1. Bayesian model averaging approach for urban drainage water quality modelling
Gabriele Freni
Università degli
Studi di Enna “Kore”

2. 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

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 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

4. Few possible model alternatives
8 modelling approaches have been analysed considering the combination of 4 build-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 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

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
Maybe not NO NO
YES
Maybe
Maybe not

10. 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

11. 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

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
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

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 averaging approach for urban drainage water quality modelling
Gabriele Freni
Università degli
Studi di Enna “Kore”

16. 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

17. 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.

18. The mathematical model
Quality module

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