< List of probability distributions < *Beckmann distribution*

## What is the Beckmann distribution?

The **Beckmann distribution** is a continuous probability distribution similar to the normal (Gaussian) distribution. Widely used in the computer science and optics literature, it arises from Gaussian roughness assumptions for microsurfaces [1].

Although it is relatively simple distribution to understand and implement, it can be computationally complex to evaluate due to the inclusion of the special function *tan *in its definition, [2].

## Beckmann distribution properties

The Beckmann distribution is defined by

where

- (θ, ϕ) is the probability density function (pdf) of the surface slope.
- θ is the zenith angle (angle between the surface normal and the line of sight).
- ϕ is the azimuth angle (angle in the plane tangent to the surface).
- θ_s is the mean slope angle.
- σ is the standard deviation of the surface slope

It can also be parameterized as

where

- where
*α*= the roughness of the macro surface, usually set to between 0.005 and 0.5, *θ* is the angle between_{h}*h*and the macro surface normal*n*.

The pdf of a Beckmann distribution is unimodal, but its shape around the peak varies depending on the arguments.

The four-argument form Beckmann distribution [μ_{1}, μ_{2}, σ_{1}, σ_{2}] is equivalent to Beckmann Distribution [μ_{1}, μ_{2}, σ_{1}, σ_{2}, 0], and is sometimes called the *uncorrelated Beckmann distribution *[3].

## Uses of the Beckmann distribution

The distribution is used in a variety of applications, including:

- Computer graphics: used to model the distribution of the reflection coefficient of a surface.
- Wireless communications: models the distribution of signal strength in a fading channel.
- Optics: can model the distribution of the intensity of light scattered by a rough surface.

## Beckmann vs binormal

The Beckmann distribution and binormal distribution are both capable of modeling the distribution of a random vector’s magnitude. However, the Beckmann distribution is more versatile, able to model a wider range of phenomena. In contrast, the binormal distribution is simpler to understand and implement, thus making it a popular choice for a variety of applications.

One reason for the Beckmann distribution’s greater flexibility is its ability to model the magnitude of a random vector with correlated components, while the binormal distribution is only useful for modeling the distribution of the magnitude of a random vector with independent components.

The Beckmann distribution can be obtained from the binormal distribution (the two dimensional form of the multivariate normal distribution) through the square root of the sum of the squares of the random vector’s two components. As a consequence, the Beckmann distribution is more peaked and has a higher probability of resulting in a smaller magnitude.

## References

- Microfacet Models for Refraction through Rough Surfaces
- Romanyuk, O. et al. Optical Fibers and Their Applications 2012. Proceedings of the SPIE, Volume 8698, article id. 86980L, 4 pp. (2013).
- Wolfram Research (2010), BeckmannDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BeckmannDistribution.html (updated 2016).