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Riccardo Rigon
Il S
ole
, F. L
elon
g, 2
00
8, V
al d
i Se
lla
Solar Radiation Physics and Geometry
for hydrologists
Monday, December 10, 12
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
R. Rigon
2
Monday, December 10, 12
R. Rigon
Educational Goals
• To recognise that the water cycle is powered by solar energy
• 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 gain knowledge of the spatial and temporal variation of the
radiation distribution on the Earth
• To introduce the concepts necessary to better understand the elements
of the energy balance needed in remote-sensing applications, the snow
balance, and evapotranspiration
1
3
Introduction
Monday, December 10, 12
R. Rigon
The Sun is the origin of the water cycle
2
4
The Sun
Monday, December 10, 12
R. Rigon
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.
Composition of the Sun
3
5
The Sun
Monday, December 10, 12
R. Rigon
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%
Other elements: 0.1%
4
6
The Sun
Monday, December 10, 12
R. Rigon
The Sun and the planets to scale
5
7
The Sun
Monday, December 10, 12
R. Rigon
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.
The internal structure of the Sun
6
8
The Sun
Monday, December 10, 12
R. Rigon
9
The Sun
Provide a relatively constant rate of radiation energy that in few minutes
from the cromosphere arrives to the Earth.
Det
ail
of
a P
elli
zza
da
Volp
edo P
ain
tin
g
Monday, December 10, 12
R. Rigon
Solar Spots
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.
7
10
The Sun
Radiation flux is regular up to a point. In reality it manifests variations.
Monday, December 10, 12
R. Rigon
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).
Variability of the Emissions
8
11
The Sun
Monday, December 10, 12
R. Rigon
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.
Variability of the Emissions
8
12
The Sun
Monday, December 10, 12
R. Rigon
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.
Variability of the Emissions
9
13
The Sun
Monday, December 10, 12
R. Rigon
R = ✏ � T 4
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
The Stefan-Boltzmann law
10
14
The Sun
Monday, December 10, 12
R. Rigon
Radiation
emitted
R = ✏ � T 4
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
The Stefan-Boltzmann law
10
14
The Sun
Monday, December 10, 12
R. Rigon
Radiation
emitted
emissivity
R = ✏ � T 4
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
The Stefan-Boltzmann law
10
14
The Sun
Monday, December 10, 12
R. Rigon
Radiation
emitted
emissivityStefan-Boltzmann constant
R = ✏ � T 4
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
The Stefan-Boltzmann law
10
14
The Sun
Monday, December 10, 12
R. Rigon
Radiation
emitted
emissivityStefan-Boltzmann constant
absolute temperatureR = ✏ � T 4
Every body with a temperature different than T=0 K emits radiation as a function
of its temperature according to the Stefan-Boltzmann law
The Stefan-Boltzmann law
10
14
The Sun
Monday, December 10, 12
R. Rigon
RSun = ✏ � T 4 = 1 ⇤ 5.67 ⇤ 10�8 ⇤ 60004 ⇡ 25.12 ⇤ 109J m�2 s�1
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
11
15
The Sun
Monday, December 10, 12
R. Rigon
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.
The Sun is nearly a “blackbody”!
12
16
The Sun
Monday, December 10, 12
R. Rigon
The area below the curves is given by the Stefan-Boltzmann law. The curves
themselves are given by Planck’s law.
The Sun is nearly a “blackbody”!
13
17
The Sun
Monday, December 10, 12
R. Rigon
The complete electromagnetic spectrum
The spectrum of solar radiation stretches far beyond the visible band where,
however, nearly half the available energy is concentrated
Figu
re 2
.9
C.B
. A
gee
16
18
The Sun
Monday, December 10, 12
R. Rigon
Planck’s Law
•Planck’s law is the general law for electromagnetic emission from the
surface of a blackbody*:
W� =2⇡c2h��5
ech
�KT � 1[Wm�2µm�1]
14
19
The Sun
* Stefan-Boltzmann law is just the integration of Plank’s law over wavelengths
Monday, December 10, 12
R. Rigon
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.
