Université De Strasbourg

Thèse Dirigée

M Garambois

Recover, AMU Pierre-André Cr Inrae

M Monnier

M Mosé

IUT Robert Pr

Robert Schumann

Mme Delenne

Mme Soares-Frazão

Sandra

M Flipo

Mme Le Ber

ENGEES, ICUBE Florence Dr

Mme Roux Hélène

OPTIMAL SYNERGY OF MULTI-SOURCE DATA AND HYDRAULIC-HYDROLOGICAL MODELS FOR THE CARTOGRAPHIC MODELING OF COMPLEX HYDROSYSTEMS

Keywords: Assimilation variationnelle, Modèles hydrauliques-hydrologiques, Équations de Saint-Venant, Altimétrie satellite, Observations multi-sourcées, Modèle Multi-dimensionnel vi Variational Data assimilation, Hydraulic-hydrological models, Shallow Water equations, Satellite altimetry, Multi-source observations, Multi-dimensional model viii Assimilation variationnelle, Modèles hydrauliques-hydrologiques, Equations de Saint-Venant, Altimétrie satellite, Observations multi-sourcées, Modèle multi-dimensionnel Variational Data assimilation, Hydraulic-hydrological models, Shallow Water equations, Satellite altimetry, Multi-sourced observations, Multi-dimensional model

This work studies the large scale hydraulic-hydrological modeling of hydrographic networks and the optimal integration of their multi-source observability. The growing wealth of information in data sets brought by hydrometeorology and high resolution remote-sensing, in complement to in situ measurements, is still insufficiently exploited to constrain fine, basin scale hydrodynamic models. Yet, such numerical representations are crucial for the accurate estimation of fluxes and states in hydrographic networks in order to answer major socio-economic issues, like flood prediction, especially in the context of climate change. The physical quantities of interest include inflow hydrographs, water depths, flow velocities and flood extents and durations. Information on such quantities is difficult to obtain, especially for floods over large areas, where multiple rivers are potentially ungauged. Several issues are related to the integration of multi-source data, heterogeneous in space, time and nature, into basin-scale hydraulic-hydrological models.

This manuscript presents modeling approaches based on hydraulic-hydrological models and using variational data assimilation methods with multi-source data. These approaches aim at the synergy of, on one hand, complex numerical models that represent physical quantities and, on the other hand, the heterogeneous observability of hydrodynamic signatures over a hydrographic network. Numerical implementation are carried out in the variational data assimilation and hydrodynamic modeling platform DassFlow.

The first part of this work deals with the variational estimation of uncertain or unknown model parameters (bathymetry-friction couple and lateral inflows), in a 1D hydraulic model at a large spatial scale, from altimetric and optic satellite observations of water surface deformations of river surfaces. The difficulty of the estimation lies in the correlated effect of parameters on hydraulic signatures and on their heterogeneous and sparse observability when compared to the physical scale of propagations and to the length of hydraulic controls. The considered inverse problems are ill-posed and regularization methods are used within the assimilation process. This work introduces source terms for the modeling of lateral inflows in a 1D Saint-Venant model and relies on the analysis of the associated hydraulic signatures. These signatures provide important information for parameter identifiability in the context of the sparse observability of the free surface. It is demonstrated that accurate inferences can be obtained when information provided by observables and a priori estimates are sufficient with regard to the nature, quantity and type of the sought parameters. A large scale model of an anastomosed Amazonian river inflowed by numerous tributaries is built from altimetric and optical satellite data and inflowed with the large scale hydrological model MGB. This effective model, calibrated using historic altimetric data, allows a fair modeling of flood signatures. Their characteristics and the visibility given by wide swath altimetry are analyzed with the help of identifiability maps. Simultaneous inferences of distributed bathymetry, friction and lateral inflow hydrographs are carried out on this model, using noisy wide-swath satellite altimetry. This synthetic observability is that of the upcoming SWOT mission, which features unprecedented spatial coverage of global rivers and sufficient vii revisit time to observe flood wave propagation. Inferences lead to accurate parameter estimations and this study paves the way towards uses of SWOT observations for the regional estimation of fluxes.

The second part of this work presents a new method for the multi-scale hydraulic-hydrological modeling of river networks implemented in the parallel computation and variational assimilation DassFlow framework. A multi-dimensional model is proposed using the full 2D shallow water equations, solved by a single finite volume solver applied to a 1Dlike-2D mesh of a complete hydrographic network. It answers the need for accurate 2D modeling in areas of interests such as floodplains and confluences through 2D meshes. A so-called "1Dlike" meshing strategy is applied to 1D reaches -it can make use of water masks database extracted from high resolution images -and allows hydrodynamic representation of these reaches at the network scale while keeping computational cost relatively low. The 2D hydraulic solver is adapted to handle interfaces between 1Dlike and 2D meshes. The parsimonious hydrological model GR4H, in its differentiable state-space version, is coupled to the multi-dimensional hydraulic model and integrated into the DassFlow assimilation tool chain. The obtained integrated chain is validated on synthetic and real cases. Effective bathymetry and friction allow fitting fine water surface elevations and signal propagation on a large scale 1Dlike model of a reach the Garonne river. Inferences of hydraulic temporal forcings are carried out on a 1Dlike model of the Adour network from in situ observability and lead to estimation of uncertain inflows. A 1Dlike-2D version of the Adour model allows the modeling of high resolution flow in a large network for a low computational cost. Inferences of hydrological parameters are carried out on a synthetic case using downstream hydraulic observable. These results demonstrate the possibility of informational feedback over a hydrographic network in the assimilation tool chain.

Résumé

Ces travaux de thèse s'intéressent à la modélisation hydrologique-hydraulique des réseaux hydrographiques et à l'intégration optimale de données multi-sources. La richesse informative croissante des jeux de données apportés par l'hydrométéorologie et la télédétection à haute résolution en complément des mesures in situ, est encore insuffisamment exploitée pour contraindre des modélisations hydrodynamiques fines à l'échelle des bassins. De telles représentations numériques sont pourtant cruciales pour l'estimation spatio-temporelle précise des états-flux, en vue de répondre à des enjeux socio-économiques majeurs comme la prévision des crues en contexte de changement climatique. Les quantités physiques d'intérêt vont des hydrogrammes de débit amonts et échanges latéraux jusqu'à leur traduction en hauteurs d'eau, vitesses et temps de submersion sur les plaines d'inondation en aval. De telles informations sont difficilement accessibles, en particulier pour les crues sur de larges territoires où de nombreux cours d'eau sont potentiellement non jaugés. Plusieurs problématiques sont reliées à la modélisation numérique hydrologique-hydraulique de bassin et à sa mise en cohérence avec des données multi-sources hétérogènes en espace, en temps et en nature.

