High-order kernels for Riemannian Wavefield Extrapolation

Mixed domain -- pseudo-screen

The pseudo-screen solution to equation (13) derives from a first-order expansion of the square-root around $a_0$ and $b_0$ which are reference values for the medium characterized by the parameters $a$ and $b$:

$$k_\tau \approx {k_\tau }_0+ \left. \done{k_\tau }{a} \right\... ...ne{k_\tau }{b} \right\vert _{a_0,b_0} \left (b-b_0\right )\;.$$ (39)

The partial derivatives relative to $a$ and $b$, respectively, are:

$$\left. \done{k_\tau }{a} \right\vert _{a_0,b_0} = \omega \frac{1}{\sqrt{1- \left ( \frac{b_0k_\gamma }{a_0 \omega }... ...ht )^2} {1-3c_2 \left ( \frac{b_0k_\gamma }{a_0 \omega } \right )^2}\right ]\;,$$ (40)

$$\left. \done{k_\tau }{b} \right\vert _{a_0,b_0} = -\omega \frac{b_0}{a_0} \left ( \frac{ k_\gamma }{ \omega } \righ... ... -\omega \frac{a_0}{b_0} \left ( \frac{b_0k_\gamma }{a_0 \omega } \right )^2\;.$$ (41)

Therefore, the pseudo-screen equation becomes

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right )+ \... ...right )^2 \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (42)

If we make the notations

$$\nu = a_0 \left [c_1 \left (\frac{a}{a_0}-1 \right )- \left (\frac{b}{b_0}-1 \right )\right ] \left (\frac{b_0}{a_0}\right )^2$$ (43)

$$\mu = 1$$ (44)

$$\rho = 3c_2 \left (\frac{b_0}{a_0}\right )^2$$ (45)

we obtain the mixed-domain pseudo-screen solution to the one-way wave equation in Riemannian coordinates:

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right )+ \... ...{\mu-\rho \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (46)

High-order kernels for Riemannian Wavefield Extrapolation