Hydrological Sciences-Journal-des Sciences Hydrologiques, 47(6) December 2002
93 ]
GIS approach to scale issues of perimeter-based
shape indices for drainage basins
ANDRAS BARDOSSY & FRIDJOF SCHMIDT
Institute of Hydraulic Engineering
D-70550 Stuttgart, Germany
(IWS), University of Stuttgart, Pfaffenwaldring
61,
andras.bardossvffiiws.uni-slnltBart.de; [email protected]
Abstract Shape indices have been in use for several decades to describe the
characteristics and hydrological properties of drainage basins. Due to the fractal
behaviour of the basin boundary, perimeter-based shape indices depend on the scale at
which they are determined. Therefore, these indices cannot objectively compare
drainage basins across a range of scales and basin sizes. This paper presents an
objective GIS-based methodology for determining scale-dependent shape indices from
gridded drainage basin representations. The scale effect is addressed by defining a
representative scale at which the indices should be determined, based on a threshold
symmetric difference between two grids representing the drainage basin at different
resolutions.
Key words drainage basin morphometry; shape indices; compactness index; fractal geometry;
fractal dimension; symmetric difference; scale dependency
Approche par SIG des problèmes d'échelle des indices de forme des
bassins versants basés sur le périmètre
Résumé Les indices de forme ont été utilisés depuis plusieurs décennies pour décrire
les caractéristiques et les propriétés hydrologiques des bassins versants. En raison des
propriétés fractales du tracé de la limite du bassin versant, les indices de forme basés
sur le périmètre dépendent de l'échelle à laquelle ils sont déterminés. Par conséquent,
ces indices ne peuvent pas permettre de comparer objectivement des bassins versants
sur une grande gamme d'échelles et de tailles. Cet article présente une méthodologie
objective basée sur un SIG, qui permet de déterminer des indices de forme dépendant
de l'échelle à partir de représentations maillées du bassin versant. L'effet d'échelle est
pris en compte en définissant une échelle représentative à laquelle les indices
devraient être déterminés, basée sur une différence symétrique de seuil entre deux
grilles représentant le bassin versant avec différentes résolutions.
Mots clefs morphométrie de bassin versant; indices de forme; indice de compacité;
géométrie fractale; dimension fractale; différence symétrique; effet d'échelle
INTRODUCTION
In geomorphology, morphometry is dedicated to the quantification of morphology.
Shape indices used in drainage basin morphometry relate to the quantification of basin
shape. Morphometric characteristics of drainage basins provide a means for describing
the hydrological behaviour of a basin. Multivariate statistical approaches may be used
to establish correlations between morphometric parameters and hydrological key
variables, such as the time of concentration, the shape of the unit hydrograph, and
discharge maxima. Morphometric parameters may provide information for hydrological modelling, especially in the stage of model calibration. However, the morphometric parameters must be well defined and able to be derived from the available data
using standardized techniques.
Open for discussion until I June 2003
932
Andrâs Bàrdossy & Fridjof Schmidt
Table 1 Shape indices expressing drainage basin shape.
Index
Compactness (C)
Formula
Source
Gravelius(1914)
C
-\ jn A
2
Form factor (F)
Horton(1932)
A
2
L
Basin circularity (c)
4 K A
r
Basin elongation (E)
E=
Lemniscate ratio (K)
K
Miller (1953)
Schumm(1956)
iil
Ly
In
Û
K
Chorleye?a/. (1957)
AA
Among the shape indices for drainage basins, the most well known are probably
the compactness index, C (Gravelius, 1914), the form factor, F (Horton, 1932), the
basin circularity, c (Miller, 1953), the basin elongation, E (Schumm, 1956), and the
lemniscate ratio, K (Chorley et al., 1957) (see Table 1). The compactness index, C is
defined as the ratio between the length of the drainage basin boundary (the perimeter)
and the perimeter of a circle with the same area. It is always greater than 1 and
approaches unity when the basin approaches a circular shape. While C and c combine
the perimeter and the area of the drainage basin, F, E, and K combine the basin length
and area. By definition, all these indices are dimensionless.
