HYDROLOGICAL PROCESSES
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
Scales and similarities in runoff processes with respect to
geomorphometry
Jochen Schmidt\ Kirsten Hennrich and Richard Dikau
Geo`raphisches Institut\ Universitat Bonn\ Meckenheimer Allee 055\ D!42004 Bonn\ Germany
Abstract]
Numerous investigations using various techniques have been carried out towards a more detailed understanding
of relationships and interactions between catchment morphometry and rainfallÐruno} processes[ Recently\
this research question has become more relevant through the need for accurate\ yet simple\ computer models
simulating the water balance of large areas[ Moreover\ advances in the analysis of landform morphometry
through the availability of high!resolution digital elevation models "DEMs# and powerful geographical infor!
mation systems "GIS# have enhanced research e}orts with this aim[
In this study several computer techniques and models were applied to investigate the e}ects of geo!
morphometry on rainfallÐruno} processes at di}erent scales[ The sensitivity of dynamic hydrological processes
to comparatively static boundary conditions requires di}erent methods for modelling\ analysis and visualization
of di}erent kinds of data appropriate to di}erent scales[ Therefore an approach integrating several geo!
computational concepts\ including spatial analysis of di}erent types of geodata\ static modelling of spatial
structures\ dynamic four!dimensional modelling of hydrological processes and statistical techniques was chosen[
Geomorphometric analysis of the study sites was carried out with GIS packages "including ARC:INFO and
GRASS#\ special purpose software and self!developed tools[ SoilÐmorphometry relationships were modelled
within a GIS environment[ Hydrological models "SAKE and TOPMODEL# were then used to simulate
rainfallÐruno} processes\ and _nally statistical tools and sensitivity analysis were applied to gain an insight
into the hydrological signi_cance of the various geomorphometric properties[
The results demonstrate the importance of small subregions of the catchment\ particularly those having low
slope angles\ low ~ow lengths and concavities[ The spatial distribution of soil types signi_cantly in~uences
modelled runo}[ Spatial distributions of soil types are partly related to morphometry and can be captured
using soilÐmorphometry models[ Further results show that catchments which di}er signi_cantly in mor!
phometry show di}erent runo} responses and di}erent hydrological sensitivity to changes in boundary
conditions[ A crude derivation of geomorphometricÐhydrological landform types could be reached[ Therefore\
geomorphometric classi_cations of catchment types could form a basis for representative hydrological mod!
elling at the large scale[ Models describing soil distribution in relation to geomorphometry could assist
regionalization of spatial heterogeneity and structure of soil parameters relevant in hydrological modelling[
Moreover\ quanti_cation of geomorphometric catchment structure\ e[g[ in terms of contributing areas\ is
needed to describe signi_cant geomorphometric catchment characteristics[ Copyright Þ 1999 John Wiley +
Sons\ Ltd[
KEY WORDS hydrological modelling^ geomorphometry^ GIS^ DEM^ soilÐmorphometry relationship
INTRODUCTION
Regionalization issues have been main research tasks in hydrology\ especially over the past decade "Gupta
et al[\ 0875^ Bloschl and Sivapalan\ 0884#[ Recently\ scaling problems have become more relevant through
Correspondence to] J[ Schmidt\ Geographische Institut\ Universitat Bonn\ Mechenheimer Allee 055\ D!42004 Bonn\ Germany[
E!mail] jochenÝslide[giub[uni!bonn[de
Copyright © 2000 John Wiley & Sons, Ltd.
Received 3 February 1998
Accepted 25 June 1999
0853
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
Figure 0[ Scales in hydrology and geomorphology[ The _gure shows in a crude way some dominant features of each discipline in a
spatial and spatio!temporal context[ Translating scale properties from one discipline to the other is an open research question "Anderson
and Burt\ 0889#
the need for hydrological computer models simulating the water balance of large areas[ These models are
useful in ~ood forecasting\ for example\ or improving atmospheric circulation models[ Large!scale models
cannot incorporate detailed and physically based description of processes\ because of unknown boundary
conditions and limited computing capacities[ Parameterization of boundary conditions and simpli_cation
of models are therefore two necessary steps towards the development of hydrological models for larger
scales[ One research objective within this broad range of topics is the de_nition of parameters describing the
e}ects of landform structure and topology on hydrological processes\ which we term {e}ective geo!
morphometric parameters|[
On a qualitative basis\ it is well known that hydrological processes are in~uenced by geomorphometric
properties such as local slope angle\ convergencies or drainage density "Gregory and Walling\ 0862#[ There
exist some approaches quantifying these relationships through drainage basin parameters "see Moore et al[\
0880# and model conceptions\ such as the geomorphological instantaneous unit hydrograph "see Moore et
al[\ 0880^ Bloschl and Sivapalan\ 0884#[ A general quanti_cation of these e}ects\ however\ is still an
unanswered research question[ Recent advances in the analysis of landform morphometry through the
availability of high!resolution digital elevation models "DEMs# and powerful geographical information
system "GIS# packages enhance such research e}orts and form the focus of this paper[
With respect to the signi_cance of geomorphometric properties in hydrology\ scaling e}ects have to be
considered\ meaning that "i# runo}Ðmorphometry relationships "which tend to be invariant over certain
spatial ranges# and "ii# spatial thresholds a}ecting changes in these relationships have to be determined
"Bloschl and Sivapalan\ 0884^ Wood\ 0884#[ In general\ local scale\ hillslope scale and catchment scale are
often used to distinguish di}erent spatial scales in hydrology "Figure 0# "Kirkby\ 0877#[ On the local scale
water ~ow!path geometries\ ~ow velocities and quantities are in~uenced directly by parameters such as slope
angle and upslope drainage area[ Additionally\ geomorphometry a}ects hydrological processes indirectly
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0854
through their dependency on several other factors "such as soil parameters#[ The hillslope scale is dominated
by runo} production mechanisms in~uenced by soil properties "partitioning of overland ~ow and subsurface
~ow# and hillslope form[ Extracting typical {hillslope stripes| has been one strategy to represent hillslope
hydrology within larger scale catchments[ On the basin scale\ the hydrograph is in~uenced by basin mor!
