A RESISTIVITY COMPUTATION METHOD FOR LAYERED EARTH MODELS

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 192-203 ◽  
Author(s):  
Harold M. Mooney ◽  
Ernesto Orellana ◽  
Harry Pickett ◽  
Leonard Tornheim
Keyword(s):  
Apparent Resistivity ◽  
Complete Separation ◽  
Computation Method ◽  
Layered Earth ◽  
Electrode Arrangement ◽  
Number Of Layers ◽  
Earth Structures ◽  
Recursion Formulas ◽  
Single Set ◽  
Numerical Integration Procedure

A procedure is given to compute apparent resistivity and induced‐polarization results for layered earth structures. The method is designed for use with large computers. Results may be obtained for any number of layers, and for any of the commonly used electrode configurations. Specific expressions are given for Schlumberger, Wenner, azimuthal‐dipole, axial‐dipole, and for the potential function. The method consists in expanding a portion of the integrand as a series in [Formula: see text] and integrating analytically term by term. Convergence of the resulting series is established. The required coefficients for each term in the series can be obtained by recursion formulas from preceding coefficients. Accuracy of the results can be estimated and can be preselected. For the Schlumberger electrode arrangement with spacing L, for example, the error produced by truncating the series after M terms will be no greater than [Formula: see text]. A more rigid bound on the error is also given. Accuracy of the method was further checked against published apparent resistivity data and against a numerical integration procedure devised for this purpose. The following characteristics make the method well suited for use with digital computers: (1) The formulation is relatively simple and easily programmed. (2) A single program will handle any number of layers. (3) The computer can be made to generate the required coefficients internally. (4) The computer can be programmed to terminate the computation as soon as any preselected accuracy has been achieved. (5) Complete separation is attained between earth structure and electrode arrangement; thus, a single set of stored coefficients can be used repeatedly for different electrode spacings and different electrode arrangements.

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.

Inversion of borehole normal resistivity logs

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1541-1548 ◽  
Author(s):  
Fang‐Wei Yang ◽  
Stanley H. Ward
Keyword(s):  
Theoretical Model ◽  
Ridge Regression ◽  
Arbitrary Number ◽  
Inverse Method ◽  
Apparent Resistivity ◽  
Random Noise ◽  
High Resistivity ◽  
Number Of Layers ◽  
Boundary Position ◽  
Resistivity Logs

This paper reports on an investigation of the inversion of borehole normal resistivity data via ridge regression. Interpretation is afforded of individual thin beds and of complicated layered structures. A theoretical solution is given for a layered model containing an arbitrary number of layers in the forward problem. Two forward model results for resistive and conductive thin beds indicate that for high‐resistivity contrasts, the departure between true and apparent resistivity may be more important than the effects caused by the variations in borehole diameter and mud resistivity. Four normal resistivity logs were chosen to test the inversion scheme. Two of the logs were theoretical logs with and without random noise added, and the remaining two were field examples. Theoretical model results and field examples indicate that the inverse method can be used to obtain the resistivity for each layer when the boundary position is known, but it also can be used to obtain the thickness and resistivity for each layer simultaneously.

Magnetotelluric response on vertically inhomogeneous earth having conductivity varying exponentially with depth

Geophysics ◽  
1982 ◽  
Vol 47 (1) ◽  
pp. 89-99 ◽  
Author(s):  
D. Kao
Keyword(s):  
Half Space ◽  
Apparent Resistivity ◽  
Bessel Functions ◽  
Homogeneous Layer ◽  
Modified Bessel Functions ◽  
Layer Model ◽  
Transitional Layer ◽  
Number Of Layers ◽  
Inhomogeneous Models ◽  
Electric And Magnetic Fields

Magnetotelluric (MT) response is studied for a vertically inhomogeneous earth, where conductivity (or resistivity) varies exponentially with depth as [Formula: see text]. Horizontal electric and magnetic fields in such an inhomogeneous medium are given in terms of modified Bessel functions. Impedance and apparent resistivity are calculated for (1) an inhomogeneous half‐space having conductivity varying exponentially with depth, (2) an inhomogeneous half‐space overlain by a homogeneous layer, and (3) a three‐layer model with the second layer as an inhomogeneous or transitional layer. Results are presented graphically and are compared with those of homogeneous multilayer models. In the case of resistivity increasing exponentially with depth, the results of the above inhomogeneous models are equivalent to those of Cagniard two‐layer models, with [Formula: see text]. In the case of resistivity decreasing exponentially with depth, the homogeneous multilayer approximation depends upon the number of layers and the layer parameters chosen; |Z/ωμ| as a function of frequency is more useful than the apparent resistivity in determining the values of p and [Formula: see text].

A resistivity computation method for layered earth models with transition boundary

1971 ◽  
Vol 85 (1) ◽  
pp. 153-160 ◽  
Author(s):  
B. P. S. Rathor ◽  
H. S. Rathor
Keyword(s):  
Computation Method ◽  
Layered Earth ◽  
Transition Boundary ◽  
Earth Models

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.

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