**2.1 Some solar radiation concepts**

Energy is transferred by electromagnetic waves characterized by wavelength and frequency. The electromagnetic spectrum ranges from high frequency cosmic radiation to low frequency radio waves. For practical purposes the solar spectrum reaching the Earth comprises mostly the ultraviolet, visible and infrared radiation. The spectral regions of the solar spectrum are listed in Table 1.


Table 1. Spectral ranges of solar radiation with wavelengths in nanometers units (1nm = 10-9 m).

The solar infrared radiation beyond 4000 nm that reaches the Earth is insignificant compared to that from 250 to 4000 nm. A few terms regarding solar radiation are considered here. The spectral region from 400 nm to 700 nm is also referred as the *photosynthetically active radiation* (PAR) as this is the region used by plants for photosynthesis. The region from 700 nm to 3000 nm is the reflected infrared, because the surfaces at the environment temperature do not emit in this part of the spectrum and from 1100 nm to 3000 nm is referred as the shortwave infrared.

The corpuscular theory of light is also of interest. It relates the amount of energy of the electromagnetic radiation in Joules (J) to a given wavelength. A quantum of energy is the amount of energy of a photon (a package of discrete energy of a single frequency) and is given by *E = h.ν*, where *h* is the Planck's constant (6.626 x 10-34 J.s) and *ν* is the frequency in Hertz. In the PAR region it is of interest to use the units of mol photon m-2s-1. If the sun is the radiation source of the PAR, 1 MJ m-2 is equivalent to 4.6 mol photon m-2 s-1 (Norman & Arkebauer, 1991).

The amount of energy that reaches a surface per unit area per time is called *irradiance* (E) and its unit is Watts per square meter (Wm-2) being 1.0W = 1.0 Js-1.The *radiance* (L) is the flux density per area and per solid angle in steradian (sr) and has units of Wm-2 sr-1. These two are related as *E=πL* where *π* is the integral of a projected solid angle over the upper hemisphere, in units of sr. The *solar constant* is the amount of energy coming from the sun that reaches the Earth outside the atmosphere for a distance sun-Earth of 1 (one) Astronomical Unit (UA, the average distance between the sun and the Earth). The solar constant was determined by the World Meteorological Organization (WMO) as 1367 Wm-². From the total radiation at the top of the atmosphere 534.7 Wm-2 is in the PAR region.

In order to illustrate the distribution of this radiation we performed calculations for the day 23rd of September 2011 at PESAGRO weather station in Seropédica (22º 45' 28.37"S and 43º41' 5.47"W, Rio de Janeiro State, Brazil), at 13 UT (Universal Time). The sun-Earth distance was 1.0031UA and *solar zenith angle* (SZA, the angle from the sun direction to the vertical direction right above) was 34.4º. Considering the correction for the SZA the actual value of incoming radiation at the top of the atmosphere on a surface parallel to the Earth's surface was 1127.7 Wm-².Using the 6S model (Vermote et al., 1997) we found that the *total global* (from 250 to 4000 nm, *direct* plus *diffuse*) irradiance on a horizontal surface on the ground was 796.1 Wm-², which corresponds to 70.6% of the total radiation at the top of the atmosphere. From this amount of radiation reaching the horizontal surface, 356.2 Wm-2was in the PAR region. The surface incoming PAR in that condition was 68.9% from direct beam and 31.1% from diffuse sky irradiance. If the surface is perpendicular to the solar beam the incoming direct irradiance in the PAR region was 298.8 Wm-2, i.e., the transmission of PAR direct beam was 67.7%. These calculations using the 6S model considered a clear sky condition, tropical atmospheric model and continental aerosol model, with horizontal visibility of 15 km and target altitude of 34 m.

