Lambertian reflectance

Lambertian reflectance is the ideal diffuse-reflection model with direction-independent outgoing radiance, the frequency-integrated specific intensity. If a fraction $0\le \rho \le 1$ of incident radiant flux is reflected, a surface receiving irradiance $E$ has reflected radiance $L=\rho E/\pi$. The reflected outward power per area is $\pi L=\rho E$ by the same hemispherical integral used for radiant exitance. Its radiant intensity still acquires the projected area factor in Lambert's cosine law.

For $E=100\mathrm{W}{\mathrm{m}}^{-2}$ and $\rho =0.80$, the reflected radiance is $80/\pi \approx 25.4648\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$. A patch of area $0.020{\mathrm{m}}^2$ reflects $1.6\mathrm{W}$ in total, and its radiant intensities at $0^{\circ }$ and $60^{\circ }$ are respectively $1.6/\pi$ and $0.8/\pi \mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. The radiance stays fixed as the view changes, provided the illumination stays fixed. Changing the incident angle can change irradiance and therefore the reflected radiance.

A spherical body of radius R observed from distance $d\gg R$ is illuminated and viewed from the same direction, with incident irradiance $F_{\star }$. For uniform Lambertian reflectance A, a patch at angle $\theta$ has outgoing radiance $A{F}_{\star }\cos \theta /\pi$. The observer receives this radiance times projected area divided by $d^2$. Since ${\int }_{\mathrm{hemisphere}}{\cos }^2\theta dS=2\pi R^2/3$, integration proves the displayed full-phase irradiance. Its geometric albedo is $2A/3$. Total reflected power alone does not specify the brightness in one direction for a non-Lambertian surface.