The historically adjusted range and the historically rescaled adjusted range

It was remarked by Hurst in 1951 that the adjusted range gives the size of the smallest reservoir capable of providing a constant discharge equal to the mean inflow. Since that time this range and its rescaled modification, the Hurst range, have been widely discussed, not however primarily with a view to applying them to reservoir design problems, but rather on account of their possible relevance to the simulation of geophysical time series.Acknowledging the well-known conceptual weaknesses of adjusted ranges and the theoretical difficulties that inhibit their direct utilisation in the design and operation of real reservoirs, the authors argue that the interest displayed on ranges during the past few decades justifies the effort of eliminating one in particular of these weakness, namely their non-implementability as operating policies, a consequence of the fact that they can only be retrospectively evaluated. The paper proposes modifications in which the unknowable mean and standard deviation of future samples are replaced by the known mean and sample standard deviation of historical data, leading to the historically adjusted range and the historically rescaled and adjusted range. The latter is produced as an implementable approximation to Hurst's (1951) solution to the optimal reservoir problem.     

Rescaled range (R/S) analysis on seismic activity parameters

IntroductionSeismicity analysis is an important branch of seismology, which focuses statistically on the study of time, space and magnitude distribution of a group of earthquakes (FU, 1997). Earthquakes as instability phenomena of the lithosphere, however, involve very complicated mechanics. The seismic activity level is not steady, but fluctuating time by time, which shows the alternative change of periods of quiescence and activity (FU, 1986; Mogi, 1984; ZHANG, 1987). Because of such com…     

Factors influencing long range dependence in streamflow of European rivers

Investigating long range dependence of river flows, especially in connection with various climate and storage related factors, is important in order to improve stochastic models for long range dependence and in order to understand deterministic and stochastic variability in long‐term behaviour of streamflow. Long range dependence expressed by the Hurst coefficient H is estimated for 39 (deseasonalized) mean daily runoff time series in Europe of at least 59 years using five estimators (rescaled range, regression on periodogram, Whittle, aggregated variances, and least squares based on variance). All methods yield estimates of H > 0.5 for all data sets. The results from the different estimators are significantly positively correlated for all pairs of methods indicating consistency of the methods used. Correlations between H and various catchment attributes are also analysed. H is strongly positively correlated with catchment area. Apparently, increasing storage with catchment area translates into increasing long range dependence. H is also positively correlated with mean discharge and air temperature and negatively correlated with the mean specific discharge and the seasonality index (maximum Pardé coefficient). No significant correlation is found between the Hurst coefficient and the length of the analyzed time series. The correlations are interpreted in terms of snow processes and catchment wetness. Copyright © 2012 John Wiley & Sons, Ltd.     

Rescaled range ( R/S ) analysis on seismic activity parameters

The rescaled range (R/S) analysis, proposed by Hurst, is a new statistical method. Being different from traditional statistical method, R/S analysis can provide the information of maximum fluctuation (range) of statistical parameters. At present paper, several modern instrumental earthquake catalogues in different spatial scale, temporal scale, and with different seismic activity background are studied, and R/S method is used to analyze the variation of range of seismic parameters such as earthquake frequency, and earthquake time interval. For different seismic parameters, the ratio of range to standard deviation — R/S is a power law function of the length of time, and the exponent H of power law is always greater than 0.5. As we know, H=0.5 is the characteristics of all ideal random processes. Our results indicate that earthquake series is not an ideal Poisson process, on the contrary, the earthquake as a phenomenon bears dual characteristics of randomicity and regularity, and the greater H departs from 0.5, the more regularity the time series will show, and vice versa. With time scale changing, one can give the conservative estimate of the fluctuation, which might occur in a relatively long time scale, only by using the limited and known time records. Foundation item: State Natural Science Foundation of China (40074023).     