From Sun to Earth
18
20
From Sun To Earth
Monday, December 10, 12
R. Rigon
T h e S u n i r r a d i a t e s
approximately at the solar
constant rate, which is, on
the average, on the top of
the atmosphere,
Solar radiation
http://en.wikipedia.org/wiki/Solar_constant
Froli
ch, 1
98
5
19
21
From Sun To Earth
Monday, December 10, 12
R. Rigon
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.
Astronomical variability of radiation
22
Copying with Earth surface
Monday, December 10, 12
R. Rigon
Seasons
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.
Figure 3.1
23
From Sun To Earth
Monday, December 10, 12
R. Rigon
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
Corrections to the solar constant
http://www.ascensionrecta.com/
20
24
From Sun To Earth
Monday, December 10, 12
R. Rigon
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
25
From Sun To Earth
Monday, December 10, 12
R. Rigon
Astronomical influences
Orbit angle
Orbit change
Orbit shape
26
From Sun To Earth
Monday, December 10, 12
R. Rigon
Therefore the solar contant must be corrected
(e.g. Corripio, 2002):
Solar radiation in hydrological models
27
S�
From Sun To Earth
Monday, December 10, 12
R. Rigon
N is the day of the year (in 1, ..., 365)
where:
Solar radiation in hydrological models
28
Therefore the solar contant must be corrected
(e.g. Corripio, 2002):S�
From Sun To Earth
Monday, December 10, 12
R. Rigon
Radiation intensity
Solar intensity governs seasonal climatic changes and the local climatic niches
which are linked to the apparent height of the Sun.
29
Copying with Earth surface
Monday, December 10, 12
R. Rigon
Insolation and latitude
Incoming solar radiation is not evenly distributed across all lines of latitude, creating a heating imbalance.
Figu
re 3
.7
30
Copying with Earth surface
Monday, December 10, 12
R. Rigon
Radiative imbalance
31
Copying with Earth surface
Monday, December 10, 12
R. Rigon
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
Radiation received from the Sun
50
Copying with Earth surface
Monday, December 10, 12
R. Rigon
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.
The geometry of radiation
33
Copying with Earth surface
Monday, December 10, 12
R. Rigon
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.
To sum up
34
Copying with Earth surface
Monday, December 10, 12
R. Rigon
The geometry of radiation
To calculate the aforementioned
quantities it is usual to use a
topocentric coordinate system,
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.
Nau
tic
Alm
anac
Off
ice,
19
74
35
Copying with Earth surface
Monday, December 10, 12
R. Rigon
The geometry of radiation
The X-axis is, therefore, tangent
to the earth and positive in a
West-East direction. The Y-axis
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.
Nau
tic
Alm
anac
Off
ice,
19
74
36
Copying with Earth surface
Monday, December 10, 12
R. Rigon
Solar Vector
The solar vector can be expressed as a
function of the angles that have been
defined. The resulting trigonometric
expression is:
Therefore, to determine the position of
the Sun one needs to know the latitude,
t h e h o u r a n g l e , a n d t h e s o l a r
declination.
⌥s =
�
⇤� sin⇥ cos �
sin⇤ cos ⇥ cos � � cos ⇤ cos �cos⇤ cos ⇥ cos � + sin⇤ sin �
⇥
⌅
37
Copying with Earth surface
X
Y
Z
Monday, December 10, 12
R. Rigon
Hour angle
The hour ang le can be eas i l y
calculated as:
⇥ = �
�t
12� 1
⇥
if t is the solar hour
38
Copying with Earth surface
Monday, December 10, 12
R. Rigon
Solar declination
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
39
Copying with Earth surface
Is the angular height of Sun from the horizon at equator at noon*
*http://en.wikipedia.org/wiki/Declination
Monday, December 10, 12
R. Rigon
X
Y
Z
40
Projection on a plane at a certain latitude
is the solar vector
⌥s =
�
⇤� sin⇥ cos �
sin⇤ cos ⇥ cos � � cos ⇤ cos �cos⇤ cos ⇥ cos � + sin⇤ sin �
⇥
⌅
If is the vertical unit row-vector
corresponding to the Z axis:
and
Copying with Earth surface
Monday, December 10, 12
R. Rigon
41
Projection on a plane at a certain latitude
with the symbols explained above
Then the projection of the solar
irradiation on the plane YX is reduced by
the factor where:
or:
Copying with Earth surface
X
Y
Z
Monday, December 10, 12
R. Rigon
42
To sum up:
Was:
Is now:
The solar constant can be modified as follows.
