IMPLEMENTATION OF LITWEQ BY FINITE DIFFERENCE

Li (1986) presents the finite difference discretization of (7) with the diffraction and reverberation terms. He didn't include the second term at the right hand side of (7), usually called thin lens term. I am going to show a full LITWEQ discretized operator using a second order finite difference scheme. This scheme should be tested with synthetic or real data that has strong lateral velocity variations to check its stability.

The finite difference scheme uses a nine point star pattern on (t₁,t₂). Before the discretization, we need to define the linear operator L which represents a convex combination of the points in the star:

$$Lu(t_{1},t_{2})={\alpha_{1}}u(t_{1},t_{2})+\cdots+{\alpha_{9}}u(t_{1}+1,t_{2}+1) ,$$ (8)

where u represents the approximate solution to the wave-field. Using L, the second and first order differential operators can be approximated as follows:

$$\begin{eqnarray} {\partial^2 P\over\partial x^2}&\approx &-{1\over(\Delta x)^2}T... ...t_{i}} & \approx & {\sqrt{2}\over\Delta t}F_{i}L u(t_{1},t_{2}) , \end{eqnarray}$$ (9) (10)

where T represents the classical centered finite difference scheme to approximate a second order derivate. F₁ and F₂ represent forward schemes to approximate the first order derivate, along the t₁ and t₂ respectively. Substituting (9) and (10) in (7) and approximating the others differential operators in the conventional way we get the discretized LITWEQ operator:

$$\begin{eqnarray} {c_1}TLu(t_1,t_2)+{c_2}({F_1}{F_2})u(t_1,t_2) & = & {c_3}({F_1}... ...+{T_2})u(t_1,t_2) \nonumber \\ & + & {c_3}({F_1}{F_2})u(t_1,t_2), \end{eqnarray}$$ (11)

where c_i are functions of $\Delta t$, $\Delta x$,$v(x,\tau)$ and $v(\tau)$ and T₁, T₂ represent centered schemes along t₁ and t₂ respectively.