Determination and integration of appropriate spatial scales for river basin modelling

Abstract

Appropriate spatial scales of dominant variables are determined and integrated into an appropriate model scale. This is done in the context of the impact of climate change on flooding in the River Meuse in Western Europe. The objective is achieved by using observed elevation, soil type, land use type and daily precipitation data from several sources and employing different relationships between scales, variable statistics and outputs. The appropriate spatial scale of a key variable is assumed to be equal to a fraction of the spatial correlation length of that variable. This fraction was determined on the basis of relationships between statistics and scale and an accepted error in the estimation of the statistic of 10%. This procedure resulted in an appropriate spatial scale for precipitation of about 20 km in an earlier study. The application to river basin variables revealed appropriate spatial scales for elevation, soil and land use of respectively 0·1, 5·3 and 3·3 km. The appropriate model scale is determined by multiplying the appropriate variable scales with their associated weights. The weights are based on SCS curve number method relationships between the peak discharge and some specific parameters like slope and curve number. The values of these parameters are dependent on the scale of each key variable. The resulting appropriate model scale is about 10 km, implying 225–250 model cells in an appropriate model of the Meuse basin meant to assess the impact of climate change on river flooding. The usefulness of the appropriateness procedure is in its ability to assess the appropriate scales of the individual key variables before model construction and integrate them in a balanced way into an appropriate model scale. Another use of the procedure is that it provides a framework for decisions about the reduction or expansion of data networks and needs. Copyright © 2003 John Wiley & Sons, Ltd.

Full text

HYDROLOGICAL PROCESSES
Hydrol. Process. 17, 2581– 2598 (2003)
Published online 11 August 2003 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.1268
Determination and integration of appropriate spatial
scales for river basin modelling
M. J. Booij*
Department of Civil Engineering, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
Abstract:
Appropriate spatial scales of dominant variables are determined and integrated into an appropriate model scale. This is
done in the context of the impact of climate change on flooding in the River Meuse in Western Europe. The objective
is achieved by using observed elevation, soil type, land use type and daily precipitation data from several sources
and employing different relationships between scales, variable statistics and outputs. The appropriate spatial scale of
a key variable is assumed to be equal to a fraction of the spatial correlation length of that variable. This fraction was
determined on the basis of relationships between statistics and scale and an accepted error in the estimation of the
statistic of 10%. This procedure resulted in an appropriate spatial scale for precipitation of about 20 km in an earlier
study. The application to river basin variables revealed appropriate spatial scales for elevation, soil and land use of
respectively 0Ð1, 5Ð3and3Ð3 km. The appropriate model scale is determined by multiplying the appropriate variable
scales with their associated weights. The weights are based on SCS curve number method relationships between
the peak discharge and some specific parameters like slope and curve number. The values of these parameters are
dependent on the scale of each key variable. The resulting appropriate model scale is about 10 km, implying 225– 250
model cells in an appropriate model of the Meuse basin meant to assess the impact of climate change on river flooding.
The usefulness of the appropriateness procedure is in its ability to assess the appropriate scales of the individual key
variables before model construction and integrate them in a balanced way into an appropriate model scale. Another
use of the procedure is that it provides a framework for decisions about the reduction or expansion of data networks
and needs. Copyright 2003 John Wiley & Sons, Ltd.
KEY WORDS river flooding; rainfall-runoff modelling; spatial scales; appropriateness; correlation length;
semi-correlation; variance reduction function; Meuse basin
INTRODUCTION
Global climate changes induced by increases in greenhouse gas concentrations are likely to increase
temperatures, change precipitation patterns and probably raise the frequency of extreme events (IPCC, 2001).
These changes may have serious impacts on society, e.g. on river deltas because of both sea level rise
and an increased occurrence of flooding events. Flooding events may cause enormous economic, social
and environmental damage and even loss of life. This necessitates the application of robust and accurate
flood estimation procedures to provide a strong basis for investments in flood protection measures with
climate change.
A broad palette of models is available for this purpose, ranging from simple, lumped black-box models
to complex, distributed models including lots of physics and mathematics. These include empirical models,
conceptual models and physically based models. These divisions are somewhat arbitrary and hybrid forms
exist in which different methods are combined. The complexity of models depends not only on the model
class to which they belong, but also on the processes incorporated, the process formulations used, and the
different space and time scales employed. In general, models should be sufficiently detailed to capture the
* Correspondence to: M. J. Booij, Department of Civil Engineering, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.
E-mail: m.j.booij@ctw.utwente.nl
Received 10 June 2002
Copyright 2003 John Wiley & Sons, Ltd. Accepted 19 November 2002

2582 M. J. BOOIJ
dominant processes and natural variability, but not unnecessarily refined that computation time is wasted
or data availability is limited. It would seem that an optimum model complexity associated with minimum
total uncertainty exists, with a balance in uncertainties from input, model structure and parameters. This
raises the question of what such an appropriate model should look like given the specific modelling objective
and research area. More specifically, which physical processes and data should be incorporated and which
mathematical process formulations should be used and at which spatial and temporal scale to obtain an
appropriate model level?
Booij (2002a) has developed a model appropriateness procedure in which subsequently dominant processes,
appropriate scales and associated appropriate process formulations are determined. This study focuses on the
determination of the appropriate spatial scales. An appropriate scale of a variable is intuitively defined as a
scale which is sufficiently detailed to capture the variability of that variable, but not more than that. First
the dominant processes, important with respect to the impact of climate change on flooding in a particular
river basin, have to be defined. Next, these dominant processes can be linked to dominant variables to be
able to assess the appropriate scales for the different processes through these variables. In this way, the
appropriate variable scales are derived before model construction, which is believed to be more reasonable
than on the basis of model sensitivities (see Booij, 2002b). Only for the integration of these appropriate
variable scales to an appropriate model scale have model sensitivities been used, which is considered to
be inevitable.
