By using the definition for a second-order exponential integral, 



f™ e-*y 



E 2 (x) = J dy 



1 y 2 



(eq. 5.1.4 of ref. 21) and the recurrence relationship, 



E 2 (x) = e" x - xE-|(x) 

 (eq. 5.1.14 of ref. 21), the power received equation can be written as 



Pr 



^[l - ( Y£ + Yf )mRe ( ^ + ^ )mR Et [( Yji + Yf )m R ]j (5) 



Since the smallest values for the effective attenuation coefficients are 



the respective absorption coefficients, af = 0.45 m -1 at 685 ran and 



a a > 0.05 m -1 between 400 ran ^ X^ ^ 650 nm (ref. 17). Thus, the value 



of Y £ + Y f must be greater than 0.50 m~\ and since m « 1.33 and R = 100 m 



for most cases, ( Y £ + Y f)mR ^ 66. The functional representation for E-|(x) 



can be shown to be approximated with less than 0.1 -percent error by the first 



two terms in its asymptotic expansion, 



when x ^ 66 (eq. 5.1.51 and table 5.2 of ref. 21). Substitution of this 

 approximation into equation (5) yields the following general equation for the 

 fluorescence power detected by the laser fluorosensor system: 



?r : 



P ^A r AXdO - pj,)(1 - p f )e- (6 ^ )R 

 4 it AXf( Y j, + Y f)m 2 R 2 



The uncertainty discussed previously in specifying the values of Y £ and Y f 

 makes the small correction factors for surface reflectivity and atmospheric 

 attenuation for ranges less than 1 km negligible. Thus, the simplified form 

 of the power received is 



10 



