1. Solar Radiation
Physics and Geometry
for hydrologists
Il Sole, F. Lelong, 2008, Val di Sella
Riccardo Rigon
Monday, December 10, 12
2. When you see the Sun rise,
do you not see a round disc of fire
somewhat like a guinea?
Oh no, no! I see an innumerable
company of heavenly host
crying
“Glory, glory, glory is the Lord God
Almighty.”
W. Blake
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3. Introduction
Educational Goals
• To recognise that the water cycle is powered by solar energy
• To gain knowledge of the spatial and temporal variation of the
radiation distribution on the Earth
• To present the ways in which radiation is produced, received by the
Earth, transmitted by the atmosphere, reflected, absorbed, and reemitted
by the Earth’s surface
• To introduce the concepts necessary to better understand the elements
of the energy balance needed in remote-sensing applications, the snow
balance, and evapotranspiration
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4. The Sun
The Sun is the origin of the water
cycle
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5. The Sun
Composition of the Sun
The Sun is mainly composed of hydrogen. The rest is prevalently He4.
Hydrogen is the fuel for the nuclear fusion that takes place inside the Sun and
produces helium. However, the He4 contained in the Sun for the most part
originates from previous stellar lives. 5
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6. The Sun
Sun Fact Sheet
The Sun is a G2 type star, one of the hundred billion stars of this type in our
galaxy (one of the hundred billion galaxies in the known universe).
Diameter: 1,390,000 km (the Earth: 12,742 km or 100 times smaller)
Mass: 1.1989 x 1030 kg (333,000 times the mass of the Earth)
Temperature: 5800 K (at the surface) 15,600,000 K (at the core)
The Sun contains 99.8% of the total mass of the Solar System (Jupiter
contains nearly all the rest).
Chemical composition:
Hydrogen 92.1%
Helium 7.8%
6
Other elements: 0.1%
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7. The Sun
The Sun and the planets to scale
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8. The Sun
The internal structure of the Sun
The Sun’s energy is created in the core by fusing hydrogen into helium. This
energy is irradiated through the radiative layer, then transmitted by convection
through the convective layer, and, finally, radiated through the photosphere,
which is the part of the Sun that we see. 8
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9. The Sun
Provide a relatively constant rate of radiation energy that in few minutes
from the cromosphere arrives to the Earth.
Detail of a Pellizza da Volpedo Painting
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10. The Sun
Solar Spots
Radiation flux is regular up to a point. In reality it manifests variations.
Solar spots appear as dark spots on the surface of the Sun and they have a
temperature of 3,700 K (to be compared to the 5,800 K of the surrounding
photosphere). A solar spot can last for may days, the most persistent lasting for
many weeks.
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11. The Sun
Variability of the Emissions
An image of the sun in X-ray
band, taken by the Yohkoh solar
observatory satellite, which
shows changes in emissions of
the solar corona from a
maximum in 1991 (left) to a
minimum in 1995 (right).
11
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12. The Sun
Variability of the Emissions
Solar radiation is subject to
fluctuations, some of which are
localised in restricted areas, while
others are more global and follow
an 11-year cycle.
Every 11 years the sun goes from
a limited number of solar spots
and flares to a maximum, and
vice versa. During this cycle the
Sun’s magnetic poles switch
orientation. The last solar
minimum was in 2006.
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13. The Sun
Variability of the Emissions
The graph shows the solar spot cycle over the last 400 years. It should be
noted that before 1700 there was a period in which very few solar spots were
observed. This period coincides with the Little Ice Age, which is why there are
suggestions that there is a connection between solar spot activity and the
climate on Earth. The most evident cycle has a period of 11 years. But there
is a second cycle which seems to have a period of 55-57 years.
