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Probabilistic post-processing models for flow forecasts for a system of catchments and several lead times
Water Resources Research
Abstract
This paper introduces a methodology for the construction of probabilistic inflow forecasts for multiple catchments and lead times. A post-processing approach is used, and a Gaussian model is applied for transformed variables. In operational situations it is a straightforward task to use the models to sample inflow ensembles which inherit the dependencies between catchments and lead times. The methodology was tested and demonstrated in the river systems linked to the Ulla-Førre hydropower complex in southern Norway, where simultaneous probabilistic forecasts for five catchments and ten lead times were constructed. The methodology exhibits sufficient flexibility to utilise deterministic flow forecasts from a numerical hydrological model as well as statistical forecasts such as persistent forecasts and sliding window climatology forecasts. It also deals with variation in the relative weights of these forecasts with both catchment and lead time. When evaluating predictive performance in original space using cross-validation, the case study found that it is important to include the persistent forecast for the initial lead times and the hydrological forecast for medium-term lead times. Sliding window climatology forecasts become more important for the latest lead times. Furthermore, operationally important features in this case study such as heteroscedasticity, lead time varying between lead time dependency and lead time varying between catchment dependency are captured.
... This produces univariate PDFs that are marginally calibrated but do not preserve any spatial or temporal (between lead times) dependencies (Schefzik et al., 2013). In turn, postprocessed predictive distributions should be also given in the form of streamflow ensemble traces to allow their use as inputs for decision support systems, including optimization models for reservoir management, waterway operations, and detailed hydrodynamic models for the prediction of flooded areas (Bellier et al., 2018;Engeland and Steinsland, 2014;Hemri et al., 2015;Herr and Krzysztofowicz, 2010). Realistic forecast trajectories, that is, time series that have correct autocorrelation and are unbiased (Bennett et al., 2016), can be drawn from predictive distributions by using multivariate postprocessing methods that account for dependence structures, such as empirical methods that follow reordering notions like the Schaake Shuffle (Clark et al., 2004) and Ensemble Copula Coupling (ECC) (Schefzik et al., 2013). ...
... Therefore, hydrological forecasts and corresponding observations must be transformed to make the distributions as nearly normal as possible. Examples of commonly used transformations in the topic of hydrological postprocessing are the normal quantile transform (e.g., Bellier et al., 2018;Verkade et al., 2017;Weerts et al., 2011), log-sinh (e.g., Pagano et al., 2013;Wang et al., 2012) and the Box-Cox power transformation (e.g., Baran et al., 2019;Duan et al., 2007;Engeland and Steinsland, 2014;Hemri and Klein, 2017;Hemri et al., 2015;Qu et al., 2017). Here the Box-Cox approach was applied according to: ...
... x is the observed/ forecasted streamflow; and λ is the Box-Cox parameter. The optimal λ value for each location can be found through a grid search with a pre-defined range, selecting the value that minimizes the Kolmogorov-Smirnov statistics (e.g., Duan et al., 2007;Hemri et al., 2015), the Shapiro-Wilk test (e.g., Engeland and Steinsland, 2014) or the CRPS value (e.g., Baran et al., 2019;Hemri and Klein, 2017). However, performing such a grid search in a continental domain that comprises a large number of locations is computationally expensive. ...
Probabilistic hydrological forecasting and ensemble techniques have leveraged streamflow prediction at regional to continental scales up to several weeks in advance. However, ensembles that only account for meteorological forecast uncertainty are typically biased and subject to dispersion errors, thus limiting their use for rational decision-making and optimization systems. Statistical postprocessing is therefore necessary to convert ensemble forecasts into calibrated and sharp predictive distributions, and it should also account for dependencies between lead times to enable realistic forecast trajectories. This work provides a continental-scale assessment of the use of statistical postprocessing on medium-range, ensemble streamflow forecasts over South America (SA). These forecasts were produced through a large-scale hydrologic–hydrodynamic model forced with a global precipitation dataset and ECMWF reforecast data. The Ensemble Model Output Statistics (EMOS) technique was used to generate conditional predictive distributions in 488 locations at each forecast lead time, while the Ensemble Copula Coupling method with the transformation scheme (ECC-T) was applied to derive ensemble traces from EMOS distributions. Postprocessed streamflow forecasts were cross-validated for the period from 1996 to 2014 using a range of verification metrics. Results showed that the skill and reliability of EMOS forecasts substantially improve over the raw ensembles, and that EMOS leads to skillful predictions relative to discharge climatology and persistence forecasts up to 15 days in advance in most locations. Furthermore, EMOS results in predictive distributions that are generally sharper than the climatology. Limitations in depicting autocorrelations of forecast trajectories were observed in rivers for which the raw ensemble spread is very low and EMOS has to largely increase dispersion, especially at short lead times. The study’s findings suggest that combining a continental-scale hydrological model with EMOS and ECC-T methods can lead to skillful predictions and realistic ensemble traces in several locations in SA, even if in situ hydrometeorological observations are not available in real time.
... These models account jointly for both rainfall and hydrological uncertainty. Examples of forecast-rainfall-based QPP models include the multisite short term (1-10 days) forecasting model of Engeland and Steinsland (2014) and the monthly/seasonal forecasting model of the Australian Bureau of Meteorology (Tuteja et al., 2011;Woldemeskel et al., 2018); 2. Observed-rainfall-based QPP models are developed and calibrated using hydrological model simulations forced with observed rainfall. As these models account only for hydrological uncertainty, a sequential approach is then used to quantify total forecast uncertainty, as follows (i) forecast rainfall uncertainty is represented by propagating rainfall replicates through the hydrological model and (ii) hydrological uncertainty is represented by applying the QPP model to each streamflow replicate. ...
... Water Resources Research (Perrin et al., 2003) used in this study, the contribution of parametric uncertainty to total uncertainty is generally small (Kavetski, 2018;McInerney et al., 2018;Sun et al., 2017). For this reason, forecasting methods typically neglect parametric uncertainty and focus on forecast rainfall and hydrological uncertainty (e.g., Engeland & Steinsland, 2014;Verkade et al., 2017). ...
... We construct a climatological distribution using a sliding window approach, similar to the approach used by Engeland and Steinsland (2014). Consider a time series of known (observed or computed) daily values, e w ¼ e w t ; t ¼ 1; …; N w ð Þ . ...
Subseasonal streamflow forecasts, with lead times of 1–30 days, provide valuable information for operational water resource management. This paper introduces the multi‐temporal hydrological residual error (MuTHRE) model to address the challenge of obtaining “seamless” subseasonal forecasts — that is, daily forecasts with consistent high‐quality performance over multiple lead times (1–30 days) and aggregation scales (daily to monthly). The key advance of the MuTHRE model is combining the representation of three temporal characteristics of hydrological residual errors: seasonality, dynamic biases, and non‐Gaussian errors. The MuTHRE model is applied in 11 Australian catchments using the hydrological model GR4J and post processed rainfall forecasts from the numerical weather prediction model ACCESS‐S, and is evaluated against a baseline model that does not model these error characteristics. The MuTHRE model provides “high” improvements (practically significant in the majority of performance stratifications) in terms of reliability: (i) at short lead times (up to 10 days), due to representing non‐Gaussian errors, (ii) stratified by month, due to representing seasonality in hydrological errors, and (iii) in dry years, due to representing dynamic biases in hydrological errors. Forecast performance also improves in terms of sharpness, volumetric bias, and CRPS skill score; these improvements are statistically but not practically significant in the majority of stratifications. Importantly, improvements are consistent across multiple time scales (daily and monthly). This study highlights the benefits of modeling multiple temporal characteristics of hydrological errors and demonstrates the power of the MuTHRE model for producing seamless subseasonal streamflow forecasts that can be utilized for a wide range of applications.
... The problem of lead-time dependence has not been considered much in the postprocessing literature. Pinson and Girard (2012); Hemri et al. (2013Hemri et al. ( , 2015, and Engeland and Steinsland (2014) are interested in the temporal dependence structure over lead times for applications to wind speed and hydrological predictions. Hemri et al. (2015) fit separate models for different lead times and then smooth postprocessing parameters using cyclic splines. ...
... Box plot of (a) energy and (b) p-Variogram score for multivariate wind-speed predictions (at all lead times) for lead-time-continuous (C-models, blue) and lead-time-separated EMOS (S-models, orange) for both models trained in a running window (RWIN-models, left) and ones including seasonality adjustments within the model (SWM-models, right). 2015; Engeland and Steinsland, 2014). Figure 15 shows the distribution across locations of the energy and p-Variogram scores for the multivariate vector of postprocessed wind-speed predictions issued at a common date. ...
Numerical weather prediction (NWP) ensembles often exhibit biases and errors in dispersion, so they need some form of postprocessing to yield sharp and well‐calibrated probabilistic predictions. The output of NWP models is usually at a multiplicity of different lead times and, even though information is often required on this range of lead times, many postprocessing methods in the literature are applied either at a fixed lead time or by fitting individual models for each lead time. However, this is (1) computationally expensive because it requires the training of multiple models if users are interested in information at multiple lead times and (2) prohibitive because it restricts the data used for training postprocessing models and the usability of fitted models. This article investigates the lead‐time dependence of postprocessing methods in the idealized Lorenz'96 system as well as temperature and wind‐speed forecast data from the Met Office Global and Regional Ensemble Prediction System (MOGREPS‐G). The results indicate that there is substantial regularity between the models fitted for different lead times and that one can fit models that are lead‐time‐continuous that work for multiple lead times simultaneously by including lead time as a covariate. These models achieve similar and, in small data situations, even improved performance compared with the classical lead‐time‐separated models, whilst saving substantial computation time.
... This correlation is likely to be higher for runoff than for meteorological variables. Engeland and Steinsland (2014) presented a method which would fit different weights to different locations and lead times, but still assuming the same number of forecasts for all lead times. ...
