## 3 Fundamentals
### Resistivity Inversion Theory  
Assume that the apparent resistivity inversion model comprises several rectangular blocks with a constant resistivity, and the apparent resistivity of each rectangular block can be determined via the iterative nonlinear optimization method. In this paper, the apparent resistivity value was inverted via the least square method under the smooth restricted condition, as expressed in Equation (7) [26].  
\[(J^{T}J+\lambda W^{T}W)\ \widetilde{P}=J^{T}\widetilde{g} \tag{7}\]  
where \(J\) is the Jacobian partial derivative matrix, \(\lambda\) is the damping coefficient, \(W\) is the smoothing filter parameter matrix and \(\widetilde{P}\) is the correction vector of model parameters, \(\widetilde{g}\) is the deviation vector of the logarithmic difference between the measured apparent resistivity and the calculated apparent resistivity.  
The time-delay resistivity inversion function describes the change in the geoelectric parameters of the underground medium at the same location at different times through time constraints, which can highlight the deformation and failure characteristics of the underground medium. The inversion results of the initial resistivity data were used as a reference for the subsequent inversion, and regularization in the space and time domains was introduced to improve the inversion speed and imaging accuracy [27]. The inversion objective function can be expressed as Equation (8), and the solution of the set of linear equations of the least squares method can be obtained as:  
\[\left(J_{i}^{T}R_{d}J_{i}+\lambda_{i}W^{T}R_{m}W\right)\ \Delta r_{i}^{k}=J_{i}^{T}R_{d}g_{i}- \lambda_{i}W^{T}R_{m}Wr_{i-1}^{k}-\beta_{i}\lambda_{i}W^{T}R_{i}V\left(r_{i-1} ^{k}-r_{i-1}^{0}\right) \tag{8}\]  
where \(J_{i}\) is the Jacobian partial derivative matrix, \(i\) is the number of iterations, \(g_{i}\) is the logarithmic difference matrix between the measured apparent resistivity and the calculated apparent resistivity, \(W\) is the smoothing filter parameter matrix, \(\lambda\) is the damping coefficient, \(R_{d}\) and \(\mathrm{R_{m}}\) are the weight matrices, \(r_{i-1}^{0}\) and \(r_{i-1}^{k}\) is the initial value of the model and the k-th time value, \(\Delta r_{i}^{k}\) is the model residual of the \(i\)th iteration, V is the weight matrix between models at different times and \(\beta\) is the weight coefficient [28].  
[PERSON] [29] introduced the residual value \(\Delta\rho\) and resistivity reflection coefficient R to explain the change in the underground medium, as expressed in Equations (9) and (10), respectively. When \(\Delta\rho\) is positive, the higher the value is, the higher the degree of crack development. When \(\Delta\rho\) is negative, the higher the value is, the higher the water content in the medium. For \(\rho_{0}\rightarrow\infty\) and \(\mathrm{R}\rightarrow-1\), the current tends to occur within the medium \(\rho_{1}\) range, while an increase in the \(\left|\mathrm{R}\right|\) value indicates an increase in the water content in the medium. For \(\rho_{0}\to 0\) and \(\mathrm{R}\to 1\), the current tends to occur within the medium \(\rho_{0}\) range, while an increase in the \(\left|\mathrm{R}\right|\) value indicates that the water content in the medium decreases or cracks develop.  
\[\Delta\rho=\rho_{i}-\rho_{0} \tag{9}\]  
\[R=\frac{\rho_{i}-\rho_{0}}{\rho_{i}+\rho_{0}}100\% \tag{10}\]  
where \(\rho_{i}\) and \(\rho_{0}\) are the resistivity value at time i and reference time, respectively.