The incoming radiation on a plant canopy can be reflected, absorbed or transmitted to the soil background. From the conservation of energy it follows that the *incident energy* (Eiλ) is equal to the sum of the *reflected* (Erλ), *absorbed* (Eaλ) and *transmitted* (Etλ) fluxes:

$$\text{Ei}\lambda = \text{Er}\lambda + \text{Ea}\lambda + \text{Et}\lambda \tag{1}$$

Dividing these by the *incident energy*:

222 Solar Radiation

monocultures to the complex interrelationships that exists among trees and grasses in silvipastoral agro-systems. In this chapter we will explore a few selected subjects within this broad chain of processes. Recently, focusing on the role of theory in plant science, Woodward (2011) noted that "the development of plant science is based on observations, the development of theories to explain these observations and the testing of these theories."Besides the need of theory to overcome empirical approaches, the theoretical basis is also functional for a better understanding of possibilities and limitations of the new available instrumentation from advances in remote sensing and other technologies for grassland monitoring and assessment. We give emphasis to the concept of sward canopy structure, discussing the central role of Leaf Area Index in the pasture trophic program via light interception. We also give some theoretical and practical emphasis on methodological aspects and procedures for measurements of canopy structure and radiation interception by

vegetation. And finally we considered the efficiency use of radiation.

practical aspects of sward canopy structure.

**2.1 Some solar radiation concepts** 

solar spectrum are listed in Table 1.

referred as the shortwave infrared.

(1nm = 10-9 m).

**2. Leaf Area Index (LAI) and the G-function: Theoretical considerations** 

Before talking more specifically about leaf area index and radiation interception as key variables of pasture ecosystems, we need to review some basic concepts about electromagnetic nature of solar radiation underlying the discussion on theoretical and

Energy is transferred by electromagnetic waves characterized by wavelength and frequency. The electromagnetic spectrum ranges from high frequency cosmic radiation to low frequency radio waves. For practical purposes the solar spectrum reaching the Earth comprises mostly the ultraviolet, visible and infrared radiation. The spectral regions of the

**Spectral region Spectral range (nm)** 

The solar infrared radiation beyond 4000 nm that reaches the Earth is insignificant compared to that from 250 to 4000 nm. A few terms regarding solar radiation are considered here. The spectral region from 400 nm to 700 nm is also referred as the *photosynthetically active radiation* (PAR) as this is the region used by plants for photosynthesis. The region from 700 nm to 3000 nm is the reflected infrared, because the surfaces at the environment temperature do not emit in this part of the spectrum and from 1100 nm to 3000 nm is

The corpuscular theory of light is also of interest. It relates the amount of energy of the electromagnetic radiation in Joules (J) to a given wavelength. A quantum of energy is the amount of energy of a photon (a package of discrete energy of a single frequency) and is

Ultraviolet 10 to 400 Visible 400 to 700 Infrared 700 to 4000 Table 1. Spectral ranges of solar radiation with wavelengths in nanometers units

$$\mathbf{1} = \mathbf{p}\lambda + \mathbf{d}\lambda + \mathbf{t}\lambda \tag{2}$$

where ρλ, αλ and τλ are respectively the spectral *reflectance*, *absorbance* and *transmittance*. For remote sensing purposes it is considered the bidirectional reflectance, which is defined as reflectance acquired from a reflected directional flux divided by the directional incoming flux (Nicodemus et al., 1977). This is attained because the radiance that reaches the sensor is directional and under clear sky condition, the incoming flux can be considered a direct radiation flux.

#### **2.2 Canopy structure definitions**

According to Campbell & Norman (1989) "plant canopy structure is the spatial arrangement of above-ground organs of plants in a plant community" and as such includes the size, shape and orientation of all aboveground plant organs, making quantitative descriptions exceedingly difficult (Nobel et al., 1993). In practice, the leaf canopy structure of a plant community can be described in terms of *Leaf Area Index* (LAI), *leaf angle distribution* (LAD) and *leaf clumpiness*. The LAI concept was originally introduced by Watson in 1947 and is the one side leaf area per unit area of soil (m2 leaf m-2 ground surface) and can be regarded as the number of leaf layers arranged above the ground. The LAD is the probability density of a leaf being in a certain angle in relation to the horizontal (or the leaf normal to the vertical). Clumpiness is related to how the leaves are distributed in the space. Area index, angle distribution and clumpiness can also be defined for stems (for stem area: SAI; for angle distribution: SAD) or any other aerial parts of the plant. Theoretical functions have been used in the literature to describe LAD for most situations present in nature, like those introduced by Wit (1965). It was expanded by Bunnik (1978) by adding the uniform and spherical types of LAD, which can be seen in Figure 1. Here a uniform distribution in azimuth is assumed, which means that the normal to the leaves are random regarding to the azimuth.

