Uncertainty assessment of streamflow simulation

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

Authors:Authors:Juan Camilo Múnera1, Félix Francés1, Ezio Todini2

(1) Technical University of Valencia (Spain)Research Institute of Water Engineering and EnvironmentResearch Group of Hydrological and Environmental Modelinghttp://lluvia.dihma.upv.eshttp://lluvia.dihma.upv.es

(2) University of Bologna (Italy)Earth Sciences and Environment Departmenthtt // i ib it/

EGU Leonardo Conference Series on the Hydrological CycleFLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS

Bratislava, Slovakia, 23-25 November 2011

http://www.geomin.unibo.it/

IntroductionIntroduction

I t t l f P di ti U t i t (PU) fl d f ti d Important role of Predictive Uncertainty (PU) on flood forecasting anddecision making: Flow simulations and forecasting made from models are not error free

G i i i i i f di i / i l i Growing interest in assessing uncertainty of predictions/simulations Severe economic and social consequences derived from flood emergencies

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

How to estimate PU at ungauged sites?



IntroductionIntroduction

Our proposal: Combining simulations of the hydrological distributed model TETIS(Vélez et al, 2001; Francés et al, 2007) with the MISP technique (Mutually InteractiveState-Parameter Estimation) (Todini, 1978) Distributed models: advantage of simulating flows at any point throughout the

spatial domainspatial domain MISP is a Kalman filter based algorithm, which:

Performs the state-parameter estimation of a discrete time dynamic systemMakes alternative use of two interacting filters in parallel both with minimum variance Makes alternative use of two interacting filters in parallel, both with minimum variance

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



Hydrological model TETISHydrological 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 Reduces spatial scale effects with regard to lumped models Allows to exploit all available physical and environmental spatial information

Flow

Separate runoff modeling on slopes and channels

Hillslope Gully Main river

Each tank drains into the topographical downstream corresponding tank. Drainage area thresholds defined for each type of tank

( )EGU Leonardo Conference Series on the Hydrological Cycle

FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONSBratislava, Slovakia, 23-25 November 20114

Nonlinear channel routing scheme (geomorphologic kinematic wave).

Hydrological model TETISHydrological model TETIS

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

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

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

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

Structure (Frances et al, 2007):

Phase I: Parameter estimation from all physical iH )(

1u RiH )(

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

Phase II: Global correction factors for each Calibration

u iH )(



1u )(

5

parameter map (effective parameters)

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

Vegetation cover index: (t) Crop factor Maximum static capacity: Hu Initial abstractions +

il ill itupper soil capillary capacity

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: kp Vertical deep soil permeabilityp y p p p y Underground losses capacity: kpp Lower aquifer permeability



Calibration ProcessCalibration Process

Parameter estimation normally through comparison between simulated andy g pobserved values of some state variables Traditionally: Discharge at the basin outlet 0

1

2

335

40

45

50

m]

Ppt MeanSimulatedObserved

Model performance assessment: Graphic comparison

4

5

6

7

8

95

10

15

20

25

30

Prec

ipita

tion

[m

Disc

harg

e[m

³/s]

Statistical Objective Functions:

Balance Error

100QQS

(%)BE ii

100

21:1

0:00

22:1

0:00

23:1

0:00

0:10

:00

1:10

:00

2:10

:00

3:10

:00

4:10

:00

5:10

:00

6:10

:00

7:10

:00

8:10

:00

9:10

:00

Time [hours]

Root mean square Error

n

1i

2ii SQ

n1RMSE

2ii SQ

Qi

Efficiency Index (Nash-Sutcliffe)

Automatic calibration of correction factors: SCE-UA (Duan et al., 1992;

2mi

ii

QQ

SQ1NSE



Sorooshian et al, 1993)

Mutually interactive parameter estimation (MISP)

The first filter performs the minimum variance state estimation given the The first filter performs the minimum variance state estimation, given theparameters set :

tttttt

vxHz

wxx

1111

where:tttt vxHz

xt: State vector (nx1)zt: Measurement vector (mx1): State transition matrix (nxn), H: Compatibility matricesvt, 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.

** w

where:

*tt

*t

*t

tttt

vHzw

111

t: Parameter vector (px1)




Equations required to accomplish a calculation cycle: Equations required to accomplish a calculation cycle:

1t1ttttt vx̂Hzv

1

State update: tttt KH

Parameters update:

1tttttt

tt1tttt

ttTt1ttt

Tt1ttt

PHKIP

vKx̂x̂

RHPHHPK

1

T

ttTt1tttt

Tt

Ttttttt

CHPK

RHPHC

HKCKHR

1 1tttttt

t1tttt1tt1t wx̂x̂ Time update:

T

1ttt1ttt1t

tTt1ttt

QKCKPP

wKˆˆ

CHPK

T1ttt1t

Tt1tttt1tt1t QPP

1t1tttt1ttt1t QKCKPP




Equations required to accomplish a calculation cycle: Equations required to accomplish a calculation cycle:

1t1ttttt vx̂Hzv

1

State update: tttt KH

Parameters update:

