Evidence of chaos in the rainfall-runoff process

Abstract

The transformation of rainfall into runoff is one of the most important processes in hydrology. In the past few decades, a wide variety of automated or computer-based approaches have been applied to model this process. However, many such approaches have an important limitation in that they treat the rainfall-runoff process as a realization of only a few parameters of linear relationships rather than the process as a whole. What is required, therefore, is an approach that can capture not only the overall appearance but also the intricate details of the nonlinear behaviour of the process. The purpose of this study is to investigate the possibility of understanding the dynamics of the rainfall-runoff process from a new perspective, as a chaotic process. The possible existence of chaotic behaviour in the rainfall-runoff process is studied by investigating the rainfall and runoff time series: (a) separately; and (b) jointly (using the runoff coefficient). Monthly rainfall and runoff observed over a period of 131 years (January 1807-December 1937) at the Göta River basin in the south of Sweden are analysed. The correlation dimension method is employed to identify the presence of chaos. The correlation dimensions obtained for the rainfall and runoff time series are 6.4 and 5.5, respectively. The finite dimensions obtained for the rainfall and runoff time series indicate the possible existence of chaos in these processes, implying that the joint rainfall-runoff process might also exhibit chaotic behaviour. The correlation dimension of about 7.8 obtained for the runoff coefficient also indicates the possible presence of chaos and supports the above results.     

Signatures of low-dimensional chaos in hourly water level measurements at coastal site of Mariupol,Ukraine

Variations of water levels in ports and estuaries are important for ship guidance and navigation and are influenced by a variety of factors. The hourly data that was collected from the coastal site at the Port of Mariupol, Ukraine during January–December 2005 were analysed with an objective to reveal features of chaotic behaviour. The concepts and methods of chaos theory (average mutual information, correlation dimension, false nearest neighbours, Lyapunov exponents) were applied. The manifestation of low-dimensional chaos was found in the time series. The possibility of nonlinear prediction was concluded.     

Is a chaotic multi‐fractal approach for rainfall possible?

An Erratum has been published for this article in Hydrological Processes 15 (12) 2001, 2381–2382. Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono‐fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi‐fractal approaches. The assumption behind multi‐fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi‐fractals. However, it has been observed recently that multi‐fractal approaches might provide positive evidence of a multi‐fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi‐fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi‐fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi‐fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi‐fractal approach for rainfall characterization. Copyright © 2001 John Wiley & Sons, Ltd.     

Searching for strange attractor in wastewater flow

 Chaos is a complex and irregular world in contrast with simple and regular natures of linear systems. Scientists and engineers have invoked low-dimensional chaos for understanding the nature of real systems. In this study, the complex behavior of a daily wastewater flow and evidence of deterministic nonlinear dynamics are investigated. The analysis involves both a metric approach of the correlation dimension and a topological technique called the close returns plot. The estimation procedure of delay time and delay time window is reviewed using a new technique called the C–C method for the state space reconstruction. And both parameters are used for estimating the correlation dimension. As a result, the daily wastewater flow shows no evidence of chaotic dynamics, which implies that stochastic models rather than deterministic chaos may be more appropriate for representing an investigated series.     

Measuring nonlinear dependence in hydrologic time series

It has been a common practice to employ the correlation dimension method to investigate the presence of nonlinearity and chaos in hydrologic processes. Although the method is generally reliable, potential limitations that exist in its applications to hydrologic data cannot be dismissed altogether. As for these limitations, two issues have dominated the discussions thus far: small data size and presence of noise. Another issue that is equally important, but less discussed in the literature, is the selection of delay time (τ _d ) for reconstruction of the phase-space, which is an essential first step in the correlation dimension method, or any other chaos identification and prediction method for that matter. It has also been increasingly recognized that fixing the delay time window (τ _w ) rather than just the delay time itself could be more appropriate, since the delay time window is the one that is of actual interest at the end to represent the dynamics. To this effect, Kim et al. (1998a) [Phys Rev E 58(5):5676–5682] developed a procedure for fixing the delay time window and demonstrated its effectiveness on three artificial chaotic series, and followed it up with the development of the C–C method to estimate both the delay time and the delay time window. The purpose of the present study is to test this procedure on real hydrologic time series and, hence, to assess their nonlinear deterministic characteristics. Three hydrologic time series are studied: (1) daily streamflow series from St. Johns near Cocoa, FL, USA; (2) biweekly volume time series from the Great Salt Lake, UT, USA; and (3) daily rainfall series from Seoul, South Korea. The results are also compared with those obtained using the conventional autocorrelation function (ACF) method.     

