## 2 Methodology
### DL-ERT Inversion  
The deep-learning network for electrical resistivity tomography (DL-ERT) inversion comprised a base model and a borehole mixer. The base model was designed to perform pure ERT inversion using apparent resistivity data, \(\mathbf{d}\), for the direct estimation of the resistivity model, \(\mathbf{m}\), as follows:  
\[\hat{\mathbf{m}}=Base(\mathbf{d};\theta), \tag{4}\]  
where \(\theta\) denotes the model parameters of the base network, \(Base()\), and \(\hat{\mathbf{m}}\) is the predicted resistivity model of the base network. Note that the base model does not require borehole information. As illustrated in Figure 6, the base model employs a modified U-Net structure, which is a simplified version of the existing U-Net [32]. The inputs \(\mathbf{d}\) and labels \(\mathbf{m}\) of this network are pseudosections of apparent resistivity and the corresponding 2D resistivity models, respectively, each having a size of 64 \(\times\) 128. In the encoding path of the base model, the first convolution block is composed of two 3 \(\times\) 3 convolutional layers and one 2 \(\times\) 2 max-pooling layer. Other convolution blocks comprise three 3 \(\times\) 3 kernel-size convolutional layers and one 2 \(\times\) 2 kernel-size max-pooling layer. Batch normalization is applied after each convolutional layer to increase the learning speed and stabilize the learning process [33]. The rectified linear unit (ReLU) activation functions [34] are added for the convolutional layers. The decoder is symmetrical to the encoder, and the transposed convolutional layers are used to perform up-sampling in the decoding path [35]. Skip connections are used to prevent the loss of data information by directly connecting the encoder layer to the decoder layer. The detailed structure of the base model is presented in Table 1.  
Figure 5: (a) Pseudosection’s setting and observation points in pseudo-elevation for dipole–dipole array, and (b) corresponding pseudosection of apparent resistivity data. The borehole mixer is designed to incorporate borehole information into the network to improve the inversion results. A borehole feature map, \(\mathbf{b}\), is created by adding the true \begin{table}
\begin{tabular}{c c} \hline \hline
**Layer** & **Output Shape** \\ \hline Input\_1 & (64,128,1) \\ Conv2d & (64,128,64) \\ Conv2d\_1 & (64,128,64) \\ Max\_pooling2d & (32,64,64) \\ Conv2d\_2 & (32,64,128) \\ Conv2d\_3 & (32,64,128) \\ Conv2d\_4 & (32,64,128) \\ Max\_pooling2d\_1 & (16,32,128) \\ Conv2d\_5 & (16,32,256) \\ Conv2d\_6 & (16,32,256) \\ Conv2d\_7 & (16,32,256) \\ Max\_pooling2d\_2 & (8,16,256) \\ Conv2d\_8 & (8,16,512) \\ Conv2d\_9 & (8,16,512) \\ Conv2d\_10 & (8,16,512) \\ Conv2d\_transpose & (16,32,256) \\ Concatenate & (16,32,512) \\ Conv2d\_11 & (16,32,256) \\ Conv2d\_12 & (16,32,256) \\ Conv2d\_13 & (16,32,256) \\ Conv2d\_transpose\_1 & (32,64,128) \\ Concatenate\_1 & (32,64,256) \\ Conv2d\_14 & (32,64,128) \\ Conv2d\_15 & (32,64,128) \\ Conv2d\_16 & (32,64,128) \\ Conv2d\_transpose\_2 & (64,128,64) \\ Concatenate\_2 & (64,128,128) \\ Conv2d\_17 & (64,128,64) \\ Conv2d\_18 & (64,128,64) \\ Conv2d\_19 & (64,128,1) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Base model structure. Figure 6: DL-ERT inversion network comprising a base model and a borehole mixer. resistivity values at the borehole locations to a map of the same size as the output of the base model, \(\hat{m}\), setting all other parts except the borehole locations to zero, as shown in Figure 7. Then, the resistivity values of the output of the base model, \(\hat{m}\), are replaced with the true resistivity values of the borehole, and the replaced output and the borehole feature map, \(\mathbf{b}\), are concatenated and fed into the borehole mixer network. The following relationship can be employed for the final estimation of \(\mathbf{m}\):  
\[\hat{m}^{*}=\text{Mixer}(\hat{m},\mathbf{b};\theta^{*}), \tag{5}\]  
where \(\theta^{*}\) denotes the model parameters of the borehole mixer network, Mixer().