Module 07 · Geometric attributes
Module 05 differentiated the shape of the reflector twice. This one differentiates something else entirely: how bright the reflector is as you walk along it. A channel can be perfectly flat and still have edges, and no amount of structural curvature will ever find them. The amplitude changes sharply right there, and that change is a bend the same arithmetic can measure.
The seismic line — click it to move the analysis point
Build the model — results update as you drag
Work through the five steps in order, then try the exercises. Method and references are there when you want them.
Walk along a reflector and two separate things can change. The reflector can go up and down — that is structure, and its second derivative is the structural curvature of module 05. Or the reflector can get brighter and dimmer while staying at exactly the same depth, and the second derivative of that is amplitude curvature.
The line in the header has both. There is a gentle fold, and there is a channel cut across it whose fill is dimmer than the surrounding rock. Notice that the channel has no structural expression at all — the reflector does not step or bend where the channel edge is, it simply dims. A structural attribute is blind to it by construction.
The two curves are independent. Drag the fold relief to zero and the structure curve flattens while the amplitude curve is untouched. Drag the channel dimming to zero and the reverse happens. Real data has both at once, and the two families of attribute exist because no single measurement sees both.
Structural curvature is computed on a surface of depth. Amplitude curvature is computed on a surface of amplitude: at every trace, follow the reflector to wherever it sits and record how strong it is there. That gives a map with the same footprint as the structure map but an entirely different quantity in it.
This is the step where the channel appears. On the structure map it is invisible. On the amplitude map it is the most obvious thing there.
1 Structure on the horizon depth, in meters
2 Amplitude on the horizon the same surface, a different measurement
Take the amplitude map, find its gradient in the inline and crossline directions, then differentiate again. The two eigenvalues of that second derivative are the most-positive and most-negative amplitude curvatures, written e_pos and e_neg.
Look at where they land. The channel interior is dim but flat, so its second derivative is near zero and the response there is close to nothing. What lights up is the pair of edges, where the amplitude turns a corner. One edge is a crest of the amplitude surface and the other is a trough, so a channel appears as one bright lineament on e_pos beside one on e_neg — the same paired signature a fault gives in structural curvature, for exactly the same reason.
1 Amplitude what we are differentiating
2 e_pos most-positive amplitude curvature
3 e_neg most-negative amplitude curvature
Module 05 made a point of fitting a circle to the reflector: curvature was one over its radius, and the principal curvatures k₁ and k₂ were corrected for the dip the reflector already had. None of that survives here, and the reason is units.
A structural surface is length against length — meters of depth against meters of distance — so a circle drawn on it is a real circle with a radius in meters. An amplitude surface is amplitude against length. There is no meaningful circle to fit to something whose two axes are not the same kind of quantity, and there is no dip to correct for, because the vertical axis is not a direction in the ground at all. The AASPI documentation says exactly this and draws the same conclusion: revert to the simpler most-positive and most-negative curvature.
So the family of attributes is smaller. You get e_pos, e_neg and their strike, and from those the curvedness and shape index of module 06. You do not get k₁ and k₂ in any meaningful sense, and two structural attributes disappear entirely: reflector convergence and rotation about the normal both describe how a surface in the ground is oriented, and an amplitude surface is not in the ground.
The whole argument is in two panels. Structural curvature is computed on the depth map; amplitude curvature on the amplitude map. Same survey, same horizon, same operator length.
The fold appears on the left and not the right. The channel appears on the right and not the left. Turn the fold relief up and down, and the channel dimming up and down, and watch each panel respond to its own quantity and ignore the other one completely.
1 Structural curvature k₁ from the depth map
2 Amplitude curvature e_pos from the amplitude map
Every attribute up to module 06 measures the shape of the reflector. That works well for faults and folds and finds nothing at all in a flat-lying stratigraphic feature, because there is nothing there to bend. Amplitude curvature addresses that case by differentiating how bright the reflector is instead of where it is.
It maps outlines, not bodies. The dim interior of a channel is flat, so its second derivative is near zero and it reads an order of magnitude below the edge. If you want the body of the channel you want an amplitude map, or an envelope map — the strength of the reflection regardless of whether it is a peak or a trough. This attribute gives you the boundary instead.
It only sees abrupt change. Soften the channel edge from 1 trace to 4 and the edge and the interior become indistinguishable, because a gradational boundary has no sharp second derivative anywhere. A facies change that takes a few hundred meters to happen will not appear, however large the total amplitude contrast across it.
Amplitude is noisier than structure, and differentiating twice amplifies that noise. Dip is a fitted, smoothed estimate; amplitude is read straight off the trace. Differentiating that twice makes amplitude curvature the most noise-sensitive attribute in this course after aberrancy, and it is equally sensitive to acquisition footprint, tuning and any amplitude-altering processing step. An anomaly here is only as trustworthy as the amplitudes underneath it, so it belongs on data whose amplitudes you actually believe.
The model is one reflector carrying a gentle fold, cut by a channel whose fill is dimmer than the surrounding rock. The channel has no structural expression whatever — the reflector runs through it at exactly the depth it would have had otherwise — so any response to it in a structural attribute would be an artifact. Depth and reflection strength are stored as two independent maps and the seismic line is built by convolving the second with a 30 Hz Ricker wavelet and shifting it by the first.
Amplitude curvature is computed exactly as structural curvature was in module 05, on a different input. The amplitude extracted along the horizon is differenced in the inline and crossline directions to give two gradients, those are smoothed along the operator length, and differenced again to give the three second derivatives. e_pos and e_neg are the two eigenvalues of that second-derivative matrix, and the strike is its eigenvector direction. There is no term in the dip components anywhere, which is the whole point of step 4.
Where this departs from AASPI, and why. Program curvature3d takes as input the inline and crossline components of the coherent energy gradients computed by program similarity3d, not amplitude read off a picked horizon. Coherent energy is the energy retained by the Karhunen–Loève filter of module 04, so it is a smoothed, dip-steered measure of reflection strength rather than a raw sample value, and using it makes the result markedly more stable. Amplitude along a horizon is used here because it is the version a beginner can see the whole of, and because the second-derivative behavior being taught is identical. AASPI also allows envelope or impedance as the input quantity.
The documentation is explicit about the two consequences drawn in step 4. Because the amplitude gradients carry units of amplitude per unit distance rather than length per unit length, it makes no sense to fit a circle tangent to the surface, so the computation reverts to most-positive and most-negative curvature, which highlight crests and troughs of amplitude without including the effects of structural dip. And reflector convergence and reflector rotation lose their meaning for amplitude curvature. There is also no vertical derivative, because the vertical axis is time or depth while the input derivatives are amplitude gradients.
Simplifications. This is one horizon rather than a volume. The operator is a smoothing length rather than the fractional-derivative operator AASPI uses, which its documentation describes as a first derivative followed by a low-pass filter; the wavelength behavior demonstrated is the same. Curvedness, shape index and the shape components are computed from e_pos and e_neg in exactly the way module 06 computes them from k₁ and k₂, and are covered there rather than repeated here.
Next up
One derivative further again. Where coherence needs a break and curvature needs a bend, aberrancy responds to the asymmetry of a flexure. It puts a single anomaly on the fault where curvature puts two beside it, and its azimuth says which way the ground dropped.
Open module 08 → All modules