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The sparseness in seismic data has been successfully used to achieve many improvements in different areas such as acquisition, regularization, filtering, and imaging. The search for a sparse description for a given practical seismic (linear) problem is often conducted via the so called Iteratively Reweighted Least-Squares Inversion with an adaptive choice of a diagonal matrix, computed with a statistically derived prescription. In this paper, we discuss the role of this matrix in the inversion and propose an alternative heuristic formula which allows one to more easily constrain the solution to physical expectations and is likely to improve sparseness and accelerate convergence. Introduction and Review Linear inversion problems are very common in geophysical applications. Usually, a set of representative vectors are combined in a weighted way so as to fit a set of measurements. Mathematically, it is to say A~x = ~b, where A is a matrix with model vectors in its columns, ~x is a vector of weights to be determined, and ~b is the data vector. The choice of the model vectors is generally guided by physical demands and is not committed to fill simple mathematical properties like completeness or uniqueness of the solution ~x. For instance, in an irregular discrete Fourier transform (IDFT), columns of A exhibits the values of sinusoidal functions taken at unevenly chosen positions. Thus, the matrix A for IDFT departs a lot from the usual discrete Fourier transform. A may not be invertible and the solution ~x may not be unique. Least square (L2) error (‖A~x−~b‖2) is an option to define ~x for overdetermined problems. Developing a linear problem to L2 minimum error norm leads to equations like AHA~x = AH~b (1). Underdetermined and/or ill-posed problems are often handled with a dumped version of the L2 error norm (here called the Dumped Least Square equation or DLSE) as (AHA + λ I)~x = AH~b where λ is ideally chosen real, positive, and small enough to allow for a solution that keeps the error as small as possible, and I is the identity. At this point, it might be noticed that extending the original linear problem as, [ A √ λ I ] |
