Khaled H. Hamed

Flood frequency analysis

INTRODUCTION Hydrologic Frequency Analysis General Aspects and Approaches Other Models Return Period, Probability, and Plotting Positions Flood Frequency Models Hydrologic Risk Regionalization Tests on Hydrologic Data SELECTION AND EVALUATION OF PARENT DISTRIBUTION: CONVENTIONAL MOMENTS Moments of Distributions and Their Sample Estimates Moment Ratio Diagrams (MRDs) Probability Plots Selection of Distributions Regional Homogeneity and Regionalization SELECTION AND EVALUATION OF PARENT DISTRIBUTIONS: PROBABILITY WEIGHTED MOMENTS AND L-MOMENTS Moments of Distributions and Their Sample Estimates L-Moment Ratio Diagrams Goodness of Fit Tests A Case Study PARAMETER AND QUANTILE ESTIMATION Introduction Parameter Estimation Quantile Estimation Confidence Intervals NORMAL AND RELATED DISTRIBUTIONS Normal Distribution Two-Parameter Lognormal (LN(2)) Distribution Three-Parameter Lognormal (LM(3)) Distribution GAMMA FAMILY Exponential Distribution Two-Parameter Gamma (G(2)) Distribution Pearson (2) Distribution Log-Pearson (3) Distribution U.S. Water Resources Council Method (WRCM) EXTREME VALUE DISTRIBUTIONS Generalized Extreme Value (GEV) Distribution The Extreme Value Type (EV(1) Distribution Weibul Distribution WAKEBY FAMILY The 5-Parameter Wakeby Distribution (WAK(5)) The 4-Parameter Wakeby Distribution (WAK(4)) The Generalized Pareto Distribution LOGISTIC DISTRIBUTIONS Logistic Distribution Generalized Logistic Distribution COMPUTER PROGRAM Introduction Description of Program REFERENCES