Besides precipitation, three processes are recognized as being responsible for the relatively fast trans-
port of precipitation to the stream network during flood events (Maidment, 1992). These are infiltration
excess overland flow (Horton overland flow), saturation excess overland flow (Dunne overland flow) and
subsurface storm flow (interflow). The relative contributions of these three flood-generating processes to
a specific flood vary heavily between different watersheds and different precipitation events, and within
a specific watershed. Additionally, the soil moisture conditions at the beginning of a flood are of impor-
tance and should be accurately determined. These initial conditions are mainly influenced by the pre-
ceding precipitation, evapotranspiration and subsurface flow in the soil matrix. Moreover, the river flow
itself contributes to the occurrence and magnitude of floods. The dominant variables were derived from
these processes by Booij (2002a) on the basis of several studies (e.g. Mauser and Sch¨
adlich, 1998;
Veihe and Quinton, 2000) and are precipitation, elevation, soil type (texture and parent material) and
land use.
The emphasis will be on the appropriate spatial scale and thus, the appropriate time scale should be deter-
mined beforehand. The variables elevation, soil and land use are assumed to be time invariant on the temporal
domain considered (¾80 years), although land use may show a seasonal dependence and long-term trends.
Obviously, precipitation is strongly variable in time and should be modelled accordingly. The appropriate
temporal scale to assess the impact of climate change on flooding in the river basin considered has been
assessed at one day (Booij, 2002a). The appropriate spatial scale for extreme daily precipitation was deter-
mined at 20 km (Booij, 2002c) and will be used in the integration of the appropriate variable scales into an
appropriate model scale. The appropriate spatial scales for elevation, soil and land use are derived in this
study.
Several studies show that the ‘appropriate’ scales for elevation (e.g. Bruneau et al., 1995; Brasington
and Richards, 1998; Wolock and McCabe, 2000), soil moisture (e.g. Farajalla and Vieux, 1995; Western
et al., 1998; Cosh and Brutsaert, 1999) and land use type (e.g. Moody and Woodcock, 1995; Walsh
et al., 2001) vary considerably depending on geographical area (climate, vegetation), extent of the area
and support scale of the data. For example, it is found that the scales for soil moisture determined
by Farajalla and Vieux and Cosh and Brutsaert compare favourably (about 1000 m), in contrast to the
35–60 m estimated by Western et al. This may be due to incorrect, large sampling distances in the
former studies resulting in an overestimation of the ‘true’ appropriate scale. The spatial scales of land
use have been mainly assessed in relation to land use change; implications of spatial scales of land use
for hydrological modelling are rarely investigated. The extent and support scale are the basics of the
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2583
so-called scale triplet (Bl¨
oschl and Sivapalan, 1995) and can have a substantial effect on the estimation
of the correlation length and related appropriate scales (see Western and Bl¨
oschl, 1999). These scale
features should be taken into account when determining and integrating the appropriate scales for the key
variables.
The objective of this paper is therefore to determine the appropriate spatial scales of the dominant variables
elevation, soil type, land use and precipitation and integrate them into an appropriate model scale. Dominant
refers to the relative importance of these variables with respect to the impact of climate change on river
flooding. The River Meuse basin in France, Belgium and the Netherlands is the geographical area chosen
for this research purpose. The objective is achieved by using observed data from several sources and then
employing different relationships between scales, variable statistics and outputs. The results are presented and
discussed and conclusions drawn.
OBSERVED DATA
Elevation, soil, land use, precipitation and discharge data are used in this study. The sources and spatial
characteristics (spacing, extent) of the data are described below.
Elevation data are from a global digital elevation model (DEM) GTOPO30 (US Geological Survey, 1996)
and a continental United States digital elevation model US7Ð5MIN (US Geological Survey, 1995). These data
are distributed by the EROS Data Center Distributed Active Archive Center (EDC DAAC), located at the US
Geological Survey’s EROS Data Center in Sioux Falls, South Dakota.
GTOPO30 is a global data set covering the full extent of latitude and longitude with elevation values
ranging from 407 to 8752 m. The horizontal grid spacing is 30 arc-sec (1/120°or approximately 1 km) and
the vertical unit is metres above mean sea level. The source for the European GTOPO30 data is the Digital
Terrain Elevation Data (DTED) set. DTED is a raster topographic database with a horizontal grid spacing of
3 arc-sec (approximately 90 m). The source for the United States GTOPO30 data are USGS 1°DEMs with a
horizontal grid spacing of 3 arc-sec. The 30 arc-sec data were obtained from the 3 arc-sec data by selecting
one representative elevation value to represent the area covered by 100 full resolution cells (10 ð10). The
absolute vertical accuracy of GTOPO30 for Europe and the United States is š30 m linear error at the 90%
confidence level (US Geological Survey, 1993).
US7Ð5MIN is a data set for the continental United States. The horizontal grid spacing is 30 m and the
vertical unit is metres above mean sea level. The data sources are digitized cartographic map contour
overlays and scanned National Aerial Photography Program (NAPP) photographs. The absolute vertical
accuracy of US7Ð5MIN derived from photographs is 7 m or better (90%) and 8–15 m (10%) (US Geological
Survey, 1995).
