Nature-based solutions (NBS) as a sustainable strategy has recently received increasing attention for urban stormwater management. Thus, an evaluation of cost-effectiveness of NBS scenarios by ...integrating hydrological impacts and life cycle costs is significant for the decision-making process. This study first investigates the hydrological responses of a 5.2 km2 semi-urban watershed under various implementation NBS scenarios and highly spatially variable rainfall fields. The fractal dimension is considered as a scale invariance indicator to quantify the heterogeneous spatial distributions of NBS in each scenario across a range of scales. The hydrological responses of NBS scenarios are assessed by the fully distributed and physically based model (Multi-Hydro) under different rainfall conditions with a high spatial resolution of 10 m. In order to assess the cost-effective NBS scenarios, the hydrologic indicator (reduction of peak flow and total runoff volume) is integrated with the economic indicator (life cycle costs). The results show that the optimal NBS scenarios are characterised with fractal dimension ranges from 1.5 to 1.6 under all studied rainfall events. Considering the NBS scenarios under the strongest rainfall event, concentrating NBS downstream of the catchment can be more cost-effective. This study can provide some guidelines for the decision-making process on sustainable urban planning and improve the flood resilience of cities.
•Evaluation of hydrological impacts of nature-based solutions scenarios.•Life cycle costs of nature-based solutions scenarios.•Assessing the cost-effectiveness of nature-based solutions scenarios.
Advances in nonlinear dynamics, especially modern multifractal cascade models, allow us to investigate the weather and climate at unprecedented levels of accuracy. Using new stochastic modelling and ...data analysis techniques, this book provides an overview of the nonclassical, multifractal statistics. By generalizing the classical turbulence laws, emergent higher-level laws of atmospheric dynamics are obtained and are empirically validated over time-scales of seconds to decades and length-scales of millimetres to the size of the planet. In generalizing the notion of scale, atmospheric complexity is reduced to a manageable scale-invariant hierarchy of processes, thus providing a new perspective for modelling and understanding the atmosphere. This synthesis of state-of-the-art data and nonlinear dynamics is systematically compared with other analyses and global circulation model outputs. This is an important resource for atmospheric science researchers new to multifractal theory and is also valuable for graduate students in atmospheric dynamics and physics, meteorology, oceanography and climatology.
•Green roof designs rely on hydrological behaviour of granular substrates.•Substrate hydraulic properties are closely related to their Grain Size Distribution.•Fractal approximations in size ...distribution models are physically unrealistic.•Pore and Grain size distributions are directly related to each other.•More realistic Multifractal-based models are proposed here.
Hydrological behaviour of granular substrates is of critical interest in Nature-based solutions (NBS) like green roofs. To simulate this behaviour in a physically realistic manner it is indispensable to model the substrate’s hydraulic conductivity (HC) as it determines infiltration rate at various degrees of saturation. Since HC is directly dependent on water content retained by the substrate, it is necessary to physically model this water retention (WR) behaviour too. Capillary water is stored or retained in pore spaces and this water content that can be retained by a substrate under different suction pressures is therefore dependent upon its pore size distribution (PSD). Since pores in any granular media are spaces where grains are absent, their size distribution too is intrinsically related to the substrate’s grain size distribution (GSD) which provides the probability of finding grains smaller than some threshold diameter dg,t. Although earlier studies have attempted to model PSD, WR and HC, they frequently use simplifying mono-fractal approximations, whereas this study proposes a more generalized multifractal-based approach. Furthermore, while it is quite usual to incorporate pore tortuosity through some indirect parameter l in the HC model, a related ink-bottle effect which even though capable of affecting WR behaviour is commonly ignored. Therefore, this paper attempts to address the aforementioned research gaps in modelling GSD, PSD, WR and HC by i) investigating the somewhat overlooked question of similarity in multifractal behaviour between grain size fields and substrate density fields, and consequently suggesting an improved method for estimating universal multifractal (UM) parameters of grain size fields in a more reliable manner from just conventional GSD measurements in order to be directly used in the multifractal GSD model, ii) proposing a new UM-based PSD model, and subsequently using it to obtain a new UM-based WR model with a parameter to directly represent ink-bottle effect - a consequence of the substrate’s pore configuration or arrangement, iii) using this UM-based WR model to suggest a new UM-based HC model without the necessity for a separate pore tortuosity parameter. Finally, the proposed models have been validated by using experimental measurements from 4 different commercially used green roof substrates.
