Offset, angle and the gather

A survey fires a source, records at a line of receivers, moves along and fires again. One firing gives a shot record: many receivers, one source position. Each trace in it reflected at a different place in the subsurface, so a shot record mixes geometry with geology.

The traces are therefore re-sorted before anything else is done to them. The midpoint of a trace is the point on the surface halfway between its source and its receiver, and for a flat reflector it is directly above the point where that trace reflected. Collecting every trace that shares a midpoint gives a common-midpoint gather, or CMP gather: several recordings of the same piece of subsurface, at different source-receiver separations.

That separation is the offset. A CMP gather is therefore a set of traces of one reflection point, ordered by offset, and it is the object every later module works on. The number of traces in it is the fold.

The right-hand panel shows this gather as recorded. The event is not flat, because a longer offset means a longer path and a later arrival. That curvature is the subject of step 3.

Everything from module 05 onward is about how the reflection changes with the angle at which the wave meets the boundary. What a survey records is offset. The two are related, and converting between them is where a good deal of trouble starts.

Module 00 used the simplest relationship: treat the ray as a straight line, and the half-offset and the depth are two sides of a right triangle, so the angle is the arctangent of one over the other. That is exact only if the velocity is constant all the way down.

It is not. Velocity generally rises with depth as sediments compact, which is the trend module 01 built into the grain frame. A ray traveling through rising velocity bends continuously, refracting away from the vertical as it goes, so it meets the reflector at a wider incidence angle — further from the normal, closer to horizontal — than a straight line from the same source would.

The left-hand panel draws both. The right-hand panel plots the two angles against offset, and the gap between them widens the whole way. It is not a small correction: at the settings this module opens with, the shortcut is 6.6 degrees low near 2000 m offset and 10.1 degrees low at 3000 m. An analysis that uses the wrong angle is fitting the right data to the wrong axis.

A longer offset is a longer path, so the reflection arrives later. The difference between the arrival time at some offset and the arrival time at zero offset is the normal moveout, and on a gather it is the curvature of the event.

To a good approximation the arrival time follows a hyperbola: t² = t₀² + x²/V², where x is the offset, t₀ the zero-offset time, and V a velocity that describes the whole overburden. The velocity that makes this work is close to the root-mean-square velocity down to the reflector, which is not the same as the average velocity and not the same as the velocity at the reflector itself. Three different velocities are in play and confusing them is a common source of error.

The moveout is large. At the settings here the longest trace arrives several hundred milliseconds after the nearest one, which is far more than the length of the wavelet. Nothing can be compared across the gather until it is removed.

Removing the moveout means shifting every sample on every trace up by the difference between its own arrival time and the zero-offset time. Do it correctly and the event becomes flat across the gather, which is what makes the amplitudes comparable.

The shift is not the same at every time on a trace, because moveout depends on t₀ as well as on offset. Shallow samples move up further than deep ones, so the trace is not translated, it is stretched. The waveform on a far trace comes out longer and lower in frequency than the one on a near trace, and the effect grows with offset and shrinks with depth.

That matters here more than it does for most purposes. An amplitude read from a stretched far trace is not measuring the same thing as one read from an unstretched near trace, and the stretch is worst exactly where the angle range is widest and the AVO information is largest. Processing normally mutes the far traces where the stretch exceeds some threshold, which removes the angles carrying the largest amplitude variation.

Once the gather is flat the usual next step is to add the traces together and divide by the fold. That is the stack, and it is the section almost every interpreter works from. It is a good thing to do: the reflection is the same on every trace and adds coherently, while random noise does not, so the signal-to-noise ratio improves roughly with the square root of the fold.

It is also where the angle information stops. The amplitude in this gather is not constant across offset — the rocks are the shale and gas sand from modules 02 and 03, and their reflection coefficient changes with angle. Adding the traces together replaces that whole trend with its average.

The left-hand panel plots the amplitude trace by trace, which is the AVO response seen as data rather than as a curve. The right-hand panel shows the stack: one number, sitting somewhere in the middle of the trend, carrying no record that a trend existed.

Nothing is wrong with the stack. It answers a different question. A measurement that depends on how the amplitude varies with angle has to be made before the stack, on the gather, which is why AVO analysis lives where it does in a processing sequence and why a stacked volume alone cannot be used for it.

Deciding whether a target is reachable. The angle a survey achieves at a target depends on the offsets recorded and the depth of the target. A deep reservoir on a short spread may never reach a wide enough angle range for the amplitude variation to be measurable, and that is a survey design question rather than a processing one.

Converting offsets to angles correctly. Every AVO analysis needs an angle for each trace. Using the straight-ray shortcut with a real velocity gradient puts every point on the wrong part of the axis, and the error grows with offset, so it distorts the shape of the trend rather than shifting it.

Reading angle stacks. A near stack and a far stack are averages over ranges of angle, and which angles fell into which stack depends on the velocity model used to sort them. Two processing runs with different velocities produce different angle stacks from identical field data.

Knowing what the mute removed. Stretch mutes are applied routinely and they remove the far offsets first. Since those carry most of the angle information, a gather that looks clean after processing may have had the most useful part of it removed.

The overburden is a linear velocity gradient, V(z) = V₀ + kz, which is a reasonable description of a compacting clastic section and has closed-form ray paths. The true incidence angle is obtained by finding the ray parameter whose path emerges at the requested offset, by bisection, and then reading the angle at the reflector. The straight-ray angle drawn beside it is arctan of the half-offset over the depth, which is the same shortcut module 00 used and said it would revisit here.

Moveout is the hyperbolic approximation, t² = t₀² + x²/V²rms, with the RMS velocity computed from the same gradient. This is the standard correction and it is an approximation: it is exact for a single homogeneous layer and degrades as offset approaches depth. The correction applied to the gather is done by resampling each trace at the moveout time for every output sample, which is how the stretch arises — it is a consequence of the arithmetic rather than something added afterward.

The reflection amplitudes are exact Zoeppritz P-wave coefficients for the module 03 rock pair: a Castagna mudrock shale over a 30% porosity gas sand from the module 01 frame and the module 02 fluid, at 23 MPa and 64 °C. They are evaluated at each trace's own true incidence angle. Traces whose angle exceeds 60° are dropped, because the coefficient changes very rapidly there and the approximations used downstream have stopped holding; the readouts report the farthest trace that survived.

Idealized rather than modeled:

• The reflector is flat and horizontal, so the midpoint really is above the reflection point. On a dipping reflector it is not, and that is what migration exists to fix.
• The fold is uniform and the offsets are regular. Real gathers are irregular, have gaps, and vary in fold across a survey.
• No geometrical spreading, transmission loss, absorption or multiples. Each of these changes amplitude with offset for reasons that have nothing to do with the rock, and correcting for them is a large part of what preparing gathers for AVO involves.
• No anisotropy. In a shale-rich section the moveout is not hyperbolic even in principle, and an anisotropic correction with a third parameter is used instead.
• No noise. What noise does to a measurement made across a gather is module 08.