On closed-form expressions for the approximate electromagnetic response of a sphere interacting with a thin sheet — Part 1: Theory in the frequency and time domain

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. E189-E198 ◽  
Author(s):  
Jacques K. Desmarais
Keyword(s):  
Closed Form ◽  
Time Domain ◽  
Mineral Exploration ◽  
Thin Sheet ◽  
Electromagnetic Response ◽  
Uniform Field ◽  
Electromagnetic Modeling ◽  
The Time Domain ◽  
Glacial Clays ◽  
Metamorphic Terranes

In mineral exploration and geologic mapping of igneous and metamorphic terranes, the background is often dominantly resistive. The most important electromagnetic interaction is between a discrete conductor and an overlying sheet of conductive overburden (e.g., glacial clays or weathering products of the basement rocks). To enable the electromagnetic modeling of these common situations, here I provide closed-form expressions for the approximate electromagnetic response of a sphere embedded in highly resistive rocks and interacting with an overlying thin sheet. The sphere is assumed to be dipolar and excited by a locally uniform field. The expressions in the time and frequency domains are represented as sums of complete and incomplete cylindrical functions. New asymptotic approximations are provided for the efficient evaluation of the required incomplete cylindrical functions. The frequency-domain formulas are validated by numerical transformation to the time domain and comparison to the time-domain solution.

Approximate semianalytical solutions for the electromagnetic response of a dipping-sphere interacting with conductive overburden

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. E265-E277 ◽  
Author(s):  
Jacques K. Desmarais ◽  
Richard S. Smith
Keyword(s):  
Mineral Exploration ◽  
Thin Sheet ◽  
Electromagnetic Response ◽  
Uniform Field ◽  
Geologic Mapping ◽  
Sphere Model ◽  
Northern Ontario ◽  
Simple Method ◽  
Convolution Integrals ◽  
Metamorphic Terranes

Electromagnetic exploration methods have important applications for geologic mapping and mineral exploration in igneous and metamorphic terranes. In such cases, the earth is often largely resistive and the most important interaction is between a conductor of interest and a shallow, thin, horizontal sheet representing glacial tills and clays or the conductive weathering products of the basement rocks (both of which are here termed the “conductive overburden”). To this end, we have developed a theory from which the step and impulse responses of a sphere interacting with conductive overburden can be quickly and efficiently approximated. The sphere model can also be extended to restrict the currents to flow in a specific orientation (termed the dipping-sphere model). The resulting expressions are called semianalytical because all relevant relations are developed analytically, with the exception of the time-convolution integrals. The overburden is assumed to not be touching the sphere, so there is no galvanic interactions between the bodies. We make use of the dipole sphere in a uniform field and thin sheet approximations; however, expressions could be obtained for a sphere in a dipolar (or nondipolar) field using a similar methodology. We have found that there is no term related to the first zero of the relevant Bessel function in the response of the sphere alone. However, there are terms for all other zeros. A test on a synthetic model shows that the combined sphere-overburden response can be reasonably approximated using the first-order perturbation of the overburden field. Minor discrepancies between the approximate and more elaborate numerical responses are believed to be the result of numerical errors. This means that in practice, the proposed approach consists of evaluating one convolution integral over a sum of exponentials multiplied by a polynomial function. This results in an extremely simple algorithmic implementation that is simple to program and easy to run. The proposed approach also provides a simple method that can be used to validate more complex algorithms. A test on field data obtained at the Reid Mahaffy site in Northern Ontario shows that our approximate method is useful for interpreting electromagnetic data even when the background is thick. We use our approach to obtain a better estimate of the geometry and physical properties of the conductor and evaluate the conductance of the overburden.

On: “Crone pulse electromagnetic response of a conducting thin horizontal sheet—Theory and application,” by S. S. Rai (GEOPHYSICS, 50, 1350–1354, August 1985).

