Geology Reference
In-Depth Information
contributions of power at all frequencies; the white noise spectrum is a simple
horizontal line. However many processes that are noise-like exhibit “memory,”
in that recent behavior influences the current behavior of the process. This
memory suppresses high frequencies and amplifies low frequencies, which
“reddens” the spectrum.
Autoregressive noise is modeled as a first-order linear Markov process
(Gilman et al. 1963):
r(n)
=−+
ρ
r(n
1)
e(n
)
where r is the “autocorrelated” random process, ρ (0 ≤ ρ < 1) relates the
previous (n - 1) to the current (n) observation of r, and e is the Gaussian
white noise. This is also known as a first-order autoregressive AR(1) model
(Jenkins & Watts 1968). Direct inversion of the above equation reveals that
ρ is the lag - 1 autocorrelation coefficient of r (Priestley 1981). The power
spectrum of r is:
2
ρ
ρπ ρ
1
−
R(f R
)
=
0
1
−
cos(
f/f
Nyquist
)
+
2
where R
0
= σ
r
2
/(1 - ρ
2
) is the average power of the spectrum, with σ
r
2
the
variance of r. R(f ) is shown in Figure 4.18 for different ρ. When ρ = 0, the
model reduces to Gaussian white noise with power level R
0
; as ρ increases,
power is progressively shifted into the low frequencies at the expense of
power in the high frequencies. A realization of AR(1) noise with ρ = 0.9 is
shown in Figure 4.17b, where it is displayed as log
10
(power) vs. linear fre-
quency combined with the test time series with the two frequencies (signal).
Power law noise, or “1/f
a
” noise, is characterized by reddened spectra with
specific slopes in log
10
(power) versus log
10
(frequency) space. White noise is
characterized by a = 0. Flicker noise is characterized by a = 1 and is observed
widely in both natural and artificial phenomena (Press 1978; Milotti 2002) and
is a “balance between randomness and correlation at all timescales” (Voss
1979). While it is the most commonly observed noise, it is the least understood.
Brownian noise is characterized by a = 2, also known as “random walk” noise
or a Wiener stochastic process, with characteristics of the well-known
Brownian motion (Gillispie 1996). The spectra of these two power law
models (Figure 4.18) show that this noise is distinctively different from the
AR(1) noise models, with comparatively greater inflation of power in the lowest
frequencies. Recently, autoregressive model approximations have been
developed for 1/f
a
noise, which may lead to new null models.
Hypothesis testing of data spectra has traditionally relied on the autore-
gressive noise for the null model, with its readily adaptable parameterization,
i.e., use of the lag - 1 autocorrelation coefficient of the data time series for
computing R(f ). An example of a hypothesis test of the “signal +
red noise
”
time series using AR(1) as the null model is shown in Figure 4.19. Only two
spectral bands exceed the 99% CL, at the signal frequencies, 0.050 and 0.055.
