Prediction of low flows in ungauged basins - Runoff Prediction in Ungauged Basins: Synthesis across Processes, Places and Scales - page 176

was almost constant. The lognormal distribution was iden-
tified as the most suitable three-parameter distribution for
both low flow indices, according to best fit. The hydro-
logical differences between the regions were reflected by
clear differences in the distribution parameters.
The index method assumes that the stream gauge data
are independent, i.e., that at different gauges different
events are observed. Because of the large spatial extension
of most low flow events this may not always be the case, so
there is a need to account for their spatial correlations
(Hosking and Wallis, 1997 ) in order to ensure that the
uncertainty of the low estimates is not underestimated. In
order to estimate the return periods of a given low flow
runoff, distribution functions are usually fitted to the
extreme value series. Tallaksen et al.( 2004 ) reviewed
methods for estimating the parameters of the distribution
functions and recommended that the choice of the distribu-
tion function should be guided by statistical judgement in
combination with the hydrological interpretation of low
flow processes in the study area. For minimum flows, the
distribution function should be skewed and have a finite
lower limit greater than or equal to zero, such as the
generalised extreme value (GEV) distribution for annual
minimum series (e.g., Zaidman et al., 2003 ; Demuth and
Külls, 1997 ) and the generalised Pareto (GP) distribution
for partial duration series (e.g., Tallaksen and Hisdal, 1997 ;
Madsen and Rosbjerg, 1998 ; Meigh et al., 2002 ). Apart
from GEV and GP distributions, different forms of Wei-
bull, Gumbel, Pearson Type III, lognormal, Gamma and
other distributions have been used (Vogel and Kroll, 1989 ;
Pearson, 1995 ; Vogel and Wilson, 1996 ; Chen et al., 2006 ;
Modarres, 2008 ). Some of them are special cases of the
two limiting distributions. Gottschalk and Perzyna ( 1989 )
suggested that the Weibull distribution for annual min-
imum series is consistent with linear low flow recessions
which, along with its flexibility, have made it a common
choice in low flow studies around the world (Tallaksen,
2000 ).
To infer the index parameter for ungauged sites, regres-
sion analyses with catchment and climatic characteristics
are often used. Tallaksen et al.( 2004 ) used the general-
ised least squares regression method in three regions in
Germany to estimate the index values for low flow dur-
ations and deficit volumes. Land use, morphometry and
soil characteristics as well as the mean annual precipita-
tion were found to be important characteristics. In cases
where at-site runoff data were available, they recom-
mended combining these data with the regional estimates
by a weighted mean, with weights chosen according to
the relative uncertainties (Madsen and Rosbjerg, 1997 ).
The resulting model was shown to perform well in esti-
mating the T-year low flow duration and deficit volume in
the region.

While the above methods focused on extreme value
distributions based on the index low flow method, it is
also possible to scale the low flow quantiles of the flow
duration curve by an index flow. Young et al.( 2003 ) used
a region of influence approach where, for each ungauged
basin, they selected a number of donor catchments that
were similar to the ungauged basins in terms of catchment
and climate characteristics. They then assumed that the low
flow variable (such as Q 95 ) scaled by the mean annual
runoff in the ungauged basin is the same as the weighted
mean of the corresponding values from the donor catch-
ments. They chose the weight on the basis of inverse
distances, which gives more weight to catchments that
are close to the ungauged basin. They obtained a perform-
ance similar to regression models for the UK catchments, if
similarity was defined on the basis of the hydrology of soil
types (HOST) classification (Boorman et al., 1995 ).

8.3.3 Geostatistical methods
Geostatistical approaches exploit the spatial correlations of
low flows based on the rationale that catchments that are
geographically close to each other may exhibit similar
processes. The low flows in the basins are estimated as
the weighted mean of the observations and the weights are
estimated from the spatial correlations as a function of
spatial distance. Traditional geostatistics have been
developed in resource exploration and meteorology (Math-
eron, 1965 ; Gandin, 1963 ) where spatial distance is clearly
defined. However, river networks exhibit a tree-like struc-
ture, which needs to be accommodated in the estimation
methods. Gottschalk ( 1993a ) was probably the first to
develop a method for calculating covariance along a stream
network based on river distance, which he used to estimate
runoff based on water balance constraints in river junctions
(Gottschalk, 1993b ). Similar methods were proposed by
Ver Hoef et al.( 2006 ) and Cressie et al.( 2006 ). An
alternative is to consider catchments as two-dimensional
objects superimposed on the steam network where runoff
generation is conceptualised as a spatially continuous pro-
cess, which is defined for any point in the landscape
(Viglione et al., 2010a , b ), and runoff is the integral of
local runoff over the catchment. Sauquet et al.( 2000a ),
Sauquet ( 2006 ) and Gottschalk et al.( 2006 ) proposed a
method where the spatial dependence of runoff of catch-
ments of different sizes is modelled by a regularised co-
variogram. The problem of nested catchments is addressed
by disaggregating observed runoff into runoff contribu-
tions from subcatchments or grid cells. Similar to
Gottschalk ( 1993b ), they included water balance con-
straints into the kriging system. A similar method, known
as top-kriging, was developed by Skøien et al.( 2006 ) ,
which does not use water balance constraints but addresses