A century of cascades and three decades of multifractals have built up a truly interdisciplinary framework that has enabled a new approach and understanding of nonlinear phenomena, in particular, in ...geophysics. Nevertheless, there seems to be a profound gap between the potentials of multifractals and their actual use. For instance, it seems ironic that multifractals have been mostly restricted to scalar‐valued fields, whereas cascades were first invoked for the wind velocity. We argue that this requires to proceed to new developments of the multifractal formalism and to the emergence of multifractal operators. This paper therefore aims to first simplify the introduction to the most recent developments based on the analysis and generation of multifractal fields with the help of the group property of the responses of a nonlinear system to a scale change. The generators of the multifractal operators are introduced with the help of symmetries as simple and basic as orthogonal rotations and mirror symmetries. This leads in a rather straightforward manner to the large class of Gauss–Clifford and Lévy–Clifford generators that combine a number of seductive properties, including universal statistical and robust algebraic properties. At the same time, we obtain new results on the entanglement of spherical and hyperbolic geometries, as well as on the existence of finite statistics of these cascades.
Plain Language Summary
It is already exceptional that a quatrain has been inspiring theoretical and empirical research for a century, furthermore on a fundamental question of mathematical physics. It is moreover ironic that we are only now becoming able to address the real content of this quatrain. But that is what this paper is about! The fundamental question is that the basic properties of the solutions of the fundamental equations of fluid mechanics remain unknown, although these equations were derived almost two centuries ago. Therefore, the observed, extreme variability of the wind velocity still remains a puzzle. The aforementioned quatrain pointed out a cascade mechanism of transfer of the velocity from large to small wind structures. The major step discussed in the paper is to elucidate this transfer for vector quantities like wind, which have directions as well as magnitude, whereas cascades have been developed so far for scalar quantities that have only magnitude and no direction. This step should have many far‐reaching, theoretical and practical impacts due the importance of vector quantities, in particular, the wind velocity. Furthermore, it should help us to answer to the longstanding, aforementioned fundamental question of mathematical physics.
Key Points
The concept of cascades has been inspiring for nonlinear geophysics for a century and has led to the breakthrough of multifractals
New developments on stochastic algebra of cascade generators were needed for vectors to bridge the gap between potentials and applications
Gauss‐Clifford and Lévy–Clifford generators combine statistical universality and robust algebraic properties
This study proposes to exploit recent advances in data mining techniques to examine the strategies published on the Internet by public authorities and implemented in Paris. We first conducted a ...qualitative review of strategic documents, released over 15 years. We then used a list of key terms concerning flood risk management to compute a semantic network and identify clusters of terms that tend to appear in similar contexts. The analysis reveals an evolution of how policy and decision-making processes are depicted on the Web. The proposed methodology can be reproduced in other cities facing climate risks.
This study proposed a scale-invariance framework within the fractal and Universal Multifractal (UM) framework to assess hydrological performances and economic dimensions of nature-based solutions ...(NBS) across various spatial scales. Firstly, a series of NBS scenarios are created by implementing NBS heterogeneously over Guyancourt city (a peri-urban catchment located in the Southwest of Paris). Then, the spatial heterogeneity and the implementation levels of NBS in the NBS scenarios are quantified by a scale-invariance indicator (fractal dimension; DF) across various spatial scales. The X-band radar rainfall data with high space-time resolution was obtained from École des Ponts ParisTech, which is used as the rainfall forcing for numerical modelling experiments. Then, the hydrological responses of the NBS scenarios are simulated by using the fully distributed and physically-based hydrological model (Multi-Hydro) under the selected spatially variable rainfall event. The renormalised maximum probable singularity indicator (RI) is developed based on the UM framework, and it is employed to quantify the hydrological effectiveness in terms of efficiency in runoff reduction of the NBS scenarios. The economic indicator is represented by the life cycle costs (LCC), which are used to estimate the economic costs of NBS scenarios. Finally, the economic dimensions of NBS across various spatial scales are quantified by integrating DF and the LCC of NBS scenarios. The results show that the permeable pavement scenarios 3 and 4 perform better than the other NBS scenarios in mitigating overland flow. The assessment of the economic dimensions of NBS suggests that a higher implementation level of NBS measures in the small-scale range is necessary. The economic dimensions of NBS at the large-scale range vary between 225 m2 and 600 m2. Overall, this study will potentially provide valuable strategies for better managing stormwater runoff in urban catchments and support the decision-making processes of implementing NBS on multiple spatial scales.
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•Quantify nature-based solutions (NBS) spatial distributions by fractal dimension•Assessing hydrologic responses of NBS scenarios by a scale-invariance indicator•A scale-invariance framework to assess the economic dimensions of NBS scenarios•Permeable pavement scenarios (3 and 4) can better mitigate overland flow.•The economic dimensions of NBS vary in different ranges of spatial scales.