Soil data are from the European Soil Database. The data are distributed by the European Soil Bureau
located at the Joint Research Centre’s Space Applications Institute in Ispra, Italy. Only data from the Soil
Geographical Database (King et al., 1994), part of the European Soil Database, are used. The horizontal scale
is 1 : 1 000 000 similar to a horizontal grid spacing of approximately 2Ð5 km, because the positional accuracy
is estimated at 0Ð5–5 km (0Ð5– 5 mm at scale 1 : 1 000 000). Two attributes from this database have been used,
namely Dominant Parent Material (MAT1) and Dominant Surface Textural Class (TEXT1).
Land use data are from the European Environmental Agency (EEA) NATLAN Database. The data are
distributed by the European Environmental Agency in Copenhagen, Denmark. Only data from the CORINE
Land Cover Database (Bossard et al., 2000), part of the EEA NATLAN Database, are used. The horizontal
grid spacing is 250 m. The CORINE database has 44 classes and is derived from Landsat and SPOT
satellite images.
Daily precipitation data from 39 stations in the Meuse basin in Belgium and France for the period 1970–99
were used (see Booij, 2002c). Daily discharge data at the basin outflow point (Borgharen) and at several
sub-basin outflow points were employed.
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2584 M. J. BOOIJ
DETERMINATION AND INTEGRATION OF APPROPRIATE SPATIAL SCALES
Variables at different scales
The spatial variance is an important statistic describing the patterns of elevation, soil type and land use
type. The variance 2for a variable Zx is estimated as:
2D
n

xD1
[Zx ]2
n11
where nis the total number of points or cells in the spatial domain and is the average of Zx over n.In
general, it applies that the larger the area over which Zx is aggregated, the smaller will be the variance.
The relation between the point variance (subscript ‘p’) and the areally averaged variance (subscript ‘A’) can
be stated as follows:
2AD2p22
where 2is the variance reduction function decreasing with increasing area A. Its magnitude depends on the
spatial correlation structure of the variable, and the size and shape of the area. Rodriguez-Iturbe and Mejia
(1974) showed for a stationary isotropic spatial random field that 2is the expected value of the correlation
coefficient between any two points randomly chosen at distance h:
2DH
0
hfhdh3
where His the maximum distance within the area, (h) is the spatial correlation function and fh is the
probability density function (pdf) of the random variable h. Assuming a square area with sides a(and area
ADa2), the pdf is as follows (Ghosh, 1951):
fh D4h
a4h 4
where
h D1
2a22ah C1
2h20<h<a
h Da2sin1a
hcos1a
hC2ah2a21
2h2C2a2 a<h<H
Assuming an exponential spatial correlation function h Dexph/,whereis the spatial correlation
length, the spatial variance associated with an aggregation scale Acan be determined from the spatial point
variance. The determination of is done using variograms from geostatistics (continuous variables) and
semi-correlation functions (categorical variables).
The starting point for a variogram is the semi-variance between two points (xand xCh) at distance h
(e.g. Kitanidis, 1997):
gx, h D1
2[Zx Zx Ch]25
This semi-variance gx, h can be determined for all pairs of points in a specific area. An exponential variogram
model is then fitted to these semi-variances:
h D21exp h
 6
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2585
The sill 2is estimated directly from the variogram and the spatial correlation length for each area can
then be determined from the relation between and h. In this way, fitted variograms h for a specific area
are obtained.
Variograms cannot be used for categorical variables such as land use. The spatial variability of these
variables is described by introducing a so-called semi-correlation rŁas:
rŁs, h D
n

xD1







Zsx Ch
M

mD1
Zmx Ch







n7
where sis a specific category of Zx resulting in Zsx, m is an arbitrary category of Zx resulting in Zmx
and Mis the total number of categories. For each category s, the semi-correlation rŁcan be plotted as a
function of the separation distance hto obtain a semi-correlogram from which the spatial correlation length
for each category can be estimated.
The correlation length will be used to determine the appropriate scale and should therefore be meaningful
for hydrological applications, in particular under extreme conditions. For example, it does not make sense
to estimate correlation lengths associated with mountain ranges or large geological structures, as they are
not assumed to play a key role in extreme hydrological behaviour. On the other hand, scales related to hill
slopes and soil types may be very important for this kind of behaviour. These considerations may arise
when non-stationary variograms or semi-correlograms with a number of preferred scales are the result of the
geostatistical analyses. In that case, the scale associated with extreme hydrological behaviour, assuming local
stationarity at that scale (see e.g. Bl¨
oschl, 1999), will be employed. If necessary, this will be clarified for the
variable concerned.
Variables at appropriate scales
The appropriate spatial scale for a variable is dependent on its correlation structure and the application
area studied. Spatial scales should be sufficiently detailed to capture natural variability, but not unnecessarily
refined that computation time is wasted. The appropriate spatial scale can be determined by means of the
relations above given a specific appropriateness criterion. This criterion is based on the bias allowed in
estimating the variance of areally averaged variables from the variance of point variables. The higher this
permitted bias, the larger the appropriate scale. The determination of the appropriate scale is illustrated in
Figure 1, where the reduction of the variance is given as a function of dimensionless aggregation scale (a/).
