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 Jacques K. Desmarais 1 ABSTRACT In mineral exploration and geologic mapping of igneous and metamorphic terranes, the background is often domi- nantly resistive. The most important electromagnetic interac- tion 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 electromag- netic 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 as- sumed 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. INTRODUCTION The plate and sphere models are commonly used for interpreting borehole, ground, and airborne electromagnetic data (Dyck et al., 1981; Dyck and West, 1984; Lamontagne et al., 1988; Macnae et al., 1998; Schaa, 2010; Smith and Wasylechko, 2012; Fullagar et al., 2015; Macnae, 2015; Vallée, 2015; Annan, 1974). These models are especially attractive in resistive environments, such as metamor- phic and igneous terranes, in which the free-space approximation can sometimes be justified. However, if the geology is better represented by a background that is conductive, free-space approaches become inappropriate. In such situations, models describing the interaction of a discrete conductor with a conductive background are more suitable. A plethora of techniques exist for calculating the electromagnetic response of interacting conductors (Wait, 1953; Singh 1973; Lee 1975, 1983; Ward and Hohmann, 1987; Bartel and Becker, 1988; Newman and Hohmann, 1988; Smith and West, 1988; Raiche and Sugeng, 1989; Walker and West, 1991; Song, 1993; San Filipo and Won, 2005; Shubitidze, 2011). These are generally based on costly procedures that involve solving integral forms of Maxwell’s equa- tions. For frequency and more specifically time-domain methods, even if analytical solutions of Maxwell’s equations exist in the fre- quency-wavenumber domain, the problem still involves calculating several integral transforms over special functions, which is still a relatively costly procedure (Vallée, 2015). Thus, these methods are not routinely used for geologic mapping and interpretation of min- eral exploration data. Moreover, many of the models in the literature involve a half-space or layered-earth background, which is not nec- essarily needed for applications in crystalline terranes, where a large part of the background is resistive. In such situations, the most im- portant interaction can be that of the discrete conductor with a thin layer of conductive overburden, comprising of, for example, glacial clay deposits or weathering products (Xie et al., 1998). For such purposes, Desmarais and Smith (2016) develop an approximate semi- analytical approach for calculating the electromagnetic response of a sphere interacting with a thin sheet. The sphere was assumed to be dipolar and embedded in a locally uniform field. Their approach fol- lows some of the concepts proposed by Liu and Asten (1993). The sphere model is attractive, due to its versatility in terms of being able to model bodies of various sizes, conductivity, position, and orienta- tion (using the so-called dipping-sphere approach), as well as its rel- ative simplicity of computation. Expressions were provided in the time domain and were called semianalytical because the only nonanalytical step was the evaluation of time-convolution integrals, Manuscript received by the Editor 30 July 2017; revised manuscript received 17 November 2018; published ahead of production 07 March 2019; published online 12 April 2019. 1 University of Saskatchewan, Earth Sciences, 114 Science Place, Saskatoon, Saskatchewan S7N 5E2, Canada and Università di Torino, Dipartimento di Chimica, Via P. Giuria 5, Torino 10125, Italy. E-mail: jkd788@mail.usask.ca; jacqueskontak.desmarais@unito.it (corresponding author). © 2019 Society of Exploration Geophysicists. All rights reserved. E189 GEOPHYSICS, VOL. 84, NO. 3 (MAY-JUNE 2019); P. E199–E198, 2 FIGS., 2 TABLES. 10.1190/GEO2017-0498.1 Downloaded 05/26/19 to 130.192.118.38. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/