High-order kernels for Riemannian Wavefield Extrapolation

Mixed domain -- Fourier finite-differences

The pseudo-screen solution to equation (13) derives from a fourth-order expansion of the square-root around $(a_0,b_0)$ and $(a,b)$:

$$k_\tau \approx \omega a \left [1+\frac{1}{2} \left ( \frac{b k_\gamma }{a \omega... ...ht )^2+ \frac{1}{8} \left ( \frac{b k_\gamma }{a \omega } \right )^4\right ]\;,$$ (47)

$${k_\tau }_0 \approx \omega a_0 \left [1+\frac{1}{2} \left ( \frac{b_0k_\gamma }{a_0 \... ...)^2+ \frac{1}{8} \left ( \frac{b_0k_\gamma }{a_0 \omega } \right )^4\right ]\;.$$ (48)

If we subtract equations (A-16) and (A-17), we obtain the following expression for the wavenumber along the extrapolation direction $k_\tau$:

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right ) + \frac{1}{2}\omega \left [a \left (\frac{b }{a }\right )^2- a_0 \l... ...frac{b_0}{a_0}\right )^2\right ] \left ( \frac{ k_\gamma }{ \omega } \right )^2$$

$$+ \frac{1}{8}\omega \left [a \left (\frac{b }{a }\right )^4- a_0 \l... ...c{b_0}{a_0}\right )^4\right ] \left ( \frac{ k_\gamma }{ \omega } \right )^4\;.$$ (49)

We can make the notations

$$\delta_1 = a \left (\frac{b }{a }\right )^2- a_0 \left (\frac{b_0}{a_0}\right )^2\;,$$ (50)

$$\delta_2 = a \left (\frac{b }{a }\right )^4- a_0 \left (\frac{b_0}{a_0}\right )^4\;,$$ (51)

therefore equation (A-18) can be written as

$$k_\tau = {k_\tau }_0+ \omega \left (a-a_0\right ) + \frac{1}... ...ga \delta_2 \left ( \frac{ k_\gamma }{ \omega } \right )^4\;.$$ (52)

Using the approximation

$$\frac{1}{2}\delta_1 u^2 + \frac{1}{8}\delta_2 u^4 \approx \... ...{1}{2}\delta_1^2 u^2} {\delta_1-\frac{1}{4}\delta_2 u^2} \;,$$ (53)

we can write

$$k_\tau = {k_\tau }_0+ \omega \left (a-a_0\right ) +\omega \f... ...}\delta_2 \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (54)

If we make the notations

$$\nu = \frac{1}{2}\delta_1^2 \;,$$ (55)

$$\mu = \delta_1 \;,$$ (56)

$$\rho = \frac{1}{4}\delta_2 \;,$$ (57)

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

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right )+ \... ...{\mu-\rho \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (58)

High-order kernels for Riemannian Wavefield Extrapolation