Prediction of floods in ungauged basins - Runoff Prediction in Ungauged Basins: Synthesis across Processes, Places and Scales - page 204

tendency in the regression model framework. They noted
that in these cases the correlation coefficient between
regression model errors may be non-zero and included
this tendency for clustering and related it to geographical
distance between catchment centroids. This concept
therefore uses spatial proximity as one of the similarity
measures.

2000

500

200

Other methods of parameter estimation Alternatives
to using generalised least square techniques for estimating
the regression model parameters are available. For
example, Kjeldsen and Jones ( 2009 ) used a maximum-
likelihood method, while Reis et al.( 2005 ) and Micevski
and Kuczera ( 2009 ) used Bayesian approaches. Pandey
and Nguyen ( 1999 ) found that using estimation methods
directly in non-linear regressions produced better results
than log-linear models for quantile estimation. Gupta et al.
( 1994 ) proposed a simple non-linear method to estimate
the parameters of relationships between flood peak CV and
quantiles with catchment area within their multi-scaling
theory (see Section 9.2.1 ). Yet another alternative is to
use artificial neural networks (ANNs) (Shu and Burn,
2004b ; Dawson et al., 2006 ; Shu and Ouarda, 2008 ) that
are able to account for non-linear relationships between
catchment characteristics and the flood peaks, and the
catchment characteristics among themselves. In a study in
Quebec, Canada, Shu and Ouarda ( 2007 ) used basin area,
mean basin slope and the fraction of the basin area covered
with lakes, which were negatively correlated with the
specific flood quantiles, and mean annual precipitation
and mean annual days over 0 C, which were positively
correlated (see Figure 9.12 ). The joint ANN

50

Sample length
n

20

< 40
n ≥ 40

10

10 20 50 200 500 2000

Q m (m 3 /s)

Figure 9.11. Observed versus estimated mean annual floods
obtained with the GLS method for the Piemonte region, Italy.
From Laio et al.( 2011 ).

importance of including the cross-correlation of flood
peaks for properly quantifying the uncertainty of flood
quantile estimates. An example of using the GLS method
for estimation of the moments of maximum annual peak
series is provided in Figure 9.11 . The regressions (Laio
et al., 2011 ) capture the variability of the mean annual
flood peak well. The selected best model is a power law
relationship ( Equation 9.1 ) and includes four characteris-
tics: mean annual flood peaks are positively related to
catchment area, to mean annual precipitation, to the mean
maximum annual precipitation intensity for one hour inter-
val, and to a permeability index of the soil. The exponent
related to catchment area is 0.8, meaning that the specific
mean annual flood (in (m 3 /s)/km 2 ) actually decreases with
catchment area, consistent with the literature.
A careful regression analysis uses diagnostics to val-
idate assumptions about the error model. Tasker and
Stedinger ( 1989 ) improved the representation of the
overall regression error for the GLS method of Stedinger
and Tasker ( 1985 ) by treating the total regression error as
a sum of the sampling error for the estimates of the flood
statistics and a modelling error in modelling the true
index floods across catchments using regression. One
drawback of the regression approach is that there is a
tendency for residuals to cluster (NERC, 1975 ). This can
also be interpreted as the existence of local flood con-
trolling factors not currently captured by the available
lumped catchment characteristics. IH ( 1999 ) and Kjeld-
sen and Jones ( 2009 , 2010 ) accounted for the clustering

CCA method
identified that mean annual precipitation and the fraction of
the basin area covered with lakes are the most important
variables, followed by the mean basin slope. In a region
such as Quebec it is not surprising that the percentage of
lakes has to be taken into account in regional analyses of
most signatures, and of floods in particular.
Non-parametric regression is an alternative method that
does not make any assumptions on the form of the regres-
sion function. Gingras and Adamowski ( 1995 ) found para-
metric and non-parametric regression relationships to
provide equally good estimates, suggesting noise in the
data is dominating the signal.

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Hydrological interpretation
In the previous sections we have argued that a relationship
can be established between flood characteristics (e.g.,
quantile peaks, CV of the maximum annual peak flows,
parameters of the flood frequency curve, etc.) and catch-
ment and climatic characteristics. The regression approach
finds correlations between flood and catchment character-
istics, but this does not ensure a causal relationship,