HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS 

 THAT PENETRATE THE BOTTOM 



This particular form of the Green's function is not genuinely 

 symmetric in source and receiver as a Green's function should be. 

 The source is singled out by being placed at the origin leading to 

 the asymmetry in z which is the location of the field point. In 

 some subsequent expansions, the order of the z term will be switched 

 because of this asymmetry and this can be easily justified by 

 appealing to physical intuition. As is well known, in the linear 

 velocity medium rays are circles whose centers lie on the plane where 

 the velocity goes to zero. R, as seen in Figure 6, is the distance 

 from the "image" source a distance z above this plane to the observa- 

 tion point. 



The view being taken of the bottom is shown in the middle portion 

 of Figure 6, where the ray enters in the first region, is refracted 

 downward, and emerges at the exit region through a different patch of 

 the ocean floor. It may also have reflected one or more times in the 

 middle region. A set of local coordinate systems is introduced in 

 the lower portion of Figure 6, where it is assumed that the ocean may 

 have a mean displacement in the entry region referred to the mean value 

 of the sea floor in the reflection region, and the exit region may 

 have yet another mean displacement. Hence, there will be phase 

 differences involved in the travel paths in the bottom associated 

 with these mean displacements. The phases can be given additional 

 statistical fluctuations associated with roughness in the local areas 

 where the sound enters the bottom, reflects, and emerges. The distance 

 L between the coordinate origins at the entry and exit regions is the 

 horizontal distance of the refracted beam in the bottom. 



At this point, it is convenient to make a number of assumptions. 

 First, the regions should be well separated, that is, they should have 

 small linear dimensions compared to L. When L is large compared to 



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