... Additionally, not all ensembles are available for all lead times, and we would prefer a method which will assure temporal continuity between lead times. The data set is considerably larger than in previous studies, so we do for example not see the method of Engeland and Steinsland (2014) as feasible. ...
For many years, meteorological models have been run with perturbated initial
conditions or parameters to produce ensemble forecasts that are used as a
proxy of the uncertainty of the forecasts. However, the ensembles are usually
both biased (the mean is systematically too high or too low, compared with
the observed weather), and has dispersion errors (the ensemble variance
indicates a too low or too high confidence in the forecast, compared with the
observed weather). The ensembles are therefore commonly post-processed to
correct for these shortcomings. Here we look at one of these techniques,
referred to as Ensemble Model Output Statistics (EMOS) (Gneiting et al.,
2005). Originally, the post-processing parameters were identified as a fixed
set of parameters for a region. The application of our work is the European
Flood Awareness System (http://www.efas.eu), where a distributed model
is run with meteorological ensembles as input. We are therefore dealing with
a considerably larger data set than previous analyses. We also want to
regionalize the parameters themselves for other locations than the
calibration gauges. The post-processing parameters are therefore estimated
for each calibration station, but with a spatial penalty for deviations from
neighbouring stations, depending on the expected semivariance between the
calibration catchment and these stations. The estimated post-processed
parameters can then be used for regionalization of the postprocessing
parameters also for uncalibrated locations using top-kriging in the
rtop-package (Skøien et al., 2006, 2014). We will show results from
cross-validation of the methodology and although our interest is mainly in
identifying exceedance probabilities for certain return levels, we will also
show how the rtop package can be used for creating a set of post-processed
ensembles through simulations.
... The errors of a hydrological model are known to be heteroscedastic, have a skewed distribution, and to be dependent (e.g. Engeland and Steinsland, 2014). To account for heteroscedasticity and skewedness a Box-Cox transformation with parameter k was used for both the predicted and true streamflow. ...
... Based on previous analysis (Engeland and Steinsland, 2014) the Box-Cox parameter k was set to 0.2. A Shapiro-Wilks test showed that the transformed observed streamflow y BC then is close to Gaussian distributed. ...
In this study, we explore the effect of uncertainty and poor observation quality on hydrological model calibration and predictions. The Osali catchment in Western Norway was selected as case study and an elevation distributed HBV-model was used. We systematically evaluated the effect of accounting for uncertainty in parameters, precipitation input, temperature input and streamflow observations. For precipitation and temperature we accounted for the interpolation uncertainty, and for streamflow we accounted for rating curve uncertainty. Further, the effects of poorer quality of precipitation input and streamflow observations were explored. Less information about precipitation was obtained by excluding the nearest precipitation station from the analysis, while reduced information about the streamflow was obtained by omitting the highest and lowest streamflow observations when estimating the rating curve. The results showed that including uncertainty in the precipitation and temperature inputs has a negligible effect on the posterior distribution of parameters and for the Nash–Sutcliffe (NS) efficiency for the predicted flows, while the reliability and the continuous rank probability score (CRPS) improves. Less information in precipitation input resulted in a shift in the water balance parameter Pcorr, a model producing smoother streamflow predictions, giving poorer NS and CRPS, but higher reliability. The effect of calibrating the hydrological model using streamflow observations based on different rating curves is mainly seen as variability in the water balance parameter Pcorr. When evaluating predictions, the best evaluation scores were not achieved for the rating curve used for calibration, but for rating curves giving smoother streamflow observations. Less information in streamflow influenced the water balance parameter Pcorr, and increased the spread in evaluation scores by giving both better and worse scores.
... However, current techniques are generally too computationally expensive for operational river flow forecasting applications (Emerton et al., 2016). For example, defining a joint distribution between the river discharge at multiple locations would allow 40 forecasts to be conditioned on observations available at specific locations (Engeland and Steinsland, 2014). However, for largescale distributed systems and multiple lead-times the size of the joint distribution quickly becomes too large. ...
Forecasting river discharge is essential for disaster risk reduction and water resource management, but forecasts of future river state often contain errors. Post-processing reduces forecast errors but is usually only applied at the locations of river gauges, leaving the majority of the river network uncorrected. Here, we present a data-assimilation-inspired method for error-correcting ensemble simulations across gauged and ungauged locations in a post-processing step. Our new method employs state augmentation within the framework of the Localised Ensemble Transform Kalman Filter (LETKF) to estimate an error vector for each ensemble member. The LETKF uses ensemble error covariances to spread observational information from gauged to ungauged locations in a dynamic and computationally efficent manner. To improve the efficiency of the LETKF we define new localisation, covariance inflation, and initial ensemble generation techniques that can be easily transferred between modelling systems and river catchments. We implement and evaluate our new error-correction method for the entire Rhine-Meuse catchment using forecasts from the Copernicus Emergency Management Service's European Flood Awareness System (EFAS). The resulting river discharge ensembles are error-corrected at every grid box but remain spatially and temporally consistent. The skill is evaluated at 89 proxy-ungauged locations to assess the ability of the method to spread the correction along the river network. The skill of the ensemble mean is improved at almost all locations including stations both up- and downstream of the assimilated observations. Whilst the ensemble spread is improved at short lead-times, at longer lead-times the ensemble spread is too large leading to an underconfident ensemble. In summary, our method successfully propagates error information along the river network, enabling error correction at ungauged locations. This technique can be used for improved post-event analysis and can be developed further to post-process operational forecasts providing more accurate knowledge about the future states of rivers.
... Hydrological model parameters are estimated using likelihood maximisation based on the BC0.2 error model (McInerney et al., 2020), implemented using a quasi-Newton optimisation algorithm run with 100 independent multistarts Note that, in this work, we do not consider parametric uncertainty (in the hydrological and residual error models), which is expected to be a (relatively) minor contributor to total forecast uncertainty, given the long data period used in the estimation; this simplification is common in contemporary forecasting implementations (e.g. Engeland and Steinsland, 2014;Verkade et al., 2017). ...
Subseasonal streamflow forecasts inform a multitude of water management decisions, from early flood warning to reservoir operation. Seamless forecasts, i.e. forecasts that are reliable and sharp over a range of lead times (1–30 d) and aggregation timescales (e.g. daily to monthly) are of clear practical interest. However, existing forecast products are often non-seamless, i.e. developed and applied for a single timescale and lead time (e.g. 1 month ahead). If seamless forecasts are to be a viable replacement for existing non-seamless forecasts, it is important that they offer (at least) similar predictive performance at the timescale of the non-seamless forecast.
This study compares forecasts from two probabilistic streamflow post-processing (QPP) models, namely the recently developed seamless daily Multi-Temporal Hydrological Residual Error (MuTHRE) model and the more traditional (non-seamless) monthly QPP model used in the Australian Bureau of Meteorology's dynamic forecasting system. Streamflow forecasts from both post-processing models are generated for 11 Australian catchments, using the GR4J hydrological model and pre-processed rainfall forecasts from the Australian Community Climate and Earth System Simulator – Seasonal (ACCESS-S) numerical weather prediction model. Evaluating monthly forecasts with key performance metrics (reliability, sharpness, bias, and continuous ranked probability score skill score), we find that the seamless MuTHRE model achieves essentially the same performance as the non-seamless monthly QPP model for the vast majority of metrics and temporal stratifications (months and years). As such, MuTHRE provides the capability of seamless daily streamflow forecasts with no loss of performance at the monthly scale – the modeller can proverbially “have their cake and eat it too”. This finding demonstrates that seamless forecasting technologies, such as the MuTHRE post-processing model, are not only viable but also a preferred choice for future research development and practical adoption in streamflow forecasting.
... As a further investigation, to validate the rationality of the proposed approach, we use Q-Q plot as a graphical tool to illustrate the reliability of the forecast [85]. If the forecast system is statistically reliable, the forecast falls inside the prediction interval [86]. As depicted in Fig. 17, the error is falling around the straight line which shows that the error between the observations and the estimations follows a normal distribution, and thus shows the suitability of our approach. ...
With increasing concerns of climate change, renewable resources such as photovoltaic (PV) have gained popularity as a means of energy generation. The smooth integration of such resources in power system operations is enabled by accurate forecasting mechanisms that address their inherent intermittency and variability. This paper proposes a novel probabilistic framework to predict short-term PV output taking into account the variability of weather data over different seasons. To this end, we go beyond existing prediction methods, building a pipeline of processes, i.e., feature selection, clustering and Gaussian Process Regression (GPR). We make use of datasets that comprise of power output and meteorological data such as irradiance, temperature, zenith, and azimuth. First, a correlation study is performed to select the weather features which affect solar output to a greater extent. Next, we categorise the data into four groups based on solar output and time by using k-means clustering. Finally, we determine a function that relates the aforementioned selected features with solar output by using GPR and Matérn 5/2 as a kernel function. We validate our method with five solar generation plants in different locations and compare the results with existing methodologies. More specifically, in order to test the proposed model, two different methods are used: (i) a 5-fold cross validation; and (ii) holding out 30 random days as test data. To confirm the model accuracy, we apply our framework 30 independent times on each of the four clusters. The average error follows a normal distribution, and with 95% confidence level, it takes values between −1.6% and 1.4%. The proposed framework decreases the normalised root mean square error and mean absolute error by 54.6% and 55.5%, respectively, when compared with other relevant works.
... As a further investigation, to validate the rationality of the proposed approach, we use Q-Q plot as a graphical tool to illustrate the reliability of the forecast [85]. If the forecast system is statistically reliable, the forecast falls inside the prediction interval [86]. As depicted in Fig. 17, the error is falling around the straight line which shows that the error between the observations and the estimations follows a normal distribution, and thus shows the suitability of our approach. ...