Influence of statistical time range on estimating seismicity parameters

Introduction At present, the current probabilistic method of seismic hazard analysis in the world is gener-ally adopted in seismic safety evaluation (Cornell, 1968). It supposed that earthquake occurrence conforms to the segmented Poisson distribution model in time and space and the proportional rela-tion among earthquakes of various magnitudes obeys the cut-exponential distribution law in a sta-tistical area in compiling Seismic Intensity Zoning Map in China (1990) owing to the temporal- sp…     

Simultaneous estimation of the parameters of the Hurst�CKolmogorov stochastic process

Various methods for estimating the self-similarity parameter (Hurst parameter, H) of a Hurst–Kolmogorov stochastic process (HKp) from a time series are available. Most of them rely on some asymptotic properties of processes with Hurst–Kolmogorov behaviour and only estimate the self-similarity parameter. Here we show that the estimation of the Hurst parameter affects the estimation of the standard deviation, a fact that was not given appropriate attention in the literature. We propose the least squares based on variance estimator, and we investigate numerically its performance, which we compare to the least squares based on standard deviation estimator, as well as the maximum likelihood estimator after appropriate streamlining of the latter. These three estimators rely on the structure of the HKp and estimate simultaneously its Hurst parameter and standard deviation. In addition, we test the performance of the three methods for a range of sample sizes and H values, through a simulation study and we compare it with other estimators of the literature.     

Conditioned ranges for independent symmetric three valued inflows

The paper considers, as inputs to a storage system, independent random variablesX ₁,X ₂,... which have a symmetric three-valued distribution. Expressions are obtained for the conditional expected values of (i) the adjusted range ofX ₁,...X _n, conditioned on the eventX ₁+...+X _n=0; (ii) the Hurst range ofX ₁,...,X _n , conditioned on essentially the same event. Numerical comparisons of these two conditioned ranges show that they are close in value unless the inputs have high kurtosis.     

The hurst effect and its implications for extrapolating process rates

In long-term records of climatic or hydrological events, persistent periods of high and low values are measured by the ‘rescaled range’, which is defined for a sequence of n consecutive values. As n increases the rescaled range for historic sequences increases as n^H for 0·5 < H < = 1. This is the Hurst effect. Sequences of independent random events give H = 0·5, while sequences in which the values are successive positions of a random walk (i.e. the sum of independent events) give H = 1. Intermediate values of H may be obtained from fractal sequences, from sequences with a steadily changing mean, or through a storage model. In all but the fractal case H exceeds 0·5 for a time related to the duration of the changing mean or the response time of the store. An exponential storage model has been explored in some detail. Storage models have the advantage of offering a plausible physical mechanism for persistence. Analysis of simulated sequences shows that the Hurst effect is evident for about 30 x the response time of the store, so that relatively modest store sizes are enough to explain observed Hurst effects over available periods of record. Comparison with empirical values shows that high H values seem to be associated with large physical stores. Whether the Hurst effect is due to storage effects or climatic or other change imposed from outside, the effect on longterm process rates is strong. Short-term measurements of process rates show compressed variances and simple extrapolation is likely to produce large errors. If it is assumed that long-term rates are strongly influenced by climatic variables, then measurements may be used to obtain parameters of a climatic model for differences in process rates. Errors in extrapolation will be least where the model has a sound physical basis. A first implication for magnitude and frequency distributions is that the sequence of events may be as important as their distribution over time in determining process impact. A second conclusion is that the geomorphic history of an area is inevitably dominated by a sequence of larger and larger events receding into the past, and that the Hurst exponent over geologic time-spans measures the extent of that domination.     

地震活动性参数的变尺度（Rem>/Sem>）分析

将改变时间标度的方法应用于地震活动性分析中,选择具有不同空间尺度范围、时间范围和地震活动背景的现代仪器地震目录作为研究对象,分析了几个地震活动性参数——地震频度和地震发生时间间隔与时间标度的关系．这些地震活动性参数的极差和均方差之比与时间标度呈现出幂指数变化方式,且幂指数都大于0.5,表明地震的发生并非无记忆的泊松过程,而是具有随机和规律的双重特性．犎指数偏离0.5的程度可以衡量随机与规律成分所占比重,偏离越大,该序列的规律性成分越多,反之则越少．通过时间标度变换,可以较短时间的观测为基础,对变量未来的发生情况做出保守的估计．     