Ce manuscrit présente des approches de modélisation des réseaux hydrographiques, basées sur des modèles hydrauliques-hydrologiques couplés et exploitant des méthodes d'assimilation variationnelle de données multi-sources. Ces approches sont basées sur la synergie entre, d'une part des modèles numériques complexes qui représentent des quantités physiques et, d'autre part, l'observabilité hétérogène de signatures hydrodynamiques à l'échelle des réseaux hydrographiques. Les développements des outils numériques sont effectués au sein de la plateforme d'assimilation variationnelle et de modélisation hydrodynamique DassFlow.

La première partie de ce travail porte sur l'estimation variationnelle de paramètres inconnus ou incertains (couple friction-bathymétrie, hydrogrammes latéraux) d'un modèle hydraulique 1D appliqué à large échelle spatiale, à partir d'observations altimétriques et optiques de déformations de la surface libre des rivières. La difficulté de cette estimation vient de l'effet corrélé des paramètres sur les signatures hydrauliques, de leur observabilité hétérogène et éparse en comparaison des échelles physiques des propagations et longueurs de contrôles hydrauliques. Les problèmes inverses considérés sont généralement mal posés et des méthodes de régularisation sont utilisées dans la méthode d'assimilation. Ce travail repose sur l'ajout de termes sources pour la prise en compte d'apports latéraux dans un modèle Saint-Venant 1D et sur l'analyse des signatures hydrauliques associées. Ces signatures apportent une information importante sur l'identifiabilité des paramètres qui les provoquent, en contexte d'observabilité éparse de la surface libre. Dans cette partie, des inférences très précises sont obtenues quand la quantité d'information contenue dans les observables et les informations a priori sont suffisantes au regard de la nature, de la quantité et du type des inconnues. Un modèle grande échelle d'un cours d'eau Amazonien anastomosé et alimenté par de nombreux affluents est construit à partir de données satellitaires optiques et altimétriques et est alimenté par des débits simulés avec le modv èle hydrologique grande échelle MGB. Le modèle effectif calé sur des chroniques altimétriques historiques permet une représentation réaliste des signatures de crues. Leurs caractéristiques fréquentielles et leur visibilité par altimétrie large fauchée SWOT sont analysées à l'aide de cartographies d'identifiabilité. Des inférences simultanées de bathymétrie, friction et hydrogrammes latéraux spatialement distribués sont effectués sur ce modèle, à l'aide d'observations altimétriques large fauchée synthétiques. Ces dernières correspondent à celles qu'apportera la future mission SWOT, qui fournira une couverture spatiale sans précédent des rivières à l'échelle du globe. Cette mission aura une période de revisite suffisamment courte pour observer la propagation d'ondes de crue sur de grands bassins. Cette étude ouvre la voie vers l'utilisation d'observations SWOT pour l'estimation régionale de flux.

La deuxième partie de ce travail présente une nouvelle méthode pour la modélisation et l'assimilation hydraulique-hydrologique multi-échelle de réseaux hydrographiques, implémentée dans l'environnement de calcul parallèle et d'optimisation variationnelle de DassFlow. Un modèle hydraulique multi-dimensionnel est proposé, sur la base des équations de Saint-Venant 2D complètes résolues par un solveur volumes finis appliqué à des maillages 1Dlike-2D de réseaux hydrographiques complets. Il permet de répondre au besoin de modélisation 2D précise dans les zones d'intérêt, comme les plaines d'inondation et les confluences, à travers l'utilisation de maillages 2D localisés. Une stratégie de maillage "1Dlike" des cours principaux des rivières, pouvant utiliser des bases de données de masques d'eau issus d'images haute résolution, permet la modélisation hydrodynamique de ces tronçons à l'échelle du réseau tout en gardant un coût de calcul relativement bas. Le solveur hydraulique 2D est adapté pour gérer les interfaces entre les maillages 1Dlike et 2D. Le modèle hydrologique parcimonieux GR4H, dans sa version "espace-temps" qui est différentiable, est couplé au modèle hydraulique multidimensionnel et inclus à la chaîne d'assimilation DassFlow. La chaîne intégrée ainsi obtenue est validée sur des cas académiques et réels. L'inférence de champs de friction et bathymétries effectives permet de représenter fidèlement des lignes d'eau et la propagation de crues avec un modèle 1Dlike d'un bief de la Garonne. Des inférences de forçages hydrauliques temporels sont effectuées sur un modèle 1Dlike du réseau hydrographique de l'Adour, à partir d'observations in situ et permettent l'estimation de flux incertains. Un modèle 1Dlike-2D de l'Adour permet la modélisation haute résolution d'écoulements dans un grand réseau hydrographique et ses plaines d'inondation pour un coût de calcul relativement faible. Des inférences de paramètres hydrologiques sont effectuées sur un cas test synthétique, à partir d'observables hydrauliques en aval. Ces résultats démontrent la possibilité de remontée d'information dans la chaîne de modélisation sur un réseau hydrographique.

Introduction

In surface hydrology, the accurate estimation of fluxes is a key scientific problem, linked to important socio-economic issues, like flood and drought prediction. Indeed, river discharge is defined as an essential physical variable by the World Meteorological (see Global Climate Observing System Organization, 2011]). Climate change and the potential intensification of the hydrological cycle it implies make this goal even more crucial. To address it, fine, accurate, dynamic and spatially distributed modeling of basin-scale river network states and fluxes are needed. The physical quantities to represent include discharge and water levels, inundation extents and submersion times. Information on these states and fluxes is difficult to obtain at appropriate spatial and temporal scales, especially along large, potentially ungauged, river networks. However, the growth in size and quality of multi-source datasets given by high resolution hydrometeorology and satellite and airborne sensors provides an opportunity that is as of yet underexploited. In complement to in situ observations, this data can be integrated into hydraulic-hydrological models to improve flux prediction capabilities. This PhD thesis studies the hydraulic-hydrological modeling of hydrographic networks and the optimal integration of multi-source observations. This integration consists in the coherent combination of, on one hand, complex numerical flow models and, on the other hand, observations of hydrodynamic signatures in river networks, heterogeneous in nature and in spatiotemporal patterns. In this work, hydraulic models solving the full Shallow Water equations, either 1D or 2D, are used in a variational data assimilation framework. Several issues are related to the hydrological-hydraulic modeling of river networks and to the fusion of heterogeneous multi-source observations. One issue is the relatively sparse observability of the water surface compared to the hydraulic signatures at play. On one hand, in situ data is limited to static stations, that provide data that is dense in time but relatively sparse in space with regards to the scales of signatures related to hydraulic controls (bathymetry-friction and lateral exchanges), and costly data acquisition campaigns, that provide rich but localized information. On the other hand, current satellite observations provide global data, but their revisit time are still relatively large with regards to signal propagation times. In view to enrich observability and to make the most of available data, satellite and in situ data can be combined, which leads to a second issue: their heterogeneity. That is an heterogeneity of spatial and temporal patterns and frequencies, of sensor/product accuracy and an heterogeneity in the type of observed quantities. Satellite radar altimetry provides water surface elevation over relatively large global rivers (over 50 to 100 m wide), while satellite optical sensors may provide river widths and flood extents. Spaceborne radars (Shuttle Radar Topography Mission, SRTM) enabled building a global Digital Elevation Model, while airborne LiDAR (Laser Imaging Detection and Ranging) can provide bathymetry over a region of interest, like a floodplain, with higher accuracy and resolution. In situ flow sensors can provide water depth and local discharge estimation, while fixed or airborne optical sensors, i.e. cameras, may provide 2D surface velocity fields. Airborne measurement of river channels bathymetry remains difficult except for shallow and clear waters via penetrating LiDAR. Therefore most parameters required to model worldwide rivers networks remain poorly observed or not observed at all. Thus, the third issue is the inference of uncertain or unknown river flow model parameters. These include channel parameters, such as bathymetry and friction, distributed inflows but also hydrological parameters for integrated models. Given hydrodynamic flow models and partial flow observability, one faces difficult ill-posed inverse problems and the introduction of a priori knowledge and regularization strategies is necessary.

