4D binning & matching filters

Section 8.1 showed why dedicated 4D acquisition is the single biggest lever on repeatability. The next biggest lever is post-acquisition matching: identify the non-repeatability remaining in the data, estimate corrections from parts of the trace where baseline and monitor should agree, apply those corrections to bring the two surveys into alignment. This section covers two pillars of that matching: 4D binning (pairing traces by geometry) and matching filters (bringing paired traces into spectral/amplitude/phase agreement).

1. 4D binning: pairing traces by geometry

A streamer baseline and a streamer monitor will never have exactly repeated shot/receiver positions. 4D binning pairs each monitor trace with the nearest baseline trace in source-receiver position space. The geometric mismatch of a pair is measured as $\\Delta S + \\Delta R$, the source-position difference plus the receiver-position difference; dedicated 4D repeats keep most pairs within a few tens of metres, and residual NRMS grows as $\\Delta S + \\Delta R$ grows. Plate (g) of the figure in part 3 shows why the search matters: with the monitor cables feathered at 5° against the baseline's 3°, pairing each trace with the same channel drifts past 100 m at 3000 m offset, while the nearest baseline trace keeps the median $\\Delta S + \\Delta R$ at 33 m, with 82 % of pairs within 50 m.

• Near-source, near-receiver matching first; if not available, match far source or far receiver second.
• Source-receiver reciprocity adds candidate pairs (a source at A with a receiver at B matched with a source at B and a receiver at A) where direct pairs are poor, though source and receiver ghost and directivity differences limit how well a reciprocal pair repeats.
• Cross-streamer matching is weaker than in-streamer matching because cross-streamer varies with feathering; track cable positions via GPS + compass data.
• Ocean-bottom nodes shrink the receiver half of the problem: ROV redeployment puts each node within a few metres of its earlier position, and permanently installed systems (PRM) make the receiver half essentially identical between surveys.

2. Matching filters: the principle

Given a binned baseline trace $B(t)$ and monitor trace $M(t)$, a matching filter is a linear operator $F$ applied to the monitor such that

in the non-reservoir parts of the trace. The filter estimates and compensates for the non-repeatability between surveys. Three common forms of $F$:

• Scalar amplitude. $F = \\alpha$, a single gain factor. Captures source-strength differences but nothing else.
• Time shift. $F$ = delay by $\\tau$. Captures bulk delays from tide and water-velocity (temperature and salinity) changes and from residual geometry.
• Convolutional matching. $F$ = convolution by a short filter $f(t)$ designed by Wiener to minimise $|B - f \* M|^2$. Captures phase, spectral, and amplitude differences in one filter. The general case.

The critical design principle: estimate F using non-reservoir samples only. If you include the reservoir zone, F will partly match out the very 4D signal you are trying to preserve. Production flows define a "reservoir exclusion window" (typically ±100-200 ms around the target horizon) and compute $F$ on the remainder.

3. The figure: matching in stages

The figure builds a baseline and a monitor line over a layered earth with a bright reservoir top at 1.50 s, gives the monitor a 4D change of the reservoir reflection coefficient and a non-repeatability you set (a gain, a time shift, a phase rotation, a high-cut and fresh noise), and matches it back in stages. Step through the stages and watch the difference in (b): outside the dashed reservoir window it should fade to the noise, inside it the 4D change should stay.

The stages add to one another, and every operator is estimated only from the design samples outside the window:

• Raw: the difference as recorded.
• Gain: estimate $\\alpha = \\text{RMS}(M\_{\\text{non-res}}) / \\text{RMS}(B\_{\\text{non-res}})$ and divide the monitor by it.
• Time: cross-correlate the gain-matched monitor with the baseline over non-reservoir samples, find the lag of the correlation peak and refine it below one sample (here by interpolating the correlation and fitting a parabola to its peak), shift the monitor back by that amount, measure the gain again, and recompute NRMS.
• Wiener: design a short least-squares filter that maps the matched monitor onto the baseline over the same samples, and apply it to the whole trace.

At the defaults ($\\Delta t = 2.6$ ms, gain $\\times 1.08$, $\\Delta R = 0.03$, 5 % noise) the non-reservoir NRMS starts at 45 %, almost all of it from the time shift: for a dominant frequency $f$ a pure shift gives $\\text{NRMS} \\approx 200 \\sin(\\pi f \\Delta t)\\,\\%$, 41 % at 25 Hz. The gain match leaves it at 45 %. The time match finds the shift within 0.05 ms and takes the non-reservoir NRMS to 4.8 %, the noise floor of 5.0 %, while the difference inside the reservoir window keeps 100 % of its true 4D amplitude, because the estimator never saw those samples. Turn on Include the reservoir in the design and the same matching keeps only 89 %: the gain estimate rises to $\\times 1.10$, because it reads the brighter monitor reservoir as non-repeatability.

4. Why the order of corrections matters

A single gain multiplies the cross-correlation $\\sum B(i)\\,M(i+\\text{lag})$ by the same factor at every lag, so it cannot move the peak, and normalised cross-correlation ignores it entirely. Order matters when corrections interact: an unremoved time shift or bandwidth difference biases an RMS gain measured in a finite window, and a phase or spectral difference biases the lag picked from the correlation peak. In the figure the gain measured with the 2.6 ms shift still in reads $\\times 1.077$ against the true $\\times 1.08$, and a 20° phase rotation pulls the picked lag from the true 2.6 ms to 0.68 ms. Production flows therefore iterate the corrections, or solve for gain, shift and phase together in one Wiener matching filter.

5. Full convolutional matching (Wiener)

For surveys with spectral differences (different sources or Q differences), a scalar amplitude is insufficient. The Wiener filter $f$ minimises

giving a short FIR filter (typically 50-200 ms long) that captures the full frequency-domain mismatch between surveys. Production 4D processing routinely uses Wiener matching filters, often after an initial amplitude + time match to reduce the problem's dynamic range. In the figure a 20° phase rotation leaves 12 % NRMS after gain and time, and the Wiener filter takes it to 4.8 %. A high-cut is different: no filter restores a band the monitor never recorded, so a 50 Hz high-cut still leaves 6.0 % after the Wiener stage.

6. Window choice

• Too narrow a matching window: $F$ is under-determined and overfits noise.
• Too wide: $F$ absorbs 4D signal if the window wraps into the reservoir.
• Multiple disjoint windows (e.g., shallow + deep non-reservoir zones) usually give the best balance.
• Taper the window edges to avoid ringing in the filter design.

7. QC after matching

1. Residual NRMS by time: plot NRMS against TWT after matching, as plate (c) does. Outside the reservoir window it should be low and flat; it should rise only inside the reservoir window, where the 4D signal lives.
2. Residual NRMS by offset: should be flat across the offset range; a rising trend indicates un-matched angle-dependent response.
3. Spatial NRMS maps: plot NRMS per CDP location as a map. Localised high-NRMS zones often correspond to surface features (rivers, roads, weather fronts during acquisition).
4. Reservoir isolation check: mask the reservoir zone from the difference and measure RMS. It should be below the expected 4D signal amplitude.

Where this goes next

Section 8.3 goes deep on the two main 4D QC metrics: NRMS and predictability. Where Section 8.1 introduced NRMS as a single number, Section 8.3 unpacks it as a spatially- and temporally-varying diagnostic, introduces the predictability metric (a cross-survey correlation), and catalogues the signatures of specific 4D failure modes.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.
• Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.