Multifractals provide a powerful framework to describe systems that exhibit variability over a wide range of scales together with strong intermittency. By encoding scale-dependent fluctuations through multiplicative cascades, multifractal models capture non-Gaussian statistics, heavy tails, and scale invariance in a compact and predictive manner. These properties have made multifractals particularly successful in the analysis of a wide variety of geophysical phenomena. From the outset, multifractal fields have been formulated on domains of arbitrary dimension, allowing to represent space, space–time, or higher-dimensional parameter spaces. In contrast, the codomain of multifractal constructions has most often been restricted to scalar-valued fields. Although simpler for modeling and inference, the scalar setting omits directional information, anisotropy, and cross-component couplings that are essential in vector observations. Recent works, such as (Schertzer and Tchiguirinskaia 2020), have explored the use of Clifford algebras for constructing cascade generators, offering a natural algebraic framework to represent vector-valued multifractals while preserving their multiscale and symmetry properties. In this work, we consider and simulate Clifford multifractal cascades as an extension of scalar models, capable of capturing directional variability and the internal geometry of multiscale fields. Rather than relying on a scalar stability exponent, we work in a framework where the stability can be encoded by algebra-valued or operator-like parameters, enabling anisotropic scaling and nontrivial coupling between different components of the Clifford field across scales. To characterize the resulting operator-scaling structure, we extended the scalar analysis methods and developed inference methods that enable the direct estimation of multifractal parameters. Numerical experiments on synthetic cascades demonstrate that the proposed approach reliably recovers these parameters. The results demonstrate that extending multifractal analysis to vector-valued fields is both feasible and essential for the characterization of complex multiscale phenomena.

Urban areas are increasingly exposed to localized extreme rainfall events, with evidence suggesting a trend toward higher precipitation volumes and more frequent short-duration, high-intensity storms, posing major challenges to infrastructure resilience and public safety. Urban hydrometeorology is characterized by highly nonlinear processes, strong interactions with geophysical systems, and pronounced variability across spatial and temporal scales, making both scientific understanding and operational management particularly demanding. Within this context, the Fresnel platform is a state-of-the-art urban hydrometeorological observatory combining conceptual modeling approaches with extensive field measurements. One of its components, RadX, is a Software-as-a-Service (SaaS) platform that provides real-time and historical data from high-resolution sensors, together with a graphical user interface (GUI) for Multi-Hydro, a fully distributed and physically based hydrological model developed at École nationale des ponts et chaussées (ENPC). Multi-Hydro relies on four open-source software components representing different processes of the urban water cycle. The RadX GUI allows users to efficiently run simulations using dedicated high-performance computing resources, configure multiple scenarios for a given catchment, modify land-use parameters, and assess their impacts on drainage system discharges. The originality of this contribution lies in the development of a new 3D isometric graphical interface based on an open-source game engine. Unlike conventional interfaces relying on the editing of raster matrices, this approach provides a more intuitive and spatially explicit visualization of land-use configurations. It enables a clearer representation and manipulation of Nature-based Solutions (NbS), such as porous pavements, whose implementation often remains abstract when expressed solely through raster data. Beyond hydrological modeling, RadX also supports integrating shared value principles into business models to enhance resilience and sustainability. Within the PIA3 TIGA-CFHF project (“Construire au futur, habiter le futur”), it promotes an integrated vision where economic activities are situated within a complex socio-environmental system, aligning economic performance with environmental and societal objectives. To support this transition, RadX aims to incorporates multifractal and advanced socio-economic analysis tools that enable organizations to assess performance and develop shared value–oriented strategies aligned with measurable environmental objectives. The RadX platform is continuously improved through an iterative development process driven by feedback from students, academic researchers, and industry practitioners, and may integrate additional visualization or forecasting components in future developments.

