## 2 Spatial Time Domain Reflectometry  
### Basic Principles of Time Domain Reflectometry  
Time domain reflectometry (TDR) has become an important monitoring technique for various civil engineering problems [21; 22; 23; 24]; the main application is the monitoring of moisture content for porous media [25; 26; 27; 28]. A TDR measurement system is composed of a pulse generator combined with an oscilloscope and a transmission line system (see Figure 1a). The transmission line includes a leading coaxial cable and a sensing waveguide, called the TDR probe, which is embedded in the medium for characterization. The characteristics (length and geometry) of the probe depend on the application. The classical geometry remains in the rod configuration. The pulse generator delivers an electromagnetic pulse along the transmission line, and the oscilloscope records returning reflection. Reflection will occur at any impedance discontinuities along the transmission line. In particular, the pulse will be reflected at the beginning and end of the TDR probes. The travel time analysis between these two reflections can be used to determine the flight time \(\Delta t\) of the pulse to travel along the probe. This flight time is generally computed based on the dual tangent method [29], where the reflection arrival is located at the intersection of the two tangents to the TDR curve (see Figure 1b). Finally, the apparent permittivity \(\varepsilon_{app}\) of the medium can be determined with the well-known relation:  
\[\varepsilon_{app}=\left(\frac{c_{0}\Delta t}{2L_{p}}\right)^{2} \tag{1}\]  
where \(c_{0}\) is the speed of the light and \(L_{p}\) is the length of the probe. Please note that for most porous media, the permittivity is complex and frequency-dependent (dispersion). The term apparent permittivity is used here because the time-domain method does not intend to compute the complete frequency-dependent complex permittivity.  
In a second step, the apparent permittivity is linked to the parameter of interest--in most cases, water content, \(\theta\). The most frequent method is to use the empirical equation. The classical example is the famous Topp equation [30]:  
\[\theta_{Topp}=-5.3\times 10^{-2}+2.92\times 10^{-2}\varepsilon_{app}-5.5\times 10^ {-4}\varepsilon_{app}{}^{2}+4.3\times 10^{-6}\varepsilon_{app}{}^{3} \tag{2}\]  
The accuracy of such an empirical model highly depends on the nature of the soil related to density, the salinity of soil water, mineral composition, etc. For example, [PERSON]'s equation is not suitable for high swelling clay mineral [31]. An alternative to this solution is to derive a material-specific calibration functions [32].  
Another approach relies on a physical model based on mixture equations [33; 34]. Mixing models consider the soil as a medium that is composed of different phases. The relative permittivity of the mixture \(\varepsilon_{m}\) is the result of the sum of the dielectric properties of each phase multiplied by its volume fraction:  
\[\varepsilon_{m}^{a}=\sum_{k=1}^{N}V_{k}.\varepsilon_{r,k}^{a} \tag{3}\]  
where \(N\) is the number of phases, \(a\) is a structure parameter, \(V_{k}\) is the volume fraction and \(\varepsilon_{r,k}\) is the relative permittivity of the solid phase. The classical configuration for the mixture equation is the three-phase mixtures (air-solid-liquid) with a structure index equal to 0.5, called the Complex Reflective Index Model (CRIM) [35]. In this configuration, \(\varepsilon_{m}\) can be expressed directly as a function of water content [36]. The main source of interest of such models is the versatility: they can take into account the temperature and frequency dependence of the single components. The main disadvantage of such a method is the lack of knowledge and understanding of the interaction between single components.