With increasing concerns of climate change, renewable resources such as photovoltaic (PV) have gained popularity as a means of energy generation. The smooth integration of such resources in power system operations is enabled by accurate forecasting mechanisms that address their inherent intermittency and variability. This paper proposes a probabilistic framework to predict short-term PV output taking into account the uncertainty of weather. To this end, we make use of datasets that comprise of power output and meteorological data such as irradiance, temperature, zenith, and azimuth. First, we categorise the data into four groups based on solar output and time by using k-means clustering. Next, a correlation study is performed to choose the weather features which affect solar output to a greater extent. Finally, we determine a function that relates the aforementioned selected features with solar output by using Gaussian Process Regression and Matern 5/2 as a kernel function. We validate our method with five solar generation plants in different locations and compare the results with existing methodologies. More specifically, in order to test the proposed model, two different methods are used: (i) 5-fold cross-validation; and (ii) holding out 30 random days as test data. To confirm the model accuracy, we apply our framework 30 independent times on each of the four clusters. The average error follows a normal distribution, and with 95% confidence level, it takes values between -1.6% to 1.4%.
... One of the primary techniques to reflect different uncertainties in hydrologic forecasts is to create a probabilistic forecast [10,11]. Probabilistic forecasts can be made using three approaches: a probabilistic pre-processing approach plus a deterministic forecast model; a probabilistic forecast model; and a deterministic forecast model plus a probabilistic post-processing approach [12][13][14]. The first two approaches quantify uncertainties in inputs and model structure while the third quantifies the overall uncertainty in model structure and parameters. ...
It is fundamentally challenging to quantify the uncertainty of data-driven flood forecasting. This study introduces a general framework for probabilistic flood forecasting conditional on point forecasts. We adopt an unscented Kalman filter (UKF) post-processing technique to model the point forecasts made by a recurrent neural network and their corresponding observations. The methodology is tested by using a long-term 6-h timescale inflow series of the Three Gorges Reservoir in China. The main merits of the proposed approach lie in: first, overcoming the under-prediction phenomena in data-driven flood forecasting; second, alleviating the uncertainty encountered in data-driven flood forecasting. Two commonly used artificial neural networks, a recurrent and a static neural network, were used to make the point forecasts. Then the UKF approach driven by the point forecasts demonstrated its competency in increasing the reliability of probabilistic flood forecasts significantly, where predictive distributions encountered in multi-step-ahead flood forecasts were effectively reduced to small ranges. The results demonstrated that the UKF plus recurrent neural network approach could suitably extract the complex non-linear dependence structure between the model’s outputs and observed inflows and overcome the systematic error so that model reliability as well as forecast accuracy for future horizons could be significantly improved.
... A number of studies have also been conducted on the use of post-processed ensemble precipitation forecasts while maintaining spatial and temporal correlations in ensemble streamflow forecast. Engeland and Steinsland (2014) introduced a method for constructing probabilistic streamflow forecasts for several catchments and lead times. They developed a Gaussian copula-based regression model for preserving the temporal correlation predictand. ...
Ensemble precipitation forecast is effective in reducing the uncertainty and providing reliable probabilistic streamflow forecast. However, for operational applications, precipitation forecasts must go through bias correction in mean and spread. Although post-processing methods, such as BMA, have demonstrated good performance in ensemble-based calibration, the spatial correlation between stations may be altered after post-processing. In this research, ensemble precipitation forecasts of four NWP models, including ECMWF, UKMO, NCEP, and CMA within the TIGGE database, was bias-corrected and post-processed using quantile mapping and BMA for a case study basin in Iran. The ECC method was then used to recover the spatial correlation of ensemble forecasts. Subsequently, probabilistic streamflow forecast was conducted using post-processed precipitation forecasts. The results showed that the errors in the mean and spread of ensemble precipitation forecasts were corrected for each of the four NWP models while the ECC method was effective in maintaining spatial correlation. Furthermore, the results of probabilistic streamflow forecast showed that the performance of the forecast models improved after post processing, with the ECMWF model providing the best forecasts. More work is recommended to improve the impact of the ECC method on NWP models’ performance.
... weather, traffic and events [103] , enabling fulfillment adjustments in real time. Second, in addition to the forecasted demand, safety stock and lead time, two important parameters in the Flowcasting/DRP system, can also be adaptive based on analyzing causal factors through probabilistic predictive models [39] with BDA [30] . Third, PA has already benefited from the feedback mechanism where demand planners flag irregular demand with the labels indicating its most probable underlying causes. ...
The vision of a well-integrated supply chain (SC) was developed as early as 1958 by Forrester, who addressed what would eventually be called the Bullwhip Effect (BWE). The Flowcasting concept, originally called Retail Resource Planning, was proposed to connect all SC upper tiers to the storefront through fulfillment logic based on the Distribution Resource Planning (DRP) system. This method can therefore be understood as fully or SC-wide integrated DRP with a focus on the role of retailers instead of that of vendors or distributors. We studied a Canadian retailer that implemented Flowcasting in order to gain insight into the benefits and operational logic of this system. Based on the data obtained from the company, we simulate Flowcasting operations across a 3-tier SC compared to the Reorder Point (ROP) system, which was previously used at the firm, as well as a combination of ROP and DRP (ROP/DRP or partially integrated DRP), which are some of the most common implementations in use. The simulation is configured based on the company's settings, including historical average demand, demand estimates, lead time, etc. Then, multivariate regression is deployed to statistically compare the efficacy of these methods in SC management using various assessment criteria, including BWE measures. The results show that the requirement calculation logic used in an SC-wide integrated DRP system (Flowcasting) generally outperforms the other two approaches, and its benefits in curtailing the BWE become more noticeable in the upper tiers of the SC. This paper indicates the enormous potential of SC-wide integrated DRP logic in rule-based replenishment planning systems.
... The CRPS integral in equation(29)is approximated numerically using samples 528 from the predictive distributions. In a given catchment, the reference distribution at day t is taken as the 529 climatology (empirical distribution of streamflow) of a 29-day window centred on day t, computed over 530 all years in the calibration window (similar toEngeland and Steinsland, 2014). A value of CRPSS 0 531 indicates better performance than the reference distribution ( CRPSS 1 indicates a perfect point 532 prediction), and value of CRPSS 0 indicates worse performance than the reference distribution.533Note ...
Plain Text Summary: In ephemeral catchments, where streams often run dry, streamflow predictions and their uncertainty estimates (“probabilistic predictions”) are a key input for risk-based environmental management. However, when the observed record has days with zero flow, producing reliable probabilistic predictions is challenging. Pragmatic assumptions are often made, with zero flows not treated explicitly when the hydrological model is calibrated to a catchment of interest. This study evaluates if explicit statistical treatment of zero flows improves probabilistic streamflow prediction in ephemeral catchments.
The pragmatic and explicit approaches are compared over 74 Australian catchments with diverse climatic and hydrological conditions. We find that for catchments with around 5-50% zero flows, the explicit approach produces more reliable predictions. However, for catchments with fewer than 5% zero flows the pragmatic approach can suffice, while for catchments with more than 50% zero flows neither approach is reliable. These findings are supported by statistical theory and analysis.
Our findings provide hydrological modellers with practical guidance about when and why it is important to explicitly treat zero flows in calibration, and which probability models are best suited for quantifying hydrological uncertainty in ephemeral catchments.
... In the case study, we consider the Osali catchment which is a part of the Ulla-Førre hydropower complex south west in Norway [17,18]. Daily inflow observations, in unit m 3 s −1 , are recorded and data are provided by Statkraft, which is the largest hydropower producer in Norway. ...
This paper contributes to forecasting of renewable infeed for use in dispatch scheduling and power systems analysis. Ensemble predictions are commonly used to assess the uncertainty of a future weather event, but they often are biased and have too small variance. Reliable forecasts for future inflow are important for hydropower operation, and the main purpose of this work is to develop methods to generate better calibrated and sharper probabilistic forecasts for inflow. We propose to extend Bayesian model averaging with a varying coefficient regression model to better respect changing weather patterns. We report on results from a case study from a catchment upstream of a Norwegian power plant during the period from 24 June 2014 to 22 June 2015.
... forecasting applications tend to focus on residual errors and often ignore posterior parameter uncertainty 157 (e.g., Engeland andSteinsland, 2014, McInerney et al., 2017). This is the strategy employed in this study, 158 ...
Probabilistic predictions from hydrological models, including rainfall-runoff models, provide valuable information for water and environmental resource risk management. However, traditional “deterministic” usage of rainfall-runoff models remains prevalent in practical applications, in many cases because probabilistic predictions are perceived to be difficult to generate. This paper introduces a simplified approach for hydrological model inference and prediction that bridges the practical gap between “deterministic” and “probabilistic” techniques. This approach combines the Least Squares (LS) technique for calibrating hydrological model parameters with a simple method-of-moments (MoM) estimator of error model parameters (here, the variance and lag-1 autocorrelation of residual errors). A case study using two conceptual hydrological models shows that the LS-MoM approach achieves probabilistic predictions with similar predictive performance to classical maximum-likelihood and Bayesian approaches—but is simpler to implement using common hydrological software and has a lower computational cost. A public web-app to help users implement the simplified approach is available.
... In addition, the individual trajectories must be realistic regarding the strong autocorrelation of streamflow, especially at the hourly time step. While hydrological postprocessing has been extensively studied in the literature, very few studies (Engeland & Steinsland, 2014;Hemri et al., 2013Hemri et al., , 2015 have concerned the generation of coherent multivariate streamflow ensembles. This multivariate extension, referred to as sampling-reordering, is the core subject of this paper. ...
Hydrological ensemble forecasts are frequently miscalibrated, and need to be statistically postprocessed in order to account for the total predictive uncertainty. Very often, this step relies on parametric, univariate techniques that ignore the between-basins and between-lead times dependencies. This calls for a procedure referred to as sampling-reordering, which generates a coherent multivariate ensemble from the marginal postprocessed distributions. The ensemble copula coupling (ECC) approach, which is already popular in the field of meteorological postprocessing, is attractive for hydrological forecasts as it preserves the dependence structure of the raw ensemble assumed as spatially and temporally coherent. However, the existing implementations of ECC have strong limitations when applied to hourly streamflow, due to raw ensembles being frequently nondispersive and to streamflow data being strongly autocorrelated. Based on this diagnosis, this paper investigates several variants of ECC, in particular the addition of a perturbation to the raw ensemble to handle the nondispersive cases, and the smoothing of the temporal trajectories to make them more realistic. The evaluation is conducted on a case study of hydrological forecasting over a set of French basins. The results show that the new variants improve upon the existing ECC implementations, while they remain simple and computationally inexpensive.
