Module 04 · Horizontal resolution
Modules 01 to 03 asked how thin a bed you can see. This one asks how small a feature you can see, looking down on a map. The answer is set by the same wavelength and by one processing step, migration.
The pulse spreads laterally as it travels down. When it reaches the reflector it illuminates a circular patch rather than a single spot, and the recorded echo is the sum of the returns from that whole patch.
The patch is as wide as it is because echoes from slightly off to the side travel only slightly further, arrive at nearly the same moment, and pile onto the same part of the trace. The patch has a name — the Fresnel zone — and at ordinary depths it is hundreds of meters across, usually bigger than whatever you are trying to map.
A single unmigrated trace therefore reports one number for an area the size of several football fields.
Migration is the processing step that undoes the spreading. It takes the energy recorded across the Fresnel zone and returns it to the position it came from. When data is described as migrated, this is what has been done to it.
The optical comparison is a defocused lens. A point source images as a broad disc, no energy has been lost, and focusing gathers it back into a small spot. Migration does the same with recorded seismic energy.
Panels 3 and 4 below show it on the section. Unmigrated, each feature draws a curved diffraction arc across many traces, and arcs from neighboring features overlap. Migration collapses each arc back to a point.
The comparison holds at the limit as well. A lens cannot focus a point below its own diffraction limit, and migration cannot shrink a feature below about a quarter of a wavelength — the same λ/4 that limited bed thickness in module 02.
Two small bright spots sitting on the same horizon, some distance apart.
Migration is the processing step that does the focusing. 3D focuses in every direction. 2D focuses along the line only, because a single line samples only one direction.
Nearly everything handed to you today is migrated. Older vintages and some legacy 2D lines are not, so if you ever work a survey from the tape library, set this to Unmigrated and see what you would be interpreting.
Migration gathers energy from traces on either side of the target, so this is the half-width of the recorded aperture rather than the full survey. The less of it there is to work with, the softer the focus.
Left: how big a patch each one smears into. Right: the map you would actually be handed.
1 The size of the blur drawn to scale
2 What the map looks like two features, or one?
A single small feature does not put its echo on one trace. It puts a little energy on every trace that can see it, which draws the curved arc on the left. The two arcs cross each other: energy from both features arrives on the same traces, mixed together. That overlap is the Fresnel zone seen in section, and it is why the unmigrated map is a smear. Migration gathers each arc back to the point that produced it.
3 Unmigrated each feature spreads into an arc
4 Migrated each arc collapses to a point
The section panels are computed rather than sketched: each small feature produces a diffraction arc at τ(x) = √(t₀² + (2Δx/V)²), convolved with a zero-phase Ricker wavelet and weighted by obliquity. That is exactly what a zero-offset section over a small feature looks like, and it is why migration has arcs to collapse.
The focused panels are not a full migration. They are the band-limited response that a successful migration produces: each feature images as a spot whose two-feature resolving distance is λ/(4 sin θ), with θ = arctan(aperture / depth) measured from the vertical at the target, so the aperture here is the distance to one side. This is the standard resolution result. It is used here in preference to running a real migration operator because a real one brings its own artifacts, which would distract from the point. The trade-off is that these panels show what migration is trying to achieve rather than the ways it can fail.
The spot shape is calibrated numerically so that two features exactly the quoted distance apart are on the point of separating — otherwise the picture and the number in the stat strip would disagree, which they did by a factor of nearly three before the calibration was added.
The Fresnel zone quoted is the first Fresnel zone at zero offset, equivalently r ≈ √(λz/2). Left out: velocity error, dip, 3D acquisition footprint, and the fact that migration only focuses correctly when the velocity model is right. A football field is taken as 100 yards, about 91 m.
Next up
You have now met the λ/4 limit twice, downward and sideways. The next module asks what happens to it as you go deeper, where velocity climbs, the earth removes the high frequencies, and the thinnest bed you can see grows by about a factor of five between the shallow section and the deep one.
Open module 05 → All modules