Eric Parent

Monte Carlo assessment of parameter uncertainty in conceptual catchment models : the Metropolis algorithm

Abstract Two Monte Carlo-based approaches for assessing parameter uncertainty in complex hydrologic models are considered. The first, known as importance sampling, has been implemented in the generalised likelihood uncertainty estimation (GLUE) framework of Beven and Binley. The second, known as the Metropolis algorithm, differs from importance sampling in that it uses a random walk that adapts to the true probability distribution describing parameter uncertainty. Three case studies are used to investigate and illustrate these Monte Carlo approaches. The first considers a simple water balance model for which exact results are known. It is shown that importance sampling is inferior to Metropolis sampling. Unless a large number of random samples are drawn, importance sampling can produce seriously misleading results. The second and third case studies consider more complex catchment models to illustrate the insights the Metropolis algorithm can offer. They demonstrate assessment of parameter uncertainty in the presence of bimodality, evaluation of the significance of split-sample tests, use of prior information and the assessment of confidence limits on hydrologic responses not used in calibration. When compared with the capabilities of traditional inference based on first-order approximation, the Metropolis algorithm provides a quantum advance in our capability to deal with parameter uncertainty in hydrologic models.

Bayesian change-point analysis in hydrometeorological time series. Part 1. The normal model revisited

A Bayesian method is presented for the analysis of two types of sudden change at an unknown time-point in a sequence of energy inflows modeled by independent normal random variables. First, the case of a single shift in the mean level is revisited to show how such a problem can be straightforwardly addressed through the Bayesian framework. Second, a change in variability is investigated. In hydrology, to our knowledge, this problem has not been studied from a Bayesian perspective. Even if this model is quite simple, no analytic solutions for parameter inference are available, and recourse to approximations is needed. It is shown that the Gibbs sampler is particularly suitable for change-point analysis, and this Markovian updating scheme is used. Finally, a case study involving annual energy inflows of two large hydropower systems managed by Hydro-Quebec is presented in which informative prior distributions are specified from regional information.

Bayesian change-point analysis in hydrometeorological time series. Part 2. Comparison of change-point models and forecasting

This paper provides a methodology to test existence, type, and strength of changes in the distribution of a sequence of hydrometeorological random variables. Unlike most published work on change-point analysis, which consider a single structure of change occurring with certainty, it allows for the consideration in the inference process of the no change hypothesis and various possible situations that may occur. The approach is based on Bayesian model selection and is illustrated using univariate normal models. Four univariate normal models are considered: the no change hypothesis, a single change in the mean level only, a single change in the variance only, and a simultaneous change in both the mean and the variance. First, inference analysis of posterior distributions via Gibbs sampling for a given change-point model is recalled. This scientific reporting framework is then generalized to the problem of selecting among different configurations of a single change and the no change hypothesis. The important operational issue of forecasting a future observation, often neglected in the literature on change-point analysis, is also treated in the previous model selection perspective. To illustrate the approach, a case study involving annual energy inflows for eight large hydropower systems situated in Quebec is detailed.

Bayesian POT modeling for historical data

Abstract When designing hydraulic structures, civil engineers have to evaluate design floods, i.e. events generally much rarer that the ones that have already been systematically recorded. To extrapolate towards extreme value events, taking advantage of further information such as historical data, has been an early concern among hydrologists. Most methods described in the hydrological literature are designed from a frequentist interpretation of probabilities, although such probabilities are commonly interpreted as subjective decisional bets by the end user. This paper adopts a Bayesian setting to deal with the classical Poisson–Pareto peak over treshold (POT) model when a sample of historical data is available. Direct probalistic statements can be made about the unknown parameters, thus improving communication with decision makers. On the Garonne case study, we point out that twelve historical events, however imprecise they might be, greatly reduce uncertainty. The 90% credible interval for the 1000 year flood becomes 40% smaller when taking into account historical data. Any kind of uncertainty (model uncertainty, imprecise range for historical events, missing data) can be incorporated into the decision analysis. Tractable and versatile data augmentation algorithms are implemented by Monte Carlo Markov Chain tools. Advantage is taken from a semi-conjugate prior, flexible enough to elicit expert knowledge about extreme behavior of the river flows. The data augmentation algorithm allows to deal with imprecise historical data in the POT model. A direct hydrological meaning is given to the latent variables, which are the Bayesian keytool to model unobserved past floods in the historical series.

Encoding prior experts judgments to improve risk analysis of extreme hydrological events via POT modeling

Abstract One of the main decisions to be made in operational hydrology is to estimate design floods for safety purposes. These floods are generally much rare events that have already been systematically recorded and consequently the results of any estimation process are subject to high levels of uncertainty. When adopting the frequentist framework of probability, the so called ‘respect of scientific objectivity’ shall forbid the hydrologists to introduce prior knowledge such as quantified hydrological expertise into the analysis. However, such an expertise can significantly improve the capability of a probabilistic model to extrapolate extreme value events. The Bayesian paradigm offers coherent tools to quantify the prior knowledge of experts. This paper develops an inference procedure for the peak over threshold (POT) model, using semi-conjugate informative priors. Such prior structures are convenient to encode a wide variety of prior expertise. They avoid recourse to Monte Carlo Markov Chain techniques which are presently the standard for Bayesian analyses, but such algorithms may be uneasy to implement. We show that prior expertise can significantly reduce uncertainty on design values. Using the Garonne case study with a sample of systematic data spanning over the period 1913–1977, we point out that: (1) the elicitation approach for subjective prior information can be based on quantities with a definite practical hydrological meaning for the expert; (2) with respect to the usual Poisson–Generalized Pareto model, a semi-conjugate prior offers a flexible structure to assess expert knowledge about extreme behavior of the river flows. In addition, it leads to quasi-analytical formulations; (3) tractable algorithms can be implemented to approximate the prior uncertainty about POT parameters into these semi conjugate distribution forms via simple Monte Carlo simulations and normal approximations; (4) the design value and its credible interval are notably changed when incorporating prior knowledge into the risk analysis.

BAYESIAN ANALYSIS OF A SIMULTANEOUS CHANGE IN BOTH THE MEAN AND THE VARIANCE

Although such an hypothesis is rarely stated explicitly, the assumption that stochastic time series are stationary plays a crucial role in water resources management. Under the assumption that tomorrow will statistically behave like yesterday, stochastic models are fitted to hydrometeorological variables such as river flow, precipitation and temperature. The estimated models are then used for many engineering purposes, in particular for simulating the operation of hydropower systems (energy planning, design of power plants, operation of reservoirs). Consequently such models, and the decisions stemming from them, are based on the assumption of stationary behavior of hydrometeorological inputs. However, many climatologists believe that there is more and more evidence that serious changes have occurred in many parts of the world, which leads to questioning the stationarity hypothesis in hydrometeorological time series analysis.