Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints

Details Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints

Feb 2010

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010geoji.180..871g&link_type=abstract

Geophysical Journal International, Volume 180, Issue 4, pp. 871-882.

Astronomy and Astrophysics

Astronomy

2

Wavelet Transform, Inverse Theory, Tomography

Scientific paper

In this paper, we deal with the solution of linear and non-linear geophysical ill-posed problems by requiring the solution to have sparse representations in two appropriate transformation domains, simultaneously. Geological structures are often smooth in properties away from sharp discontinuities (i.e. jumps in 1-D and edges in 2-D). Thus, an appropriate `regularizer' function should be constructed so that recovers the smooth parts as well as the sharp discontinuities. Sparsity inversion techniques which require the solution to have a sparse representation with respect to a pre-selected basis or frames (e.g. wavelets), can recover the regions of smooth behaviour in model parameters well, but the solution suffers from the pseudo-Gibbs phenomenon, and is smoothed around discontinuities. On the other hand, requiring sparsity in Haar or finite-difference (FD) domain can lead to a solution without generating smoothed edges and the pseudo-Gibbs phenomenon. Here, we set up a regularizer function which can be benefited from the advantages of both wavelets and Haar/FD operators in representation of the solution. The idea allows capturing local structures with different smoothness in the model parameters and recovering smooth/constant pieces of the solution together with discontinuities. We also set up an information function without requiring the true model for selecting optimum wavelet and parameter β which controls the weight of the two sparsifying operators in the inverse algorithm.
For both linear and non-linear geophysical inverse problems, the performance of the method is illustrated with 1-D and 2-D synthetic examples and a field example from seismic traveltime tomography. In all of the examples tested, the proposed algorithm successfully estimated more credible and high-resolution models of the subsurface compared to those of the smooth and traditional sparse reconstructions.

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