Amplitude against angle

AVO is a statement about angle: how far from the normal the wave meets the interface. A recording crew measures offset: how far the receiver is from the source. Those are different quantities, and the step between them is where a good deal of error enters unnoticed.

The obvious conversion draws a straight line from source down to the reflector and back up, which gives tan θ = (x/2)/z. That would be right if the earth had one velocity. It does not. Velocity rises with depth, so by Snell's law the ray is bent continuously, and the angle it finally makes at the reservoir is larger than the straight line suggests. The left panel draws both rays.

The right panel plots angle against offset for three depths, and shows two things. The curve flattens, so doubling the offset does not double the angle. And the same offset is a completely different angle at a different depth — the widest angle available falls as the target deepens. A 3000 m offset reaches 64° on a 1200 m target and 35° on a 3200 m one, which is why deep AVO is hard for reasons that have nothing to do with the rocks.

A wave arriving straight down is pure compression. It squeezes the rock and the rock pushes back, and how hard it pushes back is the impedance. Nothing about the rock's resistance to shearing comes into it, because nothing is being sheared.

Tilt the ray and that stops being true. An obliquely incident P wave sets the interface moving partly sideways, which shears it, and part of the energy converts into a shear wave. From that moment the shear velocities of both rocks appear in the answer.

How to read the curve. Every one of these plots is a gather, redrawn. A gather is a set of traces recorded at increasing distance from the source, and each trace has its own reflection strength. Lining those strengths up left to right gives the curve. So the left edge of any curve on this page is the trace recorded almost straight above the reservoir, the right edge is the one recorded from furthest away, and the height of the line is simply how strong the reflection is on that trace. The header at the top of the page shows both pictures side by side, and so does every step from here on.

The panels below isolate what offset adds. They take one rock and change only its shear velocity, by fifteen percent either way, leaving Vp and density exactly alone. Look at the left edge of the curves: all three start from the identical point, because at zero offset Vs has no effect whatsoever. Then follow them right. By thirty degrees they are far apart.

The gathers beside them show the same result in the form it appears on a workstation. The near traces of all three rocks are indistinguishable. The far traces are obviously different. That difference is the only place the shear information lives.

AVO rests on that difference. Vs responds to a gas fill differently from the way Vp does, and normal incidence does not record Vs at all.

Now the full calculation for the rock on the sliders. The curve on the left is the exact solution — Zoeppritz's equations, solved rather than approximated — and the panel beside it is the gather that curve produces once each trace is convolved with a wavelet.

Take a brine sand at 20% porosity first. It starts positive and falls, crossing zero somewhere out on the far offsets. Now switch to gas. It starts negative and gets more negative with angle, which is the response people picture when they say "a bright spot". Then set the porosity to 10% and keep the gas: it starts positive and falls steeply. Same fluid, opposite-looking gather.

The starting value and the slope are separate facts. They carry different information, and module 06 is entirely about that split. A curve that changes sign partway out appears on a gather as a reflection that fades and comes back, so an event vanishing at mid offset is a signal rather than a processing problem.

Nobody inverts Zoeppritz's equations in practice. They are exact but opaque — four coupled equations with no useful intuition in them — so the working forms are linearizations. The two-term Shuey approximation, R(θ) ≈ R₀ + G sin²θ, is the one almost all AVO analysis is built on, and the three-term version adds a curvature term that matters only at wide angles.

Both are drawn below against the exact answer, with the error underneath. Out to about twenty degrees they are all fine. Past thirty they separate, and for this rock pair the three-term approximation is further from the truth than the two-term one. Its extra term carries tan²θ, which grows without limit, so at wide angles it diverges rather than losing accuracy gradually. Whether it does so for any particular pair of rocks is something the panel below computes rather than something to assume.

The practical consequence is that the two-term form is reliable over the angle range where most surveys have usable data, and that fitting a third term to a noisy gather with a limited angle range usually costs more than it returns. Module 08 puts a number on that.

Modules 01 and 02 each ended with two rocks that produced identical zero-offset amplitudes. Now they can be tested properly, because there is a second measurement to test them with.

The first panel plots the gap between each pair against angle, beside the size of the noise. Above the shaded band the difference is measurable; inside it, it is not. One line climbs out and the other does not. The two panels beside it draw the four rocks as gathers.

The middle panel is module 03's pair: an oil sand at 30% porosity against a plain brine sand at 35%. Their zero-offset amplitudes differ by 0.0001. Their Vp/Vs values differ by 0.21, which is a lot, so the curves pull apart steadily and by thirty degrees the gap is 0.0256 — comfortably outside a clean or moderate noise level. Offset breaks this tie.