Several authors have questioned the value of the compactness and the basin circularity indices for hydrological analyses, because they only describe the basin shape and
not its orientation towards the outlet (e.g. Horton, 1932; Chorley et al., 1957). Others
have pointed out that precautions need to be taken when measuring the perimeter
because of its scale dependency (e.g. Roche, 1963; Jarvis, 1976; Gardiner, 1981).
In his groundbreaking paper, Mandelbrot (1967) asks: "How long is the coast of
Britain?", arguing that the length of coastlines increases when measured with
yardsticks (dividers) of decreasing length. Mandelbrot shows that coastlines, as well as
many other natural boundaries, have a fractal dimension greater than 1, meaning that
any halving of the yardstick length will require more than twice as many steps to span
the line. Since then, fractal analysis techniques have been applied in numerous studies
of topographic phenomena (e.g. Goodchild & Mark, 1987; Snow & Mayer, 1992; Gao
&Xia, 1996; Pike, 2000).
Breyer & Snow (1992) verified the idea of fractal geometry for the outline of
several drainage basins. Their results show that drainage basin boundaries have in fact
a fractal dimension across a range of scales. Hence, it is clear that the length of any
drainage basin boundary cannot be measured with complete precision, but will increase as a larger scale is chosen. On the other hand, changes in the perimeter value
due to scale changes are not associated with a systematic error in the estimation of
basin area, so that P/A1'2 ratios are also scale-dependent. With regard to the fractal
character of drainage basin boundaries, Bendjoudi & Hubert (2002) have suggested to
completely abandon shape indices which are based on the perimeter, although methods
GIS approach to scale issues of perimeter-based shape indices for drainage basins
933
have been proposed to account for its fractal dimension (Woronow, 1981; Breyer &
Snow, 1992).
Neither the absolute value of the perimeter, nor its fractal dimension, seem to be of
special interest in basin hydrology, which is concerned with the water mass balance of
the basin area rather than mass transfer across its horizontal boundaries. It is the
opinion of the authors that techniques for determining shape indices should address the
known scale issues rather than avoiding scale-dependent shape indices outright.
Therefore, the objective of this paper is therefore to propose a basin-dependent
reference scale at which the perimeter should be measured, so that the shapes of
drainage basins of different sizes may be compared.
Together with the ever-increasing computing power of personal computers, GIS
programs offer a variety of techniques to automatically extract topographic characteristics from high-quality digital elevation models (DEMs) which have become
available in recent years. For example, algorithms for flow direction and watershed
delineation have been provided by Jenson & Domingue (1988). Applications in terrain
analysis have been compiled by Wilson & Gallant (2000). Fractal analysis methods
have been widely used in the study of river networks (e.g. Puente & Castillo, 1996;
Beauvais & Montgomery, 1997; Willemin, 2000). Some further applications relate to
surface roughness (Outcalt et al, 1994) and landscape ecology (Nikora et al., 1999).
The latter have been largely fostered by an increasing availability of rasterized data
from remote sensing (Walsh et al., 1998).
METHODS
In hydrology, the drainage area of a basin (i.e. the area of its horizontal projection) is
needed whenever a member of the water balance equation is to be quantified in volume
units for the basin as a whole, or for parts of it. Thus, uncertainties in the basin area
will lead to uncertainties of the same order of magnitude in the water balance
calculations. Therefore, it is essential to determine the basin area as precisely as
possible. When basins are delineated automatically from a DEM, the precision depends
largely on the DEM's quality and resolution. A higher resolution will generally lead to
a more precise representation both of the basin shape and its area.
For a given shape with a fractal outline, more details will appear in the trace of its
boundary as it is mapped with an increasing resolution. Due to its fractal nature, the
measured perimeter will thereby increase systematically as a larger scale (higher
resolution) is chosen, and will approach infinity as an infmitesimally short yardstick is
applied. However, the area does not change systematically at the same time, nor does
the distance from a fixed point to the farthest point on the boundary (length). In
particular, both area and length remain finite. Consequently, the ratio between the
perimeter and the square root of the enclosed area (used in the compactness index, C)
is not constant as it is for non-fractal curves (e.g. 2-%V2 for circles), but it diverges as
the scale is modified, whereas the ratio between the length and the square root of the
area (used in E) is not divergent (Feder, 1988).