phometry\ which can be expressed by representative attributes for catchment height distribution "relief
indices#\ length and form of the basin "form indices# and parameters describing the drainage network
"Gregory and Walling\ 0862^ Cooke and Doornkamp\ 0889^ Schmidt and Dikau\ 0888#[ It is also well known
that mesoscale or macroscale landform types a}ect hydrological characteristics signi_cantly[
Regarding these scaling e}ects\ di}erent research methods have to be used in the study of this problem[
The GIS packages provide sets of useful techniques and tools for investigations on di}erent spatial levels
"given that su.cient data are available#[
In 0880\ the Deutsche Forschungsgemeinschaft "German Research Council\ Bonn# established the project
{Regionalization in Hydrology|[ The general aim of this project is to develop methods transferring hydrol!
ogical models and parameters from small spatial scales to larger spatial scales "Kleeberg\ 0881#[ One part of
this project concerns the de_nition and regionalization of geomorphometric characteristics and attributes
with hydrological relevance on di}erent spatial scales[ This part of the project was carried out by research
groups at the University of Bonn "Germany# and the University of Heidelberg "Germany# using several
computational techniques[ In this paper some results of this investigation are presented "see Schmidt et al[\
0887^ Hennrich et al[\ 0888^ Schmidt and Dikau\ 0888#\ and used to]
0[ identify relationships between runo} and catchment morphometry at di}erent scales^
1[ show possible regionalization methods transferring morphometryÐruno} relationships over di}erent
scales[
STUDY AREAS
Focusing on scale issues in hydrology and geomorphology requires that di}erent size ranges in catchments
have to be incorporated[ In our study\ we distinguished "i# local "subcatchment# scale\ "ii# catchment scale
and "iii# regional scale[ On the local scale detailed catchment and subcatchment information is necessary\
whereas for catchment and regional!scale studies a comparatively large inventory of catchments is required[
For this\ two main study areas have been chosen]
0[ the Weiherbach * a highly instrumented catchment "5=2 km1# providing detailed data relevant for small!
scale studies and test purposes^
1[ the Leineturm * a large catchment "880 km1# having similar landform\ geology\ soils and land use to the
Weiherbach catchment\ enabling comparisons at the di}erent scales "the Leineturm catchment was the
_nal test area for all projects within the project {Regionalization in Hydrology|#[
The locations of both catchments are shown in Figure 1[
The Weiherbach catchment is located in the Kraichgau loess region of south!west Germany[ The catchment
has been investigated extensively and therefore a wide variety of _eld data have been collected\ including
meteorological\ hydrological\ land use and soil data "see Merz and Plate\ 0886^ Merz and Bardossy\ 0887^
Schmidt et al[\ 0887#[ A high!resolution "01[4 m grid size# DEM was also available for this study[ These data
serve as a useful basis for small!scale morphometryÐruno} investigations[ The Weiherbach catchment shows
an asymmetry\ caused by Pleistocene processes\ between smaller\ steeper and highly eroded west!facing
subcatchments and larger\ gentler\ loess!covered east!facing subcatchments "Figure 1#[ Elevation ranges
from 039 m to 149 m[ The area has been dominated by intensive agricultural land use[ Main soil types are
loess soils[
The Leineturm catchment is located in the central part of Germany\ south of Hannover "Figure 1#[ It is
the catchment of the upper Leine River\ a part of the Weser catchment[ Its south!eastern part lies within the
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0855
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
Figure 1[ Research areas used in this study] the small Weiherbach catchment "5=2 km1# is located in the Kraichgau loess region in south!
west Germany[ The Leineturm catchment "880 km1# lies within the mountainous Harz region in the central part of Germany
mountainous Harz region[ The lower part of the catchment is dominated by agricultural land use\ and the
upper part is mainly forested[ Elevation ranges from 019 m to 419 m[ Climatological and hydrological data
are available from several stations throughout the catchment[ Moreover\ a DEM with a 20!m grid size was
also available[
METHODOLOGICAL APPROACH
Sensitivity of dynamic hydrological processes varying in time and space to comparatively static boundary
conditions\ landsurface morphometry and soil features requires di}erent methods for modelling\ analysis
and visualization of data[ Therefore an approach integrating several geocomputational concepts\ including
spatial analysis of di}erent types of data\ static modelling of spatial structures\ dynamic four!dimensional
modelling of hydrological processes\ and statistical analysis techniques was chosen[
Investigations of the e}ects of geomorphometry on rainfallÐruno} processes require\ as basic information\
a quantitative description of the topography and hydrological characteristics of the study area[ In our study
hydrological features have been produced synthetically by means of physically based hydrological catchment
models[ In order to extract relationships between morphometry and model output\ the model runs were
carried out under homogeneous boundary conditions "land use\ soil parameters\ precipitation# and the same
initial conditions "soil moisture# for both catchments[ Parameter values were chosen according to typical
measured values[ Moreover\ model runs were carried out to validate a realistic reproduction of the hydrol!