... ., H N ). A multivariate probabilistic forecast system (Krzysztofowicz andMaranzano 2004, Engeland andSteinsland 2014) would offer a joint probability distribution of (H 1 , H 2 , . . ., H N ) which would require a multivariate HUP and will be considered in further study. ...
Quantifying the uncertainty in hydrologic forecasting is valuable for water resources management and decision-making processes. The hydrologic uncertainty processor (HUP) can quantify hydrologic uncertainty and produce probabilistic forecasts under the hypothesis that there is no input uncertainty. This study proposes a HUP based on a copula function, in which the prior density and likelihood function were explicitly expressed, and the posterior density and distribution obtained using Monte Carlo sampling. The copula-based HUP was applied to the Three Gorges Reservoir, and compared with the meta-Gaussian HUP. The Nash-Sutcliffe efficiency and relative error were used as evaluation criteria for deterministic forecasts, while predictive QQ plot, reliability, resolution and continuous rank probability score (CRPS) were used for probabilistic forecasts. The results show that the proposed copula-based HUP is comparable to the meta-Gaussian HUP in terms of the posterior median forecasts, and that its probabilistic forecasts have slightly higher reliability and lower resolution compared to the meta-Gaussian HUP. Based on the CRPS, both HUPs were found superior to deterministic forecasts, highlighting the effectiveness of probabilistic forecasts, with the copula-based HUP marginally better than the meta-Gaussian HUP.
... [168][169][170] To preserve the temporal correlation between predictand at different lead times for hydrological forecasts (e.g., streamflow), Engeland and Steinsland developed a Gaussian copula-based regression model. 171 Hemri et al. applied temporal BMA and temporal EMOS for streamflow postprocessing over different lead times and obtained a better forecast skill than univariate BMA or EMOS. 116,172 Nonparametric Methods Nonparametric methods are generally reordering methods like Schaake shuffle 80 and ECC 85 , both of which could be seen as empirical copula methods. ...
Computer simulation models have been widely used to generate hydrometeorological forecasts. As the raw forecasts contain uncertainties arising from various sources, including model inputs and outputs, model initial and boundary conditions, model structure, and model parameters, it is necessary to apply statistical postprocessing methods to quantify and reduce those uncertainties. Different postprocessing methods have been developed for meteorological forecasts (e.g., precipitation) and for hydrological forecasts (e.g., streamflow) due to their different statistical properties. In this paper, we conduct a comprehensive review of the commonly used statistical postprocessing methods for both meteorological and hydrological forecasts. Moreover, methods to generate ensemble members that maintain the observed spatiotemporal and intervariable dependency are reviewed. Finally, some perspectives on the further development of statistical postprocessing methods for hydrometeorological ensemble forecasting are provided. WIREs Water 2017, 4:e1246. doi: 10.1002/wat2.1246
This article is categorized under: Science of Water > Methods
Science of Water > Water Extremes
... Madadgar et al. (2012) show the improvement of the post-processing based on a copula function over the quantile mapping technique. Other nonparametric (Van Steenbergen et al., 2012;Brown and Seo, 2013) and parametric techniques (Reggiani and Weerts, 2008;Engeland and Steinsland, 2014) have been recently proposed. The use of post-processed precipitation for ensemble streamflow forecasts has been tested (Zalachori et al., 2012;Verkade et al., 2013); the results show that the errors linked to hydrological modelling remain a key component to the total predictive uncertainty. ...
The impact of errors in the forcing, errors in the model structure and parameters, and errors in the initial conditions is investigated in a simple hydrological ensemble prediction system. The hydrological model is based on an input nonlinearity connected with a linear transfer function and forced by precipitation forecasts from the European Centre for Medium‐Range Weather Forecast (ECMWF) Ensemble Prediction System (EPS). The post‐processing of the precipitation and/or the streamflow using information from the reforecasts performed by ECMWF is tested. For this purpose, hydrological reforecasts are obtained by forcing the hydrological model with the precipitation from the reforecast data.
In the present case study, it is found that the post‐processing of the hydrological ensembles with a statistical model fitted on the hydrological reforecasts improves the verification scores better than the use of post‐processed precipitation ensembles. In the case of large biases in the precipitation, combining the post‐processing of both precipitation and streamflow allows for further improvements. During the winter, errors in the initial conditions have a larger impact on the scores than errors in the model structure as designed in the experiments. Errors in the parameter values are largely corrected with the post‐processing. Copyright © 2014 John Wiley & Sons, Ltd.
Probabilistic streamflow predictions are valuable tools for predictive uncertainty estimation, hydrologic risk management, and support for decision-making in water resources. Usually, predictive uncertainty quantification is developed and assessed using only a single hydrological model, making it difficult to generalize to other model configurations. To tackle this issue, we assess changes in the model performance ranking of diverse streamflow models by applying a residual error model post-processing approach to multiple basins and multiple models. This assessment employed 141 basins from the Great Lakes watershed covering the USA and Canada, and 13 diverse streamflow models, which are evaluated using deterministic and probabilistic performance metrics. As the first study to implement probabilistic methods to diverse streamflow models applied to a multitude of basins, the analysis here examines the dependence of probabilistic streamflow estimation quality on model quality. Our findings show that streamflow model choice influences the robustness of probabilistic predictions. It was found that moving from deterministic to probabilistic predictions using a post-processing approach does not change the streamflow model performance ranking for the best and worst deterministic models, but models of intermediate rank in deterministic evaluation do not have consistent ranking when evaluated in probabilistic mode. Post-processing residual errors of long short-term memory (LSTM) network models are consistently the best-performing model in terms of deterministic and probabilistic metrics. This study highlights the significance of combining deterministic streamflow model predictions with residual error models for improving the quality and increasing the value of hydrological predictions, quantifying uncertainty, and facilitating decision-making in operational water management. It also clarifies the degree to which probabilistic predictions depend upon good model performance and can compensate for poor model performance.
Different post-processing techniques are frequently employed to improve the outcome of ensemble forecasting models. The main reason is to compensate for biases caused by errors in model structure or initial conditions, and as a correction for under- or overdispersed ensembles. Here we use the Ensemble Model Output Statistics method to post-process the ensemble output from a continental scale hydrological model, LISFLOOD, as used in the European Flood Awareness System (EFAS). We develop a method for local calibration and interpolation of the post-processing parameters and compare it with a more traditional global calibration approach for 678 stations in Europe based on long term observations of runoff and meteorological variables. For the global calibration we also test a reduced model with only a variance inflation factor. Whereas the post-processing improved the results for the first 1-2 days lead time, the improvement was less for increasing lead times of the verification period. This was the case both for the local and global calibration methods. As the post-processing is based on assumptions about the distribution of forecast errors, we also present an analysis of the ensemble output that provides some indications of what to expect from the post-processing.
Ensemble forecasting applied to the field of hydrology is currently an established area of research embracing a broad spectrum of operational situations. This work catalogues the various pathways of ensemble streamflow forecasting based on an exhaustive review of more than 700 studies over the last 40 years. We focus on the advanced state of the art in the model‐based (dynamical) ensemble forecasting approaches. Ensemble streamflow prediction systems are categorized into three leading classes: statistics‐based streamflow prediction systems, climatology‐based ensemble streamflow prediction systems and numerical weather prediction‐based hydrological ensemble prediction systems. For each ensemble approach, technical information, as well as details about its strengths and weaknesses, are provided based on a critical review of the studies listed. Through this literature review, the performance and uncertainty associated with the ensemble forecasting systems are underlined from both operational and scientific viewpoints. Finally, the remaining key challenges and prospective future research directions are presented, notably through hybrid dynamical ‐ statistical learning approaches, which obviously present new challenges to be overcome in order to allow the successful employment of ensemble streamflow forecasting systems in the next decades. Targeting students, researchers and practitioners, this review provides a detailed perspective on the major features of an increasingly important area of hydrological forecasting.
(Hydro-) Meteorological predictions are inherently uncertain. Forecasters are trying to estimate and to ultimately also reduce predictive uncertainty. Atmospheric ensemble prediction systems (EPS) provide forecast ensembles that give a first idea of forecast uncertainty. Combining EPS forecasts, issued by different weather services, to multi-model ensembles gives an even better understanding of forecast uncertainty. This article reviews state of the art statistical post-processing methods that allow for sound combinations of multi-model ensemble forecasts. The aim of statistical post-processing is to maximize the sharpness of the predictive distribution subject to calibration. This article focuses on the well-established parametric approaches: Bayesian model averaging (BMA) and ensemble model output statistics (EMOS). Both are readily available and can be used for straightforward implementation of methods for multi-model ensemble combination. Furthermore, methods to ensure seamless predictions in the context of statistical post-processing are summarized. These methods are mainly based on different types of copula approaches. Since skill of (statistically post-processed) ensemble forecasts is generally assessed using particular verification methods, an overview over such methods to verify probabilistic forecasts is provided as well.
