Contributions en optimisation topologique : extension de la méthode adjointe et applications au traitement d'images

De nos jours, l'optimisation topologique a été largement étudiée en optimisation de structure, problème majeur en conception de systèmes mécaniques pour l'industrie et dans les problèmes inverses avec la détection de défauts et d'inclusions. Ce travail se concentre sur les approches de dérivées topologiques et propose une généralisation plus flexible de cette méthode rendant possible l'investigation de nouvelles applications. Dans une première partie, nous étudions des problèmes classiques en traitement d'images (restauration, inpainting), et exposons une formulation commune à ces problèmes. Nous nous concentrons sur la diffusion anisotrope et considérons un nouveau problème : la super-résolution. Notre approche semble meilleure comparée aux autres méthodes. L'utilisation des dérivées topologiques souffre d'inconvénients : elle est limitée à des problèmes simples, nous ne savons pas comment remplir des trous ... Dans une seconde partie, une nouvelle méthode visant à surmonter ces difficultés est présentée. Cette approche, nommée voûte numérique, est une extension de la méthode adjointe. Ce nouvel outil nous permet de considérer de nouveaux champs d'application et de réaliser de nouvelles investigations théoriques dans le domaine des dérivées topologiques.Nowadays, topology optimization has been extensively studied in structural optimization which is a major interest in the design of mechanical systems in the industry and in inverse problems with the detection of defects or inclusions. This work focuses on the topological derivative approach and proposes a more flexible generalization of this method making it possible to address new applications. In a first part, we study classical image processing problems (restoration, inpainting), and give a common framework to theses problems. We focus on anisotropic diffusion and consider a new problem: super-resolution. Our approach seems to be powerful in comparison with other methods. Topological derivative method has some drawbacks: it is limited to simple problems, we do not know how to fill holes, ... In a second part, to overcome these difficulties, an extension of the adjoint method is presented. Named the numerical vault, it allows us to consider new fields of applications and to explore new theoretical investigations in the area of topological derivative

Water temperature modeling in the Garonne River (France)

Stream water temperature is one of the most important parameters for water quality and ecosystem studies. Temperature can influence many chemical and biological processes and therefore impacts on the living conditions and distribution of aquatic ecosystems. Simplified models such as statistical models can be very useful for practitioners and water resource management. The present study assessed two statistical models – an equilibrium-based model and stochastic autoregressive model with exogenous inputs – in modeling daily mean water temperatures in the Garonne River from 1988 to 2005. The equilibrium temperature-based model is an approach where net heat flux at the water surface is expressed as a simpler form than in traditional deterministic models. The stochastic autoregressive model with exogenous inputs consists of decomposing the water temperature time series into a seasonal component and a short-term component (residual component). The seasonal component was modeled by Fourier series and residuals by a second-order autoregressive process (Markov chain) with use of short-term air temperatures as exogenous input. The models were calibrated using data of the first half of the period 1988–2005 and validated on the second half. Calibration of the models was done using temperatures above 20 ◦C only to ensure better prediction of high temperatures that are currently at stake for the aquatic conditions of the Garonne River, and particularly for freshwater migrating fishes such as Atlantic Salmon (Salmo salar L.). The results obtained for both approaches indicated that both models performed well with an average root mean square error for observed temperatures above 20 ◦C that varied on an annual basis from 0.55 ◦C to 1.72 ◦C on validation, and good predictions of temporal occurrences and durations of three temperature threshold crossings linked to the conditions of migration and survival of Atlantic Salmon

Data assimilation method for real-time flash flood forecasting using a physically based distributed model

The MARINE model (Roux et al, 2011) is a physically based distributed model dedicated to real time flash flood forecasting on small to medium catchments. The infiltration capacity is evaluated by the Green and Ampt equation and the surface runoff calculation is divided into two parts: the land surface flow and the flow in the drainage network both based on kinematic wave hypothesis. In order to take into account rainfall spatial-temporal variability as well as the various behaviours of soil types among the catchment, the model is spatially distributed, which can also help to understand the flood driving processes. The model integrates remote sensing data such as the land coverage map with spatial resolution adapted to hydrological scales. Minimal data requirements for the model are: the Digital Elevation Model describing catchment topography and the location and description of the drainage network. Moreover some parameters are not directly measurable and need to be calibrated. Most of the sources of uncertainties can be propagated thanks to variational method (Castaings et al, 2009) and finally help to determine time dependent uncertainty intervals. This study also investigates the methodology developed for real-time flash flood forecasting using the MARINE model and data assimilation techniques. According to prior sensitivity analyses and calibrations, parameters values were determined as constants or initial guess. Then a data assimilation method called the adjoint state method is used to update some of the most sensitive parameters to improve accuracy of discharges predictions. The forecast errors are evaluated as a function of lead time and discussed from an operational point of view. Multiple strategies in term of updatable parameters set, length of time window, parameters bounds and observation threshold used to trigger the assimilation method are discussed regarding accuracy, robustness and real-time feasibility

