Bayesian Model Averaging Approach for Urban Drainage Water Quality Modeling

1. Bayesian model averagingapproach for urban drainage water qualitymodelling Gabriele Freni gabriele.freni@unikore.it Università degli Studi di Enna “Kore”

2. Introduction The model approach has to be chosen in order to obtain a compromise between Complexity Accuracy • Lack in knowledge sewer quality processes • Low level of over-parameterization as well as low degree of auto-correlation and possible compensation effects among the parameters • Common available field data: TSS, BOD, COD, NH4

3. State of the art in Uncertainty Analysis • Uncertainty analysis can provide useful hints for evaluating model reliability in dependence with data availability • getting knowledge on thesources of errorsin the modelling process todefinepriorities for model improvement (Willems, 2005) • for assessing riskswhen model results are used on the basis of decisions • quantitative uncertainty analysis can provide an illuminating role to helptarget data gatheringefforts (Frey, 1992). • The evaluation of parameter uncertainties is necessary to estimate theimpact of these onmodel performance(Beck, 1987). • BUT…. • Sometimes the modeller ends up with more questions than at the beginning

4. Few possible model alternatives 8 modelling approaches have been analysed considering the combination of 4build-up models and 2 wash-off models:

5. Few possible structural alternatives Parameters variation range:

6. Fossolo (near Bologna - Italy) The experimental catchment: Fossolo Catchment Bologna (IT) • 12 recorded eventsfrom 22/4/94 to 21/8/97 • Drained area 40,71 ha, with an impervious percentage of 75% • The drainage network ends in a polycentric section pipe 144 cm high and 180 cm wide  200 m N

7. Fossolo (near Bologna) Event 21st August 1997 • ADWP = 3 [d] • Imax = 56 [mm/h] • Iaver = 16 [mm/h] • Qmax = 1 [m3/s] • Ttot = 100 [min]

8. Final Bayesian update (with all the ev.) SIM_01 SIM_02 SIM_03 SIM_04 SIM_05 SIM_06 SIM_07 SIM_08 (using all the database)

9. Which model to choose? NO Maybe Maybenot NO NO YES Maybe Maybenot

10. Which model to choose? • Best efficiency is not informative: several modelling structure have similar efficiencies (model structure equifinality)

11. Model Averaging techniques (MA) Y Y Y Y Y Y Model 4 Average Model 1 Model 3 Model 2 Model 5 t t t t t t w3 w1 w2 w5 w4

12. The Bayesian Model Average (BMA) Posterior probability is obtained as conditional sum of the posterior probabilities provided by each possible model: Each model is treated by Bayesian uncertainty analysis: Weights are computed by Bayesian uncertainty analysis

13. BMA application to Fossolo -83.4% -14.1% -38.3% with respect to the average uncertainty band

14. Conclusions • Model structure is responsible for a relevant part of the uncertainty • …. And more significantly, it is responsible of unhandled uncertainties because, usually, model selection is done before the uncertainty analysis • Model selection may depend on too many case-specific variables • BMA can help to run models safely and reduce the overall uncertainty • Computational cost is a limitation (we have to run a Bayesian analysis on each possible model)

15. Bayesian model averagingapproach for urban drainage water qualitymodelling Gabriele Freni gabriele.freni@unikore.it Università degli Studi di Enna “Kore”

16. BMA application to Fossolo

17. Net rainfall (S, f) Rainfall - runoff processes Inlet sewer hydrograph Sewer propagation outlet sewer hydrograph The mathematical model Rainfall Quantity module The choice of the cascade of two linear reservoirs in series and a linear channel allows to split the hydraulic phenomena in the catchment from those in the sewer system.

18. The mathematical model Qualitymodule

19. Bayesian update (after using 6 ev.) SIM_01 SIM_02 SIM_03 SIM_04 SIM_05 SIM_06 SIM_07 SIM_08