Bayesian Joint Probability Modelling Approach Research Articles

Calibration and quantifying uncertainty of daily water quality forecasts for large lakes with a Bayesian joint probability modelling approach

Correcting the systematic bias and quantifying uncertainty associated with the operational water quality forecasts are imperative works for risk-based environmental decision making. This work proposes a post-processing method for addressing both bias correction and total uncertainty quantification for daily forecasts of water quality parameters derived from dynamical lake models. The post-processing is implemented based on a Bayesian Joint Probability (BJP) modeling approach. The BJP model uses a log-sinh transformation to normalize the raw forecasts and corresponding observations, and uses a bivariate Gaussian distribution to characterize the dependence relationship. The posterior distribution of the transformation parameters is inferenced through Metropolis Monte Carlo Markov chain sampling; it generates unbiased probabilistic forecasts that account for uncertainties from all sources. The BJP is used to post-processing raw daily forecasts of dissolved oxygen (DO), ammonium nitrogen (NH), total phosphorus (TP) and total nitrogen (TN) concentrations of Lake Chaohu, the fifth largest lake in China with lead times from 0 to 5 days. Results suggest that an average 93.1% forecast bias has been removed by BJP. The root mean square error in probability skill scores range from 5.8% for NH to 68.2% for TP, and the non-parametric bootstrapping test suggests that 67.7% forecasts are significantly improved averaged across all sampling sites, water quality parameters and lead times. The probabilities of the calibrated forecasts are reasonably consistent with the observed relative frequencies, and have appropriate spread and thus correctly quantify forecast uncertainty. The BJP post-processing method used in this study can be a useful operational tool that help to better realize the potential of water quality forecasts derived from dynamical models.

On the Joint Calibration of Multivariate Seasonal Climate Forecasts from GCMs

Abstract Multivariate seasonal climate forecasts are increasingly required for quantitative modeling in support of natural resources management and agriculture. GCM forecasts typically require postprocessing to reduce biases and improve reliability; however, current seasonal postprocessing methods often ignore multivariate dependence. In low-dimensional settings, fully parametric methods may sufficiently model intervariable covariance. On the other hand, empirical ensemble reordering techniques can inject desired multivariate dependence in ensembles from template data after univariate postprocessing. To investigate the best approach for seasonal forecasting, this study develops and tests several strategies for calibrating seasonal GCM forecasts of rainfall, minimum temperature, and maximum temperature with intervariable dependence: 1) simultaneous calibration of multiple climate variables using the Bayesian joint probability modeling approach; 2) univariate BJP calibration coupled with an ensemble reordering method (the Schaake shuffle); and 3) transformation-based quantile mapping, which borrows intervariable dependence from the raw forecasts. Applied to Australian seasonal forecasts from the ECMWF System4 model, univariate calibration paired with empirical ensemble reordering performs best in terms of univariate and multivariate forecast verification metrics, including the energy and variogram scores. However, the performance of empirical ensemble reordering using the Schaake shuffle is influenced by the selection of historical data in constructing a dependence template. Direct multivariate calibration is the second-best method, with its far superior performance in in-sample testing vanishing in cross validation, likely because of insufficient data relative to the number of parameters. The continued development of multivariate forecast calibration methods will support the uptake of seasonal climate forecasts in complex application domains such as agriculture and hydrology.

A Bayesian modelling method for post-processing daily sub-seasonal to seasonal rainfall forecasts from global climate models and evaluation for 12 Australian catchments

