Jose D. Salas

Nonlinear dynamics, delay times, and embedding windows

In order to construct an embedding of a nonlinear time series, one must choose an appropriate delay time d. Often, d is estimated using the autocorrelation function; however, this does not treat the nonlinearity appropriately, and it may yield an incorrect value for d. On the other hand, the correct value of d can be found from the mutual information, but this process is rather cumbersome computationally. Here, we suggest a simpler method for estimating d using the correlation integral. We call this the C‐C method, and we test it on several nonlinear time series, obtaining estimates of d in agreement with those obtained using the mutual information. Furthermore, some researchers have suggested that one should not choose a fixed delay time d, independent of the embedding dimension m, but, rather, one should choose an appropriate value for the delay time window w D .m 1/ , which is the total time spanned by the components of each embedded point. Unfortunately, w cannot be estimated using the autocorrelation function or the mutual information, and no standard procedure for estimating w has emerged. However, we show that the C‐C method can also be used to estimate w. Basically w is the optimal time for independence of the data, while d is the first locally optimal time. As tests, we apply the C‐C method to the Lorenz system, a three-dimensional irrational torus, the Rossler system, and the Rabinovich‐Fabrikant system. We also demonstrate the robustness of this method to the presence of noise. c 1999 Elsevier Science B.V. All rights reserved.

Revisiting the Concepts of Return Period and Risk for Nonstationary Hydrologic Extreme Events

Current practice using probabilistic methods applied for designing hydraulic structures generally assume that extreme events are stationary. However, many studies in the past decades have shown that hydrological records exhibit some type of nonstationarity such as trends and shifts. Human intervention in river basins (e.g., urbanization), the effect of low-frequency climatic variability (e.g., Pacific Decadal Oscillation), and climate change due to increased greenhouse gasses in the atmosphere have been suggested to be the leading causes of changes in the hydrologic cycle of river basins in addition to changes in the magnitude and frequency of extreme floods and extreme sea levels. To tackle nonstationarity in hydrologic extremes, several approaches have been proposed in the literature such as frequency analysis, in which the parameters of a given model vary in accordance with time. The aim of this paper is to show that some basic concepts and methods used in designing flood-related hydraulic structures assuming a stationary world can be extended into a nonstationary frame- work. In particular, the concepts of return period and risk are formulated by extending the geometric distribution to allow for changing exceeding probabilities over time. Building on previous developments suggested in the statistical and climate change literature, the writers present a simple and unified framework to estimate the return period and risk for nonstationary hydrologic events along with examples and applications so that it can be accessible to a broad audience in the field. The applications demonstrate that the return period and risk estimates for nonstationary situations can be quite different than those corresponding to stationary conditions. They also suggest that the nonstationary analysis can be helpful in making an appropriate assessment of the risk of a hydraulic structure during the planned project-life. DOI: 10.1061/ (ASCE)HE.1943-5584.0000820.

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.

Forecasting of short-term rainfall using ARMA models

Abstract Flood forecasting depends essentially on forecasting of rainfall or snow melt. In this paper, rainfall forecasting is approached assuming that hourly rainfall follows an autoregressive moving average (ARMA) process. This assumption is based on the fact that the autocovariance structure of some point processes, such as hourly rainfall processes, is equivalent to the autocovariance structure of certain low-order ARMA processes. Two estimation and fitting procedures are investigated. The first takes all rainfall occurrences throughout the period of record as the basis for parameter estimation, and the second is an event-based adaptive procedure. These procedures are compared for rainfall data at a point and rainfall data averaged over a basin. Hourly rainfall from two gaging stations in Colorado, USA, and from several stations in Central Italy are used. Results show that the event-based estimation approach yields better forecasts.

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.

Streamflow Forecasting Based on Artificial Neural Networks

Streamflow forecasting is an important component of water resource system control and a challenging task for water resources engineers and managers. Good streamflow forecasts enables an efficient operation of water resources systems within technical, economical, legal, and political priorities. A forecasting system that accounts for all significant temporal and spatial variability of the entire streamflow field, provides a good basis for proper control and management of the water resources system. Streamflow forecasts, usually with lead times of hours and days, are often used for flood warning purposes and for real time operation of water resources systems. Forecasts, with lead times ranging from weeks to months, are generally used for water system planning and management, such as allocation of irrigation water, hydropower planning, and drought analysis and mitigation.

Nonparametric Simulation of Single-Site Seasonal Streamflows

Various parametric and nonparametric models have been suggested in literature for stochastic generation of seasonal streamflows. State-of-the-art nonparametric models are reviewed herein and their drawbacks identified. We developed a simple model that employs the k-nearest neighbor resampling algorithm with gamma kernel perturbation (denoted as KGK model), which enables generation of data that are not the same as the historical data. For preserving the annual variability two approaches are developed. The first one employs the aggregate variable concept (KGKA model), and the second one uses a pilot variable that leads the generation of the seasonal data (KGKP model). The pilot variable refers to the annual data that has been previously generated, but its role is not for disaggregation but rather for conditioning the state that guides the generation of seasonal flows. The proposed models have been compared with a currently available nonparametric model that considers the reproduction of the interannual vari...

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.