Quantifying predictive uncertainty of streamflow forecasts based on a Bayesian joint probability model
Introduction
Recent advances in hydrological modelling, weather forecasting, and hydro-climatic teleconnections have facilitated considerable improvements in streamflow forecasts (Cloke and Pappenberger, 2009, Wood et al., 2011, Schepen et al., 2012). The forecasts provide insightful information not only on hourly and daily streamflows, but also on monthly and seasonal streamflows (Maurer and Lettenmaier, 2003, Georgakakos et al., 2012a, Chen et al., 2014). As a result, streamflow forecasts have been incorporated into decision-making processes (Maurer and Lettenmaier, 2004, Georgakakos et al., 2012b, Zhao and Zhao, 2014). Operational systems for streamflow forecasting and water resources management have been established throughout the world, for example, in the Columbia River (Hamlet et al., 2002, Alemu et al., 2011), the Nile River (Block, 2011), and the Yangtze River (Kwon et al., 2009, Li et al., 2010). Adaptive management of water resources based on streamflow forecasts helps efficiently cope with operational risks due to streamflow variability and climate change (Vicuna et al., 2010, Georgakakos et al., 2012a, Georgakakos et al., 2012b).
Despite the great potential and wide use of streamflow forecasts, the uncertainty inherent in the forecasts has been a major obstacle to their applications (Maurer and Lettenmaier, 2004, You and Cai, 2008, Sankarasubramanian et al., 2009). Under-estimation of forecast uncertainty leads to operational risks, while over-estimation induces overly conservative decisions (Zhao et al., 2014). The utility of forecast information reduces as the magnitude of forecast uncertainty increases (Zhao et al., 2011, Zhao et al., 2013, Hejazi et al., 2014). Forecast information at long lead-times may be of little value to decision-makers due to the considerable forecast uncertainty (Zhao et al., 2012, Wang et al., 2014, Xu et al., 2014).
There are in general three approaches to estimating forecast uncertainty in hydrology (Montanari and Brath, 2004, Coccia and Todini, 2011). The first option is to formulate a probabilistic forecasting model that outputs a streamflow forecast along with its confidence intervals (Krzysztofowicz, 2001). For example, ensemble forecasts are generated from multiple initial conditions and hydrological forcing, and they contain multiple streamflow scenarios to represent future uncertainty (Cloke and Pappenberger, 2009). The second option is to estimate forecast uncertainty by analysing statistical properties of forecast errors (Wood and Schaake, 2008). A number of post-processing models have been proposed to infer forecast uncertainty based on archived samples of past streamflow forecasts and observations (e.g., Weerts et al., 2011, Pokhrel et al., 2013a, Pokhrel et al., 2013b). The third option is based on Monte Carlo methods that use simulation and re-sampling techniques (Montanari and Brath, 2004). In addition, expert information can also be incorporated into hydrological forecasting for empirical bias correction and uncertainty estimation (Wood and Schaake, 2008, Liersch and Volk, 2007, Pappenberger et al., 2013).
In practice, deterministic forecasts, which are periodically updated (Zhao et al., 2013), are often used for decision-making. Unfortunately, the uncertainty inherent in deterministic forecasts is not provided, limiting the power of its application. How can the corresponding forecast uncertainty be quantified? This study attempts to address this question by quantifying the predictive uncertainty based on a Bayesian joint probability model (BJP), through a case study of forecast data collected from the Three Gorges Reservoir. In comparison with traditional post-processing models which deal with forecast error between the original forecast and observation data, the BJP model employs data transformation and handles transformed data in its post-processing.
In statistics, when forecast and observation data are from non-Gaussian distributions, which is almost always the case with hydrological data, the distribution of forecast error would be more complicated and difficult to handle in model formulation. This problem is circumvented in BJP by transforming the distributions of original forecast and observation data to Gaussian before forming their association (correlation). Moreover, the Bayesian approach used in BJP produces samples of the model parameters and therefore make it easy to conduct predictive uncertainty analysis.
The BJP model was developed by Wang et al. (2009) and Wang and Robertson (2011) for seasonal streamflow forecasting. This model has since been used to calibrate precipitation and streamflow predictions and bridge the relationships between climatic indices and local precipitation and streamflow (e.g., Schepen et al., 2012, Robertson et al., 2013a, Robertson et al., 2013b, Bennett et al., 2014a, Bennett et al., 2014b). The former studies have applied the model to generate seasonal forecasts of precipitation and streamflow. Instead of making forecasts, this study explores the use of the BJP model in post-processing raw daily streamflow forecasts and estimating the predictive uncertainty. Focus is given to short-term forecast uncertainty modelling and the examination of the reliability of ensemble spread in capturing forecast uncertainty. As will be demonstrated later in this paper, the heteroscedasticity and non-Gaussianity of forecast uncertainty are effectively addressed.
The remainder of the paper is structured as follows. Section 2 illustrates the case under investigation with a focus on the observed characteristics of forecast uncertainty. Section 3 presents the BJP model, along with the prerequisite variance stabilizing transformation and the predictive uncertainty estimation procedures. Section 4 examines performances of the BJP model through a process of leave-one-year-out cross validation. Section 5 discusses results and concludes the study.
Section snippets
Case study and data description
The predictive uncertainty of real-time streamflow forecast of inflow to the Three Gorges Reservoir is analysed in this study. The Three Gorges Reservoir is one of the largest reservoirs in the world. The reservoir controls floods from 56% of the drainage area of the Yangtze River. A deterministic forecasting system exists to aid reservoir operations (Li et al., 2010, Zhao et al., 2013). Forecasts are based on two upstream streamflow gauge stations (Cuntan Station on the mainstream of Yangtze
Methods
The BJP model characterizes the relationship between predictors and predictand using a joint multivariate Gaussian distribution after transformation. In this study, the predictor is the deterministic streamflow forecast generated by the Three Gorges operational streamflow forecasting system and the predictand is the observed streamflow. Therefore, the relationship between raw forecast and observed streamflow, which indicates forecast uncertainty, is formulated. In the BJP model, the log–sinh
Results
The BJP-based predictive uncertainty estimation is evaluated through leave-one-year-out cross validation. There are in total 6 ∗ 92 = 552 data points. The forecast-observation pairs in one year are selected as testing data [] (n = 1, 2, …, 92), and the pairs in the other five years as training data [] (m = 1, 2, …, 460). The validation is conducted from 2004 to 2009 and the results are pooled to perform the evaluation.
Summary and conclusions
Uncertainty is an inherent part of any streamflow forecast. It is an important determinant of the utility of forecasts for water resources management. This study addresses the problem of estimating the uncertainty of streamflow forecasts for the Three Gorges Reservoir. A BJP model is set up to post-process deterministic forecasts to quantify predictive uncertainty. The parametric variance-stabilizing log–sinh transformation is employed to deal with heteroscedasticity and to normalize
Acknowledgement
This study is partially supported by NSFC (51409145) and MSTC (2013BAB05B03).
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