Broadband Dispersion Extraction Using Simultaneous Sparse Penalization

In this paper, we propose a broadband method to extract the dispersion curves for multiple overlapping dispersive modes from borehole acoustic data under limited spatial sampling. The proposed approach exploits a first order Taylor series approximation of the dispersion curve in a band around a given (center) frequency in terms of the phase and group slowness at that frequency. Under this approximation, the acoustic signal in a given band can be represented as a superposition of broadband propagators each of which is parameterized by the slowness pair above. We then formulate a sparsity penalized reconstruction framework as follows. These broadband propagators are viewed as elements from an overcomplete dictionary representation and under the assumption that the number of modes is small compared to the size of the dictionary, it turns out that an appropriately reshaped support image of the coefficient vector synthesizing the signal (using the given dictionary representation) exhibits column sparsity. Our main contribution lies in identifying this feature and proposing a complexity regularized algorithm for support recovery with an l1 type simultaneous sparse penalization. Note that support recovery in this context amounts to recovery of the broadband propagators comprising the signal and hence extracting the dispersion, namely, the group and phase slownesses of the modes. In this direction we present a novel method to select the regularization parameter based on Kolmogorov-Smirnov (KS) tests on the distribution of residuals for varying values of the regularization parameter. We evaluate the performance of the proposed method on synthetic as well as real data and show its performance in dispersion extraction under presence of heavy noise and strong interference from time overlapped modes.