Robust inversion of vertical electrical sounding data using a multiple reweighted least‐squares method

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 L1‐ or L2‐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 L2, L1 and the MRLS algorithm. The stabilized solutions, even though blurred, could only be obtained by using a heavy ridge regression parameter in L2‐ and L1‐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.

A Robust Two-Step Inversion of Complex Magnetotelluric Apparent Resistivity Data

“…Following we extend to the inversion of complex data the inversion procedure proposed by Porsani et al (2001) for the inversion of vertical electrical sounding. For an inverse solution we have to look for equality between the complex observed apparent …”

“…The eight resistivities and seven thicknesses obtained have been used as a starting model for the subsequent interpretation employing the MRLS method proposed in Porsani et al (2001).…”

“…The first step consists of an improved use of the low frequency asymptotic Bostick-Niblett approach, according to the procedure developed by Ramos and Sampaio (1993). The second step employs the result of the inversion obtained in the first step as a starting model, to apply an extension to complex data of the linearized inversion with a multiple re-weighted least-squares (MRLS) method proposed by Porsani et al (2001).…”

Delimitação do topo de aqüífero na região de Porto Seguro - Bahia através da inversão de sondagens elétricas verticais

“…observadas (Rijo et al, 1977;Lines & Treitel, 1984;Carrasquilla & Rijo, 1991;Ferreira & Porsani, 1977;Marinho, 1997;Monteiro & Porsani, 1999;Monteiro, 1999). A seguir apresentamos o método utilizado na interpretação quantitativa das SEVs desenvolvido com base no procedimento iterativo proposto por Porsani et al (2001). Seja d(x) o valor de resistividade aparente de uma SEV, observado na separação x=AB/2 e seja ρ a (m,x) a função não linear utilizada para cálculo dos valores teóricos de resistividade aparente (Gosh, 1971a(Gosh, , 1971bJohansen, 1977;Koefoed,1979 …”

“…Desta maneira, em cada passo k do processo iterativo, podemos pesquisar várias vezes por um melhor modelo para substituir o modelo corrente. Esta estratégia, proposta por Porsani et al (2001), permite que se explore o espaço dos modelos de forma mais extensiva, conferindo robustez e eficácia ao método de IL.…”