ogical catchment behavior by the models used[ However\ several detailed sensitivity analyses were carried
out in the small Weiherbach test catchment in order to investigate the in~uence of variable boundary
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0856
Figure 2[ Structure of methods\ tools and parameters[ The solid lines with arrows indicate data transfer and data processing steps[ The
dotted lines with arrows indicate the main analysis procedures described in the text] analysis of "0# primary geomorphometric
parameters and spatial hydrological variables\ "1# soilÐmorphometry relationship and its implications for rainfallÐruno} modelling\
"2# representative catchment parameters and runo} indices\ and "3# meso!scale landform types and runo} characteristics
and initial conditions on model output[ The modelled hydrological outputs include event!based runo}
hydrographs and overland ~ow depth[ Spatially variable model inputs and outputs were stored within
GRASS and ARC:INFO GIS software as local hydrological attributes[ Hydrographs were used to produce
a series of hydrological indices "Hennrich et al[\ 0888#\ which were stored as hydrological catchment
attributes[ These indices were used in a cluster procedure to classify hydrological catchment types[
Topography has been analysed using a collection of geomorphometric GIS tools and self!developed
algorithms "Schmidt and Dikau\ 0888#[ Additionally\ GIS tools were applied to analyse and produce soilÐ
morphometry relationships for the Weiherbach catchment\ allowing investigations of interactions between
soil type distribution and relief attributes for hydrological modelling "Schmidt et al[\ 0887^ Hennrich et al[\
0888#[
Statistical packages "mainly SAS and some self!developed tools# were used in the analysis of runo}Ð
morphometry relationships\ incorporating the above data[ Additionally\ model scenarios were produced to
investigate model sensitivity to several input parameters[ The complete structure of the methods used is
presented in Figure 2[
Hydrolo`ical models
Two hydrological models with di}erent parameter and computational requirements have been used to
cover di}erent scale issues in this study[ Moreover\ it was our aim to use results of two modelling approaches
"one detailed and physically based\ the other simpli_ed# to see if both produce comparative hydrological
behaviour at di}erent scales[
The quasi!three!dimensional model SAKE "Simulationsmodell fur Ab~usse kleiner Einzugsgebiete\ which
translated means runo} simulation model for small catchments# has been utilized for modelling rainfallÐ
runo} processes in the Weiherbach catchment[ The model SAKE works with a regular horizontal grid and
an uneven vertical discretization "soil horizons#[ The di}erent hydrological ~uxes through the cells are
modelled by di}erential equations "Merz and Plate\ 0886^ Merz and Bardossy\ 0887^ Schmidt et al[\ 0887#[
The model SAKE is event!based\ working with a _xed temporal resolution of 0 min "the internal time!steps
may vary according to the stability of numerical solutions#[ Some of the required spatial boundary conditions
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0857
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
"DEM\ land use\ soil types\ initial moisture conditions# can be easily stored\ manipulated and provided to
the hydrological model by GIS tools[ The model SAKE produces di}erent types of output[ Temporal
variation in runo} hydrographs\ soil moisture and some other hydrological variables can be produced for
each grid point in the catchment[ Moreover\ spatio!temporal distributions of hydrological variables\ such
as overland ~ow depth and soil moisture\ can be simulated for each time!step[ Calibration and validation
of SAKE was carried out in a highly instrumented subcatchment of the Weiherbach "Schmidt et al[\ 0887#[
The hydrological model\ TOPMODEL is based on the concept of variable source areas contributing to
runo} production through saturated overland ~ow "Beven and Kirkby\ 0868#[ Formation of contributing
areas is related to the topographic parameters ln"a:tanb#\ where a is the upslope area drained per unit
contour length and b is the slope angle[ Model inputs are the frequency distribution of ln"a:tanb# "easily
provided by GIS#\ daily precipitation and evapotranspiration time!series and several lumped soil and routing
parameters[ Model outputs are the runo} hydrograph\ water balances and contributing areas[ The model
was calibrated for the Leineturm catchment "Hennrich et al[\ 0888#\ and the calibrated soil parameters were
used as constant values throughout this study[
Geomorphometric analysis
Various approaches have been used and applied to describe landform surfaces quantitatively[ For the
research problem investigated in this study several morphometric concepts have been utilized to describe
landforms on di}erent spatial scales "Figure 2 and Table I# "see Schmidt and Dikau\ 0888#[ On the local
scale primary geomorphometric parameters\ such as the slope angle b\ have been extracted to investigate
morphometric in~uences on hydrological variables such as overland ~ow depth and velocity[ Moore et al[
Table I[ Geomorphometric parameters and objects on di}erent scales "see Moore et al[\ 0880^
Schmidt and Dikau\ 0888# with some examples related to this study
Primary morphometric parameters
Simple
Complex
Compound
Morphometric objects
Linear
Areal
Representative morphometric parameters
Dimension
Relief
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Slope angle b
Vertical curvature wv
Horizontal curvature wh
Flow length l
Flow accumulation a
Slope of downslope ~owpath
ln"a:tan b#
ln"tan b:l#
a×tan b×wh
Flowpaths
Thalweg networks
Slopes
Form elements
Catchments
Landform units
Object length of width\ object area
Circularity
Elongation ratio
Statistic measures of primary parameters
Slope of longest ~owpath
Hypsometric integral
Portion of high curvature "de_ned by threshold#
Average height
Index of relief thickness "average height:area#
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0858
"0880# and Schmidt et al[ "0887# have proposed several primary geomorphometry parameters with local
hydrological signi_cance[ In particular\ compound parameters are supposed to be strongly related to
hydrological processes[ Most of these parameters can be derived easily by standard GIS tools in GRASS or
ARC:INFO[ Some parameters were derived by special geomorphometric software packages or self!developed
GIS tools "Schmidt and Dikau\ 0888#[
At the catchment scale a series of drainage basin parameters "Hennrich et al[\ 0888# were tested for
hydrological relevance[ Again\ most of them could be derived by standard GIS algorithms\ but in several
cases our own algorithms had to be developed[
Derivation of landform units can be carried out using various approaches\ including classi_cation of
morphometric parameters\ _lter techniques and cluster analysis "Dikau\ 0878\ 0883^ Dikau et al[\ 0884^
Sulebak et al[\ 0886^ Gimel|farb et al[\ 0888^ Schmidt and Dikau\ 0888#[ We applied a classi_cation technique
proposed by Hammond "0853#\ and applied by Dikau et al[ "0884# within a geocomputational environment\
and cluster analysis in order to derive meso!scale homogeneous landform units for the Leineturm study area
"Hennrich et al[\ 0888#[
RESULTS
Some of the research results presented here can be found in greater detail elsewhere "Schmidt et al[\ 0887^
Hennrich et al[\ 0888#[ It is the purpose of this paper to present them embedded in a wider framework\ which
includes
0[ hierarchical methodological aspects^
1[ scale issues relevant to the research^
2[ possible scale!linkages[
Figure 3 shows the research themes central to this paper\ emphasizing both the di}erent scales and the
detailed research framework[
Local scale
The local scale approaches included investigations of the relevance of morphometry to local hydrological
processes "index 0 in Figure 2# and the utility of soilÐmorphometry relationships in hydrological modelling
"index 1 in Figure 2#[
The hydrological model SAKE simulates overland ~ow depth for each time!step "minutes# at the de_ned