Another important issue lies in the difficulty to estimate distributed hydrological inflows over a river network. This pertains to the problem of spatially distributed hydrological model calibration and regionalization, that is prediction of discharge at ungauged locations. Considering sparse observability of hydraulic signatures, is it possible to infer semi-distributed hydrological parameters in a hydrological-hydraulic model using a VDA method? In other words, which information feedback is possible from the river network to upstream hydrological components ?

The above issues are studied is this PhD. All numerical developments are implemented in the hydrodynamic modeling and assimilation platform DassFlow.

In Chapter 1, hydraulic and hydrological modeling is presented in the scope of basin-scale hydrographic network observability. In Chapter 2, the identifiabilty of model parameters and temporal forcings is studied on synthetic and real-like 1D models using satellite altimetry. In Chapter 3, a new method for the multi-scale modeling of complex river network with data assimilation is developed. This work is concluded in Chapter 4, which contains work perspectives and the general conclusion.

Chapter 1 This chapter presents the general context and scientific issues studied in this work, as well as the models and tools exploited or expanded upon. Recall that this thesis aims at the improvement of model-data synergy in river network hydraulic-hydrological models through the use of multi-source observations and data assimilation methods. Section 1 presents basin-scale hydraulic and hydrological phenomenons in the scope of their heterogeneous observations through diverse sensing methods. Section 2 presents the mathematical hydraulic models and resolution methods used in the following chapters of this work, along with a summary of state-of-the-art computational hydraulic models. Section 3 introduces hydrological modeling and contains a detailed presentation of two hydrological models used in the following chapters. Considering multi-source river flow observations and hydraulic-hydrological models, the inverse problems of interest are formulated in Section 4 and the variational data assimilation method used in the following chapters is presented.

River networks modeling and remote sensing

Fluvial hydraulics and hydrological variabilities

Water circulation on Earth, in the so-called "great water cycle", represents a dynamic equilibrium. Surface hydrology aims at studying the spatio-temporal variability of continental freshwater storage and fluxes (see e.g. [START_REF] Chow | Handbook of applied hydrology[END_REF]; [START_REF] Dingman | Fluvial hydraulics[END_REF]). In this section, large scale hydrological processes are recalled before focusing on river network flows in a multi-source observability context (in situ, spaceborne and airborne sensors).

Catchment hydrology

Given a river cross-section (XS), one can define a topographical catchment as the upstream geographical area, delineated by its water divide line ("ligne de partage des eaux" in French, see [START_REF] Musy | Hydrologie: Une science de la nature[END_REF]). The amount of freshwater within a catchment is largely driven by climate variability, through effective precipitation that feeds the compartments of hydrosystems (i.e. rivers, lakes, biosphere, aquifers and unsaturated soils [START_REF] Milly | Climate, soil water storage, and the average annual water balance[END_REF], Fig. 1.1).

Freshwater residence times can vary from hours, to days, to thousands years depending on the hydrological compartment (De [START_REF] Marsily | Les eaux continentales[END_REF]). Average residence time is estimated to be in the tens of days for river networks, of a few years in wetlands and upper soil layers, of decades in lakes and of thousands of years for aquifers and glaciers [START_REF] Anctil | Hydrologie, cheminements de l'eau[END_REF]).

Hydrological responses, in terms of water storage and fluxes, result from the complex combination of meteorological signals with catchment properties, involving soil-biosphere-atmosphere couplings, at play in the so-called continental water cycle (see e.g. [START_REF] Oki | The global water cycle[END_REF]). The heterogeneous physical properties, organization and connectivity of hydrological compartments modulate the distribution and dynamics of water flow (see e.g. [START_REF] Flipo | Continental hydrosystem modelling: the concept of nested streamndash;aquifer interfaces[END_REF]; [START_REF] Schuite | Improving the spectral analysis of hydrological signals to efficiently constrain watershed properties[END_REF]).

Complex non-linear and coupled physical processes are at play in the soil-vegetation zone and in the unsaturated superficial soil layers. They determine the partition of water between vertical and lateral flow components such as infiltration, surface and sub-surface runoff and percolation to bedrock/aquifers.

Name Average yearly precipitation (cm) Localization

Equatorial humid 200 Inland and coasts (e.g. Amazon basin) [START_REF] Fabre | Éléments d'hydrologie[END_REF])

Solid and liquid precipitation accounts for most of the water inputs in a catchment (orders of magnitudes of yearly precipitation given by climate types in Table 1.1). A significant part of this precipitation is evaporated from the soil, from open water surfaces or through plant transpiration. The remaining water integrates the surface and sub-surface storage and flows.

Soil, a porous medium, has a limited local infiltration capacity. Part of the infiltrated water remains in the upper soil layers, while another part is percolated deeper, in the aquifer, through the action of gravity. When the local infiltration capacity limit is reached, excess water participates in surface runoff [START_REF] Anctil | Hydrologie, cheminements de l'eau[END_REF]).

Surface runoff descends towards the lowest point of a catchment, forming naturally linked drainage channels (called hydrographic network). Before reaching the oceans, except in the case of endorheic basins such as the Okavango basin in sub-equatorial Africa, freshwater passes through hydrographic networks.

Sub-surface water may interact with the hydrographic network through lateral flows (e.g. [START_REF] Schuite | Improving the spectral analysis of hydrological signals to efficiently constrain watershed properties[END_REF] and references therein) and water exchanges can occur between distinct topographical catchments because of groundwater or human-induced fluxes (e.g. [START_REF] Mul | Identification of groundwater flow systems using geological mapping and chemical spring analysis in South Pare Mountains, Tanzania[END_REF]).

