Pre-stack time migration (PSTM)

Post-stack time migration (§5.2) treats the stacked section as a zero-offset record, smears each sample along a Kirchhoff hyperbola, and sums coherently at reflector positions. That works when stacking preserved the kinematics — i.e., when every CMP was flat in offset after NMO. The moment the reflector dips, stacking smears it at a wrong velocity, the zero-offset assumption breaks, and the post-stack hyperbola is looking for signal that is no longer in the right place. 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

— source-to-scatterer time plus scatterer-to-receiver time. This is the double-square-root (DSR) equation that governs PSTM. For a given trace, every sample $(m, h, t_obs)$ is a measurement of the DSR sum, and PSTM asks: for each output pixel $(x_o, t_o = 2z/V)$ in the migrated image, which trace samples could have produced that pixel?

2. The PSTM isochron is an ellipse

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

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

which is always less than $t_{obs}$ for $h > 0$ and exactly equals $t_{obs}$ when $h = 0$. That means: the PSTM isochron degenerates to the post-stack migration hyperbola at zero offset. PSTM is a generalisation; post-stack is its zero-offset corner case.

3. The widget

The widget shows the surface geometry (source at $m - h$, receiver at $m + h$, midpoint marker) and the corresponding image-space isochron. The solid yellow curve is the PSTM ellipse for the current $(h, t_{obs}, V)$; the dashed teal curve is the equivalent post-stack hyperbola at the same $t_{obs}$ (what you would get if this pre-stack trace were stacked at zero offset). Slide $h$ to 0 and the two curves overlap exactly. Slide $h$ upward and the ellipse flattens — its apex moves up while its sides spread out.

The info strip reports the apex time of each operator and the "flattening ratio" between them. For $h = 200\ \text{m}, t_{obs} = 0.8\ \text{s}, V = 2000\ \text{m/s}$: post-stack apex at 0.800 s, PSTM apex at about 0.775 s — 25 ms shallower. The full migration sums many such isochrons; the difference between them is what PSTM does for dip-dependent geometries.

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 narrower than a flat reflector at the same zero-offset time — a standard NMO (tuned for a flat velocity profile) over-corrects and residual moveout remains.

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 arrives at the same $(x_o, t_o)$ because their ellipses all pass through the reflector's true position — stacking is emergent, not imposed. PSTM therefore handles dips up to about 70–80° correctly; post-stack cannot.

5. Workflow: when to use PSTM

• Moderate dips + lateral v variation is mild. 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 preserves offset-dependent amplitudes because it never collapses the offset axis; 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 directly tells you how much to update $V_{mig}$ — see §5.9.

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 exactly — depth migration is required.
• Anisotropic targets. The isotropic DSR equation above breaks down in VTI media; an anisotropic PSTM (using effective parameters eta, delta) or full depth-domain anisotropic migration is needed.
• Steep dips above ~80°. The isochron operator above is the leading-order ellipse; at the ellipse’s turning points the travel-time derivatives misbehave and beam or full-wavefield methods do better.

Where this goes next

§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.