Abstract. Rainfall forecasts are an integral part of hydrological forecasting systems at sub-seasonal to seasonal timescales. In seasonal forecasting, global climate models (GCMs) are now the go-to source for rainfall forecasts. For hydrological applications however, GCM forecasts are often biased and unreliable in uncertainty spread, and calibration is therefore required before use. There are sophisticated statistical techniques for calibrating monthly and seasonal aggregations of the forecasts. However, calibration of seasonal forecasts at the daily time step typically uses very simple statistical methods or climate analogue methods. These methods generally lack the sophistication to achieve unbiased, reliable and coherent forecasts of daily amounts and seasonal accumulated totals. In this study, we propose and evaluate a Rainfall Post-Processing method for Seasonal forecasts (RPP-S), which is based on the Bayesian joint probability modelling approach for calibrating daily forecasts and the Schaake Shuffle for connecting the daily ensemble members of different lead times. We apply the method to post-process ACCESS-S forecasts for 12 perennial and ephemeral catchments across Australia and for 12 initialisation dates. RPP-S significantly reduces bias in raw forecasts and improves both skill and reliability. RPP-S forecasts are also more skilful and reliable than forecasts derived from ACCESS-S forecasts that have been post-processed using quantile mapping, especially for monthly and seasonal accumulations. Several opportunities to improve the robustness and skill of RPP-S are identified. The new RPP-S post-processed forecasts will be used in ensemble sub-seasonal to seasonal streamflow applications.

How Suitable is Quantile Mapping For Postprocessing GCM Precipitation Forecasts?

GCMs are used by many national weather services to produce seasonal outlooks of atmospheric and oceanic conditions and fluxes. Postprocessing is often a necessary step before GCM forecasts can be applied in practice. Quantile mapping (QM) is rapidly becoming the method of choice by operational agencies to postprocess raw GCM outputs. The authors investigate whether QM is appropriate for this task. Ensemble forecast postprocessing methods should aim to 1) correct bias, 2) ensure forecasts are reliable in ensemble spread, and 3) guarantee forecasts are at least as skillful as climatology, a property called “coherence.” This study evaluates the effectiveness of QM in achieving these aims by applying it to precipitation forecasts from the POAMA model. It is shown that while QM is highly effective in correcting bias, it cannot ensure reliability in forecast ensemble spread or guarantee coherence. This is because QM ignores the correlation between raw ensemble forecasts and observations. When raw forecasts are not significantly positively correlated with observations, QM tends to produce negatively skillful forecasts. Even when there is significant positive correlation, QM cannot ensure reliability and coherence for postprocessed forecasts. Therefore, QM is not a fully satisfactory method for postprocessing forecasts where the issues of bias, reliability, and coherence pre-exist. Alternative postprocessing methods based on ensemble model output statistics (EMOS) are available that achieve not only unbiased but also reliable and coherent forecasts. This is shown with one such alternative, the Bayesian joint probability modeling approach.

Ensemble forecasting of sub-seasonal to seasonal streamflow by a Bayesian joint probability modelling approach

The Bayesian joint probability (BJP) modelling approach is used operationally to produce seasonal (three-month-total) ensemble streamflow forecasts in Australia. However, water resource managers are calling for more informative sub-seasonal forecasts. Taking advantage of BJP’s capability of handling multiple predictands, ensemble forecasting of sub-seasonal to seasonal streamflows is investigated for 23 catchments around Australia. Using antecedent streamflow and climate indices as predictors, monthly forecasts are developed for the three-month period ahead. Forecast reliability and skill are evaluated for the period 1982–2011 using a rigorous leave-five-years-out cross validation strategy. BJP ensemble forecasts of monthly streamflow volumes are generally reliable in ensemble spread. Forecast skill, relative to climatology, is positive in 74% of cases in the first month, decreasing to 57% and 46% respectively for streamflow forecasts for the final two months of the season. As forecast skill diminishes with increasing lead time, the monthly forecasts approach climatology. Seasonal forecasts accumulated from monthly forecasts are found to be similarly skilful to forecasts from BJP models based on seasonal totals directly. The BJP modelling approach is demonstrated to be a viable option for producing ensemble time-series sub-seasonal to seasonal streamflow forecasts.

