Sparse Blind Deconvolution of Ground Penetrating Radar Data

Abstract

We propose an effective method for sparse blind deconvolution (SBD) of ground penetrating radar data. The SBD algorithm has no constraints on the phase of the wavelet, but the initial wavelet must be carefully captured from the data. The data are considered a convolution product of an unknown source wavelet and unknown sparse reflectivity series. The algorithm developed here is an alternating minimization technique that updates the reflectivity series and the wavelet iteratively. The reflectivity update is solved as an <inline-formula> <tex-math notation= LaTeX >$\\ell _{2}-\\ell _{1}$ </tex-math></inline-formula> problem with the alternating split Bregman iteration technique. The wavelet update is solved as an <inline-formula> <tex-math notation= LaTeX >$\\ell _{2}-\\ell _{2}$ </tex-math></inline-formula> problem with Wiener deconvolution. The algorithm converges to a local minimum. In order to increase the likelihood so that convergence coincides with the desired local minimum, special steps are taken to provide a proper initial wavelet. Synthetic and real data examples show that both subsurface reflectivity series and wavelet (amplitude and phase) can be estimated efficiently. The SBD method presented appears robust and compares favorably to previous studies in its resistance to noise.