Fractal analysis relies on scale invariance and the concept of
fractal dimension enables one to characterize and quantify the space filled by a
geometrical set exhibiting complex and tortuous patterns. Fractal tools have
been widely used in hydrology but seldom in the specific context of urban
hydrology. In this paper, fractal tools are used to analyse surface and sewer
data from 10 urban or peri-urban catchments located in five European countries.
The aim was to characterize urban catchment properties accounting for the
complexity and inhomogeneity typical of urban water systems. Sewer system
density and imperviousness (roads or buildings), represented in rasterized
maps of 2 m

The aim of this paper is to consistently characterize urban catchment properties accounting for the complexity and inhomogeneity typical of urban water systems. It is focused on two main properties of urban catchments, namely the geometry of the sewer system and the distribution of impervious surfaces. Such characterization is important to obtain insights in the urban catchment response behaviour at the various spatial scales that control the relation between rainfall and sewer flows; to develop convenient methods that allow for evaluation of the urban catchment characteristics implemented in urban drainage models (the ones that are of importance for obtaining reliable spatially variable urban catchment responses; e.g. spatial imperviousness structure); to develop methods that support the urban hydrological modeller in the decision making process with regard to spatial details required to obtain reliable model (impact) results. Achieving this has proved to be difficult using traditional tools, mostly based upon Euclidean geometry, due to the variability and inhomogeneity in catchment characteristics (inter alios Berne et al., 2004). An alternative to traditional tools could be the use of fractal geometry (Mandelbrot, 1983), which relies on the concept of scale invariance; i.e. similar structures are visible at all scales. The concept of fractal dimension enables one to characterize, in a scale-invariant way, the space filled by a geometrical set in its embedding space. Fractal analysis and more generally scaling analysis have been often and successfully used in geophysics, including hydrology, but seldom in the specific context of urban hydrology.

For example, fractal analyses have been used to characterize river networks, including quantification of main stream sinuosity (Nikora, 1991; Hjeimfeit, 1988), quantification of how the network fills space (La Barbera and Rosso, 1989; Takayasu, 1990; Foufoula-Georgiou and Sapozhnikov, 2001; Gangodagamage et al., 2011, 2014), and simultaneous quantification of both features (Tarboton et al., 1988; Rosso et al., 1991; Tarboton, 1996; Veltri et al., 1996). River basins have also been analysed with fractal analysis. For instance, Bendjoudi and Hubert (2002) showed that the perimeters of the Danube (eastern Europe) and Seine (France) river basins are too tortuous to be scale independent. Rainfall occurrence patterns also appear to exhibit fractal features (Lovejoy and Mandelbrot, 1985; Lovejoy and Schertzer, 1985; Olsson et al., 1993; Hubert et al., 1995). In extensions including the use of multifractal tools, i.e. for fields and not simply geometrical shapes, such tools have also been used to study river discharges and rainfall time series (see Tessier et al., 1996, or Pandey et al., 1998, for examples combining both). Such analysis was also carried out on simulated discharged in urban context (Gires et al., 2012).

Some authors relied on the same concept of fractal dimension for
characterizing land use cover in various contexts. For example Chen et al. (2001)
computed a fractal dimension for various land use classes and used it
to analyse land use change between two areal pictures taken 20 years apart
over a 4 km

Despite this wide range of applications, fractal analysis has seldom been used to specifically address the topic of urban hydrology. Initial attempts to characterize urban drainage networks (Sarkis, 2008; Gires et al., 2014) or imperviousness (Gires et al., 2014) have been carried out on limited areas. In this paper we go a step further and implement fractal analysis on 10 urban catchments with different characteristics located across five European countries. The investigation includes analysis of the sewer network geometry and distribution of imperviousness derived from available GIS data, including the way in which it is represented in operational semi-distributed hydrodynamic urban drainage models. In order to be able to use the same technique to analyse both sewer networks and maps of distributed imperviousness, we use fractal tools on them, and not multifractal ones such as the one found in De Bartolo et al. (2004, 2006) for river networks. Multifractals will be used in the characterization of the representation of imperviousness in models. This multi-catchment investigation allows for obtaining robust results that are representative of a range of hydrological characteristics. The opportunity to carry out this multi-catchment investigation arose from the Interreg north-west Europe (NWE) project RainGain, which focuses on improving rainfall estimation and pluvial flood modelling and management in urban areas across NWE.

