Target-oriented joint inversion of incomplete time-lapse seismic data sets

Discussion

From the numerical examples, we see that incomplete time-lapse seismic data-sets degrade time-lapse images [Figures 5(a) and 7(a)]. This image degradation is expected because the migration does not compensate for the resulting geometry (and hence illumination) differences in the migrated images (Figures 4(a) and 6(a)).

Normalization with the Hessian diagonal is insufficient to adequately attenuate undershoot artifacts [Figure 5(b)] and is markedly insufficient in the sparse data example [Figure 7(b)]. Although it is possible that specialized regularization methods can attenuate some of these artifacts, we suspect that most conventional cross-equalization methods will be inadequate.

Although separate inversion improves the quality of the time-lapse images relative to migration and normalization, several relatively high-amplitude artifacts persist [Figures 5(c) and 7(c)]. The high-amplitude artifacts in Figures 5(c) and 7(c) result from a mismatch of residual artifacts from the independent inversion of the data sets. Recall that the target-oriented approximation captures limited information contained in a poorly-conditioned full Hessian matrix. Although the spatial regularization improves the conditioning of the problem, residual artifacts in final results from each inversion differ and do not tend to cancel out.

In the undershoot example, joint inversion of all the data sets (using the RJMI method), improves the time-lapse image quality substantially [Figure 7(d)]. The improvement in the time-lapse images obtained via RJMI vs. separate inversion result from an inclusion of temporal constraints in the RJMI inversion. There is also significant reduction in artifacts in the RJMI sparse data results [Figure 7(d)] relative to separate inversion [Figure 7(c)]. However, in this sparse data example, there are still several residual artifacts in joint inversion results. These artifacts can be further attenuated using stronger regularization (at the cost of the data-fitting) or choosing a more robust minimization (e.g., L1-minimization by iterative re-weighting).

Target-oriented joint inversion of incomplete time-lapse seismic data sets