Nonlinear analysis of rainfall dynamics in California's Sacramento Valley

This study investigates the dynamic nature of rainfall observed at the Sustainable Agriculture Farming Systems (SAFS) site in California's Sacramento Valley, which was established to study the benefits of winter cover cropping in Mediterranean irrigated‐arid systems. Rainfall data of four different temporal scales (i.e. daily, weekly, biweekly, and monthly) are analysed to determine the dynamic nature of precipitation in time. In an arid climate with seasonal precipitation this has large implications for land and water management, both in the short term and in the long term. A nonlinear dynamic technique (correlation dimension method) that uses the phase‐space reconstruction and dimension concepts is employed. Bearing in mind the possible effects of the presence of zeros (i.e. no rain) on the outcomes of this analysis, an attempt is also made to compare the dynamic nature of all‐year rainfall and winter rainfall. Analysis of 15 years of data suggests that rainfall dynamics at this site are dominated by a large number of variables, regardless of the scales and seasons studied. The dimension results also suggest that: (1) rainfall dynamics at coarser resolutions are more irregular than that at finer resolutions; (2) winter rainfall has a higher variability than all‐year rainfall. These results are indeed useful to gain information about the complexity of the rainfall process at this site with respect to (temporal) scales and seasons and, hence, the appropriate model (high‐dimensional) type. However, in view of the potential effects of certain rainfall data characteristics (e.g. zeros, measurement errors, scale effects) on the correlation dimension analysis, the discussion also emphasizes the need for further verification, and possibly confirmation, of these results. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonlinear determinism in river flow: prediction as a possible indicator

Whether or not river flow exhibits nonlinear determinism remains an unresolved question. While studies on the use of nonlinear deterministic methods for modeling and prediction of river flow series are on the rise and the outcomes are encouraging, suspicions and criticisms of such studies continue to exist as well. An important reason for this situation is that the correlation dimension method, used as a nonlinear determinism identification tool in most of those studies, may possess certain limitations when applied to real river flow series, which are always finite and often short and also contaminated with noise (e.g. measurement error). In view of this, the present study addresses the issue of nonlinear determinism in river flow series using prediction as a possible indicator. This is done by (1) reviewing studies that have employed nonlinear deterministic methods (coupling phase‐space reconstruction and local approximation techniques) for river flow predictions and (2) identifying nonlinear determinism (or linear stochasticity) based on the level of prediction accuracy in general, and on the prediction accuracy against the phase‐space reconstruction parameters in particular (termed as the ‘inverse approach’). The results not only provide possible indications to the presence of nonlinear determinism in the river flow series studied, but also support, both qualitatively and quantitatively, the low correlation dimensions reported for such. Therefore, nonlinear deterministic methods are a viable complement to linear stochastic ones for studying river flow dynamics, if sufficient caution is exercised in their applications and in interpreting the outcomes. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamics of hourly sea level at Hillarys Boat Harbour,Western Australia: a chaos theory perspective

Water level forecasting using recorded time series can provide a local modelling capability to facilitate local proactive management practices. To this end, hourly sea water level time series are investigated. The records collected at the Hillarys Boat Harbour, Western Australia, are investigated over the period of 2000 and 2002. Two modelling techniques are employed: low-dimensional dynamic model, known as the deterministic chaos theory, and genetic programming, GP. The phase space, which describes the evolution of the behaviour of a nonlinear system in time, was reconstructed using the delay-embedding theorem suggested by Takens. The presence of chaotic signals in the data was identified by the phase space reconstruction and correlation dimension methods, and also the predictability into the future was calculated by the largest Lyapunov exponent to be 437 h or 18 days into the future. The intercomparison of results of the local prediction and GP models shows that for this site-specific dataset, the local prediction model has a slight edge over GP. However, rather than recommending one technique over another, the paper promotes a pluralistic modelling culture, whereby different techniques should be tested to gain a specific insight from each of the models. This would enable a consensus to be drawn from a set of results rather than ignoring the individual insights provided by each model.     

The role of soil moisture initialization in subseasonal and seasonal streamflow prediction – A case study in Sri Lanka

The two main contributors to streamflow predictability at subseasonal to seasonal timescales in tropical regions are: (i) the predictability of meteorologic (particularly precipitation) anomalies, and (ii) the land surface soil moisture state at the start of the forecast period. Meteorological predictions at subseasonal timescale are usually fraught with error and may not be dependable. The accurate initialization of soil moisture, as obtained through real-time land data analysis, may provide skill in subseasonal to seasonal streamflow prediction, even when the prediction skill for rainfall is small.     