Identification of distributed precipitation–runoff models for hourly runoff simulation based on transfer of full parameters (FP) and partial parameters (PP) are lacking for boreal mid-Norway. We evaluated storage–discharge relationships based model (Kirchmod), the Basic-Grid-Model (BGM) and a simplified Hydrologiska Byråns Vattenbalansavdelning (HBV) model for multi-basins (26 catchments). A regional calibration objective function, which uses all streamflow records in the region, was used to optimize local calibration parameters for each catchment and regional parameters yielding maximum regional weighted average (MRWA) performance measures (PM). Based on regional median Nash–Sutcliffe efficiency (NSE) and NSEln (for log-transformed series) for the calibration and validation periods, the Kirchmod model performed better than the others. Parsimony of the Kirchmod model provided less parameter uncertainty for the FP case but did not guarantee parameter identifiability. Tradeoffs between parsimony and performance were observed despite advantages of parsimony to reduce parameter correlations for the PP, which requires preliminary sensitivity analysis to identify which parameters to transfer. There are potential advantages of using the MRWA method for parameter transfer in space. However, temporal validation indicated marked deterioration of the PM. The tradeoffs between parameter transfers in space and time substantiate both spatial and temporal validation of the regional calibration methodology.
Hydrologic ensemble forecasts driven by atmospheric ensemble prediction systems need statistical post-processing in order to account for systematic errors in terms of both location and spread. Runoff is an inherently multivariate process with typical events lasting from hours in case of floods to weeks or even months in case of droughts. This calls for multivariate post-processing techniques that yield well calibrated forecasts in univariate terms and ensure a realistic temporal dependence structure at the same time. To this end, the univariate ensemble model output statistics (EMOS) post-processing method is combined with two different copula approaches that ensure multivariate calibration throughout the entire forecast horizon. The domain of this study covers three sub-catchments of the river Rhine that represent different sizes and hydrological regimes: the Upper Rhine up to the gauge Maxau, the river Moselle up to the gauge Trier, and the river Lahn up to the gauge Kalkofen. In this study the two approaches to model the temporal dependence structure are ensemble copula coupling (ECC), which preserves the dependence structure of the raw ensemble, and a Gaussian copula approach (GCA), which estimates the temporal correlations from training observations. The results indicate that both methods are suitable for modelling the temporal dependencies of probabilistic hydrologic forecasts. This article is protected by copyright. All rights reserved.
In Chapter 2 of this book a terminology was defined and the different steps in a modelling application was put into the framework of a modelling protocol. In the present chapter some of the elements in such protocol are addressed in more details with special reference to distributed models.
Well-validated rainfall-runoff models are able to capture the relationships between rainfall and streamflow and to reliably estimate initial catchment states. While future streamflows are mainly dependent on initial catchment states and future rainfall, use of the rainfall-runoff models together with estimated future rainfall can produce skilful forecasts of future streamflows. This is the basis for the ensemble streamflow prediction system, but this approach has not been explored in Australia. In this paper, two conceptual rainfall-runoff models, together with rainfall ensembles or analogues based on historical rainfall and the Southern Oscillation index (SOI), were used to forecast streamflows at monthly and 3-monthly scales at two catchments in east Australia. The results showed that both models forecast monthly streamflow well when forecasts for all months were evaluated together, but their performance varied significantly from month to month. Best forecasting skills were obtained (both monthly and 3 monthly) when the models were coupled with ensemble forcings on the basis of long-term historical rainfall. SOI-based resampling of forcings from historical data led to improved forecasting skills only in the period from September to December at the catchment in Queensland. For 3 month streamflow forecasts, best skills were in the period from April to June at the catchment in Queensland and in the period from October to January for the catchment in New South Wales, both of which were the periods after the rainy season. The forecasting skills are indicatively comparable to the statistical forecasting skills using a Bayesian joint probability approach. The potential approaches for improved hydrologic modeling through conditional parameterization and for improved forecasting skills through advanced model updating and bias corrections are also discussed.
When constructing a hydrological model at the macroscale (e.g., watershed scale), the structure of this model will inherently be uncertain because of many factors, including the lack of a robust hydrological theory at that scale. In this work, we assume that a suitable conceptual model structure for the hydrologic system has already been determined; that is, the system boundaries have been specified, the important state variables and input and output fluxes to be included have been selected, the major hydrological processes and geometries of their interconnections have been identified, and the continuity equation (mass balance) has been assumed to hold. The remaining structural identification problem that remains, then, is to select the mathematical form of the dependence of the output on the inputs and state variables, so that a computational model can be constructed for making simulations and/or predictions of the system input-state-output behavior. The conventional approach to this problem is to preassume some fixed (and possibly erroneous) mathematical forms for the model output equations. We show instead how Bayesian data assimilation can be used to directly estimate (construct) the form of these mathematical relationships such that they are statistically consistent with macroscale measurements of the system inputs, outputs, and (if available) state variables. The resulting model has a stochastic rather than deterministic form and thereby properly represents both what we know (our certainty) and what we do not know (our uncertainty) about the underlying structure and behavior of the system. Further, the Bayesian approach enables us to merge prior beliefs in the form of preassumed model equations with information derived from the data to construct a posterior model. As a consequence, in regions of the model space for which observational data are available, the errors in preassumed mathematical form of the model can be corrected, improving model performance. For regions where no such data are available the ``prior'' theoretical assumptions about the model structure and behavior will dominate. The approach, entitled Bayesian estimation of structure, is used to estimate water balance models for the Leaf River Basin, Mississippi, at annual, monthly, and weekly time scales, conditioned on the assumption of a simple single-state-variable conceptual model structure. Inputs to the system are uncertain observed precipitation and potential evapotranspiration, and outputs are estimated probability distributions of actual evapotranspiration and streamflow discharge. Results show that the models estimated for the annual and monthly time scales perform quite well. However, model performance deteriorates for the weekly time scale, suggesting limitations in the assumed form of the conceptual model.
This article analyses the performance of an integrated hydrological ensemble prediction system adapted to small to mid-sized catchments (100–600 km2) situated in the Cévennes-Vivarais region (Southern France) and characterized by short lag times (3–12 h). In this framework, flood forecasts need hourly Probabilistic Quantitative Precipitation Forecasts (PQPF) so as to provide early warning with 24–72 h of anticipation. Here, two sources of PQPF at daily and subdaily (6 h) meteorological time steps are considered: Ensemble Prediction Systems from the European Centre for Medium-range Weather Forecast (ECMWF) and analogy-based PQPF provided in real-time at the Laboratoire d'étude des Transferts en Hydrologie et Environnement. The two PQPF are firstly disaggregated to respect the required hydrological hourly time step, through either the use of a stochastic rainfall generator or the application of a multimodel approach. Then, disaggregated PQPF are used as input to a hydrological model, which is called TOPSIMPL, to provide hourly ensemble discharge forecasts up to 48 h ahead. Illustration and evaluation of ensemble discharge forecasts issued in near real-time conditions are given for some recently observed flash flood events. It is shown that hourly discharge forecasts are first-order conditioned by the accuracy of PQPF at daily or subdaily time step. Six-hour ensemble prediction systems correctly reproduce the rainfall temporal dynamics, whereas daily analogy-based PQPF are less underdispersive in terms of rainfall amounts. As a result, the merging of the two sources of PQPF substantially increases the performance of discharge forecasts, the contribution of a more sophisticated hourly rainfall generator becoming marginal. Copyright © 2012 John Wiley & Sons, Ltd.
In the present paper we describe some methods for verifying and evaluating probabilistic forecasts of hydrological variables. We propose an extension to continuous-valued variables of a verification method originated in the meteorological literature for the analysis of binary variables, and based on the use of a suitable cost-loss function to evaluate the quality of the forecasts. We find that this procedure is useful and reliable when it is complemented with other verification tools, borrowed from the economic literature, which are addressed to verify the statistical correctness of the probabilistic forecast. We illustrate our findings with a detailed application to the evaluation of probabilistic and deterministic forecasts of hourly discharge values.
This paper briefly discusses the nature, the causes and the role of predictive uncertainty in flood forecasting and proposes a novel approach to its estimation. Following the definition of predictive uncertainty, its importance in the decision process is highlighted in relation to the different sources of errors (model, parameter, observations, boundary conditions) that affect flood forecasting. Moreover, the paper briefly analyses the importance of using a full predictive uncertainty, obtained by marginalising the parameter uncertainty, instead of the predictive uncertainty conditional to a single parameter set. Finally, a new Model Conditional Processor (MCP) for the assessment of predictive uncertainty is then proposed as an alternative to the Hydrologic Uncertainty Processor (HUP) introduced by Krzysztofowicz as well as to the Bayesian Model Averaging (BMA) approach due to Raftery et al. The new MCP approach, which aims at assessing, and possibly reducing, predictive uncertainty, allows combination of the observations with one or several models’ forecasts in a multi‐Normal space, by transforming observations and model forecasts in a multivariate Normal space by means of the Normal Quantile Transform. The results of the new approach are shown for the case of the River Po in Italy, and compared with the results obtainable both with the HUP and the BMA.
In this paper, a technique is presented for assessing the predictive uncertainty of rainfall-runoff and hydraulic forecasts. The technique conditions forecast uncertainty on the forecasted value itself, based on retrospective Quantile Regression of hindcasted water level forecasts and forecast errors. To test the robustness of the method, a number of retrospective forecasts for different catchments across England and Wales having different size and hydrological characteristics have been used to derive in a probabilistic sense the relation between simulated values of water levels and matching errors. From this study, we can conclude that using Quantile Regression for estimating forecast errors conditional on the forecasted water levels provides a relatively simple, efficient and robust means for estimation of predictive uncertainty.
1] A method for quantifying the uncertainty of hydrological forecasts is proposed. This approach requires the identification and calibration of a statistical model for the forecast error. Accordingly, the probability distribution of the error itself is inferred through a multiple regression, depending on selected explanatory variables. These may include the current forecast issued by the hydrological model, the past forecast error, and the past rainfall. The final goal is to indirectly relate the forecast error to the sources of uncertainty in the forecasting procedure, through a probabilistic link with the explaining variables identified above. Statistical testing for the proposed approach is discussed in detail. An extensive application to a synthetic database is presented, along with a first real-world implementation that refers to a real-time flood forecasting system that is currently under development. The results indicate that the uncertainty estimates represent well the statistics of the actual forecast errors for the examined events.
