Uncertainty assessment of streamflow simulation

Uncertaintyy assessment of streamflow simulation on ungauged catchments using a distributed model and the Kalman filter based MISP algorithm

Authors: Juan Camilo Múnera1, Félix Francés1, Ezio Todini2 Technical University of Valencia (Spain) Research Institute of Water Engineering and Environment Research Group of Hydrological and Environmental Modeling http://lluvia.dihma.upv.es (1)

University of Bologna (Italy) Earth Sciences and Environment Department htt // http://www.geomin.unibo.it/ i ib it/ (2)

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Introduction 

IImportant t t role l off Predictive P di ti U Uncertainty t i t (PU) on flood fl d forecasting f ti and d decision making:  Flow simulations and forecasting made from models are not error free  Growing G i interest i i assessing in i uncertainty i off predictions/simulations di i / i l i  Severe economic and social consequences derived from flood emergencies



Developed methods and tools to assess flood forecasting uncertainty at gauging stations:  Hydrologic Uncertainty Processor (Krzysztofowicz, 1999)  Bayesian Model Averaging (Raftery, (Raftery 1993; Raftery et al, al 2003; 2005)  Model Conditional Processor (Todini, 2008)  And others statistical approaches…



2

How to estimate PU at ungauged sites?

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Introduction 

Our proposal: Combining simulations of the hydrological distributed model TETIS (Vélez et al, 2001; Francés et al, 2007) with the MISP technique (Mutually Interactive State-Parameter Estimation) (Todini, 1978)  Distributed models: advantage of simulating flows at any point throughout the spatial domain  MISP is a Kalman filter based algorithm, which:





Performs the state-parameter estimation of a discrete time dynamic system



Makes alternative use of two interacting filters in parallel, parallel both with minimum variance

Filtering Scheme:  Model predictions at an ungauged site are assumed as imperfect observations

(as random variables)  Incorporating observations and simulations at a gauging station as on-line instrumental variables correlated with flow at the ungauged site  Log transformation of all input data to improve Gaussian assumptions of the K.F.  Inclusion of a cross covariance term in the covariance matrix of the measurement error (assumption of spatially correlated errors)

3

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Hydrological model TETIS 

Developed at Technical University of Valencia since 1994



Spatially distributed in regular cells  Reproduces spatial variability of hydrological processes  Reduces spatial scale effects with regard to lumped models  Allows to exploit all available physical and environmental spatial information Flow

Hillslope



Gully

Main river

Separate runoff modeling on slopes and channels  Each tank drains into the topographical downstream corresponding tank.  Drainage area thresholds defined for each type of tank

 4

Nonlinear channel routing scheme (geomorphologic ( kinematic wave). ) EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Hydrological model TETIS 

Robust and p parsimonious model:  Adequate simulation of initial state (includes

balance at all times)  6 storage tanks (state variables)  5 external outflows (3 horizontal responses) 

Potential problem in distributed models:  Calibration of high number of parameters in each

cell from an outflow hydrograph.  Proposed solution: Split Effective Parameters

Structure (Frances et al, 2007): 



5

Phase I: Parameter estimation from all physical and environmental information at each cell

H u (i ) 

Phase II: Global correction factors for each parameter map (effective parameters)

H u (i )  R 1

Calibration

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Production parameters at each cell (v (v.7) 7)

6



Vegetation cover index: (t)

Crop factor



Maximum static capacity: Hu

Initial abstractions + upper soilil capillary ill capacity it



Overland flow velocity: v

Hill slope stationary velocity



Interflow velocity: kss

Horizontal macropore upper soil permeability



Base flow velocity: kb

Upper aquifer permeability



Infiltration capacity: ks

Vertical upper soil permeability



Percolation capacity: p y kp

Vertical deep p soil p permeability y



Underground losses capacity: kpp Lower aquifer permeability

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Calibration Process Parameter estimation normallyy through g comparison p between simulated and observed values of some state variables 50

0

Ppt Mean Simulated Observed

40





Balance Error

BE (%) 

Root mean square Error

 Si  Q i   Qi

RMSE 

9:10:00

 Statistical Objective Functions:

6:10:00

10

5:10:00

9

0

4:10:00

8

5

3:10:00

7

10

2:10:00

6

15

1:10:00

5

20

0:10:00

4

21:10:00

 Graphic comparison

3

25

23:10:00

Model performance assessment:

2

30

22:10:00



Discharge[m³/s]

35

1

Precipitation [m m]

45

8:10:00

 Traditionally: Discharge at the basin outlet

7:10:00



Time [hours]

