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EGS-AGU-EUGJoint Assembly Nice, France, 06 - 11
April 2003
HS3 - Parameter estimation and uncertainty
assessment in hydrological modelling New
approaches and future directions Uncertainty
assessment for rainfall-runoff simulations
Alberto Montanari, Armando Brath Faculty of
Engineering University of Bologna, Italy
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Uncertainty always affects rainfall-runoff models
output, which is never exact
Uncertainty in the model output (total
uncertainty) is due to
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A brief review of the past studies

• Glue approach (Beven and Binley, 1992)
• The uncertainty in the model simulation is
  obtained by operating a sensitivity analysis of
  the model output with respect to the model input
  and parameters.

Input uncertainty Parameter uncertainty
90 confidence bands of an hydrological
simulation of the flood frequency distribution -
River Wye catchment, Wales, From Cameron at al.,
1999
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A brief review of the past studies
SCEM-UA method (Vrugt et al., WRR 2003, to appear)
Confidence limits for the model parameters Input
uncertainty
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Objective of the study
To propose a technique for estimating model
uncertainty based on the analysis of the
rainfall-runoff model simulation residuals.
Requirement the technique should be applicable
with any kind of rainfall-runoff model.
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Assumption the model residuals are ergodic
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The Meta-Gaussian model

• Take the standard normal quantile transform of
  Q(t)
• compute the cumulative frequency F(t) of each
  t-th observation
• compute the standard normal quantile NQ(t) of
  each F(t)
• associate to each Q(t) the corresponding NQ(t)

• Take the standard normal quantile transform of
  e(t), i.e. associate to each e(t) the
  corresponding Ne(t)

• Assume that the joint probability distribution
  PNe(t),NQ(t) is a bivariate Gaussian
  distribution

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The Meta-Gaussian model

• Lets indicate with
• mNe mean of Ne
• sNe standard deviation of Ne
• mNQ mean of NQ
• sNQ standard deviation of NQ
• Then
• m mNe r(sNe/sNQ)(NQ(t)-mNQ)
• s sNe(t)NQ(t) s2Ne (1-r2)0.5

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The Meta-Gaussian model
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The Meta-Gaussian modelA relevant statistical
property
In the transformed space the model is LINEAR and
homoschedastic In the not-transformed space the
model is NON LINEAR and HETEROSCHEDASTIC
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Why the absolute model error?
The model error with sign is poorly
cross-correlated with the simulated river flow
The approach is conservative.
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Case study Samoggia River Basin (Italy)
Simulation of the hourly river flows of the year
1996 by using a continuous simulation,
spatially-distributed, conceptual rainfall-runoff
model Flood event of September 1972
Efficiency 81
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Case study Samoggia River Basin (Italy)
Regression of normalised residuals versus
normalised discharge (Transformed space)
The Meta-Gaussian residuals turned out to be
close to Gaussian (Kolmogorov-Smirnov test)
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Case study Samoggia River Basin (Italy)
The model was used in order to simulate 1000
years of synthetic reiver flows (syntethic
rainfall input)
Simulated flood frequency distribution derived by
fitting the 1000 annual synthetic maximum flows
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Final remarks
The model residuals are not model errors
underestimation of the confidence bands
The estimated confidence limits might result too
narrow in case the assumption of ergodicity and
joint gaussianity is not satisfied
Wallis 1974 it is a common experience for
models to have worse error variances than they
should when used in forecasting outside the
period of fit.
Chatfield 1993 the prediction model may
change, either during the period of fit or in the
future