Quantifying rainfall extremes and their temporal evolution is essential for hydrological risk analysis and infrastructure design, and is commonly based on intensity–duration–frequency (IDF) curves. In this study, we further develop the framework proposed by Bendjoudi et al. (1997), in which IDF curves are theoretically derived within the Universal Multifractals (UM) formalism. This approach is mathematically robust and parsimonious, and is grounded in the physically based concept of scale invariance inherited from the Navier–Stokes equations.Relying on either a unique scaling regime or two scaling regimes with a break at roughly 14 days, and the existence of a multifractal phase transition associated with moment divergence, IDF curves can be derived where rainfall intensity follows a power-law relationship with both return period (positive exponent) and duration (inverse exponent). The values of the exponents and of a prefactor can be directly inferred from the UM characterization of the rainfall process.The framework was tested using rain-gauge data from six stations in Northern Italy, with record lengths ranging from 30 to 38 years. The agreement between the theoretically predictions from the UM analysis and the observed values of the prefactor and the two exponents, according to the quality of the scaling, is discussed. Possible directions for further improvements of the framework will also be discussed. Authors acknowledge the France-Taiwan Ra2DW project for financial support (grant number by the French National Research Agency – ANR-23-CE01-0019-01).References:Bendjoudi H., Hubert P., Schertzer D., Lovejoy S., 1997, Interprétation multifractale des courbes intensité-durée-fréquence des précipitations, Comptes Rendus de l'Académie des Sciences - Series IIA - Earth and Planetary Science, 325, 5, 323-326,https://doi.org/10.1016/S1251-8050(97)81379-

Abstract. The ubiquity of the fundamental characteristic of turbulence, intermittency, is increasingly recognized in many fields. The multifractal analysis of various turbulence data, particularly from lab experiments and atmospheric sensed data, has rather constantly yielded a multifractality index of α ≈ 1.5 and a mean codimension of C1 ≈ 0.25, but with a given uncertainty. To reduce this uncertainty and understand the dynamical origin of these estimates, the multifractality of turbulence is investigated with the help of the deterministic Scaling Gyroscope Cascade (SGC) model. In this study, the forced SGC model is run with cascade levels of up to 14 and a duration of 2.5 × 104 large eddy turnover times. These simulations exhibit extreme spatial-temporal intermittency. Multifractal analysis confirms the empirical values α ≈ 1.5, C1 ≈ 0.25, showing almost independence on the forcing. It raises doubts about the Log-normal model, at least for hydrodynamic turbulence. In addition, the remaining uncertainty in multifractality resulting from the discrete numerical simulation method is investigated.

Intermittency is a defining characteristic of rainfall, yet it is largely overlooked in most IA nowcasting models. We emphasise its theoretical significance at various stages of the prediction process, from training to assessing its accuracy, including its dispersion relative to the intrinsic limits of predictability. Specifically, we develop a hybrid framework based on:- The generative adversarial network (GAN), a recently developed technique for training IA models through an adversarial process; Universal multifractals (UM), stochastic models of intermittency that are physically based on the cascade paradigm. They are universal in the sense that they are statistically attractive to other processes and depend only on three scale-independent parameters that are physically meaningful. In terms of physical relevance, we evaluate the nowcasting performance of the hybrid UM-GAN model and other baseline models (ConvLSTM, GAN) using continuous and categorical scores, as well as UM analysis in comparison to the observations. The results indicate that UM-GAN achieves the highest scores and accuracy, particularly demonstrating superior performance at lead times of 30 minutes and 60 minutes.

AbstractTemporal structure functions are usually defined as the q-th order statistical moment of the absolute fluctuation in a time series over a temporal lag at a given resolution. However, applying this in analyzing a temperature time series results in the possibility of simulating only a similar fluctuation over a temporal lag at a resolution and not the temperature directly. Since the aim is to simulate a temperature time series this simulated fluctuation series can be added to an assumed mean temperature to obtain a temperature time series. However, proceeding this way seems to necessitate some ad-hoc moving average technique that seems difficult to be physically reasoned. Secondly but more importantly both diurnal and seasonal periodicity have to be forcibly introduced once again in a non-rigorous manner. A drastic yet reasonably useful alternative would be to modify the definition of the structure-function instead. For order of statistical moment q the modified structure function is now defined here asSq(Δt)=⟨ΙTλ - Tλ/2,2Ιq⟩Where the scale ratio λ∝1/ΙΔtΙ; 2m/2m=1≤λ≤Λ=2m/20 and ΙΔtΙ is the time lag, whereas 2m is the largest possible scale out of the scales analyzed that can be represented as a power of 2. While Tλ is the temperature at scale ratio λ or scale l, Tλ/2,2 is the upscaled (by a scale ratio of 2) temperature at scale ratio λ/2 or scale 2l, and the subscript ‘2’ indicates that each element of Tλ/2 (upscaled temperature) is repeated twice consecutively. It should be noted that Tλ/2,2 is not the same as Tλ because the former is an upscaled series, twice repeated (consecutively) of the latter. The largest scale ratio considered in the analysis is Λ. By defining the structure-function in this way temperature at a larger scale after being repeated a sufficient number of times can be directly added to the fluctuation at a smaller scale to result in the temperature at a smaller scale. The universal multifractal parameters obtained from the modified structure-function analysis are not necessarily equal to those obtained from the usual structure-function analysis (i.e. the two different structure functions follow two different scaling laws). An iterative curve fitting technique is used to estimate the values of Universal Multifractal (UM) parameters C1, H, and a while the value of α is estimated using a normalized form of the modified structure function along with the un-normalized one. A simulation procedure that utilizes the aforementioned modified structure function definition is proposed here to generate temperature scenarios. Finally, reference evapotranspiration is estimated based on the simulated temperature using a simple empirical power law function. The actual evapotranspiration is estimated using the reference evapotranspiration and water content via a different, simpler empirical function. The tentative methodology proposed here when used along with simulated reference rainfall scenarios could help design zero-emission green roof solutions. KeywordsMultifractals, Non-linear geophysical systems, Cascade dynamics, Scaling, Hydrology, Meteorology.

