## 3 Three-Dimensional Joint Inversion Algorithm
### Three-Dimensional Joint Inversion Objective Function  
The strategy of constructing two joint inversion objective functions was adopted [22], in which the conductivity and polarizability models are updated alternately. The weighted cross-gradient constraint term was added into both objective functions. The objective function of the joint inversion is composed of the data misfit term, the model smoothing constraint and the cross-gradient constraint. The objective functions of the resistivity method and TDIP inversion are defined as follows:  
\[\left\{ \begin{array}{l}\Phi(m_{\sigma})=\Phi_{d}(m_{\sigma})+\lambda_{ \sigma}\Phi_{m}(m_{\sigma})+\beta_{\sigma}\Phi_{cg}(m_{\sigma},m_{\eta})\\ \Phi(m_{\eta})=\Phi_{d}(m_{\eta})+\lambda_{\eta}\Phi_{m}(m_{\eta})+\beta_{ \eta}\Phi_{cg}(m_{\eta},m_{\sigma})\end{array} \right. \tag{11}\]  
where \(m_{\sigma}\) and \(m_{\eta}\) are the natural logarithm of the conductivity model and polarizability model, respectively, which can avoids negative conductivity or polarizability during inversion; \(\Phi_{d}\) is the data misfit term, \(\Phi_{m}\) is the model smoothness constraint, \(\Phi_{cg}\) is the cross-gradient constraint; \(\lambda_{\sigma}\) and \(\lambda_{\eta}\) are the regularization weighting factors of the model smoothness constraints, and \(\beta_{\sigma}\) and \(\beta_{\eta}\) are the regularization weighting factors of the cross gradients. \(\Phi_{d}\), \(\Phi_{m}\) and \(\Phi_{cg}\) were expanded to obtain the specific form of the objective function:  
\[\left\{ \begin{array}{l}\Phi(m_{\sigma})=\Big{[}d_{\rho,obs}-F(e^{m_{\sigma} })\Big{]}^{T}V_{\rho}^{-1}\Big{[}d_{\rho,obs}-F(e^{m_{\sigma}})\Big{]}+\lambda _{\sigma}(m_{\sigma}-m_{\sigma,0})^{T}C_{m,\sigma}{}^{-1}(m_{\sigma}-m_{\sigma,0})+\beta_{\sigma}t^{T}(m_{\sigma},m_{\eta})t(m_{\sigma},m_{\eta})\\ \Phi(m_{\eta})=\Big{[}d_{\eta,obs}-F(e^{m_{\eta}})\Big{]}^{T}V_{\eta}^{-1} \Big{[}d_{\eta,obs}-F(e^{m_{\eta}})\Big{]}+\lambda_{\eta}(m_{\eta}-m_{\eta,0}) ^{T}C_{m,\eta}{}^{-1}(m_{\eta}-m_{\eta,0})+\beta_{\eta}t^{T}(m_{\eta},m_{\eta}) \mu(m_{\sigma},m_{\eta})\end{array} \right., \tag{12}\]where \(d_{\rho,obs}\) and \(d_{\eta,obs}\) are the apparent resistivity and apparent polarizability data, respectively; \(F(m)\) is the forward response; \(V_{e}\) and \(V_{\eta}\) are the data covariance matrices, mainly to avoid overfitting the observed data with poor quality in the inversion process; \(C_{m,\sigma}{}^{-1}\) and \(C_{m,\eta}{}^{-1}\) are the model covariance matrices of conductivity and polarizability, respectively. The partial derivative of the joint objective function with respect to model \(m\) is as follows:  
\[\left\{ \begin{array}{l}\frac{\partial\Phi(m_{e})}{\partial m_{e}}=-2J_{\sigma }^{T}V_{\sigma}^{-1}\Big{[}d_{\rho,obs}-F(e^{m_{e}})\Big{]}+2\lambda_{e}C_{m, \sigma}{}^{-1}(m_{\sigma}-m_{\sigma,0})+2\beta_{\sigma}B_{\sigma}^{T}B_{\sigma} (m_{\sigma}-m_{\sigma,0})\\ \frac{\partial\Phi(m_{\eta})}{\partial m_{\eta}}=-2J_{\eta}^{T}V_{\eta}^{-1} \Big{[}d_{\eta,obs}-F(e^{m_{\eta}})\Big{]}+2\lambda_{\eta}C_{m,\eta}{}^{-1}(m _{\eta}-m_{\eta,0})+2\beta_{\eta}B_{\eta}^{T}B_{\eta}(m_{\eta}-m_{\eta,0})\\ \end{array} \right., \tag{13}\]  
where \(J_{\sigma}\) and \(J_{\eta}\) are the sensitivity matrices of conductivity inversion and polarizability inversion, respectively, Ma derived the calculation of the sensitivity matrices of the resistivity method and TDIP [18]. [PERSON] and [PERSON] proposed a simple and efficient method to smooth the gradient of model parameters [23]. First, an affine linear transformation of model parameters is defined:  
\[\widetilde{m}=C_{m}{}^{-1/2}(m-m_{0}), \tag{14}\]  
Equation (13) can be rewritten as follows:  
\[\left\{