Robust inversion of vertical electrical sounding data using a multiple reweighted least-squares method |
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| Authors: | Milton J Porsani Sri Niwas † Niraldo R Ferreira ‡ |
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| Institution: | Centro de Pesquisa em Geofísica e Geologia, Instituto de Geociências, Universidade Federal da Bahia, Campus Universitário da Federação, Salvador, Bahia, Brazil,;Department of Earth Sciences, University of Roorkee, Roorkee 247667, India, and;Escola Politécnica, Universidade Federal da Bahia, Campus Universitário da Federação, Salvador, Bahia, Brazil |
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| Abstract: | The root cause of the instability problem of the least-squares (LS) solution of the resistivity inverse problem is the ill-conditioning of the sensitivity matrix. To circumvent this problem a new LS approach has been investigated in this paper. At each iteration, the sensitivity matrix is weighted in multiple ways generating a set of systems of linear equations. By solving each system, several candidate models are obtained. As a consequence, the space of models is explored in a more extensive and effective way resulting in a more robust and stable LS approach to solving the resistivity inverse problem. This new approach is called the multiple reweighted LS method (MRLS). The problems encountered when using the L 1- or L 2-norm are discussed and the advantages of working with the MRLS method are highlighted. A five-layer earth model which generates an ill-conditioned matrix due to equivalence is used to generate a synthetic data set for the Schlumberger configuration. The data are randomly corrupted by noise and then inverted by using L 2, L 1 and the MRLS algorithm. The stabilized solutions, even though blurred, could only be obtained by using a heavy ridge regression parameter in L 2- and L 1-norms. On the other hand, the MRLS solution is stable without regression factors and is superior and clearer. For a better appraisal the same initial model was used in all cases. The MRLS algorithm is also demonstrated for a field data set: a stable solution is obtained. |
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