Coherence and discontinuity attributes: seeing the unseen faults

Semblance

The most widely used coherence measure is semblance, applied to 3D fault detection by Marfurt and his colleagues (1998). At each sample $(i, x, t)$, take a small neighbourhood of traces, usually 3 inlines by 3 crosslines, $K = 9$ traces, and a vertical window of $W$ samples centred on $t$:

Here $k$ counts the traces of the neighbourhood, $t'$ runs over the window and $a_k(t')$ is the amplitude of trace $k$ at $t'$. The numerator is the energy of the stacked trace, the $K$ traces summed sample by sample; the denominator is $K$ times their summed energy. Dividing both by $K^2$ shows what the ratio means: semblance is the energy of the mean trace over the mean energy of a single trace. The two limits:

• Identical traces. Every sum is $K a(t')$, the numerator is $K^2 \sum a^2$, the denominator $K \cdot K \sum a^2$: $s = 1$.
• Traces that cancel. The stack is zero and the denominator is not: $s = 0$.

So semblance lies between 0 and 1 with no further scaling. Where the window is silent, above the first reflection or in a mute, both sums vanish and the ratio means nothing; implementations return a fixed value there, often 1, so silent zones look coherent. That is a convention, not a finding.

Why coherence ignores gain, and when it does not

Multiply every trace by 10. RMS and the envelope grow tenfold; semblance does not change, because the numerator and the denominator both grow a hundredfold. Coherence values can therefore be compared from place to place, and between processings of the same survey, in a way raw amplitudes cannot (Section 6.1). The cancellation needs the same gain on all the traces of the neighbourhood over the window: an automatic gain control that differs from trace to trace does change semblance.

The family: C1, C2, C3

• C1, cross-correlation (Bahorich and Farmer 1995): combines the largest cross-correlations of a trace with its inline and its crossline neighbour. The original coherence cube; fast, but noisy where the signal is weak.
• C2, semblance (Marfurt et al. 1998): the formula above, over any number of traces, steadier than C1 at similar cost and a common default.
• C3, eigenstructure (Gersztenkorn and Marfurt 1999): the largest eigenvalue of the covariance matrix of the neighbourhood over the sum of all its eigenvalues. Sharper and more robust to noise, at a higher cost.

Compared at the same times, over a flat window, a dipping reflector arrives later on one side of the neighbourhood than on the other, so steep but unbroken layers would lower the coherence too. The published algorithms allow for dip: C1 takes each cross-correlation at its best lag, and Marfurt and his colleagues compute C2 over a range of trial dips and azimuths and keep the most coherent; production versions search over dips or steer the window along the local dip of Section 6.3. Figure 6.4 uses C2 with a flat window and no search, and says so.

Why time slices show fault networks

On a section a fault is one line where the reflections are offset, and its neighbours lie on other sections. A time slice cuts across many faults at once: on coherence each is a narrow line of low values across a background of high ones, and the whole pattern shows how the faults connect, branch and end, and how they are oriented. A coherence time slice is the standard fault map of 3D interpretation.

Window size

• Short windows (5 to 10 samples) place a fault in time and show where it dies out, but chance differences between a few samples darken the map everywhere.
• Long windows (20 to 40 samples) steady the values and bring out faint discontinuities, but smear each fault over the whole window and blur its link to one stratigraphic level.
• The neighbourhood. Most implementations use 3 by 3 traces. Larger neighbourhoods are steadier in noisy data but merge faults closer together than their width.

A reasonable start is a window of about one or two cycles of the dominant frequency, 10 to 20 samples for much marine data, then shorter if the faults look smeared and longer if the map is too speckled.

Common coherence pitfalls

• Silent zones look coherent. Where the amplitude is near zero, the ratio is undefined and set by convention (1 in Figure 6.4). A bright, featureless zone may only mean that there was nothing to compare.
• Noise lowers coherence everywhere. Random noise differs from trace to trace, so a noisy survey is broadly low and the faults lose contrast. Noise attenuation and structure-oriented filtering are usual before coherence.
• Edges. At the edge of the survey the neighbourhood holds 6 traces, at a corner 4, so the values rest on less data.
• Not every low-coherence line is a fault. Channel margins, reef edges, salt walls, karst collapse and pinch-outs all lower coherence. Tie a lineament to the sections, the wells and the geological setting before naming it.
• Dip. With a flat window, steep but continuous layers lower coherence too, as above.

Where coherence works and where it struggles

• Works well on layered sequences cut by brittle faults, the setting it was developed for.
• Struggles with ductile deformation, where layers fold without breaking and coherence stays high; with chaotic zones such as salt, which are low everywhere; and with faults whose throw is small against the wavelength.
• Pairs well with dip and curvature (Section 6.3), which describe the bending that coherence cannot see, and with fault picking (Section 2.5), which turns its hints into an interpretation.