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.