SECT. 4] LIGHT 409 



a. The reflectance functions 



The refiectance functions are defined by : 



R{Z,-) = H{Z,+)IH{Z,-) 



(34) 

 B{Z,+) ^ H{Z,-)IH{Z,+). 



The physical interpretation of R{Z, — ) is straightforward : it represents the 

 ratio of the iipwelhng irradiance at depth Z to the downwelhng irradiance at 

 depth Z, so that B{Z, — ) may be thought of as the reflectance, with respect to 

 the downweUing flux, of a hypothetical plane surface at depth Z in the medium. 

 For completeness, we have included the reflectance R{Z, +) for the upwelling 

 stream. However, this is simply the reciprocal oi R{Z, — ). In actuality, E{Z, —) 

 depends on the scattering properties of the entire medium above and below 

 this level. It will also depend in part on the reflectance properties of the upper 

 and lower boiuidaries of the medium if these are within sight of the flux col- 

 lectors. R{Z, — ) is not an inherent property of the medium, for experiments and 

 theory show, in general, that for a given medium and a given depth in that 

 medium, the value R{Z, — ) changes with the external lighting conditions. In 

 optically deep homogeneous hydrosols, R{Z, — ) varies very little with depth. 

 Near the surface of these media, it shows relatively high variability with depth 

 depending on the state of the surface and incident lighting patterns, but with 

 increasing depth soon settles down and approaches a coilstant value independent 

 of depth. B{Z, ) thereby takes on the status of an apparent optical property 

 of the medium. Furthermore, in media that have no self-luminous organisms, 

 R{Z, — ) behaves as reflectance should : it is never greater than one. In fact, in 

 most natural hydrosols the values of R{Z, —) are usually found to be some- 

 where in the neighborhood of 0.02, for green light. In media containing self- 

 luminous organisms distributed throughout some layer, it is quite possible, 

 however, for the values of R{Z, — ) to approach one as this layer is approached, 

 and even become greater than one just before it enters the layer. 



b. The distribution functions 



A particularly simple means of characterizing the depth dependence of the 

 shape of radiance distributions, without resorting to an actual measurement of 

 the radiance over all directions at each depth, is given by the distribution 

 functions : 



D{Z,-) = h{Z,-)jH{Z,-) 



D{Z,+) = h{Z,+)IH{Z,+). 



It is easily seen from the definitions of h and H that if the shape of the radiance 

 distribution changes with depth, then D{Z, — ) and D{Z, +) wiU also change 

 with depth ; and conversely, if the values of the distribution functions vary 

 with depth, the radiance distributions must be changing shape with depth. It 

 is clear from the definitions that D{Z, — ) gives an index of the shape of the 



