HYDROLOGIC FORECAST PRODUCTS from BAYESIAN FORECASTING SYSTEM

Transcription

1 HYDROLOGIC FORECAST PRODUCTS from BAYESIAN FORECASTING SYSTEM Roman Krzysztofowicz University of Virginia USA Presented at the CHR-WMO Workshop-Expert Consultation on Ensemble Prediction and Uncertainty in Flood-Forecasting Bern, Switzerland 31 March 00 Acknowledgments: National Science Foundation, Grant No. ATM National Weather Service

2 BAYESIAN FORECASTING THEORY The Bayesian theory (Krzysztofowicz, 1999) provides a general methodological framework for probabilistic forecasting via any deterministic hydrologic model. Within this framework, a variety of Bayesian Forecasting Systems (BFSs) suited to different purposes can be developed. The first prototype systems were developed to produce short-term probabilistic forecasts of river stages, stage transitions, and floods based on probabilistic quantitative precipitation forecasts (PQPFs). Herein we show examples of probabilistic hydrologic forecast products.

3 NOTATION n H n h n index of times predictand: actual river stage realization of H n CASE STUDY Forecast point: Eldred, Pennsylvania Allegheny River Drainage area: 550 miles ( km ) PQPF: probability of precipitation occurrence in 4-h distribution of 4-h basin average amount occurrence expected disaggregation of amount into -h subperiods

4 PROBABILISTIC FORECAST PRODUCTS PRSF Probabilistic River Stage Forecast n h n P H n h n PSTF Probabilistic Stage Transition Forecast (Markov) Θ n h n h n 1 P H n h n H n 1 h n 1 PFF Probabilistic Flood Forecast F n h P Z n h Z n max H 1,...,H n

5 PRSF: Predictive Distribution of Hn n 1 lead time 4 h n lead time 48 h n 3 lead time 7 h (a) 1.0 Probability Ψ 1 (h 1 ) Actual River Stage h 1 [ft] (b) 1.0 Probability Ψ (h ) (c) 1.0 Probability Ψ 3 (h 3 ) Actual River Stage h [ft] Actual River Stage h 3 [ft]

6 PRSF: Predictive Density of Hn n 1 lead time 4 h (a) Density ψ 1 (h 1 ) n lead time 48 h n 3 lead time 7 h (b) Density ψ (h ) (c) Density ψ 3 (h 3 ) Actual River Stage h 1 [ft] Actual River Stage h [ft] Actual River Stage h 3 [ft]

7 PRSF: Predictive Density of Hn max = 3.81 n = 1 max =.3 n = max = 1.45 n = 3 max = 0.90 n = 4 hours 1 hours hours 4 hours Density ψ n (h n ) n = 5 n = n = 7 n = 8 hours 3 hours 4 hours 48 hours Density ψ n (h n ) n = 9 n = n = 11 n = 1 54 hours 0 hours hours 7 hours Density ψ n (h n ) Actual River Stage h n [ft] Actual River Stage h n [ft] Actual River Stage h n [ft] Actual River Stage h n [ft]

8 PROBABILISTIC FORECAST PRODUCTS PRSF Probabilistic River Stage Forecast n h n P H n h n PSTF Probabilistic Stage Transition Forecast (Markov) Θ n h n h n 1 P H n h n H n 1 h n 1 PFF Probabilistic Flood Forecast F n h P Z n h Z n max H 1,...,H n

9 PSTF: Predictive 1-Step Markov Transition n 1 lead time 4 h n lead time 48 h n 3 lead time 7 h (a) 1.0 Probability Ψ 1 (h 1 ) h 0 = 7.9 Ac tual R iv er Stage h 1 [ft] (b) 1.0 Probability Θ (h h 1 ) h 1 =, 8,, 1 Ac tual R iv er Stage h [ft] (c) 1.0 Probability Θ * 3 (h 3 h ) h =, 8,, 1 Ac tual R iv er Stage h 3 [ft]

