# Three-Dimensional Velocity Field Interpolation Based on Attention Mechanism
## 1 Introduction  
The establishment of a three-dimensional velocity field is a significant step in seismic exploration [1], and essential for understanding complex underground geological structures. An accurate velocity model is vital for most seismic imaging methods. The precise 3D velocity field model is a prerequisite for reverse time migration and other seismic imaging technologies. Moreover, it is crucial for observation system design, precise positioning of underground geological target bodies, structural interpretation, and reservoir prediction. The accuracy of velocity directly affects all aspects of seismic exploration and the final results. However, due to economic and natural constraints, only a small amount of real data can be obtained [2], and the complete 3D velocity field is often established by interpolating sparse data. Traditional three-dimensional spatial interpolation methods rely on inferring the relationships between known data points to generate estimates for the entire spatial domain. For example, [PERSON] [3] and others successfully reconstructed soft-tissue pseudoshadowdata using three-dimensional linear interpolation methods, providing additional geometric information helpful for subsequent medical research. [PERSON] [4] and others applied inverse distance weighting (IDW) to three-dimensional fluorescence spectroscopy data interpolation, expressing more details of the fluorescence spectrum. [PERSON] [5] and others applied three-dimensional spline interpolation to transmissivity estimation in the South China Sea, obtaining reasonable and accurate results. In the field of Earth Sciences, [PERSON] and [PERSON] [6] and others used the convex set projection (POCS) algorithm of Fourier transform as a simple iterative method for interpolating irregular filling grids of seismic data, producing high-quality results. [PERSON] [7] and others used the ordinary Kriging method to double the resolution of nuclear magnetic resonance (NMR) well logging data, enabling more accurate predictions of the free fluid porosity and permeability of carbonate reservoir rocks in the South Pars gas field, facilitating a better understanding of the reservoir. [PERSON]. [PERSON] [8] and others used Kriging and co-Kriging methods to analyze well logging data from five oil wells, estimating the porosity in the Bornu Basin area of northeastern Nigeria, laying the foundation for subsequent exploration strategies. Traditional interpolation methods usually rely on mathematical functions for fitting or interpolation functions for data completion, which may not effectively capture complex geological or geographical features in some scenarios, especially when dealing with the nonlinearity of real underground scenarios, such as heterogeneous underground media or terrain undulations. Therefore, more effective techniques are often needed for three-dimensional spatial interpolation. As the demand continues to grow, traditional 3D interpolation methods are gradually showing their limitations when facing complex, large-scale, or high-uncertainty data. Under such circumstances, introducing deep learning methods brings new possibilities for 3D interpolation. The application of three-dimensional interpolation technology is also used in medical imaging [9; 10; 11; 12; 13; 14], hydrological analysis [15], atmospheric sciences [16], hydrodynamics [17], virtual reality [18], and other fields. [PERSON] et al. [19] first achieved the restoration of the full picture of seismic data from 20% of actual data through a deep autoencoder. [PERSON] [20], [PERSON] [21], [PERSON] [22], [PERSON] [23], and others have also used three-dimensional interpolation methods to carry out related work in the field of geoscience. [PERSON] [24] and others used deep neural network (DNN) and feature extraction steps to interpolate and establish velocity models from seismic trace sets, reducing computational costs. [PERSON] et al.