Improving Precipitation Forecasts by Generating Ensembles through Postprocessing

Abstract This paper evaluates a postprocessing method for deterministic quantitative precipitation forecasts (raw QPFs) from a numerical weather prediction model. The postprocessing aims to produce calibrated QPF ensembles that are bias free, more accurate than raw QPFs, and reliable for use in streamflow forecasting applications. The method combines a simplified version of the Bayesian joint probability (BJP) modeling approach and the Schaake shuffle. The BJP modeling approach relates raw QPFs and observed precipitation by modeling their joint distribution. It corrects biases in the raw QPFs and generates ensemble forecasts that reflect the uncertainty in the raw QPFs. The BJP modeling approach is applied to each lead time and each forecast location separately. The Schaake shuffle is then employed to produce calibrated QPFs with appropriate space–time correlations by linking ensemble members generated by the BJP modeling approach. Calibrated QPFs are produced for 10 Australian catchments that cover a wide range of climatic conditions and hydrological characteristics. The calibrated QPFs are bias free, contain smaller forecast errors than that of the raw QPFs, reliably quantify the forecast uncertainty at a range of lead times, and successfully discriminate common and rare events of precipitation occurrences at shorter lead times. The postprocessing method is able to instill realistic within-catchment spatial variability in the QPFs, which is crucial for accurate and reliable streamflow forecasting.

Seasonal precipitation forecasts over China using monthly large-scale oceanic-atmospheric indices

Summary Forecasting precipitation at the seasonal time scale remains a formidable challenge. In this study, we evaluate a statistical method for forecasting seasonal precipitation across China for 12 overlapping seasons. We use the Bayesian joint probability modelling approach to establish multiple probabilistic forecast models using eight large-scale oceanic-atmospheric indices at lag times of 1–3 months as predictors. We then merge forecasts from the multiple models with Bayesian model averaging to combine the strengths of the individual models. Forecast skill and reliability are assessed through leave-one-year-out cross validation. The merged forecasts exhibit considerable seasonal and spatial variability in forecast skill. The merged forecasts are most skillful over west China in spring periods and over central-south China in autumn periods. In contrast, forecast skill in most wet summer and dry winter periods is generally low. Positive forecast skill is mostly retained when forecast lead time is increased from 0 to 2 months. Forecast distributions are found to reliably represent forecast uncertainty. Climate indices derived from sea surface temperature in the western Pacific and Indian Ocean tend to contribute more to forecast skill than indices of the El Nino-Southern Oscillation. Large-scale atmospheric circulation patterns, represented by the Arctic Oscillation and North Atlantic Oscillation, appear to contribute little to forecast skill.

Post-processing rainfall forecasts from numerical weather prediction models for short-term streamflow forecasting

Abstract. Sub-daily ensemble rainfall forecasts that are bias free and reliably quantify forecast uncertainty are critical for flood and short-term ensemble streamflow forecasting. Post-processing of rainfall predictions from numerical weather prediction models is typically required to provide rainfall forecasts with these properties. In this paper, a new approach to generate ensemble rainfall forecasts by post-processing raw numerical weather prediction (NWP) rainfall predictions is introduced. The approach uses a simplified version of the Bayesian joint probability modelling approach to produce forecast probability distributions for individual locations and forecast lead times. Ensemble forecasts with appropriate spatial and temporal correlations are then generated by linking samples from the forecast probability distributions using the Schaake shuffle. The new approach is evaluated by applying it to post-process predictions from the ACCESS-R numerical weather prediction model at rain gauge locations in the Ovens catchment in southern Australia. The joint distribution of NWP predicted and observed rainfall is shown to be well described by the assumed log-sinh transformed bivariate normal distribution. Ensemble forecasts produced using the approach are shown to be more skilful than the raw NWP predictions both for individual forecast lead times and for cumulative totals throughout all forecast lead times. Skill increases result from the correction of not only the mean bias, but also biases conditional on the magnitude of the NWP rainfall prediction. The post-processed forecast ensembles are demonstrated to successfully discriminate between events and non-events for both small and large rainfall occurrences, and reliably quantify the forecast uncertainty. Future work will assess the efficacy of the post-processing method for a wider range of climatic conditions and also investigate the benefits of using post-processed rainfall forecasts for flood and short-term streamflow forecasting.

