502 
the Mie extinction efficiency factor (Van de Hulst, 1957) . Moreover, the haze 
composition is integrated into the complex index of refraction m(A). 
where e 
Since most hazes are composed of small particles (0<r<10ym), the quadrature 
conversion will be arranged to emphasize this point, i.e. 
To 
mize the 
3(A) =ir /“2 10 3 “ N(10°) In 10 Q ext ’ m(A)}da (2) 
Here we have transformed the variable r by a = log^o r * Finite limits are given 
Z(f 
i 
Incorpoi 
to the integral to remove a source of singularity from the quadrature matrix. 
Limits are chosen so that N(10 a ) = 0, a<a, or a>a 2 . 
E (f 
i 
A factor 10 a 10 a , appears in the Yamamoto and Tanaka (1969) form of the 
where y 
integral (2) which tends to bring the factor In 10 Q Bxt (2TrlO a /A, m(A)} into the 
family of "well-behaved" functions the equation (2), in this form, becomes 
Mir 
8(A) = tt / a2 f(a) K(a,A)da, 
a i 
f(a) = 10 4a N(10 a ), 
— a 1 0 a 
K(O.X) = 10 “in 10 Q ,m(A)} (3). 
A 
This is an example of the Fredholm integral equation of the first kind. 
where A* 
The method of solving this integral equation given by Phillips and Twomey 
utilizes a notion of constraining the error in measurement of input data to extract 
Solving 
a member of a family of solutions. We introduce matrix notation for the equation 
(3) with the errors in 8 included: 
f = 
g + e = Af (4) 
Data Aqc 
where g = (8(Aj)}, e is the error vector, and f is the solution. A is the quadra 
ture matrix whose elements are given by 
a.. =w. K(a., A.) 
Ji i v i y 
where ok are quadrature weights for integration with respect to the ou . 
Clearly the system (4) does not generally provide a unique solution, there 
fore, by the criterion established by Phillips, we select f as a solution to satisfy 
The 
continer 
1973) ar. 
the phot 
depths v\ 
Haze opt 
braicall 
to Rayle 
gible, s 
r** d 2 f^ 2 j fT d 2 f 2 j 
¿0 57^ dr = y" Jo dr 7 dr 
f eF 
where r* = 10 2 and F is a family of solutions of the system (4). 
An additional constraint is placed on the error vector, namely that its mag 
nitude by constant or 
E e 2 = e 2 (6) 
i 
3 
Tat 
observat 
input ve 
( 6 )