Rainfall time series prediction is essential for monitoring urban hydrological systems, but it is challenging and complex due to the extreme variability of rainfall. A hybrid deep learning model (VMD-RNN) is used in order to improve prediction performance. In this study, variational mode decomposition (VMD) is first applied to decompose the original rainfall time series into several sub-sequences according to the frequency domain, where the number of decomposed sub-sequences is determined by power spectral density (PSD) analysis. To prevent the disclosure of forthcoming data, non-training time series are sequentially appended for generating the decomposed testing samples. Following that, different recurrent neural network (RNN) variant models are used to predict individual sub-sequences, and the final prediction is reconstructed by summing the prediction results of sub-sequences. These RNN variants are long short-term memory (LSTM), gated recurrent unit (GRU), and bidirectional LSTM (BiLSTM) and bidirectional GRU (BiGRU), which are optimal for sequence prediction. In addition to three common evaluation criteria, mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE), the framework of universal multifractals (UMs) is also introduced to assess the performance of predictions, which enables the extreme variability of predicted rainfall time series to be characterized. The study employs two rainfall time series with daily and hourly resolutions, respectively. The results indicate that the hybrid VMD-RNN model provides a reliable one-step-ahead prediction, with better performance in predicting high and low values than the pure LSTM model without decomposition.

Local rainfall measurements include a wide range of methods with different concepts and assumptions. Understanding the differences enables researchers to determine the most reliable device for each rain condition. Here, we compare two such devices: Parsivel2 (an optical disdrometer by OTT) and mini Doppler radar (part of the mini Meteorological Station by Thies). The performance assessment followed two foci: evaluating the scaling features using the framework of universal multifractals (UMs), and analysing the influence of physical parameters (drop size, wind velocity and rainfall rate). The devices were mounted on a meteorological mast, in Pays d'Othe wind farm, 110 km southeast of Paris. The mini Doppler radar returned very low rain measurements during conditions with bigger drops, and showed non-detection in light rain involving heavy wind. UM analysis revealed similar scaling behaviour for both devices, which extended to finer scales of mini Doppler radar data until 4 s beyond which white noise was detected.

