Section: Research Program
Methodological developments
In addition to the applicationdriven sections, the team also works on the following theoretical questions. They are clearly connected to the abovementioned scientific issues but do not correspond to a specific application or process.
Stochastic models for extreme events
State of the Art
Maxstable random fields [61], [60], [46], [26], [54] are the natural limit models for spatial maximum data and have spawned a very rich literature. An overview of typical approaches to modelling maxima is due to [28]. Physical interpretation of simulated data from such models can be discussed. An alternative to the maxstable framework are models for threshold exceedances. Processes called GPD processes, which appear as a generalization of the univariate formalism of the high thresholds exceeding a threshold based on the GPD, have been proposed [32], [65]. Strong advantages of these thresholding techniques are their capability to exploit more information from the data and explicitly model the original event data. However, the asymptotic dependence stability in these limiting processes for maximum and threshold exceedance tends to be overly restrictive when asymptotic dependence strength decreases at high levels and may ultimately vanish in the case of asymptotic independence. Such behaviours appear to be characteristic for many realworld data sets such as precipitation fields [27], [64]. This has motivated the development of more flexible dependence models such as maxmixtures of maxstable and asymptotically independent processes [71], [13] for maxima data, and Gaussian scale mixture processes [55], [45] for threshold exceedances. These models can accommodate asymptotic dependence, asymptotic independence and Gaussian dependence with a smooth transition. Extreme events also generally present a temporal dependence [67] . Developing flexible spacetime models for extremes is crucial for characterizing the temporal persistence of extreme events spanning several time steps; such models are important for shortterm prediction in applications such as the forecasting of wind power and for extreme event scenario generators providing inputs to impact models, for instance in hydrology and agriculture. Currently, only few models are available from the statistical literature (see for instance [24], [25], [44]) and remain difficult to interpret.
Four year research objectives
The objective is to extend stateoftheart methodology with respect to three important aspects: 1) adapting wellstudied spatial modelling techniques for extreme events based on asymptotically justified models for threshold exceedances to the spacetime setup; 2) replacing restrictive parametric dependence modelling by semiparametric or nonparametric approaches; 3) proposing more flexible spatial models in terms of asymmetry or in terms of dependence. This means being able to capture the strength of potentially decreasing extremal dependence when moving towards higher values, which requires developing models that allow for socalled asymptotic independence.
People
Gwladys Toulemonde, Fátima Palacios Rodríguez
External collaborations
In a natural way, the Cerise and Fraise project members are the main collaborators for developing and studying new stochastic models for extremes.

More specifically, research with JeanNoel Bacro (IMAG, UM), Carlo Gaetan (DAIS, Italy) and Thomas Opitz (BioSP, MIA, INRA) focuses on relaxing dependence hypothesis.

The asymmetry issue and generalization of some Copulabased models are studied with Julie Carreau (IRD, HydroSciences, UM).
Integrating heterogeneous data
State of the Art
Assuming that a given hydrodynamic models is deemed to perform satisfactorily, this is far from being sufficient for its practical application. Accurate information is required concerning the overall geometry of the area under study and model parametrization is a necessary step towards the operational use. When large areas are considered, data acquisition may turn out prohibitive in terms of cost and time, not to mention the fact that information is sometimes not accessible directly on the field. To give but one example, how can the roughness of an underground sewer pipe be measured? A strategy should be established to benefit from all the possible sources of information in order to gather data into a geographical database, along with confidence indexes.
The assumption is made that even hardly accessible information often exists. This stems from the increasing availability of remotesensing data, to the crowdsourcing of geographical databases, including the inexhaustible source of information provided by the Internet. However, information remains quite fragmented and stored in various formats: images, vector shapes, texts, etc.
This path of research begun with the Cart'Eaux project (20152018), that aims to produce regular and complete mapping of urban wastewater system. Contrary to drinkable water networks, the knowledge of sewer pipe location is not straightforward, even in developed countries. Over the past century, it was common practice for public service providers to install, operate and repair their networks separately [57]. Now local authorities are confronted with the task of combining data produced by different parts, having distinct formats, variable precision and granularity [23].
Four year research objectives
The overall objective of this research line is to develop methodologies to gather various types of data in the aim of producing an accurate mapping of the studied systems for hydrodynamics models.
Concerning wastewater networks, the methodology applied consists in inferring the shape of the network from a partial dataset of manhole covers that can be detected from aerial images [56]. Since manhole covers positions are expected to be known with low accuracy (positional uncertainty, detection errors), a stochastic algorithm is set up to provide a set of probable network geometries [4]. As more information is required for hydraulic modelling than the simple mapping of the network (slopes, diameters, materials, etc.), text mining techniques such as used in [47] are particularly interesting to extract characteristics from data posted on the Web or available through governmental or specific databases. Using an appropriate keyword list, thematic entities are identified and linked to the surrounding spatial and temporal entities in order to ease the burden of data collection. It is clear at this stage that obtaining numerical values on specific pipes will be challenging. Thus, when no information is found, decision rules will be used to assign acceptable numerical values to enable the final hydraulic modelling.
In any case, the confidence associated to each piece of data, be it directly measured or reached from a roundabout route, should be assessed and taken into account in the modelling process. This can be done by generating a set of probable inputs (geometry, boundary conditions, forcing, etc.) yielding simulation results along with the associated uncertainty.
Combining heterogeneous data for a better knowledge of studied systems raises the question of data fusion. What is the reality when contradictory information is collected from different sources? Dealing with spatial information, offset are quite frequent between different geographical data layers; pattern comparison approaches should be developed to judge whether two pieces of information represented by two elements close to each other are in reality identical, complementary, or contradictory.
People
Carole Delenne, Vincent Guinot, Antoine Rousseau, Gwladys Toulemonde
External collaborations
The Cart'Eaux project has been a lever to develop a collaboration with BergerLevrault company and several multidisciplinary collaborations for image treatment (LIRMM), text analysis (LIRMM and TETIS) and network cartography (LISAH, IFSTTAR).

