计算时间序列的功率谱密度函数的理论与程序实现外文翻译资料

DIFFERENCE METHOD OF SPECTRAL ANALYSIS

Let the magnetic field B(t) at a given time t be separated into mean and fluctuating parts, B(t) = B0 /iB(t), where B0 = (B), and angular brackets denote an ensemble average. In practice, magnetic field samples are available only at discrete times separated by some finite time At, which is 1 hour in the case of the omni-tape data. The two-point correlation function Rii(T) of the ith component of the field is

Assuming that the fluctuating field is stationary, the correlation function depends upon the lag T but not upon the time t at which the correlation is evaluated. Stationarity ensures, further, that the diagonal components of the correlation matrix are even, Rii(T) = Rii(-- T). By virtue of discreteness of the data, Rii can be calculated only for lags T = n/bull;T where n - 0, 1, 2, 3, ... , N. For our analysis we choose a maximum lag of 1 day, corresponding to N – 24 in a straightforward application of Blackman and Tukey [1958]spectral analysis the power spectrum Pii(f) is simply a discrete Fourier transform of Rii

where the frequency takes on discrete values given by fn =n/(2NAT) with n = 0, 1, 2, 3, ' , N. As discussed in the main text, we found that this straightforward technique yield sun satisfactory results. The problem is that the method requires an estimate of the mean field B0 in order that the fluctuations /bull;B can be inserted into (13). Unfortunately, zero-frequency quantities such as the mean field are rather ill-defined when significant variations exist at frequencies below the lowest frequency resolved by the analysis.

Another way of seeing this problem is to note that an estimate of power at a given frequency becomes increasingly accurate as more cycles of the variation are encompassed by the analysis. In estimating the mean, however, we can never have even a single cycle available because the period of the cycle is in principle infinite. In practice the mean can be estimated nonetheless if the analysis extends to frequencies so low that all important variations are resolved, but such is not the case here. Powerful solar drivers exist at periods of 27 days, 11years, 22 years, and perhaps even lower frequencies. To circumvent this problem, we define da new time series AB(t) to consist of the vector difference between successive data points in the time series representing a given sector population:

Note that taking the difference completely removes the mean field contribution and, moreover, converts linear trends to a constant term. Next, we formed the correlation function R/?)of the difference

and then took a discrete Fourier transform to obtain the power spectrum of the differences Pi?)(f). Making use of (15), it is a simple matter to show that the correlation of the differences is related to the ordinary correlation shown in (13) by

Taking the Fourier transform of this equation, we obtain a relationship between the original power spectrum and the power spectrum of the differences

his equation can now be used as a filter to correct the power spectrum for the use of differences. In effect the use of differences amounts to a 'pre-whitening' of the data in the
sense discussed by Blackman and Tukey [1958]. The filter (18) then 'post-darkens' the resulting spectrum. A second filter was applied to the spectrum to correct for the use of hourly averages as opposed to instantaneous samples spaced1 hour apart. One can view the process of taking an hourly average as performing a convolution of the data with a weighting function that has a value of unity over a 1-hourrangeanda value of zero everywhere else. Convolution in the time domain corresponds to multiplication in the frequency domain. Making use of this fact, one can derive the following frequency domain filter to correct for the use of hourly averages:

Where Piiis the corrected spectrum, P?) is the spectrum as calculated from the hourly average data, and h is the number of fine time scale data points going into the average of length AT .If the number of fine time scale points going into the averages is reasonably large, say h gt; 9, then one can make use of the approximation sin(rcfAT/h)bull; (rcfAT/h) (note that the maximum value off is the Nyquist frequency (2Abull;)-l),and h drops out of the filter. If we now identify Pi(i h) with the power spectrum P/?)calculated from hourly averaged increments, we can merge the filters (18) and (19) to obtain

We have tested the above analysis technique to demonstrate its applicability to the problem described in this paper. In doing so, we have deliberately chosen to simulate a data set with an underlying spectral form that is more difficult to resolve accurately than the spectral reported in this paper. A test data set was created having 8192 data points, as shown in the top panel of Figure 12. It was assumed that the temporal separation between successive points is 1 hour, in keeping with the temporal resolution of the data used in this analysis. To save computer time, we allowed the time series of the R and T components to be identical. This is of no consequence for our purposes, because we will compute only the energy spectrum (trace of the power spectrum matrix) of the test data.

