The Cosine Law

10.4 The Cosine Law

If a small area is exposed to a point source of radiation, so that the rays of light hitting the surface are nearly parallel, the

of the surface depends on its orientation with respect to the radiant beam. This is easily seen by considering the area on a surface covered by a beam of parallel light of fixed size as its angle with respect to a normal to the surface increases (Fig. 10.3). The radiant flux density on the area perpendicu- lar to the direction of the beam remains constant, but the beam covers

a larger and larger area as the zenith angle increases, so the flux density at the surface decreases. If the area covered by the beam at normal incidence is

and the area at angle is A then = cos 8. This leads directly to Lambert's cosine law:

(10.3) where

cos

is the flux density normal to the beam, is the flux density at the surface, and is the angle between the radiant beam and a normal to the surface, which is referred to as the zenith angle.

The only common source of parallel light in natural environments is the sun, and

law is used to calculate the direct solar irradiance of slopes, walls, leaves, or animals. To do the calculation,

and the

F IGUR E 10.3. The area covered by a beam of parallel light increases as the angle between the beam and a normal to the surface increases.

Attenuation of Radiation 157

angle the sun makes with a normal to the surface need to be known. Equation (10.3) can also be used to find the irradiance of a surface when the radiance of the surroundings is known, as shown in the following example.

Example 10.3. A unit area on the ground is illuminated by a hemi- sphere of isotropic radiation with a radiance of N W

(isotropic means that the radiance is constant for all incident directions). What is the irradiance of the surface?

Solution. The irradiance of the surface by a small increment of solid angle

which makes an angle 0 with a normal to the surface, is N cos

The product of the radiance and the solid angle gives the flux density of radiation on a surface perpendicular to the direction of that radiation. The cosine of the angle converts this to flux density on the horizontal ground. To find the total irradiance of the surface, integrate the radiance of the hemisphere over all solid angles that are visible from the surface. If is the azimuth angle, then

= sin 0 The irradiance is therefore

so the irradiance of a surface under isotropic radiation is always times the radiance. Here N is constant and can be taken out

integral. radiation were not isotropic the irradiance of the surface could be found in the same way, but the angular distribution of N would need to be part of the integration.

An ideal reflecting surface, sometimes referred to as an ideal Lambertian surface, has a radiance that is proportional to the cosine of the angle between a normal to the surface and the view direction. The directional-hemispherical reflectance of such a surface is unity. In re- mote sensing, various surfaces are used to approximate a Lambertian surface, such as molded Halon or barium sulfate. The reflection coeffi- cients of natural surfaces, such as lakes, vegetation, soils, and rocks may differ substantially from that of an ideal surface. Therefore, care must be taken when

what is meant when referring to the "reflectance" of some natural surface. Biophysicists and micrometeorologists have used the term albedo to refer to the bi-hemispherical reflectance integrated over the entire solar spectrum. This operational approach gets around some of the complexity.