Bellie Sivakumar

River flow forecasting: use of phase-space reconstruction and artificial neural networks approaches

Abstract The use of two non-linear black-box approaches, phase-space reconstruction (PSR) and artificial neural networks (ANN), for forecasting river flow dynamics is studied and a comparison of their performances is made. This is done by attempting 1-day and 7-day ahead forecasts of the daily river flow from the Nakhon Sawan station at the Chao Phraya River basin in Thailand. The results indicate a reasonably good performance of both approaches for both 1-day and 7-day ahead forecasts. However, the performance of the PSR approach is found to be consistently better than that of ANN. One reason for this could be that in the PSR approach the flow series in the phase-space is represented step by step in local neighborhoods, rather than a global approximation as is done in ANN. Another reason could be the use of the multi-layer perceptron (MLP) in ANN, since MLPs may not be most appropriate for forecasting at longer lead times. The selection of training set for the ANN may also contribute to such results. A comparison of the optimal number of variables for capturing the flow dynamics, as identified by the two approaches, indicates a large discrepancy in the case of 7-day ahead forecasts (1 and 7 variables, respectively), though for 1-day ahead forecasts it is found to be consistent (3 variables). A possible explanation for this could be the influence of noise in the data, an observation also made from the 1-day ahead forecast results using the PSR approach. The present results lead to observation on: (1) the use of other neural networks for runoff forecasting, particularly at longer lead times; (2) the influence of training set used in the ANN; and (3) the effect of noise on forecast accuracy, particularly in the PSR approach.

Chaos theory in hydrology: important issues and interpretations

Abstract The application of the concept of chaos theory in hydrology has been gaining considerable interest in recent times. However, studies reporting the existence of chaos in hydrological processes are often criticized due to the fundamental assumptions with which the chaos identification methods have been developed, i.e. infinite and noise-free time series, and the inherent limitations of the hydrological time series, i.e. finite and noisy. This paper is designed: (1) to address some of the important issues in the application of chaos theory in hydrology; and (2) to provide possible interpretations to the results reported by past studies reporting chaos in hydrological processes. A brief review of some of the past studies investigating chaos in hydrological processes is presented. An insight into the studies reveals that most of the problems, such as data size, noise, delay time, in the application of chaos theory have been addressed by past studies, and caution taken in the application of the methods and interpretation of the results. The study also reveals that the problem of data size is not as severe as it was assumed to be, whereas the presence of noise seems to have much more influence on the nonlinear prediction method than the correlation dimension method. The study indicates that the presence of noise in the data could be an important reason for the low-prediction accuracy estimates achieved in some of the past studies. These observations, with the fact that most of the past studies used the correlation dimension either as a proof or as a preliminary evidence of chaos, suggest that the hypothesis of deterministic chaos, as the basis in those studies, for hydrological processes is valid and has great practical potential.

Chaos theory in geophysics: past, present and future

Abstract The past two decades of research on chaos theory in geophysics has brought about a significant shift in the way we view geophysical phenomena. Research on chaos theory in geophysics continues to grow at a much faster pace, with applications to a wide variety of geophysical phenomena and geophysical problems. In spite of our success in understanding geophysical phenomena also from a different (i.e. chaotic) perspective, there still seems to be lingering suspicions on the scope of chaos theory in geophysics. The goal of this paper is to present a comprehensive account of the achievements and status of chaos theory in geophysics, and to disseminate the hope and scope for the future. A systematic review of chaos theory in geophysics, covering a wide spectrum of geophysical phenomena studied (e.g. rainfall, river flow, sediment transport, temperature, pressure, tree ring series, etc.), is presented to narrate our past achievements not only in understanding and predicting geophysical phenomena but also in improving the chaos identification and prediction techniques. The present state of chaos research in geophysics (in terms of geophysical phenomena, problems, and chaos methods) and potential for future improvements (in terms of where, why and possibly how) are also highlighted. Our popular views of nature (i.e. stochastic and deterministic), and of geophysical phenomena in particular, are discussed, and the usefulness of chaos theory as a bridge between such views is also put forth.

Characterization and prediction of runoff dynamics: a nonlinear dynamical view

Abstract An attempt is made in this study to characterize and predict runoff dynamics, using ideas gained from nonlinear dynamical theory. Daily runoff data observed over a period of 19 years (January 1, 1975–December 31, 1993) at the Lindenborg catchment in Denmark is studied using a variety of techniques. First, the autocorrelation function and the Fourier power spectrum are used as indicators to obtain some preliminary information regarding the runoff behavior. A comprehensive characterization is done next through the correlation integral analysis, the false nearest neighbor algorithm, and the nonlinear prediction method, all of which use the concept of phase-space reconstruction, i.e., reconstruction of the single-dimensional (or variable) runoff series in a multi-dimensional phase-space to represent its dynamics. The average mutual information method is used to estimate the delay time for the phase-space reconstruction. The exponential decay in the autocorrelation function plot and the sharp spectral lines in the Fourier power spectrum seem to provide some preliminary indication regarding the possible presence of chaos in the runoff dynamics. The (low) correlation dimension (of about 3.76) obtained from the correlation integral analysis, the (low) global dimension (of 4 or 5) obtained from the false nearest neighbor algorithm, and the (low) optimal embedding dimension (of 3) from the nonlinear prediction method are in close agreement with each other, providing convincing evidence regarding the presence of low-dimensional chaotic behavior in the runoff dynamics. The near-accurate predictions achieved for the runoff series (correlation coefficient of about 0.99 and coefficient of efficiency of about 0.98) indicate the appropriateness of the chaotic dynamical approach for characterizing and predicting the runoff dynamics at the Lindenborg catchment.