Copying with Earth surface
Monday, December 10, 12
R. Rigon
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.
43
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Atmospheric absorption
44
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
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Shortwave Radiation budget
The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.
45
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Radiation reflected
Shortwave Radiation budget
The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.
45
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Radiation transmitted
Radiation reflected
Shortwave Radiation budget
The solar radiation penetrates the
atmosphere, and it is transferred
towards the ground, after being
reflected and scattered.
45
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
the incoming radiation equals
the reflected one plus
the absorbed plus
the transmitted
46
Shortwave Radiation budget
S� It should not be forgot that
the radiation budget is an
energy budget, for which
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
the incoming radiation equals
the reflected one plus
the absorbed plus
the transmitted
46
Shortwave Radiation budget
S� It should not be forgot that
the radiation budget is an
energy budget, for which
Radiation
absorbed
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
47
S�
This budget can be apply to any slice of the atmosphere
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
47
S�
Corrected Solar constant
This budget can be apply to any slice of the atmosphere
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
47
S�
Corrected Solar constant
Solar radiation
reflected back to space
This budget can be apply to any slice of the atmosphere
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
47
S�
Transmitted
radiation
Corrected Solar constant
Solar radiation
reflected back to space
This budget can be apply to any slice of the atmosphere
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
47
S�
Energy absorbed by atmosphere
Transmitted
radiation
Corrected Solar constant
Solar radiation
reflected back to space
This budget can be apply to any slice of the atmosphere
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
• is the reflection coefficient, said atmospheric reflectivity (albedo)
• is the transmission coefficient, said atmospheric transmissivity
• is the absorption coefficient, said atmospheric absorptivity
Coefficients
The following coefficients can also be defined
48
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Energy conservation:
Which is, indeed, valid for reflectivity, transmissivity and absorptivity of any other body
implies that reflectivity, transmissivity and absorptivity sum to one:
49
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
50
S�
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
We just forget for a moment this. It will be splitted into two parts:
one depending on diffuse radiation and
another on cloud cover
50
S�
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
51
S�
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Atmosphere is pretty transparent: which means that we can, as a first approximation, neglect it (atmosphere is heated from below)
51
S�
Shortwave Radiation budget
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
In any case let’s concentrate on
the transmitted radiation
This can be decomposed into two parts:
direct and diffuse solar radiation
52
Shortwave Radiation budget
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Evidently, for simmetry
is also composed by reflected and diffuse solar radiation
53
Shortwave Radiation budget
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
5
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
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
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
55
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
•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).
•This phenomenon can be observed, for example, in the presence of clouds.
Mie Scattering
56
incident radiation
diffuse radiation
transmitted radiation
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
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.
57
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Tilt of the Earth’s axisand atmospheric effects
The tilt of the earth’s axis and atmospheric effects together affect the amount of radiation that reaches the ground.
58
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
59
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.