Fig. 1. Theoretical leaf angle distribution (LAD) of Bunnik (1978) for canopies.A) Density probability functions of leaf angle. B) Cumulative LAD.

The LAD represents the strategies that plants use to intercept radiation for plant processes. A planophile plant has most of the leaves near the horizontal and has a small variation of intercepted radiation for a range of solar zenith angles below 45 degrees. Erectophile canopies on the other hand, have large variations in interception and the gap fraction (fraction of openings in the canopy) is more frequent from close to nadir viewing. This way less radiation is intercepted when the sun is close to the zenith. Plagiophile canopies have most of the leaves around 45 degrees and conversely the extremophile canopies have most of the leaves around 0 and 90 degrees (respectively horizontal and vertical). The radiation interception by the extremophile canopy can lead to different results for the plant if the horizontal and vertical leaves are in different canopy layers.

According to Campbell & Norman (1989) "plant canopy structure is the spatial arrangement of above-ground organs of plants in a plant community" and as such includes the size, shape and orientation of all aboveground plant organs, making quantitative descriptions exceedingly difficult (Nobel et al., 1993). In practice, the leaf canopy structure of a plant community can be described in terms of *Leaf Area Index* (LAI), *leaf angle distribution* (LAD) and *leaf clumpiness*. The LAI concept was originally introduced by Watson in 1947 and is the one side leaf area per unit area of soil (m2 leaf m-2 ground surface) and can be regarded as the number of leaf layers arranged above the ground. The LAD is the probability density of a leaf being in a certain angle in relation to the horizontal (or the leaf normal to the vertical). Clumpiness is related to how the leaves are distributed in the space. Area index, angle distribution and clumpiness can also be defined for stems (for stem area: SAI; for angle distribution: SAD) or any other aerial parts of the plant. Theoretical functions have been used in the literature to describe LAD for most situations present in nature, like those introduced by Wit (1965). It was expanded by Bunnik (1978) by adding the uniform and spherical types of LAD, which can be seen in Figure 1. Here a uniform distribution in azimuth is assumed, which means that the normal to the leaves are random regarding to the

Fig. 1. Theoretical leaf angle distribution (LAD) of Bunnik (1978) for canopies.A) Density

The LAD represents the strategies that plants use to intercept radiation for plant processes. A planophile plant has most of the leaves near the horizontal and has a small variation of intercepted radiation for a range of solar zenith angles below 45 degrees. Erectophile canopies on the other hand, have large variations in interception and the gap fraction (fraction of openings in the canopy) is more frequent from close to nadir viewing. This way less radiation is intercepted when the sun is close to the zenith. Plagiophile canopies have most of the leaves around 45 degrees and conversely the extremophile canopies have most of the leaves around 0 and 90 degrees (respectively horizontal and vertical). The radiation interception by the extremophile canopy can lead to different results for the plant if the

probability functions of leaf angle. B) Cumulative LAD.

**A <sup>B</sup>**

horizontal and vertical leaves are in different canopy layers.

**2.2 Canopy structure definitions** 

azimuth.