Exchanges between surface, sub-surface and groundwater compartments can be vertical or lateral, depending on compartment levels: during low-flow periods, exchanges are directed towards the river network (a low frequency signal) while in high-flow periods, surface hydrological processes transmit a high frequency flood signal [START_REF] Flipo | Hybrid fitting of a hydrosystem model: Long-term insight into the Beauce aquifer functioning (France)[END_REF]). The propagation of upstream hydrological signals within a river network is studied in fluvial hydraulics. (b) Quantitative representation from Margat and Andréassian [2008]. Exchanges between the main hydrological compartments are represented. Green water is water evaporated from or transpired by plants (from forests, uncultivated fields). Blue water is water involved in surface (roughly 15%) and sub-surface (roughly 85%) flows.

River discharge

River discharge, resulting from the aggregation of upstream and past hydrological processes, is a crucial physical quantity for hydraulic studies and for the characterization of hydrological phenomenons. It has been defined as an essential climate variable by the World Meteorological Organization (WMO) in the Global Climate Observation System (GCOS, see int [2016]). Worldwide rivers discharges span a significant range of scales, as can be seen for the sample rivers presented in Table 1.2.

The largest river discharge module -or average inter-annual flow -on Earth is that of the Amazon River, about 209 000 m 3 /s near its estuary, and its two major upstream tributaries, the Solimoẽs and Negro rivers have a respective module of 103 000 m 3 /s and 28 400 m 3 /s and drainage areas of 2 147 700 km 2 and 279 900 km 2 (see Table 1.2, see also [START_REF] Ouillon | Modélisation mathématique de l'hydrodynamique à surface libre et du transport en suspension de sédiments non cohésifs: aide à l'interprétation d'images spatiales[END_REF]; [START_REF] Filizola | Transfert sédimentaire actuel par les fleuves amazoniens[END_REF] for solid discharges). Such large and hard-to-access basins are difficult to instrument and remote sensing provides an unprecedented observability of large rivers, as discussed later in Section 1. 1.4. Smaller rivers are also observable, such as the Garonne river (France) around Toulouse (see [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200m wide river: Evaluation over the garonne river[END_REF]) which represents the lower limit of observability of current altimetric satellites. Its module at the Gironde estuary is 1000 m 3 /s for a drainage area of 56 000 km 2 .

Let us define river discharge from a fluid kinematics approach, that is a mathematical description of a flow field without accounting for the forces and conditions that create it. Under the usual hypothesis of fluid continuum and considering fluid particles in a Cartesian coordinate system (see e.g. [START_REF] Chassaing | Fluid mechanics. Elements of a first course[END_REF]), the instantaneous volumic discharge through a section A and at a given time is defined as the integration of the flow velocity through this section as:

Q = A -→ u . -→ n dA (1.1)
with -→ u = u x , u y , u z T [m/s] the local velocity vector of the flow and -→ n the outer normal vector to the flow section A m 2 and with Q m 3 /s the volumic discharge. River flows are predominantly free surface flows, which means that the surface of the fluid, i.e. its interface with the atmosphere, is not subject to shear stress. In fluvial hydraulics studies, a XS orthogonal to the main flow vein is commonly chosen. A realistic velocity profile for a free surface channel flow is shown in figure 1.2.

Figure 1.2: Cross-sectional variability of flow velocity, as given by Acoustic Doppler Current Profiler (ADCP) measurements (from [START_REF] Negrel | Estimation du débit des fleuves à partir de mesures satellitaires des variables de surface sans mesures in situ[END_REF]).

River

Basin surface (km 2 ) Discharge module (m 3 /s) Amazon [4] 7 050 000 (at estuary) 209 000

Negro 279 900 [83](at Serrinha station) 28 400 [4] (14% of the Amazon module) Solimoẽs 2 147 700 [83] (at Manacapuru station)

103 000 [4](49% of the Amazon module) Niger 2 117 700 [266] (at estuary) 5 589 [3] (at Lokoja, historical extremes are 27 600 m 3 /s and 500 m 3 /s) Garonne [240] 56 000 (at estuary) 1 000

Rhine [56] 185 000 2 200 (with average high and low flows around 6 000 m 3 /s and 1 000 m 3 /s)

Moselle [56] 27 100 (at Cochem station) 288 (with average high and low flows of 3 740 m 3 /s and 9.7 m 3 /s) Table 1.2: Sample of world rivers observable by satellite altimetry, orders of magnitude of their discharges and drainage basin areas

Fluvial hydraulics

Fluvial hydraulics, also called open channel flows in civil engineering, governs the motion of water in river networks and is an essential discipline to the study of hydrology [START_REF] Dingman | Fluvial hydraulics[END_REF]). Indeed, river networks are generally composed of river channels and floodplains and convey the water fluxes that stem from hydrological processes occurring within a catchment. Fluvial hydraulics is an applicative sub-discipline of fluid dynamics (hydrodynamics or aerodynamics) that studies the motion of fluids (liquids or gas). Fluid dynamics offer a systematic structure based on empirical and semi-empirical laws derived from flow measurements (see e.g. [START_REF] Chassaing | Fluid mechanics. Elements of a first course[END_REF]; [START_REF] Ancey | Hydraulique à surface libre[END_REF]). Its foundational axioms, based on classical mechanics, are conservation laws such as conservation of mass and linear momentum considered in the following (see Section 1.2).

In river flows, diverse physical processes are at play at multiple scales, from very localized structures in boundary layers (e.g. [START_REF] Nikora | Velocity profiles in vegetated open-channel flows: combined effects of multiple mechanisms[END_REF]) to larger scale wave propagations (e.g. [START_REF] Dingman | Fluvial hydraulics[END_REF]; [START_REF]Hydrodynamique de l'environnement[END_REF]). The two main forces at play in open channel flows are gravitational and shear forces. Gravitational force, that is a body force acting on each fluid particle, is responsible for motion, whereas a shear force opposed to motion arises from frictional resistance of the solid boundary with viscous and turbulent effects [START_REF] Kline | The structure of turbulent boundary layers[END_REF]). Another surface force at play is the pressure force. The variabilities of free surface flows arise from the mass balance and forces balance that are reflected in free surface deformations and can be predicted by hydraulic theory (see e.g. [START_REF]Hydrodynamique de l'environnement[END_REF]; [START_REF] Ancey | Hydraulique à surface libre[END_REF]; [START_REF] Dingman | Fluvial hydraulics[END_REF]). The complexity of flows within a river reach can vary depending on lateral exchanges, channel shape and sinuosity, slopes, banks and bottom rugosity, the presence of vegetation, of confluences or diffluences, or of inline (bridges, weirs, logjams) or lateral man-made structures (lateral weirs, dykes), see [START_REF] Graf | Hydraulique fluviale: écoulement et phénomènes de transport dans les canaux à géométrie simple[END_REF]. The observation of river flows variabilities is of great importance for physics and engineering applications and is discussed hereafter.

