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.     

Nonlinear ensemble prediction of chaotic daily rainfall

The significance of treating rainfall as a chaotic system instead of a stochastic system for a better understanding of the underlying dynamics has been taken up by various studies recently. However, an important limitation of all these approaches is the dependence on a single method for identifying the chaotic nature and the parameters involved. Many of these approaches aim at only analyzing the chaotic nature and not its prediction. In the present study, an attempt is made to identify chaos using various techniques and prediction is also done by generating ensembles in order to quantify the uncertainty involved. Daily rainfall data of three regions with contrasting characteristics (mainly in the spatial area covered), Malaprabha, Mahanadi and All-India for the period 1955–2000 are used for the study. Auto-correlation and mutual information methods are used to determine the delay time for the phase space reconstruction. Optimum embedding dimension is determined using correlation dimension, false nearest neighbour algorithm and also nonlinear prediction methods. The low embedding dimensions obtained from these methods indicate the existence of low dimensional chaos in the three rainfall series. Correlation dimension method is done on the phase randomized and first derivative of the data series to check whether the saturation of the dimension is due to the inherent linear correlation structure or due to low dimensional dynamics. Positive Lyapunov exponents obtained prove the exponential divergence of the trajectories and hence the unpredictability. Surrogate data test is also done to further confirm the nonlinear structure of the rainfall series. A range of plausible parameters is used for generating an ensemble of predictions of rainfall for each year separately for the period 1996–2000 using the data till the preceding year. For analyzing the sensitiveness to initial conditions, predictions are done from two different months in a year viz., from the beginning of January and June. The reasonably good predictions obtained indicate the efficiency of the nonlinear prediction method for predicting the rainfall series. Also, the rank probability skill score and the rank histograms show that the ensembles generated are reliable with a good spread and skill. A comparison of results of the three regions indicates that although they are chaotic in nature, the spatial averaging over a large area can increase the dimension and improve the predictability, thus destroying the chaotic nature.     

Regional scale rainfall–runoff modeling using VARX–MGARCH approach

Multivariate time series modeling approaches are known as useful tools for describing, simulating, and forecasting hydrologic variables as well as their changes over the time. These approaches also have temporal and cross-sectional spatial dependence in multiple measurements. Although the application of multivariate linear and nonlinear time series approaches such as vector autoregressive with eXogenous variables (VARX) and multivariate generalized autoregressive conditional heteroscedasticity (MGARCH) models are commonly used in financial and economic sciences, these approaches have not been extensively used in hydrology and water resources engineering. This study employed VARX and VARX–MGARCH approaches in modeling mean and conditional heteroscedasticity of daily rainfall and runoff records in the basin of Zarrineh Rood Dam, Iran. Bivariate diagonal VECH (DVECH) model, as a main type of MGARCH, shows how the conditional variance–covariance and conditional correlation structure vary over the time between residuals series of the fitted VARX. For this purpose, five model fits, which consider different combinations of twofold rainfall and runoff, including both upstream and downstream stations, have been investigated in the present study. The VARX model, with different orders, was applied to the daily rainfall–runoff process of the study area in each of these model fits. The Portmanteau test revealed the existence of conditional heteroscedasticity in the twofold residuals of fitted VARX models. Therefore, the VARX–DVECH model is proposed to capture the heteroscedasticity existing in the daily rainfall–runoff process. The bivariate DVECH model indicated both short-run and long-run persistency in the conditional variance–covariance matrix related to the twofold innovations of rainfall–runoff processes. Furthermore, the evaluation criteria for the VARX–DVECH model revealed the improvement of VARX model performance.     

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.     

On the quest for chaotic attractors in hydrological processes

Abstract

In the last two decades, several researchers have claimed to have discovered low-dimensional determinism in hydrological processes, such as rainfall and runoff, using methods of chaotic analysis. However, such results have been criticized by others. In an attempt to offer additional insights into this discussion, it is shown here that, in some cases, merely the careful application of concepts of dynamical systems, without doing any calculation, provides strong indications that hydrological processes cannot be (low-dimensional) deterministic chaoti. Furthermore, it is shown that specific peculiarities of hydrological processes on fine time scales, such as asymmetric, J-shaped distribution functions, intermittency, and high autocorrelations, are synergistic factors that can lead to misleading conclusions regarding the presence of (low-dimensional) deterministic chaos. In addition, the recovery of a hypothetical attractor from a time series is put as a statistical estimation problem whose study allows, among others, quantification of the required sample size; this appears to be so huge that it prohibits any accurate estimation, even with the largest available hydrological records. All these arguments are demonstrated using appropriately synthesized theoretical examples. Finally, in light of the theoretical analyses and arguments, typical real-world hydrometeorological time series, such as relative humidity, rainfall, and runoff, are explored and none of them is found to indicate the presence of chaos.     

