NUMERICAL ANALYSIS OF RELATIVE RESISTIVITY FOR A HORIZONTALLY LAYERED EARTH

Geophysics ◽  
1963 ◽  
Vol 28 (2) ◽  
pp. 222-231 ◽  
Author(s):  
Seibe Onodera
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Kernel Function ◽  
Apparent Resistivity ◽  
Complete System ◽  
Layered Earth ◽  
Relative Resistivity

The method of calculating the relative resistivity, which is the ratio of the apparent resistivity to the resistivity of the upper layer, for a multiple‐layered earth is given by means of the expansion of the kernel function according to a complete system of normalized orthogonal polynomials. The method, which includes estimation of the accuracy to be expected, is illustrated by application to a three‐layer earth.

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

Reply by the author to Pierre Valla

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 919-919
Author(s):  
Umesh C. Das
Keyword(s):  
Asymptotic Behavior ◽  
Direct Current ◽  
Apparent Resistivity ◽  
Intermediate Step ◽  
Controlled Source ◽  
Layered Earth ◽  
Asymptotic Expressions ◽  
Higher Order Terms ◽  
Earth Structures ◽  
Subsurface Resistivity

I thank Pierre Valla for his interest in my paper (Das, 1995a). Transformation of controlled source electromagnetic (CSEM) measurements into apparent resistivities is carried out as an intermediate step in order to enhance interpretation. Duroux (1967; and hence Valla, 1984) derives, using asymptotic expressions (higher order terms are dropped out), apparent resistivities from CSEM measurements. Valla mentions, ‘those apparent resistivities do not have the nice asymptotic behavior’, and they can not be used as an intermediate step to estimate the layer resistivities and thicknesses in the subsurface. My aim in the paper has been not to work a ‘miracle’ but to derive a function to reflect the subsurface resistivity distributions of the layered earth structures directly. The calculations on a few models indicate that such a function can be derived which yields an unambiguous apparent resistivity. The apparent resistivity curves are similarly useful in interpretation as the direct current and magnetotelluric apparent resistivity curves. Inclusion of Duroux’s work would have given the readers a chance to appreciate my definition.

On: “A simple method of interpreting dipole resistivity soundings” by S. Niwas and M. Israil (GEOPHYSICS, 52, 1412–1417, October 1987)

Geophysics ◽  
1989 ◽  
Vol 54 (12) ◽  
pp. 1647-1647
Author(s):  
Edward Szaraniec
Keyword(s):  
Linear Combination ◽  
Kernel Function ◽  
Apparent Resistivity ◽  
Simple Method ◽  
Resistivity Data ◽  
The Subject

The subject paper consists in approximating the apparent resistivity data by using a linear combination of suitable functions chosen in such a way that (1) they give a good approximation up to the desired precision and (2) they allow the kernel function to be determined analytically. Surprisingly enough, no mention is made that such an approach, especially directed toward interpretation of resistivity soundings, was first proposed by Santini and Zambrano (1981). The subject was subsequently continued by Kumar and Chowdary (1982) and commented by Santini and Zambrano (1982), Straub (1984), and Szaraniec (1982, 1984).

A single apparent resistivity expression for long‐offset transient electromagnetics

Geophysics ◽  
1986 ◽  
Vol 51 (6) ◽  
pp. 1291-1297 ◽  
Author(s):  
Yang Sheng
Keyword(s):  
Apparent Resistivity ◽  
Early Time ◽  
Point Of View ◽  
Time Range ◽  
Late Time ◽  
Careful Study ◽  
Physical Point ◽  
Layered Earth ◽  
Transient Electromagnetic ◽  
Long Offset

Early‐time and late‐time apparent resistivity approximations have been widely used for interpretation of long‐offset transient electromagnetic (LOTEM) measurements because it is difficult to find a single apparent resistivity over the whole time range. From a physical point of view, Dr. C. H. Stoyer defined an apparent resistivity for the whole time range. However, there are two problems which hinder its use: one is that there is no explicit formula to calculate the apparent resistivity, and the other is that the apparent resistivity has no single solution. A careful study of the two problems shows that a numerical method can be used to calculate a single apparent resistivity. A formula for the maximum receiver voltage over a uniform earth, when compared with the receiver voltage for a layered earth, leads to the conclusion that, in some cases, a layered earth can produce a larger voltage than any uniform earth can produce. Therefore, our apparent resistivity definition cannot be applied to those cases. In some other cases, the two possible solutions from our definition do not merge, so that neither of them is meaningful for the whole time range.

On: “Apparent resistivity curves in controlled‐source electromagnetic sounding directly reflecting true resistivities in a layered earth”, by U. C. Das (January‐February 1995 GEOPHYSICS, 60, 53–60).

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 918-918 ◽  
Author(s):  
Pierre Valla
Keyword(s):  
Magnetic Dipole ◽  
Apparent Resistivity ◽  
Electromagnetic Sounding ◽  
Controlled Source ◽  
Layered Earth ◽  
Vertical Magnetic Dipole ◽  
Em Field ◽  
Depth Sounding

Using a clever mix of two components of the EM field caused by a vertical magnetic dipole, U. C. Das derives what he claims to be an exact apparent resistivity for use in EM depth sounding.

On: “Apparent resistivity curves in controlled‐source electromagnetic sounding directly reflecting true resistivities in a layered earth”, by U. C. Das (January‐February 1995 GEOPHYSICS, 60, 53–60).

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 917-917
Author(s):  
Brian R. Spies ◽  
James R. Wait
Keyword(s):  
Apparent Resistivity ◽  
Previous Literature ◽  
Electromagnetic Sounding ◽  
Controlled Source ◽  
Layered Earth

Das has made a number of fundamental errors in his paper on apparent resistivity in controlled‐source EM sounding, and has ignored the previous literature.

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