动力系统中分形波动的普适逆幂律分布：应用于印度和USA地区降水年际变化的可预报性
Universal Inverse Power law distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter - annual Variability of Indian and USA Region Rainfall
摘要：
自然界中的动力学系统在所有尺度上都表现出自相似的分形时空涨落，表明了长程相关性，因此具有隐含独立性、固定均值和标准差假设的统计正态分布不能用于分形数据集的描述和量化。作者发展了一个一般系统理论，它预测如下：（i）分形波动表示一个潜在的涡流连续体，较大的涡流是封闭的小尺度波动的综合平均值。(ii)涡振幅的概率分布和分形波动的方差（涡振幅的平方）谱遵循普遍的玻尔兹曼逆幂律表示为黄金分割的函数。预测的分布是非常接近的统计正态分布的中度事件的两个标准偏差的平均值，但表现出肥胖的长尾，与危险的极端事件。印度和美国台站1900年至2008年期间可用的GHCN年总降雨量时间序列的连续周期图功率谱分析表明，功率谱和相应的概率分布遵循模型预测的通用逆幂律形式，表明观测到的降雨年际变化的基础是涡动连续结构。与全球变暖有关的大气能量输入将导致所有尺度的波动加剧，并可立即在高频率（短期）波动中看到，例如由于年度、准两年期和其他较短周期（年）降雨周期的过多/不足而造成的毁灭性洪水/干旱。
英文摘要：
Dynamical systems in nature exhibit self-similar fractal space-time fluctuations on all scales indicating long-range correlations and therefore the statistical normal distribution with implicit assumption of independence, fixed mean and standard deviation cannot be used for description and quantification of fractal data sets. The author has developed a general systems theory which predicts the following (i) The fractal fluctuations signify an underlying eddy continuum, the larger eddies being the integrated mean of enclosed smaller-scale fluctuations. (ii) The probability distribution of eddy amplitudes and the variance (square of eddy amplitude) spectrum of fractal fluctuations follow the universal Boltzmann inverse power law expressed as a function of the golden mean. The predicted distribution is very close to statistical normal distribution for moderate events within two standard deviations from the mean but exhibits a fat long tail that are associated with hazardous extreme events. Continuous periodogram power spectral analyses of available GHCN annual total rainfall time series for the period 1900 to 2008 for Indian and USA stations show that the power spectra and the corresponding probability distributions follow model predicted universal inverse power law form signifying an eddy continuum structure underlying the observed inter-annual variability of rainfall. Global warming related atmospheric energy input will result in intensification of fluctuations of all scales and can be seen immediately in high frequency (short-term) fluctuations such as devastating floods/droughts resulting from excess/deficit annual, quasi-biennial and other shorter period (years) rainfall cycles.
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