用多通道AD分级采集扩展地震数据采集器的动态范围

地震数据采集是地震信号数字化必不可少的环节,动态范围是其一个重要的性能指标.实际地震信号的动态范围在160 dB以上,而目前普遍使用的24位地震数据采集器动态范围相对较小且在50 Hz采样率时最大只达到135 dB,致使24位地震数据采集器在实际使用中对小信号分辨率不够,不能有效提取地震信息;在大地震时又容易使数据采集器出现饱和限幅失真的现象而失去地震监测记录功能.本文针对在地震监测和地震研究中需要具有高分辨率和高动态范围的地震数据采集器这个亟待解决的问题,提出一种采用多通道AD转换器并行分级采集的方法,讨论了通道间失配及其标定.对研制实验样机的测试表明,其动态范围在50 Hz采样时可以达到157 dB以上,线性度优于0.005%.     

Studies of short and long memory in mining-induced seismic processes

Memory of a stochastic process implies its predictability, understood as a possibility to gain information on the future above the random guess level. Here we search for memory in the mining-induced seismic process (MIS), that is, a process induced or triggered by mining operations. Long memory is investigated by means of the Hurst rescaled range analysis, and the autocorrelation function estimate is used to test for short memory. Both methods are complemented with result uncertainty analyses based on different resampling techniques. The analyzed data comprise event series from Rudna copper mine in Poland. The studies show that the interevent time and interevent distance processes have both long and short memory. MIS occurrences and locations are internally interrelated. Internal relations among the sizes of MIS events are apparently weaker than those of other two studied parameterizations and are limited to long term interactions.     

Evaluation of rapid reservoir storage-yield procedures

Relatively rapid and reliable procedures are required to determine reservoir storage-yield relationships during preliminary studies or reconnaissance surveys associated with regional assessments of surface water resources. Several techniques are available, but all are limited in their application. The following eight procedures were evaluated: Hurst (empirical), Alexander (gamma), Gould (synthetic), Dincer, Gould (gamma), Hardison (carryover), Guglij and McMahon (empirical). Each procedure was evaluated by application to 33 Australian and 12 Malaysian streams. The storage estimates determined from the rapid procedures were compared with those from behaviour (or simulation) analysis and Gould's transition matrix method. Limitations in theoretical models result from the basic assumptions upon which they were developed. For empirical models, difficulties arise through the non-representativeness of the basic data. Some procedures are restricted in the range of conditions within which they are applicable. The results indicate that Gould's (gamma) and McMahon's (empirical) procedures provide satisfactory storage estimates for preliminary assessments.     

Climatic Variability Over Time Scales Spanning Nine Orders of Magnitude: Connecting Milankovitch Cycles with Hurst–Kolmogorov Dynamics

We overview studies of the natural variability of past climate, as seen from available proxy information, and its attribution to deterministic or stochastic controls. Furthermore, we characterize this variability over the widest possible range of scales that the available information allows, and we try to connect the deterministic Milankovitch cycles with the Hurst–Kolmogorov (HK) stochastic dynamics. To this aim, we analyse two instrumental series of global temperature and eight proxy series with varying lengths from 2 thousand to 500 million years. In our analysis, we use a simple tool, the climacogram, which is the logarithmic plot of standard deviation versus time scale, and its slope can be used to identify the presence of HK dynamics. By superimposing the climacograms of the different series, we obtain an impressive overview of the variability for time scales spanning almost nine orders of magnitude—from 1 month to 50 million years. An overall climacogram slope of ?0.08 supports the presence of HK dynamics with Hurst coefficient of at least 0.92. The orbital forcing (Milankovitch cycles) is also evident in the combined climacogram at time scales between 10 and 100 thousand years. While orbital forcing favours predictability at the scales it acts, the overview of climate variability at all scales suggests a big picture of irregular change and uncertainty of Earth’s climate.     

