1.5: Reflectance Functions

In the most general case of diffuse reflection, the reflectance of a surface will depend on both the direction of the incident radiation and that of the reflected radiation. The bidirectional reflectance distribution function, f_r, links the irradiance E to the reflected radiance, such that

\[L_{r}=f_{r}\left(\mu, \varphi ; \mu_{0}, \varphi_{0}\right) E\left(\mu_{0}, \varphi_{0}\right).\]

For a surface irradiated with flux density F, the irradiance is simply the component of the flux density perpendicular to the surface

\[E=\mu_{0} \mathbf{F},\]

so that, abbreviated, we can write

\[ L_{r}=f_{r} \mu_{0} \mathbf{F}\]

One of the simplest examples of a reflectance rule is that of a Lambertian reflecting surface for which the radiance is isotropic, so that

\[L_{r}=\frac{\lambda_{0}}{\pi} \mu_{0} \mathbf{F},\]

where λ₀ is sometimes referred to as the Lambertian albedo . Although it is not strictly physically correct, it is convenient (Chandrasekhar, p147) to identify λ₀ with the single scattering albedo ϖ₀, so for Lambert’s law the BRDF is

\[f_{r}=\frac{\varpi_{0}}{\pi}.\]

For the most part, we shall refer all reflectance rules used to a BRDF; alternative reflectance functions will be discussed in Section 1.7.

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