# Accurate Pseudo-Spectral Acoustic Wave Modelling with Time Dispersion Elimination
## 1 Introduction  
Numerical modeling of seismic wave propagation is one of the fundamental elements of exploration geophysics. Based on different physical assumptions and numerical realizations, many approaches have been proposed as wave equation [1]. For example, the ray tracing method [2], which is based on high-frequency approximation, has the advantage of an ability to isolate specific wave modes. Its main limitation is that wavefield information cannot be fully simulated due to its high-frequency approximation nature. Other kinds of modeling method are typically based on direct discretization and the approximation of wave equations, such as the finite difference method [3, 4], the finite element method [5, 6], the spectral element method [7, 8] or the pseudo-element method [9, 10, 11]. Presently, the finite difference method remains one of the most popular seismic wave modeling methods for its simple implementation and benefits of computational accuracy and efficiency [11]. However, the finite difference method requires rather dense spatial grid sampling to ensure computational stability and accuracy. The pseudo-spectral method, as discussed in this paper, is an alternative grid-based wavefield modeling method. The pseudo-spectral method uses the Fourier transform to calculate spatial derivatives which can be viewed as a limiting case of higher-order finite difference, with the operator size equaling the grid dimension [12, 13]. When compared with the finite difference method, the pseudo-spectral method has the advantage that it permits coarser spatial grid sampling to attain the same accuracy, which reduces the computational memory requirement, especially for 3D seismic wave modeling [9]. Two well-known issues that should be addressed during the implementation of the pseudo-spectrum method are so-called non-causal ringing artifacts and wrap-aroundboundary artifacts. The non-causal ringing artifacts arise from Nyquist discontinuities in the periodic wavenumber spectrum [10]. These artifacts contaminate wavefields, particularly in the presence of large abrupt changes in the medium. It has been well established that modeling on a staggered grid significantly reduces those artifacts [14; 15]. The other issue is the wrap-around boundary artifacts due to the periodicity implied by using the Fourier transform; the wrap-around boundary artifacts can be alleviated by zero padding of the computational domain [16]. However, to perfectly implement the pseudo-spectral method, another less-mentioned problem, which is referred to as unbalanced numerical dispersion [17; 18], should be solved. This problem arises from the fact that, although in the pseudo-spectral method the spatial derivative is accurately calculated using Fourier transform, the remaining temporal derivative is computed with the finite difference approach which typically has larger approximation errors than the Fourier transform. [18] proposed using the higher-order temporal finite difference approximation which is more time-consuming than the commonly used second order approximation in the pseudo-spectral method. [19] proposed the forward and inverse time dispersion transform to reduce the temporal dispersion in the non-staggered, grid-based modeling method. In this paper, we firstly combine the forward and inverse time dispersion transform with the staggered grid-based pseudo-spectral method. Our present method not only retains the advantages of the staggered grid-based pseudo-spectral method, such as coarser spatial sampling, higher spatial derivative approximation accuracy, and non-casual ringing artifacts removal, but also achieves higher temporal derivative approximation accuracy due to the adoption of an additional time-dispersion elimination term. Our examples indicate that temporal dispersion elimination process can be contaminated by spatial numerical dispersion. To obtain accurate acoustic simulation results, the temporal and spatial numerical dispersion should be simultaneously handled.