Specifically, given the fractal character of drainage basin boundaries, the compactness index, C, is expected to increase, and basin circularity, c, will decrease when
choosing a larger scale. The same holds for gridded basin representations: if the total
934
Andrâs Bdrdossy & Fridjof Schmidt
basin area is the same for two grid resolutions, the perimeter, and thus the compactness
index, C, is higher at the higher grid resolution. Therefore, the perimeter-based indices
are not well defined as long as their scale dependency is not accounted for.
As mentioned in the introduction, the shape indices are dimensionless by definition. Hence it might be suggested that, if one of the input variables is found to have a
fractal dimension, this circumstance should be reflected by the formula of the
respective index. As a consequence, the formula for the compactness index could be
written as C = (Pl/D)/(4Tt4)112 (cf. Woronow, 1981). This requires, however, that D is
known and P is measured at the same absolute resolution for all basins.
However, for shape comparison, measuring P at the same relative resolution is
more appropriate. In order to compare the compactness index or basin circularity of
two basins which are substantially different in size, their relative scales have to be
adjusted to each other to ensure that the respective index is a function of the overall
basin shape, rather than a function of its size. If the basin perimeters were measured at
the same absolute scale, the perimeter value would include the effects of boundary
irregularity for the larger basin to a higher degree than for the smaller one. Consequently, the larger basin would be incorrectly considered less compact (see Breyer &
Snow, 1992).
Adjusting the scales of two gridded basins generally means that the grid representing the larger basin will have to be resampled to a lower resolution so that the
number of grid cells per basin is in the same range for both basins. Resampling the
smaller basin grid higher than the DEM's resolution is not an option because this does
not add information to the output.
From the above, it is clear that by decreasing the grid resolution, the measured
basin perimeter will also decrease, but this occurs at the gain of more meaningful
compactness and circularity indices. After adjusting the scales of two differently sized
basins to match the same relative resolution, their perimeter-based indices can be compared. Since the zigzagging behaviour of the perimeter at the micro scale is irrelevant
in hydrology, so is its changing numerical value when altering the grid resolution, as
long as the basin area and its general shape remain constant.
A criterion for the change in general shape and area is the symmetric difference
between two grids representing the same basin at different resolutions. When resampling a gridded object (Fig. 1(a)) to a lower resolution (Fig. 1(b)), some regions in
the new grid appear as part of the object that were not part of it before, while other
regions that were part of the object in the previous grid disappear as such in the new
grid. Let D^ be the set of pixels in the original grid, G\ (with a cell size of dx), covered
Fig. 1 Grids (a) Gk, (b) G2k, (c) symmetric difference between two sets of pixels Dk
and D2t, and (d) smoothed vector representation of G].
GIS approach to scale issues of perimeter-based shape indices for drainage basins
935
by an object in a grid with a cell size of k-àx, where k e TV. In another grid G2k, with a
cell size of 2k-dx, the same object covers a different set of pixels D2k in G\. The two
sets Dk and D2k differ by the symmetric difference Dk A D2k which is the set of pixels
belonging to exactly one (not both) of the two sets D* and D2k (Fig. 1(c)). The area of
an object in grid Gk can be written as Ak = \Dk\, and the area of symmetric difference
between Dk and D2k can be defined as:
e^=|D,AD2i|
(1)
The relative area of symmetric difference is then defined as:
K
2 k
= ^
(2)
A
k
In this study, for each individual basin, two grids Gk and G2k were iteratively
created from the original basin grid G\ with a nearest neighbour method by increasing
k until k = kr for which:
e*.2^s ma ,-^<ew +1 >
(3)
The threshold value for 8k,2k that should be tolerated was set to 8max = 0.01. In
other words, the representative cell size kr-dx for grid Gk was defined as the highest
integer multiple of the original cell size dx at which the relative area of symmetric
difference dk,2k between Dk and D2k was less than the threshold Smax. Note that 8max
relates to differences in geometry, not to the difference in total basin area which is
much lower. The advantage of taking only integer multiples of dx into account is that
the resampled grids can be compared at the original resolution dx without modifying
the geometry of its objects, as long as the spatial origin of the grids is kept constant.
