Full waveform inversion (FWI) is an iterative nonlinear waveform matching procedure subject to wave-equation constraint. FWI is highly nonlinear when the wave-equation constraint is enforced at each iteration. To mitigate nonlinearity, wavefield-reconstruction inversion (WRI) expands the search space by relaxing the wave-equation constraint with a penalty method. The pitfall of this approach resides in the tuning of the penalty parameter because increasing values should be used to foster data fitting during early iterations while progressively enforcing the wave-equation constraint during late iterations. However, large values of penalty parameter lead to ill-conditioned problems. Here, this tuning issue is solved by replacing the penalty method by an augmented Lagrangian method equipped with operator splitting (IR-WRI as iteratively-refined WRI). It is shown that IR-WRI is similar to a penalty method in which data and sources are updated at each iteration by the running sum of the data and source residuals of previous iterations. Moreover, the alternating direction strategy exploits the bilinearity of the wave equation constraint to linearize the subsurface model estimation around the reconstructed wavefield. Accordingly, the original nonlinear FWI is decomposed into a sequence of two linear subproblems, the optimization variable of one subproblem being passed as a passive variable for the next subproblem. The convergence of WRI and IR-WRI are first compared with a simple transmission experiment, which lies in the linear regime of FWI. Under the same conditions, IR-WRI converges to a more accurate minimizer with a smaller number of iterations than WRI. More realistic case studies performed with the Marmousi II and the BP salt models show the resilience of IR-WRI to cycle skipping and noise, as well as its ability to reconstruct with high fidelity large-contrast salt bodies and sub-salt structures starting the inversion from crude initial models and a 3-Hz starting frequency.

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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 p − 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.Index Terms-Alternating split Bregman, generalized crossvalidation (GCV), non-Gaussian noise, sparse deconvolution.

S U M M A R YIn 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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