Fractional-order diffusion for image reconstruction

International audienceIn this paper, a general framework based on fractional-order partial differential equations allows to solve image reconstruction problems. The algorithm presented in this work combines two previous notions: a fractional derivative implementation by Discrete Fourier Transform and the edge detection by topological gradient. The purpose of the paper is to extend some existing results in image denoising problem with fractional-order diffusion equations and presents new results in image inpainting. The results emphasize the importance of particular fractional-orders

Characterization of process-oriented hydrologic model behavior with temporal sensitivity analysis for flash floods in Mediterranean catchments

This paper presents a detailed analysis of 10 flash flood events in the Mediterranean region using the distributed hydrological model MARINE. Characterizing catchment response during flash flood events may provide new and valuable insight into the dynamics involved for extreme catchment response and their dependency on physiographic properties and flood severity. The main objective of this study is to analyze flash-flood-dedicated hydrologic model sensitivity with a new approach in hydrology, allowing model outputs variance decomposition for temporal patterns of parameter sensitivity analysis. Such approaches enable ranking of uncertainty sources for nonlinear and nonmonotonic mappings with a low computational cost. Hydrologic model and sensitivity analysis are used as learning tools on a large flash flood dataset. With Nash performances above 0.73 on average for this extended set of 10 validation events, the five sensitive parameters of MARINE process-oriented distributed model are analyzed. This contribution shows that soil depth explains more than 80% of model output variance when most hydrographs are peaking. Moreover, the lateral subsurface transfer is responsible for 80% of model variance for some catchment-flood events’ hydrographs during slow-declining limbs. The unexplained variance of model output representing interactions between parameters reveals to be very low during modeled flood peaks and informs that model parsimonious parameterization is appropriate to tackle the problem of flash floods. Interactions observed after model initialization or rainfall intensity peaks incite to improve water partition representation between flow components and initialization itself. This paper gives a practical framework for application of this method to other models, landscapes and climatic conditions, potentially helping to improve processes understanding and representation

Modélisation thermohydraulique d’un tronçon de Garonne en lien avec l’habitat piscicole : approches statistique et déterministe

