Distribution of solar radiation in a rugged (tilted) terrain. As used in the class of Hydrology at the University of Trento

1. Riccardo Rigon
IlSole,F.Lelong,2008,ValdiSella
Solar Radiation
Coping with terrain

2. R. Rigon
Finally the residual radiation hits the terrain
The terrain is not a plane
but it is inclined. Therefore,
besides correcting radiation
for latitude, longitude and
hour, it is necessary to
account for slope and
aspect
2
Hitting the terrain

3. R. Rigon
In the presence of topographic surfaces
In the northern hemisphere, slopes that face South receive a greater insolation
and, therefore, the water in the soil evaporates more quickly or the snow melts
faster. Slopes with differing aspects are often characterized by different species
and densities of plants and trees.
3
Hitting the terrain

4. R. Rigon
Projection of radiation onto an
inclined surface
AfterCorripio,2003
First we calculate the normal to the surface 4
Hitting the terrain

5. R. Rigon
⇧nu =
1
|⇧nu|
⇧
⇧
⇧
⇧
⇤
1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1))
1/2 (z(i,j) + z(i+1,j) z(i,j+1) z(i+1,j+1))
l2
⇥
⌃
⌃
⌃
⌃
⌅
where z are the elevations of the four points used and l2 is the are of the
cell - of side l.
Projection of radiation onto an
inclined surface
Unit normal vector:
5
AfterCorripio,2003
Hitting the terrain

6. R. Rigon
AfterCorripio,2003
6
Representation of the vector normal to the surface of Mount Bianco
Hitting the terrain

7. R. Rigon
AfterCorripio,2003
Projection of radiation onto an
inclined surface
And we compare with the solar vector, indicating the direction of the Sun 7
Hitting the terrain

8. R. Rigon
8
⌥s = ⇤
sin ⇥ cos
sin ⇤ cos ⇥ cos cos ⇤ cos
cos⇤ cos ⇥ cos + sin ⇤ sin
⇥
⌅
Projection of radiation onto an
inclined surface
Where all the quantities were already defined previously
Hitting the terrain

9. R. Rigon
AfterCorripio,2003
Projection of radiation onto an
inclined surface
9Then we calculate the angle between the sun vector and the normal
s
Hitting the terrain

10. R. Rigon
We can define then the angle
of solar incidence
AfterCorripio,2003
Projection of radiation onto an
inclined surface
10
s
Hitting the terrain

11. R. Rigon
Projection of radiation onto an
inclined surface
Angle of solar incidence
cos s = ⌅s · ⌅nu
⇧nu =
1
|⇧nu|
⇧
⇧
⇧
⇧
⇤
1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1))
1/2 (z(i,j) + z(i+1,j) z(i,j+1) z(i+1,j+1))
l2
⇥
⌃
⌃
⌃
⌃
⌅
⌥s = ⇤
sin ⇥ cos
sin ⇤ cos ⇥ cos cos ⇤ cos
cos⇤ cos ⇥ cos + sin ⇤ sin
⇥
⌅
11
Hitting the terrain

12. R. Rigon
s = cos 1
nu.z
Aspect (from the North anti-clockwise)
Projection of radiation onto an
inclined surface
Slope
The above angles needs to be compared with those of the terrain:
12
Hitting the terrain

13. R. Rigon
13
Projection of radiation onto an
inclined surface
Remarkably the form of formula for the incident radiation is the same that
for a flat surface when the projection angle is accounted:
Hitting the terrain

14. R. Rigon
Solar radiation transmitted to the ground under
clear sky conditions
Therefore, for the direct
shortwave radiation:
Corripio,2002
14
S
as, it was before
Hitting the terrain

15. R. Rigon
15
However, it is not just matter of light but also of
shadows
Hitting the terrain

16. R. Rigon
Incident radiation
Topographic effects: shading
16
More schematically
shadow
light
Hitting the terrain

17. R. Rigon
Incident radiation
Topographic effects: shading
17
More schematically
light
shadow
Hitting the terrain

18. R. Rigon
Incident radiation
DetailsinCorripio,2003
18
Therefore the direct solar radiation must be corrected to include shading
Hitting the terrain

19. R. Rigon
sky view factor
diffuse
radiation due to
Rayleigh
scattering
diffuse
radiation due to
aerosols
diffuse
radiation due
multiple
scattering
What about diffuse radiation ?
Topographic effects: angle of view
19
Hitting the terrain

20. R. Rigon
Incident radiation
Topographic effects: angle of view
20
Any point in a rugged landscape see just a part of the sky sphere. Its fraction
says which portion of the sky contribute to diffuse shortwave radiation.
Hitting the terrain

21. R. Rigon
Incident radiation
Topographic effects: angle of view
21
Different points view a different sky
Hitting the terrain

22. R. Rigon
22
The sum
Hitting the terrain

23. R. Rigon
AfterCorripio,2003
23
Now it really hits the terrain
and, in part, it is reflected away
Hitting the terrain

24. R. Rigon
AfterCorripio,2003
24Insolation received by Mont Blanc at Spring Equinox
Finally a map
Hitting the terrain

25. R. Rigon
Typical albedo values
25
http://en.wikipedia.org/wiki/Albedo
Albedo

26. R. Rigon
Typical albedo values
26
http://en.wikipedia.org/wiki/Albedo
Albedo

27. R. Rigon
51
The percentage of radiation that is reflected (reflectance) depends on
wavelength of the radiation, and on the geometry, nature, and structure
of the surface under investigation.
Spectral Signature (or Response)
27
Spectral response

28. R. Rigon
•In the case of solar radiation, the spectral signature is defined
as the reflectance of the surface in function of the wavelength.
28
Spectral response

29. R. Rigon
29
•Every type of surface can be statistically characterised by a spectral signature.
Spectral response

30. R. Rigon
•The spectral signature of a specific element of a territory will
vary due to the variability of local environmental factors.
•Given a certain type of ground cover, static elements, such as
slope and exposition, and dynamic elements, such as surface
ground humidity, the phenological state of the vegetation, the
atmospheric transparence, etc., will cause variations in the
spectral signature curve.
Factors
30
Spectral response

31. R. Rigon
Radiation that hits the terrain, heats it.
Or causes changes of phase
water to vapor
ice to water
31
Spectral response