## 2 Piecewise Regularized Inversion Method  
Nonlinear inversion involves an iterative search for the optimal solution based on a strict transient electromagnetic forward formula, objective function, and initial model, following the search principle of the algorithm and the objective function [23]. The regularized objective function for transient electromagnetic data inversion includes fitting and roughness constraint functions for models. The expression is as follows:  
\[F=\phi_{d}+\lambda\psi_{m} \tag{1}\]  
where \(\phi_{d}\) represents the data fitting functional, \(\psi_{m}\) represents the roughness constraint function, and \(\lambda\) denotes the regularization factor. The appropriate data function is constructed using the L1 norm, which has both advantages and disadvantages compared to the L2 norm. While the L2 norm offers smooth and continuous characteristics, the L1 norm is more tolerant of extreme fitting errors and is more effective in describing layered boundaries [24]. The formula is as follows:  
\[\phi_{d}=\sum_{i=1}^{N}\Bigl{|}\Bigl{(}D_{i}^{obs}-G_{i}(m)\Bigr{)}/D_{i}^{ obs}\Bigr{|}/N \tag{2}\]  
the model constraint function imposes constraints on the conductivity change rate of each layer in the vertical direction, expressed as follows:  
\[\psi_{m}=\sum_{i=1}^{N-1}{(m_{i+1}-m_{i})^{2}}/h_{i}(N-1) \tag{3}\]  
where \(D^{obs}\) is the observation data vector; \(N\), \(m\), \(h\) represent the number of layers, resistivity parameters, and layer thickness parameters of the prediction model, respectively; and \(G\) is the forward operator. The initial model is established based on prior information, including model parameters such as the resistivity and thickness of each layer. The initial model is constructed using available a priori information, which is typically obtained from conductivity-depth imaging methods or borehole data. For instance, consider a low-resistivity anomaly with a small thickness in a specific geological structure, as shown in Figure 1. Figure 2 illustrates different methods for constructing the initial model. Method 1 involves the direct establishment of the initial model; however, when information about thin layers with low resistivity is not readily available from prior information, the model must use the parameters of three layers to fit the geological structure of five layers, resulting in poor fitting performance. Method 2 assigns a relatively large number of layers in the initial model, which mitigates the impact of parameter uncertainty on the inversion results to some extent; however, increasing the number of layers exacerbates the non-uniqueness of the inversion and makes it challenging to discern the model results, posing new challenges. To address the issues with existing methods, this paper proposes a new method for designing the initial model, consisting of two main parts. First, based on the initial model established using Method 1, each layer except the bottom layer is further subdivided, and the model parameters are expressed as follows:  
\[m_{i}=[\rho_{i,1},\rho_{i,2},...\rho_{i,l_{i}}],h_{i}=[\delta_{i,1},\delta_{i,2},...\delta_{i,l_{i}}],0<i<N\]  
\[h_{i,\min}\leq(\delta_{i,1}+\delta_{i,2}+...+\delta_{i,\mathrm{J}})\leq h_{i, \max}\]  
\[m_{i,\min}\leq(\rho_{i,1},\rho_{i,2},...,\rho_{i,\mathrm{J}})\leq m_{i,\max} \tag{4}\]  
where \(N\) denotes the number of segments divided according to the a priori information; \([h_{i,\min},h_{i,\max}]\) and \([m_{i,\min},m_{i,\max}]\) are the upper and lower bounds of the layer thickness and resistivity in the \(i\)-th segment, respectively; and \(J_{i}\) is the number of inner layers divided in each segment. \(\left[\rho_{i,1},\rho_{i,2},...,\rho_{i,l_{i}}\right]\) and \(\left[\delta_{i,1},\delta_{i,2},...,\delta_{i,l_{i}}\right]\), respectively, represent the resistivity parameters and layer thickness parameters after subdividing \(m\) into \(J_{i}\) layers.