Uncertainty based modeling of rainfall-runoff: Combined differential evolution adaptive Metropolis (DREAM) and K-means clustering

Simulation of rainfall-runoff process in urban areas is of great importance considering the consequences and damages of extreme runoff events and floods. The first issue in flood hazard analysis is rainfall simulation. Large scale climate signals have been proved to be effective in rainfall simulation and prediction. In this study, an integrated scheme is developed for rainfall-runoff modeling considering different sources of uncertainty. This scheme includes three main steps of rainfall forecasting, rainfall-runoff simulation and future runoff prediction. In the first step, data driven models are developed and used to forecast rainfall using large scale climate signals as rainfall predictors. Due to high effect of different sources of uncertainty on the output of hydrologic models, in the second step uncertainty associated with input data, model parameters and model structure is incorporated in rainfall-runoff modeling and simulation. Three rainfall-runoff simulation models are developed for consideration of model conceptual (structural) uncertainty in real time runoff forecasting. To analyze the uncertainty of the model structure, streamflows generated by alternative rainfall-runoff models are combined, through developing a weighting method based on K-means clustering. Model parameters and input uncertainty are investigated using an adaptive Markov Chain Monte Carlo method. Finally, calibrated rainfall-runoff models are driven using the forecasted rainfall to predict future runoff for the watershed. The proposed scheme is employed in the case study of the Bronx River watershed, New York City. Results of uncertainty analysis of rainfall-runoff modeling reveal that simultaneous estimation of model parameters and input uncertainty significantly changes the probability distribution of the model parameters. It is also observed that by combining the outputs of the hydrological models using the proposed clustering scheme, the accuracy of runoff simulation in the watershed is remarkably improved up to 50% in comparison to the simulations by the individual models. Results indicate that the developed methodology not only provides reliable tools for rainfall and runoff modeling, but also adequate time for incorporating required mitigation measures in dealing with potentially extreme runoff events and flood hazard. Results of this study can be used in identification of the main factors affecting flood hazard analysis.     

The nonlinear hydro-climatic process in the Yarkand River, northwestern China

Based on the hydrologic and meteorological data in the Yarkand River Basin during 1957–2008, the nonlinear hydro-climatic process was analyzed by a comprehensive method, including the Mann–Kendall trend test, wavelet analysis, wavelet regression analysis and correlation dimension. The main findings are as following: (1) The annual runoff, annual average temperature and annual precipitation showed an increasing trend during the period of 1957–2008, and the average increase extent in runoff, temperature and precipitation was 2.234 × 10⁸ m³/10 year, 0.223 °C/10 year, and 4.453 mm/10 year, respectively. (2) The nonlinear pattern of runoff, temperature and precipitation was scale-dependent with time. In other words, the annual runoff, annual average temperature and annual precipitation at five time scales resulted in five patterns of nonlinear variations respectively. (3) Although annual runoff, annual average temperature and annual precipitation presented nonlinear variations at different time scales, the runoff has a linear correlation with the temperature and precipitation. (4) The hydro-climatic process of the Yarkand River is chaotic dynamic system, in which the correlation dimension of annual runoff, annual average temperature and annual precipitation is 3.2118, 2.999 and 2.992 respectively. None of the correlation dimensions is an integer, and it indicates that the hydro-climatic process has the fractal characteristics.     