The MeDo project lead by N. Chahinian (HSM) in collaboration with linguists of UMR Praxiling, uses data mining and text analysis approaches to retrieve information on wastewater networks from the Web. Carole Delenne has a slight implication in this project, as domain expert to guide the text annotations and for the uncertainties definition and representation in the mapping of the data collected.

Concerning geographical data fusion for the wastewater network cartograhpy, the Phd thesis of Yassine BelGhaddar has been funded by the French Association of Research and Technology (ANRT) in collaboration with BergerLevrault company and in codirection with A. Begdouri (LSIA Fès, Morocco).
Numerical methods for porosity models
State of the Art
Porositybased shallow water models are governed by hyperbolic systems of conservation laws. The most widespread method used to solve such systems is the finite volume approach. The fluxes are computed by solving Riemann problems at the cell interfaces. This requires that the wave propagation properties stemming from the governing equations be known with sufficient accuracy. Most porosity models, however, are governed by nonstandard hyperbolic systems.
Firstly, the most recently developed DIP models include a momentum source term involving the divergence of the momentum fluxes [42]. This source term is not active in all situations but takes effect only when positive waves are involved [39], [40]. The consequence is a discontinuous flux tensor and discontinuous wave propagation properties. The consequences of this on the existence and uniqueness of solutions to initial value problems (especially the Riemann problem) are not known, or are the consequences on the accuracy of the numerical methods used to solve this new type of equations.
Secondly, most applications of these models involve anisotropic porosity fields [48], [59]. Such anisotropy can be modelled using $2\times 2$ porosity tensors, with principal directions that are not aligned with those of the Riemann problems in two dimensions of space. The solution of such Riemann problems has not been investigated yet. Moreover, the governing equations not being invariant by rotation, their solution on unstructured grids is not straightforward.
Thirdly, the Riemannbased, finite volume solution of the governing equations require that the Riemann problem be solved in the presence of a porosity discontinuity. While recent work [31] has addressed the issue for the single porosity equations, similar work remains to be done for integral and multiple porositybased models.
Four year research objectives
The four year research objectives are the following:

investigate the properties of the analytical solutions of the Riemann problem for a continuous, anisotropic porosity field,

extend the properties of such analytical solutions to discontinuous porosity fields,

derive accurate and CPUefficient approximate Riemann solvers for the solution of the conservation form of the porosity equations.
People
Vincent Guinot, Pascal FinaudGuyot
External collaborations
Owing to the limited staff of the LEMON team, external collaborations will be sought with researchers in applied mathematics. Examples of researchers working in the field are

Minh Le, Saint Venant laboratory, Chatou (France): numerical methods for shallow water flows, experience with the 2D, finite element/finte volumebased Telemac2D system.