 100

1 n

n

 Q

 Si 

2

i

i 1

 Q  S  NSE  1   Q  Q 

2



Efficiency Index (Nash-Sutcliffe)

i

2

i



7

i

m

Automatic calibration of correction factors: SCE-UA (Duan et al., 1992; Sorooshian et al, 1993) EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Mutually interactive parameter estimation (MISP) 

The first filter performs the minimum variance state estimation, estimation given the parameters set : x t   t t 1x t 1  t 1w t 1

z t  H t xt  vt where:



xt: State vector (nx1) Measurement vector (mx1) zt: : State transition matrix (nxn) , H: Compatibility matrices vt, wt: Normal independent processes

The second one performs the minimum variance parameter estimation, given the state estimate, both at the present and previous time steps.  t   t 1  t*1w *t 1 z *t  H *t  t  v *t where:

8

t: Parameter vector (px1) EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Mutually interactive parameter estimation (MISP) 

Equations required to accomplish a calculation cycle: Parameters update:

State update: v t  z t  H t xˆ t t 1  v t 1



K t  Pt t 1 H Tt H t Pt t 1 H Tt  R t t



1

xˆ t t  xˆ t t 1  K t v t

Pt t  I  K t H t Pt t 1

Time update: xˆ t 1 t   t 1 t xˆ t t  t 1 w t Pt 1 t   t 1 t Pt t  Tt 1 t  t 1 Q t t tT1

9

 t  H t K t  t R t  H t K t C t t K Tt H Tt C t   H t Ptt 1 H t T  R t t

 

K t  Ptt 1 H t T C t 

1

ˆ t  1 t  ˆ t t  1  K t  t  t 1 w  Pt 1 t  Ptt  1  K t C t  K t T  t 1 Q  t 1

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Mutually interactive parameter estimation (MISP) 

Equations required to accomplish a calculation cycle: Parameters update:

State update: v t  z t  H t xˆ t t 1  v t 1



K t  Pt t 1 H Tt H t Pt t 1 H Tt  R t t



1

xˆ t t  xˆ t t 1  K t v t

Pt t  I  K t H t Pt t 1

Time update: xˆ t 1 t   t 1 t xˆ t t  t 1 w t Pt 1 t   t 1 t Pt t  Tt 1 t  t 1 Q t t tT1

 t  H t K t  t R t  H t K t C t t K Tt H Tt C t   H t Ptt 1 H t T  R t t

 

K t  Ptt 1 H t T C t 

1

ˆ t  1 t  ˆ t t  1  K t  t  t 1 w  Pt 1 t  Ptt  1  K t C t  K t T  t 1 Q  t 1

ˆ t t ) is used as  The estimation of the state vector provided by the first filter ( x observations in order to estimate the parameter vector . observations, 

 Optimality is reached after a series of runs through the historical data, as model

residuals become less and less correlated.  At the th same time, ti it is i possible ibl to t estimate ti t the th unknown k noise i statistics: t ti ti 10

w , v, Q, R

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

MISP implementation 

System equations (matrix form):

x t   t t 1x t 1  t 1w t 1

z t  H t xt  vt



 z 1 t    1     z 2 t    0    z 3 t   0

Parameter equation:  t   t 1  t*1 w *t1 z *t  H *t  t  v *t

11

Q 2 t   11 12 Q   2   t  1   1 0 Q1t    0 0   Q1t 1   0 0 Q s1    0 0 t    Q  s1t 1   0 0

t

12   Q 2 t 1    0 0 0 0   Q 2 t  2    w 2 t 1  12  22 12  22   Q1t 1      t 1  w1t 1   1 0 0 0   Q1t  2    w s1  t 1  3 3  Q   0 0 1  2 s1t 1   Q 0 0 1 0   s1t  2   Q 2  t   Q  2 t 1    v 1 t   0 0 0 0 0   Q   1 t  0 1 0 0 0     v 2 t   Q1t 1   v3  0 0 0 1 0  t     Q s1t    Q  11   s1t 1   1 Ungauged  2  Site (2)    Gauged  3 site (1)  1  3   2 t 11

12

11

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Case study: simulation at Baron Fork 

Distributed Model Intercomparison Project (DMIP2), NOAA/NWS.  Series of experiments to guide NOAA/NWS research into advanced hydrologic

models for river and water resources forecasting. 