10 PSTF: Predictive 1-Step Markov Transition n 1 lead time 4 h (a) h 0 = 7.9 Density ψ 1 (h 1 ) n lead time 48 h n 3 lead time 7 h Ac tual R iv er Stage h 1 [ft] (b) Density θ (h h 1 ) h 1 =, 8,, 1 Ac tual R iv er Stage h [ft] (c) Density θ * 3 (h 3 h ) h =, 8,, 1 Ac tual R iv er Stage h 3 [ft]

11 PSTF: Predictive 1-Step Markov Transition n = 1 hours n = 1 hours n = 3 hours n = 4 4 hours * n = 5 n = n = 7 n = 8 hours 3 hours 4 hours 48 hours * n = 9 n = n = 11 n = 1 54 hours 0 hours hours 7 hours *

12 PROBABILISTIC FORECAST PRODUCTS PRSF Probabilistic River Stage Forecast n h n P H n h n PSTF Probabilistic Stage Transition Forecast (Markov) Θ n h n h n 1 P H n h n H n 1 h n 1 PFF Probabilistic Flood Forecast F n h P Z n h Z n max H 1,...,H n

13 PFF: Exceedance of Max River Stage F n h P Z n h Z n max H 1,...,H n Max River Stage h [ft] n = 1 hours n = 1 hours n = 3 hours n = 4 4 hours Max River Stage h [ft] n = 5 hours n = 3 hours n = 7 4 hours n = 8 48 hours Max River Stage h [ft] n = 9 54 hours n = 0 hours n = 11 hours n = 1 7 hours _ 1.0 Probability F n (h) _ 1.0 Probability F n (h) _ 1.0 Probability F n (h) _ 1.0 Probability F n (h)

14 PFF: Isoprobability Time Series of Quantiles (and Credible Intervals) of Max River Stage Max River Stage h [ft] Exceedance Probability observed Lead Time [n] [hours]

15 PFF: Distribution Function of Time to Flooding P T h t n F n h n 1,...,1 1.0 _ Probability P(T(h) t n ) = F n (h) h = ft h = ft h = ft Lead Time [n] [hours]

16 REFERENCES Krzysztofowicz, R., Bayesian Theory of Probabilistic Forecasting via Deterministic Hydrologic Model, Water Resources Research, 35(9), , Kelly, K.S. and Krzysztofowicz R., Precipitation Uncertainty Processor for Probabilistic River Stage Forecasting, Water Resources Research, 3(9), 43 53, 000. Krzysztofowicz, R. and Kelly, K.S., Hydrologic Uncertainty Processor for Probabilistic River Stage Forecasting, Water Resources Research, 3(11), , 000. Krzysztofowicz, R. and Herr, H.D., Hydrologic Uncertainty Processor for Probabilistic River Stage Forecasting: Precipitation-Dependent Model, Journal of Hydrology, 49(1 4), 4 8, 001. Krzysztofowicz, R., Integrator of Uncertainties for Probabilistic River Stage Forecasting: Precipitation-Dependent Model, Journal of Hydrology, 49(1 4), 9 85, 001. Krzysztofowicz, R., Bayesian System for Probabilistic River Stage Forecasting, Journal of Hydrology, 8(1 4), 1 40, 00. Krzysztofowicz, R., Probabilistic Flood Forecast: Bounds and Approximations, Journal of Hydrology, 8(1 4), 41 55, 00. Maranzano, C.J. and Krzysztofowicz, R., Identification of Likelihood and Prior Dependence Structures for Hydrologic Uncertainty Processor, Journal of Hydrology, 90(1 ), 1 1, 004. Krzysztofowicz, R. and Maranzano, C.J., Hydrologic Uncertainty Processor for Probabilistic Stage Transition Forecasting, Journal of Hydrology, 93(1 4), 57 73, 004. Krzysztofowicz, R. and Maranzano, C.J., Bayesian System for Probabilistic Stage Transition Forecasting, Journal of Hydrology, 99(1 ), 15 44, 004.

17 * * *