2

(2.3a)

(2.3b)

α is the ground albedo. Assuming the total transmission *T*k can be specified as the sum of

the transmission of the direct solar component *T*dir and the diffuse solar component *T*dif,

(2.4)

The direct solar flux component at the ground is given as

(2.5)

and the diffuse component as

(2.6)

the above equations for the downwelling direct and diffuse flux it is necessary to assume

(2.7)

and therefore *T*kdir can be obtained from

(2.8)

Shapiro parameterized *T*k and *R*k in terms of the atmospheric and cloud conditions as

follows:

(2.9)

(2.10)

ϕ k = *Wf*k .

(2.11)

ρk is the cloud reflectance for cloud layer *k*, *r*k the clear sky reflectance for layer *k*, τ k the

cloud transmission for layer *k*, *t*k the clear sky transmission for layer *k*, *f*k 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

set.

ρk = *a*ko + *a*1 + *a*k2 cos2 ϕo + *a*k cos3 ϕo

3

(2.12a)

2

3

(2.12b)

1

(2.12c)

τ k = *bb*ko + *bb*k + *bb*k2 cos2 ϕo + *bb*k3 cos3 ϕo .

1

(2.12d)

The coefficients (*a*k's, *aa*k's, bk's and *bb*k's) are parameterized in terms of the following

thick cirrus and cirrostratus, altostratus and altocumulus, and low clouds. The cloud

weighting factor *W *is given as

(2.13)

5