1887
Volume 18 Number 2
  • E-ISSN: 1365-2478

Abstract

A

This paper describes certain procedures for deriving from the apparent resistivity data as measured by the Wenner electrode configuration two functions, known as the kernel and the associated kernel respectively, both of which are functions dependent on the layer resistivities and thicknesses. It is shown that the solution of the integral equation for the Wenner electrode configuration leads directly to the associated kernel, from which an integral expression expressing the kernel explicitly in terms of the apparent resistivity function can be derived. The kernel is related to the associated kernel by a simple functional equation

where (λ) is the kernel and (λ) the associated kernel.

Composite numerical quadrature formulas and also integration formulas based on partial approximation of the integrand by a parabolic arc within a small interval are developed for the calculation of the kernel and the associated kernel from apparent resistivity data. Both techniques of integration require knowledge of the values of the apparent resistivity function at points lying between the input data points. It is shown that such unknown values of the apparent resistivity function can satisfactorily be obtained by interpolation using the least‐squares method. The least‐squares method involves the approximation of the observed set of apparent resistivity data by orthogonal polynomials generated by Forsythe's method (Forsythe 1956). Values of the kernel and of the associated kernel obtained by numerical integration compare favourably with the corresponding theoretical values of these functions.

Loading

Article metrics loading...

/content/journals/10.1111/j.1365-2478.1970.tb02104.x
2006-04-27
2024-03-29
Loading full text...

Full text loading...

References

  1. Bowman, Frank, 1958, Introduction to Bessel functions: New York , Dover Publications, Inc.
    [Google Scholar]
  2. Forsythe, G. E., 1957, Generation and use of orthogonal polynomials for data fitting with a digital computer: Journ. Soc. Indust. Appl. Math. 5, 74–87.
    [Google Scholar]
  3. Koefoed, O., 1965, Direct methods of interpreting resistivity observations: Geophysical Prospecting13, 568–592.
    [Google Scholar]
  4. Koefoed, O., 1966, The direct interpretation of resistivity observations made with a Wenner electrode configuration: Geophysical Prospecting14, 71–79.
    [Google Scholar]
  5. Mooney, H. M. and Wetzel, W. W., 1956, The potentials about a point electrode and apparent resistivity curves for a two‐, three‐, and four‐layer earth: Minneapolis , University of Minnesota Press.
    [Google Scholar]
  6. Paul, M. K., 1968, A note on the direct interpretation of resistivity profiles for Wenner electrode configuration: Geophysical Prospecting16, 159–162.
    [Google Scholar]
  7. Pekeris, C. L., 1940, Direct method of interpretation in resistivity prospecting: Geophysics5, 31–42.
    [Google Scholar]
  8. Ralston, Anthony, 1965, A first course in numerical analysis: New York , McGraw‐Hill Book Company.
    [Google Scholar]
  9. Sneddon, Ian N., 1955, Functional analysis. In: Encyclopedia of Physics, V. 2: Berlin , Springer‐Verlag, p., 198–347.
    [Google Scholar]
  10. Sunde, Earling D., 1949, Earth conduction effects in transmission systems: Princeton, D. Van Nostrand Company, Inc.
    [Google Scholar]
  11. Vozoff, Keeva, 1956, Numerical resistivity analysis: Geophysics23, 536–556.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1111/j.1365-2478.1970.tb02104.x
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