Coherence is not one calculation

Four pairs of traces. Look at each one and decide, before you read a single number, whether you would call them similar. Case 3 is the one to weigh: the shape is identical and only the size differs. Whether that counts as similar has no single answer, and the answer a reader gives decides which attribute suits their purpose.

Both traces are drawn on the same axis so you can see the answer rather than work it out. Where the two lines lie on top of each other the traces agree; where they come apart they do not.

Real attributes do not work on pairs of traces. They work on a window of several at once, laid along the bed using the dip field from modules 01 to 03. The question becomes: can this whole window be described as one seismic event? Two long-standing ways of answering that are below, and the sliders above let you change the window in different ways to see how each one responds.

• Amplitude changes sideways. The shape of the wavelet stays exactly the same; only how big it is changes from trace to trace. This happens constantly in real data — a bed thins, porosity changes, the reflection gets weaker.
• The shape itself changes. Now the wavelet is rotated a little more on each trace, so the traces are genuinely different shapes rather than different sizes. Watch which attributes respond to this and which responded to the slider above.
• Noise, because real traces never match perfectly, and it is worth knowing how much of the disagreement is only noise.
• Overall amplitude turns the whole window up or down together. Nothing about how well the traces agree has changed, so watch which numbers stay put and which move.
• Traces used and window height set how much data each answer rests on.

Here is the idea that ties nearly all of these attributes together. Except for the Sobel filter, every one of them is built from a single object: a covariance matrix. Entry C(m,n) is trace m multiplied by trace n, sample by sample, summed down the window. A window of J traces gives a J-by-J table of those products, and the diagonal C(m,m) is simply the energy of trace m. Two traces that rise and fall together give a large positive entry; two that are unrelated give an entry near zero. The matrix is built from the analytic trace, meaning the amplitude together with its Hilbert transform, which keeps the estimate steady where the amplitude happens to pass through zero.

What an eigenvector is, for this purpose. Ask a different question: is there a single waveform that, scaled up or down trace by trace, could account for the whole window? An eigenvector is a candidate set of those scale factors — one weight per trace. The first eigenvector is the set of weights that accounts for as much of the window's energy as any set can, and it is what panel 3 draws: read it as "trace 1 needs this much of the common waveform, trace 2 this much", and so on.

And an eigenvalue is how much that explained. Each eigenvector comes with a number saying how much of the total energy it accounts for, and the numbers add up to the total. So if one waveform really does describe the window, λ₁ takes nearly everything and the rest are near zero. If the traces are unrelated, no single waveform helps much and the energy is spread across all of them. Panel 2 shows each eigenvalue as a share of the total, which is why the bars always sum to 1.

AASPI defines outer-product similarity as c = rᵀ C r / E, where C is the covariance matrix from step 3, E is the total energy in the window, and r is a test vector: one weight per trace, the same shape as an eigenvector.

Read it as a question. You are proposing a set of weights and asking how much of the window's energy that proposal accounts for. The quantity rᵀ C r is the energy explained by your proposal; dividing by E turns it into a fraction between 0 and 1. So the answer is always "this share of the window is described by the pattern I suggested".

Several attributes that were published separately, under different names and in different decades, turn out to be this one equation with different proposals for r.

• Propose that every trace contributes equally — r a constant vector — and the answer is a semblance estimate of coherence. You are asking how much of the window is explained by the traces simply being the same as each other.
• Propose the first eigenvector, which step 3 showed is the best possible set of weights, and the answer is eigenstructure coherence. Since no proposal can explain more, this is the largest value the equation can return, and it works out to λ₁ divided by the sum of the eigenvalues.
• Propose the center sample vector and you get the third option AASPI offers. That name needs unpacking. Pick the one sample you are actually computing the attribute at, and read the amplitude of every trace in the window at that instant — five traces gives five numbers. Those numbers are the center sample vector. Using them as r asks a pointed question: how much of the whole window looks like this one instant across the traces? It costs nothing to compute, because you already have those amplitudes, whereas the first eigenvector has to be solved for.
• Try each in turn and watch the weights in panel 1 change shape. On clean data all three land in much the same place; it is when the window is noisy or the amplitudes vary that they separate.