The dotted line illustrates the determination of the appropriate scale assuming a permitted bias of 10%. This
arbitrary bias is assumed to be appropriate in this study and results in an appropriate scale for the variance
which is about 20% of the correlation length. The appropriate scale for return values was found to be 25%
of the correlation length assuming a bias of 10% (see Booij, 2002c). It thus depends on the relevant statistic
of which fraction of the correlation length represents the appropriate scale for a given bias.
Variables at the appropriate model scale
The appropriate scales for each key variable should be combined to derive an appropriate river basin model
scale. For variables and processes with smaller appropriate scales than this appropriate model scale, distribution
functions may be used to solve the subgrid scale variability or values and processes may be simply averaged.
For variables and processes with less variability, constant values may be used over several appropriate model
scales. The integration of separate appropriate variable scales towards an appropriate model scale is performed
by means of relations between key variable scales and the output variable of interest. The relative importance
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2586 M. J. BOOIJ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.001 0.01 0.1 1 10
reduction
0.21
a/
l
Figure 1. Determination of the appropriate scale for a variable accepting a 10% bias
of the separate appropriate variable scales is dependent on the sensitivity of the model output to changes in
these scales. This sensitivity can be assessed by means of a sensitivity analysis, however when no specific
model is available another method should be employed. The SCS method (see e.g. Maidment, 1992) is used
for this purpose, because an approximate estimation is required and the model output of interest is the peak
discharge.
The peak discharge qpin the SCS method is derived from a triangular approximation to the hydrograph
shown in Figure 2 resulting from a rainfall excess intensity peof duration Tp(and volume PeDpeTp). The
lag Tlfrom the centroid of rainfall excess to the peak and the time of rise Tqto the peak are illustrated as
well. The base length of the hydrograph 2Ð67Tqis based on the study of many unit hydrographs (Maidment,
1992). The volume of runoff under the hydrograph Vpis derived with the basic SCS relationship
VpD[25Ð4Pe5Ð08fCN]2
25Ð4PeC20Ð3fCNwith fCND1000
CN 10 8
Tl
Tp/2
Tp
Tq
qp
pe
1.67Tq
Figure 2. SCS triangular hydrograph
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2587
where Vpand Peare in millimetres and CN is the well-known curve number dependent on soil type, land use
type and hydrologic condition of the land surface. CN values can be found in extensive tables (e.g. Maidment,
1992) and vary between 20 and 100. Equating the volume Vpto the volume 1
2qp2Ð67Tqin Figure 2, rearranging
and adjusting for units gives qp(m3s1):
qpD5Ð28AVp
0Ð5TpCTl
9
where Ais the catchment area (km2), Tpis in hours and Tlis in hours as follows (Kent, 1972):
TlDl0Ð8[1 CfCN]0Ð7
3Ð42S0Ð5
0
10
where lis the hydraulic length of the catchment (km) and S0is the overland slope in parts per 10 000.
The sensitivity of the peak discharge to changes in variable scales is assessed by introducing into
Equation (9) relationships between its parameters (Pe,CNandS0) and variable scales, instead of constant
parameters. In this way, relations between the peak discharge and the variable scales (e.g. for elevation) are
obtained. The relationships between the SCS parameters and variable scales are empirical ones acquired by
estimating the different parameters for variable fields with different spatial resolutions. If CN and S0are used
as indicators for respectively soil type/land use and topography, relations between CN and soil type/land
use scale, and S0and elevation scale can be incorporated into Equation (9). A precipitation–scale relation
can be directly implemented into Equation (8). In this way, relationships between the key variable scale
and the output variable are established. These relationships are used to assess the weights associated with
an appropriate variable scale. Finally, the weights are multiplied with the appropriate scales to obtain the
appropriate model scale.
It should be mentioned that this hypothesis (i.e. that the appropriate model scale is a linear combination
of appropriate scales of dominant variables) has not been tested. The hypothesis may not hold for small and
fast responding catchments with strong non-linear spatial interactions in rainfall-runoff processes. However,
in large river basins (like the Meuse) the methodology may give a good indication on the appropriate model
scale, which can be used in the model design. Moreover, the weights used in the methodology have been
obtained by non-linear relations between the model output and the variable scales, through the SCS method
and its parameters.
RESULTS AND DISCUSSION
Appropriate scale for elevation
West er n a nd B l ¨
oschl (1999) presented directives on spacing, support and extent when estimating the cor-
relation length. They gave for spacing (equal to support in this case) a maximum value of 20% of the true
correlation length. Assuming true correlation lengths of about 100–1000 m corresponding to realistic hill
slope lengths, it is obvious that DEMs with a resolution of 1 km cannot be used for this purpose. Therefore,
the spatial correlation length should be determined from the US DEM with a resolution of 30 m, on condition
that the elevation pattern of this DEM is representative for the Meuse basin. Unfortunately, DEMs with a
higher resolution than 1 km were not available for the Meuse basin. The geographical area chosen is a region
in the northeast of the United States (69 °W–68°W and 46 °N–47°N) with a similar elevation pattern and
elevation distribution as the Meuse basin.
