Prediction of floods in ungauged basins - Runoff Prediction in Ungauged Basins: Synthesis across Processes, Places and Scales - page 207

period, while in Equation (9.2) the function f( ), which
defines the index flood in terms of catchment characteris-
tics, does not (see e.g., Gupta et al., 1994 ). Also, regional
regressions assume continuous variability of flood fre-
quency curve characteristics across space and/or climate/
catchment characteristics, while the index flood method
identifies groups of catchments, which share the same
growth curve (see Laio et al., 2011 ). As a rule, the GLS
regression diagnostics do not support this assumption
(Micevski and Kuczera, 2009 ). Cross-correlation between
flows at stations in a pooling group can also increase the
uncertainty of estimates of extreme flow quantiles (Rosb-
jerg, 2007 ). This can affect the evaluation of the homogen-
eity of the pooling group (Castellarin et al., 2008 ). From a
practical perspective, while the underlying assumption of an
invariant growth curve is usually rejected by the evidence,
the fact that sampling error dominates model error ensures
the index flood method is a reasonable approximation. In a
comparative study, Rosbjerg ( 2007 ) found that the index
flood method led to quantile estimates with slightly less
uncertainty than quantile regression. It should be noted that
the index flood method typically uses a regression approach
to estimate the scale factor (the index flood). Therefore, its
efficacy is contingent on the sound application of the regres-
sion approach.

All arid basins

30

Queensland

South Africa and Botswana

20

Saudi Arabia and Yemen

10

Iran

Great Britain

0

1 10 100 1000 5000

Return period (yrs)

Figure 9.14. Flood growth curves in different regions around the
world. From Farquharson et al.( 1992 ).

catchments. More importantly, however, the results pre-
sented in Figure 9.14 illustrate that one can organise the
global distribution of flood frequency curves (i.e., growth
curves) into distinct classes or regimes based on climate
(aridity). In a sense, these results parallel the organisation of
annual water balance using the Budyko framework ( Chap-
ter 5 ). There is already considerable evidence that shows
that both of these runoff signatures mirror landscape signa-
tures such as a scaled drainage density (Wang and Wu,
2012 ) and fraction of deep-rooted vegetation (Xu et al.,
2012 ), which are also strongly correlated to climate aridity.
This is further evidence of co-evolution of several catchment
characteristics and climate, resulting in emergent patterns
and associated runoff signatures, which provide alternative
predictors of a co-evolutionary kind, such as aridity, drain-
age density, vegetation cover and catchment area.

Relaxing the assumptions
There is research in the direction of weakening the main
assumption underlying the index flood method, i.e., the
requirement for a homogeneous region. For example,
Kjeldsen and Jones ( 2009 ) proposed a version of the index
flood method that relaxes the requirement for a homoge-
neous region by incorporating the difference between
basins (heterogeneity) into the weighting assigned to each
member of a pooling group. Thus, information from
gauged sites was weighted according to the degree of
similarity to the target site as well as the record length.
More generally, in the empirical Bayes method introduced
by Kuczera ( 1982 ), parameters for an ungauged site are
inferred from the prior distribution obtained either from
regional data in terms of observations or from physical
characteristics at the other sites in the region. The method
requires that a common model for the occurrence of
extreme events in the region is formulated. Similar to the
method of Kjeldsen and Jones (2009), the empirical Bayes
method does not prescribe strict homogeneity as is done by
the index flood method, which assumes that all properties
are identical after scaling. Thus, the index flood method
can be considered a special case of an empirical Bayes
model in which the prior distribution concentrates the
probability mass at a single point. Inferring the prior infor-
mation from generalised least squares (GLS) regression
ensures that

Comparison of index method with regressions
In Equation (9.2) the index flood is site dependent, while the
growth curve is assumed to be the same for the entire
homogeneous pooling group. The index flood method thus
assumes that the distribution of flood peaks at different sites
within a pooling group is the same except for a scale param-
eter, which is the index flood for a site. The procedure has
been developed to improve the estimation of the growth
curve due to the exploitation of information from the entire
homogeneous group as opposed to one-site samples. The
main difference with regression methods is that the relation-
ship of flood quantiles or model parameters with catchment
characteristics in Equation (9.1) depends on the return inter-site correlation is properly accounted