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 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.

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.

Optimum second vertical derivatives in geologic mapping and mineral exploration

Geophysics ◽  
1982 ◽  
Vol 47 (12) ◽  
pp. 1706-1715 ◽  
Author(s):  
V. K. Gupta ◽  
N. Ramani
Keyword(s):  
Power Spectrum ◽  
High Frequency ◽  
Mineral Exploration ◽  
Frequency Noise ◽  
Second Derivative ◽  
Geologic Mapping ◽  
Northern Ontario ◽  
Near Surface ◽  
Vertical Derivative ◽  
Gravity Interpretation

The second vertical derivative is often used in gravity interpretation to enhance localized near‐surface features. Since high‐frequency noise is amplified considerably in a true second derivative map, some smoothing is always necessary. One of the methods available to the interpreter is the optimum second derivative technique which utilizes Wiener theory to design filters from an analysis of the power spectrum to suit the degree of noise in the data. The power spectrum of the Bouguer gravity field obtained from over 5100 gravity stations covering an area of approximately [Formula: see text] in Archean greenstone belts of northern Ontario was computed. The radially averaged spectrum decreased monotonically with frequency and flattened at the high‐frequency end. The spectrum, separated into its signal and noise components, was used to design an optimum second vertical derivative filter which has a peak at 0.35 cycles/grid interval with low‐ and high‐frequency cut‐off at 0.12 and 0.44 cycles/grid intervals, respectively. The resultant second derivative map corresponds remarkably well with the known surface geology. In most of the mapped regions the zero contours coincide with the lithological boundaries; positive and negative anomalies match surface exposures of the mafic and felsic rock units, respectively. In the central part of the area which covers the Birch‐Uchi greenstone belt, where the geology has been well mapped, the second derivative map is quite successful in delineating the successive groups of mafic to felsic metavolcanic rocks that represent the product of cyclic volcanism. The distribution of the mineral deposits in the study area can be related to the positive or negative second derivative anomalies. For example, most of the known gold and silver mineralization occurring in mafic metavolcanics is associated with positive second derivative anomalies. Similarly, the polymetallic Zn‐Cu‐Ag deposits are mainly located in the negative second derivative anomalies caused by felsic metavolcanics. A properly designed second derivative map can thus be an important supplement to geologic mapping in the identification of lithological units, in the study of structure, and as an indirect tool in regional mineral exploration.

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.

To: “The electromagnetic response of thin sheets buried in a uniformly conducting half‐space,” by R. Clark Robertson in January 1987 GEOPHYSICS, p. 108–117

Geophysics ◽  
1987 ◽  
Vol 52 (4) ◽  
pp. 583-583
Keyword(s):  
Electric Field ◽  
Half Space ◽  
Thin Sheet ◽  
Skin Depth ◽  
Modeling Technique ◽  
Electromagnetic Response ◽  
Thin Sheets ◽  
Incident Electric Field

On p. 112, the caption of Figure 4 should read “The (a) magnitude and (b) phase in radians of the x component of the horizontal electric field obtained for a square thin sheet of integrated conductivity 1 S, 8 skin depths on a side, buried at a depth of 0.1 skin depth when the incident electric field is x polarized. Each segment is 1 skin depth on a side.” On p. 114, the last sentence of the first paragraph in the Discussion should read “It is easy to see why the surface thin sheet is a popular modeling technique for magnetotelluric applications.”

Airborne resistivity data leveling

Geophysics ◽  
1999 ◽  
Vol 64 (2) ◽  
pp. 378-385 ◽  
Author(s):  
Haoping Huang ◽  
Douglas C. Fraser
Keyword(s):  
Frequency Domain ◽  
Mineral Exploration ◽  
Primary Field ◽  
Geologic Mapping ◽  
Resistivity Data ◽  
High Magnitude ◽  
Geological Features ◽  
Absolute Measure ◽  
Tie Lines ◽  
Automated Technique