Figure 3 shows the cumulative frequency distribution of elevations from US7Ð5MIN (US 30m DEM) for five
selected subregions of about 150 km2, GTOPO30 for the whole region (US 1000m DEM), GTOPO30 for the
15 main sub-basins of the Meuse basin (Tributary 1000m DEM) and GTOPO30 for the Meuse basin (Meuse
1000m DEM). The cumulative frequency distributions for the US region and the Meuse basin are similar,
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2588 M. J. BOOIJ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100 200 300 400 500 600 700 800
Elevation (m)
Frequency (-)
US 30 m DEM
US 1000 m DEM
Tributary 1000 m DEM
Meuse 1000 m DEM
Amblève
Mehaigne
Figure 3. Cumulative frequency distribution of elevation for US 30m DEM, US 1000m DEM, Meuse tributary 1000m DEM and Meuse
1000m DEM
indicating that the US region has a similar elevation pattern and elevation distribution as the Meuse basin. The
cumulative frequency distributions for elevations derived from the US 30m DEM are steeper than those derived
from the Tributary 1000m DEM and thus in the latter DEMs larger elevation differences can be found. This can
be explained by the fact that the Tributary 1000m DEMs have been derived for larger areas (350–3200 km2)
than the US 30m DEMs (150 km2). A similar effect as for the extent has been observed for the support
scale, i.e. the smaller the support scale for a specific area, the larger the elevation differences. Large elevation
differences are particularly found for the sub-basins in the eastern part of the Meuse basin (e.g. the Ambl`
eve).
The mean, minimum and maximum elevation in the Meuse 1000m DEM are respectively 284, 43 and 676 m.
Figure 4 shows the semi-variance as a function of separation distance for the US 1000m DEM, the Meuse
1000m DEM and the US 30m DEM as double logarithmic plots. For the US 30m DEM, the semi-variance
has been derived for the five subregions for very small separation distances up to 2 km. Nine grid points
were selected in each subregion and for each grid point average semi-variances were calculated for a specific
separation distance (radius). The average semi-variance as a function of separation distance over 45 (5 ð9)
grid points is shown in Figure 4b. Similar approaches were used for the US 1000m DEM and Meuse 1000m
DEM, but then for nine grid points in the whole region and separation distances up to 20 km. The variograms
in Figure 4a are similar, although the one for the US region shows less spatial variability. Therefore, it can
be assumed that the US region is representative for the Meuse basin regarding elevation patterns and it is rea-
sonable to assume that the variogram for the US 30m DEM in Figure 4b is representative for the Meuse basin
as well. The variograms in Figure 4a are non-stationary and no clear preferred scales can be distinguished.
The variogram in Figure 4b may not be completely stationary (in particular for larger separation distances
than shown), but a correlation length can be derived if local stationarity is assumed. This correlation length
agrees well with hill slope scales important for hydrological behaviour.
The mean, minimum and maximum correlation length determined from the 45 US 30m variograms are
respectively 527, 272 and 1401 m. It can be seen that even for the minimum correlation length the resolution
of the US 30m DEM is sufficient. The mean correlation lengths computed from the US 1000m DEM and the
Meuse 1000m DEM were respectively 4138 and 4220 m. This overestimation of the correlation lengths could
be expected from the relations between spacing and apparent correlation described in Western and Bl¨
oschl
(1999). The appropriate scale for elevation can be determined with the criterion for appropriate scales given
earlier. Then, it was found that the appropriate scale for an appropriate description of the variability is about
20% of the correlation length. The appropriate scale for elevation in the Meuse basin is therefore 110 m.
This scale is in the same range as the scales recommended for DEMs in other studies (e.g. Brasington and
Richards, 1998).
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2589
1
10
100
1000
10000
100 1000 10000 100000
Separation distance (m)
Semi-variance (m2)
US 1000 m DEM
Meuse 1000 m DEM
(a)
1
10
100
1000
1 10 100 1000 10000
Separation distance (m)
Semi-variance (m2 )
US 30 m DEM
(b)
Figure 4. Semi-variance (m2) as a function of separation distance (m) for (a) US 1000m DEM and Meuse 1000m DEM and (b) US 30m
DEM
The methodology for the integration of scales requires relationships between key variable scales and the
output variable of interest. The key variable elevation is represented in Equation (10) by the slope. Therefore,
the average slope is derived from elevation maps with different resolutions. These different resolutions are
assumed to represent different data availability levels (e.g. at 1 km or 8 km resolution).
Appropriate scale for soil types
Figure 5 shows the frequency distribution of MAT1 and TEXT1 in the Meuse basin. The major dominant
parent materials are secondary limestone (18%, particularly in the southern part of the basin), residual and stony
loam (34%, in the Ardennes and central part) and loess (17%, in the northern part). The major surface textural
class is medium, which means 18% <clay <35% and sand >15% or clay <18% and 15% <sand <65%.
Variograms for the different soil types cannot be constructed, because soil data are categorical data.
Therefore, the semi-correlation is calculated and plotted versus separation distance in a semi-correlogram.
Figure 6 shows the semi-correlogram for the different classes of MAT1 and TEXT1. Obviously, the
general trend is a decreased semi-correlation with increased separation distance. This trend is approximately
exponential as confirmed by the high weighted mean (taking into account the frequencies) PPCCŁvalues of
0Ð93 for MAT1 and 0Ð98 for TEXT1. PPCCŁis defined as the correlation between minus the logarithm of the
semi-correlation and the distance, analogous to the probability plot correlation coefficient of Filliben (1975).
The semi-correlogram for secondary limestone in Figure 6a shows a slightly increasing semi-correlation with
increasing distance between 30 and 45 km. This may be attributed to large-scale spatial patterns of secondary
limestone. The semi-correlation decreases towards its ‘background’ frequency from Figure 5 at a distance
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2590 M. J. BOOIJ
(a)
0.00
0.05
0.10
0.15
0.20
0.25
River alluvium
Limestone
Secondary
limestone
Marl
Secondary
marl
Secondary
clay
Residual clay
Clay with flints
Tertiary sands
Soft quartzy
sandstone
Residual loam
Stony loam
Sandy loam
Eolian loam
Loess
Acid crystalline
rocks
Granite
Schists
Frequency (-)
(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Coarse Medium Medium fine Fine No texture
Frequency (-)
Figure 5. Frequencies of (a) MAT1 and (b) TEXT1 in the Meuse basin
larger than the corresponding correlation length. The semi-correlograms can be used to derive the spatial
correlation lengths for the different soil types.