River networks monitoring

Human societies developed in close relation with water because of irrigation and transportation needs, drinking water supply. Therefore, the observation of river flows has early become a subject of interest. Let us mention historical observations of river flows such as for the Nile River in ancient Egypt (from Arab records starting in the 7 th century, see Toussoun and international de géographie [11: 1925: Le Caire]) or more recent observations of the Rhône River, with records dating up to the 16 th century (see [START_REF] Pichard | Les crues sur le bas Rhône de 1500 à nos jours. Pour une histoire hydro-climatique[END_REF]). These historical measurements, typically of flood peak heights at local sites of importance, have evolved into distributed sensor networks. In the last decades, satellite observations have offered the opportunity to complement these in situ observations. Even more recently, airborne sensors, carried by planes and drones, have further contributed to increase the observability of river networks.

Nowadays, the monitoring of river networks involves a variety of sensors, aimed at observing rivers flows variabilities. In the following, we differentiate between local measurements, that include permanent stations and local remote-sensing, and global satellite observations.

Local measurements

In situ measurements can be divided into: i) permanent local stations gathering temporally dense data (see e.g. [START_REF] Musy | Hydrologie: Une science de la nature[END_REF] for measurement tools and techniques) and ii) impermanent data acquisition campaigns that focus on areas-of-interests over a shorter time period (see e.g. [START_REF] Hauet | Application de la méthode LSPIV pour la mesure de champs de vitesse et de débits de crue sur modèle réduit et en rivière[END_REF]; [START_REF] Altenau | AirSWOT measurements of river water surface elevation and slope: Tanana River, AK[END_REF] and details below).

Local stations provide either water depth measurements (e.g. through limnimetric physical scales or limnigraphic automated sensors, see Fig. 1.4b(a)) or discharge estimation (e.g. through stage-discharges relations, called rating curves (RC), or direct measurement, see Fig. 1.4b(b)), over large time scales and at relatively low time steps. Permanent stations are distributed heterogeneously around the surface of the Earth. In France, a network of limnimetric and current meters (the French Banque HYDRO 1 , Fig. 1.3, left)regroups 3200 current stations for 670 000 km2 of drained area (or 210 km 2 per station on average). This network finds use in hydraulic and hydrological studies but also in real-time flood warning systems such as in the French vigilance system for floods, Vigicrue 2 , and for flash floods, Vigicrue Flash3 . Other regions are less densely gauged, like the Amazon basin (Fig. 1.3, right), with a few hundreds of stations for 7 × 10 6 km 2 of drained area). Overall, permanent station number is in decline worldwide with some regional disparities (see [START_REF] Fekete | The current status of global river discharge monitoring and potential new technologies complementing traditional discharge measurements[END_REF]).

Impermanent observation means include: at-a-section measurements, e.g. through simple helix-based current meters or more advanced Acoustic Doppler Current Profilers (ADCP, see Fig. 1.6) that provide depth and width dependent water velocity measurements (see Fig. 1.5(a)), and longitudinal measurements, e.g. through on-board GPS altimeters that can provide spatially dense WS elevation and WS slope data across a river reach or river network (see Fig. 1.5(b)).

New remote-sensing methods are also deployed at the local scale. Airborne sensors, carried by plane (e.g. [START_REF] Altenau | AirSWOT measurements of river water surface elevation and slope: Tanana River, AK[END_REF]) and, more recently, by drones (e.g. [START_REF] Tauro | Assessment of drone-based surface flow observations[END_REF]; [START_REF] Strelnikova | Drone-based optical measurements of heterogeneous surface velocity fields around fish passages at hydropower dams[END_REF]), bring lower cost and more flexibility to data acquisition. Recent progress in measurement techniques, such as penetrating blue LiDAR operated from low altitude overflight (e.g. [START_REF] Lee | Effect of flying altitude and pulse repetition frequency on laser scanner penetration rate for digital elevation model generation in a tropical forest[END_REF]; [START_REF] Hopkinson | The influence of flying altitude, beam divergence, and pulse repetition frequency on laser pulse return intensity and canopy frequency distribution[END_REF]), provide high resolution and accurate bathymetry of floodplains and relatively shallow main channels. Furthermore, remote non-intrusive optical sensors (i.e. cameras), either fixed in place or mobile, can provide bi-dimensional fields of surface velocities extracted by Large Scale Particle Image Velocimetry (LSPIV) methods, see Le [START_REF] Le Coz | Performance of image-based velocimetry (LSPIV) applied to flash-flood discharge measurements in Mediterranean rivers[END_REF]. For example, [START_REF] Dramais | Advantages of a mobile lspiv method for measuring flood discharges and improving stage-discharge curves[END_REF] presents channel discharge estimations using static cameras.

Local measurements represent a large part of the available observability of rivers. Local in situ bathymetric surveys are crucially needed in river engineering and modeling, for most river channels and especially at deeper locations. In situ stage and flow measurements generally provide temporally dense observation of the water depth variations. In turn, both permanent and impermanent local data sources are necessary to constrain at a station stage-discharge laws (e.g. in [START_REF] Getirana | Hydrological modelling and water balance of the Negro River basin: evaluation based on in situ and spatial altimetry data[END_REF]; Malou et al. [2021] and references therein) or more complex hydraulic models (e.g. in [START_REF] Papanicolaou | Calibration and verification of a 2D hydrodynamic model for simulating flow around emergent bendway weir structures[END_REF]; [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF] and references therein).

(a) Map of river network and Banque HYDRO stations of the Rhin-Sarre territory. 84 gauging stations for an area (in white) of around 1.1 × 10 4 km 2 (or 1310 km 2 per station on average).

(b) Water level gauging stations given by the ANA, CotaOnline, RHN and GDH databases (available from http://www.snirh.gov.br). 82 active stations cover a drainage area around 7 × 10 6 km 2 (or 85 000 km 2 per station on average). Note that only 58% of this area is in Brazil, further upstream observability may be derived from other gauging networks, notably in Peru, Bolivia, Colombia and Venezuela. ). The color heatmap is a cross-sectional flow velocity field. The boat took 28 minutes to cross the 2950 m of river width. Estimated total flow is around 50 900 m 3 /s (for a river module of 103 000 m 3 /s, see Table 1.2).

(b) Top-down representation of an ADCP campaign results. Depth-averaged velocities of ADCP transects at the confluence of Negro and Solimões Rivers [START_REF] Trevethan | Influence of tributary water chemistry on hydrodynamics and fish biogeography about the confluence of Negro and Solimões rivers, Brazil[END_REF]).