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14. The Sun
The Stefan-Boltzmann law
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
R=✏ T4
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15. The Sun
The Stefan-Boltzmann law
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
R=✏ T4
Radiation
emitted
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16. The Sun
The Stefan-Boltzmann law
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
R=✏ T4
emissivity
Radiation
emitted
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17. The Sun
The Stefan-Boltzmann law
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
R=✏ T4
Stefan-Boltzmann constant
emissivity
Radiation
emitted
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18. The Sun
The Stefan-Boltzmann law
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
R=✏ T4 absolute temperature
Stefan-Boltzmann constant
emissivity
Radiation
emitted
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19. The Sun
The physics of Radiation
On the basis of the temperature of the Sun photosphere (~6000 K), and the
Stephan-Boltzmann law, the total energy emitted by the Sun is
RSun = ✏ T 4 = 1 ⇤ 5.67 ⇤ 10 8
⇤ 60004 ⇡ 25.12 ⇤ 109 J m 2
s 1
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20. The Sun
The Sun is nearly a “blackbody”!
The Sun is practically a blackbody. The difference between a true blackbody
and the Sun is due to the fact that the corona and the chromosphere
selectively absorb certain wavelengths. 16
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21. The Sun
The Sun is nearly a “blackbody”!
The area below the curves is given by the Stefan-Boltzmann law. The curves
themselves are given by Planck’s law.
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22. The Sun
The complete electromagnetic spectrum
Figure 2.9
C.B. Agee
The spectrum of solar radiation stretches far beyond the visible band where,
however, nearly half the available energy is concentrated
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23. The Sun
Planck’s Law
•Planck’s law is the general law for electromagnetic emission from the
surface of a blackbody*:
2⇡c2 h 5
W = ch [W m 2
µm 1
]
e KT 1
* Stefan-Boltzmann law is just the integration of Plank’s law over wavelengths 19
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24. From Sun To Earth
From Sun to Earth
The energy irradiated by the Sun passes through an imaginary disc with diameter
the same as the Earth’s. The energy flow is maximum at that point on the Earth
where the radiation is perpendicular.
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25. From Sun To Earth
Solar radiation
The Sun irradiates
approximately at the solar
constant rate, which is, on
the average, on the top of
the atmosphere,
Frolich, 1985
http://en.wikipedia.org/wiki/Solar_constant
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26. Copying with Earth surface
Astronomical variability of radiation
In its orbit around the Sun, the Earth keeps its north-south rotational axis
unvaried, causing a different angle between the Sun’s rays and the surface of the
Earth. 22
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27. From Sun To Earth
Seasons
Figure 3.1
The Earth is 5 million kilometers closer to the Sun during the northern
winter: a clear indication that temperature is controlled more by orientation
than by distance.
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28. From Sun To Earth
Corrections to the solar constant
The Earth’s orbit around the Sun is an ellipse. The shape of the ellipse is
determined by its eccentricity, which varies in time, changing the distances of
the aphelion and perihelion
24
http://www.ascensionrecta.com/
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29. From Sun To Earth
Precession of the polar axis
The axis of rotation moves with a slow period, executing
a complete precession every 26,000 years.
Polar stars behave like this for only a very short period
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30. From Sun To Earth
Astronomical influences
Orbit shape
Orbit change
Orbit angle
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31. From Sun To Earth
Solar radiation in
hydrological models
Therefore the solar contant must be corrected
S (e.g. Corripio, 2002):
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32. From Sun To Earth
Solar radiation in
hydrological models
Therefore the solar contant must be corrected
S (e.g. Corripio, 2002):
where:
N is the day of the year (in 1, ..., 365)
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33. Copying with Earth surface
Radiation intensity
Solar intensity governs seasonal climatic changes and the local climatic niches
which are linked to the apparent height of the Sun.
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34. Copying with Earth surface
Insolation and latitude
Figure 3.7
Incoming solar radiation is not evenly distributed across all lines of latitude,
creating a heating imbalance.
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35. Copying with Earth surface
Radiative imbalance
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36. Copying with Earth surface
Radiation received from the Sun
decreases towards the poles and it is reduced in areas where clouds
form frequently
For example, the complete energy balance is greater at the equator but the
greatest amount of insolation is in the subtropical deserts
Average annual radiation is
< 80 W/m2 in the cloudy parts of the arctic and the antarctic
>280 W/m2 in the subtropical deserts
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37. Copying with Earth surface
The geometry of radiation
From a subjective point of view, the Sun varies its height in the sky seasonally.
