Why stack, and why stacking is not enough

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 Sections 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 the signal-to-noise ratio by roughly $\\sqrt{N}$. For a 60-fold CMP ($N = 60$) that is $20\\log\_{10}\\sqrt{60} = 10\\log\_{10} 60 \\approx 17.8$ dB of signal-to-noise gain. Stacking also suppresses residual multiples that do not flatten under primary-velocity NMO (because they stack destructively across offset), and it averages over small lateral variations in geology 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 holds exactly only for flat layers: for a reflector dipping at $\\theta$ the moveout velocity becomes $V/\\cos\\theta$ and the reflection points smear updip across offsets. The rest of this section sets that aside and asks a sharper question: even a perfect zero-offset section, what does it show?

2. What a stack shows, and where things are

The figure puts a constant-velocity earth ($V$ = 2000 m/s) in plate (a): a flat reflector at 400 m, a reflector dipping at $\\theta$ and a point diffractor. Plate (b) is its ideal zero-offset section, what a perfect stack approximates, with two-way time drawn at $z = Vt/2$ so that both plates share one scale and an angle on screen is the angle stated. Tilt the reflector, move the diffractor and the probe trace, and compare where each echo sits in (b) with where it came from in (a). Then press Migrate on plate (b).

At the opening setting the reflector dips 30°, but the section shows it dipping only 26.6°, and the echo recorded at 1200 m comes from 560 m updip, at 970 m depth; the section hangs it straight below the trace at 1120 m. The diffractor at 900 m depth becomes a hyperbola whose flanks are already 436 m from the apex 100 ms below it. The flat reflector is the control, the one feature the section gets right. The dipping reflector and the diffractor show the first two failure modes of stacking; the third, lateral velocity variation, is described in the text below. Migrate (b) and the reflector dips 30.0° again while the hyperbola collapses to its apex at 0.90 s.

3. Failure mode 1: dipping reflectors arrive at the wrong dip and the wrong place

For a reflector dipping at angle $\\theta$ whose depth below surface point $x$ is $d(x) = d\_0 + x\\tan\\theta$, the zero-offset ray meets the reflector at right angles after a path of length $d(x)\\cos\\theta$, so the two-way time is

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 a vertical reflector (90°) the apparent dip reaches only $\\arctan(\\sin 90°) = 45°$: the time dip $dt/dx = 2\\sin\\theta/V$ can never exceed $2/V$, so no event on a zero-offset section dips more than 45° once time is converted with $V/2$. The echo is also in the wrong place: it comes from where the ray meets the reflector, $d(x)\\cos\\theta\\sin\\theta$ updip of the trace, but the section hangs it straight below the trace. Dipping reflectors appear at the wrong dip and the wrong lateral position in an unmigrated stack.

4. Failure mode 2: point diffractors spread 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

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 spread across the entire section, and the deeper the point the broader the hyperbola. Stacking does not collapse this hyperbola back to a point. The cleanest single definition of migration is that it collapses every diffraction hyperbola to its apex. For each image point, sum the stacked section along the diffraction hyperbola centred on that point; coherent energy adds up only where a real scatterer sits. (Equivalently, spread each sample over the semicircle of points that could have produced it.)

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

NMO assumes one vertical velocity profile per CMP. The 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 (Section 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 (Section 5.2): operates on the already-stacked section. Collapses diffractions and repositions dipping events assuming mild lateral velocity variation.
• Pre-stack time migration (Section 5.3): operates on CMP gathers before stacking. Captures dip-dependent NMO that post-stack migration cannot.
• Pre-stack depth migration, Kirchhoff (Section 5.4): ray-traced travel times in $v(x,z)$. Handles strong lateral velocity variation.
• Beam migration (Section 5.5): propagates energy along Gaussian beams, which handles the multiple arrivals that single-arrival Kirchhoff misses, at a cost below full wave-equation methods.
• One-way wave-equation migration (Section 5.6): extrapolates wavefields downward depth step by depth step; handles strong lateral velocity variation and multipathing, but not turning waves or dips near 90°.
• Reverse-time migration (Section 5.7): the gold standard: full two-way wave equation with imaging condition. Handles salt overburdens, sub-salt targets, turning waves.
• Artifacts & QC (Section 5.8): the characteristic artifacts of each method and the QC you need to trust a migration.
• Velocity model building (Section 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 is imaged best with RTM, which handles the steep salt flanks and multipathing that cheaper methods miss. 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.

Where this goes next

Section 5.2 introduces the simplest migration that actually works, post-stack time migration, as a Kirchhoff (diffraction-sum) operator. The diffraction hyperbola of Figure 5.1 becomes the path along which each point of the image is summed.

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.