501 
DETERMINATION OF NATURAL HAZE SIZE DISTRIBUTION FUNCTIONS 
FROM OPTICAL DEPTH OBSERVATIONS 
David A. Rainey, Warner K. Reeser, William E. Marlatt 
Colorado State University 
ABSTRACT 
Work by Phillips and Twomey has provided a technique for solving the Fred 
holm integral equation of the first kind, which arises frequently in remote sensing 
problems. This technique has been applied widely to the problem as determining 
natural haze size distribution functions. 
Determination of a distribution function is made using optical depth measure 
ments for the earth’s atmosphere observed at the ground. This process yields an 
average distribution for an aerosol layer one kilometer in height. 
INTRODUCTION 
A principle problem encountered in studies of the effect on attenuation of 
solar radiation in the atmosphere by natural hazes reduces to that of characteri 
zing the haze by composition and size distribution. The standard approach to this 
problem is to solve the integral equation which gives the extinction coefficient 
for the layer. This is equivalent to inverting the linear system given by the 
numerical quadrature form of the integral. Phillips (1962) proposed a method of 
solution in which the solution is picked from a family of solutions by placing a 
constraint of the error function of the data and subjecting the solution to a 
smoothness criterion. Twomey (1963) provided a form of Phillips solution that 
did not require the quadrature matrix to be square. 
The method was applied to horizontal extinction measurements over Chesapeake 
Bay by Yamamoto and Tanaka. Their work showed that natural hazes are fairly well 
approximated by a power law. 
The method used by Yamamoto and Tanaka is applied here to optical depth data 
where extinction is obtained by assuming that the optical depth measured is the 
extinction of a haze of one kilometer in height. 
As in other studies of this type, Mie scattering is assumed, 
METHOD 
The extinction coefficient, 3, may be given as a function of wavelength, 
A, by the integral equation 
3(A) = tt /°° r 2 N(r)Q ext (x, m(A) }dr 0-1 
^ -3 -1 
where the solution N(r) is the particle size distribution in cm Qim) , r is the 
particle radius, and x = 2-nr/X is the Mie size parameter. The factor is