Migration: why unmigrated data lies

The problem migration solves

Unmigrated seismic has three systematic distortions, all of them predictable:

1. Dipping reflectors are displaced and flattened. A reflector that dips downward to the east appears on the section shifted toward the east (downdip) of its true position, and its apparent dip is gentler than its true dip: with time scaled to depth by $V/2$, a reflector of dip $\theta$ shows the slope $\tan\alpha = \sin\theta$, so 30° reads as 26.6°. The steeper the dip, the worse both errors.
2. Point-like scatterers produce hyperbolas. An object small enough to scatter energy in all directions (the sharp edge of a faulted reflector, the tip of a pinch-out, a small karst cavity, the corner of a salt body) appears on the unmigrated section as a hyperbolic arc whose flanks bend down to later times, not as a point.
3. Tight synclines become bow ties. When a syncline is curved tightly enough that its centre of curvature lies below the surface, the echoes from each flank come up on the far side of its axis. The branches cross, and one trace records two or three reflections from a single surface.

All three stem from the same root cause. A stacked section approximates coincident source and receiver, and it plots each echo directly below the midpoint where it was recorded, at its two-way time, as if it had come from straight below. For horizontal reflectors that is where it came from. For dipping, curved or point-like features it is not: the echo arrived along a slanted path, from updip or from the side.

The diffraction hyperbola

Imagine a tiny scatterer buried at subsurface position $(x_0, z_0)$ in a constant-velocity medium of velocity $V$. A source and a receiver placed together at the surface position $x$ (the zero-offset trace that a stack approximates) record its echo after the wave has travelled down to the scatterer and back along the same slanted path, a distance $2\sqrt{(x - x_0)^2 + z_0^2}$. The echo arrives at

where $t_0 = 2 z_0 / V$ is the two-way time directly above the scatterer. Plot $t$ against $x$ and you have a hyperbola with its apex at $(x_0, t_0)$: the earliest echo, recorded directly above the scatterer. Away from the apex the flanks bend down to later times and approach the straight lines $t = 2 |x - x_0| / V$, so the slower the rock, the steeper the flanks.

Every unmigrated section you will ever see is a superposition of such hyperbolas, because by Huygens’ principle every point of every reflector acts as a scatterer. Along a flat reflector the hyperbolas’ apexes line up at the true time and their flanks cancel, which is why flat reflectors look right on unmigrated data. Along a dipping reflector the hyperbolas add up along the line that touches all of them, which lies downdip of the reflector; at a reflector’s end the last flank has nothing to cancel it and survives as a diffraction.

The three signatures you must be able to recognize

• Correct velocity ($V_{\mathrm{mig}}$ = 3000 m/s): the diffraction collapses cleanly to a focused point at its true location, marked by the cross.
• Under-migrated ($V_{\mathrm{mig}}$ < 3000 m/s): the diffraction only partly collapses and leaves a residual frown hanging below the point. It is narrower and steeper than the original, because what is left behaves like a diffraction of velocity $\sqrt{V^2 - V_{\mathrm{mig}}^2}$, 1308 m/s when $V_{\mathrm{mig}}$ = 2700 m/s.
• Over-migrated ($V_{\mathrm{mig}}$ > 3000 m/s): the energy swings past the point into an arc curving up above it, the famous migration smile. Smiles are unmistakable once you’ve seen a few.

These signatures are how a migration-literate interpreter diagnoses velocity problems on the section in front of them. Residual frowns mean the migration velocity was too low. Smiles mean it was too high. On a dipping reflector the same errors show as a wrong dip and position: at 3000 m/s the figure restores the true 30°, while 15 % too fast gives 35.1° and moves the reflector 62 m too far updip. On the syncline, a velocity well below the rock's, 9 % slow or more, still leaves a smaller bow tie at its floor, and too fast a one swings the flanks outward until, past about 12 % fast, they cross above the flat reflector. Each indicates that the velocity model needs refinement where it appears.

Time migration versus depth migration

We’ve been working in time: the output section has time on its vertical axis. A time migration gives you a clean image, laterally repositioned, but still plotted against two-way travel time. If you want to know where features are in real depth, you need a depth migration, which requires an explicit depth-domain velocity model and outputs features plotted against depth directly.

Time migration is cheaper, is routinely adequate when velocities vary smoothly and mostly with depth, and is what you’ll usually see first. Depth migration is mandatory when velocities vary strongly laterally, notably around salt bodies, complex thrust belts, and other structurally demanding settings. Always check which one you’re looking at before making structural inferences. A time-migrated section displayed against time can be mistaken for a depth section if the interpreter is not careful, a dangerous habit that leads to misplaced prospects.

Pre-stack versus post-stack migration

Migration can be applied before or after stacking. Pre-stack migration is computationally expensive but preserves offset information, handles complex velocities better, and is the workflow of choice for modern interpretation-grade data. Post-stack migration applies to the already-stacked section and is simpler and faster but assumes the stack itself is a reasonable approximation of zero-offset data, a weaker assumption in structurally complex areas. For most interpretation contexts today, the data you receive has been pre-stack time migrated (PSTM) or pre-stack depth migrated (PSDM).