HORTON: THE EFFECT OF ROUGH INTERFACES ON SIGNALS 

 THAT PENETRATE THE BOTTOM 



the parameter "a" can be related to the gradient in the sediment via 

 the approximation (Equation 2) . Given a narrow beam impinging on 

 the bottom, the refracted path in the bottom will be as shown in 

 Figure 2 with subsequent multiple reflections. A phase shift of it/2 

 occurs at each turning point in the sediment. 



Two papers treat this problem. One is by Morris (1970) in which 

 reflection bottom-loss curves are computed for the linear gradiant 

 using the refracting layer plus an additional semi- infinite layer 

 below the sediment. In the second, Brekhovskikh (1960) treats the 

 case of a continuous velocity value across the interface (that is, 

 without the step discontinuity shown in Figure 1). Both papers treat 

 plane waves and obtain a complex reflection coefficient. Brekhovskikh 

 (Equation 3) assumes no losses in the bottom and, hence, the reflection 

 coefficient has unity magnitude. Morris (1970) adds attenuation in 

 the bottom, and the refracting waves have less than unity magnitude. 



A major point of this paper is that if the problem is actually 

 for narrow beams, the result should be similar to a Rayleigh-type 

 plane-wave reflection coefficient, expandable in an infinite series 

 corresponding to the multiple bounces. This is analogous to the 

 treatment of a transmission line where the transmission loss through 

 it is calculated using a continuous wave but it can be expressed 

 as an infinite series of multiple reflections from the two ends 

 of the transmission line. When the result in Equation 3 is expanded 

 properly, it should become a reflection coefficient for the surface 

 with separate amplitudes for the successive waves corresponding to 

 the refracted and reflected paths in the bottom. 



Brekhovskikh analyzes the case where there are no losses in the 

 bottom and the velocity is continuous from the water into the bottom. 



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