Variable velocity and anisotropy

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Variable velocity and anisotropy

For inhomogeneous and anisotropic media, we can use the general form of equation 16 to introduce a general phase function, which depends on both $\mathbf{k}$ and $\mathbf{x}$ , and rewrite equation 16 as the following mixed-domain operator (Fomel et al., 2013; Wards et al., 2008):

$\displaystyle p_1(\mathbf{x},t+\Delta t) = \int P_1(\mathbf{k},t) e^{i \phi(\mathbf{x},\mathbf{k},\Delta t)} d\mathbf{k}\;.$

(17)

Equation 17 is kinematically correct if the phase function $\phi(\mathbf{x},\mathbf{k},t)$ satisfies the anisotropic eikonal equation:

$\displaystyle \frac{\partial \phi}{\partial t} = \pm V(\mathbf{x},\mathbf{k}) \left\vert\nabla \phi\right\vert\;,$

(18)

where $V(\mathbf{x},\mathbf{k})$ is the phase velocity. When velocity is constant and isotropic, the phase function reduces to

$\displaystyle \phi(\mathbf{x},\mathbf{k},t) = \mathbf{k} \cdot \mathbf{x} \pm V\vert\mathbf{k}\vert \; ,$

(19)

and corresponds to equation 16. In the more general case, assuming small time steps, $\phi$ can be expanded into the Taylor series in

(Fomel et al., 2013):

$\displaystyle \phi(\mathbf{x},\mathbf{k},t) \approx \mathbf{k} \cdot \mathbf{x}... ...{x},\mathbf{k}) t + \phi_2(\mathbf{x},\mathbf{k}) \frac{t^2}{2} + \cdots \; ,$

(20)

where

$\displaystyle \phi_1(\mathbf{x},\mathbf{k})$	$\displaystyle =$	$\displaystyle V(\mathbf{x},\mathbf{k}) \vert\mathbf{k}\vert\;,$	(21)
$\displaystyle \phi_2(\mathbf{x},\mathbf{k})$	$\displaystyle =$	$\displaystyle V(\mathbf{x},\mathbf{k}) \nabla V \cdot \mathbf{k}\;.$	(22)

Equation 17 corresponds to the one-step method (Zhang and Zhang, 2009; Fowler et al., 2010b). Fomel et al. (2013) adopted instead a two-step implementation, which uses only the $\phi_1$ term in equation 20 to cancel out the imaginary part of the wave extrapolation operator (Etgen and Brandsberg-Dahl, 2009):

$\displaystyle p(\mathbf{x},t+\Delta t) + p(\mathbf{x},t-\Delta t) \approx 2 \... ...V(\mathbf{x},\mathbf{k}) \vert\mathbf{k}\vert \Delta t\right] d\mathbf{k}\;.$

(23)

It is important to realize that the cancellation of the imaginary part relies on the fact that the phase function only contains odd-order terms of

, with the exception of the zeroth-order term which corresponds to the inverse Fourier transform. Any modification to the phase function that violates this requirement cannot be easily handled by the two-step method.

As detailed in the appendix, the unconditional numerical stability of the one-step scheme (equation 17) can be proven theoretically in the continuous cace. In practice, we have observed that the lowrank approximation preserves the stability property, as will be demonstrated in the next section with numerical examples.

Next: Higher-order terms from the Up: Theory Previous: Analytical solutions in constant

2016-11-16