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 11:3337.