## 2 EMTR and Correlation Estimator Methods
### Correlation-Estimator-Based Fault Location Method  
The correlation estimator method provides a quantitative, rather than a qualitative, interpretation. In addition, it is more of an active approach than the conventional EMTR, which can be characterised as reactive. Because there is no need to inject the time-reversed signals into the system and perform the large batches of simulations if we already have a database of the same system prior to the fault occurrence, this method is more computationally efficient. Mathematically, it can be written as follows:  
\[ \begin{split}\rho_{HH}(x_{g})=\frac{\langle H_{fos}(f)H_{fos}(fx_ {g})\rangle}{\sigma_{fos}\sigma_{fos}(x_{g})}\\ \sigma_{fos}{}^{2}=\left\langle\left|H_{fos}(f)\right|^{2}\right\rangle \\ \sigma_{fos}{}^{2}=\left\langle\left|H_{fos}(f;x_{g})\right|^{2} \right\rangle\end{split}  \tag{6}\]  
As depicted in Figure 1, when the fault occurs at some point along the line, \(H_{fos}\) is the system's single output/input response, known as the fault occurrence stage transfer function. In fact, \(V_{m}(t)\) is composed of \(V_{f}(t)\) convoluted with \(H_{fos}(t)\) in the time domain, where \(V_{f}(t)\) is modelled here as an exponential step function, which reads [8]:  
\[V_{f}(t)=V_{0}(1-\exp(-t/T_{r})) \tag{7}\]  
\(T_{r}\) and \(V_{0}\) are the rise time and peak voltage, respectively. The expression of \(V_{m}(t)\) due to a fault can be represented in the frequency domain mathematically as  
\[V_{m}(t)=\int_{BW}V_{f}(f)H_{fos}(f)e^{i2\pi ft}df \tag{8}\]  
\(V_{f}(f)\) is the fault voltage source generator in the frequency domain over the bandwidth BW, which can be defined as  
\[V_{f}(f)=V_{0}\frac{2\pi f_{c}}{\beta\pi f(2\pi f+2\pi f_{c})},\ \ f_{c}=\frac{1}{2\pi T_{r}} \tag{9}\]  
The analytical expression of \(H_{fos}\) for a single without the branch transmission line case can be obtained from the general solution of the voltage and current expressions, which was presented in detail in [23]. Hence, the \(H_{fos}\) in its mathematical form can be written as  
\[H_{fos}(f)=\frac{e^{-\gamma(x-1)}-\rho_{0}e^{-\gamma(x+1)}}{(22C(1-\rho_{0} \rho_{L}e^{-\gamma(x)})}\left[-\left(e^{\gamma(L-x_{f})}-\rho_{L}e^{-\gamma(L- x_{f})}\right)\right] \tag{10}\]  
\(\rho_{0}\) and \(\rho_{L}\) are the reflection coefficients at x = 0 and x = L. \(H_{fos}\) is the database of transfer functions computed at regular intervals along the line length. They can also be represented  
Figure 2: Single-conductor transmission line cross-section with equivalent circuit. analytically as (10). \(\sigma_{fos}\) and \(\sigma_{fcs}\) are the energy terms of the fault occurrence and fault estimation stage transfer functions, respectively. \(\rho_{HH}(x_{g})\) is the so-called correlation function, which provides the CCv corresponding to each GFL (\(x_{g}\)). The numerator in Equation (6) is the projection between the transfer functions of the fault occurrence and fault estimation stages, giving the measure of similarity, while the denominator provides the measure of intensity. Since the only information available is the fault-generated transient signal (\(V_{m}\)), to estimate the transfer function, the assumptions of \(V_{f}\) proposed in [8] are considered. \(H_{fcs}\), on the other hand, are already obtained by utilising the Baum-Liu-Tesche (BLT) formulation [24,25], which was developed in the MATLAB environment. To employ the BLT formulation, Figure 3 is taken as a reference, where JF is the fault junction, while JJ and J2 represent the terminal junctions. During the fault phenomena, the line is divided into two sections, shown as tubes P and Q.