# A Deep Learning Estimation for Probing Depth of Transient Electromagnetic Observation
## 3 Deep Learning Model and Training
The peak of the Gaussian distribution corresponds to the predicted depth. The network consists of a total of 8 layers, including 7 convolutional layers and 1 pooling layer. The RELU activation function is used in every convolutional layer to improve the nonlinear ability of the network and reduce the likelihood of the vanishing gradient. Furthermore, the sigmoid function is applied to the last convolution layer to limit the output value within the range of [0, 1]. The loss function is defined using binary cross-entropy between the true probability distribution \(p_{1}\) and prediction distribution \(q_{1}\)[16; 25]:  
\[H(p,q)=-\sum_{i=1}^{2}\sum_{x}p_{i}(x)log\ q_{i}(x) \tag{2}\]  
Here, \(p_{1}+p_{2}=1\), \(q_{1}+q_{2}=1\), and \(x\) represents the abscissa of the output distribution. Cross-entropy naturally fits with probabilistic outputs, as it directly measures the difference between the predicted probability distribution and the true distribution. The loss decreases as the predicted probabilities get closer to the true labels [25]. Three models are trained using the same dataset. The inputs, outputs, and hyperparameters of the different models are completely identical for comparison. The model's learning rate was set to 0.001, the batch size to 200, and the number of iterations to 200. Other parameters, including the kernel size, number of convolutional layers, and pooling size, are referenced in Figure 1. The model parameters of the ResNet are continuously updated through backpropagation algorithm Adam [26]. Each model was trained using a GPU (NVIDIA GeForce RTX 3070, NVIDIA Corporation, Santa Clara, CA, USA) of desktop workstation (Processor: 12 th Gen Intel(R) Core(TM) i9-12900K 3.20 GHz, Memory: 128 GB, Intel Corporation, Santa Clara, CA, USA), with the 200 epoch training process taking approximately 20 min. After training, the prediction time for 10,000 measuring data is within 1 s. In a set of 137,000 geoelectrical models, 90% are used for training, and 10% are used as a test dataset to evaluate the model's performance on unseen data. Figure S1 presents the evolution of the loss function during the training process. It can be observed that the errors  
Figure 1: The RESNET, RNN, and CNN framework used for prediction of probing depth and schematic input representation. The sizes and numbers of convolutional or pooling kernels are indicated within each layer, with green arrows representing shortcut connections. The lower-left subfigure shows a long short-term memory (LSTM) block containing a memory cell and the input, output, and forget gates (modified from [PERSON] et al., 2000 [22]). of both the training and testing sets gradually decrease with increasing iteration numbers and eventually reach a stable state. This indicates that the model successfully learns the features present in the training set. In addition, we further investigate the relationship between probing depth and observed data to demonstrate the effectiveness of the algorithm (Figure 2). The late-time IEF [8] is not only associated with the cut-off time controlling the slope of the decay curve but also serves as a comprehensive reflection of subsurface rock resistivity. Figure 2b,c represent the relationship between labeled depth and predicted depth with the IEF at the last cut-off moment, respectively. As the late-time IEF decreases (represented by negative values after taking the logarithm, thus larger values on the x-axis indicating smaller IEF), the probing depth gradually increases. This is because a faster decay of the electromagnetic field indicates higher resistivity, leading to greater penetration depth of the electromagnetic field. The significant similarity observed in Figure 2b,c demonstrates that the deep learning model can effectively learn the mapping relationship between the induced electromotive force curve and the probing depth.