A preliminary investigation on the scaling behaviour of rainfall observed in two different climates

Abstract

The problem of transformation of rainfall data from one scale to another has been gaining considerable importance in recent years. Though the application of the concept of fractal theory, in the studies conducted thus far, nearly unanimously points at the possibility of such a transformation, the suitability of the theory to the highly variable rainfall in time and space has very often been questioned. A preliminary attempt is made herein to address this issue by investigating the existence of temporal scaling behaviour in rainfall data observed in two different climatic regions: (a) a subtropical climatic region (Leaf River basin, Mississippi, USA) and (b) an equatorial climatic region (Singapore). Rainfall data of three different resolutions, six-hourly, daily, and weekly, observed over a period of 25 years, are investigated. A mono- or simple-scaling method (box dimension method) is employed. The results achieved for the different data sets clearly indicate the existence of temporal scaling in rainfall observed in the two regions, an encouraging news on the suitability of fractal theory in understanding and modelling the rainfall process. However, the insufficiency of a single dimension to characterize the rainfall behaviour is realized, as the dimension depends on the rainfall intensity level, which, in turn, may be related to the rainfall generating mechanisms. A comparison of the box-dimension results obtained for data of different resolutions, from each of the regions, seems to indicate a possible connection between them, a prospect of tremendous practical importance. Another interesting observation is the similarity between the box dimension results obtained for rainfall data from Leaf River basin and Singapore, but this is also clearly related to the intensity level. The dependence of the dimension on the intensity threshold suggests the use of a multi-dimensional fractal approach, where the process is characterized by more than one dimension (or a dimension function) instead of one single dimension. On the basis of the present results, some potential areas for further study are identified.     

A chaotic analysis on air pollution index change over past 10?years in Lanzhou, northwest China

In terms of the chaos theory, the phase-space-reconstruction method has been employed to describe the multi-dimensional phase space for the time series of air pollution index (API) during the past 10 years in Lanzhou, northwest China. The mutual information and Cao method were used to determine the reconstruction parameters, and the characteristic quantities including the Lyapunov exponent and the correlation dimension were calculated respectively. As a result, the correlation dimensions were fractioned, and the maximum Lyapunov exponent (λ ₁) > 0. It shows that these presented the obvious chaotic characteristics that resulted from the evolution of non-linear chaotic dynamic system in the time series of air pollution index over the past 10 years. In the meanwhile, three or even four main dynamic variables were discussed here that could effectively interpret the changes of air pollution index time series and their causes. Some reasonable preventive countermeasures were thus put forward. These findings might provide a scientific basis for probing further into the regional complexity and evolution of the time series of air pollution index.     

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).     

A stochastic integral equation analog of rainfall-runoff processes for evaluating modeling uncertainty

In this paper a very general rainfall-runoff model structure (described below) is shown to reduce to a unit hydrograph model structure. For the general model, a multi-linear unit hydrograph approach is used to develop subarea runoff, and is coupled to a multi-linear channel flow routing method to develop a link-node rainfall-runoff model network. The spatial and temporal rainfall distribution over the catchment is probabilistically related to a known rainfall data source located in the catchment in order to account for the stochastic nature of rainfall with respect to the rain gauge measured data. The resulting link node model structure is a series of stochastic integral equations, one equation for each subarea. A cumulative stochastic integral equation is developed as a sum of the above series, and includes the complete spatial and temporal variabilities of the rainfall over the catchment. The resulting stochastic integral equation is seen to be an extension of the well-known single area unit hydrograph method, except that the model output of a runoff hydrograph is a distribution of outcomes (or realizations) when applied to problems involving prediction of storm runoff; that is, the model output is a set of probable runoff hydrographs, each outcome being the results of calibration to a known storm event.     

A stochastic integral equation analog of rainfall-runoff processes for evaluating modeling uncertainty

In this paper a very general rainfall-runoff model structure (described below) is shown to reduce to a unit hydrograph model structure. For the general model, a multi-linear unit hydrograph approach is used to develop subarea runoff, and is coupled to a multi-linear channel flow routing method to develop a link-node rainfall-runoff model network. The spatial and temporal rainfall distribution over the catchment is probabilistically related to a known rainfall data source located in the catchment in order to account for the stochastic nature of rainfall with respect to the rain gauge measured data. The resulting link node model structure is a series of stochastic integral equations, one equation for each subarea. A cumulative stochastic integral equation is developed as a sum of the above series, and includes the complete spatial and temporal variabilities of the rainfall over the catchment. The resulting stochastic integral equation is seen to be an extension of the well-known single area unit hydrograph method, except that the model output of a runoff hydrograph is a distribution of outcomes (or realizations) when applied to problems involving prediction of storm runoff; that is, the model output is a set of probable runoff hydrographs, each outcome being the results of calibration to a known storm event.     