M.E. VazquezCendon, Univ. Santiago da Compostela (Spain): finite volume methods for shallow water hydrodynamics and transport, developed Riemann solvers for the single porosity equations.

A. Ferrari, R. Vacondio, S. Dazzi, P. Mignosa, Univ. Parma (Italy): applied mathematics, Riemann solvers for the single porosity equations.

O. Delestre, Univ. NiceSophia Antipolis (France): development of numerical methods for shallow water flows (source term treatment, etc.)

F. Benkhaldoun, Univ. Paris 13 (France): development of Riemann solvers for the porous shallow water equations.
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Inland hydrobiological systems
State of the Art
Water bodies such as lakes or coastal lagoons (possibly connected to the sea) located in high human activity areas are subject to various kinds of stress such as industrial pollution, high water demand or bacterial blooms caused by freshwater overenrichment. For obvious environmental reasons, these water resources have to be protected, hence the need to better understand and possibly control such fragile ecosystems to eventually develop decisionmaking tools. From a modelling point of view, they share a common feature in that they all involve interacting biological and hydrological processes. According to [33], models may be classified into two main types: “minimal dynamic models” and “complex dynamic models”. These two model types do not have the same objectives. While the former are more heuristic and rather depict the likelihood of considered processes, the latter are usually derived from fundamental laws of biochemistry or fluid dynamics. Of course, the latter necessitate much more computational resources than the former. In addition, controlling such complex systems (usually governed by PDEs) is by far more difficult that controlling the simpler ODEdriven command systems.
LEMON has already contributed both to the reduction of PDE models for the simulation of water confinement in coastal lagoons [34], [16] and to the improvement of ODE models in order to account for spaceheterogeneity of bioremediation processes in water resources [14].
Four year research objectives
In collaboration with colleagues from the ANRANSWER project and colleagues from INRA, our ambition is to improve existing models of lagoon/marine ecosystems by integrating both accurate and numerically affordable coupled hydrobiological systems. A major challenge is to find an optimal tradeoff between the level of detail in the description of the ecosystem and the level of complexity in terms of number of parameters (in particular regarding the governing equations for interspecies reactions). The model(s) should be able to reproduce the interannual variability of the observed dynamics of the ecosystem in response to meteorological forcing. This will require the adaptation of hydrodynamics equations to such time scales (reduced/upscaled models such as porosity shallow water models (see Section 3.2.1) will have to be considered) together with the coupling with the ecological models. At short time scales (i.e. the weekly time scale), accurate (but possibly CPUconsuming) 3D hydrodynamic models processes (describing thermal stratification, mixing, current velocity, sediment resuspension, wind waves...) are needed. On the longer term, it is intended to develop reduced models accounting for spatial heterogeneity.
The team will focus on two main application projects in the coming years:

the ANR ANSWER project (20172021, with INRA Montpellier and LEESU) focusing on the cyanobacteria dynamics in lagoons and lakes. A PhD student is coadvised by Antoine Rousseau in collaboration with Céline Casenave (INRA, Montpellier).

the long term collaboration with Alain Rapaport (INRA Montpellier) will continue both on the bioremediation of water resources such as the Tunquen lagoon in Chile and with a new ongoing project on water reuse (converting wastewater into water that can be reused for other purposes such as irrigation of agricultural fields). Several projects are submitted to the ANR and local funding structures in Montpellier.
People
Céline Casenave (INRA Montpellier), Antoine Rousseau, Vincent Guinot, Joseph Luis Kahn Casapia.
External collaborations

ANR ANSWER consortium: Céline Casenave (UMR MISTEA, INRA Montpellier), Brigitte VinçonLeite (UM LEESU, ENPC), JeanFrançois Humbert (UMR IEES, UPMC). ANSWER is a FrenchChinese collaborative project that focuses on the modelling and simulation of eutrophic lake ecosystems to study the impact of anthropogenic environmental changes on the proliferation of cyanobacteria. Worldwide the current environmental situation is preoccupying: mandriven water needs increase, while the quality of the available resources is deteriorating due to pollution of various kinds and to hydric stress. In particular, the eutrophication of lentic ecosystems due to excessive inputs of nutrients (phosphorus and nitrogen) has become a major problem because it promotes cyanobacteria blooms, which disrupt the functioning and the uses of the ecosystems.

A. Rousseau has a long lasting collaboration with Alain Rapaport (UMR MISTEA, INRA Montpellier) and Héctor Ramirez (CMM, Université du Chili).