spatial discretization[ We applied GIS visualization and map algebraic tools to analyse the spatio!temporal
Figure 3[ Di}erent levels of investigation in our study[ Small arrows indicate data processing and calculation:derivation steps[ Large
arrows indicate the main analysis procedures as described in the text "see Figure 2#] comparison and statistical analysis of primary
geomorphometric analysis and spatial hydrological variables^ statistical analysis of representative catchment parameters and runo}
indices^ comparison and statistical analysis of meso!scale landform types and runo} characteristics
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0869
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
Figure 4[ Overland ~ow depths were analysed using GIS tools "see text# for model runs of SAKE in the Weiherbach subcatchments[
The spatial pattern of runo} concentration shows strong convergence to the thalwegs[ The temporal development of overland ~ow
depth indicates fast concentration to small areas for all catchments after precipitation ceased[ Statistical analysis revealed the relevance
of the parameter ln"a:tanb# for this concentration process
development of overland ~ow depth resulting from model runs in subcatchments of the Weiherbach area
"Figure 4#[ The results show a period of fast convergence of overland ~ow to a comparatively small part
"³09)# of the catchment area directly after precipitation ended\ and a second period of slow runo}
generation from these areas of ~ow concentration "see Schmidt et al[\ 0887#[ The _rst period is related to the
hydrograph peaks and the second to the falling part of the hydrographs[ These two phases were used to
de_ne thalweg catchment areas\ which closely correlate with overland ~ow generation[ Multiple regression
of simulated overland ~ow depth and several primary geomorphometric parameters\ as well as discriminant
analysis using the de_ned contributing areas and geomorphometric parameters\ revealed the high signi_cance
of the topographic parameter ln"a:tanb# for overland ~ow concentration processes[
Firstly\ it can be stated that the hydrological signi_cance of di}erent catchment subregions is strongly
in~uenced by local morphometric properties[ In our studies\ the process of overland ~ow convergence leads
to concentration areas within a catchment "see Anderson and Burt\ 0867a\b#\ which can be related to
morphometric parameters[ These _ndings have implications for spatial patterns in other frequently used
parameters in hydrological modelling\ such as soil properties or initial moisture contents "see below#[
Soil properties used in hydrological modelling show high variability depending on measurement methods
and sampling location "Hendrickx\ 0889#[ The precise de_nition of variability in soil parameters is one
crucial problem for dynamic modelling of hydrological processes[ Additionally\ in distributed modelling\
the problem of spatial parameter distributions or {e}ective parameter values| for certain areas arises[ E}ective
parameter values are often derived by simple statistical measures\ which assume a spatial stochastic variability
of parameter values[ Often such statistical measures are not appropriate\ however\ because parameter values
show spatial patterns or structured variability in space "Merz and Plate\ 0886^ Merz and Bardossy\ 0887#[
In the Weiherbach area\ for example\ there was a clearly detectable relationship between saturated water
content and soil type[ The de_nition of appropriate units "e[g[ soil units# showing an internal stochastic
variability of parameter values and a clear external distinction from other areas is an important approach\
which underpins the concept of hydrological response units[ One method for coping with this issue is to
identify relationships between soil types and soil properties[ Another possibility is to relate soil attributes to
other information available at a better spatial resolution\ such as digital elevation models or remotely sensed
data "Moore et al[\ 0882#[ The latter approach is often appropriate\ because soil genesis is a result of
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0860
Figure 5[ A soilÐmorphometry model applied to the Weiherbach area revealed good model results for predicting colluvial soils[ The
model is based on a failure rate analysis using 09 geomorphometric parameters
geomorphological and pedological processes\ which are strongly in~uenced by several factors\ including
geology\ climate\ vegetation and\ signi_cantly\ morphometric properties "such as slope angle or local con!
vergence:divergence#[ Hence\ soil catenas are often found in the landscape[ In the Weiherbach area we found
good predictive capability of a simple soilÐmorphometry model "Figure 5# "Schmidt et al[\ 0887#[
The model SAKE uses several soil attributes\ including saturated hydraulic conductivity and saturated
water content\ as three!dimensional boundary conditions[ The model SAKE was used in a sensitivity analysis
of the in~uence of the spatial distribution of these parameters on model results[ Firstly\ di}erent spatial
distributions based on the same frequency distributions of parameter values for conductivity and saturated
water content were produced arti_cially "Figure 6#[ The parameter values were coupled with geo!
morphometric position using GIS tools in such a way that parameter patterns with di}erent degrees of
spatial autocorrelation resulted[ These arti_cial spatial patterns were used as input parameters for the model
"Merz and Plate\ 0886^ Merz and Bardossy\ 0887^ Schmidt et al[\ 0887#[ For most subcatchments\ the
simulated hydrographs show a strong dependency on the degree of spatial structuring "Figure 6#[ Further
analysis\ varying parameter values for di}erent geomorphometric positions\ showed the high signi_cance of
morphometric positions in the thalweg areas "Schmidt et al[\ 0887#[
These results suggest that runo} concentration in small catchments may de_ne areas within these catch!
ments that have varying signi_cance for hydrological catchment response[ These catchment subregions show
a strong relationship with geomorphometric properties[
This implies that other parameters relevant to hydrological modelling should be of varying signi_cance in
these subregions[ For example\ catchment runo} can be much more sensitive to variations of in_ltration
capacity of thalweg soils than to those of valley!side soils[ Sensitivity studies showed that local parameter
values of saturated hydraulic conductivity and saturated water content have varying hydrological relevance
according to geomorphometric position[ This means that spatially structured variability and\ in particular\
morphometric structured variability of parameters has to be included in hydrological modelling "Merz and
Plate\ 0886^ Merz and Bardossy\ 0887^ Schmidt et al[\ 0887#[ SoilÐmorphometry relationships are one
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0861
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
Figure 6[ Di}erent spatial patterns of model parameters show the e}ect of geomorphometric structured distributions[ The parameter
values were coupled to geomorphometric parameters with a random in~uence to vary spatial autocorrelation "degree of structure#[ The
upper diagrams show modelled hydrographs for west!facing subcatchments\ the lower diagrams represent an east!facing subcatchment[
The degree of structure shows a signi_cant in~uence on the modelled hydrograph for the east!facing subcatchment
contribution to the modelling of spatial distributions of soil units and to the derivation of spatial distribution
of soil parameters in a process!related manner[ Moreover\ morphometric modelling can help in estimating
the hydrological signi_cance of parameters[ In summary\ morphometric analysis can assist in]
0[
1[
2[
3[
de_ning landform units with relatively homogeneous "stochastic# parameter values^
modelling spatial distributions of parameter values^
calculating e}ective parameter values incorporating structured variability^
de_ning appropriate measurement programmes for soil parameters "owing to the variable model sen!
