(5ui) ; 



u-| ,u 2 ,U3,ui| 



(12) 



where 6f is the standard deviation and (6f) 2 is the variance of the func- 

 tion f which has n independent variables uj_; 6u^ is the standard devia 

 tion and (6ui) 2 is the variance of Uj_. It can be readily shown that from 

 equation (7) the following variance relationship exists: 



'6P \2 



/6a \2 /5k \ 2 

 * la-] + r 



(13) 



where k = Y& + Yf. The normalized variance of each of the variables thus con- 

 tributes equally to the normalized variance of the chlorophyll a in vivo con- 

 centration. Table 1 contains example solutions of equation (13)- It can be 

 seen that the variable with the largest normalized standard deviation dominates 

 the normalized standard deviation for the chlorophyll a concentration. Since 

 power measurements can be made to an accuracy of 2.5 percent, the concentration 

 error is due mostly to uncertainties in fluorescence cross section and effec- 

 tive attenuation coefficient. The expected magnitude of these uncertainties 

 is discussed in a subsequent section. 



Multiple-Wavelength Systems 



Two different approaches to the error analysis for a multiple-wavelength 

 system are used. In the first approach, all of the fluorescence cross sec- 

 tions aj(X^) in equation (8) are considered independent variables. The 

 second approach assumes that the fluorescence cross sections for each algal 

 color group always have the same relative magnitude and vary only in absolute 

 magnitude. 



First approach .- The variance equation which is based upon the first 

 approach can be derived from equation (9) by using the inverse matrix E" 1 

 defined in equation (11) and the general variance equation defined in 

 equation (12). The derivation of the variance equation for chlorophyll a 

 concentration (private communication from R. T. Thompson, Jr., Old Dominion 

 University, Norfolk, Virginia) results in the following relationship: 



(6nj)2 = £ 

 m=1 



(e jm ) 2 (6x m )2 



4 



£ 



i = 1 



(e imni )2(6a mi )2 



where £j m are the elements £~ 1 and the other parameters are the same as 

 those previously defined in deriving equation (11). The normalized form of 

 this equation is 



15 