Nonstationarities in hydrologic and environmental time series

1. Introduction.- 2. Data Used in the Book.- 2.1. Hydrologic and Climatic Data.- 2.2. Synthetic and Observed Environmental Data.- 2.2.1. Synthetic Data Sampling from Batchelor Spectrum.- 2.2.2. Details of Data Generated by Sampling from the Batchelor Spectrum.- 2.2.3. Synthetic Data from AR Model.- 2.3. Observed Data.- 2.3.1. Measured Temperature Gradient Profiles.- 3. Time Domain Analysis.- 3.1. Introduction.- 3.2. Visual Inspection of Time Series.- 3.3. Statistical Tests of Significance.- 3.3.1. Parametric Tests.- 3.3.2. Non-parametric Tests.- 3.4. Testing Autocorrelated Data.- 3.5. Application of Trend Tests to Hydrologic Data.- 3.5.1. Visual Inspection of Data.- 3.5.2. Statistical Trend Tests.- 3.5.3. Sub-period Trend Analysis.- 3.6. Conclusions.- 4. Frequency Domain Analysis.- 4.1. Introduction.- 4.2. Conventional Spectral Analysis.- 4.3. Multi-Taper Method (MTM) of Spectral Analysis.- 4.4. Maximum Entropy Spectral Analysis.- 4.5. Spectral Analysis of Hydrologic and Climatic Data.- 4.5.1. Results from MEM Analysis.- 4.5.2. Results from MTM Analysis.- 4.6. Discussion of Results.- 4.7. Conclusions.- 5. Time-Frequency Analysis.- 5.1. Introduction.- 5.2. Evolutionary Spectral Analysis.- 5.3. Evolution of Line Components in Hydrologic and Climatic Data.- 5.4. Evolution of Continuous Spectra in Hydrologic and Climatic Data.- 5.5. Conclusions.- 6. Time-Scale Analysis.- 6.1. Introduction.- 6.2. Wavelet Analysis.- 6.3. Wavelet Trend Analysis.- 6.4. Identification of Dominant Scales.- 6.5. Time-Scale Distribution.- 6.6. Behavior of Hydrologic and Climatic Time Series at Different Scales.- 6.7. Conclusions.- 7. Segmentation of Non-Stationary Time Series.- 7.1. Introduction.- 7.2. Tests based on AR Models.- 7.2.1. Test 1 (de Souza and Thomson, 1982).- 7.2.2. Test 2 (Imberger and Ivey, 1991).- 7.2.3. Test 3 (Davis, Huang and Yao, 1995).- 7.2.4. Test 4 (Tsay, 1988).- 7.3. A test based on wavelet analysis.- 7.4. Segmentation algorithm.- 7.5. Variations of test statistics with the AR order p.- 7.6. Sensitivity of test statistics for detecting change points.- 7.6.1. Detection results for synthetic series from model 2.1.2.- 7.6.2. Detection results for synthetic series from model 2.1.3.- 7.6.3. Detection results for synthetic series from model 2.1.4.- 7.6.4. Detection results for synthetic series from model 2.1.5.- 7.6.5. Conclusions on performances of tests 1-5.- 7.7. Performances of algorithms with and without boundary optimization.- 7.7.1. Detection of non-stationary segment.- 7.7.2. Detection of multi-segment series.- 7.8. Conclusions about the segmentation algorithm.- 8. Estimation of Turbulent Kinetic Energy Dissipation.- 8.1. Introduction.- 8.2. Multi-taper Spectral Estimation.- 8.3. Batchelor Curve Fitting.- 8.4. Comparison of Spectral Estimation Methods.- 8.5. Batchelor Curve Fitting to Synthetic Series.- 8.5.1. Batchelor curve fitting using the first error function.- 8.5.2. Batchelor curve fitting using the second error function.- 8.5.3. Batchelor curve fitting using the third error function.- 8.6. Conclusions on Batchelor curve fitting.- 9. Segmentation of Observed Data.- 9.1. Introduction.- 9.2. Temperature Gradient Profiles.- 9.2.1. Ratios of Unresolved, Bad-Fit and Good-Fit Segments.- 9.2.2. Estimated Values of ? and XT from Resolved Spectra.- 9.2.3. Estimated Values of ? and XT from Profiles in the Same Lake.- 9.2.4. Estimated Values of ? and XT from Different Lakes.- 9.3. Conclusions on Segmentation of Temperature Gradient Profiles.- 9.4. Hydrologic Series.- 9.4.1. Stationary Segments from Hydrologic Series.- 9.4.2. Change Points in Hydrologic Series.- 9.5. Conclusions on Segmentation of Hydrologic Series.- 10. Linearity and Gaussianity Analysis.- 10.1. Introduction.- 10.2. Tests for Gaussianity and Linearity (Hinich, 1982).- 10.3. Testing for Stationary Segments.- 10.3.1. Testing Temperature Gradient Profiles.- 10.3.2. Testing Hydrologic Series.- 10.4. Conclusions about Testing the Hydrologic Series.- 11. Bayesian Detection of Shifts in Hydrologic Time Series.- 11.1. Introduction.- 11.2. Data Used in this Chapter.- 11.3. A Bayesian Method to Detect Shifts in Data.- 11.3.1. Theory.- 11.3.1.1. Parameters of the distribution and the change point n1.- 11.3.1.2. The Unconditional Posterior Distributions of ?, ? and ?.- 11.3.1.3. The Conditional Posterior Distributions of ?i, ?21 and ?i.- 11.3.2. Computation Sequences.- 11.4. Discussion of Results.- 11.4.1. The Posterior Distribution of the Change point n1.- 11.4.2. The Unconditional Posterior Distributions of ?, ? and ?.- 11.4.3. The Conditional Posterior Distributions of ?i,?2i and ?i.- 11.5. Conclusions.- 12. References.- 13. Index.

Multi-taper method of analysis of periodicities in hydrologic data

Many investigators have reported the existence of periodicities, other than the annual and semi-annual, in hydrologic and climatic data. These results are based on the spectral analyses of data. Spectral analysis has several well known drawbacks, to overcome which the multi-taper method (MTM) of spectral analysis was developed. The MTM has the additional advantage that the spectral components can be tested for statistical significance. In the present study, the MTM is used to investigate the periodicities in the annual streamflow, rainfall, temperature and tree-ring data in the midwestern United States. The results indicate the existence of weak periodicities of periods less than 50 years. Considerable evidence exists for drifts in these periodic components.

