# Optimizing Finite-Difference Operator in Seismic Wave Numerical Modeling
## 1 Introduction  
Seismic wave numerical modeling is a widely used method to simulate the propagation of seismic waves in a known geological landscape. So far, it has been used in seismic exploration and seismology, e.g., earthquake disaster prediction [1], seismic performance assessment of roads, buildings, bridges, and other materials [2, 3], seismic oceanography [4], ship detection, and identification [5]. The wave equation is of fundamental importance for describing the propagation of seismic waves [6]. The methods of seismic wave numerical modeling based on the wave equation include the finite-difference (FD) method [7], the finite element method [8, 9], and the pseudo spectrum method [10]. The FD method, first presented in 1968 [7], simulates the propagation of seismic waves in various media of the earth and thus can help predicate what happens during the propagation process [11]. Compared to other techniques, the FD method is more efficient and easily implemented [6]. Generally, the wave equation of seismic wave propagation is a partial differential equation. The FD method transforms the continuous problem into the discrete problem, and one can obtain the approximate solution of the original wave equation by solving the difference equation. As the pseudospectral method is considered as the highest order finite-difference scheme [12], we shall calculate the finite-difference operator by truncating the convolution series of the pseudospectral method with the window function. However, the discretization of wave equation inevitably introduces numerical dispersion, which deteriorates the accuracy of seismic wave simulation. The window function method and optimization method are two essential ways to suppress numerical dispersion. The size of the window function grid plays an influential role in the modeling precision. Although the refined difference grids offer better performance, sometimes, coarse difference grids are preferably used to reduce the computational cost. For this reason, choosing a suitable window function is demanding in practice. The traditional window functions include [PERSON] (1973); [PERSON] (1974); [PERSON] (1992), and so on. In the past decades, some new window functions were also reported in the literature. [PERSON] et al. (2016) proposed a Chebyshev auto-convolution combined window. [PERSON] et al. (2017) developed a cosine modulated Chebyshev window. [PERSON] and [PERSON] (2018) constructed a new window function by combining the cosine function with the original window function. Optimizing the difference operator is another strategy. In this category, the Newton method, the least squares method, and the swarm intelligence algorithm are frequently used. For example, [PERSON] and [PERSON] (2019) applied the simulated annealing algorithm to search for the operator that best satisfies the constraints. [PERSON] et al. (2019) used the improved particle swarm optimization (PSO) algorithm to optimize the finite-difference operator. [PERSON] et al. (2018) developed the least-squares method to optimize the staggered-grid finite-difference operator. Both the window function and optimization method aim to find the optimal window function coefficients. The difference is that window functions are directly designed parameters, while optimization methods generate the optimal coefficients through the iteration. In this paper, we use the nature-inspired optimization (NIO) algorithms to find the fittest finite-difference coefficients. The NIO algorithms are widely used to solve the optimal solution problem in continuous and discrete space. In this paper, we implement the krill herd algorithm (KH), which is a recently proposed NIO algorithm, the PSO, and the center-decenter PSO, which is a variant of PSO. We derive the finite-difference operator through sinc function interpolation and Taylor series expansions. We define the objective function as the error function of the first and second derivatives.