Since this methodology of finding the representative scale is time consuming and
thus not very practical for application, a simplified method may be proposed. Figure 2
shows a strong relationship between the number of cells per basin and the symmetric
difference between the two grids. It can be seen that the scale at which a basin is
represented by at least 100 000 cells in Gk leads to a symmetric difference with an area
around 1% of Ak or less in most cases. Therefore, the representative cell size can be
estimated as:
krdx « <JA} /ns
(4)
where A \ is the basin area as determined from the original basin grid G\, and Ǥ is the
number of cells per basin that corresponds to a certain level of symmetric difference
Smax—here, 100 000 was chosen for «§ so that 8max = 0.01. For basins with an area of
A\ < dx2-«g, a higher 5k,2k will be expected, so choosing a DEM with a higher spatial
resolution might be considered in this case.
Obviously, the perimeter of a shape that is composed of quadratic cells cannot be
smaller than the perimeter of a rectangle with the same height and width. A circle with
diameter d represented in raster format at increasing resolution would consequently
approach a perimeter of Ad, instead of nd, so that the compactness index C approaches
4/TC = 1.273 instead of 1 as it should for a circle. Therefore, a smoothing raster-vector
conversion has to be applied prior to determining the basin perimeter and the
936
Andrâs Bârdossy & Fridjof Schmidt
ggg^
*PAf**l»»—.
10000
100000
1000000
„
10000000
Number of cells in D k
Fig. 2 Symmetric difference between two sets of pixels Dk and Z)M representing a
basin in two grids Gk and G24, and its dependency on the number of pixels in Dh based
on 153 basins in Baden-Wurttemberg.
compactness index (Fig. 1(d)). This is generally done using a Douglas-Peuker type
weeding algorithm (Douglas & Peuker, 1973). Smoothing the outline of a gridded
circle ensures that, at high resolution, the perimeter will in fact approach nd. The terms
Pr and Cr can then be defined as the representative perimeter and compactness index,
respectively, measured from the smoothed vector representation of the basin in grid Gr
which has a resolution of kr-dx so that S^ikr < 0.01.
EXAMPLES
A DEM for Baden-Wurttemberg, Germany, with quadratic cells of dx = 30 m and a
vertical resolution of dz = 1 m was used to delineate 181 basins in Arc View GIS
(Environmental Systems Research Institute, Inc.—ESRI). There were 28 basins that
had to be rejected because they were cut off at the frontiers, or because the delineation
results were not plausible (as for example in the Rhine valley for which dz was
insufficient). From the remainder, 29 basins of different size and order are displayed in
Fig. 3. They were chosen arbitrarily under the premise that, for demonstration
purposes, no overlapping (nested) basins should be present.
The compactness indices C\ and C, are compared in Table 2: C\ was determined
from the smoothed basin at the original resolution of G\ and values for Cr were
determined after applying the above method. As expected, the biggest difference
between C\ and Cr is clearly seen for the biggest basins. Although the biggest basins,
Neckar (ID 51) and Danube (Donau, ID 113), happen to be those with the highest
compactness indices, this observation cannot be ascribed to their size in general, as
might be suggested when basins of different sizes are compared at the same scale. In
GIS approach to scale issues of perimeter-based shape indices for drainage basins
50
0
50
937
100 Kilometers
Fig. 3 Some drainage basins in Baden-Wurttemberg, Germany, with ID numbers used
internally in Arc View G1S. The shading shows the compactness index C as measured
at the representative cell size k,-àx. See Table 2 for more details.
fact, for the two basins the compactness index as defined here is significantly lower
than its value as measured at the original cell size (see Table 2).
