Porosity inversion of 1-D two-phase medium with wavelet multiscale method

Abstract

Based on the idea of multiscale approximation, a wavelet multiscale method is proposed by combining the wavelet analysis and multiscale inversion strategy, and applied to the inversion of porosity in the two-phase medium. The inverse problem is decomposed to multiple scales with wavelet transform and hence the original inverse problem is re-formulated to a set of sub-inverse problem corresponding to different scales and is solved successively according to the size of scale from the smallest to the largest. On each scale, regularized Gauss-Newton method is carried out, which is stable and fast, until the optimum solution of original inverse problem is found. The wavelet multiscale method is described as the combination of three operators: the restriction operator, the relaxation operator and the prolongation operator. And then the flow of wavelet multiscale method is outlined and the restriction operator matrix and the prolongation operator matrix obtained by adapting the compactly supported orthonormal wavelet Daubechies wavelets are deduced. The inversion results obtained by wavelet multiscale method are compared with those with traditional regularized Gauss-Newton method, the results of numerical simulation demonstrated that the method is an effective and widely convergent optimization method.