远震与极远震部分震相的幅值-震级-震中距-深度建模与应用研究

不同震相具有不同的振动特性和传播规律,对应各自不同的幅值-震中距-深度衰减规律,本文基于禁核试核查国际数据中心公报,统计P,PcP,PKP,PKPbc及PKPab远震和极远震震相样本随震中距的分布,运用基于残差统计的迭代方法回归上述震相的幅值-震级-震中距-深度模型,通过震级残差标准差与均值的统计进行模型评估,结果显示:P,PcP,PKP,PKPbc及PKPab震相的震级残差标准差范围为0.30—0.36,满足一般震相相容性判断的需求;除PKPbc的残差均值大于0.03外,其余震相残差均值均为0.01左右,模型的系统偏差极小,甚至可以忽略不计。最后针对模型进行了幅值预测比对、震级相容性检测、极远震体波震级计算等三个不同场景的应用研究,验证模型可用于日常地震监测。      

Multifractal measures of synchronization of microseismic oscillations in a minute range of periods

The paper considers synchronous continuous records of microseismic background obtained within a month before the Kronotskii (Kamchatka) December 5, 1997, earthquake (M = 7.8) at six IRIS broadband stations that are located in a large region extending from central European Russia (the town of Obninsk) to the Far East (Kamchatka and Sakhalin). By averaging and downsampling, initial records were discretized at an interval of 30 s and the microseismic background was examined in the range of periods from 1 min to 2.4 h, after scale-dependent trends due to the effects of tides and temperature variations had been removed. Microseismic fluctuations were analyzed with the help of estimates of the evolution of their multifractal singularity spectra in a moving time window 12 h wide. As the criterion characterizing the background properties in a current time window, we took the values of the generalized Hurst exponent α* realizing the maximum of the singularity spectrum. Hidden synchronization effects of a microseismic field preceding a seismic event are identified by estimating the evolution of the spectral measure of coherent behavior of α* variations in a moving time window 5 days long for various combinations of jointly analyzed stations.     

Integrated evaluation of urban groundwater hydrogeochemistry in context of fractal behaviour of groundwater level fluctuations

The present study was carried out as part of a complex survey of urban groundwater quality and quantity in Szeged, southeast Hungary. The concentrations of 12 inorganic contaminants in 28 shallow groundwater monitoring wells were determined over a 2-year period (2010–2012). The evaluation of concentrations indicates remarkable contamination all over the city. Discriminant analysis (DA) was used to evaluate the spatial changes of groundwater quality. The groundwater levels were measured over a 14-year period (2000–2013). The fractal properties of water level fluctuations time series characterizing the groundwater system and Tisza River were investigated using rescaled range (R/S) analysis. The resulting Hurst exponents clearly showed the persistency and thus long memory effects of both the groundwater and the river flow. Comparison of the results of DA with the results of R/S analysis thus implies that the geological conditions and the changing groundwater quantities are not related to groundwater quality.

EDITOR A. Castellarin

ASSOCIATE EDITOR A. Fiori     

Nonparametric streamflow simulation by wavelet or Fourier analysis

Abstract

Wavelet or Fourier analysis is proposed as an alternative nonparametric method to simulate streamflows. An observed series is decomposed into its components at various resolutions and then recombined randomly to generate synthetic series. The mean and standard deviation are perfectly reproduced and coefficient of skewness tends to zero as the number of simulations increases. Normalizing transforms can be used for skewed series. Autocorrelation coefficients and the dependence structure are better preserved when Fourier analysis is used, but the mean and variance remain constant when the simulated and observed series have the same length. Monthly as well as annual flows can be simulated by this technique as illustrated on some examples. Wavelet analysis should be preferred as it generates flow series that exhibit a wider range of required reservoir capacities.     

