The Potentials about a Point Electrode and Apparent Resistivity Curves for a Two-, Three-, and Four-layered Earth. Harold M. Mooney , W. W. Wetzel

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.

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Apparent resistivity curves in controlled-source electromagnetic sounding directly reflecting true resistivities in a layered Earth

International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts ◽

10.1016/0148-9062(95)92464-s ◽

1995 ◽

Vol 32 (7) ◽

pp. 332

Keyword(s):

Apparent Resistivity ◽

Electromagnetic Sounding ◽

Controlled Source ◽

Layered Earth

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A single apparent resistivity expression for long‐offset transient electromagnetics

Geophysics ◽

10.1190/1.1442182 ◽

1986 ◽

Vol 51 (6) ◽

pp. 1291-1297 ◽

Cited By ~ 17

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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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 ◽

10.1190/1.1487025 ◽

1996 ◽

Vol 61 (3) ◽

pp. 918-918 ◽

Cited By ~ 1

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.

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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 ◽

10.1190/1.1444018 ◽

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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A RESISTIVITY COMPUTATION METHOD FOR LAYERED EARTH MODELS

Geophysics ◽

10.1190/1.1439733 ◽

1966 ◽

Vol 31 (1) ◽

pp. 192-203 ◽

Cited By ~ 39

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.

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The boundary integral method for the D.C. geoelectric problem in the 3-layered earth with a prismoid inhomogeneity in the second layer

Contributions to Geophysics and Geodesy ◽

10.2478/v10126-012-0015-6 ◽

2012 ◽

Vol 42 (4) ◽

pp. 313-343 ◽

Cited By ~ 2

Author(s):

Milan Hvoždara

Keyword(s):

Integral Method ◽

Apparent Resistivity ◽

Boundary Integral Equations ◽

Sedimentary Basins ◽

Boundary Integral ◽

Numerical Results ◽

Boundary Integral Method ◽

Layered Earth ◽

The Earth ◽

Schlumberger Array

Abstract The paper presents algorithm and numerical results for the boundary integral equations (BIE) method of the forward D.C. geoelectric problem for the three-layered earth which contains the prismoidal body with sloped faces in the second layer. This situation occurs in the sedimentary basins. Although the numerical calculations are more complicated in comparison with faces orthogonal to the x, y, z axes, the generalization to the sloped faces enables treatment of the anomalous fields for the bodies of more general shapes as rectangular prisms. The graphs with numerical results present isoline maps of the perturbing potential as well as the resistivity profiles when the source field is due to the pair of D.C. electrodes at the surface of the earth. Also presented apparent resistivity curves for the Schlumberger array AMNB sounding.

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A new formula to compute apparent resistivities from marine magnetometric resistivity data

Geophysics ◽

10.1190/1.2194513 ◽

2006 ◽

Vol 71 (3) ◽

pp. G73-G81

Author(s):

Jiuping Chen ◽

Douglas W. Oldenburg

Keyword(s):

Magnetic Field ◽

Apparent Resistivity ◽

Survey Design ◽

Radial Distance ◽

Data Interpretation ◽

Azimuthal Component ◽

Sea Floor ◽

Layered Earth ◽

New Formula ◽

The Magnetic Field

Magnetometric resistivity (MMR) is an electromagnetic (EM) exploration method that has been used successfully to investigate electrical-resistivity structures below the sea-floor. Apparent resistivity, derived from the observed azimuthal component of the magnetic field, often is used as an approximation to the resistivity of a layered earth. Two commonly used formulas to compute the apparent resistivity have their own limitations and are invalid for a deep-sea experiment. In this paper, we derive an apparent-resistivity formula based upon the magnetic field resulting from a semi-infinite electrode buried in a 1D layered earth. This new formula can be applied to both shallow and deep marine MMR surveys. In addition, we address the effects that arise from the transmitter-receiver (Tx-Rx) depth difference and the choice of the normalized range (the radial distance between transmitter and receiver, divided by the thickness of seawater) on data interpretation and survey design. The performance of the new formula is shown by processing synthetic and field data.

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INTERPRETATION OF BURIED ELECTRODE RESISTIVITY DATA USING A LAYERED EARTH MODEL

Geophysics ◽

10.1190/1.1440878 ◽

1978 ◽

Vol 43 (5) ◽

pp. 988-1001 ◽

Cited By ~ 21

Author(s):

Jeffrey J. Daniels

Keyword(s):

Direct Current ◽

Field Data ◽

Apparent Resistivity ◽

Current Source ◽

Well Logs ◽

Earth Model ◽

Layered Earth ◽

Resistivity Data ◽

True Resistivity ◽

Resistivity Logging

The layered earth model is a fundamental interpretation aid for direct current resistivity data. This paper presents a solution for the layered earth problem for a buried current source and a buried receiver. The model is developed for source and receiver electrodes buried anywhere within a horizontally stratified layered earth containing an arbitrary number of resistivity layers. Model results for the normal well‐logging array indicate that large departures between true and apparent resistivity can be caused by thin beds or highly resistant layers. A true resistivity distribution from well logs can be established by modeling when the effects from borehole rugosity and fluid resistivity are negligible. The equations derived for resistivity well logs can be used to interpret hole‐to‐hole, hole‐to‐surface, and conventional surface array data. A field example demonstrates that deviations between hole‐to‐hole field data and model results, based on well logs in the receiver hole, can be accounted for by combining the resistivity logging models in the receiver holes with information from geologic logs. Differences between the field data and the layered‐model results are attributed to lateral changes between or near the source and receiver holes.

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