The aim of the Fresnel platform of École des Ponts ParisTech is to foster research and innovation in multiscale urban resilience. Studying the hydrological response of such complex urban areas accounting also for small scale spatio-temporal precipitation variability requires adapted tools. For these reasons, RadX provides a user-friendly graphical interface to run simulations using a fully distributed and physically based model: Multi-Hydro. RadX is designed as a Software as a Service (SaaS) platform, allowing users to work with data across a wide range of space-time scales and the appropriate tools for analyzing and simulating this data.The hydrological model, developed at École des Ponts ParisTech, integrates four open-source software applications previously used and validated independently by the scientific community as well as practitionners. Its modular structure includes a surface flow module, sewer flow module, a ground flow module and a precipitation module. It is able to simulate the quantity of runoff and rainwater infiltrated into unsaturated soil layers from any space-time varying rainfall event at any location of the studied peri-urban watersheds, as well as depth and flow in all the pipes and nodes of the sewer network.Users can launch hydrological simulations using the Multi-Hydro model directly from their web browser, while they are run on dedicated servers. They can adjust two key input parameters: the land use of the studied catchment and the rainfall data. Dedicated tools have been developed to enable users to modify the land use of the catchment with the same ease as using a raster graphic editor. Users can either choose real rainfall events captured by the X-band weather radar located at École des Ponts ParisTech or utilize user-defined synthetic rainfall as input. Data from other radar can also easily be integrated. For the simulation output, the interface provides users with different tools to study in detail the impact of the chosen input parameters. For instance, by simply selecting two sewer junctions on an interactive map, users can generate a sewer path between these two points and display an interactive representation of the water level heights in sewer conduits and junctions along the user-defined sewer network path.Additional components can be integrated into RadX to meet specific requirements using visual tools and forecasting systems, including those from third parties. Developments are still in progress, with a constant loop of requests and feedback from the scientific and professional world.

Geophysical fields are extremely variable over a wide range of space–time scales. More specifically, they are intermittent in the sense that the strongest fluctuations are increasingly concentrated in sparser and sparser fractions of the space–time domain. Multifractals have been developed to analyze and simulate intermittency across scales, while climate networks can detect and characterize extreme-event synchronization. In contrast to multifractal analysis, climate networks are usually generated at a given observation scale despite displaying complex structures over larger scales and being likely to exhibit similar complexity at smaller scales. In this letter, we present how to overcome this dichotomy of approaches by analyzing in detail the effects of increasing the observation scale for climate networks as allowed by empirical data; i.e., how do they upscale? This must be understood as a preliminary step to be able to downscale them, including for practical applications such as urban geosciences that require the analysis and simulation of intermittent fields at a very high resolution. This is one of the reasons why we are using precipitation to illustrate our multifractal climate network approach.

Below cloud scavenging by rain is known to be a very efficient sinking mechanism for aerosols in atmosphere. Since this scavenging depends on interaction between aerosol particles as well as the scavening raindrops, and notably their respective size ranges, it is interesting to examine both fields together across various size ranges and across temporal scales. Towards this, a 4 month long data was used from Cherbourg-Octeville, France from 01/11/2010 to 12/03/2011 from the experimental station managed by Institut de Radioprotection et de Sûreté Nucléaire (IRSN). Here, simultaneous and continuous measurement of size resolved particle concentration (14.6 to 478.3 nm and 0.523 to 19.81 µm) range has been done using Scanning Mobility Particle Sizer (SMPS) and Aerodynamic Particle Sizer (APS), and rain measurement using a disdrometer. Variation of total aerosol concentration in nm and µm range, as well as individual number concentration in small size bins were analyzed according to rain and dry events, using the framework of Universal Multifractals (UM). UM is widely used, as a physically based scale invariant framework, for characterizing and simulating extreme variability and intermittency in geophysical fields. From initial analysis, the total concentration showed scaling properties (1 min to 1 hr), in both rain and dry events, regardless the scavenging efficiency of event. This was further explored in individual concentration ranges and they showed similar scaling properties in different rain types. However, while considering the different stages of rain, say start and end, the values of UM parameters showed some variation. To understand the behavior more clearly, few sizes were selected from nm and µm range, and efforts were made to extract the field which is devoid of scavenging by rain. Understanding the correct transformation required to extract accurate UM values and comparing the scavenging and non scavenging fields will improve understanding of particle concentration variation, and eventually understanding of scavenging coefficient.

This study investigates the variability of C-n(2) data across different spatial and temporal scales using data from a large-aperture scintillometer (LAS) installed on a 1 ha green roof, known as the Blue Green Wave, in Ecole nationale des ponts chauss & eacute;es, near Paris. Data collected at 10-min intervals during December 2019 and January 2020 were analysed using structure function and universal multifractal (UM) models. The analysis revealed strong scaling behaviour from 10 min to 2 h, with UM parameters estimated as C-1=0.1 for the mean intermittency codimension, alpha=1.8 for the multifractality index and approximately 0.44 for the non-conservative parameter H, which deviates from classical turbulence theories. Our results provide a deeper understanding of the variability in C-n(2) and offer insights into scaling properties relevant to turbulence studies in urban green roof environments.

