SECT. 4] LIGHT 413 



G. The Behavior of the Apparent Optical Properties at Great Depths 



It was emphasized repeatedly during the introduction and discussion of the 

 apparent optical properties that they exhibit certain useful, regular behavior 

 patterns. One of the most striking of these patterns occurs at great depths in 

 optically deep natural waters. We briefly summarize here some of the more 

 important of these facts. Proofs of these results, their historical background, 

 and practical consequences are given elsewhere. (Preisendorfer, 1958, 1958a, 

 1958b). 



For simplicity, we consider an infinitely deep source-free homogeneous 

 natural hydrosol. In actuality the results cited below hold in all natural hydro- 

 sols in which the ratio a/a becomes independent of depth with increasing depth. 



In analogy to K{Z, + ), K{Z, — ) and k{Z), we can define one more iT-function. 

 This is associated with radiance N{Z, 6, cf)) : 



Ki7 ft M 1 dN{Z, d, 4>) 



It can be shown that 



(i) k{Z) approaches a limit as Z -^ oo. Let this limit be denoted by kco. In 

 symbols : 



kao = Hmz-^co k{Z). (54) 



It can be shown that kao does not exceed a. In symbols : 



:^ ^00 ^ a. 



(u) For each fixed {6, ^), K{Z, 6, 0) approaches a limit as Z ^^ oo, and this 

 limit is independent of {6, (/>). This common limit for all directions {6, «/») is A^oo. 

 In symbols : 



hmz->«, K{Z, 0, (f>) = kca 



for all {d, (i>). 



{Hi) K{Z, — ) and K{Z, + ) approach limits as Z — > oo and these limits are 

 equal to A:oo. In symbols : 



hmz->oo K{Z, - ) = hmz-^oo K{Z, +) = kao 



{iv) The distribution functions D{Z,+) and D{Z,—) approach a limit as 

 Z -^ CO. Let these limits be denoted by D{ + ) and D{ — ). 



D{-) = \imz^a,D{Z,-) 



(55) 

 D{ + ) = limz^oo D{Z,+). 



(v) The reflectance function R{Z, — ) approaches a limit as Z -> oo. Let this 

 limit be denoted by Roo. In symbols : 



Rao = Hmz^oo R{Z,-). 