1] Predictive uncertainty analysis in hydrologic modeling has become an active area of research, the goal being to generate meaningful error bounds on model predictions. State-space filtering methods, such as the ensemble Kalman filter (EnKF), have shown the most flexibility to integrate all sources of uncertainty. However, predictive uncertainty analyses are typically carried out using a single conceptual mathematical model of the hydrologic system, rejecting a priori valid alternative plausible models and possibly underestimating uncertainty in the model itself. Methods based on Bayesian model averaging (BMA) have also been proposed in the statistical and meteorological literature as a means to account explicitly for conceptual model uncertainty. The present study compares the performance and applicability of the EnKF and BMA for probabilistic ensemble streamflow forecasting, an application for which a robust comparison of the predictive skills of these approaches can be conducted. The results suggest that for the watershed under consideration, BMA cannot achieve a performance matching that of the EnKF method. Citation: Vrugt, J. A., and B. A. Robinson (2007), Treatment of uncertainty using ensemble methods: Comparison of sequential data assimilation and Bayesian model averaging, Water Resour. Res., 43, W01411, doi:10.1029/2005WR004838.
Operational medium range flood forecasting systems are increasingly moving towards the adoption of ensembles of numerical weather predictions (NWP), known as ensemble prediction systems (EPS), to drive their predictions. We review the scientific drivers of this shift towards such ‘ensemble flood forecasting’ and discuss several of the questions surrounding best practice in using EPS in flood forecasting systems. We also review the literature evidence of the ‘added value’ of flood forecasts based on EPS and point to remaining key challenges in using EPS successfully.
We propose a method for post-processing an ensemble of multivariate forecasts
in order to obtain a joint predictive distribution of weather. Our method
utilizes existing univariate post-processing techniques, in this case ensemble
Bayesian model averaging (BMA), to obtain estimated marginal distributions.
However, implementing these methods individually offers no information
regarding the joint distribution. To correct this, we propose the use of a
Gaussian copula, which offers a simple procedure for recovering the dependence
that is lost in the estimation of the ensemble BMA marginals. Our method is
applied to 48-h forecasts of a set of five weather quantities using the
8-member University of Washington mesoscale ensemble. We show that our method
recovers many well-understood dependencies between weather quantities and
subsequently improves calibration and sharpness over both the raw ensemble and
a method which does not incorporate joint distributional information.
The work aims at discussing the role of predictive uncertainty in flood forecasting and flood emergency management, its relevance to improve the decision making process and the techniques to be used for its assessment.
Real time flood forecasting requires taking into account predictive uncertainty for a number of reasons. Deterministic hydrological/hydraulic forecasts give useful information about real future events, but their predictions, as usually done in practice, cannot be taken and used as real future occurrences but rather used as pseudo-measurements of future occurrences in order to reduce the uncertainty of decision makers. Predictive Uncertainty (PU) is in fact defined as the probability of occurrence of a future value of a predictand (such as water level, discharge or water volume) conditional upon prior observations and knowledge as well as on all the information we can obtain on that specific future value from model forecasts. When dealing with commensurable quantities, as in the case of floods, PU must be quantified in terms of a probability distribution function which will be used by the emergency managers in their decision process in order to improve the quality and reliability of their decisions.
After introducing the concept of PU, the presently available processors are introduced and discussed in terms of their benefits and limitations. In this work the Model Conditional Processor (MCP) has been extended to the possibility of using two joint Truncated Normal Distributions (TNDs), in order to improve adaptation to low and high flows.
The paper concludes by showing the results of the application of the MCP on two case studies, the Po river in Italy and the Baron Fork river, OK, USA. In the Po river case the data provided by the Civil Protection of the Emilia Romagna region have been used to implement an operational example, where the predicted variable is the observed water level. In the Baron Fork River example, the data set provided by the NOAA's National Weather Service, within the DMIP 2 Project, allowed two physically based models, the TOPKAPI model and TETIS model, to be calibrated and a data driven model to be implemented using the Artificial Neural Network. The three model forecasts have been combined with the aim of reducing the PU and improving the probabilistic forecast taking advantage of the different capabilities of each model approach.
A method for assimilating remotely sensed snow covered area (SCA) into the snow subroutine of a grid distributed precipitation-runoff model (PRM) is presented. The PRM is assumed to simulate the snow state in each grid cell by a snow depletion curve (SDC), which relates that cell's SCA to its snow cover mass balance. The assimilation is based on Bayes' theorem, which requires a joint prior distribution of the SDC variables in all the grid cells. In this paper we propose a spatial model for this prior distribution, and include similarities and dependencies among the grid cells. Used to represent the PRM simulated snow cover state, our joint prior model regards two elevation gradients and a degree-day factor as global variables, rather than describing their effect separately for each cell. This transformation results in smooth normalised surfaces for the two related mass balance variables, supporting a strong inter-cell dependency in their joint prior model. The global features and spatial interdependency in the prior model cause each SCA observation to provide information for many grid cells. The spatial approach similarly facilitates the utilisation of observed discharge. Assimilation of SCA data using the proposed spatial model is evaluated in a 2400 km2 mountainous region in central Norway (61° N, 9° E), based on two Landsat 7 ETM+ images generalized to 1 km2 resolution. An image acquired on 11 May, a week before the peak flood, removes 78% of the variance in the remaining snow storage. Even an image from 4 May, less than a week after the melt onset, reduces this variance by 53%. These results are largely improved compared to a cell-by-cell independent assimilation routine previously reported. Including observed discharge in the updating information improves the 4 May results, but has weak effect on 11 May. Estimated elevation gradients are shown to be sensitive to informational deficits occurring at high altitude, where snowmelt has not started and the snow coverage is close to unity. Caution is therefore required when using early images.
In the present paper we describe some methods for verifying and evaluating probabilistic forecasts of hydrological variables. We propose an extension to continuous-valued variables of a verification method originated in the meteorological literature for the analysis of binary variables, and based on the use of a suitable cost-loss function to evaluate the quality of the forecasts. We find that this procedure is useful and reliable when it is complemented with other verification tools, borrowed from the economic literature, which are addressed to verify the statistical correctness of the probabilistic forecast. We illustrate our findings with a detailed application to the evaluation of probabilistic and deterministic forecasts of hourly discharge values.
We discuss methods for the evaluation of probabilistic predictions of vector-valued quantities, that can take the form of
a discrete forecast ensemble or a density forecast. In particular, we propose a multivariate version of the univariate verification
rank histogram or Talagrand diagram that can be used to check the calibration of ensemble forecasts. In the case of density
forecasts, Box’s density ordinate transform provides an attractive alternative. The multivariate energy score generalizes
the continuous ranked probability score. It addresses both calibration and sharpness, and can be used to compare deterministic
forecasts, ensemble forecasts and density forecasts, using a single loss function that is proper. An application to the University
of Washington mesoscale ensemble points at strengths and deficiencies of probabilistic short-range forecasts of surface wind
vectors over the North American Pacific Northwest.
In this paper a method of estimating the parameters of a set of regression equations is reported which involves application of Aitken's generalized least-squares [1] to the whole system of equations. Under conditions generally encountered in practice, it is found that the regression coefficient estimators so obtained are at least asymptotically more efficient than those obtained by an equation-by-equation application of least squares. This gain in efficiency can be quite large if “independent” variables in different equations are not highly correlated and if disturbance terms in different equations are highly correlated. Further, tests of the hypothesis that all regression equation coefficient vectors are equal, based on “micro” and “macro” data, are described. If this hypothesis is accepted, there will be no aggregation bias. Finally, the estimation procedure and the “micro-test” for aggregation bias are applied in the analysis of annual investment data, 1935–1954, for two firms.
This paper evaluates a nonparametric technique for estimating the conditional probability distribution of a predictand given a vector of predictors. In the current application, the predictors are formed from a multimodel ensemble of simulated streamflows, such that the hydrologic uncertainties are modelled independently of any forcing uncertainties. The technique is based on Bayesian optimal linear estimation of indicator variables and is analogous to indicator cokriging (ICK) in geostatistics. By developing linear estimators for the conditional probability that the observed variable does not exceed several thresholds, ICK provides a discrete approximation of the full conditional probability distribution. The weights of the predictors can be chosen to minimize the expected error variance at each threshold (the Brier score) or, without loss of analytical tractability, a combination of the error variance and the expected square bias conditional upon the observation, i.e. the Type-II conditional bias (CB). The latter is referred to as CB-penalized ICK (CBP-ICK) and is an important enhancement to ICK. Indeed, the biases in atmospheric and hydrologic predictions generally increase towards the tails of their probability distributions. The performance of CBP-ICK is evaluated for selected basins in the eastern USA using a range of probabilistic verification metrics and associated confidence intervals for the sampling uncertainties. Overall, CBP-ICK produces unbiased and skillful estimates of the hydrologic uncertainties, with some sensitivity to the calibration data period at high flow thresholds. More generally, we argue that the common aim in statistical post-processing of ‘maximizing sharpness subject to reliability (Type-I CB)’ should be recast to accommodate both the Type-I and Type-II CBs, as both are important for practical applications of hydrologic predictions. Copyright © 2012 John Wiley & Sons, Ltd.
We consider the problem of forecasting future regional climate. Our method is based on blending different members of an ensemble of regional climate model (RCM) simulations while accounting for the discrepancies between these simulations, under present day conditions, and observational records for the recent past. To this end, we develop Bayesian space-time models that assess the discrepancies between climate model simulations and observational records. Those discrepancies are then propagated into the future to obtain blended forecasts of 21st century climate. The model allows for location-dependent spatial heterogeneities, providing local comparisons between the different simulations. Additionally, we estimate the different modes of spatial variability, and use the climate model-specific coefficients of the spatial factors for comparisons. We focus on regional climate model simulations performed in the context of the North American Regional Climate Change Assessment Program (NARCCAP). We consider, in particular, simulations from RegCM3 using three different forcings: NCEP, GFDL and CGCM3. We use simulations for two time periods: current climate conditions, covering 1971 to 2000, and future climate conditions under the SRES A2 emissions scenario, covering 2041 to 2070. We investigate yearly mean summer temperature for a domain in the South West of the United States. The results indicated the RCM simulations underestimate the mean summer temperature increase for most of the domain compared to our model.
