Duane C. Boes

Flood frequency analysis with systematic and historical or paleoflood data based on the two‐parameter general extreme value models

Historical and paleoflood data have become an important source of information for flood frequency analysis. A number of studies have been proposed in the literature regarding the value of historical and paleoflood information for estimating flood quantiles. These studies have been generally based on computer simulation experiments. In this paper the value of using systematic and historical/paleoflood data relative to using systematic records alone is examined analytically by comparing the asymptotic variances of flood quantiles assuming a two-parameter general extreme value marginal distribution, type 1 and type 2 censored data, and maximum likelihood estimation method. The results of this study indicate that the value of historical and paleoflood data for estimating flood quantiles can be small or large depending on only three factors: the relative magnitudes of the length of the systematic record (N) and the length of the historical period (M); the return period (T) of the flood quantile of interest; and the return period (H) of the threshold level of perception. For instance, for N = 50, M = 50 and T = 500, the statistical gain for type 2 censoring becomes significantly larger than for type 1 censoring as H becomes greater than 100 years. In addition, computer experiments have shown that the results regarding the statistical gain based on asymptotic considerations are valid for the usual sample sizes.

Population index flood method for regional frequency analysis

Regional frequency analyses based on index flood procedures have been used within the hydrologic community since 1960. It appears that when the index flood method was first suggested, the index flood was taken to be the at-site population mean, which, in turn, in the last two or three decades, has been estimated by the at-site sample mean. The objectives of this paper are to investigate the consequences of replacing a population characteristic with its sample counterpart and to propose an analytically correct regional model dubbed as the population index flood (PIF) method. In this method the homogeneity of the region is embedded in the structure of the parameter space of the underlying distribution model. Simulation experiments are conducted to test the proposed PIF method based on the generalized extreme value distribution with parameters estimated using the method of maximum likelihood (MLE) and the method of probability- weighted moments (PWM). Furthermore, in the simulation experiments the PIF method is compared with the Hosking and Wallis [1997] regional estimation scheme (HW scheme). Comparing among all index flood methods investigated herein, the PIF method with parameters estimated using MLE provides the best overall results for the 0.95 and the 0.99 quantites in terms of both bias and root-mean-square error for moderate to sufficiently large sample sizes, but for the 0.995 quantile the HW scheme seems to perform best for the investigated sample sizes.

Shifting level modelling of hydrologic series

Abstract The potential of applying shifting level (SL) models to hydrologic processes is discussed in light of observed statistical characteristics of hydrologic data. An SL model and an ARMA (1, 1) model are fitted to an actual hydrologic series. Computer simulation experiments with these models are carried out to compare maximum accumulated deficit and run properties. Results obtained indicate that the mean maximum accumulated deficit, mean longest negative run length, and mean largest negative run sum for both models are similar while there are differences in their corresponding variances.

Modeling the Dynamics of Long-Term Variability of Hydroclimatic Processes

Abstract The stochastic analysis, modeling, and simulation of climatic and hydrologic processes such as precipitation, streamflow, and sea surface temperature have usually been based on assumed stationarity or randomness of the process under consideration. However, empirical evidence of many hydroclimatic data shows temporal variability involving trends, oscillatory behavior, and sudden shifts. While many studies have been made for detecting and testing the statistical significance of these special characteristics, the probabilistic framework for modeling the temporal dynamics of such processes appears to be lacking. In this paper a family of stochastic models that can be used to capture the dynamics of abrupt shifts in hydroclimatic time series is proposed. The applicability of such “shifting mean models” are illustrated by using time series data of annual Pacific decadal oscillation (PDO) indices and annual streamflows of the Niger River.

