# 2.7: Surfaces - Hemispherical Albedo

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Also known as the directional hemispherical reflectance, the hemispherical albedo ρ refers to a point on a reflecting surface, and is defined as the ratio of the exitance M to the irradiance E, so that

$\rho \left( \mu_0, \varphi_0 \right) = \frac{M}{E \left( \mu_0, \varphi_0 \right)},$

and in terms of the BRDF, we have

$\rho \left( \mu_0, \varphi_0 \right) = \int_0^{2 \pi} \int_0^{1} f_r \left( \mu, \varphi; \mu_0, \varphi_0 \right) \mu d \mu d \varphi.$

Unlike the single scattering albedo, ρ and the other albedos that we will encounter do not necessarily have in principle a maximum possible value of unity. (See A Brief History of the Lommel-Seeliger Law). The scattering properties of the surfaces that we have studied so far are summarised in Table I, from which, for the Lommel-Seeliger law, it can be seen that the maximum possible value for ρ is ½ and 0.125 for the normal albedo.

Table I. Properties of Surfaces
Lambertian Lommel-Seeliger
fr ϖ0 $$\frac{ \varpi_0}{4 \pi} \frac{1}{ \mu_0 + \mu}$$
ρ ϖ0 $$\frac{ \varpi_0}{2} \left[ 1 - \mu_0 \ln \left( 1 + 1/ \mu_0 \right) \right]$$
pn ϖ0 $$\frac{ \varpi_0}{8}$$

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