## 2 Materials and Methods
### Apparent Resistivity Tensor Analysis
#### 2.1.2 The Tensor Invariant Departure Concept  
The subsoil is supposed to be composed of Q elementary cells with constant resistivities \(\rho_{q}\) (\(q=1\),..., \(Q\)) and volume \(\Delta V\) in accordance with [14]. When \(\mathbf{\rho}^{(a)}\) is expanded in a Taylor series, the following expression is obtained:\[\Delta\mathbf{\rho}^{(a)}=\mathbf{\rho}^{(a)}-\mathbf{\rho}_{0}^{(a)}=\sum_{q=1}^{Q}\frac{ \partial\mathbf{\rho}_{0}^{(a)}}{\partial\mathbf{\rho}_{0,q}}\Delta\mathbf{\rho}_{q}+\sum( \textit{higher}-\textit{order derivatives}), \tag{5}\]  
where \(\Delta\mathbf{\rho}^{(a)}\) is the deviation of \(\mathbf{\rho}^{(a)}\) from the apparent resistivity tensor \(\mathbf{\rho}_{0}^{(a)}\) of a reference resistivity model, \(mod_{0}\). Thus, \(\Delta\mathbf{\rho}_{q}\) is the difference between the actual resistivity \(\rho_{q}\) and the resistivity \(\rho_{q}\) at \(mod_{0}\) in the \(q\)-th cell.  
Using Equation (3), the tensor invariant departure \(\Delta\mathbf{P}\) of the real tensor invariant \(\mathbf{P}\) from the tensor invariant \(P_{0}\) associated with \(mod_{0}\) is easily derived, since the trace of a sum of matrices is equal to the total of the traces of the single matrices:  
\[\Delta P=P-P_{0}=\sum_{q=1}^{Q}\frac{\partial P_{0}}{\partial P_{0,q}}\Delta \rho_{q}+\sum(\textit{higher}-\textit{order derivatives}) \tag{6}\]  
Using Equation (3), the term \(\frac{\partial P_{0}}{\partial P_{0,q}}\) is calculated as follows:  
\[\Delta\frac{\partial P_{0}}{\partial P_{0,q}}=\frac{\left(\frac{\partial E_{0,\,1x}}{\partial\mathbf{\rho}_{0,q}}\right)J_{2y}-\left(\frac{\partial E_{0,\,2x} }{\partial\mathbf{\rho}_{0,q}}\right)J_{1y}+\left(\frac{\partial E_{0,\,2y}}{ \partial\mathbf{\rho}_{0,q}}\right)J_{1x}-\left(\frac{\partial E_{0,\,1y}}{ \partial\mathbf{\rho}_{0,q}}\right)J_{2x}}{2\left(J_{1x}J_{2y}-J_{2x}J_{1y}\right)}. \tag{7}\]  
The Frechet derivative of the electric potential for the uniform half-space can be used to analytically compute \(\frac{\partial P_{0}}{\partial P_{0,q}}\), for simplification and without compromising generality, making the assumption that \(mod_{0}\) is a uniform and isotropic half-space [21; 22]. At a point \(\mathbf{r}\equiv(x,y,0)\) on the ground's surface, the electrical potential \(\phi_{i}\) varies because of a slight change in the resistivity in a volume \(\Delta\)V submerged in a uniform half-space around the point \(\mathbf{r}_{q}\equiv\left(x_{q},\ y_{q},x_{q}\right)\), and it is expressed as  
\[\Delta\frac{\partial\phi_{i}}{\partial\rho_{0,q}}=\frac{\partial\phi_{A_{i}}}{ \partial\rho_{0,q}}+\frac{\partial\phi_{AB_{i}}}{\partial\rho_{0,q}}, \tag{8}\]  
in which  
\[\Delta\frac{\partial\phi_{A_{i}}}{\partial\rho_{0,q}}=\frac{I_{i}}{4\pi^{2}} \frac{\mathbf{r}_{A_{i}}-\mathbf{r}_{q}}{\left|\mathbf{r}_{A_{i}}-\mathbf{r}_{q}\right|^{3}}- \frac{\mathbf{r}-\mathbf{r}_{q}}{\left|\mathbf{r}-\mathbf{r}_{q}\right|^{3}}, \tag{9a}\] \[\frac{\partial\phi_{B_{i}}}{\partial\rho_{0,q}}=\frac{I_{i}}{4\pi^{2}}\frac{ \mathbf{r}_{B_{i}}-\mathbf{r}_{q}}{\left|\mathbf{r}_{B_{i}}-\mathbf{r}_{q}\right|^{3}}-\frac{ \mathbf{r}-\mathbf{r}_{q}}{\left|\mathbf{r}-\mathbf{r}_{q}\right|^{3}}. \tag{9b}\]  
Finally, the formulas are used to calculate the Frechet derivatives of the electrical field components from Equations (9a) and (9b), omitting the straightforward but laborious mathematical stages:  
\[\frac{\partial E_{0,ix}}{\partial\rho_{0,q}}=\frac{\partial}{ \partial x}\left(\frac{\partial\phi_{A_{i}}}{\partial\rho_{0,q}}+\frac{ \partial\phi_{B_{i}}}{\partial\rho_{0,q}}\right), \tag{10a}\] \[\frac{\partial E_{0,iy}}{\partial\rho_{0,q}}=\frac{\partial}{ \partial y}\left(\frac{\partial\phi_{A_{i}}}{\partial\rho_{0,q}}+\frac{ \partial\phi_{B_{i}}}{\partial\rho_{0,q}}\right). \tag{10b}\]