Encoded FWI & computational strategies

Production FWI is computationally extreme: a full-physics 3D acoustic FWI at 10 Hz on a modest survey (10 km × 10 km × 8 km, $\Delta x = 25\ \text{m}$) takes thousands of shots, each requiring a forward plus an adjoint simulation per iteration, for tens of outer iterations, for 5–10 frequency bands. Run the arithmetic and you get days-to-weeks of GPU cluster time per survey — and that is already after using every computational trick available. This section catalogues the tricks.

1. Source encoding — the central trick

The wave equation is linear in the source: if shot $i$ produces data $d_i$, the combined source $\sum_i c_i s_i$ produces data $\sum_i c_i d_i$. Pick random signs $c_i \in {-1, +1}$, build a super-source and a super-data, and run one wave simulation that collectively informs every shot's gradient:

The first term is $\sum_i g_i$ — the true FWI gradient we wanted. The second term is cross-talk: contributions from shot $i$'s forward wavefield correlated with shot $j$'s adjoint wavefield. Because $c_i c_j$ averages to zero over many random encodings, the cross-talk averages out over iterations, leaving only the correct gradient. One simulation per iteration instead of $N_{shots}$ — potentially a 100– to 1000-fold speedup, at the cost of ~3× more iterations to convergence because of the added noise.

2. The widget — cost bars

Three horizontal bars show total-simulations-to-convergence for the three strategies:

• Naive FWI: one sim per shot per iteration. For $N = 1000$ shots and 50 iterations: 50 000 sims.
• Encoded FWI: one sim total per iteration, but 3× more iterations due to cross-talk: 150 sims total — 333× cheaper than naive for N=1000.
• Mini-batch FWI: $k$ random shots per iteration, 1.5× more iterations. For $k = 0.1N = 100$: 7500 sims — 6.7× cheaper than naive.

The speedup from encoded FWI grows with $N_{shots}$: at $N = 10,000$ it is $10,000 \times 50 / 150 = 3,333\times$ faster. Production FWI over 10 000 shots that would take years naively finishes in days with encoding.

3. When encoded FWI breaks

• Irregular acquisition geometries. Encoded FWI assumes all shots contribute linearly; missing or unevenly distributed sources break the randomness assumption and cross-talk stops averaging out.
• Strong amplitude variations between shots. If one shot is 10× stronger than another, the random-sign sum is dominated by the strong shot and you effectively only invert one shot’s gradient.
• Locally correlated noise. Noise that is coherent across shots (coherent swell, electrical hum) does not average out under encoding — it stacks additively.
• Salt imaging. At sharp reflectors, the cross-talk can add coherent artefacts that never fully wash out. Hybrid strategies (encoded outer, naive inner) are used.

4. Memory — storing the forward wavefield

The adjoint-state gradient requires the forward source wavefield $U_s(x, z, t)$ at every grid point and every time step. For a 3D volume with $400 \times 400 \times 320 \times 4000$ samples (single precision), that is ~800 GB per shot. Three standard mitigations:

• Checkpointing: save $U_s$ at every $k$-th time step only. To compute the gradient at the skipped time steps, re-propagate forward from the nearest checkpoint. Memory drops by $k$ at the cost of $\log k$ extra simulations. Griewank's binomial checkpointing gives the optimal schedule.
• Random boundaries: instead of absorbing-boundary wavefield storage, randomise the velocity at the boundaries so outgoing waves return scrambled. When you reverse-time-propagate the forward wavefield (re-deriving $U_s$), the randomised boundary reproduces the original wavefield in the interior. Storage drops to zero at the cost of re-propagation.
  • Wavefield reconstruction: solve the wave equation backward in time from the last saved slab (usually just the boundary). Cheaper than re-forward, and accurate enough for L-BFGS-quality gradients.

5. Parallelism

• Shot-parallel: each shot's simulation is independent; distribute across GPU nodes. Embarrassingly parallel up to $N_{shots}$ workers.
• Domain-decomposed: split the model grid across GPUs within a node; each GPU handles its slab. Adds inter-GPU communication at slab boundaries but necessary for very large models.
• Frequency-parallel: multi-scale runs can be pipelined across independent GPU pools, one frequency per pool, with results handed off when ready.

6. Hessian, or: why L-BFGS is the default

Pure steepest-descent FWI converges slowly because $J(m)$ has very different curvature along different model directions. Newton's method $m \leftarrow m - H^{-1} g$ fixes this but requires the full Hessian $H = \partial^2 J / \partial m^2$, which costs $N_{model}^2$ simulations — infeasible for $N_{model} \sim 10^8$.

L-BFGS (limited-memory BFGS) approximates $H^{-1}$ from the last $k$ gradient–model pairs ($k = 5-20$), giving Newton-like convergence at storage cost $k \cdot N_{model}$. Combined with an approximate diagonal pre-conditioner (Gauss-Newton on the diagonal, or the Hessian of the acquisition illumination), L-BFGS is the default FWI solver in every production package. Convergence is typically 3–5× faster than steepest descent per outer iteration, paying for itself immediately.

7. A realistic computational budget

For a 3D deep-water sub-salt project, 10 km × 15 km × 8 km at $\Delta x = 25\ \text{m}$, 8 Hz, 5000 shots:

• Naive: 2 sims × 5000 shots × 200 iters × 6 bands = 12 million simulations ⇒ ~2 years on a 100-GPU cluster.
• Encoded: 2 sims × 1 super-shot × 600 iters × 6 bands = 7200 simulations ⇒ ~3 days on same cluster.
• Mini-batch $k = 500$: 2 sims × 500 shots × 300 iters × 6 bands = 1.8 M sims ⇒ ~100 days.

Only encoded FWI fits in a reasonable project budget. Every production FWI code supports at least source encoding; most support all three.

Where this goes next

§6.4 moves from acoustic to elastic/anisotropic physics — what happens when the acoustic wave equation is wrong and converted waves or anisotropy-induced travel-time errors dominate the residual.

References

• Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
• Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1. Geophysics, 64, 888.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.