Parameter inversion of the diffusive-viscous wave equation based on Gaussian process regression

The diffusive-viscous wave (DVW) equation is used to characterize the relationship between frequency-dependent seismic responses and saturated fluids by incorporating the frictional dissipation and viscous damping to the scalar wave equation. Simultaneous inversion of three model parameters in DVW equation is essential for seismic interpretations. Traditional inversion methods require continuous forward-modeling updates, resulting in low computational efficiency. Moreover, the traditional methods have limitations in simultaneously inverting multi-parameters of wave equations such as the DVW equation, usually fixing one parameter to invert the other two parameters. Gaussian process regression (GPR) is a kernel-based non-parametric probabilistic model that introduces prior variables through Gaussian processes (GP). We present a method for the inversion of the three parameters (velocity, diffusive and viscous attenuation coefficients) of the DVW equation based on GPR. The procedure consists of initially implementing the central finite difference approximation to discretize the DVW equation in the time domain. Subsequently, a Gaussian prior is provided on two snapshots of the DVW equation to obtain the corresponding kernel functions. Furthermore, the hyperparameters in kernel functions and the three model parameters are simultaneously trained by minimizing the negative logarithmic marginal likelihood with few training samples while incorporating the underlying physics in terms of encoding the DVW equation into the kernel functions. It is worth noting that it is the first time of implementing three-parameter simultaneous inversion based on the DVW equation. The numerical examples in homogeneous, layered and heterogeneous media demonstrate the effectiveness of this method.