RADIATION CALCULATIONS IN CUPID
1. GENERAL INTRODUCTION TO CUPID'S RADIATION CALCULATIONS
There are six main locations in CUPID that deal with radiation flux and
interception calculations. These are:
RADIN4: The estimation of wavelength composition of sky
radiation and its direction.
SKYIR The estimation of thermal sky radiation.
MAIN The calculation of Thick Layer properties.
RADIAT The calculation of the radiation fluxes into all
canopy layers and leaf angle classes.
MAIN The calculation of apparent radiation fluxes as
viewed from above.
MAIN The estimation of bidirectional scattering.
RADIN4 is called only if the radiation input information doesn't contain
values for radiation in the VIS and in the NIR range separately. SKYIR is
called only if there is no input information on the thermal sky radiation.
Both routines are called after the solar zenith angle calculations before
entering the hourly loop. The Thick Layer calculations in the MAIN are also
done at this location in the program.
In the hourly loop RADIAT is the first subroutine to be called. This first
call of RADIAT and other subroutines set merely first estimates for further
calculations and precede the convergence loops.
The subroutines RADIAT and LFEBAL are coupled through a common parameter,
the leaf temperature, which is as TEMPLF an input variable to RADIAT and as
TEMLF1 an output variable of LFEBAL. To reach convergence between the two (as
close as 0.2 degrees C) the program iterates through RADIAT, LFEBAL and other
subroutines, each time setting TEMPLF 10% closer towards TEMLF1. The loop
counter is NOITER.
If RADIAT is called again for convergence of leaf temperature, the
calculations in RADIAT are executed only for the thermal wavelengths as would
be for all IHR's between sunset and sunrise (discussion follows in later
chapters). To indicate reiteration or nighttime hours, the index KSTRT is set
to 3.
This leaf temperature loop is superimposed by two other convergence loops, so
that in effect the leaf temperature convergence may be run many times for one
hourly input. Note that NOITER is set back to 0, if recycled through one of
the two outer loops.
RADIAT is called once more and executed for all wavelength bands after
calling PHOTO1 towards the end of the hourly loop, after all convergence
loops have been exited. It is followed by the calculations for the apparent
radiation fluxes as viewed from above the canopy and the estimation of
bidirectional scattering.
The following page contains a simplified flowchart considering only the
radiation calculations of CUPID.
[THIS IS CURRENTLY MISSING]
2. BASIC MODEL ASSUMPTIONS AND REFERENCES
2.1 ASSUMPTIONS ON CANOPY STRUCTURE:
CUPID (1987 version) divides the canopy into layers of equal leaf area
index F. Within each layer a spherical leaf angle distribution is assumed
:
DEFINITION: If the relative frequency of leaf inclination
angles are the same as the relative frequency
of the inclination angles of the surface ele-
ments of a hemisphere, then these leaf angles
are said to have a SPHERICAL DISTRIBUTION.
It is further assumed that the leaves of every angle class are evenly
distributed (randomly) within the layer and that the area of the single
leaves are small compared to the canopy ground area.
2.2 CONSEQUENCES OF THE ASSUMED CANOPY STRUCTURE:
The model assumptions made here have the virtue of simplifying light
penetration equations in three important respects:
1) The light penetration equation for a single light incidence
angle can be formulated in a Beer's Law analogy:
(1)
where T = I(non-intercepted) / I(incident)
k : extinction coefficient
: leaf area density ( leaf area / canopy space)
s : pathway length
since with : LAI of layer j in
the direction of
the beam
and with :incidence angle
Equation (1) can be reformulated for every layer:
2) For a spherical leaf angle distribution the extinction
coefficient is constant and takes on the value 1/2, hence:
(2)
3) For a spherical leaf angle distribution, the LAI in the
direction of the incident beam is the same for all angles of
incidence. Hence F in equation (2) is constant.
2.3 TREATMENT OF RADIATION SOURCES:
The solar radiation, which is measured above the canopy, is divided into
two components: a direct beam component for which one incident zenith angle
is assumed, and a diffuse radiation component (assumed to be isotropic).
Furthermore, CUPID splits the treatment of radiation transfer into three
wavelength bands, 1) Visible (VIS), 2) Near Infrared Radiation (NIR) and
3) thermal wavelengths. (This division allows the incorporation of more
detailed information on leaf spectral properties.) While the VIS and NIR
come both as direct beam and diffuse, the thermal radiation is always
scattered.
Apart from the sky as a radiation source, the canopy layers also contribute
as a source of radiation through their ability to scatter light or to emit
thermal radiation.
There are three fates for intercepted radiation, be it direct or diffuse:It
can be absorbed, reflected or transmitted. All scattered radiation ends as
diffuse radiation and is treated in CUPID as if isotropically distributed.
The optical properties of the canopy layers are derived from leaf and soil
spectral properties for which there exists simplified input information;
namely, one value for leaf reflectance, one value for leaf transmissivity
and one value for soil reflectance.
The following scheme summarizes CUPID's assumptions on radiation transfer:
2.4 DIRECT BEAM PENETRATION:
In CUPID the light distribution equations are predominantly formulated as
fractions of the total light flux density onto a horizontal plane above
the canopy (direct and diffuse) in the corresponding wavelength band.
In a canopy divided into layers of equal LAI, the light is attenuated by
the same factor in every layer. Hence, according to equation (2):
with I : flux density of direct beam over the canopy
I : flux density going into layer j.
2.5 DIFFUSE LIGHT PENETRATION:
The light penetration function for diffuse radiation onto a horizontal
surface can be derived by integrating the direct beam penetration function
E( , , ) (that may take the form of equation (2)) over the hemisphere with
the azimuth angle between 0 and 2 and the zenith angle between 0 and /2.
In a general formulation:
with : diffuse light angular distribution
: height of canopy above ground
Assuming a spherical leaf angle distribution and isotropic diffuse radiation
the equation simplifies and can be solved by numerical approximation overa
finite number of incident zenith angles n:
(4)
2.6 LIGHT INTERCEPTION:
Intercepted radiation is either absorbed or scattered. The amount of upward
and downward scatter depends on the optical properties of the layer, its
transmissivity and reflectance. CUPID in its present version has a
simplified treatment of radiation transfer, since it assumes only two
directions for radiation scatter, to the two sides of the horizontal plane
and assumes this scatter always to take on an isotropic distribution.
In order to assess the total upward and downward diffuse radiation flux
into a layer, four radiation sources have to be taken into account:
For downward diffuse radiation flux into layer j the four components are:
1) The non-intercepted downward diffuse radiation into
layer j+1.
2) The intercepted downward diffuse radiation into layer
j+1 that has been transmitted.
3) The intercepted upward diffuse radiation into layer j
that has been reflected.
4) The intercepted direct beam radiation into layer j+1 that
has been transmitted.
Formulated as an equation this is equivalent to:
And similarly for the upward diffuse radiation into layer j:
These equations cannot be solved in a single pass through all layers, since
information about the upward radiation is required for solving the downward
radiation. Hence, first the ratio of upward to downward radiation into
layer j is solved for:
(4)
The upward and downward diffuse fluxes can then be formulated in terms of the
ratio A: