# Shallow Subsurface Wavefield Data Interpolation Method Based on Transfer Learning
## 1 Introduction
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Comparison Dimensions** & **Traditional Mathematical Methods** & **Deep Learning Methods** \\ \hline \multirow{4}{*}{Core principle} & Explicit modeling based on physical models or mathematical transformations. & Data-driven feature learning involves using neural networks to implicitly model data distributions. \\  & Rely on a small amount of & Need a large amount of annotated data. \\ \multirow{4}{*}{Data requirements} & high-quality data and do not require a training set. & Not a large amount of annotated data. \\  & Need to manually design & Not a large amount of annotated data. \\ \multirow{2}{*}{Prior knowledge dependence} & sparse transformation & Automatically learn \\  & bases or rank constraints. & optimal representations. \\ \multirow{2}{*}{Engineering applicability} & Suitable for regular & Support large-scale 3D data \\  & missing data and & and irregular missing data. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The differences between traditional mathematical methods and deep learning methods. limit the acquisition of large real-world datasets. In light of [PERSON] et al. 's [24] self-supervised approach based on the receptive field properties of CNNs, we could apply their training methodology to real-world data. Despite this, the available data for network training remained insufficient. To address this limitation, we explored the integration of simulated and real data to train the network for interpolating signals at unknown locations. Based on this framework, we used numerical simulation to generate synthetic data, which was combined with real collected data in a dual-driven approach to reconstruct the temporal signals at virtual measurement points in the subsurface vibration field. Initially, with the known subsurface structural information, we established a simulated subsurface medium model and employed elastic wave equations to simulate the propagation of subsurface vibrations, thereby constructing a pre-training dataset. Subsequently, we used finite element simulations as real-world data to further refine the trained deep interpolation model. By employing transfer learning techniques, we froze all down-sampling modules in the pre-trained deep interpolation model and fine-tuned the model using the real dataset as the target domain. This process optimized the model's interpolation capabilities for real data while completing signal interpolation at the target locations. The main contributions of our work are as follows:  
1. To address the challenge of insufficient real-world data samples for effective network training, we propose a transfer learning approach. By first training the model on a large dataset of simulated data and then performing adaptive fine-tuning using a small dataset of real-world samples, the approach effectively leverages the model parameters learned from simulated data. This allows the network to quickly adapt to the characteristics of real data, enhancing the accuracy of signal interpolation for real-world applications;
2. To overcome the lack of complete data as labels for training in the target domain, we adopted a self-supervised learning method [24]. This method constructs training samples from incomplete data using random masking. Combining this approach with the network pre-trained on simulated data enables the model to better capture the spatial patterns inherent in real-world seismic signals, resulting in a higher interpolation accuracy;
3. We developed a deep interpolation model based on the U-net architecture [25]. By introducing multi-scale convolutional modules in the encoder, we expanded the receptive field (RF) while enhancing the model's feature extraction capability. Additionally, we incorporated wavelet transform and its inverse as the down-sampling and up-sampling layers, respectively. This design preserves all information during down-sampling, reduces interpolation errors, and ensures that the reconstructed waveforms closely resemble real waveforms.