HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS 

 THAT PENETRATE THE BOTTOM 



The integral is performed in Figure 12 where m designates the 

 phase delay per unit displacement. By introducing plane-wave 

 approximations, the field (}) at the point P just under the emergent 

 area is given by the integral shown in Equation (15) over the area 

 of insonif ication. 



The last factor in Equation (15) represents the local plane 

 wave about the field point P, emerging at the exit region. 



The integral is a stochastic integral and, if the insonif led 

 area is large compared to the correlation distance of the displace- 

 ment, C, the exponential can be expanded in a convergent series 

 (Equation 16) . <^> is the average value of the displacement and 

 since the local origin is on the mean surface, <?> = 0. 



Hence, there is no phase shift associated with entry into the 



bottom. <? > is the mean square displacement and results in a loss 



o 1/2 

 of amplitude. For abyssal plains (<?;>) is of the order of 3 to 



10 centimeters and there is very small loss of amplitude associated 



with entry into the bottom. Hence, there is a coherent wave 



arriving at the exit region with very little loss. 



The same type of analysis can be repeated almost word for word 

 for the emergent ray, resulting in a second slight loss of amplitude 

 associated with the mean square displacement at exit region. 

 Typically, the entry and exit regions are far enough apart (several 

 hundred meters) that there is no statistical correlation between 

 <c}> at point Q. 



In summary , it appears that moderate roughness at the bottom- 

 water interface will produce essentially no loss of amplitude on 

 entering or leaving the bottom, and the strength of the refracted 



292 