Helicopter‐borne frequency‐domain electromagnetic (EM) data are used routinely to produce resistivity maps for geologic mapping, mineral exploration, and environmental investigations. The integrity of the resistivity data depends in large part on the leveling procedures. Poor resistivity leveling procedures may, in fact, generate false features as well as eliminate real ones. Resistivity leveling is performed on gridded data obtained by transformation of the leveled EM channel data. The leveling of EM channel data is often imperfect, which is why the resistivity grids need to be leveled. We present techniques for removing the various types of resistivity leveling errors which may exist. A semi‐automated leveling technique uses pseudo tie‐lines to remove the broad flight‐based leveling errors and any high‐magnitude line‐based errors. An automated leveling technique employs a combination of 1-D and 2-D nonlinear filters to reject the rest of the leveling errors including both long‐and short‐wavelength leveling errors. These methods have proven to be useful for DIGHEM helicopter EM survey data. However, caution needs to be exercised when using the automated technique because it cannot distinguish between geological features parallel to the flight lines and leveling errors of the same wavelength. Resistivity leveling is not totally objective since there are no absolutes to the measured frequency‐domain EM data. The fundamental integrity of the EM data depends on calibration and the estimate of the EM zero levels. Zero level errors can be troublesome because there is no means by which the primary field can be determined absolutely and therefore subtracted to yield an absolute measure of the earth’s response. The transform of incorrectly zero‐leveled EM channels will yield resistivity leveling errors. Although resistivity grids can be leveled empirically to provide an esthetically pleasing map, this is insufficient because the leveling must also be consistent across all frequencies to allow resistivity to be portrayed in section. Generally, when the resistivity looks correct in plan and section, it is assumed to be correct.

Statistical analysis of airborne gamma‐ray data for geologic mapping purposes: Crixas‐Itapaci area, Goias, Brazil

Geophysics ◽  
1989 ◽  
Vol 54 (10) ◽  
pp. 1326-1332 ◽  
Author(s):  
A. C. B. Pires ◽  
N. Harthill
Keyword(s):  
Statistical Analysis ◽  
Gamma Ray ◽  
Mineral Exploration ◽  
Aerial Survey ◽  
Geologic Mapping ◽  
Good Correspondence ◽  
Gamma Ray Spectrometer ◽  
Data Points ◽  
Spectrometer Data ◽  
Mode Technique

Q‐mode factor analysis, K‐means clustering, and G‐mode clustering were used on digitized gamma‐ray spectrometer data from an aerial survey of the Crixas‐Itapaci area, Goias, Brazil. The data points including seven variables—eU, eTh, K, total count, U/Th, U/K, and Th/K—were digitized for a 2 km square grid. For the northwest corner of the area the data were gridded at 1 km. The Q‐mode classification method supplied results that do not show a good correspondence with the known geology. The K‐means clustering procedure barely identified the main lithologic features of the area. The G‐mode technique produced results that correlate well with the known geology and identified the greenstone belts present in the area by discriminating their ultramafic and mafic components from adjacent felsic rocks. Statistical analysis of aerial gamma‐ray spectrometer data can be very helpful in mapping geologic units in poorly known areas. It can also be used for mineral exploration purposes if mineralization is known to be associated with lithologies that can be identified by the techniques used in this study.

On the electromagnetic response of a resistive cylinder buried in a conductive host

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. E341-E351 ◽  
Author(s):  
Andrei Swidinsky
Keyword(s):  
Electric Dipole ◽  
Mineral Exploration ◽  
Skin Depth ◽  
Hydrocarbon Exploration ◽  
Electromagnetic Response ◽  
Dipole Source ◽  
Additional Information ◽  
Dipole System ◽  
Single Dipole ◽  
Current Systems

The frequency-domain electromagnetic response of a confined conductor buried in a resistive host has received much attention, particularly in the context of mineral exploration. In contrast, the problem of the electromagnetic response of a confined resistor buried in a conductive host has been less thoroughly studied. However, resistive targets are important in geotechnical and hydrologic studies, archaeological prospecting, and, more recently, offshore hydrocarbon exploration. I analytically address the problem of the electromagnetic response of a completely resistive cylindrical cavity buried in a conductive host in the presence of a simplified 2D electric dipole source. In contrast to the confined conductor, which channels and induces current systems, the confined resistor deflects current and produces additional eddy current systems in the conductive host. I apply this theory to model the response of a grounded electric dipole-dipole system operating over a range of frequencies from 0 Hz to 10 kHz, in the presence of a horizontal 5-m radius insulating cylinder located 1-m beneath the surface of a uniform earth. This represents a common hazard encountered during mining and civil engineering operations. Results show that such an insulating cavity increases the recorded electric field amplitude and phase delay at all transmitted frequencies. These observations suggest that a broadband electromagnetic prospecting system may provide additional information about the location and extent of a void, over and above a standard dipole-dipole resistivity survey. When the host skin depth is much larger than all other length scales, the response can be approximated by an equivalent single dipole unless the cylinder’s radius is much larger than its distance from the transmitter. This result provids a useful rule of thumb to determine the acceptable range over which a resistive target can be modeled by a distribution of dipoles.

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