This is the subject of interest in the study of the geometry of radiation.
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38. Copying with Earth surface
To sum up
Calculations of the incident radiation onto the surface of the Earth need to
take account of the geometry of the interaction between the Sun’s rays and
the surface of the Earth, which is curved and therefore variably
exposed with respect to the direction of the Sun in function of latitude,
time of day (longitude) and, naturally, day of the year. Moreover the
Earth rotation is inclined with respect to its orbit around the Sun , and
this causes seasons to happen.
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39. Copying with Earth surface
The geometry of radiation
To calculate the aforementioned
quantities it is usual to use a
topocentric coordinate system,
Nautic Almanac Office, 1974
that is, with the origin in the
geographic position of the
observer, which is right-handed
and positioned on the plane
tangent to the Earth’s surface in
the considered point.
N.B. - A coordinate system located at the
centre of the Earth id called geocentric.
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40. Copying with Earth surface
The geometry of radiation
The X-axis is, therefore, tangent
to the earth and positive in a
West-East direction. The Y-axis
Nautic Almanac Office, 1974
is tangent in the North-South
direction and is directed towards
the South. The Z-axis lies on the
segment joining the centre of the
Earth with the point being
considered on the surface.
It is assumed that the Sun lies in
the ZY plane at the solar noon.
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41. Copying with Earth surface
Solar Vector
The solar vector can be expressed as a
function of the angles that have been
defined. The resulting trigonometric
expression is:
Z ⇥
sin ⇥ cos
⌥ = ⇤ sin ⇤ cos ⇥ cos
s cos ⇤ cos ⌅
cos⇤ cos ⇥ cos + sin ⇤ sin
Y
X Therefore, to determine the position of
the Sun one needs to know the latitude,
the hour angle, and the solar
declination. 37
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42. Copying with Earth surface
Hour angle
The hour angle can be easily
calculated as:
⇥
t
⇥= 1
12
if t is the solar hour
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43. Copying with Earth surface
Solar declination
Is the angular height of Sun from the horizon at equator at noon*
The solar declination is a function of the day of the year (and the era). It
requires complex calculations that take account of the precession movements
of the Earth. There are, however, various approximations. The one that is
presented here is due to Bourges, 1985:
where is the day of the year
*http://en.wikipedia.org/wiki/Declination
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44. Copying with Earth surface
Projection on a plane at a certain
latitude
If is the vertical unit row-vector
corresponding to the Z axis:
Z
and
⇥
sin ⇥ cos
Y ⌥ = ⇤ sin ⇤ cos ⇥ cos
s cos ⇤ cos ⌅
cos⇤ cos ⇥ cos + sin ⇤ sin
X
is the solar vector
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45. Copying with Earth surface
Projection on a plane at a certain
latitude
Then the projection of the solar
irradiation on the plane YX is reduced by
the factor where:
Z
or:
Y
X
with the symbols explained above
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46. Copying with Earth surface
To sum up:
The solar constant can be modified as follows.
Was:
Is now:
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47. Absorption and transmission of short wave radiation
Atmosphere is a gray body
• The blackbody is an ideal object that absorb all the radiative energy it receives
• Real objects (bodies, “gray bodies”) are not capable of absorbing all radiation.
• To understand the difference between a blackbody and a gray body we need to
analyse the interactions between a surface and the electromagnetic radiation
incident onto it.
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48. Absorption and transmission of short wave radiation
Atmospheric absorption
Radiation passes quite freely through the Earth’s atmosphere and it warms
the surfaces of seas and oceans. A portion of between 45% and 50% of the
incident radiation onto the Earth reaches the ground
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49. Absorption and transmission of short wave radiation
Shortwave Radiation budget
The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.
