Quantities

There are lots of common misconceptions around the different quantities used to quantify the reflectance of a surface. These are well-described in Schaepman-Strub et al. (2006), the following sections define and summarise the quantities that can be calculated using the pydirectional package.

Bidirectional Reflectance Distribution Function (BRDF)

The Bidirectional Reflectance Distribution Function (BRDF) describes how electromagnetic radiation is reflected by a surface. It is a function of the incoming and outgoing directions of the radiation, described by the incoming and outgoing zenith angles, and the relative azimuth angle between the two. In the pydirectional package, the incoming zenith angle is the solar zenith angle (\(\varphi_{s}\)), the outgoing zenith angle is the viewing zenith angle (\(\varphi_{v}\)), and the relative azimuth angle is the relative azimuth angle between the sun and the sensor (\(\phi\)). BRDF models define the surface reflectance as a function of these angles, with a set of parameters that control the shape of the function. The parameters are generally wavelength-dependent, and can be retrieved from measurements of the surface reflectance.

\[BRDF = f(\varphi_{s}, \varphi_{v}, \phi, \lambda)\]

The BRDF is an intrinsic property of the surface and, as it describes the reflection of an infinitesimal beam, it is not directly measureable.

Bidirectional Reflectance Factor (BRF)

The Bidirectional Reflectance Factor (BRF) is defined as the ratio of the BRDF of a surface to the BRDF of an ideal Lambertian surface under identical viewing and solar geometry. An ideal Lambertian surface reflects radiation equally in all directions, its BRDF is equal to \(\frac{1}{\pi}\). Hence, the BRF of a surface is given by,

\[BRF = R(\varphi_{s}, \varphi_{v}, \phi, \lambda) = \pi \cdot f(\varphi_{s}, \varphi_{v}, \phi, \lambda)\]

Similarly to BRDF, BRF cannot be directly measured.

Hemispherical Directional Reflectance Factor (HDRF)

The Hemispherical Directional Reflectance Factor (HDRF) is equal to the integral of the BRF over the solar hemisphere for a given viewing geometry. Since it involves contributions from all incoming directions, it is dependent on the atmospheric state.

Hemispherical Conical Reflectance Factor (HCRF)

The Hemispherical Conical Reflectance Factor (HCRF) is the physical quantity that most in-situ and satellite sensors’ measurements are reported as.

\[HCRF = (\varphi_{s}, \varphi_{v}, \phi, \lambda) = f_{dir} \cdot BRF + f_{diff} \cdot BRF_{sky}\]

where \(f_{dir}\) and \(f_{diff}\) are the proportion of direct and diffuse irradiances, respectively. The ratio of these two values is more easily accesible so we can rewrite HCRF using \(d = \frac{f_{dir}}{f_{diff}}\). In this derivation we also note \(f_{dir} + f_{diff} = 1\).

\[d + 1 = \frac{1}{f_{diff}}\]

\[HCRF(\varphi_{s}, \varphi_{v}, \phi, \lambda) = f_{diff} \cdot (d \cdot BRF + BRF_{sky})\]

\[HCRF(\varphi_{s}, \varphi_{v}, \phi, \lambda) = \frac{BRF \cdot d + BRF_{sky}}{1 + d}\]

where \(d\) is the ratio of direct to diffuse irradiance, and \(BRF_{sky}\) is the diffuse component of the incoming irradiance.

Note

If the HDRF is constant over the sensor’s field of view, then the HCRF is equal to the HDRF.

Directional Hemispherical Reflectance (DHR)

The Directional Hemispherical Reflectance (DHR) or black sky albedo is the integral of the BRDF over the viewing hemisphere for a given solar geometry. It is called the black sky albedo because it assumes that there is no diffuse illumination.

\[DHR(\varphi_{s}, \lambda) = \int_{2\pi} f(\varphi_{s}, \varphi_{v}, \phi, \lambda) \cdot cos(\varphi_{v}) d\Omega_{v}\]

where \(d\Omega_{v}\) is the solid angle element in the viewing hemisphere.

Bihemispherical Reflectance (BHR)

The Bihemispherical Reflectance (BHR) or white sky albedo is the integral of the BRDF over both the solar and viewing hemispheres. It is called the white sky albedo because it assumes that the surface is illuminated by a uniform diffuse light source.

\[BHR(\lambda) = \int_{2\pi} \int_{2\pi} f(\varphi_{s}, \varphi_{v}, \phi, \lambda) \cdot cos(\varphi_{v}) \cdot cos(\varphi_{s}) d\Omega_{v} d\Omega_{s}\]

Blue-Sky Albedo

The blue-sky albedo is the albedo of a surface under realistic illumination conditions, containing contributions from both the direct and diffuse components of the incoming radiation. It is a weighted average of the black-sky and white-sky albedos.

\[\alpha(\varphi_{s}, \lambda) = f_{diff} \cdot BHR(\lambda) + f_{dir} \cdot DHR(\varphi_{s}, \lambda)\]

As with HCRF, we can rewrite using the direct to diffuse ratio,

\[\alpha(\varphi_{s}, \lambda) = \frac{d \cdot DHR(\varphi_{s}, \lambda) + BHR(\lambda)}{1 + d}\]