Post-stack time migration

Post-stack time migration is the simplest migration that actually works, and the workhorse for decades of 2D and 3D seismic. It takes a zero-offset section (§5.1), applies the Kirchhoff diffraction-sum operator, and produces a section where dipping reflectors are repositioned to their correct apparent position and diffractions collapse toward their apex. The only inputs are the stacked section and a migration velocity field — no pre-stack data required.

1. The Kirchhoff operator in one line

Each output pixel $(x_o,t_o)$ in the migrated image is the sum of input-section samples along the hyperbola of all scattering points that could have produced a reflection there:

Here $D(x,t)$ is the input zero-offset section and $V_{mig}$ is the migration velocity. The summation hyperbola passes through $(x_o,t_o)$ at its apex and curves outward with slope set by $1/V_{mig}$. When the output pixel is the true apex of an input diffraction hyperbola AND $V_{mig}$ matches the earth velocity, the summation curve matches the data curve exactly — every trace contributes constructive energy and the sum peaks. Away from the apex, the same summation averages over incoherent amplitudes and cancels. This is the imaging condition in its cleanest form.

2. The widget

The input section below is the zero-offset response of a single point diffractor at $(x=500,z=900)\ \text{m}$ in a constant-velocity medium $V_{true} = 2000\ \text{m/s}$, so the apex sits at $2z/V = 0.9\ \text{s}$. The green dot marks the apex; the dashed teal hyperbola is the summation operator that current V_mig would apply at the apex pixel. The migrated section below shows the result.

Drag the velocity slider. Watch the dashed summation hyperbola change shape. When it overlays the data hyperbola perfectly ($V_{mig} = 2000\ \text{m/s}$), the output is a concentrated peak at $(500, 0.9)$ — the diffraction has collapsed to its apex. The info strip reports the apex peak amplitude in dB relative to the input, giving a quantitative focus metric.

3. The two failure modes

• Under-migration (V_mig too high). The summation hyperbola is narrower than the data hyperbola. Each trace's contribution arrives at a time that does not match, the sum does not accumulate, and the original hyperbola remains visible in the output. Migration has had almost no effect.
• Over-migration (V_mig too low). The summation hyperbola is wider than the data. The sum "spreads" the hyperbola's energy above the true apex — a tell-tale migration smile that flares up and outward. Over-migration is easy to spot on production sections because smiles have no geological counterpart.

Both artefacts respond monotonically to the velocity error. A 5 % error produces subtle defocus; a 20 % error dominates the output. Post-stack time migration is velocity-sensitive, and §5.9 explains how to iterate the velocity field to drive both artefacts toward zero.

4. What post-stack time migration assumes

• The stack equals the zero-offset response. True if NMO velocities are good, fold is high, and lateral velocity variation is mild.
• One vertical RMS velocity per CMP governs the migration. Adequate when ray bending in the overburden is mild; breaks down over salt, volcanics, heavy fault blocks.
• Dips are moderate (less than ~45°). Post-stack cannot correctly image steeply dipping events because the stack itself did not preserve their kinematics; pre-stack migration is needed.
• Amplitudes are not the deliverable. Kirchhoff summation without amplitude weighting does not preserve true amplitudes — adequate for structural interpretation, not for AVO.

5. Implementation details production cares about

A production post-stack time migration adds several refinements to the pedagogical sum above:

• Amplitude weighting. Each contributing sample is weighted by the obliquity factor $\cos\theta$ and the geometric-spreading factor $1/r$ so the operator has a defined inverse. Without these, amplitudes are scrambled.
• Anti-alias filter. The summation operator is a lowpass filter in $k_x$; at steep dips the operator must roll off above its Nyquist or spurious high-frequency bands appear as aperture noise.
• Aperture. Summing over all traces is expensive and contaminates the output with noise from distant events. Production migrations use a finite aperture, usually a few times the deepest dip-corresponding half-wavelength, and taper its edges to avoid truncation ringing.
• Variable velocity. V_mig is a function of $(x,t)$, not a scalar. The hyperbola at each output pixel is computed with the local velocity, so the summation paths bend gently with the velocity field.

6. When to stop trusting it

Post-stack time migration is cheap and fast; use it as a first-pass image on almost every project. Replace it when:

• Dips exceed ~45° (move to PSTM, §5.3).
• Lateral velocity contrasts cause visible defocus you cannot tune away (move to depth migration, §5.4+).
• You need amplitude-faithful output for AVO or inversion (move to pre-stack or depth-migration workflows with proper weighting).

Where this goes next

§5.3 moves the operator to before the stack. Pre-stack time migration maps each pre-stack trace directly to its image contribution, handling dip-dependent NMO and steep dips that post-stack cannot.

References

• Schneider, W. A. (1978). Integral formulation for migration in two and three dimensions. Geophysics, 43, 49.
• Stolt, R. H. (1978). Migration by Fourier transform. Geophysics, 43, 23.
• Gazdag, J. (1978). Wave equation migration with the phase-shift method. Geophysics, 43, 1342.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Stolt, R. H., Benson, A. K. (1986). Seismic Migration: Theory and Practice. Geophysical Press.