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.

It 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.