Application of nonlinear forecasting techniques for meteorological modeling

A nonlinear forecasting method was used to predict the behavior of a cloud coverage time series several hours in advance. The method is based on the reconstruction of a chaotic strange attractor using four years of cloud absorption data obtained from half-hourly Meteosat infrared images from Northwestern Spain. An exhaustive nonlinear analysis of the time series was carried out to reconstruct the phase space of the underlying chaotic attractor. The forecast values are used by a non-hydrostatic meteorological model ARPS for daily weather prediction and their results compared with surface temperature measurements from a meteorological station and a vertical sounding. The effect of noise in the time series is analyzed in terms of the prediction results.     

强震地面运动的混沌特性分析

引入非线性动力学理论和混沌时间序列分析方法考察强震地面运动加速度时程的非线性特征。首先采用功率谱分析法、主成份分析法和Cao方法定性判断地震动加速度时程具有混沌特性,然后应用混沌时间序列分析方法定量计算了30条地震动加速度时程的三个非线性特征参数。计算表明,这些地震动时程的关联维数为2．0～4．0的分数维,Kolmogorov熵K2为大于零的有限正值,最大Lyapunov指数在o～i．0之间。结果说明,强震地面运动具有混沌特性,地震动的高度不规则和复杂性是地震过程强非线性的反映。     

Nonlinear dynamics of level variations in the Caspian Sea

Caspian Sea level variations, which have several equilibrium states, are studied by the methods of the theory of nonlinear dynamic systems. Normal monthly values of sea level according to data collected at Makhachkala gauge from 1900 to 2000 are considered. The diagnostic characteristics of dynamic chaos are used to show that sea level variations have some properties of series with chaotic behavior. A model of level variations in the Caspian Sea, comprising a system of water balance equations for the sea basin, the dynamics of river runoff, and water balance of the sea itself, is proposed. Equation of a nonlinear oscillator is derived and shown to have solutions with chaotic regimes at some combinations of parameters.     

地震学中的某些非线性现象的初步分析

本文对地震中的一些现象,如小震调制、共维时间、共维面积以及嵌套调制和混沌的预报尺度进行了初步的分析和探讨,对我们关于前兆的复杂性以及对地震预报的进一步认识有一定的参考价值。     

Analysis of cross-correlated chaotic streamflows

Abstract

A trial is made to explore the applicability of chaos analysis outside the commonly reported analysis of a single chaotic time series. Two cross-correlated streamflows, the Little River and the Reed Creek, Virginia, USA, are analysed with regard to the chaotic behaviour. Segments of missing data are assumed in one of the time series and estimated using the other complete time series. Linear regression and artificial neural network models are employed. Two experiments are conducted in the analysis: (a) fitting one global model and (b) fitting multiple local models. Each local model is in the direct vicinity of the missing data. A nonlinear noise reduction method is used to reduce the noise in both time series and the two experiments are repeated. It is found that using multiple local models to estimate the missing data is superior to fitting one global model with regard to the mean squared error and the mean relative error of the estimated values. This result is attributed to the chaotic behaviour of the streamflows under consideration.     

The magnetosphere as a nonlinear system

Summary In order to study the nonlinear physical processes connected with substorm activity we analyse time series of local geomagnetic field variations. The concepts of deterministic chaos and magnetospheric chaotic attractors are examined. The general objective of this article is to detect low dimensional magnetosphere chaos and to properly interpret it as a consequence of magnetosphere — ionosphere informational — energetic coupling.     

行星际空间系统的低维迹象

利用１９７５年７月至１９７６年７月、１９８５年和１９８７年由行星际闪烁（ＩＰＳ）测量得到的太阳风速度资料，应用非线性动力学技术重构了这些时间序列在相空间的吸引子，求得吸引子的分维数３＜Ｄ＜４，最大Ｌｖａｐｕｎｏｖ指数总为正值．这些结果初步表明，行星际空间可能是一个低维的混沌系统．     

Incorporating the spatio-temporal distribution of rainfall and basin geomorphology into nonlinear analyses of streamflow dynamics

Many recent studies have been devoted to the investigation of the nonlinear dynamics of rainfall or streamflow series based on methods of dynamical systems theory. Although finding evidence for the existence of a low-dimensional deterministic component in rainfall or streamflow is of much interest, not much attention has been given to the nonlinear dependencies of the two and especially on how the spatio-temporal distribution of rainfall affects the nonlinear dynamics of streamflow at flood time scales. In this paper, a methodology is presented which simultaneously considers streamflow series, spatio-temporal structure of precipitation and catchment geomorphology into a nonlinear analysis of streamflow dynamics. The proposed framework is based on “hydrologically-relevant” rainfall-runoff phase-space reconstruction acknowledging the fact that rainfall-runoff is a stochastic spatially extended system rather than a deterministic multivariate one. The methodology is applied to two basins in Central North America using 6-hour streamflow data and radar images for a period of 5 years. The proposed methodology is used to: (a) quantify the nonlinear dependencies between streamflow dynamics and the spatio-temporal dynamics of precipitation; (b) study how streamflow predictability is affected by the trade-offs between the level of detail necessary to explain the spatial variability of rainfall and the reduction of complexity due to the smoothing effect of the basin; and (c) explore the possibility of incorporating process-specific information (in terms of catchment geomorphology and an a priori chosen uncertainty model) into nonlinear prediction. Preliminary results are encouraging and indicate the potential of using the proposed methodology to understand via nonlinear analysis of observations (i.e., not based on a particular rainfall-runoff model) streamflow predictability and limits to prediction as a function of the complexity of spatio-temporal forcing relative to basin geomorphology.     

