Kinematic extrapolation in Riemannian coordinates

We can simplify our Riemannian wavefield extrapolation method by dropping the first order terms in equation (12). According to the theory of second-order hyperbolic equations, these terms affect only the amplitude of the propagating waves. To preserve the kinematics, it is sufficient to keep only the second order terms of equation (12):

 

$$k_\zeta= \pm \sqrt{ \frac{\left(\omega\ss\right)^2}{\c_{\zet... ..._\eta^2 - \frac{\c_{\xi\eta}}{\c_{\zeta\zeta}}k_\xi k_\eta }\;.$$ (24)

Figure 13 illustrates the difference between wavefield extrapolation using equation (12) (panel b) and wavefield extrapolation using equation (24) (panel c). Kinematically, the two images are equivalent and the main changes are related to amplitudes.

 

RCga2.kin.ps
Figure 13 The effect of neglecting the first order terms in Riemannian wavefield extrapolation. From left to right the velocity model with an overlay of the ray coordinate system (a), extrapolation with equation (12) including the first order terms (b), and extrapolation with the simplified equation (24) (c).