Urban environments are very complex systems due to their intrinsic extreme variability over a wide range of spatio-temporal scales, and the interaction between human activities and natural processes. A notable illustration is the ongoing urbanization process that changes land cover and strongly influences the hydrological behaviour of urban catchments. Urban hydrological models were developed over the years and used to simulate the portion of the water cycle in urban environments (). They can be classified according to either the nature of the employed algorithms (empirical, conceptual or physically based; ), or their spatial resolution and how they represent the complexity of urban hydrology processes (lumped, semi-distributed and fully-distributed models).

Lumped () and semi-distributed () models are conceptual ones and rely on a simplified representation of urban catchment's heterogeneity. Indeed the whole catchment is considered as a single unit with homogeneous features for the lumped ones, while a catchment is divided into a limited number of homogeneous sub-catchments for the semi-distributed models. These two approaches were widely developed and used for modelling applications because they require limited amount of data for their implementation, and exhibit fast computation time. They often rely on a calibration step that '“forces” the model to represent the observed data. However, these models give output information at the sub-catchment scale, which is too coarse for meeting urban water managers' requirements in their need to understand some very local flooding problems or to evaluate management strategies at very small scales. Hence, the need has arisen to change the spatial resolution of hydrological models, and several works in the literature have investigated this approach for semi-distributed models (). It appears that the aggregation and disaggregation of sub-catchments changes the model output, which reflects a “scale effect issue”, and that the complex calibration step must be performed again to obtain performance similar to the previous configuration.

The influence of catchment scale on hydrological response is more pronounced for fully distributed () and gridded-based models, because of their modelling approach that usually consists in representing the high heterogeneity of urban catchment in a gridded format. The choice of an appropriate spatial resolution is always critical and the obtained model performance strongly depends on the chosen implementation scale (). The appropriate spatial resolution is obviously linked to the quality and resolution of data available as well as the modelling goal (). A more accurate representation of the land cover heterogeneity is obtained using a high-resolution grid (small pixel size). However, given current computational capabilities and data availability, high-resolution modelling is feasible only for small areas. Therefore, it is important for modellers to understand the effects of spatial resolution in urban hydrological simulations.

Scale effects and scaling in urban hydrology have been investigated by researchers – for example . This topic was also reviewed by . discussed temporal and spatial scaling issues in the context of urban storm water modelling. proposed a spatial discretization methodology applied for distributed hydrological models to get an efficient representation of land cover heterogeneity. investigated the effects of spatial resolution on predictions of peak flow and total outflow volume in an urban catchment. analysed the Digital Elevation Model (DEM) grid size and land cover representation. They found that the effect of grid size on the model performance is not linear; a 10 m grid size provides a substantial improvement over 30 and 90 m data, whereas 2 or 4 m data provide only marginal additional improvement. investigated the effects of scale in urban hydrology by trying to identify a threshold scale called “Representative Elementary Area (REA)”. The REA is strongly influenced by the topography.

Fractal tools will be used in this work to characterize scale effects in environmental data. They are widely used in several science domains including geology, medicine, meteorology and finance (). In hydrology, the fractal dimension concept has been used in many studies in the past for various purposes, ranging from catchment geometrical characterization to flow analysis (), but has seldom been used in urban hydrology ().

This work was motivated by the fact that on the one hand the inputs of the hydrological models exhibit scale-invariant features while on the other hand distributed models are implemented at a single resolution. Hence the question we seek to investigate in this paper is “at which resolution should we implement the model?” – bearing in mind practical constraints such as missing data at high resolution or longer computation time. The main goal of the paper is to investigate the existence and try to identify the appropriate resolution (or a range of resolution) for a Multi-Hydro model (Sect. 3) implementation over a peri-urban area close to Paris (Sect. 4). We first use fractal tools to analyse the features of the model's inputs and then we perform multi-scale modelling work. Methodology is presented in Sect. 4 and results are discussed in Sect. 5.

Multi-Hydro (Fig. , ) is a fully distributed and physically based model developed at Ecole des Ponts ParisTech which has been used by several authors (). It is an interacting core between four open source software packages, each of them representing a portion of the water cycle in urban environments. Multi-Hydro involves a modelling approach that consists in rasterizing the urban domain at a specific spatial resolution chosen by the user. A unique land use class for which hydrological and physical properties are specified is then assigned to each pixel.

The modelling approach involved in Multi-Hydro model relies on solving physical equations that describe the catchment behaviour. Seven processes are generally simulated; (1) precipitation, (2) interception and storage, (3) infiltration, (4) overland flow, (5) sewer flow, (6) infiltration into the subsurface zone and (7) sewer overflow.

The four modules that make up the core of Multi-Hydro are presented in Fig. :

The surface component (MHSC) is based on the existing TREX model (Two-dimensional Runoff, Erosion, and Export model) developed by Colorado State University and used in Multi-Hydro only for rainfall–runoff modelling (). The surface module computes interception, storage and infiltration occurring at each pixel according to the properties of its land cover class. The overland flow can occur after exceeding the depression storage threshold, and it is governed by equations ensuring the conservation of mass (continuity) and momentum. This flow depends on the surface properties as well as the elevation, and is computed using the diffusive wave approximation of Saint-Venant equations ().

