Problems%20of%20Inference%20and%20Uncertainty%20Estimation%20in%20Hydrologic%20Modelling

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Problems of Inference and Uncertainty Estimation
in Hydrologic Modelling
Peter Reichert Eawag Dübendorf and ETH Zürich
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Contents
Motivation Errors in Hydro-logic
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• Motivation
• Errors and Uncertainties in Hydrologic Watershed
  Modelling
• Suggested Problem Solutions
• Working Group Opportunities

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Motivation
Motivation Errors in Hydro-logic
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Motivation
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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.
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Motivation
Phytoplanktion biomass
Walensee
Examples (1) Aquatic ecosystemmodelling
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Zürichsee
Greifensee
Mieleitner et al. 2006
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Motivation
Examples (2) Climate modelling
Motivation Errors in Hydro-logic
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Tomassini et al. 2006
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Motivation
Examples (3) Hydrologicmodelling
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Yang et al. 2006
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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
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• 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.

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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.
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Motivation
Hydrologic Modelling Watershed hydrologic
modelling is a particularly good study area for
these problems as
Motivation Errors in Hydro-logic
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• 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.
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Errors in Hydrological Modelling
Motivation Errors in Hydro-logic
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Errors and Uncertainties in Hydrologic Watershed
Modelling
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Errors and Uncertainties in Hydrologic Watershed
Modelling
Motivation Errors in Hydro-logic
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• Overview of Hydrologic Processes
• A Simple Hydrologic Watershed Model
• More Complex Watershed Models
• Sources of Error in Watershed Modelling

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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.

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Errors in Hydrologic Modelling
A Simple Hydrologic Watershed Model (1)
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Kuczera et al. 2006
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Errors in Hydrologic Modelling
A Simple Hydrologic Watershed Model (2)
Motivation Errors in Hydro-logic
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Kuczera et al. 2006
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Errors in Hydrologic Modelling
A Simple Hydrologic Watershed Model (3)
Motivation Errors in Hydro-logic
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Kuczera et al. 2006
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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).

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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.
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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).

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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.

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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.

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Suggested Problem Solutions
Motivation Errors in Hydro-logic
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Suggested Problem Solutions
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Suggested Problem Solutions

• Suggested Problem Solutions
• Ad-hoc ApproachesApproaches based on
  increasing parameter uncertainty.(GLUE, SUFI,
  SUNGLASSES, etc.)
• Improvement of Output Error ModelAutoregressive
  output error models.
• Input and Model Structure Error Models
• Storm multipliers.
• Bayesian model averaging.
• Use of a stochastic hydrological model.
• Stochastic, time-dependent parameters.
• Multi-criteria optimization.

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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.
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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.
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Suggested Problem Solutions
2. Improvement of Output Error Model (Example)
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residuals, no transformation
residuals, Box-Cox transf.
residuals, Box-Cox tr., var. sd.
innovations, Box-Cox tr., var. sd.var. corr.
time
Yang et al. 2006
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Suggested Problem Solutions
2. Improvement of Output Error Model (Example)
residuals, no transformation
residuals, Box-Cox transf.
Auto-correlation
Motivation Errors in Hydro-logic
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residuals, Box-Cox tr., var. sd.
innovations, Box-Cox tr., var. sd., var. corr.
time
Yang et al. 2006
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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.

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Working Group Opportunities
Motivation Errors in Hydro-logic
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Working Group Opportunities
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Working Group Opportunities

• Research Questions / Options for Projects (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.)
• Develop a better statistical description of
  rainfall uncertainty.(Option for a collaboration
  with climate/weather working groups.)
• Explore alternative options for making parameters
  time-dependent.(Suggestions so far
  storm-dependent parameters, time-dependent
  parameter as an Ornstein-Uhlenbeck process.)

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Working Group Opportunities

• Research Questions / Options for Projects (2)
• 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?)
• Develop uncertainty estimates when using
  multi-objective optimization.(How to use
  information on Pareto set for uncertainty
  estimation of parameters and results?)
• 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?)

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Working Group Opportunities

• Research Questions / Options for Projects (3)
• 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.)
• More questions will come up during discussions.

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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.

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Working Group Opportunities

• How to Proceed?
• 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.
• Discuss and prioritize (according to expected
  chance of success) the collection of
  suggestions developed under point 1 above.
• Use preliminary results of project 1 to stimulte
  the discussions.
• Decide on research plans for projects to work on.
• Organise a workshop for discussing research plans
  and preliminary results with experts in the
  field.
• Plan the group activities for the remaining part
  of the subprogram that lead to results to be
  published and presented at a closing workshop.

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Thank you for your attention