阻尼修正条件均值谱及其在时程分析地震波选择中的应用

目前时程分析选波常采用目标谱法,即选择反应谱与目标谱有较好匹配的地震波,条件均值谱(CMS)已成为目前国内外广受关注及认可的目标谱。通常的CMS是基于5%阻尼比建立的,当结构阻尼比并非5%时,则需要对CMS进行阻尼修正。本文旨在建立1种CMS阻尼修正方法和计算公式,以便于不同阻尼比结构抗震时程分析的选波工作。以美国SAC Steel Project计划提出的针对洛杉矶地区设计的9层抗弯钢框架为场地和结构实例,建立了3种地震危险性水平(50年超越概率50%、10%和2%)下2%阻尼比的CMS目标谱,并进行了时程分析地震波选择及结构反应的计算。研究表明:提出的CMS的阻尼修正方法可不受衰减关系影响,适用的阻尼比范围(0.5%~30%)及周期范围(0.01s^10s)均较广泛,可用于各种类型的结构的抗震分析。     

Review of Gould–Dincer reservoir storage–yield–reliability estimates

The Gould–Dincer suite of techniques (normal, log-normal and Gamma), which is used to estimate the reservoir capacity–yield–reliability (S–Y–R) relationship, is the only known available procedure in the form of a simple formula, based on annual streamflow statistics, that allows one to compute the S–Y–R relationship for a single storage capacity across the range of annual streamflow characteristics observed globally. Several other techniques are available but they are inadequate because of the restricted range of flows on which they were developed or because they are based on the Sequent Peak Algorithm or are not suitable to compute steady-state reliability values. This paper examines the theoretical basis of the Gould–Dincer approach and applies the three models to annual streamflow data for 729 rivers distributed world-wide. The reservoir capacities estimated by the models are compared with equivalent estimates based on the Extended Deficit Analysis, Behaviour analysis and the Sequent Peak Algorithm. The results suggest that, in the context of preliminary water resources planning, the Gould–Dincer Gamma model provides reliable estimates of the mean first passage time from a full to empty condition for single reservoirs. Furthermore, the storage estimates are equivalent to deficits computed using the Extended Deficit Analysis for values of drift between 0.4 and 1.0 and the values are consistent with those computed using a Behaviour simulation or a Sequent Peak Algorithm. Finally, a sensitivity analysis of the effect on storage of the four main streamflow statistics confirms that the influential ones are mean and standard deviation, while effects of skew and serial correlation are orders of magnitude lower. This finding suggests that the simple reduced form of the Gould–Dincer equation may profitably be used for regional studies of reservoir reliability subject to climate change scenarios based on regional statistics, without having to perform calculations based on time series, which may not be easily obtained.     

Seasonality of low flows and dominant processes in the Rhine River

Low flow forecasting is crucial for sustainable cooling water supply and planning of river navigation in the Rhine River. The first step in reliable low flow forecasting is to understand the characteristics of low flow. In this study, several methods are applied to understand the low flow characteristics of Rhine River basin. In 108 catchments of the Rhine River, winter and summer low flow regions are determined with the seasonality ratio (SR) index. To understand whether different numbers of processes are acting in generating different low flow regimes in seven major sub-basins (namely, East Alpine, West Alpine, Middle Rhine, Neckar, Main, Mosel and Lower Rhine) aggregated from the 108 catchments, the dominant variable concept is adopted from chaos theory. The number of dominant processes within the seven major sub-basins is determined with the correlation dimension analysis. Results of the correlation dimension analysis show that the minimum and maximum required number of variables to represent the low flow dynamics of the seven major sub-basins, except the Middle Rhine and Mosel, is 4 and 9, respectively. For the Mosel and Middle Rhine, the required minimum number of variables is 2 and 6, and the maximum number of variables is 5 and 13, respectively. These results show that the low flow processes of the major sub-basins of the Rhine could be considered as non-stochastic or chaotic processes. To confirm this conclusion, the rescaled range analysis is applied to verify persistency (i.e. non-randomness) in the processes. The estimated rescaled range statistics (i.e. Hurst exponents) are all above 0.5, indicating that persistent long-term memory characteristics exist in the runoff processes. Finally, the mean values of SR indices are compared with the nonlinear analyses results to find significant relationships. The results show that the minimum and maximum numbers of required variables (i.e. processes) to model the dynamic characteristics for five out of the seven major sub-basins are the same, but the observed low flow regimes are different (winter low flow regime and summer low flow regime). These results support the conclusion that a few interrelated nonlinear variables could yield completely different behaviour (i.e. dominant low flow regime).