Pre-stack time migration (PSTM)

Post-stack time migration (Section 5.2) treats the stacked section as a zero-offset record and sums each output point along a Kirchhoff diffraction hyperbola (equivalently, spreads each input sample along a semicircle) so that energy adds coherently at reflector positions. That works when stacking preserved the kinematics, that is, when every CMP was flat in offset after NMO. Once reflectors of different dip meet under one midpoint, no single NMO velocity flattens them all, the stack is no longer a true zero-offset section, and the post-stack operator looks for signal that NMO and stack have already weakened. Pre-stack time migration (PSTM) solves this by migrating the pre-stack traces directly, before the stack is ever formed.

1. The double-square-root travel-time

Consider a single pre-stack trace recorded by a source at $x\_s = m - h$ and a receiver at $x\_r = m + h$, midpoint $m$, half-offset $h$. A scatterer at image point $(x\_o, z)$ produces a reflection at observed two-way time

that is, the source-to-scatterer time plus the scatterer-to-receiver time. This is the double-square-root (DSR) equation that governs PSTM. For a given trace, every sample $(m, h, t\_{\\mathrm{obs}})$ is a measurement of the DSR sum, and PSTM asks: for each output pixel $(x\_o, \\tau = 2z/V)$ in the migrated image, which trace samples could have produced that pixel?

2. The PSTM isochron is an ellipse

Fix $(x\_s, x\_r, V, t\_{\\mathrm{obs}})$ and ask: for which scattering points does the DSR sum equal $t\_{\\mathrm{obs}}$? In the depth plane, the answer is the set of points whose distances to $x\_s$ and $x\_r$ sum to $V t\_{\\mathrm{obs}}$. That is the defining property of an ellipse with foci at the source and receiver, major semi-axis $a = V t\_{\\mathrm{obs}}/2$, minor semi-axis $b = \\sqrt{a^2 - h^2}$.

Converting depth to vertical two-way time $\\tau = 2z/V$, the image-space isochron is a half-ellipse in $(x\_o, \\tau)$ coordinates with semi-axes $(a, 2b/V)$. Its apex sits at the midpoint $m$ at time

which is always less than $t\_{\\mathrm{obs}}$ for $h > 0$ and equals it when $h = 0$. That means: at zero offset the PSTM isochron degenerates to the post-stack semicircle, the zero-offset migration impulse response. The apex time is exactly the NMO-corrected time, so PSTM carries out NMO implicitly; what it adds beyond NMO is the width of the ellipse, which stays $a = V t\_{\\mathrm{obs}}/2$ however large $h$ grows.

3. Follow one sample to an image

The figure below holds a flat reflector and a dipping one in a constant-velocity earth. Choose a trace on the CMP gather at $m$ = 1500 m by its half-offset $h$ and follow its sample from the dipping reflector: PSTM spreads it over an ellipse, and the NMO, stack and post-stack route spreads it over a semicircle about $m$ whose apex is the NMO time. Set $h$ to 0 and the two operators coincide; tilt the reflector and find where it touches each one. Then compare the two migrations of the whole survey, and watch the image gather when the velocity is wrong.

At the opening setting (dip 30°, $h$ = 500 m, $V$ = 2000 m/s) the sample at 1000 m offset and 0.892 s spreads over an ellipse of half-width $a$ = 892 m with its apex at the NMO time $t\_0$ = 0.738 s, while the post-stack semicircle from the same apex has half-width $b$ = 738 m. The dipping reflector touches the ellipse 510 m updip of $m$, in the shaded band the semicircle never reaches, and there the semicircle falls 48 ms short of it. A flat reflector would touch both operators at the apex; a dipping one touches the ellipse where the two disagree. Across the whole survey the damage is done in the stack: NMO at the flat reflector's velocity leaves the dipping reflection 182 ms early at 2000 m offset, its offsets no longer add in phase, and its post-stack peak at $x = m$ is 20 % of the PSTM peak, against 98 % for the flat reflector.

4. Why PSTM captures dip-dependent moveout

Standard NMO assumes the reflector is flat under the CMP. For a dipping reflector, the reflection point does not sit at the midpoint; it sits up-dip of it, and far-offset traces see a slightly different reflection location than near-offset traces. The moveout curve of a dipping reflector is therefore flatter than that of a flat reflector at the same zero-offset time, because its stacking velocity is $V/\\cos\\varphi$ (Levin, 1971). NMO at the flat-reflector velocity $V$ over-corrects it and leaves residual moveout, and no single velocity can flatten two conflicting dips at the same CMP: stack for the dipping reflector in exercise 4 of the figure (dip 35°) and the flat one falls to 18 % of its PSTM peak.

PSTM bypasses the NMO/stack chain entirely. Each pre-stack trace is deposited directly onto its isochron ellipse. At a dipping reflector, energy from near-offset and far-offset traces lands at the same $(x\_o, \\tau)$ because the reflector is tangent to every one of their ellipses at its true position. Stacking is emergent, not imposed. PSTM therefore images conflicting dips correctly within the accuracy of its RMS velocity; post-stack migration can do so only after DMO has turned the stack into a true zero-offset section.

5. Workflow: when to use PSTM

• Moderate dips, mild lateral velocity variation. PSTM is the natural upgrade from post-stack time migration whenever dips exceed ~30° or when the velocity structure shows lateral ramps of a few percent per km. Cost is 10-50× post-stack depending on maximum dip and offset range.
• AVO and pre-stack amplitude work. PSTM can preserve offset-dependent amplitudes because it never collapses the offset axis, provided each offset is weighted for its own geometry (an unweighted Kirchhoff sum images far offsets brighter); in the figure the weights make a reflector of constant reflectivity image at one amplitude across the image gather in (d). Offset gathers in the migrated domain are the input to AVO.
• Velocity analysis over migrated gathers. PSTM output gathers (one per CMP, with the offset axis retained) are flat-if-you-had-the-right-velocity. Residual moveout on a PSTM gather tells you directly how much to update $V\_{\\mathrm{mig}}$ (Section 5.9): in the figure, the image gather in (d) curls up when the velocity is too slow and droops when it is too fast.

6. When PSTM is not enough

• Large lateral velocity contrasts. PSTM uses a time-domain RMS velocity that smooths ray bending. Over salt, sub-volcanic or thrust zones ray bending must be modelled explicitly, so depth migration is required.
• Anisotropic targets. The isotropic DSR equation above breaks down in VTI media; an anisotropic PSTM (using the effective parameters $V\_{\\mathrm{nmo}}$ and $\\eta$) or full depth-domain anisotropic migration is needed.
• Very steep dips and turning waves. For constant velocity the ellipse is exact; in real media the straight-ray RMS traveltime degrades at steep dip and large aperture, operator aliasing grows, and overturned energy has no straight-ray path at all, so beam or full-wavefield depth methods (Section 5.5, Section 5.7) do better.

Where this goes next

Section 5.4 moves from time domain to depth domain. Instead of an ellipse parameterised by RMS velocity, depth migration uses ray-traced travel times in an explicit $v(x,z)$ model, correctly handling the lateral velocity contrasts that PSTM can only smear.

References

• Schneider, W. A. (1978). Integral formulation for migration in two and three dimensions. Geophysics, 43, 49.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Stolt, R. H., Benson, A. K. (1986). Seismic Migration: Theory and Practice. Geophysical Press.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.