IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.47, NO.8, AUGUST 2009 2627 Modeling the Response of Electromagnetic Induction Sensors to Inhomogeneous Magnetic Soils with Arbitrary Relief Pascal Druyts, Yogadhish Das, Fellow, IEEE, Christophe Craeye, Member, IEEE and Marc Acheroy, Member, IEEE Abstract—A general model to compute the response of an Electromagnetic Induction (EMI) sensor to a magnetic soil, in both time and frequency domains, is developed. The model requires modest computational resources and can be applied to arbitrary soil inhomogeneities and relief, and to arbitrary sensor coil shapes, orientations and positions. Central to the model is the concept of a head sensitivity map which can be used to characterize the sensor head as a function of the shape, size and position of the sensor coils. Two further concepts related to the head sensitivity are presented, the zero equi-sensitivity surface and the volume of influence. We demonstrate that these concepts aid understanding of the detector behavior. The general model is based on the Born approximation, which is valid if the soil magnetic susceptibility is sufficiently small. A simpler model which is only valid for homogeneous Half-Space (HS) soils but does not require the Born approximation is also developed. The responses predicted by both models are shown to be in good agreement with each other and also with available analytic solutions. Comparing the two models also enabled an expression for the error incurred when using the Born approximation to be established. We shown that, for most soils of relevance to mine clearance, the corresponding error is negligible. Index Terms—Born approximation, electromagnetic induction (EMI), humanitarian demining, magnetic susceptibility, metal detector (MD), reciprocity, sensitivity map. I. I NTRODUCTION M ETAL Detectors (MDs) working on the principle of EMI are widely used in mine action. Many factors must be taken into account to understand the performance of a MD including the detector characteristics, the target response, the soil response and the effect of the soil on the target response. Detector characteristics include factors such as internal processing, geometry of the detector head and transmit current. Many authors [1]–[8] have proposed methods to assess the target response but the influence of the soil has received less attention until recently. Nevertheless it was known that the soil can significantly affect metal This work was supported by the Belgian Defence. P. Druyts and M. Acheroy are with the Royal Military Academy, Signal and Image Centre, Av. de la Renaissance 30, B-1000 Brussels, Belgium (e-mail: Pascal.Druyts@elec.rma.ac.be). Y. Das is with Defence R&D Canada – Suffield, Alberta, Canada. C. Craeye is with Universit´ e Catholique de Louvain, Electrical Engineering department, Place du Levant 3, B-1348 Louvain-la-Neuve, Belgium. c 2009 IEEE. Personal use of this material is permitted. However, per- mission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. detectors and, as a result, manufacturers provide most high- end MDs with soil compensation capabilities. Unfortunately, due to competition among manufacturers, details of the soil compensation techniques implemented are often proprietry, although some information is available in patents [9]. The soil can effect the target response and it can provide an additional contribution to the detector response [10]–[14]. Here we concentrate on the response of the soil in the absence of target. This response is of prime importance as it can be the source of false alarms or require the use of a lower detector sensitivity setting which in turn will reduce the target detectability. The soil response is influenced by many factors such as the current waveform in the Transmit (TX) coil, the detector head geometry, the coil electrical characteristics, the detector electronics and the soil electromagnetic properties. Some of these factors have been studied [12], [14]–[18], but many open questions still remain. For example, in many cases analytic models have been used and, as a result, the analysis was restricted to simple coil arrangements such as concentric circular coils, and to homogeneous soils with a flat air-soil interface. However, the shape and relative position of the coils have a major impact on detector performance which is indicated by the amount of different coil arrangements used in practice [19]. The soil relief and soil inhomogeneities also have a major impact on detector performance. We show that the soil response can be expressed for general head geometries and for general soil inhomogeneities and reliefs by resorting to a general form of reciprocity and we directly derive that reciprocity expression for the Magneto- Quasi-Static (MQS) regime relevant to MDs. The resulting expression for the soil response includes the magnetic field produced by the TX coil in the presence of soil. Computing that field for a general soil and for a general head geometry requires the use of computationally intensive numerical meth- ods. However, several publications [12], [17] suggest that for most soils of interest the response is mainly due to the soil magnetic susceptibility and therefore effects of soil conduc- tivity may be neglected. Furthermore, the magnetic suscepti- bilities of most natural soils are quite low [20]–[26]. In this paper we show that for such soils, with negli- gible electric conductivity and low magnetic susceptibility, major simplifications can be achieved by using the Born approximation. This also allows more realistic scenarios to