Advanced 4D: joint processing

Sections 8.2-8.3 covered post-acquisition matching: process baseline and monitor independently, then apply matching filters and QC the difference. That workflow has a ceiling: non-repeatability accumulated during processing can only be partly recovered by matching at the end. Joint 4D processing pushes the matching inside the processing chain: baseline and monitor share their estimated parameters at every stage where that is right, so the processing stops adding differences of its own.

1. The principle

Every processing stage estimates parameters from the data: demultiple filters, migration velocity models, wavelets for AVO. In an independent flow, baseline estimates its own parameters, monitor estimates its own. Any noise in the estimation is independent between surveys, so the two parameter sets disagree, and the disagreement shows up in the difference as artefact. In a joint flow, both surveys contribute to the estimation of a single parameter set that is applied to both. By construction, there is no estimation disagreement between surveys.

Joint processing does not require the earth to be unchanged: the data values themselves differ where the reservoir changes (the 4D signal). But it requires that any assumed invariants (velocities, wavelets, filter coefficients) are literally the same numbers for both surveys. The 4D signal is then isolated in the data differences, not the parameter differences.

2. Stages that benefit from joint processing

• Demultiple. SRME + Radon parameters estimated jointly reduce the small differences in multiple prediction between surveys that would otherwise leak into the 4D difference.
• Migration velocity. A single velocity model for both surveys removes the largest processing-induced source of non-repeatability; acquisition geometry and ambient noise remain. Any tomography or FWI updates use both surveys' data jointly.
• Wavelet estimation for AVO. Once both surveys are designatured to a common output wavelet, one wavelet serves both; a joint estimate uses well ties and non-reservoir zones from both.
• Simultaneous inversion. Baseline and monitor inversions share the low-frequency model, the wavelet, and the regularisation. Differences in the output elastic attributes are then structurally consistent.
• Near-offset conditioning. Reconstruct both surveys' near-offset gaps using the same model assumptions.
• Q compensation. Apply the same Q field to both surveys; disagreements between measured Q per survey are small compared to their absolute values.

3. The figure: independent against joint

In the figure below each survey is processed with its own estimation errors: a demultiple residual, a velocity error and a wavelet phase error, with noise attenuation of the monitor's swell. Choose which stages estimate one parameter set from both surveys and compare the difference of your flow, plate (d), with the fully independent one in (c). Then raise the acquisition non-repeatability and see what sharing cannot remove.

At the default setting, with demultiple, velocity and wavelet shared, NRMS outside the reservoir falls from 49 % in the independent flow to 14 % in yours, and all of what remains is acquisition non-repeatability: once the estimates are shared, the difference holds only the reservoir change and the acquisition noise. The budget in (e) shows where the independent 49 % comes from. Velocity is the largest single term. The sections are depth-migrated images shown in two-way time, where a migration-velocity error of only 0.1 % moves a reflector at 1.5 s by about 1.5 ms, and NRMS is very sensitive to such shifts: 1 ms on this 30 Hz Ricker already gives about 21 % (the single-frequency estimate $200\\,|\\sin(\\pi f\_c \\Delta t)|$ gives 19 %). How much sharing gains depends on the size of the processing errors against the acquisition floor. Raise $\\Delta S + \\Delta R$ to 80 m and the two values close to 57 % and 34 %; the gain is not a fixed percentage.

4. What joint processing is NOT

Joint processing is NOT "one survey's parameters applied to both"; that would bias the result toward the reference survey. It is "parameters estimated from the joint data": both surveys contribute to the estimation, weighted by their data quality. The mathematics is formally a single optimization over one parameter set, with two residual functions (one per survey) summed in the objective.

Joint processing also does NOT require exact geometry matching between surveys. It requires that whatever parameters are shared make sense for both geometries. For example, a joint velocity model can be estimated from two surveys with different shot layouts because the velocity describes the subsurface, not the acquisition.

5. Where independent still wins

• Noise attenuation. Each survey has its own noise. A denoiser whose noise model is estimated from both surveys fits neither: it under-attenuates the noise in the survey that has it and removes signal from the survey that does not. Share noise attenuation in the figure and the monitor's swell reaches the difference.
• Survey-specific artefacts. If one survey has a weather-related coherent artefact at a certain depth, it needs to be attenuated in that survey only. Joint processing would either mis-attenuate or mis-propagate.
• Different acquisition systems. Comparing a 1998 streamer baseline with a 2020 OBN monitor uses a joint velocity model, and a joint wavelet estimated after each survey's own designature, but not joint pre-processing: the acquisition-specific noise and deconvolution differ too much.

6. Joint 4D inversion

The culmination of Part 8's ideas is joint 4D inversion: solve simultaneously for the baseline and monitor elastic-attribute volumes with a regularisation that keeps the 4D difference blocky and zero except at actual reservoir changes. Formally:

The first two terms fit each survey to its own data with a shared wavelet $w$ and reflectivity operator $R$. The third term penalises the total variation of the difference $m\_M - m\_B$, which favours a blocky difference that is zero except where the data demand a change; an $\\ell\_1$ penalty $\\lambda\\|m\_M - m\_B\\|\_1$ would instead favour a sparse one. Joint 4D inversion usually gives cleaner, more structurally consistent differences than the "invert each, subtract" workflow, with fewer side lobes and less leakage above and below the reservoir.

7. A realistic production 4D flow

1. Joint binning (Section 8.2): pair traces by source-receiver proximity.
2. Survey-specific noise attenuation: per-survey noise decisions.
3. Joint pre-processing (spherical-divergence correction, surface-consistent amplitude scaling) with shared parameters.
4. Joint migration velocity building: one $v(x, z)$ for both surveys.
5. Joint true-amp migration: same operator applied to both datasets.
6. Post-migration cross-equalisation + matching filters (Section 8.2): final per-trace matching.
7. QC pack: NRMS, predictability, spatial maps (Section 8.3).
8. Joint 4D inversion: simultaneous elastic-attribute volumes with TV regularisation on the difference.
9. 4D interpretation: fluid fronts, compaction, pressure changes.

Part 8 closes here

The time-lapse toolkit is complete: repeatability concepts + NRMS (Section 8.1), matching filters post-acquisition (Section 8.2), the two-metric NRMS + predictability QC (Section 8.3), and the joint-processing workflow that keeps non-repeatability out of the flow from the start (Section 8.4). Part 9 turns to Machine Learning, the modern statistical tools now woven through seismic processing, from noise attenuation to FWI initialisation to fault interpretation.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
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• Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.
• Kragh, E., Christie, P. (2002). Seismic repeatability, normalized rms, and predictability. The Leading Edge, 21, 640-647.
• Lumley, D. E. (2001). Time-lapse seismic reservoir monitoring. Geophysics, 66, 50-53.
• Johnston, D. H. (2013). Practical Applications of Time-lapse Seismic. SEG Distinguished Instructor Short Course.