Efficient solution of large-scale matrix of acoustic wave equations in 3D frequency domain

In 3D frequency domain seismic forward and inversion calculation, the huge amount of calculation and storage is one of the main factors that restrict the processing speed and calculation efficiency. The frequency domain finite-difference forward simulation algorithm based on the acoustic wave equation establishes a large bandwidth complex matrix according to the discretized acoustic wave equation, and then the frequency domain wave field value is obtained by solving the matrix equation. In this study, the predecessor’s optimized five-point method is extended to a 3D seven-point finite-difference scheme, and then a perfectly matched layer absorbing boundary condition (PML) is added to establish the corresponding matrix equation. In order to solve the complex matrix, we transform it to the equivalent real number domain to expand the solvable range of the matrix, and establish two objective functions to transform the matrix solving problem into an optimization problem that can be solved using gradient methods, and then use conjugate gradient algorithm to solve the problem. Previous studies have shown that in the conjugate gradient algorithm, the product of the matrix and the vector is the main factor that affects the calculation efficiency. Therefore, this study proposes a method that transform bandwidth matrix and vector product problem into some equivalent vector and vector product algorithm, thereby reducing the amount of calculation and storage.