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)
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EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011
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
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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:
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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 tT1
9
t H t K t t R t H t K t C t t K Tt H Tt C t H t Ptt 1 H t T R t t
K t Ptt 1 H t T C t
1
ˆ t 1 t ˆ t t 1 K t t t 1 w Pt 1 t Ptt 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 tT1
t H t K t t R t H t K t C t t K Tt H Tt C t H t Ptt 1 H t T R t t
K t Ptt 1 H t T C t
1
ˆ t 1 t ˆ t t 1 K t t t 1 w Pt 1 t Ptt 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
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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 *t1 z *t H *t t v *t
11
Q 2 t 11 12 Q 2 t 1 1 0 Q1t 0 0 Q1t 1 0 0 Q s1 0 0 t Q s1t 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 Q1t 1 t 1 w1t 1 1 0 0 0 Q1t 2 w s1 t 1 3 3 Q 0 0 1 2 s1t 1 Q 0 0 1 0 s1t 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 Q1t 1 v3 0 0 0 1 0 t Q s1t Q 11 s1t 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
EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011
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
EGU Leonardo Conference Series on the Hydrological Cycle FLOODS IN 3D: PROCESSES, PATTERNS, PREDICTIONS Bratislava, Slovakia, 23-25 November 2011
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
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
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
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, 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
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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
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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
25
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