Scaling analysis of subjectively defined fields, which are often expressed as a range of values based on the field of application, is not straightforward. To avoid potential biases in statistical analysis, extracting the actual underlying field and understanding the effect of ranged values are important. One such example is atmospheric visibility which is often estimated as a range in meteorological context, as meteorological optical range (MOR). This is estimated from the extinction coefficient a-e, an objective measurement of light attenuation by constituent gases and aerosols in the atmosphere, and expressed as a range depending on the application needs. Accurate estimation of visibility and its variability is required for the safe functioning of various domains such as transport sectors and free optic communication or for understanding regional variations in air quality and climate. Since MOR is a subjective range, here we attempt to characterize it using the objectively measured a-e. In this context, we identify and illustrate the effect of a lower threshold in the data, which is not exclusive to the current problem, and examine its consequences in the multifractal characterization of the field. Here, a-e was extracted from visibility data by a present weather sensor located in the Paris region (France). This was then compared with a-e extracted from MOR measured at Paris Charles de Gaulle airport during the same period. Variability in a-e was investigated under the framework of universal multifractals (UMs), which is widely used to characterize geophysical fields that exhibit extreme variability across scales. With direct data analysis and numerical simulations mimicking the behavior, it was found that the multifractal properties exhibited by a-e are influenced by the upper limit of visibility range in the data. The biases are identified within the theoretical framework of UM, thus expanding the general understanding on the retrieval of the underlying unbiased stochastic field. SIGNIFICANCE STATEMENT: Data obtained from measurement campaigns are often influenced by instrumental limitations such as upper and lower detection limits. One such example is atmospheric visibility which is usually represented as a range, from objective measurement of light attenuation by constituent particles. Here, we examine the consequences of characterizing visibility using the extinction coefficient, by considering the presence of a lower limiting value, using two datasets in the Paris region forward scattering sensor and airport visibility measurement. Through data analysis, numerical simulation, and theoretical formulation, the biases are identified and the underlying unbiased stochastic field is retrieved. This enables a more accurate analysis and simulation of visibility, as well as other geophysical fields with similar instrumental or application limitations.

The cooling efficacy of green roofs in mitigating the urban heat island (UHI) effect within dense cities is largely attributed to evapotranspiration (ET) processes. Hence, accurate understanding and quantification of ET are pivotal for optimizing this cooling effect. ET estimation can be achieved either directly (weighing lysimeters) or indirectly (e.g., Penman-Monteith equation). Micro-meteorological approaches have been developed in recent years. Among which scintillometer can evaluate ET by its measurement parameter which corresponds to the fluctuations of air refractive index ) in combination with surface energy balance and Monin-Obukhov similarity theory. Hence, improvement in data as well as understanding of its variability across wide range of space-time scale would result in better ET estimation and ultimately optimization. Yet it is often overlooked, and little research has focused on it and notably its variability.This study explores the ET estimation on a wavy and vegetated green roof covering an area of 1 ha, known as the Blue Green Wave, which is located in Ecole des Ponts Paristech campus. Data from a large aperture scintillometer with 10-minute timestep during December 2019 and January 2020 is adopted. data variability across scales was analysed with the help of structure function and Universal Multifractal model (UM). The UM framework, widely employed for characterizing and simulating geophysical fields extremely variable across wide range of space-time scales, relies on two parameters with physical interpretation: the mean intermittency codimension and multifractality index (, indicates monofractal; , indicates log-normal model.) An additional one, which is needed for non-conservative fields such as ET is the non-conservativeness parameter H.Both structure function and UM approaches reveal good scaling behaviour on scales ranging from 10 min to 2h, confirming the relevance of the framework and demonstrating the potential for upscaling and downscaling. UM analysis conducted through Trace Moment and Double Trace Moment methods, provided similar values for UM parameters around H is approximately 0.44 in our case, which deviates from traditional scaling laws due to the intricate composition of the fluxes and requires further investigations. Indeed is influenced by temperature, humidity, air pressure and wind speed. To interpret properly structure function analysis from UM analysis, it is necessary to introduce a parameter denoted a. It corresponds to the power to which the assumed conservative underlying field should be raised before fractional integration to account for non-conservativeness to retrieve the studied field. Here, we observed that a is around 0.76 to ensure the highest consistency of the outcome from both the structure function and UM analyses. A better understanding of the underlying complexity and variability of Cn2 is achieved by our analysis. This, in turn, improves our understanding of the underlying physical processes generating variability and temporal-spatial dynamics in ET, which paves the way for future applications.

