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Eawag: Swiss Federal Institute of Aquatic Science and Technology
Problems of Inference and Uncertainty Estimation in Hydrologic Modelling
Peter Reichert
Eawag Dübendorf and ETH Zürich
SAMSI meetingOct. 16, 2006
Contents
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
Motivation
Errors and Uncertainties in Hydrologic Watershed Modelling
Suggested Problem Solutions
Working Group Opportunities
SAMSI meetingOct. 16, 2006
Motivation
Motivation
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Motivation
Practice of Environmental Modelling:
Mechanistic, deterministic description of system behaviour with a simple, additive, independent (measurement) error model – Strong autocorrelation of residuals, if temporal resolution of data is high.
This severe violation of statistical assumptions leads to unreliable error estimates.
The problem is aggravating, as temporal resolution of data and measurement accuracy are increasing.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Motivation
Examples (1):
Aquatic ecosystemmodelling
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
0
0.2
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0.6
0.8
1
1.2
1.4
1.6
1988 1989 1990 1991
bio
ma
ss [g
WM
/m3
]
0
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0.6
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1
1.2
1.4
1.6
1988 1989 1990 1991
bio
ma
ss [g
WM
/m3
]
0
1
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3
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7
8
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10
1988 1989 1990 1991
bio
ma
ss [g
WM
/m3
]
Phytoplanktion biomass
Walensee
Zürichsee
Greifensee
Mieleitner et al. 2006
SAMSI meetingOct. 16, 2006
Motivation
Examples (2):
Climate modelling
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
Tomassini et al. 2006
SAMSI meetingOct. 16, 2006
Motivation
Examples (3):
Hydrologicmodelling
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
2040
6080
100
flowm
3s
1/1/1985 12/31/1985 12/31/1986
150
100
500
rain
fall(
mm
)
2040
6080
100
flowm
3s
1/1/1987 1/1/1988 12/31/1988
150
100
500
rain
fall(
mm
)
2040
6080
100
flowm
3s
1/1/1989 12/31/1989 12/31/1990
150
100
500
rain
fall(
mm
)
Yang et al. 2006
SAMSI meetingOct. 16, 2006
Motivation
Cause of the Problem and Challenges:
The cause of this problem is not the inadequatemodel of the measurement process, but the neglection of input and model structure errors that are propagated through the model and dominate prediction uncertainty.
Both input and model structure errors lead to very similar pattern in the residuals. The challenges are
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
• to find good statistical descriptions of the random contributions of both error sources,
• to find procedures to support finding model structure improvements, and
• to separate the two error contributions.
SAMSI meetingOct. 16, 2006
Motivation
Universality of the Problem:
This problem is typical for nearly all fields of dynamic modelling in the environmental sciences.
The causes and techniques for problem analysis can be expected to be the same for different application areas, despite application-field specific interpretations and identified error models.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Motivation
Hydrologic Modelling:
Watershed hydrologic modelling is a particularly good study area for these problems as
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
• Data at high temporal resolution are available.
• Essentially the same problems occur with complex and very simple watershed models.(see next part of the talk for a justification of this statement.)
It seems to be a reasonable strategy to analyse the problem and test solutions with simple watershed models and transfer the promising solutions to the more complex case.