Evidence for Using Lagged Climate Indices to Forecast Australian Seasonal Rainfall

AbstractLagged oceanic and atmospheric climate indices are potentially useful predictors of seasonal rainfall totals. A rigorous Bayesian joint probability modeling approach is applied to find the cross-validation predictive densities of gridded Australian seasonal rainfall totals using lagged climate indices as predictors over the period of 1950–2009. The evidence supporting the use of each climate index as a predictor of seasonal rainfall is quantified by the pseudo-Bayes factor based on cross-validation predictive densities. The evidence strongly supports the use of climate indices from the Pacific region with weaker, but positive, evidence for the use of climate indices from the Indian region and the extratropical region. The spatial structure and seasonal variation of the evidence for each climate index is mapped and compared. Spatially, the strongest supporting evidence is found for forecasting in northern and eastern Australia. Seasonally, the strongest evidence is found from August–October to November–January and the weakest evidence is found from March–May to May–July. In some regions and seasons, there is little evidence supporting the use of climate indices for forecasting seasonal rainfall. Climate indices derived from sea surface temperature anomalies in the Pacific region show stronger persistence in the relationship with Australian seasonal rainfall totals than climate indices derived from sea surface temperature anomalies in the Indian region. Climate indices derived from atmospheric variables are also strongly supported, provided they represent the large-scale circulation. Many climate indices are found to show similar supporting evidence for forecasting Australian seasonal rainfall, leading to the prospect of combining climate indices in multiple predictor models and/or model averaging.

A Bayesian Approach to Predictor Selection for Seasonal Streamflow Forecasting

AbstractStatistical methods commonly used for forecasting climate and streamflows require the selection of appropriate predictors. Poorly designed predictor selection procedures can result in poor forecasts for independent events. This paper introduces a predictor selection method for the Bayesian joint probability modeling approach to seasonal streamflow forecasting at multiple sites. The method compares forecasting models using a pseudo-Bayes factor (PsBF). A stepwise expansion of a base model is carried out by including the candidate predictor with the highest PsBF that exceeds a selection threshold. Predictors representing the initial catchment conditions are selected on their ability to forecast streamflows and predictors representing future climate influences are selected on their ability to forecast rainfall. The final forecasting model combines selected predictors representing both initial catchment conditions and future climate influences to jointly forecast seasonal streamflows and rainfall. Applications of the predictor selection method to two catchments in eastern Australia show that the best predictors representing initial catchment conditions and future climate influences vary with location and forecast date. Antecedent streamflows are the best indicator of the initial catchment conditions. Predictors representing future climate influences are only selected for forecasts made between July and January. Indicators of El Niño dominate the selected predictors representing future climate influences. The skill of streamflow forecasts varies considerably between locations and throughout the year. Skill scores for the perennial streams of the Goulburn River catchment exceed 40% for several seasons, while for the intermittent streams in the Burdekin River catchment, the skill scores are lower.

Multisite probabilistic forecasting of seasonal flows for streams with zero value occurrences

Skillful and reliable forecasts of seasonal streamflows are highly valuable to water management. In a previous study, we developed a Bayesian joint probability (BJP) modeling approach for seasonal forecasting of streamflows at multiple sites. The approach has been adopted by the Australian Bureau of Meteorology for seasonal streamflow forecasting in Australia. This study extends the applicability of the BJP modeling approach to streams with zero flow occurrences. The aim is to produce forecasts for these streams in the form of probabilities of zero seasonal flows and probability distributions of above zero seasonal flows. We turn a difficult mathematical problem of mixed discrete‐continuous multivariate probability distribution modeling into one of continuous multivariate probability distribution modeling by treating zero flow occurrences as censored data. This paper presents the mathematical formulation and implementation of the modeling approach, methods for forecast verification, and results of a test application to the Burdekin river catchment in northern Queensland, Australia.

A Bayesian joint probability modeling approach for seasonal forecasting of streamflows at multiple sites

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.