Sparsity penalized reconstruction framework for broadband dispersion extraction

We propose a novel broadband method to extract the dispersion curves for multiple overlapping dispersive modes from borehole acoustic data. 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. These broadband propagators can be 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. We evaluate the performance of the proposed method on synthetic data with known dispersions and show its accuracy in extraction and robustness to the presence of heavy noise and strong interference from time overlapped modes.