410 TYLER AND PREISENDORFER [CHAP. 8 



radiance distribution in the upper hemisphere (i.e. for the downwelhng flux), 

 and D{Z, +) does a similar job of characterizing the shape of the radiance 

 distribution in the lower hemisphere (i.e. for the upwelling flux). 



Detailed experimental studies of the light fleld in Lake Pend Oreille show 

 that both D{Z,+) and D{Z,—) exhibit relatively httle change with depth 

 (Tyler, 1958). Furthermore, this independence of depth is found whether the 

 external lighting conditions are sunny or overcast (see Table XIV). 



In addition to characterizing the depth dependence of the angular structure 

 of radiance distributions, D{Z, — ) and D{Z, + ) play indispensable roles in the 

 equations of applied radiative transfer theory, particularly in those equations 

 which link the inherent and apparent optical properties of a medium. These 

 roles will be illustrated in the course of the discussion below. 



c. The K-functions 



In this section we now discuss the quantities which characterize the in- 

 dividual depth dependence of the up- and downwelling irradiances and of the 

 scalar irradiance. These are called the iC-functions. The motivations for the 

 definitions of these functions are supplied by both theoretical and experimental 

 precedent extending back over at least fifty years of applied radiative-transfer 

 theory. 



The experimental motivation for the A'-functions rests in early empirical 

 relations of the kind 



Iz = he-^^, (36) 



which simultaneously were to characterize the depth dependence of Iz and 

 define its depth-rate of decay, K.ln the above relation, Iz took many forms: 

 in some studies it was downwelling irradiance, in others it was a scalar 

 irradiance-like quantity ; in still others, its exact nature was not quite clear. 

 It was not until 1938 (Atkins et al., 1938) that there was agreement as to what 

 should really be measured. A plot of Iz on semi-log paper with depth as abscissa 

 yielded —K as the slope of the straight line. K could thus be defined opera- 

 tionally as 



A-=^l„^. (37) 



These early theoretical and experimental approaches to characterize a 7i-like 

 optical property of natural hydrosols were not sufficiently detailed to permit 

 precision and completeness in modern hydrological optics. In current basic 

 research, Iz is replaced bj^ the three precisely defined irradiances H{Z,—), 

 H{Z,-\-) and }i{Z). Furthermore, it has become necessary to distinguish not 

 only between the magnitudes H{Z,—), H{Z,+) and A(Z), but also their 

 logarithmic rate of change with depth. Careful measurements show that their 

 logarithmic rates of change are generally different, and the difference far 

 exceeds the range of experimental error. In general, semi-log plots oi H{Z, —), 



