Theoretical Justification for the One-Dimensional Geolocation Method

In this article we will discus the usage feature of the ground penetrating radar (GPR) for the solution underground geotechnologies problems. One of the main problems by the usage GPR method is that the surface of the workings is shielded by metal elements of support (frames, fittings, tightening and other). In this article we suggest to use one- dimensional GPR-scanning method instead of traditional GPR-profiling method. We assume that the scanning will be performed on the development contour in areas free from shielding. For justification one- dimensional GPR method we propose a mathematical model for the propagation of an electromagnetic signal in an inhomogeneous medium based on classical equations of electrodynamics. We also present a numerical implementation of it, which confirms the validity of the accepted problem statement.


Introduction
Let us remind how a ground penetrating radar (GPR) works. A GPR unit consists of two objects: a transmitter and a receiver. The transmitter generate electromagnetic (EM) pulse which moves into the studied medium. If the medium contains objects (or layers) with different permittivity, then part of the wave will be reflected and move back to receiver. If we know the EM wave velocity and know the total travel time we can calculate the object distance. The GPR is applied in earth, geological, archaeology science. In each of these application the GPR unit used on the earth's surface and never the under the earth. The full use of GPR is not possible for the following reasons: lack of intrinsically safe equipment; screening of the work-out contour with metal support elements (frames, fittings, tightening); high level of electromagnetic interference in the workings due to the presence of cable power lines and electric power plants; lack of methods and software for interpretation of underground GPR profiles.
Encouraging results were obtained in IGD of North SB RAS [1][2][3][4][5][6][7][8][9]. It is proposed to use wavelet transformations that include analysis of the signal measured by the GPR at different frequencies and scales when interpreting geo-radar data [1]. This approach is implemented for the Triton GPR with a main frequency of 30 MHz. The developed theoretical foundations and physical modeling in the specified frequency range allowed us to implement this method in the study of a frozen rock mass, a distinctive feature of which is the filling of voids and pores of the rock with both ice that exhibits dielectric properties and an electrically conducting melt solution [2][3][4]. Also tests of the GPR OKO-2 with an antenna unit at a main frequency of 400 MHz were performed in the conditions of the Taldinskaya-Zapadnaya mine. Stratifications were recorded in the roof of the preparatory workings. However, it should be noted that when interpreting GPR profiles, the authors did not take into account the above factors that significantly affect the informative value of the GPR method [5,6]. In [7,8] the results of using the OKO-2 GPR with the AB-1200 antenna unit are presented. The main problem was the strong influence on the results of probing cavities filled with water with high conductivity. The use of a set of corrective operations (amplitude equalization, Hilbert transformation, horizontal filtering) allowed to increase the information content of monitoring. The authors of [9] proposed to evaluate the quality of compaction of soils by pressure injection of cement-containing rasvor to use integrated GPR profiles that allow monitoring the processing of the bases of technical structures and the degree of strength set by an artificial array according to the final numerical parameters determined using special algorithms.

Formulation of the problem
It should be noted that the development of the theoretical aspects of GPR is important not only from scientific but also from a practical point of view, as it will allow in the future to automate the processing of GPR.
Let us consider the following model problem. Let us assume that at the initial moment of time on the earth's surface, the GPR transmitter emits an electromagnetic signal that fades over time. The signal reflected from the boundaries of sections of geological media and various foreign inclusions is received and recorded by the GPR receiver. It is necessary to establish a pattern of propagation of an electromagnetic wave deep into space and track the time of return of the wave reflected from the boundaries of layers and inclusions. Next, we will assume that both the transmitter and receiver are located at the same point, that is, the wave propagation occurs along the same coordinate axis. Under this assumption, both geological layers and foreign inclusions will be displayed as one-dimensional segments that have certain physical properties. With this in mind, let's consider a simplified model of the geological environment in the form of a three-layer homogeneous isotropic space within each layer with flat borders. The parameters of the layers are known: h -permittivity, specific electrical conductivity, Sm/m, layer power, m (picture 1), where 3 h   .
It is known that the most general way to describe any electromagnetic processes is the system of Maxwell's equations, on the basis of which all possible special cases can be obtained.
Let us consider the system of Maxwell's equations where  -3d-density of third-party current, Кл/м Then, after simplifications and assuming that 1   , we obtain a partial differential equation of the form (3) In the case of one-dimensional vertical propagation of an electromagnetic wave, the stress vector will contain only one non-zero component: ) (4) Equation (4) Let us use the explicit calculation scheme with second-order accuracy of the "cross" type. Here are the difference formulas for partial derivatives of the wave equation (the lower index corresponds to the number of the spatial grid, and the upper index corresponds to the time): E E E (7) To demonstrate the results of using the constructed calculation schemes, let us consider the following test example. Note for readers that a simpler modification of this example is provided on the site www.geo-radar.ru, where a two-layer model was considered without taking into account the absorbing conditions at the lower boundary.
The emitted electromagnetic signal is set using the following formula To specify the electrical parameters of the layers, we will assume the following model scheme of the host space: the moistened second layer is located between two drier layers, but the third layer will be wetter than the first. Based on this, we assume that the relative permittivity k  and specific conductivities k  of layers are: 5 1   , 20 2   , 10 3   and 001 , 0 1   , 004 , 0 2   , 003 , 0 3   См/м. Figure 2 shows the results of calculations in the form of a map isolines of function ) , ( t z E E  and the graph received on the surface of the signal reflected from the interface of layers.

Conclusion
The considered test example shows the principle operability of the proposed design scheme: 1) the map of isolines shows graphs of reflected and refracted waves by the boundaries of layers; 2) the efficiency of absorbing boundary conditions is visible -there are no graphs of reflected waves from the lower border; 3) the graph of the received signal in the form of bursts shows reflected waves returned in the opposite phase from both layer boundaries.