Abstract: This paper investigates the propagation patterns and characteristics of wave equations in inverse problems. Using the two-dimensional wave equation as the core theoretical framework, we systematically develop and numerically validate an efficient numerical method for solving the forward problem and a robust reconstruction method for the inverse problem. First, the finite difference method is employed to discretize the two-dimensional wave equation in time and space, revealing the dynamic response characteristics of wave field propagation under specific parameter configurations. Subsequently, regularization is applied to the nonlinear wave equation, transforming the inverse problem into a well-posed minimization problem. By integrating total variation regularization with the finite volume method, iterative inversion algorithms are constructed based on nonlinear optimization techniques, including the steepest descent method, Newton's method, and the conjugate gradient method. Finally, numerical simulations of the parameters in the two-dimensional wave equation are conducted using a three-layer medium model. The results demonstrate the feasibility and computational efficiency of the proposed algorithm.

Key words: Total variation regularization, Finite volume method, Inverse problem, Two-dimensional wave equation, Parameter inversion