5.1 Application of the Radiative Transfer Equations Inside the Oasis

A vegetation canopy is a porous medium for the radiation. In more specific term it is considered as semi-transparent to solar radiation. At macro-scale, the plants intercept a part of the incoming radiation and diffuse (ascendant flux and descendant flux) the others. This is done by the participation of all the vegetation components (leaf, fruit, branch, trunk). At micro-scale (molecule level) the plant canopy absorb a share of the incident radiation and diffract (reflection and transmission) the rest. The farming canopy is accounted as participating environment for solar radiation. The oasis that is expanded over a vast region can be treated as homogeneous, participating and fluid medium where the solar radiation arrives at any point in all directions from the atmosphere. For such a medium the most used equation to describe the radiative transfer is (Lopez and Semel 1999; Kryzhevoi et al 2001):

^)+Kv(s,t)®Q,v(s,t)+ov(s,t)$Q,v(s,t)=0Vv(s,t)^av(s,t) j>(a^Q)Oa,,v(s,t)Q

®Qv is the spectral specific intensity of radiation v is the frequency of the radiation Q is the direction of propagation s is the position t is the time

Kv is the spectral volumetric absorption ov is the spectral diffusion terms 0Jv(s , t) the spectral emission terms

P(Q) is the scattered radiation angular distribution function (phase function)

If we consider the solar radiation as source, the second term on the left represents the amount of solar radiation absorbed by the vegetation at any position s and any time t. It is analogues to the part of radiation intercepted. The third term on the left represents the radiation scattered out the oasis, it is equivalent to the solar radiation reflected by a vegetation layer or to that ascendant at any level of the oasis. The first term on the right express the spectral radiation emitted by the components of the oasis canopy (leaves, branch, trunk). The second term on the right show the scattering phenomenon inside the studied environment. For the oasis, an important part of the incident solar radiation will undergo a multiple re-diffusion to be arrested later by the vegetation elements. This radiation is putted on by the phase function. It can be considered as the radiation path or the optical depth.

If we assume some hypothesis related to the vegetation properties (homogeneous or pseudo-homogeneous), integrating the above equation from the entrance of the canopy to a point inside located by leaf area index, and after summing on all direction in space, we can express the solar radiation intensity by the following general expression (Lopez and Semel 1999; Kryzhevoi et al 2001):

Ov=J <&o,v(&,ç>,t)exp - j^s,t)ds q dQdv

O0v(Q,t) is the radiation intensity at the top of the farming

Q(0,9) is the solid angle over which we have integrated for all possible directions (0, 9) of the incoming radiation from the atmosphere.

s=sr

This radiation must be absorbed or intercepted by a material point or a vegetation layer inside the oasis. Consequently, it represents the amount of spectral incident radiation really used by plants in photosynthetic activity (agricultural productivity and biomass production) or for transpiration (water consumption). Many empirical formulas exist in the literature to make the conversion. The knowledge of the radiative climate inside the oasis by modelling solar radiative transfer within is important for more than one practical implications. First, to evaluate intercropping performances, identification of environmental resources and different planting configurations may be tested when we think to renew the oasis. Second, models can be used when tools for studying mono-crops are inadequate or in order to infer radiation variables difficult to obtain from field measurements.