Les espèces de poissons migrateurs (saumon atlantique, Salmo salar, en particulier) requièrent des conditions thermiques bien spécifiques. Ils sont très sensibles aux températures de l’eau et aux fortes variations estivales. Sur les trente dernières années, l’étude menée sur la Garonne (France) révèle une augmentation des températures estivales associée à un allongement de la durée des périodes chaudes. L’impact de cette modification du régime thermique sur la survie et la reproduction des espèces migratoires est également mis en évidence. Cette étude est menée sur un tronçon de Garonne, situé entre l’amont de Toulouse et l’amont de la retenue deMalause. Ce secteur est fortement touché par cette problématique avec en moyenne 2°C d’écart entre l’amont et l’aval et des températures supérieures à 25°C régulièrement atteintes. Le régime hydrologique de ce tronçon est fortement déficitaire (selon le SDAGE Adour-Garonne), la sensibilité au flux de surface est forte à cause de son lit large et peu profond, les pressions anthropiques sont importantes, ce sont autant de pistes dont l’impact sur le régime thermique est étudié. Une large collection de données hydrologiques et climatiques est exploitée afin de déterminer les processus en jeu dans l’évolution du régime thermique de ce tronçon de fleuve. Des études en tendances et corrélations et des modèles statistiques permettent de mettre en évidence d’une part la relation forte qui existe entre les températures de l’air et les températures de l’eau et d’autre part l’importance des faibles débits durant les périodes estivales. L’estimation des moyennes journalières de température de l’eau à Malause au moyen de modèles statistiques et déterministes donne de bons résultats pour les températures élevées ainsi que pour les franchissements de seuils liés aux conditions de migrations des amphilalins. Enfin un modèle numérique monodimensionnel de résolution de l’équation de transport thermique et des équations de St-Venant est développé. La physique du modèle tant au niveau hydraulique (prise en compte de fortes variabilités de pente, d’ouvrages, etc.) que thermique (apports latéraux, flux de surfaces, flux de conduction avec le lit) permet d’analyser l’évolution des différents flux qui participent au réchauffement du cours d’eau. Une évolution future à l’aide des sorties des modèles de l’IPCC est explorée et des méthodes éventuelles de restauration des conditions de températures favorables pour les espèces piscicoles sont analysées. ABSTRACT : Fish species with strong thermal requirements (i.e. Atlantic salmon) are very sensitive to temperature evolution and particularly to large increases. An investigation conducted on the Garonne River (France) during the last three decades revealed global water warming along with an increase of the high temperature period duration. Large impact of this evolution on the survival and breeding of migrating fish species was also reported. Study was thus conducted on a specific reach of the Garonne River located between the immediate upstream of Toulouse and the upstream of the Malause dam. The issue of water temperature warming is particularly relevant on this reach, with an average increase of 2°C between upstream and downstream and temperatures above 25°C frequently reported. Potential causes are numerous: drastic low-flow regime (quoting SDAGE Adour-Garonne), impacts of surface fluxes that are important due to bed shape (wide and shallow), anthropogenic impacts, etc. Large amount of climatic and hydraulic data are used to make a clear determination of the processes involved in the thermal regime evolution of this reach. Trend and correlation analyses and use of statistical models indicate the strong relation between stream temperature and climate. Low flows also seem to be related to water temperatures during summer periods. Statistic and deterministic models give good results in estimating high daily mean water temperatures (RMSE ranging from 0.99°C to 1.22°C) and predicting water temperatures threshold crossings related to the migrating conditions of Atlantic salmon.Finally, a one-dimensional numerical model that solves both shallow water and thermal equations is developed. Both the formulation of the St-Venant equations (high variability in slope, gates …) and the phenomena taken into account in the water temperature model (lateral influx, surface fluxes, bed conduction …) allows studying the evolution of fluxes driving water temperature evolution. Future evolution of the water temperature at the 2050 horizon is also evaluated using IPCC models output and potential solutions to restore favorable stream temperatures conditions for fishes are analyzed

Flash flood modelling for ungauged catchments

Flash flood is a very intense and quick hydrologic response of a catchment to rainfall. This phenomenon has a high spatial-temporal variability as its generating storm, often hitting small catchments (few km2). Data collected by (Gaume et al. 2009) about 500 flash floods over the last 50 years showed that they could occur everywhere in Europe and more often in the Mediterranean regions, Alpine regions and continental Europe. Given the small spatial-temporal scales and high variability of flash floods, their prediction remains a hard exercise as the necessary data are often scarce. Flash flood prediction on ungauged catchments is one of the challenges of hydrological modelling as defined by (Sivapalan et al. 2003). Several studies have been headed up with the MARINE model (Modélisation de l’Anticipation du Ruissellement et des Inondations pour des évèNements Extrêmes) for the Gard region (France), (Roux et al. 2011), (Castaings et al. 2009). This physically based spatially distributed rainfall runoff model is dedicated to flash flood prediction. The study aims at finding a methodology for flash flood prediction at ungauged locations in the Cévennes-Vivarais region in particular. The regionalization method is based on multiple calibrations on gauged catchments in order to extract model structures (model + parameter values) for each catchment. Several mathematical methods (multiple regressions, transfer functions, krigging. . . ) will then be tested to calculate a regional parameter set. The study also investigates the usability of additional hydrologic indices at different time scales to constrain model predictions from parameters obtained using these indices, and this independently of the model considered. These hydrologic indices gather information on hydrograph shape or catchment dynamic for instance. Results explainingglobal catchments behaviour are expected that way. The spatial-temporal variability of storms is also described through indices and linked with hydrograph shape descriptors in order to constrain model at ungauged locations. In a multi scale point of view, regional characteristics about catchments geomorphology or rainfall fields’ statistics should provide useful insight to find pertinent hydrologic response indices. These considerations with physically based distributed modelling may bring better understanding on flash floods generating mechanisms and catchment responses

Relations between streamflow indices, rainfall characteristics and catchment physical descriptors for flash flood events