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50. Absorption and transmission of short wave radiation
Shortwave Radiation budget
Radiation reflected
The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.
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51. Absorption and transmission of short wave radiation
Shortwave Radiation budget
Radiation reflected
The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.
Radiation
transmitted 45
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52. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S It should not be forgot that
the radiation budget is an
energy budget, for which
the incoming radiation equals
the reflected one plus
the absorbed plus
the transmitted
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53. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S It should not be forgot that
the radiation budget is an
energy budget, for which
the incoming radiation equals
the reflected one plus
the absorbed plus
Radiation
the transmitted
absorbed
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54. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
This budget can be apply to any slice of the atmosphere
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55. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
Corrected Solar
constant
This budget can be apply to any slice of the atmosphere
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56. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
Solar radiation
reflected back to space
Corrected Solar
constant
This budget can be apply to any slice of the atmosphere
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57. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
Transmitted
radiation
Solar radiation
reflected back to space
Corrected Solar
constant
This budget can be apply to any slice of the atmosphere
47
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58. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
Transmitted
radiation
Solar radiation
reflected back to space Energy
absorbed by
atmosphere
Corrected Solar
constant
This budget can be apply to any slice of the atmosphere
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59. Absorption and transmission of short wave radiation
Coefficients
The following coefficients can also be defined
• is the transmission coefficient, said atmospheric
transmissivity
• is the reflection coefficient, said atmospheric
reflectivity (albedo)
• is the absorption coefficient, said atmospheric
absorptivity
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60. Absorption and transmission of short wave radiation
Shortwave Radiation budget
Energy conservation:
implies that reflectivity, transmissivity and absorptivity sum to one:
Which is, indeed, valid for reflectivity, transmissivity and absorptivity of any other body
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61. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
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62. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
We just forget for a
moment this. It will be
splitted into two parts:
one depending on
diffuse radiation and
another on cloud cover
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63. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
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64. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
Atmosphere is pretty transparent: which
means that we can, as a first approximation,
neglect it (atmosphere is heated from below)
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65. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
In any case let’s concentrate on
the transmitted radiation
This can be decomposed into two parts:
direct and diffuse solar radiation
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66. Absorption and transmission of short wave radiation
Shortwave Radiation budget
S
Evidently, for simmetry
is also composed by reflected and
diffuse solar radiation
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67. Absorption and transmission of short wave radiation
Diffuse radiation comes from scattering
Incident solar radiation strikes gas molecules, dust particles, and
pollutants, ice, cloud drops and the radiation is scattered. Scattering
causes diffused radiation.
Two types of light diffusion can be distinguished:
Mie scattering
Rayleigh scattering
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68. Absorption and transmission of short wave radiation
Rayleigh Scattering
•The impact of radiation with air molecules smaller than λ/π causes
scattering (Rayleigh scattering) the entity of which depends on the
frequency of the incident wave according to a λ-4 type relation.
•In the atmosphere, the wavelengths corresponding to blue are scattered
more readily than others.
incident radiation
diffuse radiation
transmitted radiation
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69. Absorption and transmission of short wave radiation
Mie Scattering
•When in the atmosphere there are particles with dimensions greater than 2 λ/π
(gases, smoke particles, aerosols, etc.) there is a scattering phenomenon that
does not depend on the wavelength, λ, of the incident wave (Mie scattering).
incident radiation
diffuse radiation
transmitted radiation
•This phenomenon can be observed, for example, in the presence of clouds.
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70. Absorption and transmission of short wave radiation
Diffused Light
Scattering selectively eliminates the shorter visible wavelengths, leaving the
longer wavelengths to pass. When the Sun is on the horizon, the distance
travelled by a ray within the atmosphere is five or six times greater than
when the Sun is at the Zenith and the blue light has practically been
completely eliminated.
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71. Absorption and transmission of short wave radiation
Tilt of the Earth’s axis
and atmospheric effects
The tilt of the earth’s axis and atmospheric effects together affect the amount of
radiation that reaches the ground.