The paper is organized as follows. In Sect. 2 the available dataset over the 10 pilot catchments is described. The concept of fractal dimension and the methodology used to compute it are explained in Sect. 3. Results are presented and discussed in Sect. 4. In Sect. 5, the main conclusions are presented and future work is discussed.

In total, 10 urban catchments, with areas in the range of 2–8 km

General characteristics of the pilot urban catchments and their semi-distributed urban drainage models.

Location of the pilot urban catchments.

Sewer system (left), distributed imperviousness map with pixels a size of 2 m (middle), and maps of the imperviousness (%) as assigned to each sub-catchment in the semi-distributed models (right) of the pilot catchments. The axes correspond to metres (m). The black squares (visible in the middle column) correspond to the studied areas in the fractal analysis.

For each pilot catchment, three types of data are analysed in this paper and
Fig. 2 displays them for all the catchments:

The sewer system is considered as a network of linear pipes (left column in Fig. 2). The level of precision of available data is not the same for all the catchments. Indeed for the Morée-Sausset and Torquay catchments, only the main pipes are taken into account, whereas for the others all pipes down to street level (not the connections from building or houses to the network) are available.

An imperviousness map at a resolution of 2 m

A map of imperviousness is derived from catchment representation in
semi-distributed hydrodynamic models (right column in Fig. 2). A validated
operational semi-distributed hydrodynamic model was available for each of
the pilot catchments, except for Jouy-en-Josas. In this type of model, the
whole catchment is split into a number of sub-catchment, an independent
hydrological block corresponding to a portion of the full catchment. The
models are not the same for all the pilot sites but they all function with
the same underlying principles. Each sub-catchment contains a mix of
pervious and impervious surfaces, whose runoff drains to a common outlet
point, which could be either a node of the drainage network or another
sub-catchment (Rossman, 2010). Each sub-catchment is characterized by a
number of parameters, including total area, length, slope, proportion of
each land use, and soil type characteristics. Rainfall is inputted as
homogeneous in space within each sub-catchment, and based on the
sub-catchment's characteristics, the total runoff is estimated with the help
of a lumped model and routed to the outlet point. The flow in pipes is then
represented with the help if numerical approximation of one-dimensional (1-D) shallow-water
equations. The size and distribution of sub-catchments depend on the
modeller's choices according to the local features, the available data, and
desired level of precision. Based on the percentage of impervious areas
assigned to each sub-catchment within each pilot catchment, a raster map
with pixels of size 2 m

As explained in Sect. 1, the concept of fractal dimension was used in this
paper to characterize various geometrical sets (namely the sewer network
and imperviousness), embedded in a 2-D space. Let us consider such
a bounded set

This means that the outer scale of the studied set will necessarily be the original pixel size multiplied by a power of 2, closest to the maximum catchment scale (pixels are merged 4 by 4 in order to maximize the number of points in the following linear regression; less reliable results would be obtained with by merging pixels 9 by 9 or 25 by 25). As a consequence, square areas are extracted from the studied catchments to be analysed with the help of fractal analysis. Their size is chosen as a balance between achieving the greatest possible coverage (which increases the range of available scales) and limiting the portion of the square extending outside the catchment boundary (given that the artificial zeros in these portions might bias the analysis due to side effects). The studied areas within each catchment are shown in Fig. 2 for all catchments. In four catchments (Cranbrook, Ghent, Herent and Torquay) two areas are studied, sometimes slightly overlapping (Cranbrook and Ghent).

The sewer network of the Herent west study area observed with the help of pixels of various sizes. The axes correspond to metres (m).

Now that the methodology to change the resolution of the dataset has been
explained, it is possible to describe the computation of its fractal
dimension with the help of the box-counting method (Hentschel and Proccacia,
1983; Lovejoy et al., 1987). Let

The notion of fractal dimension is well suited for studying binary fields
such as a sewer network or map of imperviousness. However, when the field can
have more than two states, as it is the case in this paper for the maps of
representation of imperviousness inputted in semi-distributed hydrodynamics
models, multifractal tools might be needed. Intuitively such fields are
characterized with the help of various fractal dimensions; i.e. for each
threshold, the geometrical set of the areas where the field exceeds it
exhibits a different fractal dimension. More rigorously the notion of
threshold, which is scale dependent, is replaced by the scale-invariant one
of singularity,

Figure 4 shows a log–log plot of

Estimated fractal dimensions of the sewer system and impervious areas for all the studied areas.