Despite efforts to obtain consistent results, the prediction of patterns in the behavior of geophysical fields still faces many uncertainties. However, these analyses are important for studying the effects of human action on the environment and the effects reflected in climate change. There is much evidence that Multifractals are capable of describing intermittent behavior and statistical data of all orders and over a wide range of scales. Therefore, this work consists of using the multifractal framework to analyze recent precipitation projection data in France, verifying the evolution of its parameters over a relatively long period of time (from 1951 to 2100) and over space, using 12 points on French territory with a resolution of 2.8º x 2.8º. For this, the Double Trace Moment technique was applied to determine the mean intermittency codimensions, the multifractality indexes and the maximum probability singularities. These results were compared to the article by J.-F. Royer et al., C. R. Geoscience 340 (2008) to verify if projections remained consistent with changes in data and economic scenarios. Despite the differences found in the range of parameter values ​​and scaling behavior, recent data also indicated an increase in intermittency over time and presented spatial behavior similar to old projections, which reinforces the expectation of an increase in precipitation extremes in the coming decades.

Rainfall is extremely variable in both space and time, which makes its analysis complex. A widely used framework to properly handle these features is Universal Multifractals (UM), which is a physically based and mathematically robust framework. It relies on three parameters, meaning it is parsimonious. Two types of multifractal phase transitions can affect the analysis of a series: (i) the divergence of moments, which is related to the singular limit of the underlying cascade process at small scales and notably explains the power law fall-off observed on numerous geophysical fields, (ii) sampling limitations, which is related to the fact that great moments cannot be observed on finite series. This study employs UM to analyse the time series of rainfall intensities observed by the Parsival2 disdrometer at the 10-second resolution from three distinct typhoons over the period of July to October 2022, revealing differences and limitations in their statistical characteristics. It enables us to illustrate the two previously mentioned concepts of divergence of moments and sampling limitations and their impact on the analysis of rainfall data.The analysis of typhoon Hinnamnor exhibited limitations due to the sampling dimension, indicating that the current data length was insufficient to capture the multifractal nature of the rainfall events for large moments, reducing the robustness of the analysis for moments greater than 5.43. It reflects sampling limitations, leading to a constrained understanding of extreme events.For typhoon Nalgae, our analysis highlighted the occurrence of divergence of moments, i.e. a limitation associated with a critical moment. As higher-order moments were calculated, we observed statistical values tending towards infinity, suggesting that extreme rainfall events significantly influenced this typhoon and pointing out the inadequacy of traditional statistical methods in such scenarios. Such multifractal phase transition is seldom observed on individual series, highlighting the interest of studying this typhoon series. Finally, the analysis of the typhoon Nesat presented a behavior affected by both multifractal phase transitions resulting in a more complex interpretation. Our findings provide new insights into the multifractal analysis of typhoon rainfall intensities, emphasising the importance of considering multifractal theory and its associated phase transitions when dealing with natural phenomenon data. These discoveries lay a crucial methodological foundation for more accurate prediction and response to extreme weather events. In case of rainfall, it then has some hydrological consequences notably in terms of stormwater management or optimization of dams for hydraulic production.