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.     

Design hydrograph estimation in small and ungauged watersheds: continuous simulation method versus event‐based approach

The proper assessment of design hydrographs and their main properties (peak, volume and duration) in small and ungauged basins is a key point of many hydrological applications. In general, two types of methods can be used to evaluate the design hydrograph: one approach is based on the statistics of storm events, while the other relies on continuously simulating rainfall‐runoff time series. In the first class of methods, the design hydrograph is obtained by applying a rainfall‐runoff model to a design hyetograph that synthesises the storm event. In the second approach, the design hydrograph is quantified by analysing long synthetic runoff time series that are obtained by transforming synthetic rainfall sequences through a rainfall‐runoff model. These simulation‐based procedures overcome some of the unrealistic hypotheses which characterize the event‐based approaches. In this paper, a simulation experiment is carried out to examine the differences between the two types of methods in terms of the design hydrograph's peak, volume and duration. The results conclude that the continuous simulation methods are preferable because the event‐based approaches tend to underestimate the hydrograph's volume and duration. Copyright © 2011 John Wiley & Sons, Ltd.     

Assessing spatially dependent errors in radar rainfall estimates for rainfall-runoff simulation

Weather radar been widely employed to measure precipitation and to predict flood risks. However, it is still not considered accurate enough because of radar errors. Most previous studies have focused primarily on removing errors from the radar data. Therefore, in the current study, we examined the effects of radar rainfall errors on rainfall-runoff simulation using the spatial error model (SEM). SEM was used to synthetically generate random or cross-correlated errors. A number of events were generated to investigate the effect of spatially dependent errors in radar rainfall estimates on runoff simulation. For runoff simulation, the Nam River basin in South Korea was used with the distributed rainfall-runoff model, Vflo?. The results indicated that spatially dependent errors caused much higher variations in peak discharge than independent random errors. To further investigate the effect of the magnitude of cross-correlation among radar errors, different magnitudes of spatial cross-correlations were employed during the rainfall-runoff simulation. The results demonstrated that a stronger correlation led to a higher variation in peak discharge up to the observed correlation structure while a correlation stronger than the observed case resulted in lower variability in peak discharge. We concluded that the error structure in radar rainfall estimates significantly affects predictions of the runoff peak. Therefore, efforts to not only remove the radar rainfall errors, but to also weaken the cross-correlation structure of the errors need to be taken to forecast flood events accurately.     

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.     

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。     

Complexity analysis of rainfall and runoff time series based on sample entropy in different temporal scales

This study applied sample entropy (SampEn) to rainfall and runoff time series to investigate the complexity of different temporal scales. Rainfall and runoff time series with intervals of 1, 10, 30, 90, and 365 days for the Wu-Tu upstream watershed were used. Thereafter, SampEn was computed for the five rainfall and runoff time series. The results show that for the various temporal scales, comparisons of the complexity between the rainfall and runoff time series based on the SampEn are inconsistent. Calculating the dynamic SampEn further elucidated variations of the complexity in the rainfall and runoff time series. In addition, the results show that SampEn measures of the rainfall and runoff time series are typically higher than the approximate entropy measures of the rainfall and runoff time series for a specific temporal scale. The complexity increases when the sample size increases for a specific temporal scale. Furthermore, temporal scales with low complexity and high predictability are obtained from the variations of SampEn for the rainfall and runoff time series with different temporal scales, thereby providing a reference for determining the appropriate temporal scale for rainfall and runoff time series forecasting.     

Hydrological considerations for dam siting in arid regions: a Saudi Arabian study

Abstract

In any dam siting study in arid regions, rainfall records, runoff measurements and their greatest magnitudes are very important. Unfortunately, the data are scarce and, therefore, empirical approaches and charts obtained from similar regions in other parts of the world are necessary for complete applications. The lack of observed data presents the major problem for runoff modelling in arid regions. These regions have characteristically high rainfall intensity and consequent flash floods with large amounts of sediments. Occurrence of rainfall is sporadic, both temporally and spatially, which makes the interpretation of the rainfall-runoff relationship quite difficult. Flood estimations play a significant role in dam siting from the point of view of water availability. This paper presents the basic calculations of floods and sediment amounts that are necessary in dam siting and construction in an arid area by considering the southwestern part of the Kingdom of Saudi Arabia.