sitivity#^
4[ estimating errors in hydrological modelling resulting from parameter uncertainties[
However\ the relationships established above showed a trend with the catchment characteristics] there
was an obvious relationship between general catchment morphometry and the hydrological sensitivity to
local parameter variations[ Initial conclusions suggest that there are catchments with generally steep gradients
and small thalwegs "west!facing subcatchments in the Weiherbach area# that are less sensitive to variation
in the spatial patterns of soil parameters and to total changes of parameter values "compare diagrams in
Figure 6#[ This raises the question of geomorphometric catchment characterization addressed in the following
sections\ and shows its signi_cance for modelling hydrological processes at the local scale[
Catchment scale
Relationships between drainage basin parameters and hydrological indices have been investigated in
numerous studies[ Cooke and Doornkamp "0889# and Gregory and Walling "0862# provide useful discussion
of these issues[ Quantitative description of basin morphology and land use and the establishment of
functional relationships between these parameters and hydrological catchment characteristics\ have been
found to be useful for modelling "e[g[ Rodriguez!Iturbe and Valdes\ 0868^ Gupta et al[\ 0879# and prediction
"e[g[ Sauer et al[\ 0872^ Acreman and Sinclair\ 0875#[
In our studies\ we investigated the in~uence of basin morphometry on runo} in order to identify geo!
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0862
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
Table II[ Correlation matrix of selected morphometric catchment parameters and simulated hydrograph indices for
model runs of SAKE in the Weiherbach subcatchments[ The hydrograph indices are grouped into attributes describing
hydrograph rise\ volume and fall[ Additionally the peak value of calculated unit hydrograph is included[ The table
presents some of the best correlation results of the analysis[ Italic _gures are signi_cant at the 88) signi_cance level
Hydrograph indices
Peak time
Time of rise
Time of centroid
Runo} percentage
Discharge of centroid
Peak discharge
Time of fall
Discharge end
Peak of unit hydrograph
Geomorphometric parameters
Average
slope
angle
Relief
thickness
Average
o/ow
length
−9=42
−9=41
−9=56
9=61
9=69
9=67
−9=43
−9=59
9=64
−9=45
−9=51
−9=46
9=27
9=37
9=52
−9=12
−9=28
9=69
9=40
9=40
9=51
−9=40
−9=48
−9=58
9=38
9=44
−9=63
Slope of Percentage ln"a:tan b#\ ln"tan b:l#\
longest
of high
harmonic
average
~ow path
vertical
mean
curvature
−9=38
−9=38
−9=50
9=51
9=69
9=79
−9=36
−9=42
9=74
−9=50
−9=43
−9=57
9=56
9=79
9=79
−9=51
−9=58
9=77
9=40
9=34
9=38
−9=63
−9=60
−9=56
9=36
9=43
−9=43
−9=42
−9=35
−9=65
9=73
9=72
9=74
−9=60
−9=63
9=70
morphometric parameters on the catchment scale "termed {representative geomorphometric parameters|\ see
Table I#\ that have hydrological relevance "{e}ective parameters|#[ This part of our study relates to index 2
in Figure 2[ The simulated hydrographs were used to extract a series of hydrological indices "Hennrich et
al[\ 0888#[ In order to identify relationships between morphometry and hydrology\ the model runs were
carried out under the same homogeneous soil\ land use and initial moisture conditions[ As described earlier\
catchment parameters were extracted using a collection of tools\ including standard GIS tools\ special
software and specially developed algorithms[ Regression techniques were applied to obtain relationships
between hydrological indices and catchment parameters "Table II# "see Schmidt et al[\ 0887^ Hennrich et al[\
0888#[ The following conclusions can be drawn]
0[ parameters describing drainage basin form "e[g[ elongation ratio\ circularity# showed weak correlations
with hydrological indices^
1[ statistical measures of slope angle and geomorphometric parameters with direct hydrological relevance
"e[g[ ~ow accumulations or ~owlength\ which are derived by {~ow algorithms|# show strong correlations\
as one would expect^
2[ statistical measures of the geomorphometric parameters ln"a:tanb# and slope angle "b# show good
correlations with indices describing runo} volume^
3[ statistical measures of the compound morphometric parameter ln"tanb:l# show good correlations with
indices describing runo} volume as well as temporal dimensions of hydrographs[ The parameter ln"tanb:l#
could be a useful index to describe runo} behaviour in terms of in_ltration excess runo} "Schmidt et al[\
0887#] it is a function of local slope angle "b#\ which controls local runo} velocity and runo} production\
and of length of the downslope ~owpath "l#\ which is a measure of the probability of loss of ~owing water
on its way to the gauge[ Therefore\ the contribution of a point to catchment runo} "q# can be seen in a
simple way as a function of b and l
q½f "b#
and
q½f "0:l#
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0863
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
Table III[ Correlation coe.cients of selected relationships for model runs of SAKE and
TOPMODEL in almost 0999 subcatchments in the Leineturm area[ Because of the larger
database the correlation coe.cients decline\ although some of the principal relations identi_ed
in Table II remain stable[ All relations are highly statistically signi_cant
Relationship
Correlation for model
Average of ln"a:tan b# * peak discharge
Minimum of ln"tan b:l# * time of centroid
Average of slope angle b * peak discharge
Relief thickness * peak discharge
SAKE
TOPMODEL
−9=10
−9=11
9=20
9=39
−9=80a
−9=14
*
9=17
a
As the frequency distribution of ln"a:tan b# is a direct TOPMODEL input\ the correlation of ln"a:tan b#
is very high[
4[ the importance of curvature can be seen from the high correlation between the percentage of area having
high pro_le curvature "which in~uences ~ow acceleration# and several hydrograph indices[
The relationships derived for the 12 subcatchments in the Weiherbach area were tested in the Leineturm
area[ Almost 0999 GIS!derived subcatchments were used to check whether the relationships remain stable
with a larger data base[ In this case\ model runs with SAKE and TOPMODEL were used to produce
hydrograph indices[ The results show the signi_cance of the slope angle\ ln"tanb:l#\ ln"a:tanb# and the index
for relief thickness for simulations in the Leineturm area "Table III#[
Various subcatchment sizes have been produced for the Leineturm area[ Therefore\ the question is whether\
and how\ the derived relationship is in~uenced by catchment size[ Threshold analysis of hydrological
indices and morphometric parameters revealed that there are signi_cant changes in attribute variances with
catchment size "see Wood et al[\ 0889^ Dikau\ 0883^ Wood\ 0884^ Hennrich et al[\ 0888#[ We analysed
variation in the derived relationships with changing catchment size by comparing correlation coe.cients for
certain ranges of catchment size "Table IV#[ These size ranges were based on derived relationships between
variance of hydrological indices and geomorphometric parameters and catchment size as used by Dikau
"0883# and Wood "0884# "see Hennrich et al[\ 0888#[ For this procedure\ the results of TOPMODEL for the
Leineturm catchment were used[ The results show that on di}erent spatial scales\ signi_cance and {e}ec!