Prediction of plume migration in heterogeneous media using artificial neural networks

Because of the many uncertainties associated with most flow and transport parameters, studies often implement the numerical simulations within a Monte Carlo frame of work. The large numbers of realizations needed to achieve convergence for the statistics of concern make the Monte Carlo approach computationally demanding. In this study we attempt to develop an empirical-numerical approach to generate Lagrangian particle trajectories in two-dimensional domain given a certain input heterogeneity model without repeatedly solving the flow equation for each realization, with the purpose of evaluating ensemble plume statistics. Artificial neural networks are used to map the relationship between the particle trajectories and the physical properties of the formation. This is achieved by training the neural network through a set of sufficient examples derived from a few realizations using an exponential log-K covariance. The trained network is then used to predict particle trajectories in heterogeneity characterized by an exponential, a Gaussian, a hole-type, or a fractal covariance model. The network that is trained using the exponential model successfully predicts transport results for other models with high accuracy and low computational effort. Accuracy of prediction of the trajectory (percent of explained variance) reaches 94% in the direction of mean flow and 91% normal to it. For the cases studied here, the speed of calculation is ∼7.8 times faster than the traditional approach for 1000 realizations. As the number of realizations increases, the speed factor approaches 12.8. Solute mass flux (mean and standard deviation) and mean-plume concentrations predicted by the trained network agree very closely with the target result obtained via traditional Monte Carlo simulations.

The distribution of Kendall's tau for testing the significance of cross-correlation in persistent data

Abstract Kendalls tau (τ) has been widely used as a distribution-free measure of cross-correlation between two variables. It has been previously shown that persistence in the two involved variables results in the inflation of the variance of τ. In this paper, the full null distribution of Kendalls τ for persistent data with multivariate Gaussian dependence is derived, and an approximation to the full distribution is proposed. The effect of the deviation from the multivariate Gaussian dependence model on the distribution of τ is also investigated. As a demonstration, the temporal consistency and field significance of the cross-correlation between the North Hemisphere (NH) temperature time series in the period 1850–1995 and a set of 784 NH tree-ring width (TRW) proxies in addition to 105 NH tree-ring maximum latewood density (MXD) proxies are studied. When persistence is ignored, the original Mann-Kendall test gives temporally inconsistent results between the early half (1850–1922) and the late half (1923–1995) of the record. These temporal inconsistencies are largely eliminated when persistence is accounted for, indicating the spuriousness of a large portion of the identified cross-correlations. Furthermore, the use of the modified test in combination with a field significance test that is robust to spatial correlation indicates the absence of field significant cross-correlation in both halves of the record. These results have serious implications for the use of tree-ring data as temperature proxies, and emphasize the importance of utilizing the correct distribution of Kendalls τ in order to avoid the overestimation of the significance of cross-correlation between data that exhibit significant persistence. Citation Hamed, K. H. (2011) The distribution of Kendalls tau for testing the significance of cross-correlation in persistent data. Hydrol. Sci. J. 56(5), 841–853.

DISCUSSION of “To prewhiten or not to prewhiten in trend analysis?”

Bayazit & Önöz (2007) present a simulation study to determine when prewhitening can be applied with no real loss of power. They conclude that: “prewhitening should be avoided when the test has a high power, i.e. when the coefficient of variation is very low, the slope of trend is high, and the sample size is large”. While this statement is generally acceptable, quantifying “low” and “high” trend slopes, which appears on the lower half of page 615, is prone to misinterpretation when the results are applied to observed data. The problem stems from the use of the coefficient of variation, Cv, as an indicator. The coefficient of variation is a false indicator, since it involves the mean of the time series, a quantity of which trend test results are independent. Completely removing the mean of a time series, for example, changes very “low” Cv into infinitely “high” Cv, while the result of a trend test remains unchanged. On the other hand, the significance of the same trend slope imposed on two series having different standard deviations will be different, even if they happen to have the same Cv. As an illustrative example, consider the time series yt and zt in equations (1) and (2) below, both of length n = 25 observations, where xt is uncorrelated N(0,1): 10 0.005( 13) t t y x t = + + − (1)

