Linear inversion via the discrete wavelet transform pseudoinverse

Classical least-squares techniques (Moore-Penrose pseudoinverse) are covariance based and are therefore unsuitable for the solution of very large-scale linear systems in geophysical inversion due to the need of diagonalisation. In this paper, we present a methodology to perform the geophysical inversion of large-scale linear systems via the discrete wavelet transform. The methodology consists of compressing the linear system matrix using the interesting properties of covariance-free orthogonal transformations, to design an approximation of the Moore-Penrose pseudoinverse. We show the application of the discrete wavelet transform pseudoinverse to well-conditioned and ill-conditioned linear systems. We applied the methodology to a general-purpose linear problem where the system matrix has been generated using geostatistical simulation techniques and also to a synthetic 2D gravimetric problem with two different geological set-ups, in the noise-free and noisy cases. In both cases, the discrete wavelet transform pseudoinverse can be applied to the original linear system and also to the linear systems of normal equations and minimum norm. The results are compared with those obtained via the Moore-Penrose and the discrete cosine transform pseudoinverses. The discrete wavelet transform and the discrete cosine transform pseudoinverses provide similar results and outperform the Moore-Penrose pseudoinverse, mainly in the presence of noise. In the case of well-conditioned linear systems, this methodology is more efficient when applied to the least-squares system and minimum norm system due to their higher condition number that allows for a more efficient compression of the system matrix. Also, in the case of ill-conditioned systems with very high underdetermined character, the application of the discrete cosine transform to the minimum norm solution provides very good results. Both solutions might differ on their regularity, depending on the wavelet family that is adopted. These methods have a general character and can be applied to solve any linear inverse problem arising in technology, particularly in geophysics, and also to non-linear inversion by linearisation of the forward operator.