SAMSI meetingOct. 16, 2006
Errors in Hydrological Modelling
Errors and Uncertainties in Hydrologic Watershed Modelling
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Errors and Uncertainties in Hydrologic Watershed Modelling
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
Overview of Hydrologic Processes
A Simple Hydrologic Watershed Model
More Complex Watershed Models
Sources of Error in Watershed Modelling
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
Overview of Hydrologic Processes
The water balance in a watershed is affected by:
• rainfall,
• runoff,
• infiltration into the soil,
• evapotranspiration,
• transport through the soil (vertically and laterally),
• transport to shallow ground water,
• lateral transport in ground water,
• transport to deep ground water,
• exfiltration from soil and groundwater to surface water,
• transport in surface water.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
A Simple Hydrologic Watershed Model (1): Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
gwlatetrunoffrains )(
d
dqqqqq
t
h
bfgwgw
d
dqq
t
h
rbflatrunoffr
d
dqqqq
t
h
Kuczera et al. 2006
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
A Simple Hydrologic Watershed Model (2): Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
)(rainrain trainfq
)(rainsatrunoff trainffq
)()exp(1 petet tpetfhkq set
maxlat,satlat qfq
gwbfbf hkq
maxgw,satgw qfq
rrr hkq
rwr qAQ
100
1
)exp()99(1
1
FssFsat
shks
f
Kuczera et al. 2006
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
A Simple Hydrologic Watershed Model (3): Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
0 500 1000 1500
0.0
0.2
0.4
0.6
0.8
1.0
hs [mm]
f sat
[-]
ks=0.02/mm, sF=2300ks=0.01/mm, sF=2300ks=0.04/mm, sF=2300ks=0.02/mm, sF=1150ks=0.02/mm, sF=4600
100
1
)exp()99(1
1
FssFsat
shks
f
Kuczera et al. 2006
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
More Complex Watershed Models• Parameterization by soil properties (soil thickness,
porosity, texture, conductivity, etc.).
• Higher vertical resolution of soil profile (layers, continuous vertical resolution).
• Higher horizontal resolution of watershed (accounting for variation in soil properties, land use, etc. within the watershed).
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
More complex models (with a higher spatial resolution) are primarily required for the prediction of the effect of land use change, not to improve the quality of the fit.
These models are usually highly overparameterized, but do nevertheless not very much improve the fit.
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
Sources of Error in Watershed Modelling
• Input uncertaintyPoint measurements from rain gauges and potential evapo-transpiration measurements are extrapolated to the watershed area despite high local variation in rain intensity.
• Model structure uncertainty• Many different „storage systems“ in parallel are
represented by an „average storage“ or by storage systems parameterized using soil properties.
• All storage systems within a sub-basin are subject to the same input.
• Parameterization of „storage“ function.
• Output uncertaintyMeasurement error of stream flow (gauging curve and random error).
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
Difference – Simple vs. Complex Models
As simple and complex models usually use the same input, they face the same problems outlined above.
Only the use of higher (spatial) resolution in input could reduce some of these problems, not increase in model complexity (which was introduced for other reasons).
It is a trend in real-time hydrologic modelling to do this with the aid of radar data. But still most of the hydrologic modelling studies must be based on rain gauge data.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Errors in Hydrologic Modelling
Results for Simple Error Model
When using an independent error model the result will usually be a small prediction uncertainty for the mean and a large standard deviation of the error term.
The resiudals will show strong deviations from the indepencence assumption.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
Suggested Problem Solutions
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
Suggested Problem Solutions
1. „Ad-hoc“ ApproachesApproaches based on increasing parameter uncertainty.(GLUE, SUFI, SUNGLASSES, etc.)
2. Improvement of Output Error ModelAutoregressive output error models.
3. Input and Model Structure Error Models• Storm multipliers.• Bayesian model averaging.• Use of a stochastic hydrological model.• Stochastic, time-dependent parameters.• Multi-criteria optimization.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
1. „Ad-hoc“ Approaches
Approaches such as GLUE, SUFI, SUNGLASSES, etc. increase parameter uncertainty to cover most of the observations with a prediction uncertainty band.
This is either done by introducing a „generalized“ likelihood function, the values of which are normalized and then interpreted as probabilities or by „ad-hoc“ selection of parameter subsets that lead to an adequate coverage of observations.
Despite the poor statistical foundation, such techniques are quite popular in hydrology.
This is not the approach I would like to follow in the working group.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
2. Improvement of Output Error Model
Use of an autoregressive error model instead of the independent error model.
This approach is quite successful in the fulfilment of statistical assumptions (see example).
However, it describes only the effect and not the cause of the errors and may lead to „statistical description“ of „physical phenomena“ (description of recession curves from „storages“ by the auto-regressive error model).