The extinction of solar radiation by interception was initially described by Monsi & Saeki (1953, in English translation 2005) for horizontal leaf layers and generalized by Ross & Nilson in 1966 (Ross, 1981) to include leaf orientations other than horizontal. The rate of change of a downward direct flux of radiation (ID) with the cumulative LAI at any height from the top of canopy is given by:

$$d\mathbf{I}\_{\rm D} \;/\; d\mathbf{L} \mathbf{A} \mathbf{I} = -\lambda\_0 \mathbf{K}\_{\rm D} \mathbf{I}\_{\rm D} \tag{3}$$

where λ0 is the leaf spatial distribution parameter, which is equal to 1 for leaf elements randomly distributed and leaf position is independent of other leaf positions, also referred as a Poisson canopy (Nilson, 1971, Baret et al., 1993). KD is the *interception coefficient*, and is given by (Ross, 1981):

$$\mathbf{K}\_{\rm D} = \mathbf{G}(\emptyset, \emptyset, \{\bullet\}, \emptyset, \emptyset) \;/\; \cos\theta\_{\rm i} \tag{4}$$

where G(θi ,θl, φi , φl) is the G-function, θi is the solar zenith angle, φi is the solar azimuth, φ<sup>l</sup> is the leaf azimuth in relation to the solar azimuth, and θl is the leaf normal angle from the zenith direction, i.e., a vertical leaf has θl=*π*/2 and a horizontal leaf has θl=0.The G-function represents the projected fraction of leaf area in the solar direction and is equal to the average value of the cosine of the angle between the solar zenith angle and the leaf normal angle (θl) from the vertical.

Assuming that leaves have a random azimuth orientation (Ross, 1981), meaning that normal to the leaves are random regarding to the azimuth, the G-function is calculated as (Ross, 1981, Antunes, 1997):

$$G(\theta\_l, \theta\_l, \varphi\_l, \varphi\_l) = \int\_0^{\pi} g'(\theta\_l) \int\_0^{2\pi} \frac{|\cos \delta|}{2\pi} d\theta\_l d\varphi\_l \tag{5}$$

where g'(θl) is the density probability of leaf angle between 0 and *π*/2 and δ is the angle between leaf normal and the sun. The cosine of δ is calculated as (Ross, 1981):

$$\cos \mathfrak{G} = \cos \mathfrak{H} \dot{\mathfrak{c}} \cos \mathfrak{h}\_{\mathfrak{l}} \text{ - } \sin \mathfrak{H} \dot{\mathfrak{l}} \sin \mathfrak{h}\_{\mathfrak{l}} \text{cos} \text{sp}\_{\mathfrak{l}} \tag{6}$$

The spherical LAD is of special interest because the leaves are arranged in such a way in the canopy that the leaves from one square meter of soil can fit a surface of a sphere with the same area as the LAI. As a result, at any solar zenith angle, the fraction of leaf area projected towards the sun is always the same, which is 0.5. This means that the G-function for such a canopy is always 0.5 regardless of the sun's orientation. Maize canopy LAD has been measured in the field and found to be spherical (Antunes et al., 2001). Plants with a spherical LAD intercept the same amount of radiation regardless the direction of the solar beam and can be regarded as a good characteristic for a high productivity canopy.

Equation 3 can be solved by integrating for an entire layer of canopy yielding:

$$\mathbf{I}\_{\rm D} = \text{I}\_{\rm D} \exp(\cdot \mathbf{\hat{\upbeta}} \, \text{G} (\mathbf{\hat{\upbeta}}, \mathbf{\hat{\upbeta}}, \mathbf{\hat{\upeta}}, \mathbf{\hat{\upeta}}) \, \text{LAI} \, / \cos \theta) \tag{7}$$

where I0 is the direct beam flux intensity at the top of the canopy, which can be set to be the fraction of direct beam above the canopy. In this equation the term G(θi,θl,φi,φl) LAI/cosθ<sup>i</sup> defines the mean number of contacts of the direct beam with the canopy elements (Nilson, 1971). Although the Equation 7 is similar to the Bouguer law (also referred as Beer's law) for radiation transmission in a turbid medium, in which K is the attenuation coefficient, the concept as applied here is different from Beer's law, since this equation defines only the amount (or fraction) of direct beam left after passing through a canopy layer with a defined leaf area, LAD and clumpiness.
