = (syn - obs_batch).pow(2).sum()
(loss_sum / normalization_elements).backward()

Replacing it with any sweep_loss misfit is a two-line change:

from sweep_loss import HuberLoss          # or any other misfit in this report
loss_fn = HuberLoss(delta=0.5, reduction="sum")

# inside the training loop:
loss_value = loss_fn(syn, obs_batch)   # canonical (ns, nt, nr, nc)
loss_value.backward()

For misfits that need extra parameters (dt, epsilon, freq_band, gamma, ...), see the per-loss docs under docs/losses/.

How to cite

Wang, S. (2026). sweep-loss: a PyTorch library of misfit functions for Full Waveform Inversion. https://github.com/DeepWave-KAUST/sweep-loss

Per-loss citations: see the DOI links above.