Space-domain solution

The space-domain finite-differences solution to equation (9) is derived based on the square-root expansion, first introduced to Geophysics by Muir Claerbout (1985):  

$$k_\tau\approx \omega a + \omega\frac{ \nu \left (\frac{ k_\g... ... )^2} {\mu-\rho\left (\frac{ k_\gamma}{ \omega}\right )^2} \;,$$ (10)

where the coefficients $\mu$, $\nu$ and $\rho$ take the form:

$$\left\{ \begin{array} {l} \nu = - c_1a \left (\frac{b }{a }\... ...\\ \rho= b\left (\frac{b }{a }\right )^2\;. \end{array}\right.$$ (11)

In the special case of Cartesian coordinates, a=s and b=1, equation (10) takes the familiar form

$$k_\tau\approx \omega s - \omega\frac{ \frac{c_1}{s} \left (\... ... -\frac{b}{s^2} \left (\frac{ k_\gamma}{ \omega}\right )^2} \;,$$ (12)

where the coefficients c₁ and b take different values for different orders of the Muir expansion: c₁₅=(c₁,b)=(0.50,0.00) for the $15^\circ$ equation, and c₄₅=(c₁,b)=(0.50,0.25) for the $45^\circ$ equation.