英语翻译
a.Fourier red noise spectrum
Many geophysical time series can be modeled as
either white noise or red noise.A simple model for red
noise is the univariate lag-1 autoregressive [AR(1),or
Markov] process:
（公式不用翻译）
where a is the assumed lag-1 autocorrelation,x0 = 0,
and zn is taken from Gaussian white noise.Following
Gilman et al.(1963),the discrete Fourier power spectrum
of (15),after normalizing,is
公式
where k = 0 … N/2 is the frequency index.Thus,by
choosing an appropriate lag-1 autocorrelation,one can
use (16) to model a red-noise spectrum.Note that a = 0
in (16) gives a white-noise spectrum.
The Fourier power spectrum for the Niño3 SST is
shown by the thin line in Fig.3.The spectrum has been
normalized by N/2s2,where N is the number of points,
and s2 is the variance of the time series.Using this
normalization,white noise would have an expectation
value of 1 at all frequencies.The red-noise background
spectrum for a = 0.72 is shown by the lower dashed
curve in Fig.3.This red-noise was estimated from (公式） where a1 and a2 are the lag-1 and lag-2
autocorrelations of the Niño3 SST.One can see the
broad set of ENSO peaks between 2 and 8 yr,well
above the background spectrum.
b.Wavelet red noise spectrum
The wavelet transform in (4) is a series of bandpass
filters of the time series.If this time series can be
modeled as a lag-1 AR process,then it seems reasonable
that the local wavelet power spectrum,defined
as a vertical slice through Fig.1b,is given by (16).To
test this hypothesis,100 000 Gaussian white-noise
time series and 100 000 AR(1) time series were constructed,
along with their corresponding wavelet power
spectra.Examples of these white- and red-noise wavelet
spectra are shown in Fig.4.The local wavelet spectra
were constructed by taking vertical slices at time
n = 256.The lower smooth curves in Figs.5a and 5b
show the theoretical spectra from (16).The dots show
the results from the Monte Carlo simulation.On average,
the local wavelet power spectrum is identical
to the Fourier power spectrum given by (16).
Therefore,the lower dashed curve in Fig.3 also
corresponds to the red-noise local wavelet spectrum.
A random vertical slice in Fig.1b would be expected
to have a spectrum given by (16).As will be shown in
section 5a,the average of all the local wavelet spectra
tends to approach the (smoothed) Fourier spectrum of
the time series.