Hydrological models are extensively used in urban water management, development and evaluation of future scenarios and research activities. There is a growing interest in the development of fully distributed and grid-based models. However, some complex questions related to scale effects are not yet fully understood and still remain open issues in urban hydrology. In this paper we propose a two-step investigation framework to illustrate the extent of scale effects in urban hydrology. First, fractal tools are used to highlight the scale dependence observed within distributed data input into urban hydrological models. Then an intensive multi-scale modelling work is carried out to understand scale effects on hydrological model performance. Investigations are conducted using a fully distributed and physically based model, Multi-Hydro, developed at Ecole des Ponts ParisTech. The model is implemented at 17 spatial resolutions ranging from 100 to 5m. Results clearly exhibit scale effect challenges in urban hydrology modelling. The applicability of fractal concepts highlights the scale dependence observed within distributed data. Patterns of geophysical data change when the size of the observation pixel changes. The multi-scale modelling investigation confirms scale effects on hydrological model performance. Results are analysed over three ranges of scales identified in the fractal analysis and confirmed through modelling. This work also discusses some remaining issues in urban hydrology modelling related to the availability of high-quality data at high resolutions, and model numerical instabilities as well as the computation time requirements. The main findings of this paper enable a replacement of traditional methods of model calibration by innovative methods of model resolution alteration based on the spatial data variability and scaling of flows in urban hydrology.

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 2m × 2m pixels, were analysed to quantify their fractal dimension, characteristic of scaling invariance. The results showed that both sewer density and imperviousness exhibit scale-invariant features and can be characterized with the help of fractal dimensions ranging from 1.6 to 2, depending on the catchment. In a given area consistent results were found for the two geometrical features, yielding a robust and innovative way of quantifying the level of urbanization. The representation of imperviousness in operational semi-distributed hydrological models for these catchments was also investigated by computing fractal dimensions of the geometrical sets made up of the sub-catchments with coefficients of imperviousness greater than a range of thresholds. It enables one to quantify how well spatial structures of imperviousness were represented in the urban hydrological models.

Urban hydrological applications require high resolution precipitation and catchment information in order to well represent the spatial variability, fast runoff processes and short response times of urban catchments (Berne et al., 2004). Although fast progress has been made over the last few decades in high resolution measurement of rainfall at urban scales, including increasing use of weather radars, recent studies suggest that the resolution of the currently available rainfall estimates (typically 1 × 1 km² in space and 5 min in time) may still be too coarse to meet the stringent requirements of urban hydrology (Gires et al., 2012). What is more, current evidence is still insufficient to provide a concrete answer regarding the added value of higher resolution rainfall estimates and actual rainfall input resolution requirements for urban hydrological applications. With the aim of providing further evidence in this regard, a collaborative study was conducted which investigated the impact of rainfall input resolutions on the outputs of the operational urban drainage models of four urban catchments in the UK and Belgium (Figure 1).