Cycle skipping: detection and remedies

§6.4 cured cycle skipping by changing the DATA (low-pass filtering). This section cures it by changing the MISFIT FUNCTION. Both ideas compose in production: filter the data AND use a robust misfit, layered curriculum-style.

What cycle skipping looks like

The §6.1 widget showed the gradient sign flipping at extreme c₂ values. The §6.2 widget showed the line search stalling. The §6.4 widget showed the freq-continuation rescue. Here is the canonical signature in three pictures:

• Predicted vs observed seismograms. The two waveforms look very similar in shape but one is shifted in time by more than half the dominant period. Visually: peaks of $d_{\mathrm{pred}}$ align with TROUGHS of $d_{\mathrm{obs}}$.
• The L² residual $r = d_{\mathrm{pred}} - d_{\mathrm{obs}}$. Has a "double pulse" shape — positive lobe at the predicted arrival, negative lobe at the observed arrival, both of comparable magnitude. The gradient $\partial J/\partial m$ correlates this residual with the time-reversed adjoint, giving CONSTRUCTIVE cancellation between the two lobes — wrong sign.
• Convergence trace. $J$ stalls at a non-zero plateau. The line search reduces step size to zero and gives up. The model error $|c - c_{\mathrm{true}}|$ stays at its initial value (or grows).

The mathematical condition

Let $T_0 = 1/f_0$ be the dominant period of the source. Let $\Delta t$ be the traveltime mismatch between the predicted and observed arrivals at a receiver. Cycle skipping happens when

For a Ricker pulse with $\sigma$-parameterisation $s(t) = (1 - 2u^2) e^{-u^2}$, $u = (t-t_0)/\sigma$, the dominant frequency is $f_0 = 1/(\pi \sigma)$ and the period is $T_0 = \pi \sigma$. So the cycle-skipping threshold is

For our §6.4 widget: $\sigma = 0.012$ s, so the threshold is $\sim 19$ ms. A 1D problem with travel-time differences exceeding 19 ms across the model is cycle-skipped at single-frequency.

Three modern alternative misfits

The trick: replace the $L^2$ misfit $J = \tfrac{1}{2} \int (d_{\mathrm{pred}} - d_{\mathrm{obs}})^2$ with something that is SMOOTH in time-shift, so the cycle-skipping non-convexity disappears. Three contenders:

• Envelope misfit (Wu, Luo, Wu 2014). Replace each seismogram by its envelope $\mathrm{env}(s)(t) = \sqrt{s(t)^2 + \epsilon^2}$ before computing the residual. The envelope strips the carrier oscillations — peaks and troughs become bumps — so cycle-skipping is impossible at the envelope level. The price: resolution loss; the envelope smooths over the high-frequency content that distinguishes nearby reflectors. Production workflow: invert the envelope first, then refine with L².
• Wasserstein / optimal-transport misfit (Engquist & Froese 2014; Métivier et al. 2016). Treat the seismograms as probability distributions (after positivity-shift normalisation) and compute the Wasserstein-1 or Wasserstein-2 distance between them. The Wasserstein metric is convex in time-shift by construction, so it cures cycle skipping cleanly. Mathematically elegant; computationally intensive — Métivier 2016's SOT (Sliced Optimal Transport) approximation made it production-tractable.
• Adaptive Waveform Inversion (AWI) (Warner & Guasch 2016). Instead of comparing $d_{\mathrm{pred}}$ to $d_{\mathrm{obs}}$ directly, find the Wiener filter $w$ that best matches $d_{\mathrm{pred}} \ast w$ to $d_{\mathrm{obs}}$, then minimise the deviation of $w$ from a $\delta$-function. AWI absorbs time-shift residuals into the filter $w$; the misfit only sees what is left. Provably immune to first-order cycle skipping; a workhorse misfit at the major service companies (CGG, BP, etc.).

Try it: misfit landscapes

The widget above computes both misfits as a function of the middle-layer velocity $c_2$ on the §6.2 problem, with $c_1 = 1.0$ and $c_3 = 1.2$ fixed at truth. 60 sampled values of $c_2 \in [0.6, 2.4]$ each take one FDTD forward solve. Plotted side-by-side on log y-axis, with local minima circled.