Rainfall and wind exhibit extreme variability over wide range of space-time scales. Such features are naturally transferred to wind turbine torque and ultimately to wind energy production. Improving our understanding of wind power production requires better accounting for the impact of these small scale fluctuations. This is much needed in order to achieve UN’s (United Nations) Sustainable Development Goal 7 (affordable and clean energy for all) and in a context of increasing global transition towards renewable and carbon neutral energy. The project RW-Turb (https://hmco.enpc.fr/portfolio-archive/rw-turb/; supported by the French National Research Agency, ANR-19-CE05-0022) was developed to address this challenge and to understand better the correlation across scales between rainfall and wind fields and its impact on wind power production. A high resolution measurement campaign was set up between 12/2020 and 07/2023 with two 3D sonic anemometers (manufactured by Thies), two mini meteorological stations (manufactured by Thies), and two disdrometers (Parsivel2, manufactured by OTT) installed on a meteorological mast at 75 and 45 m respectively in the wind farm of Pays d’Othe (110 km south-east of Paris, France; operated by Boralex). The framework of Universal Multifractals (UM) is used to carry out this analysis. It is a physically based and mathematically robust framework that enables to characterize and simulate the extreme variability of geophysical fields across scales. It is furthermore parsimonious since it relies on the use of only three parameters. In a first step multifractal analysis of the available fields (wind velocity, power available at the wind farm, power produced by wind turbines, air density, and rainfall) is implemented. Event based analysis enabled to observe differences in UM parameters depending on whether it is raining or not. In general, a slightly stronger variability is found when it rains. In a second step, a joint multifractal analysis is implemented to further quantify correlation across scales between the studied fields. An increase in correlation exponent of the various fields with increase in rain rate is found. Numerical simulations are then used as a complement to data analysis. More precisely, 3D space plus time vector fields which realistically reproduce observed spatial and temporal variability of wind fields are generated with multifractal tools. Then, they are used as input into three modeling chains of increasing complexity to simulate wind turbine torque. The simplest model uses average wind field over swept area, while a more realistic one computes the torque as an integral over the blades of the turbine enabling to account for the space-time variability of wind. Finally, OpenFAST, which is widely used by researchers and practitioners is implemented. UM analysis on the simulated torque time series were performed to quantify the impact of small scale fluctuations on wind power production, as well as the ability of the various models to account for it.

This work has benefited from a multidisciplinary scientific and technical contributions geared by the HM&Co Lab of the Ecole des Ponts ParisTech (hmco.enpc.fr) towards the sustainable, desirable, and resilient city. The deepening of the Universal Multifractal (UM) concepts and the encouragement of their operational applications have been linked to several initiatives launched in recent years to better integrate the heterogeneity/intermittency into public policy practices. Considering the complex, dynamic interactions between geophysical and anthropogenic fields within a conurbation such as the Ile de France region, a transition towards the shared value economy has been considered to best stimulate sober and collaborative development, and there exist at least 3 ways to approach today’s discussions about future transformations. Their intercomparison is the core of this presentation. Following the United Nations 2030 Agenda, the first most conventional approach is based on notions of sustainable development, supported by appropriate adaptation and mitigation of climate change. Combining the notions of extreme variability and complexity would require linking together geophysical and urban scales within extreme variability, and therefore considering geosciences, and not just geophysics! Such a synergistic and integrative approach would help move beyond traditional silo thinking, addressing the complexity of data- and/or theory-driven urban geosciences. Finaly, combining the notions of scaling and nonlinear variability would ultimately require linking cascades, multiplicative chaos, and multifractals. This would initiate a break with linear stochastic models towards stronger heterogeneity / intermittency, which would in turn lead to a plausible clustering of field and activity fluctuations. The appearance of multifractal phase transitions then becomes possible, considerably amplifying the impact of any action, and would make future transformations fully efficient, effectively imitating the way in which Nature acts. This will be finally illustrated using several examples of so-called Nature Based Solutions (NBS).

Precipitation nowcasting, referring to short-term forecasting ahead for 0 to 6 hours, is an important aspect of many urban meteorological and hydrological studies. This is due to the fact that reliable nowcasting can serve as an early warning of massive flooding and a guide for water-related risk management, making it highly significant in urban areas from a socio-economic perspective. Precipitation exhibits extreme variability over a wide range of space-time scales, so nowcasting is essentially a spatiotemporal sequence forecasting. Convolutional long short-term memory (ConvLSTM) models are frequently used to capture the spatiotemporal correlation, but they often struggle with an issue that produces blurry predictions. Therefore, generative adversarial network (GAN) architecture is employed to achieve more detailed and realistic predictions. The framework of universal multifractal (UM) with only three scale-independent parameters (α, C1, H) is also introduced in the deep learning model to characterize the extreme variability of precipitation. The developed hybrid approach using stochastic models physically based on the cascade paradigm ensures that intermittency is directly taken into account, including in the generation of uncertainty. In addition to the common evaluating metrics, like mean absolute error (MAE), root mean squared error (RMSE), critical success index (CSI), probability of detection (POD), power spectral density (PSD) and UM are also introduced to evaluate nowcasting performance in the spectrum space. This ongoing work is based on the previous research about combining recurrent neural networks with variational mode decomposition and multifractals to predict rainfall time series in Paris area.

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.