High-order kernels for Riemannian Wavefield Extrapolation

Space-domain finite-differences

Starting from equation (13), based on the Muir expansion for the square-root (Claerbout, 1985), we can write successively:

$$k_\tau = \omega a \sqrt{1- \left ( \frac{b k_\gamma }{a \omega } \right )^2}$$ (32)

$$\approx \omega a \left [1- \frac{ c_1 \left ( \frac{b k_\gamma }{a \omega } \right )^2} {1-c_2 \left ( \frac{b k_\gamma }{a \omega } \right )^2} \right ]$$ (33)

$$\approx \omega a - \omega \frac{ c_1a \left (\frac{b }{a }\right )^2 \lef... ...ft (\frac{b }{a }\right )^2 \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (34)

If we make the notations

$$\nu = - c_1a \left (\frac{b }{a }\right )^2\;,$$ (35)

$$\mu = 1 \;,$$ (36)

$$\rho = c_2 \left (\frac{b }{a }\right )^2\;.$$ (37)

we obtain the finite-differences solution to the one-way wave equation in Riemannian coordinates:

$$k_\tau \approx \omega a + \omega \frac{ \nu \left ( \frac{ k... ...{\mu-\rho \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (38)

High-order kernels for Riemannian Wavefield Extrapolation