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72. Absorption and transmission of short wave radiation
One way to take into account of absorption
Would be to run a full model of atmospheric transmission (e.g. Liou, 2002).
However hydrologists prefer to use parameterizations, and the
concept of atmospheric transmissivity.
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73. Absorption and transmission of short wave radiation
Solar radiation transmitted to the ground under
clear sky conditions
S
Finally:
Corripio, 2002
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74. Absorption and transmission of short wave radiation
Solar radiation transmitted to the ground under
clear sky conditions
S
Finally:
Corripio, 2002
Fraction of direct solar radiation
included between the considered
60
wavelengths
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Monday, December 10, 12
75. Absorption and transmission of short wave radiation
Solar radiation transmitted to the ground under
clear sky conditions
S
Finally:
Corripio, 2002
Transmittance of the
atmosphere
Fraction of direct solar radiation
included between the considered
60
wavelengths
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76. Absorption and transmission of short wave radiation
Solar radiation transmitted to the ground under
clear sky conditions
Correction due to
S elevation of the site
Finally:
Corripio, 2002
Transmittance of the
atmosphere
Fraction of direct solar radiation
included between the considered
60
wavelengths
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Monday, December 10, 12
77. Absorption and transmission of short wave radiation
Solar radiation transmitted to the
ground under clear sky conditions
S
We do not enter in the details of how
and
are determined. Please look, for
instance, at Formetta et al., 2012
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78. Considering Clouds
Hydrologists (and not only them) treat the
influence of clouds separately
It is assumed that the effects of
clouds is an attenuation of the
transmitted solar radiation
Transmitted direct
radiation at the surface
after clouds correction
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79. Considering Clouds
Hydrologists (and not only them) treat the
influence of clouds separately
It is assumed that the effects of
clouds is an attenuation of the
transmitted solar radiation
Transmitted direct
Transmitted direct radiation at the surface
radiation at the surface before clouds correction
after clouds correction
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80. Considering Clouds
Hydrologists (and not only them) treat the
influence of clouds separately
An analogous formulation holds for
diffuse radiation:
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81. Considering Clouds
Hydrologists (and not only them) treat the
influence of clouds separately
An analogous formulation holds for
diffuse radiation:
Correction coefficient for
diffuse radiation
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82. Considering Clouds
Estimation of the reduction coefficients
(decomposition model)
These reduction coefficients can be
determined when we have ground
measurements of total radiation,
diffuse plus direct:
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83. Considering Clouds
Estimation of the reduction coefficients
(decomposition model)
These reduction coefficients can be
determined when we have ground
measurements of total radiation,
diffuse plus direct:
Measured total radiation
at the ground station i
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84. Considering Clouds
Estimation of the reduction coefficients
(decomposition model)
These assumption that is often
made is that, the diffuse solar
radiation measured at the station is
proportional to the total radiation:
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85. Considering Clouds
Estimation of the reduction coefficients
(decomposition model)
These assumption that is often
made is that, the diffuse solar
radiation measured at the station is
proportional to the total radiation:
reduction coefficient for
diffuse radiation
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86. Considering Clouds
Estimation of the reduction coefficients
(decomposition model)
Therefore when substituting this
diffuse radiation expression in the
total radiation equation of previous
slides, it results at stations:
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87. Considering Clouds
Estimation of the reduction coefficients
(decomposition model)
And, for the direct radiation, at
stations:
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88. Considering Clouds
The key factor is the to determine the above coefficient, on which the
procedure followed so far has moved all the unknown.
Its estimation pass through various parameterizations:
Among the most known:
•Erbs et al., 1982
•Reindl et al. 1990
•Boland et al. 2001
please find the details in Formetta et al., 2012
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89. Considering Clouds
One more issue
With the help of the parameterizations above, the correction facotrs are
determined for the stations. Which are a few points in a rugged terrain.
How do you solve the problem to transport it everywhere ?
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90. Considering Clouds
We need to use
some interpolation
technique
Like Kriging* or the Inverse distance weighting method** which is not
the matter of the present slides.
