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.

Ground roll, which is characterized by low frequency and high amplitude, is an old seismic data processing problem in land‐based seismic acquisition. Common techniques for ground roll attenuation are frequency filtering, f‐k or velocity filtering and a type of f‐k filtering based on the time‐offset windowed Fourier transform. These techniques assume that the seismic signal is stationary. In this study we utilized the S, x‐f‐k and t‐f‐k transforms as alternative methods to the Fourier transform. The S transform is a type of time‐frequency transform that provides frequency‐dependent resolution while maintaining a direct relationship with the Fourier spectrum. Application of a filter based on the S transform to land seismic shot records attenuates ground roll in a time‐frequency domain. The t‐f‐k and x‐f‐k transforms are approaches to localize the apparent velocity panel of a seismic record in time and offset domains, respectively. These transforms provide a convenient way to define offset or time‐varying reject zones on the separate f‐k panel at different offsets or times.

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S U M M A R YIn this paper, a new approach is introduced to solve ill-posed linear inverse problems in geophysics. Our method combines classical quadratic regularization and data smoothing by imposing constraints on model and data smoothness simultaneously. When imposing a quadratic penalty term in the data space to control smoothness of the data predicted by classical zeroorder regularization, the method leads to a direct regularization in standard form, which is simple to be implemented and ensures that the estimated model is smooth. In addition, by enforcing Tikhonov's predicted data to be sparse in a wavelet domain, the idea leads to an efficient regularization algorithm with two superior properties. First, the algorithm ensures the smoothness of the estimated model while substantially preserving the edges of it, so, it is well suited for recovering piecewise smooth/constant models. Second, parsimony of wavelets on the columns of the forward operator and existence of a fast wavelet transform algorithm provide an efficient sparse representation of the forward operator matrix. The reduced size of the forward operator makes the solution of large-scale problems straightforward, because during the inversion process, only sparse matrices need to be stored, which reduces the memory required. Additionally, all matrix-vector multiplications are carried out in sparse form, reducing CPU time.Applications on both synthetic and real 1-D seismic-velocity estimation experiments illustrate the idea. The performance of the method is compared with that of classical quadratic regularization, total-variation regularization and a two-step, wavelet-based, inversion method.