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Solar radiation transmitted to the ground under clear sky conditions
Finally:
Cor
rip
io, 2
00
2
60
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Solar radiation transmitted to the ground under clear sky conditions
Finally:
Fraction of direct solar radiation included between the considered
wavelengths
Cor
rip
io, 2
00
2
60
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Solar radiation transmitted to the ground under clear sky conditions
Finally:
Fraction of direct solar radiation included between the considered
wavelengths
Transmittance of the atmosphere
Cor
rip
io, 2
00
2
60
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Solar radiation transmitted to the ground under clear sky conditions
Finally:
Fraction of direct solar radiation included between the considered
wavelengths
Transmittance of the atmosphere
Correction due to elevation of the site
Cor
rip
io, 2
00
2
60
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
We do not enter in the details of how
and
are determined. Please look, for instance, at Formetta et al., 2012
Solar radiation transmitted to the ground under clear sky conditions
61
S�
Absorption and transmission of short wave radiation
Monday, December 10, 12
R. Rigon
Transmitted direct radiation at the surface after clouds correction
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
62
Considering Clouds
Monday, December 10, 12
R. Rigon
Transmitted direct radiation at the surface before clouds correction
Transmitted direct radiation at the surface after clouds correction
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
62
Considering Clouds
Monday, December 10, 12
R. Rigon
Hydrologists (and not only them) treat the
influence of clouds separately
An analogous formulation holds for diffuse radiation:
63
Considering Clouds
Monday, December 10, 12
R. Rigon
Correction coefficient for diffuse radiation
Hydrologists (and not only them) treat the
influence of clouds separately
An analogous formulation holds for diffuse radiation:
63
Considering Clouds
Monday, December 10, 12
R. Rigon
Estimation of the reduction coefficients(decomposition model)
These reduction coefficients can be
determined when we have ground
measurements of total radiation,
diffuse plus direct:
64
Considering Clouds
Monday, December 10, 12
R. Rigon
Measured total radiation at the ground station i
Estimation of the reduction coefficients(decomposition model)
These reduction coefficients can be
determined when we have ground
measurements of total radiation,
diffuse plus direct:
64
Considering Clouds
Monday, December 10, 12
R. Rigon
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:
65
Considering Clouds
Monday, December 10, 12
R. Rigon
reduction coefficient for diffuse radiation
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:
65
Considering Clouds
Monday, December 10, 12
R. Rigon
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:
66
Considering Clouds
Monday, December 10, 12
R. Rigon
Estimation of the reduction coefficients(decomposition model)
And, for the direct radiation, at
stations:
67
Considering Clouds
Monday, December 10, 12
R. Rigon
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
68
Considering Clouds
Monday, December 10, 12
R. Rigon
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.
Fig. 7. River Piave area, (Italy).
35
How do you solve the problem to transport it everywhere ?
69
Considering Clouds
Monday, December 10, 12
R. Rigon
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, 196870
Considering Clouds
Monday, December 10, 12
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
71
Hitting the terrain
Monday, December 10, 12
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. 72
Hitting the terrain
Monday, December 10, 12
R. Rigon
Projection of radiation onto an inclined surface
Aft
er C
orr
ipio
, 20
03
First we calculate the normal to the surface 73
Hitting the terrain
Monday, December 10, 12
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:
74
Aft
er C
orr
ipio
, 20
03
Hitting the terrain
Monday, December 10, 12
R. Rigon
Aft
er C
orr
ipio
, 20
03
75
Representation of the vector normal to the surface of Mount Bianco
Hitting the terrain
Monday, December 10, 12
R. Rigon
Aft
er C
orr
ipio
, 20
03
Projection of radiation onto an inclined surface
And we compare with the solar vector, indicating the direction of the Sun 76
Hitting the terrain
Monday, December 10, 12
R. Rigon
77
⌥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
Monday, December 10, 12
R. Rigon
Aft
er C
orr
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, 20
03
Projection of radiation onto an inclined surface
78Then we calculate the angle between the sun vector and the normal
s
Hitting the terrain
Monday, December 10, 12
R. Rigon
We can define then the angle
of solar incidence
Aft
er C
orr
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, 20
03
Projection of radiation onto an inclined surface
79
s
Hitting the terrain
Monday, December 10, 12
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 �
⇥
⌅
80
Hitting the terrain
