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 $b$:

$$k_\tau \approx {k_\tau }_0+ \left. \frac{\partial k_\tau }{\... ... }{\partial b } \right\vert _{a_0,b_0} \left (b-b_0\right)\;.$$ (17)

The partial derivatives relative to $a$ and $b$, respectively, are: . k_a |_a_0,b_0 &=& 11-( b_0k_a_0 )^2 [1+ c_1( b_0k_a_0 )^2 1-3c_2( b_0k_a_0 )^2],
. k_b |_a_0,b_0 &=& -b_0a_0( k_ )^211-( b_0k_a_0 )^2 -a_0b_0 ( b_0k_a_0 )^2. Therefore, the pseudo-screen equation becomes

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right )+ \... ...right )^2 \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (18)

If we make the notations &=& a_0[c_1(aa_0-1 )- (bb_0-1 )](b_0a_0)^2
&=& 1
&=& 3c_2(b_0a_0)^2 we obtain the mixed-domain pseudo-screen solution to the one-way wave equation in Riemannian coordinates:

$$k_\tau \approx {k_\tau }_0+ \omega \left (a-a_0\right )+ \... ...{\mu-\rho \left ( \frac{ k_\gamma }{ \omega } \right )^2} \;.$$ (19)

High-order kernels for Riemannian Wavefield Extrapolation