The known power spectrum employed in the creation of the test data set possesses a -5/3 power law form from the fundamental frequency, (8192 hours) -1 or3.4x 10-8 Hz, to the400thspectral component (with frequency 1.36x 10-5 Hz).In addition thef-5/3 spectral form was continued for1 decade below the fundamental frequency (using randomly phased contributions at 0.1, 0.2, 0.3,..., 0.9 times the fundamental frequency) to produce a data set with a high degree of unresolved low-frequency power, as is evident from Figure 12. Finally, the spectrum possesses a -3 power law form at frequencies higher than the 400th spectral El

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使磁场B(t)在一个给定的时间t被分割成平均项和波动项两个部份，,其中尖角括号表示一个系综平均，所以表示磁感应强度系统的集平均。在实践中，磁场样品只能存在于被一些有限的时间分割后的离散的时间点，当在全带数据的情况下时为1小时。磁场的第i个场分量的二点相关函数的是

(3.1)

假定波动字段是固定的，相关函数取决于延迟，而不是该相关性被评估的时间t。更重要的是，平稳性保证了该相关矩阵的对角线分量为偶数，即。由于数据是离散的，所以只有在延迟时间项其中n=0,1,2,3hellip;.,N时，可以计算。在我们的分析中，我们选择1天的最大时间延迟项即N=24。

在布莱克曼-杜克频谱分析的直接应用中，功率谱仅仅是的离散傅立叶变换。

(3.2)

其中频率被赋值为由给定的离散值，其中n=0,1,2,3hellip;N。正如在正文中讨论的，我们发现，这种简单的技术得出一个不太令人满意的结果。问题是，该方法需要平均场的估计值，以使得波动项可以被代入到上式(3.1)中。然而，当存在显著差异的频率低于由分析得出的最低频率时，零频数量就像平均磁场一样是相当不明确的。

发现这个问题的另一种方法是要注意，每当分析过程涵盖越多个周期，一个在给定频率的功率估计变得越精确。但是，在估计平均值时，我们永远不可能用到甚至一个周期，因为周期时间段在原则上是无限的。在实践中，平均场可估计。然而如果分析延伸到频率很低的层次，所有重要的变化都解决了，但这里不是这样的情况。绕过这个问题中，我们定义一个新的时间序列，使其包含一个时间序列的上连续数据点之间的矢量差，而这个时间序列代表一个给定扇区的总集。

(3.3)

要注意的是，采用差分法会完全消除平均场贡献，此外，线性趋势也会转换为一个常数项。接下来，我们就创造一个差分相关函数

(3.4)

然后，将式(3.4)作离散傅里叶变换来得到功率谱的差分。利用(3.3)式，我们可以很简单的得到一个结论，那就是差的相关性与由(13)中所示的普通的相关性是相互联系的，它们的关系可以用以下的式子表示：

(3.6)

将这一方程作傅里叶变换，然后我们得到原始功率谱和差分功率谱之间的关系：

(3.7)

这个方程现在可以用作过滤器以校正差分功率谱。实际上使用的差异相当于一个由布莱克曼杜克所讨论的数据“预白化”。然后过滤器（3.7）式 会“后暗化”所产生的频谱。

第二滤波器应用于频谱主要是来校正每小时平均值的用途而不是时间间隔为1小时的瞬时样本的用途。当用加权函数来处理数据的卷积时，这个权重函数包含一个超过1小时的范围的集的值和其他任意地方的一个零值，我们可以观察到取每小时平均的过程。卷积在时域相当于频域中的乘法，利用这一事实，我们可以运用下列频域滤波器来校正每小时平均值的用途。