Study basins: Baron Fork river and Peacheater Creek (tributary)  Complete description (Smith et al, 2004)  Availability of cartographic information of physical and environmental parameters  Concurrent time series (1995-2002) of discharges, radar precipitation (NEXRAD),

temperature and ETP from Reanalysis (NCEP-NCAR) 

Basin areas  Baron Fork: 795 km2  Peacheater: 65 km2

12

g g Site ((2)) Ungauged Peacheater Creek Gauged Site (1) Baron Fork at Eldon

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TETIS model calibration ď Ž

Calibration results, Baron Fork at Eldon (oct/2001-sep/2002) 500

Observed Flow

450

Statistical Index BE (%):

-14.6

400

RMSE

6.61

350

NSE:

Flow (m3/s)

Simulated Flow (Tetis model)

((m3/s): )

0.91

300 250 200 150 100 50

Date

13

02/10/2002 2

01/09/2002 2

02/08/2002 2

02/07/2002 2

02/06/2002 2

02/05/2002 2

02/04/2002 2

02/03/2002 2

31/01/2002 2

31/12/2001 1

01/12/2001 1

31/10/2001 1

01/10/2001 1

0

B.F. Eldon

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TETIS model calibration ď Ž

14

Calibrated correction factors (cell parameters) for the DMIP2 case study

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

TETIS model validation ď Ž

Temporal validation results, Baron Fork at Eldon (oct/1996-sep/2001) 1000

Statistical Index

Observed Flow 900

Simulated Flow (Tetis model)

800

BE (%):

-11.2

RMSE

14.89

((m3/s): )

NSE:

0.83

Flow (m3//s)

700 600 500 400 300 200 100

01/10/99

01/08/99

01/06/99

01/04/99

30/01/99

01/12/98

01/10/98

01/08/98

01/06/98

01/04/98

30/01/98

30/11/97

01/10/97

01/08/97

01/06/97

01/04/97

30/01/97

30/11/96

01/10/96

0

B.F. Eldon

Date

15

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TETIS model validation ď Ž

Spatial validation results, (Peacheater Cr. At Christie (oct/1996-sep/2002) 50.0

Statistical Index

Observed Flow

45.0

Simulated Flow (Tetis model) 40.0

9.3

RMSE

0.88

((m3/s): )

NSE:

35.0

0.67

30.0

3

Flow (m m /s)

BE (%):

25.0 20.0 15.0 10.0 5.0

01/10/2001

31/08/2001

01/08/2001

01/07/2001

01/06/2001

01/05/2001

01/04/2001

01/03/2001

30/01/2001

30/12/2000

30/11/2000

30/10/2000

30/09/2000

31/08/2000

31/07/2000

01/07/2000

31/05/2000

01/05/2000

31/03/2000

01/03/2000

30/01/2000

31/12/1999

30/11/1999

31/10/1999

01/10/1999

0.0 Peacheater Cr.

Date

16

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Results application of MISP Results, ď Ž

MISP results, Peacheater Cr. (oct/1996-sep/2002) 50.0

Observed flow unknown)

(assumed

Updated flow (MISP K.F.) 40.0

Statistical Index BE (%)

9.3

RMSE

0.83

((m3/s))

NSE

0.71

Flow (m 3/s)

30.0

20.0

10.0

Peacheater Cr.

02/10/01

01/09/01

02/08/01

02/07/01

02/06/01

02/05/01

02/04/01

02/03/01

31/01/01

31/12/00

01/12/00

31/10/00

01/10/00

31/08/00

01/08/00

01/07/00

01/06/00

01/05/00

01/04/00

01/03/00

31/01/00

31/12/99

01/12/99

31/10/99

01/10/99

0.0

Date

17

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Results application of MISP Results, Peacheater Creek (15/03/2002 – 29/04/2002) 20 0 20,0

500 0 500,0 Observed flow Peacheater (assumed unknown)

18,0 16,0

400,0

Updated flow Peach Peach. (MISP K K.F.) F)

14,0

Flow (m3/s)) Peacheater C Cr.

450,0

Simulated flow Peach. (Tetis)

350,0

Observed flow (Baron Fork)

12,0

300,0

10,0

250,0

8,0

200,0

6,0

150,0

4,0

100,0

2,0 ,

50,0 ,

Flow (m3//s) Baron Fo ork



Peacheater Cr.

29/04/2002

24 4/04/2002

19/04/2002

4/04/2002 14

09/04/2002

4/04/2002 04

30/03/2002

25/03/2002

20/03/2002

0,0

15/03/2002

0,0

B.F. Eldon

Date

18

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Results application of MISP Results, Peacheater Creek, uncertainty bounds (15/03/2002 – 29/04/2002)



30 0 30,0

Observed flow Peacheater (assumed unknown) Updated flow Peach. (MISP K.F.)