This paper presents a comparison between C-band and X-band radar data over an instrumented and regulated catchment of the Paris region. We study the benefits of polarimetry and the respective ...hydrological impacts with the help of rain gauge and flow measurements using a semi-distributed hydrological model. Both types of radar confirm the high spatial variability of the rainfall down to their space resolution (1 km and 250 m, respectively). Therefore, X-band radar data underscore the limitations of simulations using a semi-distributed model with sub-catchments of an average size of 2 km. The use of the polarimetric capacity of the Météo-France C-band radar was limited to corrections of the horizontal reflectivity, and its rainfall estimates are adjusted with the help of a rain gauge network. On the contrary, no absolute calibration and scanning optimisation were performed for the polarimetric X-band radar of the Ecole des Ponts ParisTech (hereafter referred to as the ENPC X-band radar). In spite of this and the fact that the catchment is much closer to the C-band radar than to the X-band radar (average distance of 15 km vs. 35 km, respectively), the latter seems to perform at least as well as the former, but with a higher spatial resolution. This was best highlighted with the help of a multifractal analysis, which also shows that the X-band radar was able to pick up a few rainfall extremes that were smoothed out by the C-band radar.
•Measured rainfall is strongly biased by the fractality of the measuring networks.•Rain gauge networks’ distributions lead to partial information on the rainfall fields.•Semi-distributed models ...statistically reduce rainfall fields into virtual rain gauges.•The size of the sub-catchments should be comparable to the rainfall data resolution.•A rain gauges’ conditioning may be rather counterproductive for some rainfall events.
Precipitation risk and water management is a key challenge for densely populated urban areas. Applications derived from high spatio-temporal resolution observation of precipitations are to make our cities more weather-ready. Finer resolution data available from dual polarised X-band radar measurements enhance engineering tools as used for urban planning policies as well as protection (mitigation/adaptation) strategies to tackle climate-change related weather events. For decades engineering tools have been developed to work conveniently either with very local rain gauge networks, or with mainly C-band radars that have gradually been set up for space-time remote sensing of precipitation. Most of the time, the C-band radars continue to be calibrated by the existing rain gauge networks. Inhomogeneous distributions of these networks lead to only a partial information on the rainfall fields. Here we show that the statistics of measured rainfall is strongly biased by the fractality of the measuring networks and that this fractality needs to be properly taken into account to retrieve the original properties of the rainfall fields, in spite of the radar data calibration. In this work, we use the semi-distributed hydrological modelling over the Bièvre catchment to generate a virtual rain gauges’ network. And, firstly, performing a fractal analysis of this network distribution, we demonstrate that the semi-distributed hydrological models statistically reduce the distributed (weather radar) rainfall fields into rainfall measured by a much scarcer network of virtual rain gauges. Then, with the help of the Intersection Theorem and multifractal theory, we statistically compare the virtual rain gauges’ data with the rainfall data measured by the dual-polarimetric X-band radar operated at Ecole des Ponts with a spatial resolution of 250 m, providing pre-factors that indicate the need of a proper re-normalisation of rain gauge rainfall data when comparing (or calibrating) with radar data and the possible counter productivity of this conditioning.
Recent studies have highlighted the need for high resolution rainfall measurements for better modelling of urban and peri-urban catchment responses. In this work, we used a fully-distributed model ...called “Multi-Hydro” to study small-scale rainfall variability and its hydrological impacts. The catchment modelled is a semi-urban area located in the southwest region of Paris, an area that has been previously partially validated. At this time, we make some changes to the model, henceforth using its drainage system globally, and we investigate the influence of small-scale rainfall variability by modelling three rainfall events with two different rainfall data inputs: the C-band radar data provided by Météo-France at a 1 km × 1 km × 5 min resolution, and the new X-band radar (recently installed at Ecole des Ponts, France) data at a resolution of 250 m × 250 m × 3.41 min, thereby presenting the gains of better resolution (with the help of Universal Multifractals). Finally, we compare the Multi-Hydro hydrological results with those obtained using an operational semi-distributed model called “Optim Sim” over the same area to revalidate Multi-Hydro modelling, and discuss the model’s limitations and the impacts of data quality and resolution, observing the difficulties associated with semi-distributed models when accounting the spatial variability of weather radar data. This work concludes that it may be useful in future to improve rainfall data acquisition, aiming for better spatio-temporal resolution (now achieved by the weather dual-polarized X-band radars) and data quality when considering small-scale rainfall variability, and to merge deterministic, fully-distributed and stochastic models into a hybrid model which would be capable of taking this small-scale rainfall variability into account.
The standard method for characterizing the variability of overland flow focuses primarily on a single scale, usually the smallest scale available (i.e. the highest resolution). However, the extremes ...of overland flow are generally variable over a wide range of scales. Thus, the smallest scale has no specific hydrological significance, and a scale-independent characterization is more physically relevant. This study investigated the spatial variability of overland flows and evaluated the hydrological performance of the Nature-Based Solutions scenarios in terms of a possible improvement of the morphological functioning of the catchment at several scales.
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