# Three-Dimensional Inversion of the Time-Lapse Resistivity Method on the MPI Parallel Algorithm
## 2 Rationale and Theory
### Three-Dimensional Time-Lapse Inversion Objective Function  
In this study, we use the regularized constrained inversion method, which adds model terms to the objective function to constrain, avoiding excessive data fitting and reducing the multiplicity of geophysical inversion solutions. The objective function of regularization constraint inversion is as follows:  
\[\mathbf{\sigma}(\mathbf{M}_{\mathbf{\sigma}})=\Phi_{d}(\mathbf{M}_{\mathbf{\sigma}})+\lambda_{m}^{ \sigma}\Phi_{m}(\mathbf{M}_{\mathbf{\sigma}}), \tag{8}\]  
The objective function consists of two parts: the data function \(\Phi_{d}(\mathbf{M}_{\mathbf{\sigma}})\) and the model function \(\Phi_{m}(\mathbf{M}_{\mathbf{\sigma}})\). \(\Phi_{d}(\mathbf{M}_{\mathbf{\sigma}})\) is used to calculate the difference between the observation data and the response data of the inversion result, \(\Phi_{m}(\mathbf{M}_{\mathbf{\sigma}})\) is used to calculate the smoothness of the model, and \(\lambda_{m}^{\sigma}\) is the weight factor of the model function, which is used to adjust the proportions of the data function and the model function in the objective function. By introducing the time-lapse function calculated by (6) into (8), the time-lapse inversion objective function can be obtained:  
\[\Phi(\mathbf{M}_{\mathbf{\sigma}})=\Phi_{d}(\mathbf{M}_{\mathbf{\sigma}})+\lambda_{m}^{\sigma }\Phi_{m}(\mathbf{M}_{\mathbf{\sigma}})+\lambda_{t}^{\sigma}\Phi_{t}(\mathbf{M}_{\mathbf{ \sigma}}), \tag{9}\]  
where \(\lambda_{t}^{\sigma}\) is the weight factor of the time-lapse function. In this study, we use the method of L-curves to choose the \(\lambda_{m}^{\sigma}\) and \(\lambda_{t}^{\sigma}\)[17], which can not only ensure the minimum of \(\Phi_{d}(\mathbf{M}_{\mathbf{\sigma}})\), but also ensure the minimum of \(\Phi_{m}(\mathbf{M}_{\mathbf{\sigma}})\) and \(\Phi_{t}(\mathbf{M}_{\mathbf{\sigma}})\). Expanding (9) yields the specific form of the time-lapse inversion objective function:  
\[\Phi(\mathbf{M}_{\mathbf{\sigma}})=\big{[}\mathbf{D}_{\mathbf{\rho},\mathbf{obs}}-\mathbf{F}(\mathbf{M}_{ \mathbf{\sigma}})\big{]}^{T}\mathbf{V}_{\mathbf{\sigma}}^{-1}\big{[}\mathbf{D}_{\mathbf{\rho},\bm {obs}}-\mathbf{F}(\mathbf{M}_{\mathbf{\sigma}})\big{]}+\lambda_{m}^{\sigma}(\mathbf{M}_{\mathbf{ \sigma}}-\mathbf{M}_{\mathbf{\sigma},0})^{T}\mathbf{C}_{\mathbf{M}\mathbf{\sigma}}^{-1}(\mathbf{M}_{ \mathbf{\sigma}}-\mathbf{M}_{\mathbf{\sigma},0})+\lambda_{t}^{\sigma}\mathbf{M}_{\mathbf{\sigma}} ^{T}\mathbf{C}^{T}\mathbf{C}\mathbf{M}_{\mathbf{\sigma}}, \tag{10}\]where \(D_{\mathbf{p},\mathbf{obs}}\) represents the apparent resistivity data observed at all times, \(\mathbf{F}(\mathbf{M_{\sigma}})\) represents the forward response function, \(\mathbf{M_{\sigma,0}}\) represents the initial model of inversion, and \(\mathbf{V_{\sigma}}\) represents the covariance matrix of the data. The gradient of the objective function can be obtained by calculating the partial derivative of the objective function with respect to \(\mathbf{M_{\sigma}}\):  
\[g(\mathbf{M_{\sigma}})=-2\mathbf{I_{\sigma}^{T}}\mathbf{V_{\sigma}^{-1}}\big{[}\mathbf{D_{\mathbf{ p},\mathbf{obs}}}-\mathbf{F}(\mathbf{M_{\sigma}})\big{]}+2\lambda_{m}^{\sigma}\mathbf{C}_{\mathbf{M _{\sigma}}}^{-1}(\mathbf{M_{\sigma}}-\mathbf{M_{\sigma,0}})+2\lambda_{t}^{\sigma}\mathbf{ M_{\sigma}^{T}}\mathbf{C^{T}}\mathbf{C}\mathbf{M_{\sigma}}, \tag{11}\]  
where \(J_{\sigma}\) represents the sensitivity matrix of the inversion. Owing to the complexity of the underground model and observation data, it is necessary to smooth the gradient of the objective function to improve the computational efficiency and stability of the inversion. [PERSON] and [PERSON] proposed a simple and efficient method to smooth the gradient of model parameters [18]. The specific transformation formula is derived as follows:  
\[