Ecological Archives M074-011-A6

Mark W. Denny, Brian Helmunth, George H. Leonard, Christopher D. G. Harley, Luke J. H. Hunt, and Elizabeth K. Nelson. 2004. Quantifying scale in ecology: lessons from a wave-swept shore. Ecological Monographs 14:513–532.

Appendix F. 1/f Noise: A review.

Because 1/f noise is pervasive among our spatial variables, its basic attributes deserve a brief review. (Eq. 8) typically ranges from 0 to 2. When  = 0, the autospectral density S(f) is the same at all spatial frequencies.  This property is characteristic of processes known collectively as white noise.  In such a process, the value of a variable at any given location in time or space is independent of (and uncorrelated with) the variable at all other locations. In contrast, when > 0, the lower the frequency (that is, the larger the spatial or temporal scale of variation), the larger the variance, and individual measurements are correlated to some extent. One particular case is worthy of note. When  = 1, the spectrum is known as pink noise, and has the property that it is "the natural result of a mixture of different phenomena acting impartially on different scales" (Halley 1996).  Although exactly how phenomena interact to yield this type of spectrum remains an area of active research, Halley (1996) makes a compelling argument that pink noise should be considered the null model for environmental and ecological variability. 

Pink noise forms an important boundary within 1/f-noise processes. For  1 and a maximal measurable frequency, fg, the integral

(F.1)

is improper; it does not converge. In other words, for  1, the variance increases without limit as the scale of measurement is increased. This characteristic poses a problem for any definitive measurement of scale. In contrast, if 0 < < 1, the integral in Eq. F.1 does converge. In this case, as the scale of measurement increases, the variance of the process approaches a defined value, and a definitive measure of scale is feasible, although it still may be sensitive to the grain and extent of measurement (see Appendix B).

 

LITERATURE CITED

Halley, J. M. 1996. Ecology, evolution and 1/f noise. Trends in Ecology and Evolution 11:33–37.



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