## 4 Lateral Constraint Inversion Theory
### Objective Function  
The data utilized in this study comprise induced electromotive force measurements, with the observation data at the \(i\)th measurement point represented in vector form as follows:  
\[\mathbf{d}_{\text{obs},i}=\left[d_{i,1},d_{i,2},\dots,d_{i,i},\dots d_{i,N_{c}} \right]^{\text{T}} \tag{1}\]  
Let \(d_{i,j}\) denote the electromagnetic response of the \(i\)th measurement point at \(j\)th time channel, where \(N_{c}\) represents the total number of time channels. If the number of survey points to be inverted is denoted as \(Ns\), the observation data \(\mathbf{d}_{\text{obs}}\) for the entire survey line can be expressed collectively as follows:  
\[\mathbf{d}_{\text{obs}}=[\mathbf{d}_{\text{obs},1},\mathbf{d}_{\text{obs},2},\dots,\mathbf{d }_{\text{obs},i},\dots\mathbf{d}_{\text{obs},N_{i}}]^{\text{T}} \tag{2}\]  
The OCCAM inversion principle is applied by representing the model parameters for the \(i\)th measuring point in vector notation:  
\[\mathbf{m}_{i}=[\rho_{i,1},\rho_{i,2},\dots,\rho_{i,j},\dots,\rho_{i,N},h_{i,1},h _{i,2},\dots,h_{i,j},\dots,h_{i,N-1}]^{\text{T}} \tag{3}\]  
Let \(N\) denote the number of model layers \(\rho_{i,j}\) and \(h_{i,j}\) represent the earth resistivity and thickness of the \(j\)th layer in the model at the \(i\)th measurement point. Let all inversion parameters be denoted as follows:  
\[\mathbf{m}=[\mathbf{m}_{1},\mathbf{m}_{2},\dots,\mathbf{m}_{i,},\dots,\mathbf{m}_{N_{i}}]^{\text{ T}} \tag{4}\]  
Figure 3: Construction layout of TEM.  
Construct the objective function as follows:  
\[\varphi(\mathbf{m})=\lambda\varphi_{m}(\mathbf{m})+\varphi_{d}(\mathbf{m}) \tag{5}\]  
Here \(\varphi(\mathbf{m})\) represents the objective function, \(\lambda\) the regularization factor, \(\varphi_{m}(\mathbf{m})\) the model constraint objective function, \(\varphi_{d}(\mathbf{m})\) the observed data objective function, and \(\mathbf{m}\) the model vector (i.e., resistivity).  
The observed data objective function \(\varphi_{d}(\mathbf{m})\) can be expressed as follows:  
\[\varphi_{d}(\mathbf{m})=\|\mathbf{W}(\Delta\mathbf{d})\|^{2}\approx\|\mathbf{W}[\mathbf{d}-F(\mathbf{ m})-\mathbf{J}\Delta\mathbf{m}]\|^{2} \tag{6}\]  
The discrepancy between the actual data and the modeled data response is denoted as \(\Delta d\), the weighted roughness matrix is represented as \(\mathbf{W}\), the forward operator is denoted as \(F\), and the Jacobian matrix, which is the partial derivative matrix of the forward response with respect to resistivity, is denoted as \(\mathbf{J}\). In the context of the transient electromagnetic system, the components in the \(i\)th row and \(j\)th column of the Jacobian matrix at the \(k(k=1,2,\ldots,N)\)th measurement point along the direction of the measurement line can be mathematically formulated as follows:  
\[j_{i,j}^{k}=\frac{\partial d_{k}(i)}{\partial m_{k}(j)} \tag{7}\]  
The model constraint objective function \(\varphi_{m}(\mathbf{m})\) in Equation (5) can be expressed as follows:  
\[\varphi_{m}(\mathbf{m})=\|\mathbf{R}_{m}\mathbf{m}\|^{2} \tag{8}\]  
Here, \(\mathbf{m}\) represents the model vector, \(\mathbf{R}_{m}\) denotes the roughness matrix of a single measurement point, and the model's first-order roughness is utilized, expressed as follows:  
\[\mathbf{R}_{m}=\begin{bmatrix}0&&&&0\\ -1&1&&&\\ &-1&1&&\\ &&\ddots&\ddots&\\ 0&&&-1&1\end{bmatrix} \tag{9}\]  
The lateral constraint inversion method utilized in this study involves inverting the amalgamation of multiple measurement points, necessitating the integration of the roughness matrix from an individual measurement point. The comprehensive roughness matrix is presented below.  
\[\mathbf{R}_{mz}=\begin{bmatrix}\mathbf{R}_{m}&&&&0\\ &\mathbf{R}_{m}&&\\ &&\ddots&\\ 0&&&\mathbf{R}_{m}\end{bmatrix} \tag{10}\]  
The quantity of elements within the matrix \(\mathbf{R}_{m}\) corresponds to the total measurement points utilized for global inversion.