Advanced 4D: joint processing

Part 8 — Time-Lapse (4D) Processing

Learning objectives

State the principle of joint 4D processing: shared parameters across baseline and monitor
Identify which processing stages benefit most from joint parameter estimation
Quantify the NRMS improvement joint processing typically delivers vs independent flows
Recognise when joint processing is and is not applicable

§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 partially recovered by matching at the end. Joint 4D processing pushes the matching inside the processing chain: baseline and monitor share parameters at every stage, so non-repeatability is absorbed into the estimation rather than left as a residual.

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 geology 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 biggest single source of non-repeatability. Any tomography or FWI updates use both surveys' data jointly.
Wavelet estimation for AVO. The same source wavelet applies to 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.

Per-stage NRMS contributions for three processing stages (pre-processing, migration, AVO/inversion). The slider controls divergence between the independent flows’ parameter estimates — how much the baseline and monitor processors disagree. Joint values are invariant: they are the unavoidable minimum NRMS from each stage even with perfectly matched parameters. Independent values grow linearly with divergence. The total NRMS at the bottom (root-sum-square of the three stages, approximately how NRMS compounds across independent processing steps) shows the gap clearly: at typical production divergence (0.3–0.5 on the slider), joint flows deliver 40–60 % lower total NRMS than independent flows.

4. What joint processing is NOT

Joint processing is NOT "one survey's parameters applied to both" — that would be biased 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 sources; running a shared denoiser would attenuate survey-specific noise in the wrong survey.
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 joint velocity/wavelet 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 constraint that the 4D difference is sparse (non-zero only at actual reservoir changes). Formally:

\min \sum \| d_B - W R(m_B) \|^2 + \| d_M - W R(m_M) \|^2 + \lambda\, \| m_M - m_B \|_{TV}

The first two terms fit each survey to its own data with a shared wavelet $W$ and forward operator $R$ . The third term is a total-variation (or L1) regularisation on the difference $m_M - m_B$ , which pushes it toward sparsity — zero except where the data demand a change. Joint 4D inversion routinely outperforms the "invert each, subtract" workflow by factor 2–5 on NRMS of the final difference volume.

7. A realistic production 4D flow

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

**The one sentence to remember**

Joint 4D processing estimates shared parameters from both surveys simultaneously, so baseline and monitor can only differ where the earth actually changes — 40–60 % lower NRMS than independent processing for the same acquisition quality, and the only route to state-of-the-art 4D imaging.

Part 8 closes here

The time-lapse toolkit is complete: repeatability concepts + NRMS (§8.1), matching filters post-acquisition (§8.2), the two-metric NRMS + predictability QC (§8.3), and the joint-processing workflow that keeps non-repeatability out of the flow from the start (§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.
Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
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.