عنوان مقاله [English]
The aim of this paper is to present a new numerical method to solve the wave equation with a good accuracy and high stability using the Leapfrog symplectic integrator and rapid expansion method (REM). It can be used for seismic modeling and reverse time migration (RTM). Using the REM with Fourier transform method for spatial derivative, the Leapfrog-rapid expansion method (L-REM) can be even used for larger time steps. The L-REM provides the solution of the wave equation and its first time derivative at the current time step. In addition to the very low error for the small time steps, increasing the time step also lead to more accurate results and high stability in comparison with the similar methods such as Störmer-Verlet, Leapfrog and Störmer-Verlet-rapid expansion method which will also be discussed in this paper.
Wave-field extrapolation is implemented by solving the wave equation through various mathematical methods. The finite difference method is a well-known and popular numerical tool to discretize the wave equation, and its use has been common in the approximation of the spatial and time derivatives for a wave-field. Originally, the time operator was approximated by a second-order scheme, whereas the spatial derivatives were approximated by a fourth-order scheme. Approximating the time derivative in this way may introduce numerical error, leading to distortion of the pulse and numerical dispersion, which can be avoided with small time steps at the expense of increasing the computational time. Also some other methods such as Störmer-Verlet (SV), Leapfrog (L) and Störmer-Verlet-rapid expansion method (SV-REM) have been presented to improve the solution of wave equation. In the current study, a symplectic scheme based on Leapfrog integrator and rapid expansion method (L-REM) is proposed to extrapolate the wave-field and its first derivatives in time for the same time step which can be used to calculate Poynting vectors for wave-field separation and to calculate the reflection angles.
Methodology and Approaches
In order to verify the numerical accuracy and behaviour of the error associated with Leapfrog-REM scheme a numerical example has been presented to be solved using different time sampling values. For implementation, an explosive source used in the centre of the computational domain having a Ricker wavelet with a maximum frequency of 25 Hz.
Results and Conclusions
The presented L-REM scheme provides the solution of the wave equation and its first time derivative for different time steps. In addition to the very low error for the small time steps, increasing the time step also lead in the more accurate results and high stability in comparison with the similar methods.