Empirical result on this problem (60-sample landscape):

• L² landscape: typically 10 local minima — one near $c_2 = 1.5$ (truth) plus 9 spurious minima at multiples of the Ricker period offset. Each spurious minimum is a "cycle-skipping island" where the predicted-vs-observed traveltime mismatch is an integer number of half-periods. Gradient descent converges to whichever island the starting model lies in.
• Envelope landscape: 1 minimum at $c_2 = 1.5$. The smoothed-$s^2$ envelope $\sqrt{\mathrm{smooth}(s^2, \sigma) + \epsilon^2}$ removes the carrier oscillations that produce the L² wiggles. The result is a smooth bowl with a global minimum at truth.

This is the Wu, Luo, Wu 2014 figure-of-merit, made interactive. Their published version uses Hilbert-transform envelopes (computed via FFT) which produce essentially the same convex landscape.

Why we don't run iterative descent in this widget

You might expect a "race two FWI runs" widget like §6.4's. The mathematical claim of the convex landscape is that envelope-FWI gradient descent converges from any starting point. In practice, the simple $\sqrt{s^2 + \epsilon^2}$ envelope has a known limitation: the gradient adjoint source

still contains the carrier $s_{\mathrm{pred}}(t)$ as a multiplicative factor, so the adjoint wavefield $\lambda(x, t)$ inherits the same phase oscillations as L²'s. The correlation $\int u_{tt} , \lambda , dt$ then suffers the same phase-cancellation issues at the gradient level even though the misfit landscape is convex. The Wu 2014 paper carefully derives the Hilbert-envelope adjoint chain rule (which requires FFT in the backprop step too) to fix this — beyond this widget's in-browser scope.

Production envelope FWI codes use Hilbert envelopes; production frequency-continuation codes use band-pass filtering; production AWI codes use Wiener filter convolution. All three converge from cycle-skipped starting models because their misfits + their adjoint chain rules together preserve the convex landscape all the way to the gradient.

The production workflow: misfit continuation

Real production FWI runs a CASCADE of misfits:

• First, low-frequency envelope inversion (immune to cycle skipping; resolves only large-scale structure).
• Then, low-frequency L² inversion (warm-started from envelope; recovers mid-scale structure).
• Then, frequency continuation through the §6.4 stages (refining successively higher-frequency content).
• Finally, full-band L² inversion at the highest frequencies (high-resolution detail).

Each stage starts from the previous stage's converged model. This is the standard recipe in CGG's Geovation, Schlumberger's WesternGeco, and BP's reservoir-FWI workflows. The §6.4 frequency-continuation widget builds curriculum on the data dimension; this section's envelope idea builds curriculum on the misfit-function dimension; production stacks both.

What about PINN-FWI?

The same misfit-replacement trick works for PINN-FWI. Replace the $L_{\mathrm{data}}$ term in the joint loss

by an envelope or Wasserstein version. Auto-diff handles the gradient automatically. Sun & Alkhalifah (2021) and Yang et al. (2023 Geophysics) both report PINN-FWI with envelope or Wasserstein misfits successfully recovering velocity from cycle-skipped starting models that defeated L²-PINN-FWI. The §6.4 frequency-continuation idea also applies — just bandpass-filter the data input to the PINN-FWI joint loss.

What §6.6 will do

§6.6 frees a SECOND layer parameter (the top-layer velocity $c_1$) in addition to the middle-layer $c_2$, and shows the resulting 2D misfit landscape $J(c_1, c_2)$. The cross-talk between layer velocities — the inability of single-shot transmission data to independently constrain both — appears as an elongated valley in the heatmap. The same valley structure shows up at much larger scale in production multi-parameter FWI (P-wave velocity vs S-wave velocity vs density); the §6.6 prose covers the full $v_p$/$v_s$/$\rho$ generalisation and its remedies (multi-shot illumination, joint inversion with auxiliary data, impedance reparameterisation).

References

• Wu, R.-S., Luo, J., Wu, B. (2014). Seismic envelope inversion and modulation signal model. Geophysics 79(3), WA13–WA24.
• Engquist, B., Froese, B.D. (2014). Application of the Wasserstein metric to seismic signals. Communications in Mathematical Sciences 12(5), 979–988.
• 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.
• Warner, M., Guasch, L. (2016). Adaptive waveform inversion: Theory. Geophysics 81(6), R429–R445.
• Sun, B., Alkhalifah, T. (2021). The application of optimal transport for FWI in seismic exploration. Geophys. J. Int.
• Yang, F., Ma, J., Lu, Y., Liu, X. (2023). Physics-informed deep learning for full-waveform inversion via Wasserstein loss. Geophysics.