⎡

sin ϕ0 sin ϕ cos ∆ϑ ⎤

⎥+

cos ϕ0

⎣

⎦

(4.3)

(π - ∆ϑ )(*S*c + cos ϕ ) + ∆ϑ (1 + *S*c cos ϕ )

α (1 - cos ϕ )

+ *I*s ↓

π (1 + *S*c )

2

the ratio of the average diffuse radiance from the solar and anti-solar quadraspheres

(Jordan 1991). The last term in Equation (4.3) represents solar radiation reflected back

onto the surface due to the slope.

Up

ϕ0

N

ϕ

ϑ0

ϑ

E

All objects radiate energy over the entire spectrum proportional to their surface properties

and temperature according to the Planck's function, *I*i↑,*emit *= εσ T 4 where, *ε *is the broad

band surface emissivity, *T *(*K*) is the surface temperature, and *σ *is the Stefan-Boltzmann

calculated according to Chapter 3.

The outgoing longwave radiation is composed of an emitted component, which follows

the Stefan-Boltzmann behavior, and a reflected component, which is proportional to the

incoming longwave radiation. According to Kirchoff's law, the emissivity, transmis-

sivity, and reflectance of a surface must sum to 1.0. The transmissivity of soil surfaces is

approximately zero, therefore,

(4.4)

30