High frequency WKBJ approximation for the Greens function

Numerically, the above ray equation can be solved by a set of ray-tracing algorithms (Cerveny, 2001). 

Expression (13.80) is called WKBJ (ray theoretic) Green s function, because it is associated with the names of several physicists, G. Wentzel, H. A. Kramers, L. Brillouin, and H. Jeffreys, who independently introduced this approximation in connection with the solution of different physical problems (Morse and Feshbach, 1953). I would also recommend an excellent book by Bleistein et al. (2001), where the interested reader can find a more thorough mathematical analysis of the WKBJ approximation. 

In the special case of a 1-D model, the WKBJ approximation of Green s function takes the form (Bleistein et al., 2001, p. 69) 

Note that the 1-D formula (13.83) contains one additional negative power of u in comparison with a 3-D WKBJ approximation (13.80). This term was introduced by analogy with the expression (13.79). Certainly, in the model with a constant wavespeed, c z) = cq, formula (13.83) naturally reduces to the expression for a 1-D Green s function in a homogeneous medium  

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