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2005 AGU Fall Meeting San Francisco, 5 - 9
December 2005
Toward Reduced Uncertainty in Hydrologic and
Environmental Predictions Bringing
Experimentalists and Modelers Together On the
use of the Whittle likelihood for calibrating
hydrological models in scarcely gauged catchments

Alberto Montanari Faculty of Engineering Universit
y of Bologna, Italy http//www.costruzioni-idrauli
che.ing.unibo.it/people/alberto
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Goal of the work

• To introduce a maximum likelihood estimator for
  calibrating hydrological models in scarcely
  gauged catchments.
• The estimator is based on the use of the
  Whittles 1953 approximation to the Gaussian
  likelihood function.
• This kind of estimator is popular in time series
  analysis (long-memory stochastic processes
  Montanari 2003).

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How the estimator works??

• If the basin is gauged, estimation of the model
  parameters is carried out by matching the
  periodograms of observed and simulated series.
  Therefore it is a kind of indirect estimation.
• The periodogram is equivalent to the
  autocovariance function of the data.
• The periodogram is an estimate for the spectral
  density of the data. Therefore, we can roughly
  say that we compare the spectral density (or
  autocovariance function) of the process with the
  one of the data simulated by the model.

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Whittles approximation to the Gaussian maximum
likelihood function

• The Whittles likelihood can be computed through
  the relationship

• lj are the Fourier frequencies (lN/2 is the
  Nyquist frequency)
• J is the periodogram of the observed sample
• JM is the periodogram of the simulated series
  that depends on the parameter vector q of the
  hydrological model.

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Properties of the estimator

• Under the assumption of zero mean and i.i.d.
  residuals of the hydrological model, Whittles
  likelihood provides asymptotically consistent and
  normally distributed estimates in the case of
  non-Gaussian and linear models Giraitis and
  Surgailis, 1990.
• Asymptotical normality is no more guaranteed for
  the case of non-linear models.
• It is necessary to ensure that the model
  residuals have mean near to zero. Reject the
  parameter combinations that do not respect such
  condition.

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Testing the estimator (1)

• Lets refer to the case of the HYMOD
  rainfall-runoff model (lumped, 5 parameters).
• First test synthetic data, gauged basin
  (contemporaneous rainfall and river flow
  observations are available).
• 50 estimations are performed by using samples of
  hourly rainfall and river flows data covering a
  time span of 2 years (total observation period of
  synthetic data is 100 years).
• Results are compared with direct estimation that
  is obtained by maximizing the Nash-Sutcliffe
  1970 efficiency.

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Testing the estimator (2)

• The best parameter set given by Whittles
  likelihood was searched by using a genetic
  algorithm.
• Goodness of fit tests were computed by referring
  to river flows greater than different threshold
  levels.

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Summary of the test

• Whittles likelihood seems to be a little bit
  less efficient than the direct estimation
  technique that was used as term of comparison
  (maximization of the Nash-Sutcliffe 1970
  efficiency).
• Whittles likelihood is slower than direct
  estimation(a transformation of the data is
  needed).
• Therefore the question is
• Why should we use this kind of estimator for
  calibrating hydrological (rainfall-runoff)
  models???

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Whittles likelihood possesses some interesting
properties

• The estimation of the model parameters can be
  carried out by focusing on selected Fourier
  frequencies only. If we are interested in drought
  more than in high flows, we can perform a
  fit-to-the-purpose calibration.
• More interesting all we need to estimate model
  parameters is the periodogram of the river flows.
  We do not need the observed river flows!
• If we were able to estimate the periodogram
  basing on an alternative information, we can fit
  the model even in absence of observations.
• Remember that the spectral density of river
  flows can be assumed to be stationary, in absence
  of climate change, land-use change and other
  disturbances.

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How to estimate the periodogram of a process
without observed data?

• Many river flows series can be (roughly)
  approximated by an autoregressive process of
  order 1.Under this assumption, all we need for
  estimating the periodogram (at least at high
  frequencies) is mean and variance of the data and
  the autoregressive coefficient.
• The periodogram can be estimated by using sparse
  or old data (see following slides)

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Example of applicationReno River Basin (Italy)
8 years of mean areal rainfall and temperature
over the basin at hourly time scale
(1993-2000)(Suppose we do not have the river
flows)
HYMOD model Lumped, 5-parameter model
Simulated river flowsPeriodogram of
thesimulated river flows
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Example of applicationReno River Basin (Italy)
(2)
Term of comparisonDirect estimation HYMOD direct
estimation carried out by maximizing the
Nash-Sutcliffe efficiency.

• Observed input data to HYMOD are rainfall and
  temperature in the period 1993-2000.
• Observed output data are hourly river flows in
  the same period 1993-2000.
• Nash efficiency in calibration is 0.61 (HYMOD
  is not performing very well on the Reno River
  basin, especially on large flows).

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Example of applicationReno River Basin (Italy)
(3)
Results of indirectestimation with
Whittle HYMOD indirect estimation carried out
by using todays hourly rainfall and daily river
flows observed 50 years ago
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Example of applicationSynthetic data
Indirect Whittles estimation was further tested
by using the 100 year long synthetic sample
(again by using not contemporaneous rainfall and
river flows). We use the hourly river flow data
of years 51-100 to estimate the periodogram of
the river flow process and the rainfall data of
years 1-50 (10 subsamples of 5 years) to perform
10 HYMOD estimations.
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Final remarks
At this point, one may observe that such
estimation method is highly uncertain (as usual
when dealing with ungauged catchments). Is this
method useful in practical applications?
At this point, one may observe that such
estimation method is highly uncertain (as usual
when dealing with ungauged catchments). Is this
method useful in practical applications?
This method should not be considered in an
optimality context. When indirect estimations are
performed, the concept of optimality may be (even
more?) inconsistent. The estimator proposed here
can be useful in order to identify a set of
behavioral models.
This method should not be considered in an
optimality context. When indirect estimations are
performed, the concept of optimality may be (even
more?) inconsistent. The estimator proposed here
can be useful in order to identify a set of
behavioral models.
In the case of the Reno River shown before, the
behavioral models are identified by selecting
those-models that provide simulations whose
spectrum is consistent with the spectrum of past
river flows data. The set of behavioral models
can be further reduced by eliminating such models
that are non-behavioral accordingly to modelers
perception (for instance those models which gives
not realistic simulations in term of high flows
can be rejected)
In the case of the Reno River shown before, the
behavioral models are identified by selecting
those-models that provide simulations whose
spectrum is consistent with the spectrum of past
river flows data. The set of behavioral models
can be further reduced by eliminating such models
that are non-behavioral accordingly to modelers
perception (for instance those models which gives
not realistic simulations in term of high flows
can be rejected)
A final point about the Whittles estimator. The
distributional properties of the estimator are
not known. Not possible to estimate confidence
limits for the model parameters analytically (but
possible by applying bootstrapping).
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Announcement
European Geosciences Union General Assembly
Wien 2-7 April 2006 SPECIAL SESSION Water
Management in mountain basins (including
environmental flow) Please contribute!Deadline
for abstract submission 13 January 2006
For any detail please contact me
alberto.montanari_at_unibo.it http//www.costruzioni-
idrauliche.ing.unibo.it/people/alberto