Bayesian model averaging approach for urban drainage water quality modelling Gabriele Freni gabriele.freni@unikore.it Università degli Studi di Enna “Kore”

Introduction The model approach has to be chosen in order to obtain a compromise between Accuracy Complexity 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

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 the sources of errors in the modelling process to define priorities for model improvement (Willems, 2005) for assessing risks when model results are used on the basis of decisions quantitative uncertainty analysis can provide an illuminating role to help target data gathering efforts (Frey, 1992). The evaluation of parameter uncertainties is necessary to estimate the impact of these on model performance (Beck, 1987). BUT…. Sometimes the modeller ends up with more questions than at the beginning

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

Few possible structural alternatives Parameters variation range:

Fossolo (near Bologna - Italy) The experimental catchment: Fossolo Catchment Bologna (IT) 12 recorded events from 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

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]

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)

Which model to choose? NO Maybe Maybe not NO NO YES Maybe Maybe not

Average uncertainty band width for TSS concentration [mg/l] Which model to choose? Best efficiency is not informative: several modelling structure have similar efficiencies (model structure equifinality) Model Average uncertainty band width for TSS concentration [mg/l] 1 2 3 4 5 6 7 8 9 10 11 12 SIM_01 543.4 673.5 652.4 834.2 542.2 632.5 765.4 543.5 657.3 643.2 413.3 SIM_02 488.2 601.2 511.2 406.5 405.4 354.5 313.4 698.6 531.2 332.3 411.2 365.4 SIM_03 437.3 566.3 401.2 322.7 311.2 300.2 214.5 665.6 489.7 320.2 326.9 846.1 SIM_04 645.6 768.7 896.5 554.3 324.3 767.4 876.3 564.3 675.3 875.3 923.1 513.2 SIM_05 603.4 753.2 611.2 743.2 456.3 534.2 769.3 1002.4 420.5 SIM_06 223.5 324.5 356.4 453.8 275.2 109.2 265.4 278.6 365.5 217.3 163.2 SIM_07 214.8 254.3 403.4 539.4 304.5 543.2 453.9 322.0 304.3 344.2 301.2 245.6 SIM_08 238.3 278.5 421.9 512.2 224.5 435.7 510.2 455.2 403.3 244.7 257.7

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

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

BMA application to Fossolo Model 12 SIM_01 413.3 SIM_02 365.4 SIM_03 846.1 SIM_04 513.2 SIM_05 420.5 SIM_06 163.2 SIM_07 245.6 SIM_08 257.7 BMA 140.5 -83.4% -14.1% -38.3% with respect to the average uncertainty band

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)

Bayesian model averaging approach for urban drainage water quality modelling Gabriele Freni gabriele.freni@unikore.it Università degli Studi di Enna “Kore”

BMA application to Fossolo Model Weights 1 2 3 4 5 6 7 8 9 10 11 12 SIM_01 0.06 0.07 0.12 0.09 0.08 0.11 SIM_02 0.1 0.13 0.21 0.16 0.18 0.04 0.17 0.15 0.14 SIM_03 0.28 0.05 SIM_04 0.03 0.02 SIM_05 SIM_06 0.24 0.19 0.25 0.26 0.2 SIM_07 0.23 SIM_08 BANDS 175.68 219.47 238.5 244.89 174.47 90.67 206.86 261.56 241.31 239.09 212.32 168.97

The mathematical model Rainfall Net rainfall (S, f) Quantity module Rainfall - runoff processes Inlet sewer hydrograph Sewer propagation outlet sewer hydrograph 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.

The mathematical model Quality module

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