This is a nice intermediate step, but the effort must be on a description of the actual error sources.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
2. Improvement of Output Error Model (Example)Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
-6-4
-20
24
6 Calibration 1
std.
res
idua
l
1/1/1985 1/1/1986 1/1/1987 12/31/1987 12/30/1988
-6-4
-20
24
6 Calibration 2
std.
res
idua
l
1/1/1985 1/1/1986 1/1/1987 12/31/1987 12/30/1988
-6-4
-20
24
6 Calibration 3
std.
res
idua
l
1/1/1985 1/1/1986 1/1/1987 12/31/1987 12/30/1988
-6-4
-20
24
6 Calibration 4
std.
inno
vatio
n
1/1/1985 1/1/1986 1/1/1987 12/31/1987 12/30/1988Yang et al. 2006
residuals, no transformation
residuals, Box-Cox transf.
residuals, Box-Cox tr., var. sd.
innovations, Box-Cox tr., var. sd.var. corr. time
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
2. Improvement of Output Error Model (Example)Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
Yang et al. 2006
residuals, no transformation
residuals, Box-Cox transf.
residuals, Box-Cox tr., var. sd.
innovations, Box-Cox tr., var. sd., var. corr. time
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
Calibration 1
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
Calibration 2
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
Calibration 3
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
Calibration 4
Auto-correlation
SAMSI meetingOct. 16, 2006
Suggested Problem Solutions
3. Input and Model Structure Error Model
Only recently better error models have been suggested. The essential elements are that
the high input uncertainty in total rainfall and potential evapotranspiration over the watershed must be considered explicitly,
a deterministic description is not adequate due to stochastic distribution of input over the watershed („the different storage systems“),
model structure (systematic) errors must be distinguished from random errors.
It would be an interesting SAMSI activity to discuss how to best do this and compare results of different approaches.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Working Group Opportunities
Working Group Opportunities
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Working Group Opportunities
Research Questions / Options for Projects (1)
1. Compare results when making different model parameters stochastic and time-dependent. (Ongoing with a postdoc in Switzerland extending earlier work with continuous-time stochastioc parameters.)
2. Develop a better statistical description of rainfall uncertainty.(Option for a collaboration with climate/weather working groups.)
3. Explore alternative options for making parameters time-dependent.(Suggestions so far: storm-dependent parameters, time-dependent parameter as an Ornstein-Uhlenbeck process.)
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Working Group Opportunities
Research Questions / Options for Projects (2)
4. Investigate on how to learn from state estimation of stochastic hydrological models.(Can the pattern of state adaptations lead to insights of model structure deficits or input errors?)
5. Develop uncertainty estimates when using multi-objective optimization.(How to use information on Pareto set for uncertainty estimation of parameters and results?)
6. Analyse differences in results of suggested approaches when using different models.(Is there a generic behaviour of different techniques when they are applied to different models/data sets?)
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Working Group Opportunities
Research Questions / Options for Projects (3)
7. Improve the efficientcy of posterior maximisation and posterior sampling.(Efficiency becomes important when having complex watershed models in mind. Efficient global optimizers and sampling from multi-modal posterior distributions becomes then important.)
8. More questions will come up during discussions.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Working Group Opportunities
Practical Considerations
• State estimation of time-dependent OU-parameters as well as the simple hydrological model are implemented in the UNCSIM package by PR.This package also provides a simple interface to complex hydrological models.
• Jasper Vrugt (LANL) can provide implementations of several simple hydrological models and analysis techniques in Matlab.
• The simple hydrological models can also easily be implemented in any other computing environment.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
SAMSI meetingOct. 16, 2006
Working Group Opportunities
How to Proceed?1. Initiate a reading group for discussing key papers
and suggestions of how to attack the problems.This could be a separate working group or a subgroup of the methodology working group.
2. Discuss and prioritize (according to expected chance of success) the „collection of suggestions“ developed under point 1 above.
3. Use preliminary results of project 1 to stimulte the discussions.
4. Decide on research plans for projects to work on.
5. Organise a workshop for discussing research plans and preliminary results with experts in the field.
6. Plan the group activities for the remaining part of the subprogram that lead to results to be published and presented at a closing workshop.
Motivation
Errors in Hydro-logic Modelling
Suggested Pro-blem Solutions
Working GroupOpportunities
Eawag: Swiss Federal Institute of Aquatic Science and Technology
Thank you for your attention