Modellering af afløbssystemer, Usikkerheder, stokastiske modeller ecipient og Sundhed • Udvikling af metoder til at assimilere målinger fra afløbssystemet i fysisk baserede modeller som Mike Urban samt at undersøge, hvorledes de assimilerede målinger påvirkes af diverse fejlkilder. • Stokastisk forecasts af regnvandsmængder i forbindelse med afløbsmodellering, herunder real time control. Dette gøres med Greybox modellering. ræsentation af det ambitiøse forskningsprojekt Storm- and Waste water Informatics SWI..

1. Online deterministic models: Updating with
EnKF
Morten Borup
DTU Environment & Krüger A/S, Veolia VWS Denmark
Morten Grum
Krüger A/S, Veolia VWS Denmark
Peter Steen Mikkelsen
DTU Environment

2. Distributed models
Mike Urban model
of Avedøre WWTP catchment
1707 sub-catchments
6601 Manholes
7749 Pipe sections
40 Pumps
40 Basins
5 km

3. Why online distributed
models?
We have the models
Best system description we have
Can tie all system data together
Water on terrain, status of gauges,
pollution fluxes
5 km

4. Existing DA-metods for MIKE URBAN
Deterministic local updating: MOUSE UPDATE
Water levels at single points are corrected
Deterministic upstream updating: RDII updating
Upstream hydrological lumped conceptual models are updated
No Hydrodynamic global updating method exists

5. Optimal updating of distributed models
N states -> NxN crosscorrellations
(States in MIKE URBAN models: 104–106
-> 108–1012 crosscorrelations)
Minimizing the error from gauge and model uncertainty
Correction is distributed from calculated/expected spatial
correlation

6. z
Kalman Filter
q
Value
Best estimate of s knowing z: 𝑠 = 𝑠̂ +
𝜎𝑠,𝑞
𝜎 𝑞
2 (z − q)
s=x1
𝜎𝑞
2
𝜎𝑠,𝑞
𝜎𝑧
2
When z is uncertain: 𝑠 = 𝑠̂ +
𝜎𝑠,𝑞
𝜎 𝑞
2+𝜎 𝑧
2 (z − q)
When s is part of vector x: 𝒙 = 𝒙� + 𝐾(z − q)
𝑥 =
𝑥1
𝑥2
𝑥3
6
Kalman gain K determine
Δ𝑥1/Δε
Δ𝑥2/Δε
Δ𝑥3/Δε
x3x2s

7. z
Kalman Filter
q
Value
s=x1
𝜎𝑞
2
𝜎𝑧
2
x3x2
Error covariance is the cornerstone of KF
Estimated by KF using full error covariance matrix P
This requires:
1. Normal distributed Errors (including any input uncertainty)
2. Linearity
3. Limited number of states (size of P = n*n)
𝜎𝑠,𝑞
𝑷 =
𝜎 𝑥𝑥
2
𝜎 𝑥𝑥,𝑥𝑥 𝜎 𝑥𝑥,𝑥3
𝜎 𝑥2,𝑥1 𝜎 𝑥2
2
𝜎 𝑥2,𝑥3
𝜎 𝑥𝑥,𝑥𝑥 𝜎 𝑥3,𝑥𝑥 𝜎 𝑥3
2
7

8. Ensemble Kalman Filter
q
Value
Ensembles of models used to represent state uncertainty
Xi,3Xi,2Xi,1
X1
8

9. Ensemble Kalman Filter
q
Value
Ensembles of models used to represent state uncertainty
Covariance can be calculated directly from ensemble:
No need for full covariance matrix for finding K
Xi,3Xi,2Xi,1
X4
X5
X1
X2
X3
σ1,3 =
1
𝑁 − 1
�(𝑋𝑖,1 − 𝑋,1)(𝑋𝑖,3 − 𝑋,3)
𝑁
𝑖
9

10. Ensemble Kalman Filter
q
Value
Xi,3Xi,2Xi,1
𝜎𝑧
2
z
EnKF step-by-step
1.Calculate K using the relevant covariances from ensemble
2.Correct each ensemble member using K(observed-modelled)
K
10

11. Ensemble Kalman Filter
q
Value
Xi,3Xi,2Xi,1
𝜎𝑧
2
z
EnKF step-by-step
1.Calculate K using the relevant covariances from ensemble
2.Correct each ensemble member using K(observed-modelled)
3.Run model until next measurements arrives
K
11

12. EnKF vs. KF
• EnKF
Computational efficient for LARGE models
Can handle any kind of noise
Handles non-linearity
• KF
If gauss-linearity then much more efficient for not
too big models
12

13. EnKF vs. KF
• Uncertainty descriptions
13

14. Testing EnKF
• 2010 flow data from
Ballerup
• Simple 3 linear reservoir WA
model
• Only noise on first reservoir
(state proportional)
• Std. measured flow = 10%
14
0.5 h
2.5 h
2.5 h

15. Ballerup 2010
Measured and
modelled flow
when no update
15
12/7 13/7 14/7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Measured
Model

16. 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
forecast
EnKF (N=50)
Measured and
modelled flow
when no update
+
One step
prediction from
using EnKF
including 95%
prediction interval
Number of
ensemble
members N = 50
16
12/7 13/7 14/7
5

17. 12/7 13/7 14/7
50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
forecast
EnKF (N=5)
Measured and
modelled flow
when no update
+
One step
prediction from
using EnKF
including 95%
prediction interval
Number of
ensemble
members N = 5
17

18. Kalman gain K for different N
18
0 100 200 300 400 500 600 700 800
-0.5
0
0.5
1
1.5
2
2.5
x 10
4
0 100 200 300 400 500 600 700 800
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
x 10
4
K for first (red) and
second (blue)
reservoirs.
Kalman gain K much
more stable when N
is high.
N=50
N=5

19. EnKF and MU
1. Distributed models tend to be big
2. And non-linear
3. Ensembles of distributed rain estimates
Challenges
1. Manipulation of governing equations
(St. Venant eq.)
2. Slow hydrological response time

20. Updating MIKE URBAN
• New computational engine (Mike1D)
• Interface to states (h and Q)
20
𝑥 =
𝑥1
𝑥2
𝑥3
.
. .

21. Example of water level only update in
MIKE URBAN
21
Rapid change of H means Q-H
relationship instantaneously changed
-> large gradients ->instability ??

22. Summary
Ensemble based data assimilation methods are
suitable for urban runoff
The states in the new MIKE URBAN
hydrodynamic engine can be manipulated
without instability
HD data assimilation for MIKE URBAN is on the
way
22

23. Questions
23