Reverse-Time Migration (RTM)

Reverse-time migration is the gold standard of imaging. Unlike Kirchhoff (§5.4) which sums along ray paths or one-way WE (§5.6) which keeps only downgoing waves, RTM propagates the full two-way wave equation for both the source and the receiver wavefields, and images via cross-correlation of the two. It handles turning waves, severe multipathing, and strong lateral velocity contrasts — the cases where every other method either mis-positions or miss-images the target entirely. It is also the most expensive migration in routine production.

1. The RTM recipe

• Forward-propagate the source. Solve the wave equation $\frac{\partial^2 U_s}{\partial t^2} = V^2 \nabla^2 U_s$ starting from the source wavelet at the source location. Step forward in time for the recording length $T$. Save the entire wavefield $U_s(x, z, t)$ for every grid point and every time step (or, in memory-optimised schemes, save checkpoints and replay).
• Reverse-propagate the receivers. Inject the recorded traces at the receiver locations as a time-reversed boundary condition at $t = T$ and run the same wave equation backward in time. The resulting wavefield $U_r(x, z, t)$ is where the receiver energy came from at each earlier time, threaded correctly through the velocity model.
• Apply the imaging condition. The image is the zero-lag cross-correlation:

Where both wavefields are simultaneously nonzero at some point $(x, z)$, that is a reflector. Where only one is nonzero (or neither), the sum averages to zero. The sum over all time captures every reflector the pair (source, receiver) can illuminate.

2. The widget

Constant-V medium with a source S, a receiver R, and a single reflector point. The yellow circle expanding from S is the source wavefront at time $t$. The teal circle expanding from R is the time-reversed receiver wavefront — radius $V \cdot (t_R - t)$ where $t_R$ is the total reflection time. The dashed purple ellipse is the isochron (the same one PSTM uses, from §5.3).

Slide the time slider. At early times the source wavefront is small and close to S; the receiver wavefront is large. As time advances, the source wavefront grows and the receiver wavefront shrinks (since $t_R - t$ decreases). They cross at the reflector exactly when $t = t_R / 2$ for a symmetric geometry, or at the appropriate split for asymmetric. A white flash ring lights up at the reflector when both wavefronts are passing through it simultaneously — that is the imaging condition firing at that time step. Accumulate over all times and a full image emerges.

3. Why the isochron ellipse reappears

The condition $V \cdot t + V \cdot (t_R - t) = V \cdot t_R = |SP| + |PR|$ defines exactly the ellipse with foci at S and R with major semi-axis $V \cdot t_R/2$. This is the same isochron that appeared in PSTM §5.3. RTM and PSTM agree on the isochron in homogeneous media — what RTM gains is the ability to correctly compute both wavefields (and hence the isochron) in heterogeneous media where rays bend, reflect internally, and multi-path. In salt, sub-salt, thrust belts, sub-basalt: the isochron is no longer an ellipse but a complicated surface that only two-way wave propagation can compute.

4. What RTM handles that others cannot

• Turning waves. Waves that propagate down, bend back upward due to a high-velocity layer, reflect, and return. One-way methods drop the upgoing half; RTM keeps both directions at every time step.
• Prismatic reflections. A wave that reflects off a dipping interface (e.g., a salt flank) and then reflects off a horizontal interface below. The intermediate propagation segment is not captured by Kirchhoff first-arrival travel times but emerges automatically in RTM.
• Severe lateral velocity contrasts. Salt (4500 m/s) adjacent to sediment (2000 m/s). Ray tracing struggles with shadow zones and multi-pathing; the full wave equation handles both naturally.
• Steep and overhanging dips. RTM can image reflectors dipping past 90° (salt overhangs) because the wavefield propagation does not assume a dominant direction.

5. The price

• Compute. A 3D acoustic RTM at 25 Hz peak for a 10×10 km survey at $\Delta x = 12.5\ \text{m}$ depth $\Delta z = 10\ \text{m}$ is ~20–50 TFLOPs per shot. With thousands of shots per survey, RTM runs on GPU clusters for days to weeks.
• Memory / I/O. Saving the source wavefield at every time step is impractical; checkpointing and random boundary methods reduce the need but I/O still dominates.
• Velocity sensitivity. RTM is exquisitely sensitive to velocity-model errors. A 1 % velocity error at depth maps to 1 % lateral mis-positioning. Velocity model building (§5.9) is not optional.

6. RTM-specific artefacts

• Low-frequency backscatter noise. Where the source wavefield and the reverse-time receiver wavefield are both downgoing (or both upgoing), their cross-correlation produces long-wavelength artefacts above real reflectors. Standard fix: a Laplacian filter on the image (removes low wavenumbers) or a Poynting-vector imaging condition that restricts cross-correlation to opposed-direction wavefields.
• Random-boundary reflections. Open-boundary implementations absorb outgoing waves imperfectly; "random-boundary" schemes re-use stored random-velocity wavefields at the boundary to avoid spurious reflections during reverse-time playback.
• Imaging-condition instability in low-velocity zones. Where V is small, $U_s$ and $U_r$ can both accumulate noise; deconvolution imaging conditions (divide instead of correlate) can help but introduce their own stability issues.

Where this goes next

§5.8 surveys the artefacts each migration method introduces — stretch, smile, aperture, migration operator noise, RTM low-frequency backscatter — and the QC steps that let you distinguish a real event from a migration artefact. §5.9 covers velocity model building, which all of §5.4–5.7 depend on.

References

• Baysal, E., Kosloff, D. D., Sherwood, J. W. C. (1983). Reverse time migration. Geophysics, 48, 1514.
• McMechan, G. A. (1983). Migration by extrapolation of time-dependent boundary values. Geophys. Prosp., 31, 413.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Claerbout, J. F. (1985). Imaging the Earth’s Interior. Blackwell.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.