Figure 7 shows the spatial correlation lengths for the different classes of MAT1 and TEXT1. The correlation
lengths for MAT1 vary between 1 and 10 km for residual clay, clay with flints and schists to more than 30 km
for secondary limestone and secondary marl. The correlation lengths for TEXT1 vary between 15 and 35 km.
The horizontal grid spacing of the soil data is sufficient for most of the soil types, because their correlation
lengths are more than five times the grid spacing. The grid spacing is insufficient for the above-mentioned three
MAT1 classes with small correlation lengths and limestone. However, these MAT1 classes occupy less than
5% of the Meuse basin and therefore possible errors in estimated correlation lengths due to grid spacing can
be neglected. The weighted mean spatial correlation length for dominant parent material is about 25Ð1km.
Then, the appropriate scale for a good description of the parent material variability is about 20% of this
correlation length or 5Ð3 km. These figures are for the dominant surface textural class respectively 31Ð5km
and 6Ð6 km. The appropriate scale with respect to the parent material has been chosen to be representative
for the soil data, because it is the smallest one and it is supposed to be more suitable in the methodology for
the integration of appropriate scales. The correlation lengths for parent material and dominant surface textural
class are much larger than the correlation lengths for soil moisture found in the literature (50–1200 m). These
differences can be attributed to the additional variability for soil moisture introduced by e.g. topography and
vegetation on top of the variability of the parent material and soil texture itself.
The key variable soil is represented in Equations (8) –(10) by the curve number CN. The average CN will
be derived from soil (MAT1) maps with different resolutions.
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2591
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10000 20000 30000 40000 50000 60000 70000 80000
Separation distance (m)
0 10000 20000 30000 40000 50000 60000 70000 80000
Separation distance (m)
Semi-correlation (-)
River alluvium Limestone Secondary limestone Marl Secondary marl
Secondary clay Residual clay Clay with flints Tertiary sands Soft quartzy sandstone
Residual loam Stony loam Sandy loam Eolian loam Loess
Acid crystalline rocks Granite Schists
(b)
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Semi-correlation (-)
Coarse (clay < 18 % and sand > 65 %)
Medium (18% < clay < 35% and sand > 15%/ clay < 18% and 15% < sand < 65%)
Medium fine (clay < 35 % and sand < 15 %)
Fine (35 % < clay < 60 %)
No texture (histosols, ...)
Figure 6. Semi-correlation as a function of separation distance (m) for (a) MAT1 and (b) TEXT1
Appropriate scale for land use types
Figure 8 shows the frequency distribution of land use types in the Meuse basin. The land use types are
described in Bossard et al. (2000). In the Meuse basin, 29 of the 44 land use types in the CORINE database
are present. Urban, industrial and mining areas occupy about 9% of the space, agricultural areas 54%, forests
and scrubs about 36% and water-related areas about 1%. The Meuse north sub-basin has the largest fraction
of urban, industrial and mining areas (29%, not shown here), the northwestern Jeker and Mehaigne sub-basins
have the largest fraction of agricultural areas (80%) and the Viroin sub-basin has the largest fraction of forests
and scrubs (57%).
Land use data are categorical data as well and therefore semi-correlograms are computed for the different
land use types as shown in Figure 9. The correlograms can be approximated by exponential functions as
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2592 M. J. BOOIJ
0
5000
10000
15000
20000
25000
30000
35000
River alluvium
Limestone
Secondary limestone
Marl
Secondary marl
Secondary clay
Residual clay
Clay with flints
Tertiary sands
Soft quartzy sandstone
Residual loam
Stony loam
Sandy loam
Eolian loam
Loess
Acid crystalline rocks
Granite
Schists
Spatial correlation
length (m)
0
5000
10000
15000
20000
25000
30000
35000
40000
Coarse Medium Medium fine Fine No texture
Correlation length (m)
(a)
(b)
Figure 7. Spatial correlation length (m) for (a) different MAT1 and (b) different TEXT1
confirmed by the reasonable weighted mean PPCCŁvalue of 0Ð89. Figure 9 shows expected features such as
the slow decrease of the semi-correlation with separation distance for broad-leaved forest and the fast decrease
of the semi-correlation with separation distance for continuous urban fabric. The ‘background’ frequency from
Figure 8 is reached at a large distance compared to the correlation length as shown for some land use types
with small correlation lengths (−20 km).
Figure 10 shows the spatial correlation lengths for the different land use types in the Meuse basin. The
correlation lengths vary between 100 and 500 m for port areas, airports, rice fields and sclerophyllous veg-
etation to about 20 km for discontinuous urban fabric, broad-leaved forest and mixed forest. The horizontal
grid spacing of the CORINE land use data is sufficient for most of the land use types, except for the above-
mentioned four land use types with small correlation lengths and construction sites, sport and leisure facilities
and natural grassland. However, these seven land use types occupy less than 1% of the Meuse basin and
therefore possible errors in estimated correlation lengths due to grid spacing can be neglected. The weighted
mean spatial correlation length for land use is 15Ð7 km which gives an appropriate scale for a good description
of the land use variability of about 3Ð3 km. This appropriate scale is in the middle of the range of spatial
scales recommended for land use mapping and used in land use models.
