Tomographic velocity inversion

Part 3 — Velocity Analysis & NMO

Learning objectives

Set up a tomographic inverse problem: G m = d, where G is the ray-path matrix, m is slowness, d is travel-time residuals
Explain why tomography solves for the full model simultaneously instead of iterating per-CMP picks
Describe the straight-ray and bent-ray (iterative) variants and when each is appropriate
Identify the resolution limits and common failure modes (uneven illumination, regularization trade-offs)

Conventional velocity analysis (§§3.1–3.5) picks V(t₀) at each CMP independently. That works when lateral variation is smooth. When it is not — foothills, reefs, karst, buried river valleys — neighboring picks can be inconsistent, and the per-CMP approach fails to reconstruct the full velocity field. The answer is tomography: simultaneously invert the travel-time residuals everywhere in the survey for the velocity model everywhere.

1. The inverse problem

Discretize the subsurface into cells (the model grid) with per-cell slowness s = 1/V. For each measured ray, its travel-time residual (observed − predicted) is

\Delta t_{\text{ray}} = \sum_{\text{cells}} \ell_{\text{ray, cell}} \cdot \Delta s_{\text{cell}}

where ℓ is the ray length inside a given cell. Writing this for every ray gives a linear system

\mathbf{G}\,\Delta\mathbf{s} = \Delta\mathbf{t}

where G is the (N_rays × N_cells) ray-path length matrix, Δs is the slowness update vector, and Δt is the travel-time residual vector. This is exactly the Ax = b shape from §0.7 — massively overdetermined in well-illuminated areas, underdetermined elsewhere.

The demo uses a 16×16 grid. The true model has a high-velocity 4×4 anomaly in the center. 10 shots along the top and 10 receivers along the bottom give 100 ray paths — every cell has several rays crossing it. Each iteration:

Ray-trace through the current model and compute predicted travel times.
Compute residuals Δt = t_obs − t_pred.
Back-project: each cell accumulates Δt · ℓ over all rays crossing it, weighted by ℓ.
Apply the update with a small step size.
Repeat.

The RMS residual panel on the right tracks convergence — it should decay over iterations.

Click Step to see one update; Run 20 iters to watch it animate. After ~10 iterations the inverted model clearly shows the central anomaly; by 20, the RMS residual is below ~5 ms. Click Reset to start again.

3. Straight-ray vs bent-ray

The widget uses straight rays — shot and receiver are connected by a line regardless of velocity. This is:

Exact for constant-velocity model.
Approximately correct when velocity contrasts are small and refraction is mild.
Wrong when contrasts are strong — e.g., salt body, steep velocity gradient.

Production tomography uses bent rays: ray-trace through the CURRENT velocity model, using Snell’s law or the eikonal equation. At each iteration, you re-trace rays with the updated model. More accurate, more expensive — a few-hundred CPU-hours per 3D field is typical.

4. Resolution and illumination

A cell with many rays crossing at diverse angles is well-illuminated — tomography resolves it crisply. A cell with only one or two rays at similar angles is poorly illuminated — the solution there is ambiguous, and the inversion either leaves it at the starting value or over-fits noise.

Common illumination gaps:

Below the deepest ray. No rays go below the deepest turning point; the model below is unconstrained.
Shadow zones. Strong velocity contrasts bend rays away from certain cells.
Survey edges. Corners and edges see only one-sided illumination.

5. Regularization

Because of the poorly-illuminated cells and noise in travel-time picks, the raw least-squares update is unstable. Tomography solves a regularized problem:

\min_{\Delta s}\; \|G\,\Delta s - \Delta t\|_2^{2} + \lambda\,\|L\,\Delta s\|_2^{2}

where L is a smoothness operator (discrete Laplacian or gradient) and λ is the regularization weight. Low λ → fits data closely, noisy-looking model. High λ → smooth model, fits data less well. Picking λ is the key art of tomographic inversion.

6. Inputs to production tomography

First-break picks — refraction-tomography for near-surface velocity.
Reflection residuals — CMP gather residuals after NMO, projected through the overburden model.
Well-to-seismic ties — checkshot or VSP velocity constraints added as equations.
Walkaway VSPs — dense tomographic data around a well.

Most production velocity models come from combining refraction-tomography (near surface) and reflection tomography (deeper) iteratively with migration — the output of tomography feeds migration, and migrated gathers feed residual tomography. The loop closes after 2–4 iterations.

7. Tomography vs FWI

Tomography uses travel times only. FWI (§§6) uses the full waveform — amplitudes, phases, everything. FWI can resolve finer structure but needs a good starting model (usually from tomography) or it falls into local minima. Production workflow: tomography first, then FWI on the tomography output for the final kinematic tuning.

**The one sentence to remember**

Tomography solves G·Δs = Δt for the whole velocity model simultaneously from ray paths and travel-time residuals, with regularization to tame poorly-illuminated cells; it produces the velocity model every other velocity-dependent algorithm downstream depends on.

Part 3 closes here

You can now: compute RMS from interval velocities and invert via Dix, apply NMO and understand stretch, pick velocities from semblance, account for VTI with the η parameter, refine with residual-velocity + HOMO, and build a full V(x, y, t₀) model by tomography. Part 4 starts the migration-era tools with multiple attenuation — SRME, Radon, adaptive subtraction — the final clean-up before imaging.

References

Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.
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.