A bootstrap approach to assess parameter uncertainty in simple catchment models

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Abstract

Catchment models simulate water and solute dynamics at catchment scales and are invaluable tools for natural resource management. Parameters for catchment models can provide useful information about the importance of the hydrological processes involved. We propose and demonstrate a bootstrap approach to assess parameter uncertainty in dynamic catchment models. This approach, which is non-Bayesian and essentially non-parametric, requires no distributional assumptions about parameters and only weak assumptions about the distributional form of the model residuals. It is able to handle autocorrelated model errors which are very common in the application of dynamic hydrological models at catchment scales. The ability of our bootstrap approach to assess parameter uncertainty is demonstrated using numerical experiments with the abc hydrological model and an application of a conceptual model of salt load from an irrigated catchment in southeastern Australia.

Introduction

Catchment models simulate water and solute dynamics at catchment scales and are invaluable tools for natural resource management. Parameters for catchment models can provide useful information about the importance of the hydrological processes involved. However, these parameters are typically not directly measurable and must be estimated indirectly from observed input and calibration data such as rainfall and pollutant loads. The derived parameter estimates only make sense under the assumption that the model provides an adequate representation of the hydrological processes. If the model is physically plausible, describes a substantial amount of the variation in the data, and no systematic departures of the model from the data can be detected, then this assumption is defensible. In order to interpret the estimates and to infer the importance of processes, parameter uncertainty then needs to be assessed. In such a situation, estimates of uncertainty of model parameters quantify the reliably of our hydrologic process understanding for the catchment under investigation. Uncertainty analysis of model parameters is usually considered to be one of the fundamental steps in the development and evaluation of models used for natural resource management (Jakeman et al., 2006). Therefore, parameter uncertainty in catchment modelling has received considerable attention over a long period of time (Beck, 1987, Jakeman and Hornberger, 1993, Gallagher and Doherty, 2007).

Catchment models are often dynamic in form and consequently, when fitted to data, model residuals (differences between observed and modelled data) are typically autocorrelated as various sources of model error may be propagated in time through internal model states (e.g soil water storage). While a number of methods have been proposed for assessing parameter uncertainty for these dynamic catchment models, the vast majority of these techniques rely on either parametric frequentist methods (e.g. Gallagher and Doherty, 2007 and references therein) or Bayesian methods (Kuczera, 1983, Beven and Binley, 1992, Kuczera and Parent, 1998, Yang et al., 2007) with often untestable distributional assumptions. Whilst the Bayesian approach has considerable modelling and computational flexibility (e.g. via Markov Chain Monte Carlo methods), it requires that fixed but unknown model parameters (constants) be regarded as random variables. This represents an arguable philosophic position that departs from the strict relative-frequency interpretation of probability. The Bayesian approach necessitates the (usually data-free) specification of prior distributions for model parameters. In addition, both Bayesian and parametric frequentist methods require the specification of the form of the error distribution for response variables. In contrast to these methods, non-parametric, or semi-parametric, bootstrap represents a frequentist approach that neither requires prior parameter distributions, nor specification of the form of the error distribution for response variables. It therefore involves fewer assumptions whilst guaranteeing a relevant error distribution that is taken directly from the data (or model residuals) (Efron and Tibshirani, 1993).

While bootstrap has been extensively used for time independent data with a wide range of documented applications in hydrology (e.g. Zucchini and Adamson, 1989, Schaap and Bouten, 1996, Parasuraman et al., 2007), its use for time dependent data appears to be quite narrow mostly focusing on generation of synthetic streamflow sequences that are used in simulation studies. For this purpose, sophisticated approaches have been developed (e.g. Lall and Sharma, 1996, Srinivas and Srinivasan, 2005). To the best of the authors’ knowledge, bootstrap has not been applied yet to assess parameter uncertainty for dynamic catchment models.

Our paper presents a bootstrap for assessing model parameter uncertainty which is able to handle autocorrelated errors commonly found in the application of dynamic catchment models (World Meteorological Organisation, 1992). The validity of our bootstrap approach to assess parameter uncertainty for catchment models is firstly demonstrated in a known controlled situation using a synthetic case study with the abc hydrological model (Fiering, 1967, p. 69). Secondly, its utility is described in a practical situation through application to a conceptual model of salt load from an irrigated catchment in southeastern Australia. By demonstrating the validity and utility of the proposed bootstrap for a practically important type of environmental models, i.e. dynamic catchment models, it is hoped that this paper will be useful to the broader environmental modelling community.

Section snippets

Bootstrap approach

The essential idea of the non-parametric bootstrap approach is that the data provide the best estimate of the distribution from which the data were drawn (Efron and Tibshirani, 1993). Pseudo-replicate samples (bootstrap samples) drawn at random with replacement from the data can be used to furnish information about the uncertainty of quantities estimated from the data. The model can be re-fitted to each bootstrap dataset, yielding ‘bootstrap estimates’ of model parameters. The process repeated

Numerical experiments with the abc hydrological model

We demonstrate the ability of model based bootstrap approach to assess parameter uncertainty of dynamic catchment models using numerical experiments with the abc hydrological model. Firstly, hypothetical streamflow data were generated using the abc hydrological model and a specified AR(1) normal error. Secondly, uncertainty (exact variance–covariance) of model parameters was determined using standard linear regression theory. Finally, parameter uncertainty was evaluated using model based

Conceptual model

The proposed bootstrap was applied to a conceptual model of salt load from an irrigated catchment in southeastern Australia which is a major contributor of salt load to the Murray River. The conceptual model represents, at a lumped catchment scale, monthly groundwater discharge into deep drains, which is the dominant process mobilising salt from the catchment. The parameters of this model provide information on groundwater recharge and discharge processes. For a detailed description of the

Concluding remarks

We proposed and demonstrated a bootstrap approach capable of assessing parameter uncertainty for dynamic catchment models. The proposed approach has fewer assumptions than Bayesian methods as the bootstrap neither requires prior parameter distributions, nor the error distribution specification for response variables. It rests on the classical frequentist framework, its strict relative-frequency interpretation of probability and definition of a random variable. It is non-parametric and robust

Acknowledgements

This work was funded by the Department of Primary Industries, the Department of Sustainability and Environment, North Central Catchment Management Authority and the Goulburn Broken Catchment Management Authority. We would like to thank Kym Butler (Department of Primary Industries) for his suggestions on the presentation of our bootstrap approach. We would like to thank Prof. Tony Jakeman and two anonymous reviewers for their valuable comments that have improved the presentation of our paper to

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