* Goovaerts, 1997
**Shepard, 1968
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91. Hitting the terrain
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
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92. Hitting the terrain
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.
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93. Hitting the terrain
Projection of radiation onto an
inclined surface
After Corripio, 2003
First we calculate the normal to the surface 73
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94. Hitting the terrain
Projection of radiation onto an
inclined surface
Unit normal vector:
⇥
After Corripio, 2003
1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1) )
⇧ ⌃
1 ⇧ ⇧ 1/2 (z(i,j) + z(i+1,j)
⌃
⇧u =
n ⇧ z(i,j+1) z(i+1,j+1) ) ⌃
⌃
|⇧ u | ⇤
n ⌅
l2
where z are the elevations of the four points used and l2 is the are of the
cell - of side l.
74
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Monday, December 10, 12
95. Hitting the terrain
After Corripio, 2003
Representation of the vector normal to the surface of Mount Bianco
75
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Monday, December 10, 12
96. Hitting the terrain
Projection of radiation onto an
inclined surface
After Corripio, 2003
And we compare with the solar vector, indicating the direction of the Sun 76
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Monday, December 10, 12
97. Hitting the terrain
Projection of radiation onto an
inclined surface
⇥
sin ⇥ cos
⌥ = ⇤ sin ⇤ cos ⇥ cos
s cos ⇤ cos ⌅
cos⇤ cos ⇥ cos + sin ⇤ sin
Where all the quantities were already defined previously
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Monday, December 10, 12
98. Hitting the terrain
Projection of radiation onto an
inclined surface
s
After Corripio, 2003
Then we calculate the angle between the sun vector and the normal 78
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Monday, December 10, 12
99. Hitting the terrain
Projection of radiation onto an
inclined surface
We can define then the angle
s of solar incidence
After Corripio, 2003
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Monday, December 10, 12
100. Hitting the terrain
Projection of radiation onto an
inclined surface
Angle of solar incidence
cos s = ⌅ · ⌅u
s n
⇥
1/2 (z(i,j) z(i+1,j) + z(i,j+1) z(i+1,j+1) )
⇧ ⌃
1 ⇧ ⇧ 1/2 (z(i,j) + z(i+1,j)
⌃
⇧u =
n z(i,j+1) z(i+1,j+1) ) ⌃
|⇧ u | ⇧
n ⇤ ⌃
⌅
l2
⇥
sin ⇥ cos
⌥ = ⇤ sin ⇤ cos ⇥ cos
s cos ⇤ cos ⌅
cos⇤ cos ⇥ cos + sin ⇤ sin
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Monday, December 10, 12
101. Hitting the terrain
Projection of radiation onto an
inclined surface
The above angles needs to be compared with those of the terrain:
Slope
s = cos 1
nu.z
Aspect (from the North anti-clockwise)
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Monday, December 10, 12
102. Hitting the terrain
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:
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Monday, December 10, 12
103. Hitting the terrain
Solar radiation transmitted to the ground under
clear sky conditions
S
Therefore, for the direct
shortwave radiation:
Corripio, 2002
as, it was before
83
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Monday, December 10, 12
104. Hitting the terrain
However, it is not just matter of light but also of
shadows
84
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Monday, December 10, 12
105. Hitting the terrain
Incident radiation
Topographic effects: shading
More schematically
light
shadow
85
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Monday, December 10, 12
106. Hitting the terrain
Incident radiation
Topographic effects: shading
More schematically
86
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Monday, December 10, 12
107. Hitting the terrain
Incident radiation
Topographic effects: shading
More schematically
shadow
86
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Monday, December 10, 12
108. Hitting the terrain
Incident radiation
Topographic effects: shading
More schematically
light
shadow
86
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Monday, December 10, 12
109. Hitting the terrain
Incident radiation
Therefore the direct solar radiation must be corrected to include shading
Details in Corripio, 2003
87
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Monday, December 10, 12
110. Hitting the terrain
What about diffuse radiation ?
