Physics-Informed Neural Networks for multi-frequency surface wave tomography

Surface wave tomography based on dispersion is an important approach for resolving the velocity structure of the crust and upper mantle. Traditional surface wave tomography methods based on dispersion data typically require first construction of 2D phase/group velocity maps, followed by a point-wise inversion of dispersion data to derive 1D profiles of shear wave velocity as a function of depth at each grid point, and finally forming the 3D velocity model. However, the 2D tomography method based on ray theory has a strong dependence on the selection of the initial velocity model and regularization parameters. Furthermore, the eikonal tomography method requires dense observations. Therefore, we propose a surface wave tomography method based on a physics-informed neural network, which can construct the phase/group velocity maps of multiple frequencies simultaneously, eliminating the need for repeated separate inversion for each frequency. The network comprises two branches, one branch takes in the coordinates of the virtual source and station as well as period as input to fit the observed surface wave travel times, and the other branch takes in the station coordinates and period to predict the phase/group velocity. The two branches are constrained by the eikonal equation. After the training is completed, the velocity of each grid point in each period can be quarried using the neural network and form the group/phase velpcity maps for each period. We tested the new method using data from the Feidong dense array and the Weifang dense array, and compared the tomography results with those of the traditional method. The test results demonstrate that the new method is a meshless tomography method with data adaptive resolution. In addition, this method does not require an initial velocity model or explicit regularizations. It is highly automatic, simple, and easy to use, with potential to combined with existing dispersion curve automatic extraction methods for automatic tomography without human intervention.