Flash flood is a very intense and quick hydrologic response of a catchment to rainfall. This phenomenon has a high spatial-temporal variability as the generating storm often hits small catchments (few km²). Given the small spatial temporal scales and high variability of flash floods, their prediction remains a hard exercise as the necessary data are often scarce. This study investigates the potential of hydrologic indices at different scales to improve understanding of flash floods dynamics and characterize catchment response in a model independent approach. These hydrologic indices gather information on hydrograph shape or catchment dynamic for instance and are useful to examine catchment signature in function of their size. Results show that for middle-size (>100 km²) catchments response shape can be correlated to storm cell position within the catchment contrarily to smaller catchments. In a multi-scale point of view, regional characteristics about catchment geomorphology or rainfall field statistics should provide useful insight to find pertinent hydrologic response indices. The combined use of these indices with a physically-based distributed modelling could facilitate calibration on ungauged catchments

Impact of rainfall spatial variability on Flash Flood Forecasting

According to the United States National Hazard Statistics database, flooding and flash flooding have caused the largest number of deaths of any weather-related phenomenon over the last 30 years (Flash Flood Guidance Improvement Team, 2003). Like the storms that cause them, flash floods are very variable and non-linear phenomena in time and space, with the result that understanding and anticipating flash flood genesis is far from straightforward. In the U.S., the Flash Flood Guidance (FFG) estimates the average number of inches of rainfall for given durations required to produce flash flooding in the indicated county. In Europe, flash flood often occurred on small catchments (approximately 100 km2) and it has been shown that the spatial variability of rainfall has a great impact on the catchment response (LeLay and Saulnier, 2007). Therefore, in this study, based on the Flash flood Guidance method, rainfall spatial variability information is introduced in the threshold estimation. As for FFG, the threshold is the number of millimeters of rainfall required to produce a discharge higher than the discharge corresponding to the first level (yellow) warning of the French flood warning service (SCHAPI: Service Central d’Hydrométéorologie et d’Appui à la Prévision des Inondations). The indexes δ1 and δ2 of Zoccatelli et al. (2010), based on the spatial moments of catchment rainfall are used to characterize the rainfall spatial distribution. Rainfall spatial variability impacts on warning threshold and on hydrological processes are then studied. The spatially distributed hydrological model MARINE (Roux et al., 2011), dedicated to flash flood prediction is forced with synthetic rainfall patterns of different spatial distributions. This allows the determination of a warning threshold diagram: knowing the spatial distribution of the rainfall forecast and therefore the 2 indexes δ1 and δ2, the threshold value is read on the diagram. A warning threshold diagram is built for each studied catchment. The proposed methodology is applied on three Mediterranean catchments often submitted to flash floods. The new forecasting method as well as the Flash Flood Guidance method (uniform rainfall threshold) are tested on 25 flash floods events that had occurred on those catchments. Results show a significant impact of rainfall spatial variability. Indeed, it appears that the uniform rainfall threshold (FFG threshold) always overestimates the observed rainfall threshold. The difference between the FFG threshold and the proposed threshold ranges from 8% to 30%. The proposed methodology allows the calculation of a threshold more representative of the observed one. However, results strongly depend on the related event duration and on the catchment properties. For instance, the impact of the rainfall spatial variability seems to be correlated with the catchment size. According to these results, it seems to be interesting to introduce information on the catchment properties in the threshold calculation

Modelling river discharge at sub-daily time-step: comparison of the performances of the conceptual SWAT model and the process-oriented MARINE model

Due to global change, the frequency of intense rainfall events and consequent flash floods are expected to increase in the next decades across the Mediterranean coastal basins. To date, few distributed models are able to simulate hydrological processes at basin-scale at a reasonable time scale to describe these flash events with accurate details. The MARINE model is one of them: it is a process-oriented fully distributed model operating dynamically at the rainfall event time-scale. Both infiltration and saturation excess are represented along with subsurface, overland and channel flows. It does not describe ground-water processes since the model's purpose is to simulate individual flood events during which ground-water processes are considered negligible. The SWAT model is a conceptual semi-distributed model assuming several simplifications in equations that dynamically simulates above- and below-ground processes. It has been recently upgraded to sub-daily time-step calculations. Considering the 1400 km² Têt Mediterranean river basin (southwestern France) as a case-study, the objective of this study was to assess and compare the performances of these two models when simulating the discharge at sub-daily time-step. We first calibrated the two models based on the same input dataset (topography, land-use, soil classes, and meteorological stations’ grid). We then compared the performances of the two models on a number of selected flood events. This ongoing work will contribute to assess the ability of the SWAT model to simulate discharge at sub-daily time-step