Analysing the solar radiative transfer inside a vegetation canopy, like the oasis( an intercropping system that has three production levels with different trimming) and determining the amount of all the radiation flux exchanged (intercepted flux, ascendant flux, descendant flux, absorbed flux, reflected flux, transmitted flux) at any point in space (vertical and horizontal) and at hour scale for every day and for all the seasons of the year is beneficial at all sides when considering the agricultural planning for the oasis of North Africa. In fact from estimating solar radiative flux we can deduce the amount of evapotranspiaration at any point of the oasis and at any height in the canopy so the quantity of water really needed by every species can be calculated (Monteith and Unsworth, 1990). Also determining the solar radiative flux reflected by the leaves of a plant inside a vegetation storey, by a layer in or by all the storey ( market gardening, fruit trees, date palm) is very helpful to detect the eventual diseases that can attack the plants and to follow their photo-sanitary state (Grancher et al , 1993; Jones 1992). The reasoning is as follow: If we have a library of data for the reflectivity at the level of leaf, plant or layer of plants when the plant itself is in good health, this can be done by direct measurement in situ (photo satellite, radar in plane, portative radar) or ex situ ( spectroscopy measurement). Simulating this reflectivity (ratio between the reflected solar radiation and the incident solar radiation) for each hour, every day and all the season of the year by a model that we can elaborate. Any difference recorded between the simulating values and those measured for a plant in good health, is an indicator for the photo-sanitary specialists to intervene. Finally the biomassproduction and the harvest of a farming can be deduced and forecasted by modelling the solar radiation absorbed or intercepted by every species, any farming storey and all the oasis. This is because the physiological properties of plants, their geometrical characteristics and their densities can be the input or the out put of the model. We will present now a set of applied equations for the oasis emerged from the radiative transfer equation cited above.

5.2 Reasoning to Model the Oasis Architecture for an Optimal Use of Resources

5.2.1 Formulating Solar Radiative Transfer Inside the Oasis

The global solar radiation is defined as the sum of direct solar radiation and diffuse solar radiation. The first is that arriving from the sun directly and received by a unity of surface perpendicular to the radiation. The second is that received from the vault of heaven by the surface. The celestial vault is chatted into solid angle sectors and hemispherical fluxes are computed by numerical integration of the directional fluxes. When traversed a vegetable layer, the global incident solar radiation at depth f inside the vegetation suffered a diminution that expresses the amount of solar radiation intercepted (Grancher et al., 1993; Berbigier and Bonnefond, 1995; Tournebize and Sinoquet, 1995.). This depth is located with the Leaf Area Index accounted from the top of the canopy. The subtraction is the combination of the removal from the direct radiation and from the diffuse radiation. The direct beam radiation received at depth f can be written like this:

The diffuse radiation received from the celestial vault at depth f can be expressed by ®rd =Ord,o exp[-^(a,hr) ]

The global solar radiation received at the level f inside the oasis is:

^(a ,hs) extinction coefficient for the direct solar radiation hs is the sun elevation f the leaf area index accounted from the top of the oasis O rs 0 is the direct solar radiation received above the canopy

Ord,o is the diffuse solar radiation received above the oasis ^ (a, hr) is the mean extinction coefficient for the diffuse flux density a is the mean inclination of leaves on the trees hr the mean elevation of the radiation sector

After penetrates inside the vegetation, the incident beam will suffered a multiple rediffusion under the effect of leaves, trunks and branches. The beam will be scattered in upward and downward directions. The study of the multiple scattering process at the scale of all the canopy needed to elaborate the radiative balance for a thin canopy layer inside the vegetation and the use of some hypothesis. We can found the following 2nd order linear differential equations:

d20-df2

<df2

R2 _ (1 _ T2)]o- = Rß( -1)0rs,0 exp(-ßf) + Rß'(ß - 1)0rd,o exp(-ßf)

R2-(1-T)2 ]+ =-4(l-T)r+R2+ßT ] ra,oexp(-ßf ) ßR-T)+R2+ßT ] ,*oexp(-ß f )

0+ is the ascendant rescattred flux density ( upward direction).

O- is the descendant rescattred flux density (downward direction) T is the transmittance factor of the leaves or the plant in a stand R is the reflectance factor of the leaves or the plant in a stand

| is the extinction coefficient for rescattered radiation

The general analytical solutions of the 2nd order linear differential equations are: 0+ (f ) — Xi exp(nf ) + X2 exp(-nf ) + Y1O rs,o exp(-|f ) + ¥2$^ exp(-|'f )

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