For the scaling regime associated with small scales (i.e. right portion of
the graph), a fractal dimension basically equal to 1 is found for all the
study areas. This does not contain information on the network's features but
simply reflects the linear structure of the pipes at these scales. It also
means that the maximum resolution of the available data (2 m pixels here) is
not critical to the analysis and does not introduce a potential bias. Indeed,
increasing or decreasing it would simply yield to extending or shrinking the
widths of the scale range of this regime but will not affect the values at
larger scales discussed below. The break is located at roughly 64 m for most
of the areas, which is consistent with the distance between two streets. It
is at 32 m in Coimbra and Rotterdam-Centrum, which correspond to densely
urbanized city centres. The break at 128 m for the Morée-Sausset sewer
is due to the fact that only major sewer pipes are available and included in
the numerical network model meaning small-scale features simply extend over
wider range of scales. Including more pipes would likely lead to shifting
the scale break to smaller scales. It appears that for all the catchments
the break is observed at roughly the approximate inter-pipe distance of the
portion of network taken into account. For the large-scales regime
(

Sewer system (left) and computation of the corresponding fractal dimension, i.e. Eq. (1) in log–log plot (right), for the Torquay north study area. For the left figure, the axes correspond to metres (m).

These results are consistent with values found in similar studies for
drainage networks. Sarkis (2008) found a fractal dimension equal to 1.67 for
the pluvial drainage network of the Val-de-Marne County (south-east of
Paris), based on an analysis at scales of 290 m to 18 km, only considering
the main pipe network. Typical values for natural river network fractal
dimensions (computed with the box-counting technique) are usually smaller
than those found here for urban catchments. For instance Takayasu (1990)
found

Figure 5 displays the impervious pixels (in blue), along with the
computation of the fractal dimension of the corresponding geometrical set
for the Torquay north area. It appears that a unique scaling regime on the
whole range of available scales is identified (single straight line),
resulting in fractal dimension 1.81. Unique scale regimes are also found for
impervious surface distributions in all the other studied areas. The scaling
regime is robust with visible straight lines as in Fig. 5 (right) and

For a given catchment, numerical values of fractal dimension for distributed imperviousness are similar to the ones found at large scales in the sewer system analysis. Discrepancies are usually smaller than 0.1; smaller than the differences between the various catchments. Areas of similar urban density have similar fractal dimensions and lower density urban areas are consistently characterized by lower fractal dimensions. These numerical similarities are worth noting and actually one of the main finding of this analysis, confirmed on a wide set of study areas. Indeed it suggests that the scaling behaviours observed on sewer networks and distributed land use have the same physical basis and reflect a unique underlying level of urbanization. The only difference being that it stops at the inter-pipe distance for the sewer network, whereas it expands down to 2 m scale for the imperviousness. Contrary to other formalisms, such as the use of a single percentage of imperviousness defined with data at an arbitrary scale, this fractal dimension is quantity valid across scales and furthermore based on the characterization of two aspects related to urbanization (namely the sewer network and the distributed imperviousness), which makes it robust.

After having investigated the fractal behaviour of sewer system and
imperviousness with the help of distributed data, the imperviousness
distribution used in operational semi-distributed hydrodynamic models is
studied in this section. A given threshold

As expected, at higher thresholds, the remaining impervious areas are
smaller and the associated fractal dimensions are also smaller. It should be
noted that the quality of the scaling also tends to diminish for increasing
imperviousness thresholds. This effect is significant for some areas such as
Moree-Sausset, Herent, and Sucy-en-Brie and hence limits the possible
interpretation of this analysis. In these cases, there is a very limited
(one or sometimes even zero) number of remaining sub-catchments at high
imperviousness thresholds, which is likely to bias the analysis. This
phenomenon is due the smaller number of sub-catchment in these cases. The
most critical one is that of Sucy-en-Brie, for which the model consists of
only eight sub-catchments (see Fig. 2). Such low spatial resolution hampers
implementation of fractal analysis and this is reflected in the low

Impervious pixels at a 2 m resolution (left) and computation of the fractal dimension of the corresponding geometrical set, i.e. Eq. (1) in log–log plot, (right) for the Torquay north study area. For the left figure, the axes correspond to metres (m).