(3.8)

其中 被校正过的频谱，是从每小时平均数据计算出的频谱，并且h是当大规模的数据点的进入平均长度时的精细时间数字，如果大规模数据进入点的平均值时的精细时间数目是多少相当大的，比如说hgt;9,那么就可以利用近似 （注意，f的最大值为Nyquist频率 ,h从过滤器中漏出。如果我们现在用每小时平均增量算出来的功率谱来定义，我们可以合并滤波器(3.7)和(3.8)获得

(3.9)

图9 连续实验数据

这个测试数据集一共有8192个数据点，如图9所展示的，为了保证分析中使用的数据的时间分辨率，我们假定连续点之间的时间间隔为1小时。为了节省计算时间，我们允许R和T的时间系列中的元素可以是相同的。这对我们的运算目的没有影响，因为我们将只计算测试数据的能量谱（功率谱矩阵的迹线）。

在创建测试数据集时所采用的已知的功率谱具有基本频率的次幂的形式, 或 HZ到第400个频谱分量（频率 HZ）。另外，继续进行十年基波频率以下的 的光谱形式（在随机相位的贡献0.1，0.2，0.3，...，0.9倍基波频率），以产生一个数据具有高度未解决的低频功率的设置，就如图9所显示的。最后，在频率高于奈奎斯特频率功率的第400个频谱分量时，该频谱具有-3次幂形式。这种相对陡峭的功率谱被用来演示了这种技术来解决在事件中复杂的频谱形式时的能力，它们可以存在于在本文中所描述的分析中使用的数据集的。更浅的频谱不太受到未解决或正待解决的低频波动导致的泄露功率的影响。

图10 布莱克曼-杜克和差分法（预白化的布莱克曼-杜克方法）频谱分析

圆圈表示正功率谱估计，加号代表负功率谱的大小，实线表示准确的输入频谱。

图10显示了这个测试数据集的分析的结果。左边的表示进行产品分析时的标准平均延迟计算出的能量谱。不使用零来填充相关函数，相关函数也没有受到任何窗口函数的限制。在相关函数的计算中使用的最大时间延迟为24小时，或是时间序列长度的0.3％。未解决的低频功率的长度和急剧下降的频谱形式计算功率谱（由圆圈表示），显著变化被用于创建数据集中的平滑频谱形状（用实线表示）。而负功率是物理上可实现，它是布莱克曼-杜克分析的一个可能的结果。右边的图显示差分预白化布莱克曼-杜克方法分析的结果。不同的是使用通过(3.3)给出的预先白化滤波器和由(3.6)给出的交变暗滤波器，用于预白化分析的分析参数是相同的，与上述的标准分析中使用。即使是频谱突破HZ准确地得到解决。在所计算的频谱的估计中用于创建数据集的平滑功率谱表现的很明显，并且从分析可以得到的其中没有负功率。

图11 不连续的数据

图12：两种方法对比结果图

如图11，用该数据集重复前面的分析。标准布莱克曼-杜克分析负功率水平相对之前的分析增加了，正功率谱估计承受小，如果有的话，到真实基础谱形式的关系。预白化的分析再次与再现良好的精度频谱。

总结，我们获得IMF功率谱的方法包括四个步骤：

(1)构成 (3.3)中指定的时间系列。

(2)计算不同 的相关函数。如图(3.4)显示的（在实际运算中我们假设遍历性然后替代统计平均值通过统计值随时间t的变化）

(3) 对于离散傅立叶变换，按照(3.2)执行获得不同的功率谱。

(4) 通过在(3.8)的组合过滤器来对 进行相乘这么做是为了使用不同的数据和每小时的平均值，经过相乘我们得到光谱 ，这正是我们基础分析的形式。

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