# Control and Analysis of Layered Soil Structure to Ensure Railway Roadbed Stability
## 2 A Description of the Laboratory Experiment
The NPM is widely used in many software packages for analyzing electrical circuits, which provides a convenient and accessible platform for modeling geophysical processes. As a result, we can consider complex interactions between elements of the medium and, consequently, the possibility of studying various scenarios and impacts on the model. The representation of the transfer function of the studied section of the geological environment in the form of a geoelectric model of series-connected complex resistances [17] makes it possible to use the model of an N-layer imperfect dielectric. The given model contains \(N\) elements with layer thickness \(d\) and electrical parameters of the \(i\)-th element: dielectric constant, \(\varepsilon_{i}\), and specific electrical resistance, \(\rho_{i}\). The transfer function of a soil foundation fragment can be described by a set of series-connected RC (Resistance-Capacity) circuits with parameters as follows [17]:  
\[C_{i}=\varepsilon_{i}S(j\omega,\Delta u_{i})/d(\Delta u_{i}),\ R_{i}=\rho_{i}d( \Delta u_{i})/S(j\omega,\Delta u_{i}), \tag{1}\]  
where \(S(j\omega)\) is the effective area of a soil base fragment, determined considering the skin effect, d is the layer thickness, \(\varepsilon_{i}\) is the dielectric constant, and \(\rho_{i}\) is the specific electrical resistance. If we neglect the influence of grounding parameters, the transfer function of the geoelectric section can be equivalently represented by the model of a layered imperfect dielectric:  
\[\dot{H}(j\omega,\Delta u)=\sum_{i=1}^{N}\frac{R_{i}}{1+x_{i}^{2}}-j\sum_{i=1} ^{N}\frac{R_{i}x_{i}}{1+x_{i}^{2}}, \tag{2}\]  
where \(\mathrm{x_{i}}=\omega\mathrm{R_{i}C_{i}}=\omega\varepsilon_{i}\rho_{i}\). In the absence of defects and deformations in the system \"soil base--roadbed--railway track\", the geoelectric signal recorded by the geodynamic control system in operator form is defined as  
\[Y(p)=\left(\sum_{i=1}^{n}\prod_{j=1}^{m}[N_{ij}(p,\Delta u)]\right)X(p), \tag{3}\]  
where \(n\) is the number of horizontal layers, and \(m\) is the number of vertical sections in each layer. Changing the characteristics of the layers of the geological section causes variations in the transfer function \(H(p)\) and changes the recorded response:  
\[\overset{\sim}{Y}(p)=Y(p)+\Delta Y(p)=\left(\sum_{i=1}^{n}\prod_{j=1}^{m}[N_{ ij}(p,\Delta u)+\Delta N_{ij}(p,\Delta u)]\right)X(p) \tag{4}\]  
From this expression, it is evident that the amplitude-phase characteristics of the recorded geoelectric signals will contain complete information about the transfer function and characteristics of the hydrogeological environment section. At the same time, from the theory and experience of practical use [18], it is known that the control of phase characteristics will have greater sensitivity, accuracy, and noise immunity, which makes it possible to diagnose geodynamic indicators of the initial stage of occurrence of defects and deformations of the roadbed of the railway. To assess the prospects of using the phaseometric method in the task of obtaining information about the properties of the soil, laboratory modeling of the natural-technical system \"soil-probing drill\" was carried out. For this purpose, a laboratory setup was created, the diagram of which is shown in Figure 1. Electrodes A and B are for input signals, with a phase difference of 90\({}^{\circ}\), and electrodes M\({}_{1}\), M\({}_{2}\), M\({}_{3}\), N\({}_{1}\), N\({}_{2}\), and N\({}_{3}\) are for the resulting output signals.