Linear geophysical inversion via the discrete cosine pseudo‐inverse: application to potential fields

In this paper we present a methodology to perform geophysical inversion of large scale linear systems via a covariance-free orthogonal transformation: the Discrete Cosine Transform (DCT). The methodology consists in compressing the matrix of the linear system as a digital image and using the interesting properties of orthogonal transformations to define an approximation of the Moore-Penrose pseudo-inverse. This methodology is also highly scalable since the model reduction achieved by these techniques increases with the number of parameters of the linear system involved due to the high correlation needed for these parameters to accomplish very detailed forward predictions, and allows for a very fast computation of the inverse problem solution. We show the application of this methodology to a simple synthetic 2D gravimetric problem for different dimensionalities and different levels of white Gaussian noise, and to a synthetic linear system whose system matrix has been generated via geostatistical simulation to produce a random field with a given spatial correlation. The numerical results show that the DCT pseudoinverse outperforms the classical least-squares techniques, mainly in presence of noise, since the solutions that are obtained are more stable and fit the observed data with a lowest RMS error. Besides, we show that the model reduction is a very effective way of parameter regularization when the conditioning of the reduced DCT matrix is taken into account. We finally show its application to the inversion of a real gravity profile in the Atacama Desert (north Chile) obtaining very successful results in this nonlinear inverse problem. The methodology presented here has a general character and can be applied to solve any linear and nonlinear inverse problems (through linearization) arising in technology and particularly in geophysics, independently of the geophysical model discretization and dimensionality. Nevertheless, the results shown in this paper are better in the case of ill-conditioned inverse problems for which the matrix compression is more efficient. In that sense, a natural extension of this methodology would be its application to the set of normal equations. This article is protected by copyright. All rights reserved