D2 = d3 ( d1d2 - R1R2T22 ) - d1R2αT32 - R1α (T2T3 )
d j = 1 - R j R j-1 .
α is the ground albedo. Assuming the total transmission Tk can be specified as the sum of
the transmission of the direct solar component Tdir and the diffuse solar component Tdif,
Tk can be written as
Tk = Tkdir + Tkdif .
The direct solar flux component at the ground is given as
I sd↓r = T1dirT2dirT3dir Io↓
and the diffuse component as
I sd↓ff = I s↓ - I sd↓r .
Io↓ (=1369.3 W/m2) is the shortwave flux at the top of the atmosphere. In order to solve
the above equations for the downwelling direct and diffuse flux it is necessary to assume
Tkdif = Rk
and therefore Tkdir can be obtained from
Tkdir = Tk - Tkdif .
Shapiro parameterized Tk and Rk in terms of the atmospheric and cloud conditions as
Rk = ϕ k ρk + (1 - ϕ k ) rk
Tk = ϕ kτ k + (1 - ϕ k )tk
ϕ k = Wfk .
ρk is the cloud reflectance for cloud layer k, rk the clear sky reflectance for layer k, τ k the
cloud transmission for layer k, tk the clear sky transmission for layer k, fk the fractional
cloud amount for layer k, and W is a cloud weighting factor. ρ, r, t, τ , and W are
parameterized in terms of the cosine of the solar zenith angle using the SOLMET data
ρk = ako + a1 + ak2 cos2 ϕo + ak cos3 ϕo
rk = aak + aa1 + aak cos2 ϕo + aak cos3 ϕo
tk = bko + bk + bk2 cos2 ϕo + bk3 cos3 ϕo
τ k = bbko + bbk + bbk2 cos2 ϕo + bbk3 cos3 ϕo .
The coefficients (ak's, aak's, bk's and bbk's) are parameterized in terms of the following
atmospheric and cloud categories: clear, smoke and haze, thin cirrus and cirrostratus,
thick cirrus and cirrostratus, altostratus and altocumulus, and low clouds. The cloud
weighting factor W is given as
W = co + c1 cos ϕo + c2 fk + c3 fk cos ϕo + c4 cos2 ϕo + c5 fk2 .