SECT. 4] TTNDEUWATER VISIBILITY 453 



wherein the depth of the target 2^ = 21 and the depth of the observer 2 = 22, 

 then (2), (3) and (4) can be combined and integrated throughout the ])ath of 

 sight to show that the apparent spectral radiance of the target. iNr{z, 6, (/>), is 

 related to the inherent spectral radiance of the target, t^oizi, 6, </>), by the 

 equation 



tNr{z, e, <f>) = tNoizt, e, </.) exp {-a(2)r} 



+ N{zt, d, <i>) exp { + K{z, d, <f>)r cos ^} [ 1 - exp { - a{z)r + K{z, d,(f>)r cos d}], (5) 



wherein the first term on the right represents the residual image-forming light 

 from the target and the second term represents radiance contributed by the 

 scattering of ambient light in the sea throughout the path of sight. 



If the target is seen against a background of inherent spectral radiance, 

 bNo{zt, 6, (f>), the apparent spectral radiance, bNr{z, 6, (f>), of the background will 

 be given by an equation identical with (5) after replacing the presubscripts t 

 by b. This equation and equation (5) can be combined with the defining relations 

 for inherent spectral contrast, Co{zt, 6, ^), and apparent spectral contrast 

 Cr{z, 6, (f)), which are, respectively, 



Coizt, d, <t>) = [tNoizt, e, ct>)-bNo{zt, e, cf>)]lbNo{zt, d, <f>), 



and 



Cr{z, e, «/.) = [tNriz, d, <f>)-bNr{z, 6, <f>)]lbNr{z, 6, </>). 



When this is done, the ratio of inherent spectral contrast to the apparent 

 spectral contrast is found to be 



Coizt, 6, cf>)ICr{z, d, <f>) 



= 1 - [iV(2,, e, (f>)lbNo{zt, e, <^)][1 - exp {a{z)r - K{z, 6, cf>) r cos d}]. (6) 



In the special case of an object suspended in deep water, bNo{zt, 6, <j)) = N{zt, 6, cj)) 

 so that 



Cr{z, d, cf>) = Co{zt, e, </.) exp { - a{z)r + K{z, d, cf>)r cos d}. (7) 



When the observer's path of sight to an object seen against a background of 

 water is horizontal the apparent spectral contrast is 



Cr{z) = Co(2)exp{-a(2)r}, (8) 



which indicates that the apparent contrast is independent of azimuth and 

 depends only on the total attenuation coefficient, a(2), for image-forming light. 

 For many practical purposes it is a sufficient approximation to assume 

 K{z, 6, (f)) to be independent of direction and to be of the same magnitude as 

 the irradiance X-functions described in the previous chapter, i.e. to assume 

 K{z, 6, (f>) = k{z, - ) = K{z, - ) = K{z, + ) = K{z) in all of the foregoing equations, 

 indicating thereby that the reduction of contrast in image transmission through 

 water is virtually independent of azimuth. The principles described by the 

 foregoing equations were first discovered in the course of experiments with an 