1tttttt

tt1tttt

ttTt1ttt

Tt1ttt

PHKIP

vKx̂x̂

RHPHHPK

1

T

ttTt1tttt

Tt

Ttttttt

CHPK

RHPHC

HKCKHR

1 1tttttt

t1tttt1tt1t wx̂x̂ Time update:

T

1ttt1ttt1t

tTt1ttt

QKCKPP

wKˆˆ

CHPK

The estimation of the state vector provided by the first filter ( ) is used asobservations in order to estimate the parameter vector

T1ttt1t

Tt1tttt1tt1t QPP

ttx̂

1t1tttt1ttt1t QKCKPP

observations, in order to estimate the parameter vector . Optimality is reached after a series of runs through the historical data, as model

residuals become less and less correlated.At th ti it i ibl t ti t th k i t ti ti RQ



At the same time, it is possible to estimate the unknown noise statistics: R,Q,v,w

MISP implementationMISP implementation

System equations (matrix form):

12

11

22

12

22

21

22

21

12

11

12

11

12

11

1

12

2

00000001

tt

t

t

t

t

t

wQQQ

QQQ

1111 tttttt wxx

11

111

21

11

21

32

31

2121

11

1

11

0100000000

000100ts

tt

ts

ts

t

ts

ts

t ww

QQQ

QQQ

tt

t

t

t

vv

QQQ

zz 1

1

12

2

1

000100000001

tttt vxHz

Parameter equation:

t

t

ts

ts

tt

t

vv

QQQ

zz

3

2

11

1

113

2

010000000100tttt

11

*tt

*t

*t

*t

*ttt

vHz

w

111 t

3

31

12

Gauged site (1)

Ungauged Site (2)



t

32

Case study: simulation at Baron ForkCase 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 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 Ungauged Site (2) Baron Fork: 795 km Peacheater: 65 km2

Gauged Site (1)Baron Fork at Eldon

g g ( )Peacheater Creek



TETIS model calibrationTETIS model calibration

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

400

450

500

Observed Flow

Simulated Flow (Tetis model)

Statistical IndexBE (%): -14.6RMSE (m3/s): 6.61

250

300

350

m3 /s

)

( )NSE: 0.91

150

200

250

Flow

(

0

50

100

1 1 1 1 2 2 2 2 2 2 2 2 2

01/1

0/20

01

31/1

0/20

01

01/1

2/20

01

31/1

2/20

01

31/0

1/20

02

02/0

3/20

02

02/0

4/20

02

02/0

5/20

02

02/0

6/20

02

02/0

7/20

02

02/0

8/20

02

01/0

9/20

02

02/1

0/20

02

Date B.F. Eldon



TETIS model calibrationTETIS model calibration

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



TETIS model validationTETIS model validation

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

900

1000

Observed Flow

Simulated Flow (Tetis model)

Statistical IndexBE (%): -11.2RMSE (m3/s): 14.89

600

700

800

/s)

( )NSE: 0.83

300

400

500

Flow

(m3 /

100

200

300

0

01/1

0/96

30/1

1/96

30/0

1/97

01/0

4/97

01/0

6/97

01/0

8/97

01/1

0/97

30/1

1/97

30/0

1/98

01/0

4/98

01/0

6/98

01/0

8/98

01/1

0/98

01/1

2/98

30/0

1/99

01/0

4/99

01/0

6/99

01/0

8/99

01/1

0/99

DateB.F. Eldon



TETIS model validationTETIS model validation

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

Statistical IndexBE (%): 9.3RMSE (m3/s): 0.8840.0

45.0

50.0

Observed Flow

Simulated Flow (Tetis model)( )

NSE: 0.67

30.0

35.0

m3 /s

)

15.0

20.0

25.0

Flow

(m

0.0

5.0

10.0

Peacheater Cr.

01/1

0/19

99

31/1

0/19

99

30/1

1/19

99

31/1

2/19

99

30/0

1/20

00

01/0

3/20

00

31/0

3/20

00

01/0

5/20

00

31/0

5/20

00

01/0

7/20

00

31/0

7/20

00

31/0

8/20

00

30/0

9/20

00

30/1

0/20

00

30/1

1/20

00

30/1

2/20

00

30/0

1/20

01

01/0

3/20

01

01/0

4/20

01

01/0

5/20

01

01/0

6/20

01

01/0

7/20

01

01/0

8/20

01

31/0

8/20

01

01/1

0/20

01

Date



Results application of MISPResults, application of MISP

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

Statistical IndexBE (%) 9.3RMSE (m3/s) 0.8340.0

50.0Observed flow (assumedunknown)

Updated flow (MISP K.F.)

( )NSE 0.71

30.0

40.0

)

20.0Flow

(m3 /s

10.0

0.0

01/1

0/99

31/1

0/99

01/1

2/99

31/1

2/99

31/0

1/00

01/0

3/00

01/0

4/00

01/0

5/00

01/0

6/00

01/0

7/00

01/0

8/00

31/0

8/00

01/1

0/00

31/1

0/00

01/1

2/00

31/1

2/00

31/0

1/01

02/0

3/01

02/0

4/01

02/0

5/01

02/0

6/01

02/0

7/01

02/0

8/01

01/0

9/01

02/1

0/01

Peacheater Cr.



