High-order kernels for Riemannian Wavefield Extrapolation

Space-domain extrapolation

The space-domain finite-differences solution to equation (13) is derived based on a square-root expansion as suggested by Francis Muir (Claerbout, 1985):

$$k_\tau \approx \omega a + \omega \frac{ \nu \left ( \frac{ k... ...{\mu-\rho \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;,$$ (16)

where the coefficients $\mu$, $\nu$ and $\rho$ take the form derived in Appendix A:

$$\nu = - c_1a \left (\frac{b }{a }\right )^2\;,$$ (17)

$$\mu = 1 \;,$$ (18)

$$\rho = c_2 \left (\frac{b }{a }\right )^2\;.$$ (19)

In the special case of Cartesian coordinates, $a=s$ and $b=1$, equation (16) takes the familiar form

$$k_\tau \approx \omega s - \omega \frac{ \frac{c_1}{s} \left ... ...c_2}{s^2} \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;,$$ (20)

where the coefficients $c_1$ and $c_2$ take different values for different orders of Muir's expansion: $(c_1,c_2)=(0.50,0.00)$ for the $15^\circ$ equation, and $(c_1,c_2)=(0.50,0.25)$ for the $45^\circ$ equation, etc. For extrapolation in Riemannian coordinates, the meaning of $15^\circ$, $45^\circ$ etc is not defined. We use this terminology here to indicate orders of accuracy comparable to the ones defined in Cartesian coordinates.

High-order kernels for Riemannian Wavefield Extrapolation