The observation of high C values for the bigger basins may be partly attributed to
the topographic characteristics of Baden-Wurttemberg. For example, the Neckar and
Danube river basins share a common boundary along the tortuous escarpments of the
Jurassic Swabian Alb cuesta landscape. In contrast, the Enz basin (ID 49), situated on
morphologically more homogenous Triassic sediments of the northern Black Forest, is
relatively compact, while the Schussen basin (ID 171), which drains an undulating
moraine landscape at the foothills of the Alps, is rather convoluted despite its smaller
size.
Beside such visual plausibility checks, relationships between compactness, C, and
the scale-independent shape indices, F, E and K were examined. For the modified
compactness index, Cr, increased correlation coefficients, r, of 0.31, 0.35 and 0.40
were obtained between Cr and F, E and K, respectively, compared with 0.26, 0.28 and
0.30 for C\. At the same time, the correlation of C and basin area decreased from 0.50
to 0.32, and for C and basin length, r changed from 0.73 to 0.56, also supporting the
scale independence of Cr.
938
Andrâs Bârdossy & Fridjof Schmidt
Table 2 Shape indices for the drainage basins shown in Fig. 3.
A (km l) P (km) L (km) k,.-àx
1 Wertheim
Tauber
894
205
60
90
2 Riedern
132
Erfa
72
19
30
550
12 Neckargemund Elsenz
139
36
60
14 Obrigheim
Elz
158
77
21
30
21 Jagstfeld
Jagst
1821
361
91
90
26 Friedrichshall
Kocher
1965
278
86
120
35 Neckarsulm
Sulm
116
60
20
30
39 Neckargartach
Lein
119
67
23
30
41 Ubstadt-Weiher Kraichbach
163
73
23
30
46 Laufen
Zaber
113
63
22
30
2224
49 Besigheim
Enz
255
76
150
51 Besigheim
Neckar
5591
616
117
180
67 Utzmemmingen Eger
126
65
19
30
75 Ballmertshofen Egau
278
84
26
60
92 Sontheim
756
152
Brenz
38
90
504
130
Blau
36
60
110 Ulm
5409
620
140
Donau
150
113 Ulm
115 Erlenbach
Biberach
105
51
16
30
128 Biberach
Kinzig
813
161
33
90
141 Riegel
Dreisam
556
135
39
60
533
148 Riegel
Elz
127
37
90
168 Grimmelshofen Wutach
436
136
37
60
170 Ludwigshafen
Stockacher Ach 217
97
15
30
204
171 Friedrichshafen Schussen
808
44
60
172 Lôrrach
Wiese
429
112
41
60
174 Albbruck
91
32
Hauensteiner Alb 242
30
175 Seefelden
Seefelder Ach
309
89
21
60
177 Friedrichshafen Rotach
111
71
26
30
181 Ôflingen
119
67
Wehra
23
30
ID Location
River
D
c, c,
F
cr
E
K
1.05
1.05
1.06
1.07
1.07
1.07
1.06
1.06
1.05
1.05
1.06
1.07
1.06
1.06
1.08
1.07
1.06
1.05
1.05
1.04
1.04
1.06
1.06
1.06
1.04
1.04
1.06
1.05
1.05
2.04
1.76
1.72
1.74
2.52
1.91
1.57
1.73
1.62
1.67
1.67
2.61
1.64
1.47
1.67
1.69
2.63
1.40
1.66
1.65
1.62
1.90
1.86
2.10
1.57
1.66
1.48
1.89
1.74
0.25
0.38
0.44
0.35
0.22
0.27
0.29
0.23
0.31
0.24
0.39
0.41
0.34
0.41
0.53
0.39
0.28
0.42
0.74
0.36
0.39
0.31
0.92
0.42
0.26
0.23
0.73
0.17
0.22
0.27
0.32
0.36
0.33
0.18
0.32
0.41
0.33
0.38
0.36
0.43
0.19
0.37
0.50
0.41
0.38
0.18
0.51
0.40
0.38
0.42
0.29
0.29
0.24
0.43
0.36
0.49
0.28
0.33
0.56
0.69
0.75
0.67
0.53
0.58
0.61
0.54
0.63
0.55
0.70
0.72
0.66
0.72
0.82
0.71
0.59
0.73
0.97
0.68
0.70
0.63
1.08
0.73
0.57
0.54
0.96
0.46
0.53
3.17
2.09
1.80
2.26
3.53
2.95
2.67
3.42
2.51
3.33
2.02
1.93
2.32
1.92
1.48
1.99
2.83
1.87
1.06
2.19
2.03
2.53
0.85
1.87
3.06
3.40
1.08
4.66
3.55
1.94
1.76
1.67
1.74
2.38
1.77
1.57
1.73
1.62
1.67
1.52
2.32
1.64
1.41
1.56
1.63
2.38
1.40
1.59
1.61
1.55
1.84
1.86
2.03
1.52
1.66
1.43
1.89
1.74
A: area; P: perimeter (determined at k,.-dx after smoothing); L: length; k,-Ax: representative cell size as
defined in the text; D: estimated fractal dimension of the basin boundary; C,: compactness index measured
at the original cell size dr; Cr: compactness index measured at the representative cell size kr-dx;
F: form factor; cr: circularity index measured at the representative cell size k,.-dx; E: elongation factor;
K: lemniscate ratio.
