Migration: why unmigrated data lies

Part 1 — Foundations of Seismic

Learning objectives

Understand why unmigrated seismic data misrepresents dipping and point-like features
Recognize the diffraction hyperbola as the footprint of a subsurface point scatterer
Describe what migration does and why it depends on velocity
Identify under-migration and over-migration from their visual signatures

Of all the processing steps we summarized in §1.3, migration is the one an interpreter must understand viscerally. It is the difference between a picture that reflects where things actually are and a picture that systematically lies about where things are. Unmigrated data is not "almost right" — it is wrong in predictable ways that an interpreter who understands migration can anticipate.

The problem migration solves

Unmigrated seismic has two systematic distortions:

Dipping reflectors are displaced. A reflector that dips downward to the east appears on the section shifted toward the east compared to its true position — and its apparent dip is gentler than its true dip. The steeper the dip, the worse the distortion.
Point-like scatterers produce hyperbolas. A single object small enough to scatter energy in all directions (a tight salt body, a localized fault, a gas chimney) appears on the unmigrated section as a downward-opening hyperbolic arc, not as a point.

Both distortions stem from the same root cause: unmigrated data plots the reflection at the midpoint of the source-receiver pair, not at the actual subsurface reflection point. For horizontal reflectors these coincide. For dipping or point-scattering features they do not.

The diffraction hyperbola

Imagine a tiny scatterer buried at subsurface position $(x_0, z_0)$ in a constant-velocity medium. A source at position x fires and the scatterer returns energy to a receiver somewhere at the surface. The two-way travel time is the source-to-scatterer distance plus the scatterer-to-receiver distance, all divided by the velocity. For a source-receiver pair whose midpoint is at x, the energy is recorded at travel time:

t(x) = \sqrt{ t_0^2 + (2 (x - x_0) / V)^2 }

where $t_0 = 2 z_0 / V$ is the two-way time directly above the scatterer. Plot this t-versus-x relationship and you have a hyperbola with its apex at $(x_0, t_0)$ . The apex points upward (toward shallower time). The sides ride asymptotically toward $t = 2 |x - x_0| / V$ — a steep line whose slope is controlled by the medium velocity.

Every unmigrated section you will ever see is a composition of hyperbolas: one per point scatterer, or more precisely, one per horizontal-wavenumber component of every reflector. Even a nominally-flat reflector consists of overlapping hyperbolas whose apexes line up along the true reflector depth — which is why flat reflectors look "right" on unmigrated data. They only look right because the hyperbolas constructively interfere along the horizontal line.

Migration is the operation that reverses this hyperbolic smear. Every sample on the unmigrated section is asked: "what single subsurface point would have produced this?" and the energy is moved back to that point. When done correctly, diffraction hyperbolas collapse to focused points; dipping reflectors shift to their true position and recover their true dip. When done incorrectly, these things get worse, not better.

The widget above shows a single point scatterer in a 3000 m/s medium. The left panel is the unmigrated section — a textbook diffraction hyperbola. The right panel is the result of migrating with whatever velocity you choose on the slider. Drag through the full range and pay attention to three regimes:

The three signatures you must be able to recognize

Correct velocity (Vm = 3000): the hyperbola collapses cleanly to a focused point at its true location. The crosshair marks where the scatterer actually is.
Under-migrated (Vm < 3000): the hyperbola isn’t fully collapsed. The image still shows a residual arc opening downward, just less pronounced than the unmigrated version. Energy is smeared along what looks like the same curve, shorter.
Over-migrated (Vm > 3000): the hyperbola has been "over-collapsed" — it folds past the scatterer and inverts into an upward-opening curve, the famous migration smile. Smiles are unmistakable once you’ve seen a few.

These three signatures are how a migration-literate interpreter diagnoses velocity problems on the section in front of them. Residual hyperbolas mean the migration velocity was too low. Smiles mean it was too high. Both indicate that the velocity model needs refinement for the area where they appear.

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 only vertically, 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).

One last practical note. Migration produces characteristic artifacts — spurious features that are not real geology. Migration smiles from inadequate aperture near the edge of the survey, migration "sags" in steeply-dipping zones where velocity was underestimated, noise amplified into coherent curves in low-fold areas. An interpreter who recognizes these artifacts avoids the trap of picking them as real events. We’ll return to specific artifact signatures in Part 2.

References

Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). Society of Exploration Geophysicists.
Sheriff, R. E., & Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Bacon, M., Simm, R., & Redshaw, T. (2003). 3-D Seismic Interpretation. Cambridge University Press.
Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.