A Fast and Automatic Sparse Deconvolution in the Presence of Outliers

We present an efficient deconvolution method to retrieve sparse reflectivity series from seismic data in the presence of additive Gaussian and non-Gaussian noise. The problem is first formulated as an unconstrained optimization including a mixed
lp
-
l
₁
measure for the data misfit and for the model regularization term, respectively. An efficient algorithm based on the alternating split Bregman technique is developed, and a numerical procedure based on the generalized cross-validation (GCV) technique is presented for the selection of the corresponding regularization parameter. To circumvent excessive computations of multiple optimizations to determine the minimizer of GCV curve, we formulate the deconvolution problem in the frequency domain as a basis pursuit denoising and solve it using the split Bregman algorithm with computational complexity
O
(
N
log(
N
)). Apart from significant stability against outliers in the data, the main advantage of such formulation is that the GCV curve can be generated during the iterations of the optimization procedure. The minimizer of the GCV curve is then used to properly determine the error bound in the data and hence the optimum number of iterations. Numerical experiments show that the proposed method automatically generates high-resolution solutions by only a few iterations needless of any prior knowledge about the noise in the data.