Monday, December 10, 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:
81
Hitting the terrain
Monday, December 10, 12
R. Rigon
82
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
Monday, December 10, 12
R. Rigon
Solar radiation transmitted to the ground under clear sky conditions
Therefore, for the direct shortwave radiation:
Cor
rip
io, 2
00
2
83
S�
as, it was before
Hitting the terrain
Monday, December 10, 12
R. Rigon
84
However, it is not just matter of light but also of shadows
Hitting the terrain
Monday, December 10, 12
R. Rigon
Incident radiationTopographic effects: shading
85
More schematically
shadow
light
Hitting the terrain
Monday, December 10, 12
R. Rigon
Incident radiationTopographic effects: shading
86
More schematically
Hitting the terrain
Monday, December 10, 12
R. Rigon
Incident radiationTopographic effects: shading
86
More schematically
shadow
Hitting the terrain
Monday, December 10, 12
R. Rigon
Incident radiationTopographic effects: shading
86
More schematically
light
shadow
Hitting the terrain
Monday, December 10, 12
R. Rigon
Incident radiationD
etai
ls i
n C
orr
ipio
, 20
03
87
Therefore the direct solar radiation must be corrected to include shading
Hitting the terrain
Monday, December 10, 12
R. Rigon
What about diffuse radiation ?Topographic effects: angle of view
88
Hitting the terrain
Monday, December 10, 12
R. Rigon
sky view factor
What about diffuse radiation ?Topographic effects: angle of view
88
Hitting the terrain
Monday, December 10, 12
R. Rigon
sky view factor
diffuse radiation due to
Rayleigh scattering
What about diffuse radiation ?Topographic effects: angle of view
88
Hitting the terrain
Monday, December 10, 12
R. Rigon
sky view factor
diffuse radiation due to
Rayleigh scattering
diffuse radiation due to
aerosols
What about diffuse radiation ?Topographic effects: angle of view
88
Hitting the terrain
Monday, December 10, 12
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
88
Hitting the terrain
Monday, December 10, 12
R. Rigon
Incident radiationTopographic effects: angle of view
89
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
Monday, December 10, 12
R. Rigon
Incident radiationTopographic effects: angle of view
90
Different points view a different sky
Hitting the terrain
Monday, December 10, 12
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91
The sum
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Monday, December 10, 12
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Aft
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92
Now it really hits the terrainand, in part, it is reflected away
Hitting the terrain
Monday, December 10, 12
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Aft
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, 20
03
93Insolation received by Mont Blanc at Spring Equinox
Finally a map
Hitting the terrain
Monday, December 10, 12
R. Rigon
Typical albedo values
94
http://en.wikipedia.org/wiki/Albedo
Albedo
Monday, December 10, 12
R. Rigon
Typical albedo values
95
http://en.wikipedia.org/wiki/Albedo
Albedo
Monday, December 10, 12
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)
96
Spectral response
Monday, December 10, 12
R. Rigon
•In the case of solar radiation, the spectral signature is defined as the reflectance of the surface in function of the wavelength.
97
Spectral response
Monday, December 10, 12
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98
•Every type of surface can be statistically characterised by a spectral signature.
Spectral response
Monday, December 10, 12
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
99
Spectral response
Monday, December 10, 12
R. Rigon
Radiation that hits the terrain, heats it.
Or causes changes of phase
water to vapor
ice to water
100
Spectral response
Monday, December 10, 12
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101
Or is used for photosynthesis
or other chemical reactions
Spectral response
Monday, December 10, 12
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102
Earth “is” a gray body
Having a temperature emits radiation
A. A
dam
s -
Par
t of
the
snak
e ri
ver
pic
ture
Long wave radiation
Monday, December 10, 12
R. Rigon
Gray Bodies
• Plank’s Law for gray bodies:
• The Stefan-Boltzmann equation for gray bodies:
W� = ✏�2⇡c2h��5
ech
�KT � 1[Wcm�2µm�1]
W = ✏�T 4[Wcm�2]
103
where ε is the average emissivity calculated over the entire electromagnetic
spectrum.
Long wave radiation
Monday, December 10, 12
R. Rigon
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:
Kirchhoff (1860) demonstrated that a good “radiator” is also a good “absorber”, that is to say:
✏� =W�(real body)W�(black body)
↵ = ✏ ⇢ + ⌧ + ✏ = 1
104
Long wave radiation
Monday, December 10, 12
R. Rigon
Comparison of blackbody and gray body
105
In reality emissivity depends, at least, on wavelength. Earth should be probably defined a selective radiator
Long wave radiation
Monday, December 10, 12
R. Rigon
See the Earth as gray body
a n d g i v e n t h a t t h e
temperature of the Earth’s
surface is, on average,
about 288 K, it obviously
e m i t s a s p e c t r u m o f
radiation in the infrared
band.