Hurst phenomenon as a pre-asymptotic behavior

Interpretation of the Hurst phenomenon has been controversial in statistical hydrology ever since the Hurst publication indicating that the expected rescaled adjusted range of certain geophysical time series apparently does not behave as n12. One interpretation has been that the expected rescaled range is asymptotically proportional to nh with h > 0.50. Another interpretation is that series exhibiting this phenomenon have Hurst slope h greater than 0.50 for small or moderate values of n but still 0.50 as the asymptotic value. Computer simulations using skewed variables, ARMA (1,1) models and shifting level models, support the interpretation that the Hurst phenomenon is a pre-asymptotic behavior. It was found in these simulations that: (1) skewness has some (but small) effect; (2) certain shifting level models and a suitable ARMA model having the same correlation structure yield a pre-asymptotic behavior of the expected rescaled range, similar to the Hurst phenomenon; and (3) re-analysis of the Hurst data suggests that this range is within the pre-asymptotic or transient region. Departure from normality and the dependence structure of series act, either individually or in combination, to further accentuate the transient behavior inherent to normal independent random variables.

Product Periodic Autoregressive Processes for Modeling Intermittent Monthly Streamflows

Several attempts have been made in the past to model hydrological processes such as monthly streamflows in dry regions. One of the crucial problems in modeling this type of process is the handling of zero flows. A stochastic model is presented herein which enables the reproduction of the percentage of zero flows in each month, the monthly mean and variance, and the month-to-month correlation of the intermittent flows. The model considers the intermittent monthly flow process as a product of a periodic binary discrete process and a periodic continuous process. Both the discrete and the continuous processes are periodic first-order autoregressive. Parameter estimation has been developed based on the method of moments, method of transition probability, and method of maximum likelihood.

ON THE EXPECTED RANGE AND EXPECTED ADJUSTED RANGE OF PARTIAL SUMS OF EXCHANGEABLE RANDOM VARIABLES

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables. In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed. EXPECTED ADJUSTED RANGE OF PARTIAL SUMS OF EXCHANGEABLE RANDOM VARIABLES; ADJUSTED RANGE IN STORAGE THEORY; HURST PHENOMENON; STABLE DISTRIBUTIONS

Regional flood frequency analysis based on a Weibull model : Part 2. Simulations and applications

Regional flood frequency analysis based on an index flood assumption and a two-parameter Weibull distribution was studied and simulation experiments were performed to compare the sample properties of quantile estimates based on the maximum likelihood (ML), moments (MOM), and probability weighted moments (PWM) methods, and to determine the applicability of the asymptotic variances of quantile estimators obtained for each method for finite samples. Results of these experiments showed biases and mean square errors vary with sample size, number of sites, nonexceedance probability, shape parameter, and estimation method. The ratio of the asymptotic variance (Avar) and the mean square error has been examined to see how well Avar represents the variability of quantile estimators for finite samples. In general, asymptotic formulas are quite good even for samples of size 25. The proposed regional model and estimation procedures are illustrated by analyzing some actual flood data from Illinois and Wisconsin. q 2001 Elsevier Science B.V. All rights reserved.

Regional flood frequency analysis based on a Weibull model: Part 1. Estimation and asymptotic variances

Abstract Parameter estimation in a regional flood frequency setting, based on a Weibull model, is revisited. A two parameter Weibull distribution at each site, with common shape parameter over sites that is rationalized by a flood index assumption, and with independence in space and time, is assumed. The estimation techniques of method of moments and method of probability weighted moments are studied by proposing a family of estimators for each technique and deriving the asymptotic variance of each estimator. Then a single estimator and its asymptotic variance for each technique, suggested by trying to minimize the asymptotic variance over the family of estimators, is obtained. These asymptotic variances are compared to the Cramer–Rao Lower Bound, which is known to be the asymptotic variance of the maximum likelihood estimator. A companion paper considers the application of this model and these estimation techniques to a real data set. It includes a simulation study designed to indicate the sample size required for compatibility of the asymptotic results to fixed sample sizes.

Modeling Road Racing Times of Competitive Recreational Runners Using Extreme Value Theory

Abstract Times of competitive recreational distance runners in a 10K road race can be modeled using extreme value theory. The derivative of the function fit to the times across age using this theory will express the rate of change in time per year that the runners are experiencing.