Intercept, gradient and the classes

Module 05's curve bends. That is inconvenient, because storing a bent curve for every sample in a survey is not practical, and because a bend is hard to interpret. So AVO analysis does something simple: it plots the amplitude not against angle but against sin²θ, on which the curve straightens out almost completely, and then fits a straight line to it.

A straight line has two numbers in it. Where it crosses the left-hand edge is the intercept, written R₀ — the amplitude the reflection would have at zero offset, which is exactly the number module 03 was about. How steeply it tilts is the gradient, G. Together they are the whole of what most AVO work uses.

The gradient is not a property of the rock. It is a property of the rock and of the angle range fitted over, because the real curve is not quite straight. Moving the angles used for the fit slider moves G while the rock stays where it is. Two people can analyze the same gather, do nothing wrong, and report different gradients because one of them muted at 30° and the other at 40°.

Once a rock is two numbers it can be plotted as a point. How to read the picture on the left: it is a graph rather than a map or a cross-section. Each dot is one rock. Its position left to right is that rock's intercept, which is what its near trace looks like. Its position up and down is the gradient, which is how much the reflection changes out to the far traces. The circle is the rock currently on the sliders.

The gray dots are brine-filled rocks, at every porosity, clay content and shale velocity this page can build. The colored ones hold oil or gas. The picture shows a whole population at once, which is also what a crossplot of a real survey shows — one dot per sample, hundreds of thousands of them.

The three short arrows on the current rock give the picture a sense of direction. Each one shows where the dot moves under a single change: a higher porosity, a swap of the fluid to gas, or a faster shale on top. They are computed by actually making that change and seeing where the point goes, so they point wherever the rock physics says they should rather than wherever a textbook figure says.

Two rocks can share an intercept and differ in gradient. The panel on the right shows exactly that pair — the current rock, and the wet rock whose near trace matches it most closely. The near traces are the same and the far traces are not. Everything the gradient contributes is in that difference.

The gray dots are not scattered at random. Wet rocks fall along a line, because their Vp, Vs and density are related to each other in a fairly consistent way, whatever their porosity or clay content. That line is the background trend, and it is drawn on the crossplot below.

Once the trend is there, the useful question stops being "where is this dot" and becomes "how far off the trend is it". A rock sitting on the trend behaves like the wet rocks around it, whatever its intercept happens to be. A rock a long way off the trend behaves in a way the wet rocks do not, and a pore fluid is the usual reason.

This is why an absolute gradient threshold is a poor detector and distance from the trend is a good one. It also explains a common frustration: a trend fitted in one interval does not transfer to another, because the rocks are different, so an anomaly cut-off carried across a survey quietly stops meaning the same thing.

Class I, Class II, Class III and Class IV are the most quoted vocabulary in AVO, and the crossplot is where they come from. Rutherford and Williams sorted gas sands by where their intercept sat: strongly positive, near zero, or strongly negative. Castagna later added a fourth for soft sands whose reflection gets weaker with offset rather than stronger.

The regions are shaded on the plot below. Notice what they are: rectangles on a graph. Nothing physical happens as a dot crosses from one into the next. The boundary near zero intercept is drawn at ±0.03 here and different books put it in different places, because there is no natural place to put it.

The middle panel summarizes the four classes. Each row gives the signs of the intercept and gradient, what the gather does out to the far offsets, and a small sketch of the curve. Every row is generated by going and finding a real rock in this page's own parameter space that lands in that class, so the sketches cannot drift away from the physics.

The panel on the right makes the point another way. Both cards are gas sands at full saturation under the same shale. One is Class I and one is Class III. If the classes described kinds of rock, that could not happen.

Set a gas sand at 20% porosity, then move the shale velocity slider from one end to the other. The reservoir does not change. Its porosity, its clay content, its fluid and its saturation all stay exactly where they are. The dot travels across the crossplot and the class label under it changes.

Under a slow shale the sand is harder than its surroundings, so the near trace is a peak and the response is Class I. Under the default shale the two impedances nearly match, the near trace almost vanishes and it is Class II. Under a fast shale the sand is the softer rock, the near trace is a clear trough, and it is Class III — the textbook bright spot.

The three gathers on the right are the same gas sand every time. So a class is not a description of a reservoir. It is a description of a contrast, and half of that contrast belongs to a rock nobody is trying to characterize — which is what module 07 finds from the other direction, where an unmeasured shale widens the admissible porosity by nine units.

