# Shallow Subsurface Wavefield Data Interpolation Method Based on Transfer Learning
## 3 Results  
### Construction of Pre-Training Dataset (Source Domain)  
The composition of the training dataset is crucial for the successful training of neural networks, as the quality of the training dataset determines the interpolation performance of the network post-training. The deep interpolation model designed in this invention takes seismic data as both input and output, with these data implicitly linked to the underlying geological model. Given the context of this study, which focuses on the reconstruction of shallow subsurface seismic wave signals, it is imperative that the simulated data adequately account for the complex propagation processes of reflection and transmission in the shallow subsurface wavefield. The reflection and refraction of elastic waves conform to the propagation laws of subsurface wavefields, and since the soil medium structure is consistent and the medium velocity model can be simulated, using the elastic wave equation is more suitable for forward modeling in typical shallow subsurface media. The specific procedure is as follows:  
Design a geological structure model based on the subsurface airport runway model as the experimental subject. This involves simulating a structure consisting of a layer of concrete, a layer of compacted gravel, and a layer of semi-infinite compacted clay. A two-dimensional spatial velocity matrix is established to simulate the three-layer subsurface media, with the matrix size being 120 \(\times\) 126. The first dimension represents depth, denoted as \(z\), with direction from top to bottom; the second dimension represents surface length, denoted as \(x\). Each element in the matrix represents a grid H with a depth of 1 m and length of 1 m. The depth is set as 0-39 m with 1000 m/s, 40-79 m with 1500 m/s, and 80-120 m with 2000 m/s. The excitation function of the seismic source used for data simulation is set to be a Ricker Wavelet with the following expression:  
\[S(t)=(1-2\pi^{2}f_{m}^{2}t^{2})e^{-f_{m}^{2}\pi^{2}t^{2}} \tag{9}\]  
where \(f_{m}\) represents the main frequency of the wave, usually set to 30 Hz, and t represents the time. Using the finite difference method, t is replaced by k\(\Delta\)t to bring it into the calculation. Numerical simulations of transverse and longitudinal waves are performed separately using the acoustic wave equation. The general form of sound waves is as follows:  
\[\frac{\partial^{2}u}{\partial t^{2}}=v(x,z)^{2}(\frac{\partial^{2}u}{\partial x ^{2}}+\frac{\partial^{2}u}{\partial z^{2}})+S(t) \tag{10}\]The equation involves the second-order time derivative of the wavefield \(u\), as well as second-order spatial derivatives with respect to the coordinates \(x\) and \(y\). For the second-order time derivative term, the finite difference method is used, employing a three-point difference scheme (FDM-2). The formula is given as follows:  
\[ \begin{cases}\frac{\partial^{2}u}{\partial t^{2}}=\frac{u_{ij}^{k+1}-2u_{ij}^{ k}+u_{ij}^{k-1}}{4u_{ij}^{2}}\\ \frac{\partial^{2}u}{\partial x^{2}}=\frac{u_{ij}^{k+1}-2u_{ij}^{k}+u_{ij}^{k- 1}}{4u_{ij}^{2}}\\ \frac{\partial^{2}u}{\partial z^{2}}=\frac{u_{ij}^{k+1}-2u_{ij}^{k}+u_{ij}^{k- 1}}{\Delta z^{2}}\end{cases}  \tag{11}\]  
where \(v(x,z)\) denotes the velocity at discrete spatial points; the subscript of \(u\) represents the index of the discrete spatial point, and the superscript represents the time difference index, where \(k\) denotes the current time step, \(k+1\) the next time step, and \(k-1\) the previous time step. In space, due to \(\Delta x=\Delta z\), it is collectively set as \(\Delta x=\Delta z=\Delta h\).