Topographic effects: angle of view
88
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Monday, December 10, 12
111. Hitting the terrain
What about diffuse radiation ?
Topographic effects: angle of view
sky view factor
88
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Monday, December 10, 12
112. Hitting the terrain
What about diffuse radiation ?
Topographic effects: angle of view
sky view factor
diffuse
radiation due to
Rayleigh
scattering
88
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Monday, December 10, 12
113. Hitting the terrain
What about diffuse radiation ?
Topographic effects: angle of view
sky view factor
diffuse
radiation due to diffuse
Rayleigh radiation due to
scattering aerosols
88
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Monday, December 10, 12
114. Hitting the terrain
What about diffuse radiation ?
Topographic effects: angle of view
sky view factor
diffuse
diffuse radiation due
radiation due to diffuse
multiple
Rayleigh radiation due to
scattering
scattering aerosols
88
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Monday, December 10, 12
115. Hitting the terrain
Incident radiation
Topographic effects: angle of view
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.
89
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Monday, December 10, 12
116. Hitting the terrain
Incident radiation
Topographic effects: angle of view
Different points view a different sky
90
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Monday, December 10, 12
118. Hitting the terrain
Now it really hits the terrain
and, in part, it is reflected away
After Corripio, 2003
92
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Monday, December 10, 12
119. Hitting the terrain
Finally a map
After Corripio, 2003
Insolation received by Mont Blanc at Spring Equinox 93
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Monday, December 10, 12
120. Albedo
Typical albedo values
http://en.wikipedia.org/wiki/Albedo
94
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Monday, December 10, 12
121. Albedo
Typical albedo values
http://en.wikipedia.org/wiki/Albedo
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Monday, December 10, 12
122. Spectral response
Spectral Signature (or Response)
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.
51
96
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Monday, December 10, 12
123. Spectral response
•In the case of solar radiation, the spectral signature is defined
as the reflectance of the surface in function of the wavelength.
97
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Monday, December 10, 12
124. Spectral response
•Every type of surface can be statistically characterised by a spectral signature.
98
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Monday, December 10, 12
125. Spectral response
Factors
•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.
99
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Monday, December 10, 12
126. Spectral response
Radiation that hits the terrain, heats it.
Or causes changes of phase
water to vapor
ice to water
100
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Monday, December 10, 12
127. Spectral response
Or is used for photosynthesis
or other chemical reactions
101
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Monday, December 10, 12
128. Long wave radiation
Earth “is” a gray body
Having a temperature emits radiation
A. Adams - Part of the snake river picture
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Monday, December 10, 12
129. Long wave radiation
Gray Bodies
• Plank’s Law for gray bodies:
2⇡c h 2 5
W =✏ ch [W cm 2
µm 1
]
e KT 1
• The Stefan-Boltzmann equation for gray bodies:
W = ✏ T [W cm
4 2
]
where ε is the average emissivity calculated over the entire electromagnetic
spectrum.
103
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Monday, December 10, 12
130. Long wave radiation
Gray Bodies
The behavior of a real (gray) body is related to that of a black body by
means of the quantity ελ, known as the emission coefficient or
emissivity, which is defined as:
W (real body)
✏ =
W (black body)
Kirchhoff (1860) demonstrated that a good “radiator” is also a good
“absorber”, that is to say:
↵=✏ ⇢+⌧ +✏=1
104
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Monday, December 10, 12
131. Long wave radiation
Comparison of blackbody and gray body
In reality emissivity depends, at least, on wavelength. Earth should be
probably defined a selective radiator
105
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Monday, December 10, 12
132. Long wave radiation
See the Earth as gray body
and given that the
temperature of the Earth’s
surface is, on average,
about 288 K, it obviously
emits a spectrum of
radiation in the infrared
band.
106
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Monday, December 10, 12
133. Long wave radiation
Radiation emitted by the Sun and the Earth
Yochanan Kushnir
107
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Monday, December 10, 12
134. Long wave radiation
See the Earth as gray body
and given that the
temperature of the Earth’s
surface is, on average,
about 288 K, it obviously
emits a spectrum of
radiation in the infrared
band.
