Rayleigh scattering code
King-Fai Li, Vijay Natraj and Yuk L Yung
Literature Review
Coulson et al. (1960) calculated Chandrasekhar’s X and Y functions for Rayleigh
scattering with Lambert’s surface underneath.
Sweigart (1970) calculated Chandrasekhar’s  and  functions for Rayleigh
scattering with absorption.
Caldwell (1971) calculated Chandrasekhar’s X and Y functions for isotropic
scattering by non-iterative method.
Caldwell (1982) calculated Chandrasekhar’s X and Y functions for isotropic
scattering by iterative method using improved trial functions.
Statement of Problem
Rayleigh scattering can be characterized by Chandrasekhar’s X and Y functions,
which are defined by
X i     1    i      X i    X i      Yi    Yi     
1

d 
  
(*)
1

 
Yi     e
   i     Yi    X i      X i    Yi     
d 
0
  
where  is the optical thickness,   cos  ,  being solar zenith angle, i  e, r
0
denotes two perpendicular directions in the plane normal to the direction of
3
propagation of the electromagnetic radiation, and  e     1   2  and
4
3
 r     1   2  .
8
The integrand of Yi has a singular point at     , especially at   1 .
Methods used in previous studies include L’Hospital’s rule [Coulson et al., 1960],
transforming (*) into integro-differential equations [Caldwell, 1971] so that the
singularity disappears, and iterative method in open intervals [Caldwell, 1982]. In
this work, Romberg’s integration modified to open intervals [Numerical Recipes 4.4]
will be used with the integrator MIDPNT. The integration of Yi is split into two
well-behaved regimes for   1 :
   1 ,  1
 0 
.
Yi      1
 ,
 1
 0
References
Schulz, F. M. VDISORT: An improved and generalized discrete ordinate method for
polarized (vector) radiative transfer, J. Quan. Spec Rad. Trans. 61(1) (1999):105-122.
Herman, B. M., COMPARISON OF THE GAUSS-SEIDEL SPHERICAL POLARIZED
RADIATIVE-TRANSFER CODE WITH OTHER RADIATIVE-TRANSFER CODES,
Applied Optics 34.21 (1995):4563-4572.
K. F. Evans and G. L. Stephens, A new polarized atmospheric radiative transfer model,
Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 46, Issue 5,
November 1991, Pages 413-423, doi:10.1016/0022-4073(91)90043-P