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Robust inversion of vertical electrical sounding data using a multiple reweighted least‐squares method
Author(s) -
Porsani Milton J.,
Niwas Sri,
Ferreira Niraldo R.
Publication year - 2001
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1046/j.1365-2478.2001.00249.x
Subject(s) - inverse problem , matrix (chemical analysis) , sensitivity (control systems) , algorithm , computer science , mathematics , mathematical optimization , mathematical analysis , materials science , electronic engineering , engineering , composite material
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.