Atmosphere is not
anymore transparent to at
these wavelengths.
108
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Monday, December 10, 12
135. Long wave radiation
The atmosphere is
warmed from below
Therefore the temperature is
higher at ground level than it
is at higher altitudes.
109
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Monday, December 10, 12
136. Long wave radiation
Greenhouse Effect
In the absence of atmospheric absorption the average temperature of the Earth’s
surface would be about -170C.
110
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Monday, December 10, 12
137. Long wave radiation
Greenhouse Effect
Instead the average temperature is about 15 0C
111
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Monday, December 10, 12
138. Long wave radiation
Radiative heating
is completed by convective heat transfer, and by water vapor fluxes (latent and
sensible heat).
But this you can see better on the energy budget slides. 112
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Monday, December 10, 12
139. Long wave radiation
But now concentrate on the surroundings of a point
After Helbig, 2009
Any point being at a certain temperature emits long wave radiation
which must be accounted for
113
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Monday, December 10, 12
140. Long wave radiation
The atmosphere emits infrared itself
bacause of its temperature
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Monday, December 10, 12
141. Long wave radiation
All the contributions
Long-wave radiation is given by the
balance of incident radiation from
the atmosphere and the radiation
emitted by the ground. Both values
are calculated with the Stefan-
Boltzmann law.
115
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Monday, December 10, 12
142. Long wave radiation
Longwave (infrared) raditation
Topographic effects: angle of view
116
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Monday, December 10, 12
143. Long wave radiation
Longwave (infrared) raditation
Topographic effects: angle of view
Longwave radiation
coming from sky
116
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Monday, December 10, 12
144. Long wave radiation
Longwave (infrared) raditation
Topographic effects: angle of view
Longwave radiation Longwave radiation
coming from sky coming from
surrounding
116
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Monday, December 10, 12
145. Long wave radiation
Longwave (infrared) raditation
Topographic effects: angle of view
Longwave radiation Longwave radiation Radiation losses
coming from sky coming from by the area under
surrounding exam
116
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Monday, December 10, 12
146. Long-wave radiation
The first component should be
calculated by integrating the formula
over the entire atmosphere, but,
given how complex this process is,
typically an empirical formula is
used that uses the value of air
temperature as measured near
ground level (2m) and a value of the
atmospheric emissivity based on
specific humidity, temperature, and
cloudiness. The second component,
on the other hand, is function of the
surface temperature and its
emissivity.
117
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Monday, December 10, 12
147. Long wave radiation
Long-wave radiation
The real process:
The hydrological parameterisation:
118
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Monday, December 10, 12
148. Long wave radiation
Long-wave radiation
The real process:
The hydrological parameterisation:
Global emissivity of the
atmosphere
118
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Monday, December 10, 12
149. Long wave radiation
Long-wave radiation
The real process:
The hydrological parameterisation:
Temperature at 2 m
from ground
Global emissivity of the
atmosphere
118
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Monday, December 10, 12
150. Long wave radiation
Parameterisation of Long-wave radiation
The hydrological parameterisation:
6 4
εatm = εBrutsaert (1− N ) + 0.979N Brutsaert (1975) + Pirazzini et al. (2000)
εatm = εBrutsaert (1+ 0.26N) Brutsaert (1975) + Jacobs (1978)
6 4
εatm = εIdso (1− N ) + 0.979N Idso (1981) + Pirazzini et al. (2000)
6 4
εatm = εIdso,corr (1− N ) + 0.979N Hodges et al. (1983) + Pirazzini et al.
(2000)
where N is the fraction of sky covered by clouds
119
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Monday, December 10, 12
151. Net Radiation
The sum of longwave and shortwave ratio
is called net radiation
120
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Monday, December 10, 12
152. 1Thank you for your attention !
G.Ulrici - 2000 ?
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Monday, December 10, 12