The practical consequence is that "we are looking for a Class III response" is not a prospecting criterion on its own. The class a target produces depends on the seal above it, and that can change across a field faster than the reservoir does.

Reducing a curve to two numbers is what makes AVO practical at survey scale. Two volumes instead of a hundred angle stacks, one dot per sample on a crossplot, and anomalies that can be found by a computer rather than by eye. All of that is real value.

• Crossplot the interval, not the sample. A dot means something only against the population it came from. Extract a window that contains enough wet rock to define a trend, or there is nothing to be anomalous with respect to.
• Record the angle range with the gradient. Step 1 shows the gradient moving by about five percent when the fit range changes and the rock does not. A gradient volume without its angle range attached cannot be compared with anything.
• Prefer distance from the trend to absolute thresholds. A cut-off on gradient alone will follow lithology and depth around a survey. Distance from a locally fitted trend will not, or at least not as much.
• Treat the class as vocabulary. It is a useful way to tell a colleague what the gather looks like. It is not a measurement, and it is not a property of the reservoir.

The error this module is built against is the one in step 5. A prospect gets described as "a Class III opportunity", the phrase gets repeated, and it hardens into a property of the reservoir. Then the seal changes across the field, the response moves to Class II, and the same sand stops looking like the thing everyone agreed to look for. Nothing about the reservoir changed. The class was never about the reservoir.

Step 1 contains a quieter one. The intercept and the gradient are usually treated as two independent measurements, and they are not — they come from one fit to one set of points, so an error in one is compensated by an error in the other. That correlation is why gradients are so much less stable than intercepts, and it is the subject of the next module.

The rock physics is modules 01 and 02's, unchanged, and the reflection coefficients are exact Zoeppritz as in module 05. The Method tabs of modules 01 and 02 list the assumptions in the rock model and they all apply here.

The intercept and gradient are fitted, not derived. They come from an ordinary least-squares straight line through the exact reflection coefficients plotted against sin²θ, sampled every degree from zero to the angle on the fit-range slider. That is what an AVO processor does with real amplitudes. It is not the same as reading Shuey's analytic R₀ and G off the rock properties: for the default rock the fitted gradient is −0.4169 against Shuey's −0.5270, a difference of more than a fifth. Shuey's expression is the tangent to the curve at zero offset, while a fit is a compromise across the whole range, and the two only agree in the limit of a very narrow range of angles. Both are called "the gradient".

The background trend is fitted live, by least squares through every brine-filled rock this page can build: porosity from 5 to 35%, clay from 0 to 30%, and shale velocity from 2300 to 3100 m/s. It comes out as G = −0.141 − 1.171·R₀ with an R² of 0.94 and a scatter of 0.039 about the line. Real background trends are fitted from well logs or from the seismic itself over an interval, and are usually less tidy than this one because real wet rocks vary in ways this model does not include.

The class boundaries are a convention and are drawn here at ±0.03 in intercept, with the III/IV split on the sign of the gradient. Rutherford and Williams defined the first three classes descriptively, by where gas sands of different consolidation happened to sit, rather than by numeric cut-offs; Castagna and Swan added Class IV later. Different texts and different companies draw the lines in different places, and no value is standard. Any dot near a boundary here should be read as "near a boundary", not as a member of one side.

Where this departs from production practice.

• The fit is to noise-free amplitudes. The noise control shades the gathers but the intercept and gradient are fitted to the exact values, so this page shows the effect of the angle range in isolation. What noise does to the fit — and it does a great deal, mostly to the gradient — is module 08.
• Every angle is weighted equally. Real gathers have fewer usable traces at wide angles, variable fold, and a mute that removes the widest ones, all of which weight the fit unevenly.
• One interface, one wavelet, thickness fixed at 45 m, so tuning does not vary across the exercises and the amplitudes are the interface response rather than an event amplitude.
• Angles are exact for the module 04 velocity model, and the fit is done against true incidence angle. Fitting against a straight-ray angle instead, which is common, biases the gradient in the direction module 04 quantified.
• No anisotropy. In a shale-dominated overburden this is a significant omission: anisotropy contributes a term that looks exactly like a gradient and can be comparable in size to the fluid effect being sought.
• The crossplot shows a modeled population, not a survey. A real crossplot has one dot per sample, hundreds of thousands of them, with a density that carries information this page's evenly-spaced grid of models does not reproduce.