The presence of nonlinear dependence and chaos has strong implications for predictive modeling and the analysis of dominant processes in hydrology and climate. Analysis of extremes may aid in developing predictive models in hydro-climatology by giving enhanced understanding of processes driving the extremes and perhaps delineate possible anthropogenic or natural causes. This dissertation develops and utilizes different set of tools for predictive modeling, specifically nonlinear dependence, extreme, and chaos, and tests the viability of these tools on the real data. Commonly used dependence measures, such as linear correlation, cross-correlogram or Kendall's tau, cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic or monotonic. Mutual information (MI) has been frequently utilized for capturing the complete dependence structure including nonlinear dependence. Since the geophysical data are generally finite and noisy, this dissertation attempts to address a key gap in the literature, specifically, the evaluation of recently proposed MI-estimation methods to choose the best method for capturing nonlinear dependence, particularly in terms of their robustness for short and noisy data. The performance of kernel density estimators (KDE) and k-nearest neighbors (KNN) are the best for 100 data points at high and low noise-to-signal levels, respectively, whereas KNN is the best for 1000 data points consistently across noise levels. One real application of nonlinear dependence based on MI is to capture extrabasinal connections between El Nino-Southern Oscillation (ENSO) and river flows in the tropics and subtropics, specifically the Nile, Amazon, Congo, Parana, and Ganges rivers which reveals 20-70% higher dependence than those suggested so far by linear correlations. For extremes analysis, this dissertation develops a new measure precipitation extremes volatility index (PEVI), which measures the variability of extremes, is defined as the ratio of return levels. Spatio-temporal variability of PEVI, based on the Poisson-generalized Pareto (Poisson-GP) model, is investigated on weekly maxima observations available at 2:50 grids for 1940-2004 in South America. From 1965-2004, the PEVI shows increasing trends in few parts of the Amazon basin and the Brazilian highlands, north-west Venezuela including Caracas, north Argentina, Uruguay, Rio De Janeiro, Sao Paulo, Asuncion, and Cayenne. Catingas, few parts of the Brazilian highlands, Sao Paulo and Cayenne experience increasing number of consecutive 2- and 3-days extremes from 1965-2004. This dissertation also addresses the ability to detect the chaotic signal from a finite time series observation of hydrologic systems. Tests with simulated data demonstrate the presence of thresholds, in terms of noise to chaotic-signal and seasonality to chaotic-signal ratios, beyond which the set of currently available tools is not able to detect the chaotic component. Our results indicate that the decomposition of a simulated time series into the corresponding random, seasonal and chaotic components is possible from finite data. Real streamflow data from the Arkansas and Colorado rivers do not exhibit chaos. While a chaotic component can be extracted from the Arkansas data, such a component is either not present or can not be extracted from the Colorado data.
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