Richard W. Katz

Statistics of extremes in hydrology

The statistics of extremes have played an important role in engineering practice for water resources design and management. How recent developments in the statistical theory of extreme values can be applied to improve the rigor of hydrologic applications and to make such analyses more physically meaningful is the central theme of this paper. Such methodological developments primarily relate to maximum likelihood estimation in the presence of covariates, in combination with either the block maxima or peaks over threshold approaches. Topics that are treated include trends in hydrologic extremes, with the anticipated intensification of the hydrologic cycle as part of global climate change. In an attempt to link downscaling (i.e., relating large-scale atmosphere– ocean circulation to smaller-scale hydrologic variables) with the statistics of extremes, statistical downscaling of hydrologic extremes is considered. Future challenges are reviewed, such as the development of more rigorous statistical methodology for regional analysis of extremes, as well as the extension of Bayesian methods to more fully quantify uncertainty in extremal estimation. Examples include precipitation and streamflow extremes, as well as economic damage associated with such extreme events, with consideration of trends and dependence on patterns in atmosphere–ocean circulation (e.g., El Ni~ phenomenon). 2002 Elsevier Science Ltd. All rights reserved.

Extreme events in a changing climate: Variability is more important than averages

Extreme events act as a catalyst for concern about whether the climate is changing. Statistical theory for extremes is used to demonstrate that the frequency of such events is relatively more dependent on any changes in the variability (more generally, the scale parameter) than in the mean (more generally, the location parameter) of climate. Moreover, this sensitivity is relatively greater the more extreme the event. These results provide additional support for the conclusions that experiments using climate models need to be designed to detect changes in climate variability, and that policy analysis should not rely on scenarios of future climate involving only changes in means.

Extreme high-temperature events: changes in their probabilities with changes in mean temperature

Abstract Most climate impact studies rely on changes in means of meteorological variables, such as temperature, to estimate potential climate impacts, including effects on agricultural production. However, extreme meteorological events, say, a short period of abnormally high temperatures, can have a significant harmful effect on crop growth and final yield. The characteristics of daily temperature time series, specifically mean, variance and autocorrelation, are analyzed to determine possible ranges of probabilities of certain extreme temperature events [e.g., runs of consecutive daily maximum temperatures of at least 95°F (35°C)] with changes in mean temperature of the time series. The extreme temperature events considered are motivated primarily by agricultural concerns, particularly, the effects of high temperatures on corn yields in the U.S. Corn Belt. However, runs of high temperatures can also affect, for example, energy demand or morbidity and mortality of animals and humans. The relationships betw...

Time Series Models to Simulate and Forecast Wind Speed and Wind Power

Abstract A general approach for modeling wind speed and wind power is described. Because wind power is a function of wind speed, the methodology is based on the development of a model of wind speed. Values of wind power are estimated by applying the appropriate transformations to values of wind speed. The wind speed modeling approach takes into account several basic features of wind speed data, including autocorrelation, non-Gaussian distribution, and diurnal nonstationarity. The positive correlation between consecutive wind speed observations is taken into account by fitting an autoregressive process to wind speed data transformed to make their distribution approximately Gaussian and standardized to remove diurnal nonstationarity. As an example, the modeling approach is applied to a small set of hourly wind speed data from the Pacific Northwest. Use of the methodology for simulating and forecasting wind speed and wind power is discussed and an illustration of each of these types of applications is presen...

Precipitation as a chain-dependent process

Abstract A probabilistic model for the sequence of daily amounts of precipitation is proposed. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation. Methods are given for computing the distribution of the maximum amount of daily precipitation and the distribution of the total amount of precipitation. The application of this model is illustrated by an example, using State College, Pennsylvania, precipitation data.

Monitoring and Understanding Changes in Heat Waves, Cold Waves, Floods, and Droughts in the United States: State of Knowledge

Weather and climate extremes have been varying and changing on many different time scales. In recent decades, heat waves have generally become more frequent across the United States, while cold waves have been decreasing. While this is in keeping with expectations in a warming climate, it turns out that decadal variations in the number of U.S. heat and cold waves do not correlate well with the observed U.S. warming during the last century. Annual peak flow data reveal that river flooding trends on the century scale do not show uniform changes across the country. While flood magnitudes in the Southwest have been decreasing, flood magnitudes in the Northeast and north-central United States have been increasing. Confounding the analysis of trends in river flooding is multiyear and even multidecadal variability likely caused by both large-scale atmospheric circulation changes and basin-scale “memory” in the form of soil moisture. Droughts also have long-term trends as well as multiyear and decadal variability...

On Some Criteria for Estimating the Order of a Markov Chain

Tong (1975) has proposed a procedure for estimating the order of a Markov chain based on Akaikes information criterion (AIC). In this paper, the asymptotic distribution of the AIC estimator is derived and it is shown that the estimator is inconsistent. As an alternative to the AIC procedure, the Bayesian information criterion (BIC) proposed by Schwarz (1978) is shown to be consistent. These two procedures yield different estimated orders when applied to specific samples of meteorological observations. For parameters based on these meteorological examples, the AIC and BIC procedures are compared by means of simulation for finite samples. The results obtained have practical implications concerning whether, in the routine fitting of precipitation data, it is necessary to consider higher than first-order Markov chains.

Overdispersion Phenomenon in Stochastic Modeling of Precipitation

Abstract Simple stochastic models fit to time series of daily precipitation amount have a marked tendency to underestimate the observed (or interannual) variance of monthly (or seasonal) total precipitation. By considering extensions of one particular class of stochastic model known as a chain-dependent process, the extent to which this “overdispersion” phenomenon is attributable to an inadequate model for high-frequency variation of precipitation is examined. For daily precipitation amount in January at Chico, California, fitting more complex stochastic models greatly reduces the underestimation of the variance of monthly total precipitation. One source of overdispersion, the number of wet days, can be completely eliminated through the use of a higher-order Markov chain for daily precipitation occurrence. Nevertheless, some of the observed variance remains unexplained and could possibly be attributed to low-frequency variation (sometimes termed “potential predictability”). Of special interest is the fact...

Use of conditional stochastic models to generate climate change scenarios

Stochastic models have been proposed as one technique for generating scenarios of future climate change. One particular daily stochastic weather generator, termed Richardsons Model or WGEN, has received much attention. Because it is expressed in a conditional form convenient for simulation (e.g., temperature is modeled conditional on precipitation occurrence), some of its statistical characteristics are unclear. In the present paper, the theoretical statistical properties of a simplified version of Richardsons model are derived. These results establish that when its parameters are varied, certain unanticipated effects can be produced. For instance, modifying the probability of daily precipitation occurrence not only changes the mean of daily temperature, but its variance and autocorrelation as well. A prescription for how best to adjust these model parameters to obtain the desired climate changes is provided. Such precautions apply to conditional stochastic models more generally.

Extreme value theory for precipitation: Sensitivity analysis for climate change

Abstract Extreme value theory for the maximum of a time series of daily precipitation amount is described. A chain-dependent process is assumed as a stochastic model for daily precipitation, with the intensity distribution being the gamma. To examine how the effective return period for extreme high precipitation amounts would change as the parameters of the chain-dependent process change (i.e., probability of a wet day, shape and scale parameters of the gamma distribution), a sensitivity analysis is performed. This sensitivity analysis is guided by some results from statistical downscaling that relate patterns in large-scale atmospheric circulation to local precipitation, providing a physically plausible range of changes in the parameters. For the particular location considered in the example, the effective return period is most sensitive to the scale parameter of the intensity distribution.