A two‐step wavelet‐based regularization for linear inversion of geophysical data

Regularization methods are used to recover a unique and stable solution in ill‐posed geophysical inverse problems. Due to the connection of homogeneous operators that arise in many geophysical inverse problems to the Fourier basis, for these operators classical regularization methods possess some limitations that one may try to circumvent by wavelet techniques. In this paper, we introduce a two‐step wavelet‐based regularization method that combines classical regularization methods with wavelet transform to solve ill‐posed linear inverse problems in geophysics. The power of the two‐step wavelet‐based regularization for linear inversion is twofold. First, regularization parameter choice is straightforward; it is obtained from a priori estimate of data variance. Second, in two‐step wavelet‐based regularization the basis can simultaneously diagonalize both the operator and the prior information about the model to be recovered. The latter is performed by wavelet‐vaguelette decomposition using orthogonal symmetric fractional B‐spline wavelets. In the two‐step wavelet‐based regularization method, at the first step where fully classical tools are used, data is inverted for the Moore‐Penrose solution of the problem, which is subsequently used as a preliminary input model for the second step. Also in this step, a model‐independent estimate of data variance is made using nonparametric estimation and L‐curve analysis. At the second step, wavelet‐based regularization is used to partially recover the smoothness properties of the exact model from the oscillatory preliminary model. We illustrated the efficiency of the method by applying on a synthetic vertical seismic profiling data. The results indicate that a simple non‐linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods.