英语翻译a.Fourier red noise spectrumMany geophysical time series

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.

一.傅立叶红色的噪音光谱
许多地球物理学的时间系列能被做模型当做
白色噪音或红色的噪音.一个简单的模型为红色
噪音是单变数落后-1 autoregressive[AR(1),或
Markov] 程序:
（公式不用翻译）
哪里一是假装的落后-1 自相关,x 0=0,
而且 zn 从高斯白色噪音被拿.下列各项
Gilman 等人 (1963) ,不连续的傅立叶使光谱有力量
(15),在使常态化之后,是
公式
哪里 k=0 … N/2 是频率索引.因此,被
选择一个适当的落后-1 自相关,一能
使用 (16) 做模型一种红色-噪音光谱.注意那一 =0
在 (16) 给一种白色噪音光谱.
傅立叶为 Ni 使光谱有力量?o 3 SST 是
在图 3 被瘦线显示.光谱是
由 N 是点的数字的 N/2 年代 2 使常态化,
而且 s 2 是时间系列的不一致.使用这
常态化,白色噪音会有期待
在所有的频率 1 价值.红色-噪音背景
光谱为一 =0.72 被比较低人显示猛掷
在图 3 弯.这红色者-噪音被估计了从 (公式）哪里一 1 和一 2 是落后-1 和落后-2
Ni 的自相关?o 3 SST.一能见到那
在 2 和 8 yr 之间的 ENSO 峰巅的宽广组,涌出
在背景光谱上面.
b.小浪红色的噪音光谱
小浪变换在 (4) 是一系列的通带
时间系列的过滤器.如果这次哪一系列能是
做模型当做一个落后-1 AR 程序,然后它似乎合理
那当地的小浪力量光谱,定义
当做一个垂直的薄切片,经过图 1 b ,有被.(16) 到
测试这一个假设,100000个高斯白色噪音
时间系列和 100000 AR(1) 时间系列被构造,
连同他们的对应小浪力量一起
频谱.这些白色者的例子- 和红色-噪音小浪
频谱在图 4 被显示.当地的小浪频谱
藉由在时间拿垂直的薄切片被构造
n=256.在无花果树的比较低的平滑曲线.5 一和 5 b
表示理论上的频谱从.(16) 点表示
来自蒙地卡罗模拟的结果.一般说来 ,
当地的小浪力量光谱是同一的
对被给的傅立叶力量光谱被.(16)
因此,在图 3 的较低的被猛掷的曲线也
符合红色-噪音地方小浪光谱.
在图 1 b 的一个任意的垂直薄切片会被期望
给一种光谱被.(16) 当做意志被显示在
第 5 节一,所有的当地小浪频谱的平均
容易接近傅立叶光谱(使)光滑
时间系列.