It is also observed that the variation in the profile in Figure 1 need 

 not take place at a single location, but may take place anywhere in the 

 vertical cross-section of the ocean between the source and the range value 

 U^, We are, therefore, including in this hypothesis the influence of hori- 

 zontal gradients. The hori7ontal gradients being introduced are the radial 

 gradients about the axis which passes through the source and is perpendicu- 

 lar to the surface. The distribution of sound velocity is then cylindri- 

 cally symmetric about this axis. 



Variation of Profiles 



The variation in the profile from ?-, to Po in Figure 1 generates an 

 infinite set of sound velocity profiles all contained in the shaded region. 

 To each of these profiles corresponds a ray in the shaded region on the 

 right of Figure 1, which extends to the range H^. Each of these rays has 

 associated with it a value of Snell's Constant. It is observed then, that by 

 holding the range fixed, we have established a correspondence between the 

 sound velocity profiles and the value of Snell's Constant. Although little 

 can be said about the general aspects of this correspondence, enough specific 

 information can be obtained to determine the extreme values of the acoustic 

 parameters. 



The difficulty in attempting to describe a sound velocity profile arises 

 because the profile is, in a sense, infinitely dimensional. That is, an in- 

 finite set of values of sound velocity can occur at an infinite number of 

 depth points. However, this paper is concerned with a description of the en- 

 vironment so as to attain the extreme values of the acoustic parameters. 

 That is, our description of the environment does not have to be specific for 

 the intermediate values of the acoustic parameters. This simplifies the 

 first of the above three problem areas with which the evolution of a predic- 

 tion system must cope. 



Environmental Inputs 



The next task is to determine the state of the environment which 

 gives rise to the extreme values of the acoustic parameters. This state 

 turns out to be the extreme sound-velocity profiles that can exist in a 

 given region. In addition to the two extreme profiles we must also know the 

 depths of the source, receiver, and ocean bottom, and, in add: tion, the 

 velocity of sound at the source. 



This description of the environment has several operational implications. 

 The extreme profiles can be accurately established from historical data. In 

 that case, the extreme profiles are valid over a large oceanographic area, 

 and, in most cases, are valid for the entire year. It is therefore unneces- 

 sary to measure the sound velocity profile during Fleet operations. A de- 

 termination of the source and bottom depths can be easily made. The same is 

 true of the sound velocity at the source. It is emphasized that the entire 

 scope of this paper is restricted to bottom-reflected sound in water, suf- 

 ficiently deep, to eliminate bottom-limited profiles. 



48 