The four modules are connected via the Multi-Hydro core, which groups together a set of codes allowing interaction, retro-action (feedback) and data exchange between these modules. In this case study, these interactions are performed after each time loop of 5 min. More precisely, the surface module outputs are used as inputs for the soil and the drainage modules, and in the same way the sewer overflow is taken into account in the overland depth for the next step. Multi-Hydro produces a large set of outputs that describe the catchment response. For example, overland water depth maps are available at each time step as well as overland discharge flow maps and velocity profiles at any point of the catchment. Saturation profile of the subsurface zone and sewer flows are also computed. The model also provides a detailed volume balance at each time step.

Multi-Hydro is highly demanding on data quality and resolution. Distributed data (usually available in GIS format) describing the topography and land use over the catchment must be collected at a high resolution. Precise information about all the components of the sewer network are also necessary for the drainage module. Such information is usually available for urban areas and can be obtained from the local authority in charge of the water management. Details about all pipes (geometry, length, diameter as well as inlet and outlet nodes) should be carefully validated, as well as all the system nodes (coordinates and elevation). The subsurface structure should be described as well if there is a need to simulate the infiltration through the unsaturated zone. The rasterization of the urban domain is the first step of Multi-Hydro implementation. During this process a unique class of land use is attributed to each individual pixel. This attribution can be done following at least two methodologies, illustrated in Fig. . The first one is based on a priority order defined by the user to attribute land use class. The second method is based on a majority rule, which means that each pixel will be affected by the majority class of land observed within it, with an exception of the gully class which remains a priority regardless of the method applied, to ensure the connection between the surface and the drainage system.

The possibility to implement other rules was investigated in the framework of , but the model formulation allows only one land use class (characterized by a few parameters such as the conductivity) per pixel. This means that implementing other approaches (such as a fractional approach) would require to a great increase in the number of classes as well as the development of a multifractal spatial characterization of key parameters such as conductivity. Those are possible motivating future investigation paths but they are outside the scope of the current study. Hence it was chosen to limit the study to two rules for affecting pixels' class, while keeping the number of classes reasonable.

The two rules considered here were tested and compared by . The main findings are reported in Sect. 5 (see Figs.  and ) and indicate that considering the majority rule methodology leads to a better representation of the catchment heterogeneity. Consequently, the majority rule was used here during the rasterization step to attribute a unique land cover class to each pixel.

Multi-Hydro has already been implemented in several locations for different purposes: in the cities of Villecresnes (France) and Manchester (UK) for flood mitigation by using barriers and retention basins (), in Sucy (France) for retention basin management (), in Sevran (France) to study the impact of small-scale rainfall variability in urban areas (), and in Villepinte and Champs-sur-Marne (France) for quantifying the impact of large-scale implementation of blue and green infrastructures on storm water management ().

This work was motivated by the fact that on the one hand the inputs of the hydrological models exhibit scale-invariant features while on the other hand distributed models are implemented at a single resolution. Hence the question we tried to investigate in this paper is “at which resolution should we implement the model?” – bearing in mind practical constraints such as missing data at high resolution or longer computation time. The main goal of the paper is to investigate the existence and try to identify the appropriate resolution (or a range of resolutions) for Multi-Hydro implementation.

In the first part of the paper, fractal tools were used to characterize the scale dependence observed within distributed data (available in commonly used GIS formats) used to configure urban storm models. Both the structure of the sewer network and the distribution of impervious areas were analysed. Then multi-scale modelling investigations were carried out using the fully distributed model to analyse the effect of this scale dependence on the model performance.

The model was implemented at 17 spatial resolutions ranging from 100 to 5 m. The case study area is a 2.45 km2 urban catchment located southeast of Paris, in Val-de-Marne County.

Results coming from this work confirm the scale dependence of the obtained model outputs. In fact, model performance indeed increases with decreasing spatial scales. This is due to a better representation of the catchment small-scale heterogeneity and scaling behaviour, notably for the impervious areas which are immediately active during a rainfall event. At large scales (100–40 m), the model shows a fast computation time (only a few minutes) and also reproduces well the overall flow dynamic, but the flow volumes remain largely overestimated. At small scales (10–5 m) the urban catchment configuration, including the overall imperviousness, becomes scale independent, without any further improvement, and one can notice a possible decline of the model performance. The small fluctuations of the model performance at this range of scales are in fact related to specific issues taking place at high resolution: mainly data problems, such as GIS data quality and missing information, as well as model numerical instabilities, without ignoring the computation time constraints essential for urban hydrology applications.

Over the remaining medium range of scales (30–15 m) for our case study, the model exhibits high performance and fast computation time since the increase in data resolution creates sufficient spatial variability among the grid-based parameters of the model. Such variability becomes somewhat representative (i.e. up to the selected precision) for the geophysical variability of the studied urban catchment.

Due to a tremendous increase in number of data pixels for grid-based models, one easily understands the difficulty of applying the classical methods for model parameter calibration. Analysis performed here demonstrates that forcing the model to give a better performance by changing its parameters is simply not reasonable for grid-based models because of their strong scale dependence. In turn, such scaling dependence induces an alternative to the classical model calibration. As we have demonstrated here, a better consideration of such scale dependence makes it possible to define an optimum range of scales – over which the model performs much better with respect to the measurements. This can be seen as a proposed alternative to the classical parameter calibration of grid-based models.