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Proceedings of Symposium on Remote Sensing and Photo Interpretation

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 conversion will be arranged to emphasize this point, i.e.
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
to the integral to remove a source of singularity from the quadrature matrix.
Limits are chosen so that N(10 a ) = 0, aa 2 .
E (f
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
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).
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
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
1973) ar.
the phot
depths v\
Haze opt
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)
input ve
( 6 )