Satellite observations

Global observation of the oceans dates back to the mapping of oceans for sailing purposes, through not only coast but also wind and current maps. In 1957, the first satellite launch opened up a new way to observe the Earth, first through orbiting Laser and Doppler sensors for geodesic measurements 4 . In 1978, the first satellite mission aimed at ocean surface observation, notably using nadir altimetry, is launched. Since then, increased efforts to observe the Earth with spaceborne sensors have been made, bringing unprecedented datasets on ocean, atmosphere and continental surfaces.

Recent satellite missions provide an increasing amount of observations related to the global hydrological cycle and basin properties: precipitation, surface soil moisture, surface temperatures, ice sheet surfaces, large scale water storage, oceanic and continental water surfaces elevation and extent, topography and soil occupation (see summary of satellite missions in Table 1.3, see also freely available Sentinel mission imagery in Fig. 1.8). The satellite-based remote sensing of hydraulic and hydrological signatures (see Subsection 1.1.1) can provide a global complement to in situ hydrometric networks (review in [START_REF] Balsamo | Satellite and in situ observations for advancing global earth surface modelling: A review[END_REF]), which are declining in some regions worldwide (see [START_REF] Fekete | The current status of global river discharge monitoring and potential new technologies complementing traditional discharge measurements[END_REF] and references therein). Notably, higher vertical accuracy of altimetric data has enabled a larger observability of the hydrological cycle (e.g. altimetry on rivers down to 50 m in width expected with the forthcoming SWOT mission, see [START_REF] Pavelsky | Assessing the potential global extent of SWOT river discharge observations[END_REF]). Furthermore, advances in high-performance computing have enabled representation of the Earth at unprecedentedly fine scales and exploitation of this new wealthof-information. For example, in [START_REF] Mizielinski | High-resolution global climate modelling: the UPSCALE project, a large-simulation campaign[END_REF], high resolutions satellite data have been used to build a finer global climate model (25 × 25 km 2 tiles, 85 vertical levels, worldwide) run on supercomputers.

In the present work, we focus on radar altimetry as a rich source of information on catchment scale hydrodynamic responses (see relevant satellite missions in Table 1.4, see sample nadir and large swath observability in Fig. 1.7). Initially, satellite altimetry over continental surfaces enabled measurements over ice sheets [START_REF] Ridley | A model of satellite radar altimeter return from ice sheets[END_REF]; [START_REF] Remy | Mapping of the topography of continental ice by inversion of satellite-altimeter data[END_REF]) and large inland water surfaces such as over the the Amazon river [START_REF] Koblinsky | Measurement of river level variations with satellite altimetry[END_REF]). [START_REF] Koblinsky | Measurement of river level variations with satellite altimetry[END_REF] suggests that an observational Root Mean Square Error (RMSE) of 10 cm would allow the monitoring of large river water levels, although measurements at the time (from the Geosat mission) reached only 50 cm in RMSE and had insufficient spatio-temporal coverage. Since then, decades of nadir altimetry (see [START_REF] Frappart | Preliminary results of ENVISAT RA-2-derived water levels validation over the Amazon basin[END_REF]; [START_REF] Birkett | Contribution of the TOPEX NASA Radar Altimeter to the global monitoring of large rivers and wetlands[END_REF]; Da Silva et al.

[2012]; [START_REF] Calmant | Monitoring continental surface waters by satellite altimetry[END_REF][START_REF] Calmant | Principles of Radar Satellite Altimetry for Application on Inland Waters[END_REF]) and imagery (e.g. [START_REF] Allen | Global extent of rivers and streams[END_REF]) of inland waters have been exploited and detection capabilities have been improved: observable river widths were around 1 km in [START_REF] Birkett | Contribution of the TOPEX NASA Radar Altimeter to the global monitoring of large rivers and wetlands[END_REF] (TOPEX/POSEIDON mission) and are currently around 100 to 50 m (Cryosat, Jason, Sentinel and SWOT missions, see Table 1 .4). This enables global observation of many worldwide rivers.

Notably, the upcoming Surface Water and Ocean Topography (SWOT, see [START_REF] Rodríguez | SWOT Surface Water and Ocean Topography Mission (SWOT)[END_REF]; [START_REF] Fu | SWOT: the surface water and ocean topography mission: wide-swath altimetric elevation on Earth[END_REF]) mission (to be launched 2022) will use a new wide-swath altimeter (see Table 1.4 for resolution). This would offer extensive global observations with a relatively dense spatiotemporal sampling and decimetric vertical accuracy (discussed below, see also [START_REF] Rodríguez | Wide-Swath Altimetry: A Review[END_REF][START_REF] Rodríguez | Observing rivers with varying spatial scales[END_REF]). [START_REF] Pavelsky | Assessing the potential global extent of SWOT river discharge observations[END_REF] estimates that observing 50 m wide rivers would allow dis-charge estimation on more than 60% of catchments with an area of 10 000 km 2 and above. In the case of the Amazon basin SWOT observations enables to depict hydrological processes such as regional rainfall-evapotranspiration patterns, inter-basin fluxes or floodplain storage (see review in [START_REF] Fassoni-Andrade | Amazon hydrology from space: scientific advances and future challenges[END_REF]).

Thus, multi-satellite observations contain important information on WS deformation observations, from a combination of diverse spatio-temporal patterns and resolutions. An aggregated multi-satellite product, especially including the most recent and upcoming missions, would provide extensive observations of global hydrodynamic signatures to be analyzed, learned and assimilated into fine river network models. Spatially distributed observations of river surfaces deformations can provide interesting information on local geometric-friction parameters (see [START_REF] Hostache | Assimilation of spatially distributed water levels into a shallow-water flood model. Part II: Use of a remote sensing image of Mosel River[END_REF]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF] and references therein), while temporally dense observations may carry information on the local hydrodynamic response of the global hydrographic network (see [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] and references therein). The latter scientific issue is studied in Chapter 2. Table 1.3: Main satellite missions used in hydrology (adapted from Observatoire Midi-Pyrénées (OMP)).

Observed variable

Mission

Instruments

Spatial resolution

Revisit period

Precipitation

Signal scales and measurement error

We define hydraulic signatures as information on hydrological cycle components carried by water surface deformations. From a hydraulic modeling perspective, model state variables carry the hydraulic signature of model parameters and temporal forcings (see Section 1.4). To inform on hydraulic signatures, observations of water surface deformations must have sufficient i) accuracy and ii) spatial and temporal sampling, with regards to the phenomenons at play.

