2D point-source ray coordinates

For the case of 2D point-source ray coordinates the acoustic wave weqrc.3d takes the form  

$$\frac{1}{\AA J} \lb \eone{\lp \frac{J}{\AA} \done{\UU}{\qt} ... ...one{\UU}{\qg} \rp}{\qg} \rb = - \frac{\ww^2}{ v^2\oft} \UU \;,$$ (77)

where, by definition,

Å& = & ^2 + ^2 =v ,
J& = & ^2 + ^2 .

The extrapolation axis is $\qt$ (one-way traveltime from the source) and $\qg$ is the shooting angle at the source.

We can expand the parentheses in weqrc.2d.t

$$\frac{1}{v^2} \dtwo{\UU}{\qt} + \frac{1}{v J} \done{\lp J/... ... \frac{1}{J^2} \dtwo{\UU}{\qg} = - \frac{\ww^2}{v^2\oft} \UU \;$$ (78)

and make the notations

&=& 1Å^2 ,
&=& 1ÅJ JÅ ,
&=& 1ÅJ ÅJ ,
&=& 1 J^2 ,

from which the acoustic wave equation for 2D point-source ray coordinates becomes:

$$\ctt \dtwo{\UU}{\qt} + \ct \done{\UU}{\qt} + \cg \done{\UU}{\qg} + \cgg \dtwo{\UU}{\qg} = - \frac{\ww^2}{v^2} \UU \;.$$ (79)

The 2D dispersion relation is

$$- \ctt \kqt^2 +i\ct \kqt +i\cg \kqg - \cgg \kqg^2 = - \ww^2 s^2 \;,$$ (80)

from which we can obtain the one-way wave equation for 2D point-source ray coordinates:

$$\kqt = i \frac{\ct}{2\ctt} \pm \sqrt{ \frac{\lp\ww s\rp^2}{\... ...}\rp^2 +i \frac{\cg }{\ctt}\kqg - \frac{\cgg}{\ctt}\kqg^2 }\;.$$ (81)