tiveness| of parameters change[ The parameter ln"a:tanb# shows high correlation on all scales\ but correlation
strength declines at smaller scales[ At small catchment scales "³0 km1#\ the index of relief thickness shows
Table IV[ Correlation coe.cients for selected relationships for model runs of TOPMODEL in di}erent size ranges of
the Leineturm subcatchments[ The correlation coe.cients of ln"a:tan b# decline with decreasing catchment size[ At small
catchment scales "³0 km1#\ the index of relief thickness shows high correlations[ The parameter ln"tan b:l# shows
intermediate correlations at several scales
Correlation for catchment sizes A
Relationship
³0 km1
Harmonic mean of ln"a:tan b# to centroid
Skewness of ln"tan b:l# to centroid
Harmonic mean of ln"tan b:l# to centroid
Sum of ln"tan b:l# to centroid
Variance of ln"tan b:l# to centroid
Index relief thickness to runo} percentage
Copyright Þ 1999 John Wiley + Sons\ Ltd[
0 km1
09 km1
19 km1
³ A ³ 09 km1 ³ A ³ 19 km1 ³ A ³ 39 km1
9=46
9=21
9=48
9=85
×39 km1
9=86
−9=43
9=26
9=30
9=29
9=15
9=06
−9=31
−9=96
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0864
high correlations[ Intermediate correlations could be found for ~owlength l and the parameter ln"tanb:l# on
all scales[
There are three problems that have to be mentioned in the context of the results presented in this chapter[
First\ there still remains a considerable amount of scatter in the data and it is questionable whether it is
possible and:or useful to improve the derived relationship by more sophisticated methods\ such as mul!
tivariate and non!linear statistics[ Second\ there is the issue of whether the relationships remain constant
under variable boundary conditions "see Conclusions#[ Moreover\ the question of how these correlations
can improve hydrological modelling is unsolved[ From our point of view\ one way of using the identi_ed
parameters and relationships could be to classify catchments and:or landform units on the basis of likely
hydrological response[ The simulations in the Weiherbach area revealed the existence of two types of
subcatchment\ one with steeper gradients and side slopes and smaller thalwegs\ the other with thalwegs that
are more broad and extended "see previous section#[ The former tend to produce simple hydrographs "quick
runo} response# and tend to be less sensitive to spatial variations in input parameter values[ The latter tend
to produce more complicated "e[g[ double peaked# hydrographs and show a considerable dependency on
spatial variations in input parameters "see Figure 6#[ Based on these _ndings\ the next step is to establish a
quantitative measure for di}erent catchment types[
Re`ional scale
A general aim of the regional!scale studies was to de_ne meso!scale geomorphometric landform types\
re~ecting similarities in their hydrological behaviour[ This part of the paper relates to index 3 in Figure 2[
The methodological basis of the approach and the connections to the previous sections can be seen as a
sketch in Figure 3[
Principal component analysis and cluster analysis\ using the statistical package SAS\ were applied to
produce regional units with similarities in their hydrological behaviour[ Hydrological indices produced by
SAKE and TOPMODEL from model runs on subcatchments of the Leineturm area "see previous section#
served as a database for this procedure[ Discriminant analysis using these hydrological units and rep!
resentative geomorphometric parameters calculated for the subcatchments suggests possible geo!