A chaotic approach to rainfall disaggregation

The importance of high-resolution rainfall data to understanding the intricacies of the dynamics of hydrological processes and describing them in a sophisticated and accurate way has been increasingly realized. The last decade has witnessed a number of studies and numerous approaches to the possibility of transformation of rainfall data from one scale to another, nearly unanimously pointing to such a possibility. However, an important limitation of such approaches is that they treat the rainfall process as a realization of a stochastic process, and therefore there seems to be a lack of connection between the structure of the models and the underlying physics of the rainfall process. The present study introduces a new framework based on the notion of deterministic chaos to investigate the behavior of the dynamics of rainfall transformation between different temporal scales aimed toward establishing this connection. Rainfall data of successively doubled resolutions (i.e., 6, 12, 24, 48, 96, and 192 hours) observed at Leaf River basin, in the state of Mississippi, United States of America, are studied. The correlation dimension method is employed to investigate the presence of chaos in the rainfall transformation. The finite and low correlation dimensions obtained for the distributions of weights between rainfall data of different scales indicate the existence of chaos in the rainfall transformation, suggesting the applicability of a chaotic model. The formulation of a simple chaotic disaggregation model and its application to the Leaf River rainfall data provides encouraging results with practical potential. The disaggregation model results themselves indicate the presence of chaos in the dynamics of rainfall transformation, providing support for the results obtained using the correlation dimension method.

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.

A systematic approach to noise reduction in chaotic hydrological time series

Recent studies have shown that the noise limits the performance of many techniques used for identification and prediction of deterministic systems. The extent of the influence of noise on the analysis of hydrological (or any real) data is difficult to understand due to the lack of knowledge on the level and nature of the noise. Meanwhile, a variety of nonlinear noise reduction methods have been developed and applied to hydrological (and other real) data. The present study addresses some of the potential problems in applying such methods to chaotic hydrological (or any real) data, and discusses the usefulness of estimating the noise level prior to noise reduction. The study proposes a systematic approach to additive measurement noise reduction in chaotic hydrological (or any real) data, by coupling a noise level determination method and a noise reduction method. The approach is first demonstrated on noise-added artificial chaotic data (Henon data) and then applied on real chaotic hydrological data, the Singapore rainfall data. The approach uses the prediction accuracy as the main diagnostic tool to determine the most probable noise level, and the correlation dimension as a supplementary tool. The results indicate a noise level between 9 and 11% in the Singapore rainfall data, providing a possible explanation for the low prediction accuracy achieved in earlier studies for the (noisy) original rainfall data. Significant improvement in the prediction accuracy achieved for the noise-reduced rainfall data provides additional support for the above.

A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers

The use of phase-space reconstruction approach for understanding and predicting suspended sediment concentration dynamics in rivers is investigated. According to this approach, the dynamic changes of the suspended sediment concentration phenomenon are represented by reconstructing (or embedding) the single-dimensional (or variable) suspended sediment concentration series in a multi-dimensional phase-space. After representing the dynamics in the phase-space, a local approximation method is employed for making predictions. The approach is employed for representing and predicting the dynamics of the daily suspended sediment concentration observed in the Mississippi River basin (at St. Louis, MO) in the United States of America. The predicted suspended sediment concentrations are found to be in very good agreement with the observed ones; not only are the major trends well captured but the minor (noisy) fluctuations reasonably preserved as well. The near-accurate predictions indicate the appropriateness of the phase-space reconstruction approach for understanding the suspended sediment concentration phenomenon. The results (i.e. optimal embedding dimension) also reveal that the suspended sediment concentration dynamics are dominantly influenced by three variables, suggesting that the suspended sediment concentration (and other sediment transport related) phenomena could be viewed from a low-dimensional chaotic dynamic perspective.

Monthly runoff prediction using phase space reconstruction

Abstract A nonlinear prediction method, developed based on the ideas gained from deterministic chaos theory, is employed: (a) to predict monthly runoff; and (b) to detect the possible presence of chaos in runoff dynamics. The method first reconstructs the single-dimensional (or variable) runoff series in a multi-dimensional phase space to represent its dynamics, and then uses a local polynomial approach to make predictions. Monthly runoff series observed at the Coaracy Nunes/Araguari River basin in northern Brazil is studied. The predictions are found to be in close agreement with the observed runoff, with high correlation coefficient and coefficient of efficiency values, indicating the suitability of the nonlinear prediction method for predicting the runoff dynamics. The results also reveal the presence of low-dimensional chaos in the runoff dynamics, when an inverse approach is adopted for identification, as: (a) an optimal embedding dimension exists, and (b) the prediction accuracy decreases with an increase in prediction lead time.

Is correlation dimension a reliable indicator of low-dimensional chaos in short hydrological time series?

The reliability of the correlation dimension estimation in short hydrological time series is investigated using an inverse approach. According to this approach, first predictions are made using the phase-space reconstruction technique and the artificial neural networks. The correlation dimension is estimated next independently and is compared with the prediction results. A short hydrological series, monthly runoff series of 48 years (with a total of only 576 values) observed at the Coaracy Nunes/Araguari River watershed in northern Brazil, is studied. The correlation dimension results are in reasonably good agreement with the optimal embedding dimension obtained from the phase-space method and the optimal number of inputs from the neural networks. No underestimation of the correlation dimension is observed due to the small data size, rather there seems to be a slight overestimation due to the presence of noise in the data. The results indicate that the accuracy of the correlation dimension may not be judged on the basis of the length of the time series but on whether the time series is long enough to reasonably represent the dynamical changes in the system. Such an observation suggests that the correlation dimension could indeed be a reliable indicator of low-dimensional chaos even in short hydrological time series, which is certainly encouraging news for hydrologists who often have to deal with short time series. (Less)