HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS 

 THAT PENETRATE THE BOTTOM 



the size of the entry area, it will be shown that the Green's function 

 can be expanded about the origin, O , and will be locally a plane wave 

 emerging at the surface. 



The approximations used are listed in Figure 7. The increments 



6x, 5y, 5z, relate to the differences between the variable points in 



the two little areas, and the z coordinate then contains the difference 



in the local mean depths, 6d. Expanding all quantities to linear 



terms in 6x, 6y, 6z, and 6d, the Green's function reduces to Equation 



(6) in Figure 8, where the phase consists of two terms. $ is the 



phase length between and , and $ contains local departures 



from associated with entry and exit points (^-j ' ^ ) and (x , z ), 



respectively. is quite accurately approximated by a local plane 



wave (Equation 9) of emergent angle 6 . 



e 



Note that $ is not symmetric in (x , z ) and (x , z ). This is 

 the point alluded to earlier. If the field point (x , z ) is taken 

 as a new source, then the behavior near the origin has the wrong 

 sign. To remedy this, the first term is always the field point 

 and the second term is always the source coordinate. 



This result is summarized in Figure 9. A source ray enters the 



bottom at point Q at some angle 9 , emerging at point P at the same 



e 



angle. The variable phase delays associated with roughness at points 

 Q and P can then be added to the geometric phase delay via $ . 



The Helmholtz formula (Equation 11) in Figure 10 is used to 

 calculate the field at the point P integrated over the area of 

 insonification. For the Green's function, the linear approximation 

 is used which simplifies the normal gradient in the integral leading 

 to Equation 12 . 



285 