The distribution of Spearman’s rho trend statistic for persistent hydrologic data

Abstract Spearman’s rho, a distribution-free statistic, has been suggested in the literature for testing the significance of trend in time series data. Although the use of the test based on Spearman’s rho (also known as the Daniels test) is less widespread than that based on Kendall’s tau (the Mann-Kendall test), the two tests have been shown in the literature to be equivalent for time series with independent observations. The distribution of the Mann-Kendall trend statistic for persistent data has been previously addressed in the literature. In this paper, the distribution of Spearman’s rho as a trend test statistic for persistent data is studied. Following the same procedures used for Kendall’s tau in earlier work, an exact expression for the variance of Spearman’s rho for persistent data with multivariate Gaussian dependence is derived, and a method for calculating the exact full distribution of rho for small sample sizes is also outlined. Approximations for moderate and large sample sizes are also discussed. A case study of testing the significance of trends in a group of world river flow station data using both Kendall’s tau and Spearman’s rho is presented. Both the theoretical results and those of the case study confirm the equivalence of trend testing based on Spearman’s rho and Kendall’s tau for persistent hydrologic data. Editor Z. W. Kundzewicz; Associate editor S. Grimaldi

Harold Edwin Hurst: The Nile and Egypt, past and future

ABSTRACT H.E. Hurst spent some 60 years studying the Nile for the Egyptian government, and laid the foundation for a monumental set of hydrological records and investigations. His studies of the size of over-year reservoirs needed to maintain a given yield from Nile flows showed that this was greater than that based on random series. This finding, known as the Hurst phenomenon, was confirmed by other natural series and led to important advances in practical and theoretical statistics. His work led to the design of the Aswan High Dam and to continued research in Egypt. Editor D. Koutsoyiannis; Guest editor E. Eris

Time-Scale Analysis

In conventional spectral analysis (Fourier analysis), the signal is compared with a number of basis functions which are composed of sines and cosines of different frequencies. A sinusoid with a given frequency ω can be represented in the frequency domain as a delta function at ±ω, and is therefore well localized in frequency. However, a sinusoid in the time domain has an infinite extent, and is therefore poorly localized in time. In order to improve the performance of Fourier analysis when time localization is desired, the Short Time Fourier Transform (STFT) is used. STFT effectively reduces the extent of the sinusoids to the size of the window used in the analysis. However, the choice of a window size that is suitable for a given signal involves a tradeoff. On the one hand a short time window will capture high frequency components and allow for more time localization. On the other hand, a longer window is required to precisely capture low frequency information. Accordingly, a more effective method of analysis would require a variablesize window to be applied to the data.

Stochastic Investigation of the GERD-AHD Interaction Through First Impoundment and Beyond

The Grand Ethiopian Renaissance Dam (GERD) is currently being constructed on the Blue Nile. In the short term, water inflow to the Aswan High Dam (AHD) reservoir will be reduced as water is abstracted during GERD’s first impoundment. In the long-term, the inflow to AHD will be affected due to flow regulation and additional evaporation losses from GERD. This chapter presents a stochastic analysis of the impacts of GERD on AHD. Synthetic Nile flow series that preserve the Hurst exponent of the flow are generated using a Fractional Gaussian Noise (FGN) model. Results from the simulation of 1,000 equally probable Nile flow series using a simplified GERD-AHD system model are analyzed. The results indicate a very high downstream risk when GERD is operated as an annual storage reservoir, which questions the economic attractiveness of GERD in a regional context when operated solely for hydropower energy maximization. Operating GERD as a long-term storage reservoir results in reduced, yet still considerable impacts. Optimal GERD filling and operation policies aimed at minimizing downstream risks through a comprehensive regional economic, environmental, and social analysis are urgently needed.