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_&=& a 1- ( b k_a )^2
&& a [1- c_1( b k_a )^2 1-c_2( b k_a )^2 ]
&& a - c_1a (b a )^2( k_ )^2 1-c_2(b a )^2( k_ )^2 . If we make the notations &=& - c_1a (b a )^2,
&=& 1 ,
&=& c_2(b a )^2. we obtain the finite-differences solution to the one-way wave equation in Riemannian coordinates:

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