Energy-ratio coherence is the energy of the Karhunen–Loève filtered data over the total energy. The Karhunen–Loève filter rebuilds the window from its first L eigenvectors alone and throws the rest away, so it is a way of keeping only the part of the window that several traces have in common. Because the eigenvectors are orthonormal, keeping L components retains exactly the first L eigenvalues, so with L = 1 it is numerically identical to eigenstructure coherence. Raise L and everything starts to look coherent, because you are keeping more of the window as "signal".

This one is the odd one out, and it is worth knowing why. The Sobel filter comes from image processing, not from seismic statistics. AASPI describes its version as the magnitude of the inline and crossline derivatives of amplitude along structural dip, normalized by the same derivatives taken on absolute values. It never builds a covariance matrix and it has no concept of a common waveform. It simply asks how fast amplitude is changing sideways, which makes it respond strongly to a change in size that the eigenvalue-based attributes ignore entirely.

One synthetic line, flat-lying so that steering is not a factor, divided into five zones of forty traces each. The zones are boxed and named along the top. Every zone has a sharp edge down its middle — vertical in zones 2, 3 and 4, and a realistically dipping fault plane in zone 5 — and every edge is a different kind of problem: zone 1 has no problem at all and is the control, zone 2 changes amplitude across the edge without changing the waveform, zone 3 changes the waveform without changing its size, zone 4 has noise added to its right-hand half, and zone 5 is a fault with 16 ms of throw. Every attribute is computed at every sample with the same window.

A note on the L slider before you reach for it. The window here holds five traces, so five principal components describe it completely. Keeping all five reconstructs the window exactly and the energy ratio is 1.000 everywhere by definition; keeping four is already within half a percent of that across the whole line. The panel goes uniformly white, and that is the attribute reporting that it has stopped distinguishing anything rather than the display failing. The readout under the panel gives the range, so the collapse can be watched as a number rather than guessed at from a blank rectangle. L = 1 and L = 2 are where the attribute does something.

Coherence is the most heavily used discontinuity attribute in structural interpretation. It is what turns a volume into a fault map, and on a time slice it does something no vertical section can: it shows a fault's trace, the line it makes in map view, so you can follow it, see where it branches, and see how it relates to its neighbors.

What it is used for

• Faults. Mapping them, counting them, and feeding them to an automatic fault extractor. Most fault-surface software takes a discontinuity volume as its input, so the quality of your coherence sets the ceiling on the quality of the fault framework.
• Channels, fans and reef margins. A channel edge is a discontinuity even where there is no structural relief at all, which is why a coherence time slice can show a meander belt in a single picture.
• Data quality control. Several processing and acquisition problems show up as coherence anomalies: acquisition footprint (a regular pattern left by the shot and receiver geometry), bad traces, migration smiles (arcs left where migration was run with the wrong velocity) and statics problems (trace-to-trace time shifts from the near surface). They show up in patterns — regular stripes, survey-edge effects, a single dark trace — that geology does not make.
• As an input to other things. Coherent energy from the same calculation drives amplitude curvature in module 07, and coherence is a standard input to fault-likelihood volumes and to structure-oriented filtering.

Which one to run

The practical question is which one to run. The dividing line is amplitude. Energy ratio and eigenstructure are blind to a pure change of amplitude; semblance and Sobel are not. So:

• Mapping faults, especially where reflection strength fades into the fault zone: the eigenvalue-based attributes hold the edge together where semblance blurs it.
• Mapping internal architecture in a thin channel, where the waveform is constant and only the amplitude changes: energy ratio will show you nothing and semblance will show you the edges. Step 6 has energy ratio reading exactly 1.000 at an amplitude edge that semblance calls about 0.8.
• Not sure: run two and compare. Where they agree you have a discontinuity in both waveform and amplitude; where they disagree, the disagreement itself tells you which one it is.

And the discipline that goes with it

Low coherence means one thing only: the traces in this window stopped looking alike. Zone 4 in step 6 shows why: every attribute goes dark there, and there is no fault, no stratigraphic edge and no geology of any kind, only noise. Nothing in the number distinguishes it from the fault two zones along.

So a dark lineament earns its interpretation from context rather than from its value: whether it is planar, whether it persists across many time slices, whether it lines up with structure you can see on the section, and whether it behaves the way the different algorithms predict it should. A number without its algorithm, its window and its dip field is not interpretable.

AASPI notes that the coherence family acquired many of its names to avoid conflicts with the original Amoco patents rather than because the algorithms differ. Check which definition your software means before comparing two volumes or two papers.