In terms of accuracy, satellite altimetry remains less accurate than most in situ measurements. For spaceborne sensor products, vertical altimetric uncertainties range in the tens of centimeters. The SWOT mission will provide data over its 3 years planned mission lifetime, over a 120 km large swaths, and its requirements are a 10 to 25 cm vertical accuracy and down to 1.7 cm/km slope accuracy [START_REF] Rodríguez | SWOT Surface Water and Ocean Topography Mission (SWOT)[END_REF]). To achieve this accuracy, wide swath data will be treated and averaged over large surfaces, i.e. accuracy may be lower or observations may be less dense in smaller river. For in situ station measurements of water depth, uncertainties tend to be under 10 cm, although in situ network quality may vary globally. [START_REF] Lang | Incertitudes sur les débits de crue[END_REF] estimates that piezoresistive (resp. bubble-based) sensors, used in the French observation network, have an accuracy of 0.1 % (resp. 0.3 %) of their total range, which translates to less than decimetric accuracy. In situ sensors tend to be sensitive to sediment load and water and air temperature, whereas radar sensors are insensitive to temperature.

In terms of hydraulic signature temporal scales, Allen et al. [2018] estimates that the global median flood wave propagation time across a network is around 6 days, with a majority of networks having wave propagation time of less than a day. Allen et al. [2018] uses simple geomorphological considerations and approximates the flood wave to a kinematic wave. Temporally dense data from in situ stations would provide good local observability of such signatures. Missions with relatively low revisit time, like SWOT (1 to 4 pass every 21 days "repeat" cycle depending on latitude, see also Fig. 1.9), would provide data relevant to the scale of river discharge variations. For example, regional flood wave propagations, that can take place over the course of a week or more over large basins, would be observable.

Then arises the question of the spatial dimension of the observed signatures. Several spatial scales can be considered in the study of fluvial hydraulics, from the very local to the regional scale:

• Near the flow boundary layer, the localized impact of basal friction (from banks, bottom friction) or drag effects on obstacles, can be seen in the water surface. Satellite observations generally do not yield meaningful information on these phenomenons. These localized phenomenons are studied in laboratory-scale experiment (e.g. [START_REF] Nikora | Velocity distribution in the roughness layer of rough-bed flows[END_REF]).

• At the local scale, the longitudinal scale of phenomenons is lower than their lateral scale. Phenomenons include local backwater effects, small singularities of section or slope and change of flow regimes. High resolution satellite sensors and some impermanent measurements, e.g. using LSPIV and ADCP, may provide observability at this scale.

• At the reach scale, the longitudinal scale of phenomenons is higher or equal to their lateral scale. For this scale, we define fluvial hydraulic controls (HC) as characterized by a maximal deviation of the water depth from the normal depth following [START_REF] Montazem | Wavelet-based river segmentation using hydraulic controlpreserving water surface elevation profile properties[END_REF]. They can be exerted by control sections (e.g. reduced width, in-line structure, confluence) over up to hundreds of kilometers upstream (see Allen andPavelsky [2015, 2018]). Multi-satellite altimetry offers coverage at this scale and, depending on station density, permanent in situ networks may too.

• At the network scale, hydrological and hydrogeological phenomenons impact water surface deformations.

The observability of hydrodynamic signatures given by current available sensors networks, that is in situ stations and campaigns and satellite sensors, is extensive, yet heterogeneous. The combination of observations from satellite and local measurements, each providing specific and crucial information at the basin scale, should be strived for with the goal to enable the estimation of hydraulic parameters and hydrological fluxes. This thesis work aims at proposing methods to achieve this synergetic combination. Note that the number of passes per cycle period varies based on latitude and satellite trajectory.

Mission

Figure 1.9: Spatio-temporal scale and satellite observability of hydrological cycle components (see [START_REF] Tavakolifar | Evaluation of climate change impacts on extreme rainfall events characteristics using a synoptic weather typing-based daily precipitation downscaling model[END_REF], adapted from Uhlemann [2013]; [START_REF] Hirschboeck | Flood hydroclimatology[END_REF]). In green, spatio-temporal domain of hydrological phenomenons. In blue, river network and floodplain domain of variability. In red, orange and purple, observation windows of the upcoming SWOT wide-swath altimetry mission and of the Jason and Sentinel-3 nadir altimetry missions.

Hydraulic models

In this work, we use hydraulic models as a mean to represent flow states in river networks. They are a simplified representation of real flow physics and consist in partial differential equations describing the motion of water in a free surface flow. More precisely, shallow water equations (SWE) are considered. The 2D SWE equations (Eq. 1.6, presented below) are obtained by vertical integration of the 3D Navier-Stokes equations which are based on the principles of mass and momentum conservation applied to a volume of fluid (see e.g. [START_REF]Hydrodynamique de l'environnement[END_REF]). The 1D SWE equations (Eq. 1.2, presented below) are obtained from integration of the 3D Navier-Stokes equations over a flow section. Both versions are widely used in fluvial hydraulics; their demonstration from the Navier-Stokes equations is presented in Appendix A. Classical hypotheses for establishing those flow models are (see e.g. [START_REF] Chow | Open-channel Hydraulics[END_REF]; [START_REF] Carlier | Hydraulique générale et appliquée[END_REF]; [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF]; [START_REF] Ancey | Hydraulique à surface libre[END_REF]):

• The fluid is of constant volumic mass ρ, which implies it is incompressible and that no sediment transport is considered.

• The flow is gradually varied, which implies that the flow surface is close to a plane and varies slowly in space, and that there is no sharp change in the flow section. Thus, flow lines are parallel to each other and close to parallel to the river bottom, and vertical velocities are negligible. This is also called the long wave assumption, as it means that with regards to the water depth, signal wavelength is high [START_REF] Boussinesq | Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire[END_REF]).

• The bottom slope (and thus the water surface slope) is low, such that the water depth can be measured as perpendicular to the river bottom.

Figure 1.10: Proposed modeling method for flood modeling using a 1D model (from [START_REF] Tayefi | A comparison of one-and two-dimensional approaches to modelling flood inundation over complex upland floodplains[END_REF]). A 1D river channel (modeled with HEC-RAS) is laterally coupled to a series of interconnected storage cells through weir laws. Storage cells are linked between themselves through transfer laws as well. The article compares a computationally costly 2D approach to adapted, less computationally costly 1D methods for flood modeling.

Furthermore, since the Navier-Stokes equations are obtained far-from-everything, i.e. outside of the influence of liquid domain borders and without interaction between fluid layers, it does not account for shear stress and turbulence [START_REF] Dingman | Fluvial hydraulics[END_REF]). Thus, to account for this important phenomenon in fluvial hydraulics applications, a frictional source term is added and classically given by the empirical Manning-Strickler formulation established for uniform flows (or other similar variants of the original analytical Darcy-Weisbach formula). As an empirical relation, the friction source term accounts for all sources of momentum loss that are not specifically modeled (e.g. through an additional drag force term). Therefore, friction coefficient values may differ between SW models.

