Wavefield extrapolation in pseudodepth domain

Appendix C: Stability analysis for $\tau$ domain extrapolation

For homogeneous velocity and regularly spaced grids, the stability condition is often studied by Von Neumann analysis (Trefethen, 1994). Consider $1$ -D second-order wave equation,

$$u_{tt} = v^2 u_{zz} ,$$ (34)

approximated by central difference in space and time

$$\frac{u_i^{n+1} - 2 u_i^n + u_i^{n-1}}{\Delta t^2} = v^2 \frac{u_{i+1}^n - 2 u_i^n + u_{i-1}^n}{\Delta z^2}$$ (35)

where superscripts indicate time index and subscripts indicate space index. Substitute ansatz $u_i^n = U^n\exp\left(\mathrm{i} k z_j\right)$ into Equation H-2 yields

$$U^2 + \left( 4 \frac{v^2\Delta t^2}{\Delta z^2} \sin^2\frac{k\Delta z}{2} - 2 \right) U + 1 = 0$$ (36)

where amplification factor $U$ characterizes the growth of the numerical solution during iteration. Stability of the numerical solution requires $\lvert U\rvert \leq 1$ for all wavenumbers $k$ . It can be shown that in order for the quadratic equation

$$U^2 + \beta U + 1 = 0 <tex2html_comment_mark>$$ (37)

to have bounded roots $\lvert U\rvert \leq 1$ , it is necessary that $\lvert\beta\rvert \leq 2$ . This is equivalent to

$$0 \leq \frac{v^2\Delta t^2}{\Delta z^2} \sin^2\frac{k\Delta z}{2} \leq 1 \quad \forall \; k$$ (38)

thus yields the well-known CFL condition

$$\Delta t \leq \frac{\Delta x}{v} .$$ (39)

In $\tau$ domain, the $1$ -D problem can be extracted from the vertical component of Equation 19

$$u_{tt} = \frac{v^2}{v_m^2} u_{\tau\tau} .$$ (40)

Following the same analysis for H-2, we can obtain $\tau$ domain CFL condition

$$\Delta t \leq \frac{v_m}{v} \Delta \tau$$ (41)

Wavefield extrapolation in pseudodepth domain