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A geometric ray has a phase kz z, where z is the depth of the ray. The WKB approximation of a normal mode depth function has a phase ± ∞ z′ z kz (z, km) dz for upward and downward traveling waves where kz = k2 (z) - km2; z′ is a constant. See, for example, "," J. Acoust. Soc. Am.,. Hence for the path integral, ei kz z is replaced by the mode depth function, and for the second moment, one inserts the mode depth-function squared, integrated over depth.
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