25,0

Lower Uncert. Limit (5%)

Flow (m3/s)

20,0

Upper Uncert. Limit (95%)

15,0

10,0

5,0

Peacheater Cr.

29/04/2002

24/04/2002

19/04/2002

14/04/2002

09/04/2002

04/04/2002

30/03/2002

25/03/2002

20/03/2002

15/03/2002

0,0 B.F. Eldon

Date

19

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Results application of MISP Results, 

Peacheater Creek (06/03/1999 – 17/05/1999) 15 0 15,0

300

Observed flow Peacheater (assumed unknown) Simulated flow Peach. (Tetis) 12,0

240

U d t d fl Updated flow P Peach. h (MISP K K.F.) F)

9,0

180

6,0

120

3,0

60

Flow (m3/s) Baron Fo ork

Flow (m3/s) Peacheater C Cr.

Observed flow Baron Fork

Peacheater Cr.

0,0 15/05/1999 1

10/05/1999 1

05/05/1999 0

30/04/1999 3

25/04/1999 2

20/04/1999 2

15/04/1999 1

10/04/1999 1

05/04/1999 0

31/03/1999 3

26/03/1999 2

21/03/1999 2

16/03/1999 1

11/03/1999

06/03/1999 0

0 B.F. Eldon

Date

20

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Results application of MISP Results, 

Peacheater Creek, uncertainty bounds (06/03/1999 – 17/05/1999) 25,0

Observed flow Peacheater (assumed unknown) Updated flow (MISP K.F.) 20,0

Lower Uncert.

Uncert Limit (5%)

Flow (m3/s s) Peacheaterr Cr.

Upper Uncert. Limit (95%) 15,0

10,0

5,0

Peacheater Cr.

15/05/1999

10/05/1999

05/05/1999

30/04/1999

25/04/1999

20/04/1999

15/04/1999

10/04/1999

05/04/1999

31/03/1999

26/03/1999

21/03/1999

16/03/1999

11/03/1999

06/03/1999

0,0 B.F. Eldon

Date

21

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Results application of MISP Results, 

Peacheater Creek (15/06/2000 – 10/07/2000) 60

1800 Observed flow (unknown)

Simulated flow (Tetis)

50

1500

Updated flow (MISP K.F.)

Qobs3_BF

30

900

20

600

10

300

Baron Fork

1200

Flow (m3//s)

Flow (m3/s) Peacheaterr Cr.

40

Peacheater Cr.

22

10//07/2000

05//07/2000

Date

30//06/2000

25//06/2000

20//06/2000

0 15//06/2000

0

B.F. Eldon

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Results application of MISP Results, 

Peacheater Creek, uncertainty bounds (15/06/2000 – 10/07/2000) 120 0 120,0

Observed flow Peacheater (assumed unknown) Updated flow Peach.

(MISP K.F.)

100,0

Lower Uncert. Limit (5%)

Flow (m3//s) Peacheaterr Cr.

80,0

Upper Uncert. Limit (95%)

60,0

40,0

20,0

Peacheater Cr.

10 0/07/2000

05 5/07/2000

30 0/06/2000

25 5/06/2000

20 0/06/2000

15 5/06/2000

0,0 B.F. Eldon

Date

23

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Conclusions

24



A methodology gy based on the Kalman filter MISP algorithm g is p presented,, aimed to improve predictions performed with a distributed hydrological model at ungauged sites and to estimate the related predictive uncertainty.



The observed and simulated discharges at the basin outlet are used as instrumental variables in the Kalman filter implementation.



Importance of including a cross covariance error term in matrix R.



The results of the proposed approach are considered very satisfactory. The NSE was improved from 0.67 to 0.71 for the Peacheater Creek (assumed as ungauged site).



The Kalman filter based MISP approach allows assess the predictive uncertainty associated i d to the h model d l prediction di i (simulation ( i l i or forecasting f i mode). d )



The uncertainty band is strongly affected by model performance, and is expected that any improvement i t off the th model d l would ld reduce d predictive di ti uncertainties. t i ti EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011

Thank you for your attention!

Contact:

Juan Camilo MĂşnera E-mail: juamues1@doctor.upv.es juancmunera@yahoo.es

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Technical University of Valencia (Spain) Research Institute of Water Engineering and Environment Research Group of Hydrological and Environmental Modeling http://lluvia.dihma.upv.es

EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011