In a further step, the fractal dimension was estimated for the basin boundaries. For
this purpose, each drainage basin was resampled at different cell sizes 2'-dx with /
varying from 0 to 10. These grids were converted to polygons as described above. The
results were plotted in logarithmic plots of perimeter length vs cell size. This is
analogous to the so-called Richardson plot (Richardson, 1961), in which the perimeter
is plotted against the step length of a walking divider (e.g. Andrle, 1992). From a
Richardson plot, both evidence for the fractal character and an estimate for the fractal
dimension D can be derived, since curves with fractal geometry will plot with slope
values equal to I - D. Here, the fractal dimension was estimated from a straight line
fitted through the points derived from the cell sizes 2°-dx to 24-dx, that is 30-480 m
(Fig. 4). All representative resolutions k,-dx found for the sample data lie within this
scale range (indicated by the grey circles in Fig. 4). Although cell size was used here
GIS approach to scale issues ofperimeter-based shape indices for drainage basins
939
D = 1.07
5.6
51
. «21
5.4
. "49
Q_ 5.2
• 171
• 12
D = 1.05
1.5
'A%
2.5
3
\oq(kdx [m])
3.5
4.5
Fig. 4 Drainage basin boundary geometry for eight basins expressed as logarithmic
plots of perimeter vs cell size. The fractal dimension D was estimated from the slope
of the solid lines fitted through the points derived for cell sizes from 30 to 480 m. The
circles relate to the representative cell size k,.-dx.
instead of step length as in the divider method, and despite the small number of sample
points, the result reveals the fractal character of the basin boundaries and serves as a
first approximation to D that is consistent with the above methodology of varying the
grid resolution. The "surficial divider" method (Klinkenberg, 1994), whereby the "step
length" is expressed as the total length divided by the number of line segments (arcs)
that span a line, was also tried and yielded very similar though slightly higher D values
(about +0.007).
With an estimate for D, it is possible to calculate the perimeter, P, at any resolution
kfàx (within the scale range where D is valid) from a perimeter value, P, measured at a
different resolution kràx according to the equation:
P^P^/kf-
(5)
With equation (4) it is then also possible to estimate the representative perimeter, Pr
from which a modified compactness index value, C, can be computed (see Table 1):
Pr^Pi(kiàx^jAif
(6)
C =
(7)
2 ^7t Ai
DISCUSSION AND CONCLUSIONS
For the scale-dependent, perimeter-based indices describing the shape of drainage
basins, the above method provides a precise definition that can easily be implemented
940
Andrâs Bârdossy & Fridjof Schmidt
in GIS routines. These routines can be applied to gridded representations of drainage
basins which can be derived by means of watershed delineation algorithms for square
grid DEMs. By defining a representative scale at which the indices should be measured
and which is relative to the size of a basin, the methodology addresses the scale
problem which arises from the fractal dimension of the basin boundary and for which
the perimeter-based indices have been denounced (Bendjoudi & Hubert, 2002).
Simultaneously, a critical level for the relative area of symmetric difference between
two representations of a basin at different scales provides for the constancy of the area
and general shape of the basin.