106
Long wave radiation
Monday, December 10, 12
R. Rigon
Radiation emitted by the Sun and the Earth
Yochanan Kushnir
107
Long wave radiation
Monday, December 10, 12
R. Rigon
See the Earth as gray body
a n d g i v e n t h a t t h e
temperature of the Earth’s
surface is, on average,
about 288 K, it obviously
e m i t s a s p e c t r u m o f
radiation in the infrared
band.
A t m o s p h e r e i s n o t
anymore transparent to at
these wavelengths.
108
Long wave radiation
Monday, December 10, 12
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The atmosphere is warmed from below
Therefore the temperature is higher at ground level than it is at higher altitudes.
109
Long wave radiation
Monday, December 10, 12
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Greenhouse Effect
In the absence of atmospheric absorption the average temperature of the Earth’s surface would be about -170C.
110
Long wave radiation
Monday, December 10, 12
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Instead the average temperature is about 15 0C
Greenhouse Effect
111
Long wave radiation
Monday, December 10, 12
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Radiative heating
is completed by convective heat transfer, and by water vapor fluxes (latent and
sensible heat).
112But this you can see better on the energy budget slides.
Long wave radiation
Monday, December 10, 12
R. Rigon
But now concentrate on the surroundings of a point
113
Aft
er H
elb
ig, 2
00
9
Any point being at a certain temperature emits long wave radiation
which must be accounted for
Long wave radiation
Monday, December 10, 12
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114
The atmosphere emits infrared itself
bacause of its temperature
Long wave radiation
Monday, December 10, 12
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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
All the contributions
Long wave radiation
Monday, December 10, 12
R. Rigon
Longwave (infrared) raditationTopographic effects: angle of view
116
Long wave radiation
Monday, December 10, 12
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Longwave radiation coming from sky
Longwave (infrared) raditationTopographic effects: angle of view
116
Long wave radiation
Monday, December 10, 12
R. Rigon
Longwave radiation coming from surrounding
Longwave radiation coming from sky
Longwave (infrared) raditationTopographic effects: angle of view
116
Long wave radiation
Monday, December 10, 12
R. Rigon
Longwave radiation coming from surrounding
Radiation losses
by the area under exam
Longwave radiation coming from sky
Longwave (infrared) raditationTopographic effects: angle of view
116
Long wave radiation
Monday, December 10, 12
R. Rigon
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
s u r f a c e t e m p e r a t u r e a n d i t s
emissivity.
117
Monday, December 10, 12
R. Rigon
The real process:
The hydrological parameterisation:
Long-wave radiation
118
Long wave radiation
Monday, December 10, 12
R. Rigon
The real process:
The hydrological parameterisation:
Long-wave radiation
118
Long wave radiation
Global emissivity of the atmosphere
Monday, December 10, 12
R. Rigon
The real process:
The hydrological parameterisation:
Long-wave radiation
118
Long wave radiation
Global emissivity of the atmosphere
Temperature at 2 m from ground
Monday, December 10, 12
R. Rigon
The hydrological parameterisation:
€
εatm = εBrutsaert (1− N6) + 0.979N 4 Brutsaert (1975) + Pirazzini et al. (2000)
€
εatm = εBrutsaert (1+ 0.26N)
€
εatm = εIdso(1− N6) + 0.979N 4
€
εatm = εIdso,corr(1− N6) + 0.979N 4
Brutsaert (1975) + Jacobs (1978)
Idso (1981) + Pirazzini et al. (2000)
Hodges et al. (1983) + Pirazzini et al. (2000)
Parameterisation of Long-wave radiation
119
Long wave radiation
where N is the fraction of sky covered by clouds
Monday, December 10, 12
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120
Net Radiation
The sum of longwave and shortwave ratio
is called net radiation
Monday, December 10, 12
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121
1Thank you for your attention !
G.U
lric
i -
20
00
?
Monday, December 10, 12
R. Rigon
Table of symbols
122
Monday, December 10, 12
R. Rigon
123
Table of symbols
Monday, December 10, 12
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124
Table of symbols
Monday, December 10, 12
R. Rigon
Projection of radiation onto an inclined surface
125
Monday, December 10, 12
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The geometry of radiation
126
Monday, December 10, 12