Ground Penetrating Radar
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Frequency domain deterministic-stochastic analysis of the transient current induced along a ground penetrating radar dipole antenna over a lossy half-space
Anna Šušnjara, Dragan Poljak, Vicko Dorić, Sébastien Lalléchère, Khalil El Khamlichi Drissi, Pierre Bonnet, and Françoise Paladian
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Abstract: This paper deals with the stochastic analysis of transient current induced along a ground penetrating radar (GPR) antenna. The antenna is modelled as a horizontal dipole and is placed over a lossy half-space. The electromagnetic formulation of the problem is based on the Pocklington’s integro-differential equation in the frequency domain, which is solved by means of the Galerkin-Bubnov indirect boundary element method. The transient solution is obtained by using the inverse fast Fourier transform. The paper aims to investigate the variability of the current due to key uncertain parameters, such as the soil permittivity and conductivity, and the wire distance from the half-space. Stochastic assumptions are incorporated in the model by means of the stochastic collocation technique. Computational examples present the mean value of current distributed along the wire with the confidence margins. Sensitivity analysis is obtained, i.e., the uncertainty in the output is apportioned to different sources of uncertainty in the model input thus giving a better insight into model reliability.
Keywords: Ground Penetrating Radar (GPR); electromagnetic modelling; Galerkin-Bubnov indirect boundary element method; stochastic collocation technique; antennas; lossy half-space.
Ground penetrating radar (GPR) is used in civil engineering, archaeology, and many other areas. GPR antennas are moved over the surface of the inspected soil or structure, while emitting and receiving electromagnetic (EM) waves. In order to extract accurate and useful information from the received EM field, it is important to have as much a priori information as possible . Such information includes a good understanding of the electromagnetic properties of the involved media and used antennas . However, the knowledge about these properties is inevitably stochastic in its nature.
Many researchers have studied the EM behaviour of GPR antennas, by using different techniques that can be classified in two main categories: frequency domain (FD)  and time domain (TD) - techniques. A stochastic analysis of the transient response of a GPR antenna has been presented in -. In  the unknown current along the wire above the lossy-half space is governed by the space-time Hallen integral equation. The deterministic solution is featured by GB-IBEM method. The stochastic response is obtained with respect to uncertain antenna position (height) and uncertain ground conductivity. The work done in  and  present the stochastic current response for the wire buried in the lossy ground which may be found useful not only in GPR purposes but in other areas, for example in the design of lighting protection for electrical settlements.
As a counterpoise to time domain analysis, the stochastic analysis of frequency domain response is presented in the present paper. Stochastic Collocation (SC) method is combined with a direct EM solver to assess the variability of the current induced on a GPR dipole antenna, due to the uncertain nature of the soil and antenna height. The dipole is assumed to be thin and is placed above a lossy half-space, with its axis parallel to the air-soil interface: such simple geometry is especially convenient for testing new computational approaches and methods. The formulation of the problem, implemented in our deterministic EM solver, is based on a FD solution of Pocklington's integro-differential equation, by means of Galerkin-Bubnov Indirect Boundary Element Method (GB-IBEM) ; the transient response is then obtained via inverse Fast Fourier's transform .
The paper is organized as follows. Section 2 outlines the employed FD integral equation approach and related numerical solution (Sub-section 2.1); the theoretical basis of the Stochastic Collocation method are also presented (Sub-section 2.2). Section 3 brings computational examples, while in Section 4 general conclusions are given.
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Unrestricted use, distribution, and reproduction in any medium of this article is permitted, provided the original article is properly cited. Please cite this article as follows: A. Šušnjara, D. Poljak, V. Dorić, S. Lalléchère, K. El Khamlichi Drissi, P. Bonnet, and F. Paladian, "Frequency domain deterministic-stochastic analysis of the transient current induced along a ground penetrating radar dipole antenna over a lossy half-space," Ground Penetrating Radar, Volume 1, Issue 2, Article ID GPR-1-2-2, July 2018, pp. 37-51, doi.org/10.26376/GPR2018008.
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