Why stack, and why stacking is not enough

Part 5 — Imaging (migration)

Learning objectives

Explain why stacking improves S/N and what implicit earth model it assumes
Describe the three canonical ways stacking fails: dip, diffractions, lateral velocity variation
Read a zero-offset section and predict where each feature would sit in the true earth
State what migration is supposed to do, setting up the rest of Part 5

Parts 2–4 have focused on cleaning individual CMP gathers: correct the moveout, pick velocities, anisotropy, de-multiple, adaptive-subtract the residuals. After Part 4 each CMP is as clean as it can be. The next question is what to do with a collection of clean CMPs — how to turn them into a picture of the subsurface. The easy answer is to stack: sum each NMO-corrected CMP along offset, one zero-offset trace per CMP, line them up in order, and call the result the image. That answer is correct for a flat, layered earth and wrong for everything else. This section explains why, so that the migration tools in §§5.2–5.7 have a reason to exist.

1. What stacking buys you

Stacking along offset combines N traces that all carry the same primary reflection (after NMO correction). Random noise averages down as $\sqrt{N}$ while the coherent signal adds linearly — so stacking an N-fold CMP improves signal-to-noise ratio by roughly $\sqrt{N}$ . For an N = 60 streamer that is almost 18 dB of free gain. Stacking also suppresses residual multiples that do not flatten under primary-velocity NMO (because they stack destructively across offset) and it averages across small geology-to-geology variations within a CMP bin, producing a smoother section.

Mechanically, each stacked trace sits at the CMP midpoint and is treated as if it had been recorded by a single coincident source + receiver at that location. The full stacked section is therefore the zero-offset section: what you would record with one coincident shot/receiver pair at every surface CMP position. That equivalence is the key to everything that follows.

The top panel is a simple earth model with three toggleable features: a flat reflector at 500 m depth, a dipping reflector starting at 800 m, and a point diffractor at 900 m depth. The bottom panel is the zero-offset section the stack would produce in this earth (constant velocity $V = 2000\ \text{m/s}$ for transparency). Toggle each feature and compare the two panels. The mismatches are the three canonical failure modes of stacking.

3. Failure mode 1 — dipping reflectors arrive at the wrong apparent dip

For a reflector dipping at angle $\theta$ , the zero-offset ray hits the reflector perpendicular to it at distance $d\cos\theta$ from the surface midpoint, so the two-way time is

t(x) = (2/V)\,(d_0 + x \tan\theta)\,\cos\theta.

The dip of the event in the stacked section — in metres per metre, converting time back to depth using $V/2$ — equals $\sin\theta$ , not $\tan\theta$ . For a 30° true dip, apparent time-dip corresponds to only $\arctan(\sin 30°) = 26.6°$ . For 45°, apparent 35.3°. For vertical (90°), the dip on the stack caps out at 90° but the reflector in time is not where it should be in space. Dipping reflectors appear at the wrong dip and the wrong lateral position in an unmigrated stack.

Mnemonic. "Stack underestimates dip." Real 30° reads as ~26.6° on the section. Real 45° reads as ~35.3°. The steeper the dip, the worse the underestimate.

4. Failure mode 2 — point diffractors smear into hyperbolas

A point scatterer at $(x_d, z_d)$ sends energy back to every surface location, not just the one directly above it. The zero-offset arrival time from surface point $x$ is

t(x) = (2/V)\,\sqrt{(x - x_d)^2 + z_d^2},

which is a hyperbola with apex at $x = x_d$ , $t = 2z_d/V$ . A single point — a fault edge, a pinch-out, a salt flank, a truncated reflector — is smeared across the entire section. Stack does NOT collapse this hyperbola back to a point. What migration does — its single cleanest definition — is collapse every diffraction hyperbola to its apex. Sum each sample in the stacked section onto the hyperbola of possible scattering points that could have produced it, and coherent energy accumulates at the true scatterer location.

5. Failure mode 3 — lateral velocity variation leaves residual moveout

NMO assumes one vertical velocity profile per CMP. Real earth has lateral velocity contrasts — salt bodies, shallow gas, channel sands — that bend rays sideways. Different offsets within the same CMP sample different velocity paths, so NMO cannot flatten them all. The residual moveout left over does not stack constructively. Strong lateral velocity variation turns a high-fold CMP into a noisy, smeared stack. This is a major reason production processing moves from time migration (which handles gentle lateral velocity variation) to depth migration (§5.4 onwards) when lateral velocity contrasts are large.

6. What migration promises

Every migration method in Part 5 is solving one or more of the failure modes above:

Post-stack time migration (§5.2) — operates on the already-stacked section. Collapses diffractions and repositions dipping events assuming mild lateral velocity variation.
Pre-stack time migration (§5.3) — operates on CMP gathers before stacking. Captures dip-dependent NMO that post-stack migration cannot.
Pre-stack depth migration, Kirchhoff (§5.4) — ray-traced travel times in $v(x,z)$ . Handles strong lateral velocity variation.
Beam migration (§5.5) — adds controlled amplitude decay along ray bundles; cheaper than full wave-equation methods, more faithful than Kirchhoff.
One-way WE migration (§5.6) — extrapolates wavefields downward level by level, correctly propagating energy even in mildly complex media.
Reverse-time migration (§5.7) — the gold standard: full two-way wave equation with imaging condition. Handles salt overburdens, sub-salt targets, turning waves.
Artifacts & QC (§5.8) — the characteristic artifacts of each method and the QC you need to trust a migration.
Velocity model building (§5.9) — none of the above works without a good velocity model; how residual moveout drives iterative velocity refinement.

7. Not every project needs every method

A gentle-dip 2D line over a tabular sedimentary basin can be imaged acceptably with post-stack time migration. A sub-salt target in the Gulf of Mexico cannot be imaged without RTM. Production economics drive the choice: migration cost per km² can vary by four orders of magnitude between the cheapest time migration and the most thorough anisotropic elastic RTM. Picking the right tool is half the job.

**The one sentence to remember**

Stacking gives you a zero-offset section; migration turns a zero-offset section into a picture of the earth by repositioning dipping events and collapsing diffraction hyperbolas back to their apex.

Where this goes next

§5.2 introduces the simplest migration that actually works — post-stack time migration — as a Kirchhoff (diffraction-sum) operator. You will see the diffraction hyperbola in the widget above become the summation path for reconstructing each point in the image.

References

Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Claerbout, J. F. (1985). Imaging the Earth’s Interior. Blackwell.
Stolt, R. H., Benson, A. K. (1986). Seismic Migration: Theory and Practice. Geophysical Press.
Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.