Reverse-Time Migration (RTM)

Reverse-time migration is the gold standard of imaging. Unlike Kirchhoff (Section 5.4), which sums along ray paths, or one-way wave-equation migration (Section 5.6), which extrapolates each wavefield in only one direction of depth, 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 without special treatment; these are the cases where ray-based and one-way methods need extra machinery, and can still mis-position the target. It is also the most expensive migration in routine production.

1. The RTM recipe

1. Forward-propagate the source. Solve the wave equation $\\frac{\\partial^2 U\_s}{\\partial t^2} = V^2 \\nabla^2 U\_s + f(t)\\,\\delta(\\mathbf{x}-\\mathbf{x}\_s)$, where $f(t)$ is the source wavelet injected at the source location $\\mathbf{x}\_s$. 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).
2. 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.
3. Apply the imaging condition. The image is the zero-lag cross-correlation:

Where both wavefields are nonzero at the same time at a point $(x, z)$, that point could have produced the recorded energy. For a single source and a single receiver those points form a whole curve, the isochron $|SX| + |XR| = V t\_R$, where $t\_R$ is the recorded time. Where only one wavefield is nonzero the product is exactly zero. Summing over all receivers and all shots adds the isochrons in phase only at the true reflector, and they cancel elsewhere; that is how the image forms.

2. Watch the imaging condition work

Figure 5.7 runs RTM on the simplest earth there is: a constant velocity of 2000 m/s and one point scatterer P, a diffractor rather than a reflector, recorded by 81 receivers. It shows the recorded shot, the two wavefields and their product at one instant, and the image summed so far, starting from a single trace and adding receivers and shots until P comes into focus.

The figure opens on a single trace, the receiver at 1700 m. In (b) the source wavefront is a ring growing from S. In (c) that one trace, run backward in time, is a ring centred on R of radius $V (t\_R - t)$, which shrinks back toward R as $t$ advances. The two rings cross at one point in the earth, and that is where the product in (d) lights up. As $t$ runs, the crossing slides along the dashed ellipse, and the sum in (e) paints the whole ellipse rather than the scatterer: once the sum is complete, only 3 % of the image energy lies within 50 m of its brightest point. The fields meet on P at 390 ms, which is $|SP|/V$, the source’s own travel time to P. That equals $t\_R/2$ only when S and R sit symmetrically about P; here the trace time is 851 ms, so $t\_R/2$ would be 426 ms.

Switch to all 81 receivers and each trace adds its own ellipse. They add in phase only near P, so 68 % of the image energy now lies within 50 m of the brightest point, which sits on P; what remains are smile tails from the ends of the spread. Summing 19 shots, each seeing P from another angle, cancels most of the tails and lifts that share to 93 %. The same 19 shots migrated with a velocity 10 % too fast put the brightest point 50 m below P and drop the share to 35 %: the imaging condition is only as good as the velocity that carries both wavefields.

3. Why the isochron ellipse reappears

A point $X$ lies on both wavefronts at some time $t$ when $|SX| = V t$ and $|XR| = V (t\_R - t)$. Adding the two gives $|SX| + |XR| = V t\_R$: the ellipse with foci at S and R and semi-major axis $a = V t\_R/2$, the dashed curve in (d). This is the same isochron that appeared in PSTM Section 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 without special treatment

• Turning waves. Waves that propagate down, bend back upward due to a high-velocity layer, reflect, and return. A one-way extrapolator carries each wavefield in only one direction of depth, so the part of the path that turns back upward needs special treatment, and Kirchhoff migration images such waves only with turning-ray travel times (Hale, Hill and Stefani, 1992). 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. RTM images it without special treatment, but only when the migration velocity model contains the sharp interface that makes the second bounce; with the usual smooth model it does not.
• 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 over a 10×10 km aperture with $\\Delta x = 12.5\\ \\text{m}$ and $\\Delta z = 10\\ \\text{m}$ costs of order $10^{14}$ to $10^{15}$ floating-point operations 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 maps to roughly a 1 % depth error, defocuses diffractions into residual smiles or frowns, and shifts dipping events laterally. Velocity model building (Section 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 (Clapp, 2009) replace the absorbing layer with a randomised-velocity padding that scatters outgoing energy incoherently. The source wavefield stays time-reversible and can be recomputed backward alongside the receiver wavefield without storing snapshots, and the scattered noise does not correlate coherently into the image.
• Uneven illumination. Under salt or in shadow zones the source wavefield is weak, so the cross-correlation image is dim. A source-normalised (deconvolution) imaging condition $I = \\sum\_t U\_s U\_r / (\\sum\_t U\_s^2 + \\epsilon)$ balances the amplitudes, but needs the damping $\\epsilon$ to stay stable where $U\_s$ is nearly zero.

Where this goes next

Section 5.8 surveys the artefacts each migration method introduces (stretch, smiles like the tails in Figure 5.7, aperture truncation, operator noise, RTM low-frequency backscatter) and the QC steps that let you distinguish a real event from a migration artefact. Section 5.9 covers velocity model building, which all of Sections 5.4-5.7 depend on.

References

• Baysal, E., Kosloff, D. D., Sherwood, J. W. C. (1983). Reverse time migration. Geophysics, 48, 1514.
• Hale, D., Hill, N. R., Stefani, J. (1992). Imaging salt with turning seismic waves. Geophysics, 57, 1453.
• 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.