The key variable land use is represented in Equations (8) –(10) by the curve number CN as well. The
average CN will be derived from land use maps with different resolutions.
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2593
0
0.05
0.1
0.15
0.2
Continuous urban fabric
Discontinuous urban fabric
Industrial and commercial units
Road and rail networks
Port areas
Airports
Mineral extraction sites
Dump sites
Construction sites
Green urban areas
Sport and leisure facilities
Non-irrigated arable land
Rice fields
Fruit trees and berry plantations
Pastures
Annual crops
Complex cultivation patterns
Principally agriculture
Broad-leaved forest
Coniferous forest
Mixed forest
Natural grassland
Moors and heathland
Sclerophyllous vegetation
Transitional woodland-shrub
Inland marshes
Peatbogs
Water courses
Water bodies
Frequency (-)
Figure 8. Frequencies of land use types in the Meuse basin
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5000 10000 15000 20000
Separation distance (m)
Semi-correlation (-)
Continuous urban fabric Discontinuous urban fabric Industrial and commercial units Road and rail networks
Port areas Airports Mineral extraction sites Dump sites
Construction sites Green urban areas Sport and leisure facilities Non-irrigated arable land
Rice fields Fruit trees and berry plantations Pastures Annual crops
Complex cultivation patterns Principally agriculture Broad-leaved forest Coniferous forest
Mixed forest Natural grassland Moors and heathland Sclerophyllous vegetation
Transitional woodland-shrub Inland marshes Peatbogs Water courses
Water bodies
Figure 9. Semi-correlation as a function of separation distance (m) for different land use types
Integration of appropriate variable scales
Here, the appropriate scales for precipitation (20 km, see Booij, 2002c), elevation, soil and land use are
integrated to one appropriate model scale. This is achieved by applying the methodology described previously.
First, the relationships between the parameters from Equations (8)–(10) (e.g. the slope) and the spatial scales
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2594 M. J. BOOIJ
0
5000
10000
15000
20000
25000
Continuous urban fabric
Discontinuous urban fabric
Industrial and commercial units
Road and rail networks
Port areas
Airports
Mineral extraction sites
Dump sites
Construction sites
Green urban areas
Sport and leisure facilities
Non-irrigated arable land
Rice fields
Fruit trees and berry plantations
Pastures
Annual crops
Complex cultivation patterns
Principally agriculture
Broad-leaved forest
Coniferous forest
Mixed forest
Natural grassland
Moors and heathland
Sclerophyllous vegetation
Transitional woodland-shrub
Inland marshes
Peatbogs
Water courses
Water bodies
Spatial correlation length (m)
Figure 10. Spatial correlation length (m) for land use types in the Meuse basin
of the key variables (e.g. the elevation) are considered. Second, these relationships are integrated into the
equations to obtain relations between the output and the spatial scales of the different key variables. The
output of interest is the peak discharge with a return period of 20 years, RV(20).
The key variable precipitation in Equation (8), Pe, should be the extreme precipitation with a return period
of 20 years, because a return period of 20 years is assumed for the discharge. The relation between precipi-
tation return values and scales has been theoretically considered and applied by Booij (2002c). This relation
can be used directly in Equation (8) to obtain a relationship between the output and spatial scales of extreme
precipitation.
0.0001
0.001
0.01
0.1
1
10 100 1000 10000 100000
Spatial scale (m)
Slope (-)
US 30 m DEM (5X)
US 1000 m DEM
Meuse 1000 m DEM
Figure 11. Average slope as a function of scale (m) for US 30m DEM, US 1000m DEM and Meuse 1000m DEM
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2595
The key variable elevation is represented in Equation (10) by the slope S0. The average slope as a func-
tion of scale derived from US 30m DEM, Meuse 1000m DEM and US 1000m DEM elevation maps and
their aggregated versions is given in Figure 11. Main features are the similarity of the US 1000m DEM and
Meuse 1000m DEM slope versus scale relationships and the apparent continuity of the relationships over
a broad range of scales (from 30 m to more than 30 km). The combination of these two aspects leads to
the assumption that the slope versus scale relationships from the US 30m DEM can also be used for the
Meuse basin and thus the appropriate scale for elevation (represented by slope) of about 100 m is covered by
these relationships. This coverage is necessary for the methodology for the integration of appropriate scales.
Namely, the scale–variable relation should at least cover the range between the appropriate variable scale
and the appropriate model scale or vice versa. This requirement has been checked after the determination of
the appropriate model scale. Another interesting feature in Figure 11 is the clear distinction between relations
for flat US regions (lower curves for US 30m DEM) and hilly US regions (upper).
The key variables soil and land use are represented in Equations (8)– (10) by the curve number CN. The
average, overall curve number for the Meuse basin was derived with Equations (8)– (10) and the variables
and parameters in Table I. The values for Peand qpwere derived keeping in mind the 20-year return period
and the scale for Pe, and the value for S0is estimated from the elevation field with the highest resolution
(1000 m). This results in an average CN of 77Ð1 for the soil and land use map with the highest resolution.
This average CN and standard tables (Maidment, 1992) have been used to estimate curve numbers for the
different soil and land use types. Finally, the average curve numbers as a function of map resolution for soil
parent material and land use could be determined (not shown).
