Spectral Methods For Wave Equation And A Study Of Source Processing

Spectral method is a fast, high precision and low numerical dispersion numerical method, which can be successfully applied to seismic exploration. This dissertation reviews the numerical scheme and numerical properties of spectral method for solving seismic wave equations firstly. Then we combine the spectral method with perfectly matched layer(PML) absorbing boundary conditions. Numerical experiment results show that the absorption e?ect is very good.During the numerical simulation of seismic exploration, pulse source is used widely, and its mathematical formulation is f(t)δ(x- xs). Traditionally, people add the source on a single point(or multipoint) or use a gaussian function to approximate the delta function, which will lead to a false fluctuation. For spectral method, the above treatment for source will lead to failed computation.The green function solution of wave equation has a integral form, therefor we can discretize the wave equation as an integral. In this sense, we can discretize delta function in the same way. We have referred to Dr. Wen Xin’s work[1]about numerical quadratures to one dimensional delta function integrals, and successfully apply it to solve the wave equation with delta source function.Numerical results show that the above-mentioned delta function discrete method can eliminate the false fluctuations. However, people need to manually select truncation parameter m when discretizing the delta function. Moreover, inappropriate parameter’s selection can cause calculation error. If we use the high-order discrete method for δ function proposed by Dr. Wen Xin, parameter’s selection scope can be extended,which ensure the validity of the calculation.It’s worth emphasizing that the pulse source f(t)δ(x- xs) includes the source location information xs. When the earthquake source location is uncertain, we need to di?erentiate δ(x- xs) with respect to xs. However the traditional method obviously can’t deal with the derivative of δ function e?ectively. As we have verified validity of calculation by delta function discrete method, we can put forward a high-precision method for source location.