Symplectic scheme and the Poynting vector in reverse-time migration

Abstract

We developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method (REM). This solution can be used for seismic modeling and reverse-time migration (RTM). In seismic modeling and RTM, spatial derivatives are usually calculated by finite differences (FDs) or by the Fourier method, and the time evolution is normally obtained by a second-order FD approach. If the spatial derivatives are computed by higher order FD schemes, then the time step needs to be small enough to avoid numerical dispersion, therefore increasing the computational time. However, by using REM with the Fourier method for the spatial derivatives, we can apply the proposed method to propagate the wavefield for larger time steps. Moreover, if the appropriate number of expansion terms is chosen, this method is unconditionally stable and propagates seismic waves free of numerical dispersion. The use of a symplectic numerical scheme provides the solution of the wave equation and its first time derivative at the current time step. Thus, the Poynting vector can also be computed during the time extrapolation process at very low computational cost. Based on the Poynting vector information, we also used a new methodology to separate the wavefield in its upgoing and downgoing components. Additionally, Poynting vector components can be used to compute common gathers in the reflection angle domain, and the stack of some angle gathers can be used to eliminate low-frequency noise produced by the RTM imaging condition. We numerically evaluated the applicability of the proposed method to extrapolate a wavefield with a time step larger than the ones commonly used by symplectic methods as well as the efficiency of this new symplectic method combined with REM to successfully handle the Poynting vector calculation.

Seismic Reverse Time Migration Using A New Wave-Field Extrapolator and a New Imaging Condition

“…Originally, the time operator was approximated by a second-order scheme, whereas the spatial derivatives were approximated by a fourth-order scheme. This form of time derivative approximating may introduce numerical error, leading to distortion of the pulse and numerical dispersion, which can be avoided with small time steps (Araujo et al 2014); (ii) The two-step explicit marching method, which makes use of a polynomial expansion (Soubaras andZhang 2008, Zhang andZhang 2009). Soubaras and Zhang (2008) presented a new method for RTM.…”

“…Besides that, using the only two terms of Chebyshev expansion, they obtained the same equations used for the second-order FD time approximation; (iv) Symplectic integrators are another class of methods that can also be used to solve the wave equation in RTM. Araujo et al (2014) presented a symplectic integrator for wave-field extrapolation called Störmer-Verlet-REM which extrapolates the wave-field and its derivative with a good accuracy and stability.…”

“…Fletcher et al (2005a, b) introduced a directional damping factor into the wave equation to suppress undesired reÀection. Yoon and Marfurt (2006) introduced a Poynting-vector imaging condition that was used by several researchers to suppress the artifacts (Costa et al 2009, Jin et al 2014, Araujo et al 2014, Du and Qin 2009, Chen and Huang 2014. Chattopadhyay and McMechan (2008) explicitly compared the most common imaging conditions, excitation time, cross-correlation and receiver/source-wave-¿eld amplitude ratio to clarify which are viable for recovering accurate amplitudes and which are not.…”

“…According to Araujo et al (2014), by adding the term -2P(t) on both sides of Eq. 13 and multiplying by 1/ǻ 2 , a second order central finite differences operator is calculated as follows:…”

“…where J is the Poynting vector. Araujo et al (2014) presented an approach to separate the wave-field to down-going and up-going components. They calculated the direction of propagation of the wave-fields through the Poynting vector using Eq.…”