Relation To Irradiation

 Although we have focused on radiation emitted by a surface, the foregoing concepts may be extended to incident radiation (Figure 12.9). Such radiation may originate from emission and reflection occurring at other surfaces and will have spectral and directional distributions determined by the spectral intensity I_λ,i (λ, θ, ϕ). This quant ity is defined as the rate at which radiant energy of wavelength λ is incident from the (θ, ϕ) direction, per unit area of the intercepting surface normal to this direction, per unit solid angle about this direction, and per unit wavelength interval dλ about λ.

The intensity of the incident radiation may be related to an important radiative flux, termed the irradiation, which encompasses radiation incident from all directions. The spectral irradiation Gλ (W/m².μm) is defined as the rate at which radiation of wavelength λ is incident on a surface, per unit area of the surface and per unit wavelength interval dλ about λ. Accordingly,

where sin θ dθ dϕ is the unit solid angle. The cos θ factor originates because Gλ is a flux based on the actual surface area, where as I_λ,i is defined in terms of the projected area. If the total irradiation G (W/m²) represents the rate at which radiation is incident per unit area from all directions and at all wavelengths, it follows that

Or from equation 12.13

If the incident radiation is diffuse, I_λ,I is independent of θ and ϕ and it follow that 

             And

Comments :generally, radiation sources do not provide such a regular spectral distribution for the irradiation. However, the procedure of computing the total irradiation from knowledge of the spectral distribution remains the same, although evaluation of the integral is likely to involve more detail.