morphometric parameters for the hydrological classi_cation[ An alternative method involved classi_cation
of geomorphometric landform units by implementing an algorithm proposed by Hammond "0853# "see
Dikau et al[\ 0884^ Gimel|farb et al[\ 0888^ Hennrich et al[\ 0888# within GRASS[ Again discriminant analysis
was used to check hydrological signi_cance of the classi_cation against the hydrograph indices simulated by
SAKE and TOPMODEL[ Finally\ a hydrological morphometric landform classi_cation was produced from
the previous results[
The units in Figure 7 were generated using clustering methods based on the hydrographs produced by
SAKE and TOPMODEL[ The hydrological units produced by the two di}erent models show certain
similarities] the mountainous areas and the plains near the River Leine can be distinguished by visual
comparison with the DEM for the Leineturm area[ Discriminant analysis using geomorphometric catchment
parameters revealed high explanatory values for only some of the geomorphometric parameters already
identi_ed in the previous section "ln"a:tanb##[ Within this study\ much simpler parameters\ mainly statistical
measures for relief height and slope "e[g[ average height or slope# could be identi_ed as explanatory variables
for hydrological clusters[ Therefore the question is whether simpler landform descriptions could be useful
in relating landform morphometry to hydrological characteristics[
The geomorphometric landform classi_cation scheme of Hammond "0853# classi_es areas within a DEM
according to the parameters relief\ percentage of low slope angle and percentage of low slope angle at lower
elevation using a moving window algorithm[ This comparatively simple classi_cation procedure produces
units describing the meso!scale character of landform "see Figure 8#[ However\ the landform units produced
for the Leineturm area show similarities with the produced hydrological clusters "Figure 7#\ which was
quantitatively determined by discriminant analysis[ Based on this discriminant analysis\ a combination of
the TOPMODEL clusters and the Hammond units was produced[ The result "Figure 8# leads to a simple
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
Figure 7[ Results of cluster analysis based on the hydrograph indices simulated by SAKE and TOPMODEL[ The modelled areas are landform units similar in their
hydrological response[ The results for the two di}erent models show similarities in their spatial structure and correspondence to landform characteristics
0865
Copyright Þ 1999 John Wiley + Sons\ Ltd[
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0866
Figure 8[ Derivation of geomorphometricÐhydrological landform types[ An algorithm for landform classi_cation "Hammond\ 0853^
Dikau et al[\ 0884# was applied to generate geomorphometric landform units[ Comparing this map and the hydrological clusters in
Figure 7 reveals signi_cant similarities[ A discriminant analysis based on the factors used in the hydrological cluster procedure showed
that the hydrological factors could explain the morphometric classi_cation[ Reclassifying the hydrological factors using these results
generated a map with two hydrologicalÐgeomorphometric landform units] "0# plains and plains with hills with lower discharge and "1#
hills and mountains with higher peak discharge and overland ~ow
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
0867
J[ SCHMIDT\ K[ HENNRICH AND R[ DIKAU
segmentation into hilly:mountainous areas with high discharge values and plains with low hydrograph
characteristics[
CONCLUSIONS
Geomorphometry provides a theoretical basis\ a methodological approach and a technique for landform
analysis on di}erent scales[ Although there are still some basic problems in quantitative analysis of terrain
"Schmidt and Dikau\ 0888#\ recent computer capabilities\ especially GIS tools\ provide a su.ciently wide!
ranging framework for deriving geomorphometric objects and attributes on di}erent scales\ ranging from
local parameters to landform units[ The availability of these tools and the data they require "DEMs# poses
the question of whether\ and how\ these tools could help in modelling landsurface!related processes "Moore
et al[\ 0880#[ This study presents an approach quantifying relationships between geomorphometric attributes
and hydrological variables by linking hydrological models with GIS modelling techniques[ The coupled use
of spatial modelling tools "GIS#\ statistical packages and hydrological models provides a powerful com!
bination for investigating landform structure in relation to landform processes[ However\ basic problems
within this framework are]
0[ storage and exchange of various data types used in geomorphology and hydrology^
1[ a lack of interoperability of di}erent methods and investigation techniques[
The methodological approach used in this study su}ers from two fundamental di.culties[ First\ it remains
open as to whether the derived relationships between morphometry and hydrology remain constant under
variable boundary conditions[ Although sensitivity studies of the in~uence of parameter variations on the
proposed relationships were carried out\ it was not feasible to check all possible parameter combinations[
Generally speaking\ the complex system of hydrological catchment response to precipitation input was
simpli_ed in this study to a morphometryÐhydrology interaction\ leaving all other in~uences constant[
Second\ the analysed interaction was based purely on model results\ and the question of whether it is possible
to transfer these _ndings into real situations remains to be veri_ed[
Nevertheless\ we see several challenges arising from this work[ In investigating the spatio!temporal
structure of runo} concentration processes in small catchments\ it can be shown that there are catchment
areas of di}erent hydrological signi_cance according to geomorphometric positions[ Quantifying geo!
morphometry in terms of its hydrological in~uence on small scales requires capturing geomorphometric
catchment structure "Schmidt et al[\ 0887#[ The de_nition of areas with di}erent hydrological e.ciency is a
useful and process!related method to quantify geomorphometric structure\ because overland ~ow tends to
concentrate very rapidly in small catchment areas[ Hydrological e.ciency as a function of morphometric
position implies that other process!relevant boundary conditions have di}erent e.ciency according to their
morphometric position[ This means morphometry can help to estimate the e}ects of uncertainties in input
parameters[ SoilÐmorphometry models are helpful tools in terms of estimating spatial variations in soil type
and soil parameter distributions\ given that soil genesis is strongly related to gravitational and:or erosional
processes "e[g[ colluvial soils#[ These models should be extended to other predictor variables available at a
high spatial resolution[ Combining soilÐmorphometry models with morphometric descriptors for hydrol!
ogical e.ciency could be helpful in regionalizing point measurements of soil parameters[ The e}ect of
catchment morphometry has been investigated in numerous studies and several catchment parameters have
been proposed that re~ect hydrological behaviour "Gregory and Walling\ 0862^ Cooke and Doornkamp\
0889#[ However\ at this stage it is di.cult to generate stringent and general relationships in this respect[ We
propose the use of these parameters in the de_nition of areas showing similar hydrological response[ This
paper has shown that a crude hydrologicalÐgeomorphometric landform classi_cation is possible using a
prede_ned model of terrain classi_cation[ Hence\ quantifying hydrological similarity on the catchment scale
could be a helpful tool for hydrological regionalization\ and potentially useful for applying process!based
modelling approaches on a number of scales[
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#
SCALES AND SIMILARITIES IN RUNOFF PROCESSES
0868
ACKNOWLEDGEMENTS
This research has been supported by the Deutsche Forschungsgemeinschaft "German Research Council\
Bonn# within the project {Regionalization in Hydrology|[ We are indebted to the reviewers for their help in
improving this contribution[
REFERENCES
Acreman MC\ Sinclair CD[ 0875[ Classi_cation of drainage basins according to their physical characteristics^ an application for ~ood
frequency analysis in Scotland[ Journal of Hydrolo`y 73] 254Ð279[
Andersron MG\ Burt TP[ 0867a[ The role of topography in controlling through~ow generation[ Earth Surface Processes 2] 220Ð233[
Anderson MG\ Burt TP[ 0867b[ Toward more detailed _eld monitoring of variable source areas[ Water Resources Research 03] 0012Ð
0020[
Anderson MG\ Burt TP[ 0889[ Process studies in hillslope hydrology] an overview[ In Process Studies in Hillslope Hydrolo`y\ Anderson
MG\ Burt TP "eds#[ Wiley] Chichester^ 0Ð7[
Beven K\ Kirkby MJ[ 0868[ A physically based\ variable contributing area model of basin hydrology[ Bulletin of Hydrolo`ic Sciences
13] 32Ð58[
Bloschl G\ Sivapalan M[ 0884[ Scale issues in hydrological modelling] a review[ In Scale Issues in Hydrolo`ical Modellin`\ Kalma JD\
Sivapalan M "eds#[ Wiley] Chichester^ 8Ð36[
Cooke RU\ Doornkamp JC[ 0889[ Geomorpholo`y in Environmental Mana`ement\ 1nd edn[ Oxford University Press] Oxford[
Dikau R[ 0878[ The application of a digital relief model to landform analysis in geomorphology[ In Three Dimensional Application in
Geo`raphic Information Systems\ Raper J "ed[#[ Taylor + Francis] London^ 40Ð66[
Dikau R[ 0883[ Computergestutzte Geomorphographie und ihre Anwendung in der Regionalisierung des Reliefs[ Petermanns Geo!