The increase in r values between C and the scale-independent shape indices
demonstrates that the meaningfulness of the compactness index was improved while
its scale dependency was reduced through the scale adjustment operations. All the
correlation coefficients are significant. Although the r values are not very high (and
they were not expected to be because the shape indices measure different shape
properties), the results are consistent with the aim of quantifying drainage basin shape
more correctly.
The methodology presented here follows the approach of measuring P at the same
relative resolution. Fractal analysis is not necessary and was used here only for demonstration purposes. This approach is analogous to that of Breyer & Snow (1992) who
suggested to measure P by walking a divider around the polygon with a step length of
0.1/412. However, in the present study the focus is on rasterized data and a precise
representation of drainage basin geometry which allows to determine perimeter and
area simultaneously.
By estimating the fractal dimension of 153 drainage basin boundaries studied here,
the narrow range of D values found by Breyer & Snow (1992) could be confirmed. In
the present study, D values ranged from 1.04 to 1.10 (Breyer & Snow; 1992: 1.081.12). It must be stressed that the method used here for estimating D is not commonly
acknowledged, so further comparison with results of different methods is desirable. A
slightly convex upward shape of the curve in the log-log plots could also be reproduced for most of the basins, again raising the question whether the assumption of
constant fractal dimensions of drainage basin boundaries is valid across scales.
Given the very narrow range of D values, a further conclusion can be drawn. As
can be seen from equation (6), for any two basins X and Y with the same value for D,
the ratio of the perimeters, PrJPr,n and hence the ratio of the compactness indices,
CrJCr.v, are independent of the threshold, «g (or 8max), as long as the samerag(or ômax)
is chosen for both basins. This means that, irrespective of the allowed tolerance for the
symmetric difference, the compactness indices of different basins with the same D
remain proportional to each other, as long as the perimeters are measured at the same
relative resolution. By neglecting the small differences in D between different basins,
this statement can be extended to conclude that in linear models (e.g. linear multiple
regression) the role of perimeter-based shape indices derived at a common relative
resolution is independent both of the chosen threshold ômax and of the actual fractal
dimension value D found for the basin boundaries. Although the index values still
depend on the choice of a threshold 8max, the ratio between the indices of two basins is
independent of 8max.
The results of this study show that plausible values for the compactness and basin
circularity indices can be achieved with this method. The index values, as defined here,
GIS approach to scale issues of perimeter-based shape indices for drainage basins
941
differ from the original index values, which do not take scale effects into account, by
an amount that depends on the basin size, which is directly attributable to the fractal
geometry of the drainage basin boundary. It was shown that by measuring the
perimeter at a common relative resolution, the shape indices can be compared across
scales and ranges of basin sizes, which is not possible when scale effects are neglected.
It is therefore unnecessary to abstain from the use of perimeter-based shape indices in
respect of the fractal character of the drainage basin boundary. As it was intended to
show above, the meaning of the compactness and circularity indices can be vindicated
if scale issues are addressed appropriately. Irrespective of these improvements, the
usefulness of the indices for hydrological analyses remains yet to be further
investigated.
The feasibility of the suggested approach should be examined more rigorously,
including other fields of application. With the advent of high resolution DEMs from
laser scanner data (e.g. Ackermann, 1999; Mcintosh & Krupnik, 2002), the range of
scales should be extended to include smaller cell sizes. This could also reveal the
validity of drainage basin fractal geometry concepts at large scales and elucidate the
question whether constant fractal dimensions can be assumed.
It would be interesting to see whether the above methodology can be transferred to
other topographic basin characteristics such as the flow length of streams and rivers for
which a fractal dimension can be postulated. Implications for surface roughness
measures should also be discussed. Furthermore, future studies might include a
resampling of the DEM to investigate the sensitivity of basin geometry to the
resolution of the underlying elevation data, although there remain unresolved issues
linked to hydrological DEM corrections and drainage network extraction.
Acknowledgements The authors would like to acknowledge the support by the
Gimolus project (www.gimolus.de) and the suggestions by anonymous reviewers.
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Received 11 March 2002; accepted 2 September 2002