Compressive Seismic Recon - Is it Interpolation?

Most in the seismic industry are familiar with 5D interpolation. What makes CS-Recon different from interpolation? It's all about the math.

Interpolation

Traditional interpolation assumes that the sampled data is smooth and spatially band-limited and that consecutive samples are not significantly different. In the case of 5D interpolation, this assumption is made in each of the five dimensions. For spatially band-limited data, the results are reliable, and the signal can be recovered accurately.

To satisfy the band-limited assumption with seismic data, first bin the data to a regular grid in each interpolation dimension (typically time, inline, crossline, offset, and azimuth) and then approximate every trace to the bin center before interpolation. Because of this bin-centering mechanism, the original traces are not fully preserved after interpolation.

There exist many interpolation approaches relying on a variety of methods and assumptions, but none use the unique survey design and incoherent sampling assumption that characterizes Compressive Seismic Reconstruction.

Compressive Seismic (CS) Reconstruction

The underlying assumption in Compressive Seismic is that the sampled data is sparse in some transform domain (e.g. Tau-p) due to possessing a high amount of spatial incoherency. This spatial incoherency ensures that aliased energies are widely scattered in the transform domain, enabling a sparsity-promoting filter to remove these energies while preserving the true signal. Once these conditions are met, the sampled data can be decomposed into linear equations solvable by L1 inversion to obtain reliable and accurate results.

There are other reconstruction methods that utilize sparsity-promoting solvers, but CS-Reconstruction uniquely uses a special incoherency-promoting acquisition design in order to maximize the sparsity of the acquired data. This approach is also highly applicable to datasets that are spatially incoherent by coincidence, such as surveys with many scattered permit holes. All seismic datasets satisfy the underlying assumptions of Compressive Seismic to some degree, as there is always some level of spatial incoherency due to random positioning errors or deviations. However, a well designed CS-Acquisition (CS-A) can ensure that this assumption is fully validated.

With CS-Recon, there is no need for seismic data binned or approximated to the bin center before applying reconstruction. Our CS-Acquisition designs are optimized for a particular reconstruction transform domain with minimal spatial coherency, ensuring ideal reconstruction. An existing or conventional survey will still be reconstructed impressively (shown below), albeit not as well as a survey designed for CS.

5DR and CS-Recon were applied to conventionally acquired Permian Field data: time (0.63s) and IL view. (Tao Jiang, “Compressive seismic reconstruction with extended POCS for arbitrary irregular acquisition,” SEG Technical Program Expanded Abstracts 2017.)

Importantly, with our CS-Recon techniques, all original traces are retained as-is, with no modifications due to bin-centering mechanisms. With interpolation, no matter the bin-centering mechanism, what you end up with is not an original trace.

Conclusion

Compressive Seismic Reconstruction (CS-R) mechanics are very different from traditional interpolation. A significant advantage of CS-R over many interpolation methods for seismic processing is that all original traces are preserved. Additionally, CS-R uses the power of sampling incoherency and sparsity to remove aliased energies from the data, improving the overall reconstruction result.