Amplitude equivalence

IKO is amplitude equivalent if the amplitude $A^{(\pm)}(r)$ satisfies transport equation of the ray method (see equation (36), Chapter 4). It follows from the definition that equations (86) and (90) describe amplitude equivalent operators for homogeneous 3D and 2D media. And it is important to notice that the omitted terms (compare equations (85) and (86) as well (88) and (86)) do not influence the behavior of the senior discontinuity.

In order to construct the amplitude equivalent IKO for continuous inhomogeneous media, we may use the integral representation (82) of the solution of equation (81). It was mentioned above that the explicit expression of Green's function for inhomogeneous media is unknown. But it is sufficient to find the senior discontinuity of the Green's function by the ray method.

Ray approximation of Green's function for the equation (81) with mixed boundary conditions is

$$\begin{array} {cl} g^{(\pm)}({\bf r}, {\bf r}_0, t) \stackre... ... r}_0^-)] \over \sqrt{{J} ({\bf r},{\bf r}_0^-)} }\end{array}$$

where q=-(n-1)/2, n-dimension of the problem $(n=2\ or\ 3), \sqrt{{J}({\bf r}, {\bf r}_0)}$-geometrical spreading, ${\bf r}_0^{(-)}$-the point which is symmetrical to ${\bf r}_0$ relatively to plane z=0. The medium at z<0 is symmetrical to the medium at z>0.

Finally one can receive that the weight function of the amplitude equivalent IKO is

$$w({\bf r}, {\bf r}_0)= w_E({\bf r}, {\bf r}_0)\equiv {\cos\... ...({\bf r}) v({\bf r})} {1\over \sqrt{{J} ({\bf r}, {\bf r}_0)}}$$

where $\theta$ is the angle between the ray from ${\bf r}$ to ${\bf r}_0$and the vertical axis.