Directional Distribution

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A directional distribution measures spatial characteristics such as central tendency, dispersion and directional trends. These distributions can be applied to a wide variety of phenomena including reflected flux [1] neutrons [2], sea or wind waves [3], highway traffic [4] and spatial fibre processes [5].

In ArcGIS and other spatial statistics tools, it measures whether a distribution of geographic features shows a directional trend.

The directional distribution function

The directional distribution function (also called the spreading function, the angular distribution function or directional spreading function) is a function that represents the directional distribution of wave energy. It is dimensionless and varies with frequency [3]. “Dimensionless” means that the function does not have any units of measurement.

The function is part of the process of estimating frequency-direction spectra. The frequency-direction spectrum (also called the directional spectrum or wave spectrum) represents the distribution of wave energy in terms of frequency and direction. It is a two-dimensional function, with frequency on the x-axis and direction on the y-axis.

A directional distribution function D(f_n, θ_m) and its frequency-direction spectrum S(f_n, θ_m) are related in the following way [6]:

Where:

• f = frequency
• θ_m = a discrete angle showing wave energy arrival direction (counterclockwise from true north) with values from θ₁ = -180 degrees to θ₁₈₁=180 degrees in steps of dθ = 2 degrees.
• m = a direction index with values m=1 to m=M=181.
• S(f_n) = sea surface displacement variance spectral density.

References

1. NBS Technical Note, Vol. 439. 1968. U.S. Department of Commerce, National Bureau of Standards.
2. Pyshkina, M. et al. (2021). Study of neutron energy and directional distribution at the Beloyarsk NPP selected workplaces. Nuclear Engineering and Technology Volume 53, Issue 5, May 2021, Pages 1723-1729
3. Goda, Y. (2000). Random Seas and Design of Maritime Structures. World Scientific.
4. Sharma, S. & Singh. A. (1992). Reexamination of Directional Distribution of Highway Traffic. Journal of Transportation Engineering. Vol. 118, Issue 2.
5. Mattfeldt, T. et al. (1994). Estimation of the directional distribution of spatial fibre processes using stereology and confocal scanning laser microscopy. Journal of Microscopy, Vol 173, Issue 2, p. 87-101.
6. Long, C. (1995). Directional Wind Wave Characteristics at Harvest Platform. U.S. Army Engineer Waterways Experiment Station.