多维分形奇异功率谱分析与应用

地球物理信号中普遍含有噪声，消除噪声是地球物理信号处理中的关键技术之一.奇异功率谱分析（SSA）是在状态空间（又称相空间）中研究（系统）动力学、非线性科学与混沌现象的方法.本文在状态空间中通过SSA分解，研究、应用地球物理序列的尺度不变性进行多维分形滤波：通过在状态空间的SSA分解，构造了经验正交函数系（EOF）；在EOF子空间中定义了两种尺度与测度后，发现了两种测度与尺度皆在多个尺度范围内存在尺度不变性；利用这种尺度~测度的尺度不变性，设计、实现了多维分形奇异功率谱(MSSA)滤波模型；处理解释了大洋钻探（ODP）1143A孔岩芯自然反射性（NGR）资料；Fourier功率谱分析结果证明，MSSA能有效地压制噪声，提取有用信号.研究得出，嵌入维数对MSSA基本无影响（小于1/1000），多维分形滤波器（MSSA）能有效压制噪声或提取有用信号.     

Correlation dimension estimation of hydrological series and data size requirement: myth and reality/Estimation de la dimension de corrélation de séries hydrologiques et taille nécessaire du jeu de données: mythe et réalité

Abstract

The issue of data size (length) requirement for correlation dimension estimation continues to be the nucleus of criticisms on the (low) correlation dimensions reported for hydrological series. The present study addresses this issue from the viewpoints of both the existing theoretical guidelines and the practical reality. For this purpose, correlation dimension analysis is carried out for various data sizes from each of three types of series: (a) stochastic series (artificially generated using a random number generation technique); (b) chaotic series (artificially generated using the Henon map equation); and (c) hydrological series (real flow data observed on the Göta River in Sweden). The outcomes of the analysis of the (artificial) stochastic and chaotic series are used as a basis for interpreting the outcomes of the hydrological series. It is found that reliable dimension results for the stochastic and chaotic series are obtained even when the data size is only a few hundred points (i.e. no underestimation of dimension for small data sizes is visible), with no significant change in the scaling regimes (of the dimension plots) with respect to data size. This implies that the dimension results obtained for the hydrological series even with a few hundred points are also close to the actual ones. The insignificant difference in the scaling regimes for the various data sizes further supports this point. These results lead to the conclusions that: (1) the issue of data size requirement for correlation dimension estimation is more of a myth than reality; (2) the dimension estimates reported thus far for hydrological series could indeed be close to the actual ones (unless influenced by factors other than data size, e.g. delay time, noise, zeros, intermittency).     

Phase Space Structure, Attractor Dimension, Lyapunov Exponent and Nonlinear Prediction from Earth’s Atmospheric Angular Momentum Time Series

--On a short time scale, Atmospheric Angular Momentum (AAM) has been demonstrated to be essentially the sole excitation source of LOD variations. The LOD variation, therefore, merely reflects the AAM variation (LOD as proxy for AAM). The study of the nonlinear nature of AAM variability (e.g., its orbital complexity, dimensionality and extreme sensitivity to the initial conditions) may provide a physical premise for theoretical modelling of the earth-atmosphere-ocean system. Analysis of the high quality of detailed daily LOD/AAM variations time series, spanning the period of 1962-1992, reveals a non-zero and low positive Lyapunov exponent value which suggests possible evidence of deterministic chaos in the underlying dynamics. Application of modern nonlinear prediction techniques capable of distinguishing chaos and random fractals to the data set, further support the above findings and render a predictive time limit of approximately 12-15 days. A low dimensional strange attractor and a low average Lyapunov exponent suggest a low level of unpredictability and stability in the system dynamics. It is argued here that a possible source of the raised entropy in LOD/AAM systems possibly stems from a conceivable nonlinear interaction between the seasonal cycle and inter- or intra-annual fluctuations due to thermodynamics properties of the atmosphere-ocean system.