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Higher-order terms from the phase function

In this paper, we adopt the one-step scheme due to its superior stability and ability to handle complex-valued phase functions. Because the two-step scheme depends on a real-valued phase function, it cannot include higher-order terms from the expansion (equation 20) and implement more accurate phase functions. By switching to a one-step scheme, we can easily incorporate the second-order term ($ \phi_2$ ) in equation 20 to achieve a more accurate wave extrapolation operator. As defined in equation 22, $ \phi_2$ involves the gradient of the velocity model, and can become significant when either the time step size is large or the velocity model changes rapidly. Substituting the first three terms from the Taylor series (equation 20) into equation 17, the second-order operator takes the form

$\displaystyle p_1(\mathbf{x},t+\Delta t) \approx \int P_1(\mathbf{k},t) e^{i ...
...,\mathbf{k}) \nabla V \cdot \mathbf{k} {\Delta t}^2/2 \right]} d\mathbf{k}\;.$     (24)

This modification helps increase the accuracy of wave extrapolation, especially when $ \Delta t$ or $ \nabla V$ are large. Note that the introduction of the velocity gradient term does require the velocity model to be smoothly varying, which conforms to the usual requirement of RTM. The term $ \nabla V \cdot \mathbf{k}$ also leads to a certain degree of anisotropy in the phase function, which in practice may increase the numerical rank of lowrank approximation.


next up previous [pdf]

Next: Direction-dependent absorbing boundary conditions Up: Theory Previous: Variable velocity and anisotropy

2016-11-16