Date


Peacheater Creek (15/03/2002 – 29/04/2002)500 020 0

400,0

450,0

500,0

16,0

18,0

20,0Observed flow Peacheater (assumed unknown)

Simulated flow Peach. (Tetis)

Updated flow Peach (MISP K F )

300,0

350,0

12,0

14,0

/s)

ork) Cr.

Updated flow Peach. (MISP K.F.)

Observed flow (Baron Fork)

200,0

250,0

8,0

10,0

Flow

(m3 /

Bar

on F

o

Flow

(m3 /s

)Pe

ache

ater

C

50,0

100,0

150,0

2,0

4,0

6,0

B.F. Eldon

Peacheater Cr.

0,0

,

0,0

,

5/03

/200

2

0/03

/200

2

5/03

/200

2

0/03

/200

2

4/04

/200

2

9/04

/200

2

4/04

/200

2

9/04

/200

2

4/04

/200

2

9/04

/200

2



1 2 2 3 04 0 14 1 24 2

Date


Peacheater Creek, uncertainty bounds (15/03/2002 – 29/04/2002)30 0

25,0

30,0

Observed flow Peacheater (assumed unknown)


20,0

3 /s)

Lower Uncert. Limit (5%)

Upper Uncert. Limit (95%)

10,0

15,0

Flow

(m

5,0

B.F. Eldon

Peacheater Cr. 0,0

15/0

3/20

02

20/0

3/20

02

25/0

3/20

02

30/0

3/20

02

04/0

4/20

02

09/0

4/20

02

14/0

4/20

02

19/0

4/20

02

24/0

4/20

02

29/0

4/20

02



Date


Peacheater Creek (06/03/1999 – 17/05/1999)30015 0

240

300

12,0

15,0Observed flow Peacheater (assumed unknown)

Simulated flow Peach. (Tetis)

U d t d fl P h (MISP K F )

1809,0

3 /s)

ork

) Cr.


Observed flow Baron Fork

1206,0

Flow

(m3

Bar

on F

o

Flow

(m3 /s

Peac

heat

er C

603,0

B.F. Eldon

Peacheater Cr.

00,0

06/0

3/19

99

11/0

3/19

99

16/0

3/19

99

21/0

3/19

99

26/0

3/19

99

31/0

3/19

99

05/0

4/19

99

10/0

4/19

99

15/0

4/19

99

20/0

4/19

99

25/0

4/19

99

30/0

4/19

99

05/0

5/19

99

10/0

5/19

99

15/0

5/19

99EGU Leonardo Conference Series on the Hydrological Cycle

FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONSBratislava, Slovakia, 23-25 November 2011

20

0 1 2 2 3 0 1 1 2 2 3 0 1 1

Date


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

20,0

25,0

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

Lower Uncert Limit (5%)

15,0

s) r Cr.



10,0Flow

(m3 /s

Peac

heat

er

5,0

B.F. Eldon

Peacheater Cr.

0,0

06/0

3/19

99

11/0

3/19

99

16/0

3/19

99

21/0

3/19

99

26/0

3/19

99

31/0

3/19

99

05/0

4/19

99

10/0

4/19

99

15/0

4/19

99

20/0

4/19

99

25/0

4/19

99

30/0

4/19

99

05/0

5/19

99

10/0

5/19

99

15/0

5/19

99



Date


Peacheater Creek (15/06/2000 – 10/07/2000)180060

1500

1800

50

60

Observed flow (unknown)

Simulated flow (Tetis)

120040

/s)

rks) r Cr.

Updated flow (MISP K.F.)

Qobs3_BF

600

900

20

30

Flow

(m3 /

Bar

on F

o

Flow

(m3 /

Peac

heat

e r

300

600

10

20

B.F. Eldon

Peacheater Cr.

00

/06/

2000

/06/

2000

/06/

2000

/06/

2000

/07/

2000

/07/

2000



15/

20/

25/

30/

05/

10/

Date


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

100,0

120,0

Observed flow Peacheater (assumed unknown)


80,0

/s)

r Cr.



40,0

60,0

Flow

(m3 /

Peac

heat

er

20,0

B.F. Eldon

Peacheater Cr.

0,0

5/06

/200

0

0/06

/200

0

5/06

/200

0

0/06

/200

0

5/07

/200

0

0/07

/200

0



15 20 25 30 05 10

Date

ConclusionsConclusions

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

The observed and simulated discharges at the basin outlet are used as instrumentalvariables 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 wasimproved from 0.67 to 0.71 for the Peacheater Creek (assumed as ungauged site).

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

The uncertainty band is strongly affected by model performance, and is expected thati t f th d l ld d di ti t i ti



any improvement of the model would reduce predictive uncertainties.

Thank you for your attention!Thank you for your attention!

Contact: Technical University of Valencia (Spain)Juan Camilo MúneraE-mail: [email protected]

[email protected]

Research Institute of Water Engineering and EnvironmentResearch Group of Hydrological and Environmental Modelinghttp://lluvia.dihma.upv.es



Documents

Uncertainty assessment of streamflow simulation