This Special Issue stems from the recent emergence of a distinct field of researchers and operational forecasters who predict river flow with hydrological ensemble prediction systems (HEPSs)...
A Bayesian methodology is developed to evaluate parameter uncertainty in
catchment models fitted to a hydrologic response such as runoff, the
goal being to improve the chance of successful regionalization. The
catchment model is posed as a nonlinear regression model with stochastic
errors possibly being both autocorrelated and heteroscedastic. The end
result of this methodology, which may use Box-Cox power transformations
and ARMA error models, is the posterior distribution, which summarizes
what is known about the catchment model parameters. This can be
simplified to a multivariate normal provided a linearization in
parameter space is acceptable; means of checking and improving this
assumption are discussed. The posterior standard deviations give a
direct measure of parameter uncertainty, and study of the posterior
correlation matrix can indicate what kinds of data are required to
improve the precision of poorly determined parameters. Finally, a case
study involving a nine-parameter catchment model fitted to monthly
runoff and soil moisture data is presented. It is shown that use of
ordinary least squares when its underlying error assumptions are
violated gives an erroneous description of parameter uncertainty.
The uncertainty in a given hydrologic prediction is the compound effect of the parameter, data, and structural uncertainties associated with the underlying model. In general, therefore, the confidence in a hydrologic prediction can be improved by reducing the uncertainty associated with the parameter estimates. However, the classical approach to doing this via model calibration typically requires that considerable amounts of data be collected and assimilated before the model can be used. This limitation becomes immediately apparent when hydrologic predictions must be generated for a previously ungauged watershed that has only recently been instrumented. This paper presents the framework for a Bayesian recursive estimation approach to hydrologic prediction that can be used for simultaneous parameter estimation and prediction in an operational setting. The prediction is described in terms of the probabilities associated with different output values. The uncertainty associated with the parameter estimates is updated (reduced) recursively, resulting in smaller prediction uncertainties as measurement data are successively assimilated. The effectiveness and efficiency of the method are illustrated in the context of two models: a simple unit hydrograph model and the more complex Sacramento soil moisture accounting model, using data from the Leaf River basin in Mississippi.
The performance of ensemble prediction systems (EPSs) is investigated by examining the probability distribution of 500-hPa geopotential height over Europe. The probability score (or half Brier score) is used to evaluate the quality of probabilistic forecasts of a single binary event. The skill of an EPS is assessed by comparing its performance, in terms of the probability score, to the performance of a reference probabilistic forecast. The reference forecast is based on the control forecast of the system under consideration, using model error statistics to estimate a probability distribution. A decomposition of the skill score is applied in order to distinguish between the two main aspects of the forecast performance: reliability and resolution. The contribution of the ensemble mean and the ensemble spread to the performance of an EPS is evaluated by comparing the skill score to the skill score of a probabilistic forecast based on the EPS mean, using model error statistics to estimate a probability distribution. The performance of the European Centre for Medium-Range Weather Forecasts (ECMWF) EPS is reviewed. The system is skillful (with respect to the reference forecast) from +96 h onward. There is some skill from +48 h in terms of reliability. The performance comes mainly from the contribution of the ensemble mean. The contribution of the ensemble spread is slightly negative, but becomes positive after a calibration of the EPS standard deviation. The calibration improves predominantly the reliability contribution to the skill score. The calibrated EPS is skillful from +72 h onward. The impact of ensemble size on the performance of an EPS is also investigated. The skill score of the ECMWF EPS decreases steadily with reducing numbers of ensemble members and the resolution is particularly affected. The impact is mainly due to the ensemble spread contributing negatively to the skill. The ensemble mean contribution to the skill decreases marginally when reducing the ensemble size up to 11 members. The performance of the U.S. National Centers for Environmental Prediction (NCEP) EPS is also reviewed. The NCEP EPS has a lower skill score (vs a reference forecast based on its control forecast) than the ECMWF EPS especially in terms of reliability. This is mainly due to the smaller spread of the NCEP EPS contributing negatively to the skill. On the other hand, the NCEP and ECMWF ensemble means contribute similarly to the skill. As a consequence, the performance of the two systems in terms of resolution is comparable. The performance of a poor man's EPS. consisting of the forecasts of different NWP centers, is discussed. The poor man's EPS is more skillful than either the ECMWF EPS or the NCEP EPS up to +144 h, despite a negative contribution of the spread to the skill score. The higher skill of the poor man's EPS is mainly due to a better resolution.
Some time ago, the continuous ranked probability score (CRPS) was proposed as a new verification tool for (probabilistic) forecast systems. Its focus is on the entire permissible range of a certain (weather) parameter. The CRPS can be seen as a ranked probability score with an infinite number of classes, each of zero width. Alternatively, it can be interpreted as the integral of the Brier score over all possible threshold values for the parameter under consideration. For a deterministic forecast system the CRPS reduces to the mean absolute error. In this paper it is shown that for an ensemble prediction system the CRPS can be decomposed into a reliability part and a resolution/uncertainty part, in a way that is similar to the decomposition of the Brier score. The reliability part of the CRPS is closely connected to the rank histogram of the ensemble, while the resolution/uncertainty part can be related to the average spread within the ensemble and the behavior of its outliers. The usefulness of such a decomposition is illustrated for the ensemble prediction system running at the European Centre for Medium-Range Weather Forecasts. The evaluation of the CRPS and its decomposition proposed in this paper can be extended to systems issuing continuous probability forecasts, by realizing that these can be interpreted as the limit of ensemble forecasts with an infinite number of members.
Seasonal forecasting of streamflows can be highly valuable for water resources management. In this paper, a Bayesian joint probability (BJP) modeling approach for seasonal forecasting of streamflows at multiple sites is presented. A Box-Cox transformed multivariate normal distribution is proposed to model the joint distribution of future streamflows and their predictors such as antecedent streamflows and El Niño-Southern Oscillation indices and other climate indicators. Bayesian inference of model parameters and uncertainties is implemented using Markov chain Monte Carlo sampling, leading to joint probabilistic forecasts of streamflows at multiple sites. The model provides a parametric structure for quantifying relationships between variables, including intersite correlations. The Box-Cox transformed multivariate normal distribution has considerable flexibility for modeling a wide range of predictors and predictands. The Bayesian inference formulated allows the use of data that contain nonconcurrent and missing records. The model flexibility and data-handling ability means that the BJP modeling approach is potentially of wide practical application. The paper also presents a number of statistical measures and graphical methods for verification of probabilistic forecasts of continuous variables. Results for streamflows at three river gauges in the Murrumbidgee River catchment in southeast Australia show that the BJP modeling approach has good forecast quality and that the fitted model is consistent with observed data.
A recently developed technique for identifying continuous-time, time-dependent, stochastic model parameters is embedded in a general framework for identifying causes of bias and reducing bias in dynamic models. In contrast to the usual approach of considering bias in model output with an autoregressive error model or a stochastic process, we make the attempt to correct for bias within the model or even in model input. This increases the potential of learning about the causes of bias and of subsequently correcting deficits of the deterministic model structure. The time-dependent parameters as formulated in our approach can also consistently be used for adding stochasticity to the model without losing precise fulfilment of conservation laws used for deriving the model equations. An additional advantage of the suggested procedure is that it makes it possible to derive more realistic uncertainty bounds of internal model variables than is the case when bias is only considered for measured model output. This is important for mechanistic models in which internal variables have a direct physical meaning. The concept is illustrated by an application to a simple eight-parameter conceptual hydrological model. This application demonstrates the feasibility of the proposed approach and gives an impression of its potential for application to a large class of nonlinear, dynamic models.
Rational decision making (for flood warning, navigation, or reservoir systems) requires that the total uncertainty about a hydrologic predictand (such as river stage, discharge, or runoff volume) be quantified in terms of a probability distribution, conditional on all available information and knowledge. Hydrologic knowledge is typically embodied in a deterministic catchment model. Fundamentals are presented of a Bayesian forecasting system (BFS) for producing a probabilistic forecast of a hydrologic predictand via any deterministic catchment model. The BFS decomposes the total uncertainty into input uncertainty and hydrologic uncertainty, which are quantified independently and then integrated into a predictive (Bayes) distribution. This distribution results from a revision of a prior (climatic) distribution, is well calibrated, and has a nonnegative ex ante economic value. The BFS is compared with Monte Carlo simulation and ``ensemble forecasting'' technique, none of which can alone produce a probabilistic forecast that meets requirements of rational decision making, but each can serve as a component of the BFS.
Parameter estimation in rainfall-runoff models is affected by uncertainties in the measured input/output data (typically, rainfall and runoff, respectively), as well as model error. Despite advances in data collection and model construction, we expect input uncertainty to be particularly significant (because of the high spatial and temporal variability of precipitation) and to remain considerable in the foreseeable future. Ignoring this uncertainty compromises hydrological modeling, potentially yielding biased and misleading results. This paper develops a Bayesian total error analysis methodology for hydrological models that allows (indeed, requires) the modeler to directly and transparently incorporate, test, and refine existing understanding of all sources of data uncertainty in a specific application, including both rainfall and runoff uncertainties. The methodology employs additional (latent) variables to filter out the input corruption given the model hypothesis and the observed data. In this study, the input uncertainty is assumed to be multiplicative Gaussian and independent for each storm, but the general framework allows alternative uncertainty models. Several ways of incorporating vague prior knowledge of input corruption are discussed, contrasting Gaussian and inverse gamma assumptions; the latter method avoids degeneracies in the objective function. Although the general methodology is computationally intensive because of the additional latent variables, a range of modern numerical methods, particularly Monte Carlo analysis combined with fast Newton-type optimization methods and Hessian-based covariance analysis, can be employed to obtain practical solutions.
