⎡
sin ϕ0 sin ϕ cos ∆ϑ ⎤
I s ↓slope = I s ↓dir ⎢cos ϕ +
⎥+
cos ϕ0
⎣
⎦
(4.3)
(π - ∆ϑ )(Sc + cos ϕ ) + ∆ϑ (1 + Sc cos ϕ )
α (1 - cos ϕ )
I s ↓dif
+ Is ↓
π (1 + Sc )
2
where all angles are in radians and ∆ϑ ≡ ϑ - ϑ0 and Sc = 1.0 + 0.5 sin ϕ0 + 2.0 sin(2ϕ0 ) is
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
Figure 4.2 Sloped
Solar Radiation Geometry.
4.1.2 Longwave Radiation
All objects radiate energy over the entire spectrum proportional to their surface properties
and temperature according to the Planck's function, Ii↑,emit = εσ T 4 where, ε is the broad
r
band surface emissivity, T (K) is the surface temperature, and σ is the Stefan-Boltzmann
constant (5.669e-08 W/m2K4). The incoming longwave radiation is either measured or
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,
Ii↓ = εσ T 4 + (1 - ε ) Ii↓ .
(4.4)
r
r
30