`raphische Mitteilun`en 027] 88Ð003[
Dikau R\ Brabb EE\ Mark RK\ Pike RJ[ 0884[ Morphometric landform analysis of New Mexico[ In Advances in Geomorphometry *
Proceedin`s of the Walter F[ Wood Memorial Symposium[ Zeitschrift fur Geomorpholo`ie\ Supplement Band 090] 098Ð015[
Gimel|farb GL\ Schmidt J\ Braunmandl A[ 0888[ Gibbs _elds with multiple pairwise interactions as a tool for modeling grid!based
data[ In Process Modellin` and Landform Evolution\ Hergarten S\ Neugebauer HJ "eds#[ Springer!Verlag] Berlin\ Heidelberg\ New
York^ 36Ð62[
Gregory KJ\ Walling DE[ 0862[ Draina`e Basin[ Form and Process[ Edward Arnold] London[
Gupta VA\ Waymire E\ Wang CT[ 0879[ A representation of an instantaneous unit hydrograph from geomorphology[ Water Resources
Research 05"4#] 744Ð751[
Gupta VK\ Rodriguez!Iturbe I\ Wood EF "eds#[ 0875[ Scale Problems in Hydrolo`y[ D[ Reidel] Dordrecht[
Hammond EH[ 0853[ Analysis of properties in landform geography] an application to broad scale landform mapping[ Annals of the
Association of American Geo`raphers 43] 00Ð08[
Hendrickx JMH[ 0889[ Determination of hydraulic soil properties[ In Process Studies in Hillslope Hydrolo`y\ Anderson MG\ Burt TP
"eds#[ Wiley] Chichester^ 32Ð81[
Hennrich K\ Schmidt J\ Dikau R[ 0888[ Regionalization of geomorphometric parameters in hydrologic modelling using GIS[ In
Re`ionalization in Hydrolo`y\ Diekkruger B\ Kirkby M\ Schroder U "eds#[ IAHS Publication 143\ Proceedings of the Conference on
Regionalization in Hydrology at Braunschweig\ March 0886[ International Association of Hydrological Sciences] Wallingford^ 070Ð
080[
Kirkby MJ[ 0877[ Hillslope runo} processes and models[ Journal of Hydrolo`y 099] 204Ð228[
Kleeberg H!B[ "ed[#[ 0881[ Re`ionalisierun` in der Hydrolo`ie[ VCH Verlagsgesellschaft mbH\ Weinheim[
Merz B\ Bardossy A[ 0887[ E}ect of spatial variability on the rainfallÐruno} process in a small loess catchment[ Journal of Hydrolo`y
102"3#] 293Ð206[
Merz B\ Plate EJ[ 0886[ An analysis of the e}ects of spatial variability of soil and soil moisture on runo}[ Water Resources Research
22"01#] 1898Ð1811[
Moore ID\ Grayson RB\ Ladson AR[ 0880[ Digital terrain modelling] a review of hydrological\ geomorphological\ and biological
applications[ Hydrolo`ical Processes 4"2#] 2Ð29[
Moore ID[ Gessler PE\ Nielsen GA\ Peterson GA[ 0882[ Soil attribute prediction using digital terrain analysis[ Soil Science Society of
America Journal 46] 332Ð341[
Rodriguez!Iturbe I\ Valdes JB[ 0868[ The geomorphologic structure of hydrologic response[ Water Resources Research 04"5#] 0398Ð
0319[
Sauer VB\ Thomas Jr WO\ Strickler V\ Wilson K[ 0872[ Flood characteristics of urban watersheds in the United States[ US Geolo`ical
Survey Water Supply Paper 1196] 0Ð51[
Schmidt J\ Dikau R[ 0888[ Extracting geomorphometric attributes and objects from digital elevation models * semantics\ methods\
future needs[ In GIS for Earth Surface Systems * Analysis and Modellin` of the Natural Environment\ Dikau R\ Saurer H "eds#[
Schweizbart|sche Verlagsbuchhandlung] Berlin\ Stuttgart^ 042Ð062[
Schmidt J\ Merz B\ Dikau R[ 0887[ Morphological structure and hydrological process modelling[ Zeitschrift fur Geomorpholo`ie
Supplement Band 001] 44Ð55[
Sulebak JR\ Etzelmuller B\ Sollid JL[ 0886[ Landscape regionalization by automatic classi_cation of landform elements[ Norsk
Geo`ra_sk Tidsskrift 40"0#] 24Ð34[
Wood EF[ 0884[ Heterogeneity and scaling landÐatmospheric water and energy ~uxes in climate systems[ In Space and Time Scale
Variability and Interdependencies in Hydrolo`ical Processes\ Feddes RA "ed[#[ Cambridge University Press] Cambridge^ 2Ð08[
Wood EF\ Sivapalan M\ Beven K[ 0889[ Similarity and scale in catchment storm response[ Reviews of Geophysics 17"0#] 0Ð07[
Copyright Þ 1999 John Wiley + Sons\ Ltd[
Hydrol[ Process[ 03\ 0852Ð0868 "1999#