Albedo

The expression for the reflected surface radiance can be further simplified if the BRDF is taken as being independent of both the direction of the incident radiance and the direction of reflected radiance. In this case, equation (2.83) becomes,

$$I^{\uparrow}(\tau_1)= R_{11} \int_0^{2\pi} \int_0^1 I(\tau_1;-\mu',\phi') \mu' d\mu' d\phi' = R_{11} E^{\downarrow}(\tau_1)$$ (3.84)

where the integral over solid angle is identified as the downwelling irradiance and the upward surface radiance, no longer a function of direction, is denoted as $I^{\uparrow}(\tau_1)$ for clarity. Recognizing that $E^{\uparrow}(\tau_1)=\pi I^{\uparrow}(\tau_1)$, equation (2.84) can be reorganized as,

$$\frac{E^{\uparrow}(\tau_1)}{E^{\downarrow}(\tau_1)} = \pi R_{11} \equiv \Lambda$$ (3.85)

where the quantity $\Lambda$ is defined as the surface albedo. Physically, the surface albedo represents the fraction of energy incident on a plane surface which is reflected back into the atmosphere. Conservation of energy requires $0 \leq \Lambda \leq 1$. In the model the surface albedo is required as input. Once the downwelling irradiance is known, the upward radiance streams can be determined using,

$$I^{\uparrow}(\tau_1)= \frac{\Lambda E^{\downarrow}(\tau_1)}{\pi}$$ (3.86)

which is the boundary condition of equation (2.78), necessary for a complete solution.

Values of albedo are known to be quite variable. Albedo can be as low as 0.01 at 520 nm for coniferous forests (Koepke and Kriebel, 1978) and 0.02 at 700 nm for ocean waters (see Appendix A) while fresh snow can have an albedo as large as 0.9 (Blumthaler and Ambach, 1988). Real surfaces are hardly ever spatially homogeneous. Often, an `effective' albedo must be used which takes into account the relative contribution from the constituent surfaces. For example, when modeling partially-cloudy conditions, the pixel fraction which contains clouds is used as the weighting factor. In addition, most surfaces have an albedo which varies with solar zenith angle and wavelength and rarely are they truly Lambertian. Yet this assumption is made as the reflection properties of most surfaces are unknown or too complicated to incorporate into a model. In the visible, the use of a Lambertian surface can lead to errors as large as 40-50% (Appendix B; Weihs and Webb, 1997). Finally, assuming a surface to be depolarizing may also be incorrect. Model calculations in the visible revealed that polarization emerging from the top of the atmosphere strongly mirrored the polarization reflected from the underlying surface (Fitch, 1982). As polarization measured in the nadir by the ER-2 was typically non-zero and sometimes large, this suggests that the underlying surface is not completely depolarizing.

A discussion of two types of non-Lambertian surfaces, clouds and ocean, and how they can be modelled, is given in Appendix B.