Geophysics ◽  
1987 ◽  
Vol 52 (3) ◽  
pp. 373-374
Author(s):  
David C. Bartel
Keyword(s):  
Time Domain ◽  
Simple Formula ◽  
Thin Sheet ◽  
Step Response ◽  
Electromagnetic Response ◽  
The Subject ◽  
The Time Domain ◽  
Airborne Em

Rai uses a simple formula for the step response of a conducting, horizontal thin sheet in the time domain and applies it to the Crone pulse electromagnetic (PEM) system. He also uses this formulation to interpret some field results. The idea of an infinite, horizontal, conductive thin sheet is valid in some cases for both ground and airborne EM systems. However, I disagree with some of the derivations of the thin‐sheet equation as presented in the subject paper. The applicability of the study is not questioned; but the interpretation of the field example may be different.

Crone pulse electromagnetic response of a conducting thin horizontal sheet—Theory and field application

Geophysics ◽  
1985 ◽  
Vol 50 (8) ◽  
pp. 1350-1354 ◽  
Author(s):  
S. S. Rai
Keyword(s):  
Thin Sheet ◽  
Horizontal Layer ◽  
Field Application ◽  
Step Response ◽  
Electromagnetic Response ◽  
Attractive Feature ◽  
Time Domain Response ◽  
The Time Domain ◽  
Interpretation Model ◽  
Response Formula

The horizontal, conducting thin‐sheet model represents a special interest in interpretation of electromagnetic field data since it is a suitable interpretation model for the surficial conductive layer, a common occurrence in many terrains. For small thicknesses of overburden layers [Formula: see text]separation) the resolution of layer thickness and conductivity is not possible and interpretation needs to be carried out in terms of the layer conductance. An attractive feature of the thin‐sheet model is the simplicity with which the time‐domain response [Formula: see text] can be calculated. The step response of an infinitely thin layer was derived by Maxwell (1891). In this paper I derive the Crone pulse electromagnetic (PEM) response of a conducting infinitely thin horizontal layer. Applicability of the study is demonstrated by means of a field example.

On closed-form expressions for the approximate electromagnetic response of a sphere interacting with a thin sheet — Part 2: Theory in the moment domain, validation, and examples

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. E199-E207 ◽  
Author(s):  
Jacques K. Desmarais
Keyword(s):  
Time Domain ◽  
Thin Sheet ◽  
Test Site ◽  
Electromagnetic Response ◽  
Asymptotic Formulas ◽  
Attractive Alternative ◽  
Northern Ontario ◽  
Synthetic Test ◽  
The Moment ◽  
Analytical Time

Fully analytical formulas are derived for the approximate electromagnetic response of a sphere interacting with a thin sheet in the moment domain. The moment-domain expressions are found to be expressed as simple polynomials of hyperbolic functions. These are significantly simpler to evaluate than the frequency- and time-domain expressions and therefore provide an attractive alternative for modeling. An efficient procedure is outlined for generalizing the moment-domain expression to bipolar-repetitive waveforms. This procedure is validated on a synthetic test example and field data from the Reid-Mahaffy test site, in northern Ontario. Here, results are found in agreement with the work of previous studies. The analytical time-domain procedure is validated through synthetic test examples. The asymptotic formulas for the time-domain expressions are found to significantly reduce the number of required function evaluations, especially for models in which the sphere is not too shallow or not too big or conductive. For example, for a sphere (and overburden) of conductivity (and conductance) of [Formula: see text] (and [Formula: see text]), if the sphere radius is three times smaller than the depth to the top, the amount of required function evaluations is halved; when the ratio is two or less, the asymptotic formulas do not reduce the amount of function evaluations, for the tolerances chosen here. Less-strict tolerances will lead to a further reduction of the required function evaluations.

scholarly journals Modeling ferrite electromagnetic response in the time-domain

2003 ◽  
Author(s):  
J. Johnson ◽  
J.F. DeFord ◽  
G.D. Craig
Keyword(s):  
Time Domain ◽  
Electromagnetic Response ◽  
The Time Domain
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