This paper presents a new methodology to generate a tree from an ensemble. The reason to generate a tree is to use the ensemble in multistage stochastic programming. A correct tree structure is of critical importance because it strongly affects the performance of the optimization. A tree, in contrast to an ensemble, specifies when its trajectories diverge from each other.
A tree can be generated from the ensemble data by aggregating trajectories over time until the difference between them becomes such that they can no longer be assumed to be similar, at such a point, the tree branches.
The proposed method models the information flow: it takes into account which observations will become available, at which moment, and their level of uncertainty, i.e. their probability distributions (pdf). No conditions are imposed on those distributions.
The method is well suited to trajectories that are close to each other at the beginning of the forecasting horizon and spread out going on in time, as ensemble forecasts typically are. Copyright © 2012 John Wiley & Sons, Ltd.
In this paper a method of estimating the parameters of a set of regression equations is reported which involves application of Aitken's generalized least-squares [1] to the whole system of equations. Under conditions generally encountered in practice, it is found that the regression coefficient estimators so obtained are at least asymptotically more efficient than those obtained by an equation-by-equation application of least squares. This gain in efficiency can be quite large if “independent” variables in different equations are not highly correlated and if disturbance terms in different equations are highly correlated. Further, tests of the hypothesis that all regression equation coefficient vectors are equal, based on “micro” and “macro” data, are described. If this hypothesis is accepted, there will be no aggregation bias. Finally, the estimation procedure and the “micro-test” for aggregation bias are applied in the analysis of annual investment data, 1935–1954, for two firms.
Enhanced ability to forecast peak discharges remains the most relevant non-structural measure for flood protection. Extended forecasting lead times are desirable as they facilitate mitigating action and response in case of extreme discharges. Forecasts remain however affected by uncertainty as an exact prognosis of water levels is inherently impossible. Here, we implement a dedicated uncertainty processor, that can be used within operational flood forecasting systems.The processor is designed to support decision-making under conditions of uncertainty. The scientific approach at the basis of the uncertainty processor is general and independent of the deterministic models used. It is based on Bayesian revision of prior knowledge on the basis of past evidence on model performance against observations. The revision of the prior distributions on water levels and/or flow rates leads to posterior probability distributions that are translated into an effective decision support under uncertainty. The processor is validated on the operational real-time river Rhine flood forecasting system.
Three statistical models for the forecast errors for inflow into the Langvatn reservoir in Northern Norway have been constructed and tested according to the agreement between (i) the forecast distribution and the observations and (ii) median values of the forecast distribution and the observations. For the first model observed and forecasted inflows were transformed by the Box–Cox transformation before a first order auto-regressive model was constructed for the forecast errors. The parameters were conditioned on weather classes. In the second model the Normal Quantile Transformation (NQT) was applied on observed and forecasted inflows before a similar first order auto-regressive model was constructed for the forecast errors. For the third model positive and negative errors were modeled separately. The errors were first NQT-transformed before conditioning the mean error values on climate, forecasted inflow and yesterday’s error. To test the three models we applied three criterions: we wanted (a) the forecast distribution to be reliable; (b) the forecast intervals to be narrow; (c) the median values of the forecast distribution to be close to the observed values. Models 1 and 2 gave almost identical results. The median values improved the forecast with Nash–Sutcliffe Reff increasing from 0.77 for the original forecast to 0.87 for the corrected forecasts. Models 1 and 2 over-estimated the forecast intervals but gave the narrowest intervals. Their main drawback was that the distributions are less reliable than Model 3. For Model 3 the median values did not fit well since the auto-correlation was not accounted for. Since Model 3 did not benefit from the potential variance reduction that lies in bias estimation and removal it gave on average wider forecasts intervals than the two other models. At the same time Model 3 on average slightly under-estimated the forecast intervals, probably explained by the use of average measures to evaluate the fit.
Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection.
Personal, or subjective, probabilities are used as inputs to many inferential and decision-making models, and various procedures have been developed for the elicitation of such probabilities. Included among these elicitation procedures are scoring rules, which involve the computation of a score based on the assessor's stated probabilities and on the event that actually occurs. The development of scoring rules has, in general, been restricted to the elicitation of discrete probability distributions. In this paper, families of scoring rules for the elicitation of continuous probability distributions are developed and discussed.
A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.
The purpose of this analytic-numerical Bayesian forecasting system (BFS) is to produce a short-term probabilistic river stage forecast based on a probabilistic quantitative precipitation forecast as an input and a deterministic hydrologic model (of any complexity) as a means of simulating the response of a headwater basin to precipitation. The BFS has three structural components: the precipitation uncertainty processor, the hydrologic uncertainty processor, and the integrator. A series of articles described the Bayesian forecasting theory and detailed each component of this particular BFS. This article presents a synthesis: the total system, operational expressions, estimation procedures, numerical algorithms, a complete example, and all design requirements, modeling assumptions, and operational attributes.
Hydrologic models are twofold: models for understanding physical processes and models for prediction. This study addresses the latter, which modelers use to predict, for example, streamflow at some future time given knowledge of the current state of the system and model parameters. In this respect, good estimates of the parameters and state variables are needed to enable the model to generate accurate forecasts. In this paper, a dual state–parameter estimation approach is presented based on the Ensemble Kalman Filter (EnKF) for sequential estimation of both parameters and state variables of a hydrologic model. A systematic approach for identification of the perturbation factors used for ensemble generation and for selection of ensemble size is discussed. The dual EnKF methodology introduces a number of novel features: (1) both model states and parameters can be estimated simultaneously; (2) the algorithm is recursive and therefore does not require storage of all past information, as is the case in the batch calibration procedures; and (3) the various sources of uncertainties can be properly addressed, including input, output, and parameter uncertainties. The applicability and usefulness of the dual EnKF approach for ensemble streamflow forecasting is demonstrated using a conceptual rainfall-runoff model.
The development of automated (computer-based) calibration methods has focused mainly on the selection of a single-objective measure of the distance between the model-simulated output and the data and the selection of an automatic optimization algorithm to search for the parameter values which minimize that distance. However, practical experience with model calibration suggests that no single-objective function is adequate to measure the ways in which the model fails to match the important characteristics of the observed data. Given that some of the latest hydrologic models simulate several of the watershed output fluxes (e.g. water, energy, chemical constituents, etc.), there is a need for effective and efficient multi-objective calibration procedures capable of exploiting all of the useful information about the physical system contained in the measurement data time series. The MOCOM-UA algorithm, an effective and efficient methodology for solving the multiple-objective global optimization problem, is presented in this paper. The method is an extension of the successful SCE-UA single-objective global optimization algorithm. The features and capabilities of MOCOM-UA are illustrated by means of a simple hydrologic model calibration study.
This paper describes a methodology for calibration and uncertainty estimation of distributed models based on generalized likelihood measures. the GLUE procedure works with multiple sets of parameter values and allows that, within the limitations of a given model structure and errors in boundary conditions and field observations, different sets of values May, be equally likely as simulators of a catchment. Procedures for incorporating different types of observations into the calibration; Bayesian updating of likelihood values and evaluating the value of additional observations to the calibration process are described. the procedure is computationally intensive but has been implemented on a local parallel processing computer. the methodology is illustrated by an application of the Institute of Hydrology Distributed Model to data from the Gwy experimental catchment at Plynlimon, mid-Wales.
Meaningful quantification of data and structural uncertainties in conceptual rainfall-runoff modeling is a major scientific and engineering challenge. This paper focuses on the total predictive uncertainty and its decomposition into input and structural components under different inference scenarios. Several Bayesian inference schemes are investigated, differing in the treatment of rainfall and structural uncertainties, and in the precision of the priors describing rainfall uncertainty. Compared with traditional lumped additive-error approaches, the quantification of the total predictive uncertainty in the runoff is improved when rainfall and/or structural errors are characterized explicitly. However, the decomposition of the total uncertainty into individual sources is more challenging. In particular, poor identifiability may arise when the inference scheme represents rainfall and structural errors using separate probabilistic models. The inference becomes ill-posed unless sufficiently precise prior knowledge of data uncertainty is supplied; this ill-posedness can often be detected from the behavior of the Monte Carlo sampling algorithm. Moreover, the priors on the data quality must also be sufficiently accurate if the inference is to be reliable and support meaningful uncertainty decomposition. Our findings highlight the inherent limitations of inferring inaccurate hydrologic models using rainfall-runoff data with large unknown errors. Bayesian total error analysis (BATEA) can overcome these problems using independent prior information. The need for deriving independent descriptions of the uncertainties in the input and output data is clearly demonstrated.
The lack of a robust framework for quantifying the parametric and predictive uncertainty of conceptual rainfall-runoff (CRR) models remains a key challenge in hydrology. The Bayesian total error analysis (BATEA) methodology provides a comprehensive framework to hypothesize, infer, and evaluate probability models describing input, output, and model structural error. This paper assesses the ability of BATEA and standard calibration approaches (standard least squares (SLS) and weighted least squares (WLS)) to address two key requirements of uncertainty assessment: (1) reliable quantification of predictive uncertainty and (2) reliable estimation of parameter uncertainty. The case study presents a challenging calibration of the lumped GR4J model to a catchment with ephemeral responses and large rainfall gradients. Postcalibration diagnostics, including checks of predictive distributions using quantile-quantile analysis, suggest that while still far from perfect, BATEA satisfied its assumed probability models better than SLS and WLS. In addition, WLS/SLS parameter estimates were highly dependent on the selected rain gauge and calibration period. This will obscure potential relationships between CRR parameters and catchment attributes and prevent the development of meaningful regional relationships. Conversely, BATEA provided consistent, albeit more uncertain, parameter estimates and thus overcomes one of the obstacles to parameter regionalization. However, significant departures from the calibration assumptions remained even in BATEA, e.g., systematic overestimation of predictive uncertainty, especially in validation. This is likely due to the inferred rainfall errors compensating for simplified treatment of model structural error.
Quantile Regression, Econometric Soc. Monogr
- R. Koenker