The derived relationships between the SCS parameters and the scales of extreme precipitation, slope and
curve number (such as in Figure 11) can be implemented in Equations (8) –(10) to obtain relationships between
the output and key variable scales. Figure 12 gives the dimensionless peak discharge [qp(spatial scale)/qp
(smallest spatial scale)] as a function of the spatial scale of the key variables. The relationships between SCS
parameters and variable scales were used as an intermediate step to obtain this relation between the peak
discharge (a function of SCS parameters) and variable scales. The peak discharge increases slightly with scale
for soils in contrast to the other key variables, which may be due to the spatial distribution of soil types.
Figure 12 has been used to assess the weights associated with an appropriate variable scale by comparing the
slopes of the different relationships. The larger the slope, the larger the weight which should be attributed to
a specific appropriate variable scale. The sum of the four weights (precipitation, elevation, soil type and land
use) is obviously equal to 1. The slope was determined for at least the range between the appropriate variable
scale and the appropriate model scale (checked a posteriori). The appropriate variable scales, the associated
weights and the resulting appropriate model scale are summarized in Table II. The appropriate model scale
for modelling the impact of climate change on flooding in the River Meuse is about 10 km. This scale will
result in 225–250 model cells or sub-basins for the Meuse basin.
Table I. Values for variables and parameters in
Equations (8)–(10)
Variable/parameter Unit Value
Pemm 44Ð6
qpm3/s 2390
Akm221 ð103
Tph24
lDpAkm 145
S0parts/10 000 176
CN – 77Ð1
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2596 M. J. BOOIJ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5000 10000 15000 20000 25000 30000 35000
Spatial scale (m)
Dimensionless peak discharge
q
p
'
Precipitation
Elevation-Slope
Landuse-CN
Soil-CN
Figure 12. Dimensionless peak discharge q0
pas a function of the spatial scale (m) of the key variables
Table II. Appropriate variable scales, associated weights and appro-
priate model scale
Variable Appropriate variable scale Weight
(km) (–)
Precipitation 19Ð90Ð39
Elevation 0Ð10Ð26
Soil 5Ð30Ð21
Land use 3Ð30Ð14
Integrated 9Ð51Ð00
CONCLUSIONS
The idea of model appropriateness is introduced in a search for an optimum model complexity for a specific
research objective and area. The definition of an appropriate model is based on a consistency criterion implying
that a model should have scales and formulations at a balanced level. This criterion has been implemented in
a model appropriateness procedure, which was applied to assess the appropriate model scale. The processes
to be incorporated and the process formulations to be used were determined qualitatively by Booij (2002a).
The usefulness of the appropriateness procedure is thus in its ability to assess the appropriate scales
of the individual key variables before model construction and integrate them in a balanced way into an
appropriate model scale. The procedure will not result in a model for a specific situation with a prescribed
maximum uncertainty. Obviously, the model output uncertainty can be determined afterwards and compared
with accuracy criteria formulated by model users or policy makers. This output uncertainty should consist of
several uncertainties at a balanced level. Another use of the procedure is that it provides a framework for
decisions about the reduction or expansion of data networks and needs. Application of the framework to the
dominant variables in a specific situation may reveal where possible data inconsistencies (e.g. a too sparse
network) exist. Also, discrepancies between observations and model results may be explained with the help
of the procedure, as has been done for example by Booij (2002c). Obviously, the adequacy of the integration
function for a specific situation (the curve number method in this case) should be considered when applying
the appropriateness procedure.
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SPATIAL SCALES FOR RIVER BASIN MODELLING 2597
The results may be generalized to other geographical areas and research problems. The appropriateness
procedure can in principle be applied to every arbitrary area in the world. The dominance of processes is area
dependent, but in general, similar variables play an important role in flood generating rainfall-runoff processes.
The appropriate scales of these variables depend on the region. For example, the appropriate spatial scale
for precipitation will be much smaller for tropical, mountainous areas with large orographical effects than
for temperate, relatively flat areas like the Meuse basin. Furthermore, the appropriate scales are influenced
by the size of the basin. Small catchments require smaller time scales than large basins and consequently
associated spatial scales will differ between small and large basins. The appropriate spatial scales were based
on a daily time scale in this research for the Meuse basin, whereas the determination of appropriate spatial
scales for a similar study in a research catchment of several square kilometres should be based on an hourly
or even minute time scale leading to smaller appropriate spatial scales. Similarly, scale analysis in a climate
change study for the Amazon may be based on a weekly or monthly time scale, probably resulting in larger
appropriate spatial scales. Appropriate process formulations will change with these changes in scale as they
are often scale dependent.
The type of research problem may influence the importance of processes and associated variables. For
example, when studying the impact of climate change on low flows, evapotranspiration and groundwater
flow become more important, whereas overland flow and subsurface storm flow may be less significant.
Consequently, other variables are dominant and other appropriate spatial and temporal scales will prevail.
Therefore, based on the results obtained here, only very general guidelines with respect to appropriate models
for other research objectives can be given.
ACKNOWLEDGEMENTS
The daily station climatological data for Belgium and France have been provided respectively by Luc
Debontridder from the KMI (Belgian Royal Meteorological Institute) and Christophe Dehouck from M´
et´
eo
France. Bob Jones of the European Soil Bureau and Malene Bruun of the European Environmental Agency
kindly supplied the soil and land use data, respectively. Eric Sprokkereef of RIZA gave the essential Meuse
basin data and Joop Gerretsen of Rijkswaterstaat Limburg made the discharge data available. The comments
of two anonymous reviewers and discussions with Kees Vreugdenhil of the University of Twente helped to
improve the manuscript substantially.
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