The right panel is module 07's pair: a 30% porosity sand full of gas against a 34% porosity sand with only a tenth as much gas in it. Here Vp/Vs differs by 0.07 rather than 0.21, and the curves stay close. At thirty degrees the gap is 0.0123, which is smaller than the moderate noise level. Offset does not break this tie, and no amount of processing will, because the information is not in the data.

Adding offset converts one measurement into two, and the second one responds to shear. Ambiguities that involve a change in Vp/Vs are resolved by it. Ambiguities that do not are left where they were. Which kind an ambiguity is can be established before an AVO study starts.

Almost every practical AVO decision reduces to a question about angle range, and most of them get answered by habit instead.

• Survey design. Whether a proposed cable length gives useful AVO on the target is a step-1 calculation, and it has to be done at the target's depth rather than in general. The same acquisition that was excellent on a shallow prospect can be useless two seconds deeper.
• Deciding whether AVO is worth doing. Step 5 is the test. If the ambiguity to be resolved involves a change in Vp/Vs, offset will help. If it does not — and fizz gas does not — an AVO study will produce a number that looks like an answer and is not one.
• Reading angle stacks. A near stack and a far stack are two samples of the curve in step 3. Knowing what angles actually went into each one, after ray bending and after whatever mute was applied, is the difference between comparing two measurements and comparing two arbitrary averages.
• Choosing a formulation. The choice between two and three terms is not about rigor, as step 4 shows. It is about whether the angle range and the noise can support a third parameter.

One common error is treating angle as a detail of the processing rather than as the physical variable the whole method depends on. Amplitudes get extracted against offset, plotted against sin²θ using a straight-ray conversion, fitted, and interpreted — and the systematic angle error from step 1 propagates straight into the gradient, in a known direction, without ever announcing itself.

A second error is treating the exact solution as the rigorous one and the approximation as the compromise. Step 4 shows a case where the more complete approximation is further from the truth over the range that matters. Exactness in one part of a calculation earns nothing if the inputs to it are uncertain, and on a real gather the angles, the wavelet and the amplitudes all are.

The rock physics is modules 01 and 02's, unchanged. Voigt-Reuss-Hill mineral mixing, a Dvorkin-Nur soft-sand dry frame at 40% critical porosity and 23 MPa, Batzle-Wang fluids mixed by the Reuss average, Gassmann for the saturated modulus, and Castagna-Gardner for the bounding shale. Module 01's Method tab lists the assumptions and they all apply here.

Reflection coefficients are exact. The curves are the Zoeppritz solution for the plane-wave PP reflection, obtained by solving the four-by-four system rather than by any approximation. The Shuey curves in step 4 are drawn only to be compared against it. The base of the reservoir is computed as its own interface rather than assumed to be the negative of the top, because that symmetry holds only at normal incidence.

Ray geometry is exact too, for the velocity model assumed. The overburden is taken as V(z) = 1600 + 0.6z m/s, a compacting clastic section. For a linear gradient the rays are circular arcs with a closed-form offset and traveltime, so the incidence angle at the target is found by solving for the ray parameter that arrives at the requested offset, not by an approximation. The straight-ray comparison curve is tan θ = (x/2)/z. The gradient is fixed at 0.6/s here rather than exposed as a control, which is steeper than the 0.5/s module 04 opens with, so the true angle quoted for a given offset and depth is a little wider on this page than on that one.

Depth changes only the geometry here. Moving the depth slider changes the traveltime, the RMS velocity and therefore the angle at a given offset. It deliberately does not change the effective pressure acting on the rock, which stays at 23 MPa. In reality a deeper reservoir is also a stiffer one, and its anomaly would weaken for that reason as well. Both effects are real; separating them is what makes the angle lesson legible.

Where this departs from production practice.

• The gathers are already flattened. Normal moveout has been removed and no stretch has been applied. Real NMO stretches the far-offset wavelet to lower frequency, which changes the amplitude of a tuned event with offset for reasons that have nothing to do with the rocks. Module 04 works the stretch out; it is one of the largest sources of false AVO.
• Amplitudes are plane-wave coefficients only. No geometric spreading, no transmission loss through the overburden, no attenuation, and no source or receiver directivity — every one of which varies with offset and has to be corrected before the remaining variation means anything.
• The overburden is isotropic. Real shales are not. Anisotropy changes the moveout and the angle-dependent reflectivity, and ignoring it produces a gradient error that can be as large as the fluid effect being looked for.
• One interface pair, one wavelet, no interference from above or below, and the reservoir thickness is fixed at 45 m so that tuning does not vary across the exercises. The wedge module in the resolution set covers what tuning does to an amplitude.
• Converted waves are computed but not shown. The Zoeppritz solve returns the PS coefficients as well; only PP is used here.
• Angles beyond critical are left blank rather than guessed. Where the transmitted wave turns, the reflection coefficient becomes complex and the real-arithmetic solution stops being meaningful. The curve simply ends.