1887
  1. Home
  2. A-Z Publications
  3. Geophysical Prospecting
  4. Volume 49, Issue 2
  5. Article
Volume 49, Issue 2
  • E-ISSN: 1365-2478

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

Loading

Article metrics loading...

/content/journals/10.1046/j.1365-2478.2001.00249.x
2001-12-21
2024-04-24

  • From This Site

    /content/journals/10.1046/j.1365-2478.2001.00249.x
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. ChaveA.D., ThomsonD.J., AnderM.E.1987. On the robust estimation of power spectra, coherences and transfer functions. Journal of Geophysical Research92B, 633–648.
    [Google Scholar]
  2. ClaerboutJ.F. & MuirF.1973. Robust modelling with erratic data. Geophysics38, 826–844.
    [Google Scholar]
  3. ConstableS.E., ParkerR.L., ConstableC.G.1987. Occam's inversion: a practical algorithm for generating smooth models for electromagnetic sounding data. Geophysics52, 289–300.
    [Google Scholar]
  4. EgbertG.D. & BookerJ.R.1986. Robust estimation of geomagnetic transfer function. Geophysical Journal of the Royal Astronomical Society87, 173–194.
    [Google Scholar]
  5. Gai‐shanZ.1985. Asymptotic formula of the transform function for layered earth potential and its applications to interpretation of resistivity sounding data. Geophysics50, 1513–1514.
    [Google Scholar]
  6. GhoshD.P.1971a. The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements. Geophysical Prospecting19, 192–217.
    [Google Scholar]
  7. GhoshD.P.1971b. Inverse filter coefficients for the computation of apparent resistivity standard curves for a horizontally stratified earth. Geophysical Prospecting19, 769–777.
    [Google Scholar]
  8. GuptaP.K., Sri Niwas, GaurV.K.1997. Straightforward inversion of vertical electrical sounding data. Geophysics62, 775–785.
    [Google Scholar]
  9. InmanJ.R.1975. Resistivity inversion with ridge regression. Geophysics40, 798–817.
    [Google Scholar]
  10. InmanJ.R., RyuJ., WardS.H.1973. Resistivity inversion. Geophysics38, 1088–1108.
    [Google Scholar]
  11. JohansenH.K.1977. A man/computer interpretation system for resistivity sounding over a horizontally stratified earth. Geophysical Prospecting25, 667–691.
    [Google Scholar]
  12. JuppD.L.B. & VozoffK.1975. Stable iterative methods for the inversion of geophysical data. Geophysical Journal of the Royal Astronomical Society42, 957–976.
    [Google Scholar]
  13. KoefoedO.1979. Geosounding Principles 1, Resistivity Sounding Measurements . Elsevier Science Publishing Co.
  14. LangerR.E.1933. An inverse problem in differential equations. American Mathematical Society Bulletin39, 814–820.
    [Google Scholar]
  15. Le BrasR. & ClaytonR.1988. An iterative inversion of back‐scattered acoustic waves. Geophysics53, 501–508.
    [Google Scholar]
  16. MoharirP.1996. Response function in electromagnetism. In: Natural Source Electromagnetic Induction in Earth (eds B.R.Arora and Sri Niwas ). New Age International (P) Ltd, New Delhi.
  17. ParkerR.L.1984. The inverse problem of resistivity sounding. Geophysics49, 2143–2158.
    [Google Scholar]
  18. PekerisC.L.1940. Direct method of interpretation in resistivity prospecting. Geophysics5, 31–46.
    [Google Scholar]
  19. PorsaniM.J. & RijoL.1993. Estudos Geofísicos Aplicados à Prospeçcão de Água Subterrânea na Região do Lago Arari – Ilha de Marajó. Revista Brasileira de Geofísica11, 101–123.
    [Google Scholar]
  20. PorsaniM.J., StoffaP.L., ChunduruR.K., SenM.1993. Evaluation of measures of error using a genetic algorithm. 3rd Annual International Meeting of the Brazilian Geophysical Society, Expanded Abstracts1, 85–90.
  21. ScalesJ. & GersztenkornA.1988. Robust methods in inverse theory. Inverse Problem4, 1071–1091.
    [Google Scholar]
  22. SenM. & StoffaP.L.1995. Global Optimization Methods in Geophysical Inversion . Elsevier Science Publishing Co.
  23. SimsJ.E. & MorganE.D.1992. Comparison of four least‐squares inversion schemes for studying equivalence in one dimensional resistivity interpretation. Geophysics57, 1282–1293.
    [Google Scholar]
  24. SlichterL.B.1933. The interpretation of resistivity prospecting methods for horizontal structures. Physics4, 307–322.
    [Google Scholar]
  25. Sri Niwas & IsrailM.1986. Computation of apparent resistivities using an exponential approximation of kernel function. Geophysics51, 1594–1602.
    [Google Scholar]
  26. Sri Niwas & IsrailM.1987. A simple method of interpreting dipole resistivity sounding. Geophysics52, 1412–1417.
    [Google Scholar]
  27. StefanescuS.S.1930. Sur la distribution electrique autour d'une prise de terre ponctuelle dans un terrain a couches horizontales homogenes et isotropes. Le Journal de Physique et le Radium 7, series 1.
  28. StrangG.1986. Introduction to Applied Mathematics . Wellesley, Cambridge.
  29. VozoffK.1958. Numerical resistivity analysis – horizontal layers. Geophysics23, 536–556.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1046/j.1365-2478.2001.00249.x
Loading
Robust inversion of vertical electrical sounding data using a multiple reweighted least‐squares method
Geophysical Prospecting 49, 255 (2001); https://doi.org/10.1046/j.1365-2478.2001.00249.x
/content/journals/10.1046/j.1365-2478.2001.00249.x
/content/journals/10.1046/j.1365-2478.2001.00249.x
Loading

Data & Media loading...

  • Article Type: Research Article

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error