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This paper proposes a deep learning model for the dispersion relation prediction of Rayleigh waves and shear wave velocity inversion. The proposed deep learning method includes the forward model and the inversion model. The forward model is built by the convolution neural networks, which are used to predict the dispersion relation by the shear wave velocity; the inversion model is designed by the transformer to inversely estimate the shear wave velocity in the subsurface layers. The synthetic data generated by a constrained Markov decision process and the fast vector transfer method are used for model training and testing. 90,000 and 10,000 samples are used for the training and testing of the model, respectively.
Rayleigh waves have been widely used for subsurface investigation because of their intense energy and slight attenuation. Traditionally, the process of subsurface investigation based on Rayleigh waves can be divided into three steps. The first step is dispersion relation extraction. Different methods can be used to extract the dispersion relation from the experimental data through the multi-channel analysis of surface waves (MASW) or the spectral analysis of surface waves (SASW). The MASW requires a series of geophones placed on the ground surface, while the SASW only needs two geophones. The second step is the forward model building to determine the theoretical dispersion relation. Various methods for the forward model have been developed such as the Haskell-Thomson method, the delta matrix method, the Schwac-Knopoff method, the fast vector transfer method, the spectral element method and so on. The third step is the inversion process based on the extracted experimental dispersion relation and the generated dispersion relation by the forward model. The disadvantage of the traditional method includes: first, the forward model is not always stable to develop the theoretical dispersion relation; when the dispersion relation curve is complex, the root-searching process may not find a good solution. This may bring uncertainties in the inversion process during the variable updating. Second, convergence cannot be guaranteed in the inversion process because the inversion process for the subsurface characterization is an ill-conditioned non-linear problem, and it is easy to trap into the local optimization. The third disadvantage is the initial value for the inversion process is important; this requires prior knowledge.
With the development of artificial intelligence, machine learning techniques unfold opportunities to handle dispersion relation determination and subsurface characterization. In particular, the deep learning method aims to use various kinds of neural networks to map the complex non-linear relationship between the input and the output. An example of such a complex non-linear problem is the forward model and the inversion process for the subsurface characterization based on the dispersion relation of Rayleigh waves. There are different deep learning models such as fully connected neural networks (FCNNs), convolution neural networks (CNNs), recurrent neural networks (RNNs), transformers, and so on. Therefore, a deep learning-based method is proposed for both the forward model and the inversion process in this work.