Illustration of the computation of the fractal dimension of the
area covered by the sub-catchments, whose imperviousness is greater than a
threshold

Fractal dimension analysis of the area covered by the
sub-catchments with imperviousness greater than a threshold

Functional box-counting analysis of the map of sub-catchments
imperviousness for four selected catchments. Triangles: for each threshold

Interestingly, the fractal dimension estimates are in overall agreement with
the level of urbanization discussed in the previous section, i.e. the most
urbanized areas exhibit the greatest fractal dimension for all thresholds.
This is especially true for thresholds lower than 60 %. For greater ones,
whose estimates are less reliable, more differences are noted. For instance

For four study areas:

The percentages of distributed imperviousness (%) at the
highest data resolution

Empirical relation between the fractal dimensions of the total
impervious area and of buildings only. The continuous line indicates the
first bisector, while dotted line is given by

Given that we found that the fractal dimension of sub-catchments'
imperviousness of semi-distributed models was dependent on the threshold
used to define it, we naturally investigated the possibility of using a
multifractal framework to analyse this dependency. This is achieved by
checking the adequacy of the empirical co-dimension function

Finally, fractal dimensions of the imperviousness computed for the
semi-distributed models were compared to those derived from fully
distributed GIS data (Sect. 4.1). This is done in Fig. 9 for three studied
areas.

The percentages of distributed imperviousness (%) at the highest resolution

Such analysis could support validation of the representation of catchments in semi-distributed models; the smaller the difference, the better catchment imperviousness is represented by the model. It should be mentioned that this interpretation assumes that data available for analysing distributed imperviousness are accurate and complete, which is generally supported by the scaling behaviour of the data.

In this sub-section we discuss the results of the comparison of fractal
dimensions computed on two different geometrical sets: the total
imperviousness areas as roads and buildings (

This analysis was made to investigate the relationships between the fractality of building distributions, as a source for potential green roofs implementation for water flow management, within fractality of the whole imperviousness areas. Indeed green roofs are one of the available tools that can be used to optimize (if needed) water flows in urban and peri-urban areas, hence the need to better understand their potential distribution. More precisely, to increase the functionality of green roofs over the full range of catchment scales (Versini et al., 2016), an optimization of green roof locations could be made to increase their fractal dimension up to the fractal dimension of the total imperviousness area. The fractal tools could also be used to evaluate the potential impact of green roofs.

In this paper we implemented (multi-)fractal analysis in the context of urban hydrology on 10 catchments located in five European countries. The results have consequences both in terms of urban catchment characterization and representation in urban hydrological models.

First, it appears that the fractal dimension of either the sewer network or the impervious pixels (roads or houses) on a 2 m pixels map can be used to characterize the level of urbanization of a given area. In fact, for a given area similar estimates are obtained for both geometrical sets. The main difference is that the scale invariance is valid from one or few kilometres down to only approximately inter-pipe distance for the sewer network, whereas it extends down to 2 m for imperviousness, which matches with the spatial resolution of the imperviousness datasets. This tool is innovative in the context of urban hydrology, because it provides a quantitative estimate of a level of urbanization, which is valid across scales and not only at the scale at which it is defined as for other tools. These findings open new practical perspectives that should be explored in future work. An example is the possibility of identifying consistent – across scales – areas that should be modelled separately. Another one is the possibility of relying on the scale-invariance features to fill gaps of missing data in a realistic way. This issue is increasingly visible as one goes toward higher-resolution model. It is furthermore an acknowledgment of the complexity of the notion of imperviousness which is usually simplified in state-of-the-art urban hydrological models in which it is often represented as a mere percentage, thus neglecting without taking into account its heterogeneous distribution. Using scale-invariant concepts able to handle more appropriately these features is a lead that should used to innovatively improve distributed hydrological models.

Second, the representation of imperviousness in operational semi-distributed models was analysed. It appears that, by analysing the geometrical set made of sub-catchments with imperviousness greater than a given threshold, it is possible to retrieve urbanization patterns. In this study, it was found that fractal dimension values decrease from 1.9–2.0 for imperviousness degrees above 10 % down to 1.4–1.6 for imperviousness degrees above 90 %. Results for higher imperviousness degrees were subject to larger uncertainty as a result of data scarcity; findings should be verified in studies based on larger datasets.

It was also shown that comparing fractal dimension values related to modelled imperviousness to imperviousness represented in high-resolution GIS datasets allows one to quantify how well imperviousness is represented in urban hydrological models. These results open perspectives for the development of tools to verify whether a hydrological model properly represents the degree of imperviousness in a catchment and also to study urbanization patterns emerging at different degrees of imperviousness. Such insights could latter be used in support of hydrological analysis as well as other urban development analyses.

The data and more precisely the matrices over which fractal
and multifractal analysis carried out in this paper have been made
available on a public repository. It can be accessed through:

The authors declare that they have no conflict of interest.

The authors greatly acknowledge partial financial support from European
Union INTERREG IV NWE RainGain Project (