Solar Radiation on the Ground
Global Radiation, Shortwave Radiation: (300-3000nm)
UVR: 300-400 nm
PAR: 400-700 nm (plants use for photosynthesis)
NIR : 700-3000nm
Measurement Units: w/m
2
, cal/cm
2 /min, μmol/m 2
/s
1 w/m
2
=4.55 μmol/m
2
/s for PAR
1 w =1 J/s
1 cal = 4.18 J
Energy for a photo at a given wavelength:
E
 m/s) hc

, where h=6.63×10 -34
Js; wavelength (λ) is in meters, c is speed of light (3.0×10
8
At the top of the atmosphere, the wavelength distribution can be calculated by Planck’s
Equation:
M


2
 hc
2

5
( e hc

KT

1 )

C
1

5
( e
C
2

T

1 )
Where: c= speed of light in vacuum,
λ= wavelength, meters
T= temperature in Kelvin degrees.
K=Boltzman constant, 1.38054×10 -23 ws/K
C
1
=3.74151×10 -16
w/m
2
C
2
=1.43879×10 2
mK
M
λ
= blackbody spectral emitance at T (watts per square meter area per meter wavelength)
Blackbody: An idealized object that perfectly absorbs all incident EMR and radiate the maximum amount of energy that a medium at its temperature is capable of radiating.
Solar Constant: The radiation flux density on a unit area perpendicular to the sun beam at the top of the atmosphere. It is primarily a function of Sun-Earth distance. The WMO recommended value is 1367 w/m
2
. However, due to the variation of Sun-Earth distance from day to day, It can not be considered as constant for scientific applications, it fact it varies as
S
0

1367 ( 1 .
0

0 .
033 cos( 2

( J

10 ) / 365 )
Q: Can you tell the percent of variation in S
0
with a year from this model?

Where J is Julian date. (J=1, …, 365 within a year). Solar radiation at the top of the atmosphere is composed on 3.4% ultraviolet, 43.9% of PAR and 52.7% of NIR. Many literature assume 50/50 for PAR/NIR, not mentioning UVR. This is fine if you do not intend to validate you model with measured data.
S
V

0 .
439 S
0
S
N

0 .
527 S
0
Solar radiation has to travel through a thick layer of atmosphere before reaching the surface. During the process, the atmosphere will reflect, scatter, and absorb solar radiation.
Absorption: Water vapor, O
3
Selective, i.e. some spectrum are highly absorbed, and some spectrum are barely absorbed.
Reflection: deflect of radiation in the direction opposite to the incidence angle by large particles and clouds.
Reflection
Scattering: the redirection of incoming solar radiation hitting a small particles in all directions.
Scattering

Depending on the size of the particle, there are two types of scattering: the Raleigh
Scattering and Mie Scattering.
Raleigh Scattering refers to the scattering by atmospheric molecules. It is selective, the shorter the wavelength, the stronger the scattering. Within the visible wavelength, the blue light has the shortest wavelength, thus on a clear day when molecular scattering dominates, the sky looks blue.
Mie Scattering refers to the scattering by atmospheric aerosols, the floating particles, such as dust, pollen, pores, whose diameters are much larger than the wavelength. Mie scattering is not selective, i.e. it scattering radiation in all wavelength equation. Thus when the atmosphere is loaded with aerosols, the sky looks grey.
Because of these processes, solar radiation on the ground is usually much smaller than solar constant, and the solar radiation on the ground is composed of two components: direct and diffuse. We need to figure out the proportions for direct and diffuse radiation for PAR and NIR respectively (i.e. there is a lack of account for UVR in the literature due to its small proportion in the global radiation.)
Why?
(1) Plants only use PAR for photosynthesis, but total radiation is used for temperature and water use.
(2) Plants use diffuse radiation more efficiently than direct radiation.
(3) Leaf spectral properties are very different in PAR and NIR.
(4) Direct and Diffuse radiation penetrate canopy very differently.
Using Beer’s law, the potential direct visible radiation on a horizontal surface is
R
DV

S
V exp(

0 .
185 ( P / P
0
) m ) cos(

)
Where P is the actual air pressure, P
0
is the standard air pressure at see level (101325P a
).
The optical air mass, m, can be estimated as m

1 / cos(

)
The potential diffuse Visible reaching the horizontal surface is considered 40% of the scattered radiation:
R dV

0 .
4 ( S
V

R
DV
) cos(

)
The potential direct NIR reaching the horizontal surface is
R
DN

S
N exp(

0 .
06 ( P / P
0
) m
 w ) cos(

)

And the potential diffuse NIR radiation on the horizontal surface is
R dV

0 .
6 (( S
N

R
DN
)
 w ) cos(

)
ω is the water absorption in the near infrared for 10 mm of precipitation water w

S
N
 anti log
10
(

1 .
195

0 .
4459 log
10 m

0 .
0345 (log
10 m )
2
)
Let,
R
V

R
N

R
DV
R
DN


R dV
R dN
Then, the fraction of direct beam in PAR and NIR are f
V

R
R
DV
V


1

A RATIO
B
2
3 f
N

R
DN
R
N
1

RATIO
C
2
3
Where A=0.9, B=0.7, C=0.88, and D=0.68. RATIO is the measured to potential solar radiation, i.e. RATIO=R
T
/(R
V
+R
N
). Often we do not have R
T
measure, but R
V
only, we can convert R
V to R
T as R
T
=R
V
/0.439.
Calculating solar zenith angle by time of day: cos(

)
 sin( h )
 sin(

) sin(

)
 cos(

) cos(

) cos(

)
Where τ is local time hour angle. τ=15×(t-12) degrees, t is in hours; φ is latitude; δ is sun declination angle.
 
23 .
5 sin
2

( J

284 )
365
A second way of finding diffuse radiation
According to Liu and Jordan (1960), we get the transmittance for diffuse light (τ d
) from the transmittance for the total radiation (τ
T
), which the RATIO above.

 d

0 .
3840

0 .
4160

T
Then we can get the transmittance for direct light

D
 
T
  d
Using Leuning (1995): f d

1

1

2 .
0 (

D
0 .
2

0 .
3 ) 0 .
3


D


D
0 .
3


D

0 .
7
0 .
7
This is for total radiation, not separated for Visible and NIR.