Hydraulic models are physically-based, deterministic and spatially distributed. The resolution of a hydrodynamic model requires data to describe river geometry and friction, but also to set boundary conditions for upstream, lateral and downstream flow conditions. The amount of data required depends on the modeling goals and the underlying hypotheses. Examples of applications taken from current bibliography include flood modeling with a 1D hydraulic model and connected lateral storage cells [START_REF] Tayefi | A comparison of one-and two-dimensional approaches to modelling flood inundation over complex upland floodplains[END_REF], Fig. 1.10) or with a 2D model [START_REF] Mulatu | Alteration of the Fogera Plain flood regime due to Ribb Dam construction, Upper Blue Nile Basin, Ethiopia[END_REF], Fig. 1.11(a)) and 2D modeling of within-banks flows with adaptative mesh for morphodynamic models [START_REF] Langendoen | Improved numerical modeling of morphodynamics of rivers with steep banks[END_REF], Fig. 1.11(b)).

(a) Simulated flooding extents for different events using a 2D model with a Cartesian grid (from [START_REF] Mulatu | Alteration of the Fogera Plain flood regime due to Ribb Dam construction, Upper Blue Nile Basin, Ethiopia[END_REF]). This recent applied engineering article investigates the impact of a dam project on flooding events using a weak coupling of the hydrological model HEC-HMS and the 2D solver of the hydraulic model HEC-RAS. [START_REF] Langendoen | Improved numerical modeling of morphodynamics of rivers with steep banks[END_REF]). The article presents numerical methods for adaptative mesh in morphodynamic models using Telemac-Mascaret. Figure 1.11: Hydraulic models: 2D hydraulic models used for river reach and floodplain modeling (2) This section presents the 1D and 2D SWE (see demonstration in Appendix A), as well as an overview of numerical schemes used to solve these equations in the current work. The section also presents some widely used hydraulic models and discusses their design, intended uses and the integration of data assimilation methods.

Mathematical flow models

Mathematical flow models describe movement in a continuous medium. In this work, the medium is an incompressible Newtonian fluid modeled from an Eulerian perspective. The fundamental equations are the 3D Navier-Stokes equations that stem from applying conservation laws of mass and linear momentum to an infinitesimally small fluid volume [START_REF] Chassaing | Fluid mechanics. Elements of a first course[END_REF]; [START_REF]Hydrodynamique de l'environnement[END_REF]; [START_REF] Dingman | Fluvial hydraulics[END_REF], see Fig. A.1 in Appendix A). In fluvial hydraulics, these equations integrated over water depth to lead to a 2D formulation (Eq. 1.6), and to the 1D formulation (Eq. 1.2) when integrated over a XS. Both integrated forms are known as Shallow Water equations. Adhémar Barré de Saint-Venant, a French physicist and mathematician, was the first to mathematically describe quasi-monodimensional flows in 1871 (Barré de Saint-Venant [1871]), from a heuristic perspective. Hence 1D SW equations are also called Saint-Venant equations.

This section presents the classically used 1D and 2D SW equations, models and the resolution methods performed in the DassFlow computational platform that is used in this work.

Rivers shape and notations

Let us define model parameters and state variables in 1D and 2D frameworks. Let t denote the time and x the spatial location.

In 1D models, x ∈ R denotes a curvilinear abscissa on a filar network representation. River reaches are represented as a series of XS (defined over separate y axes, see Fig. 1.12(b)), where XS profiles b (x, y) and friction parameters are defined. The water depth h (x, t) is given by Z (x, t)b (x), where b (x) = min y b (x, y) . The top width w (x, t) is given by the position of the banks and the water depth at t. The state variables are A (x, t) = y D (x,t) y G (x,t) Z(x,y,t) b(x,y) dzdy m 2 , the flow section, and Q (x, t) = y D (x,t) y G (x,t) Z(x,y,t) b(x,y) u x dzdy m 3 /s , the flow crossing A (see Fig. 1.12(a)). Note that Q /A = U, where U is the section-averaged flow velocity.

In 2D models, x ∈ R 2 denotes the spatial coordinates. The state variables are the water depth h (x, y, t) [m] and the depth-averaged velocity u = (u, v) T [m/s]. They are evaluated at each model cell, see Fig. 1.12(b)).

(c) Left: Longitudinal and cross-sectional view. q l is a lateral inflow, see e.g. Eq. (1.2). y G and y D denote the position in a XS of the left and right banks respectively. z re f is the reference altitude. Right: Top-down view. In 1D models, the curvature is neglected, hence Q (u x ).

1D Saint-Venant equations

We consider a 1D river domain Ω 1D ⊂ R and denote by t ∈ ]0, T] the physical time. Let A (x, t) m 2 be the flow cross sectional area and Q (x, t) m 3 /s the discharge such that Q = U A with U (x, t) defined as the longitudinal XS averaged velocity [m/s] (see Fig. 1.12(a)). The 1D SWE or Saint-Venant equations in (A, Q) variables with lateral flows writes as follows (see also Appendix A):

       ∂A ∂t + ∂Q ∂x = q l ∂Q ∂t + ∂ ∂x Q 2 A + gA ∂Z ∂x = -gAS f + Uq l (1.2) (1) (2) (3) (4) (5)
where Z (x, t) is the WS elevation [m] and Z = (b + h) with b (x) the river bed level [m] and h (x, t) [m] the water depth, g m/s 2 is the gravity magnitude and q l (x, t) is the lineic lateral discharge m 2 /s . A description of the five momentum equation terms is made at the end of this subsection.

The non-linear friction term S f is classically parameterized with the empirical Manning-Strickler law established for uniform flows:

S f = |Q| Q K 2 A 2 R 4/3 h (1.3)
with K the Strickler friction coefficient in m 1/3 /s , R h (x, t) = A /P h [m] the hydraulic radius and P h (x, t) [m] the wetted perimeter, .

In order to deal with the non conservative term gA ∂b ∂x , as proposed in Vila [1986b], one can introduce a "relative pressure term" P such that:

P x, S, t = g h(x,t) 0 h -h wd h (1.4)
with S and w respectively corresponding to the flow area and width at depth h ∈ [0, h]. As done e.g. in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF], without lateral flows, the 1D SWE in (A, Q) variables can be written as:

∂ ∂t U + ∂ ∂x F (U) = S(U) U = A Q , F(U) =   Q Q 2 A + P   , S(U) =   0 g h 0 h -h ∂ w ∂x d h -gA ∂b ∂x -gAS f   (1.5)
As shown in [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF], the jacobian matrix of F(U) has two eigenvalues U + c and Uc, with U = Q /S and c = ∂P ∂S . Given that A is also strictly positive, the system Eq. (1.5) is strictly hyperbolic.

In order to obtain a well-posed problem, it is necessary to add initial and boundary conditions. Upstream, lateral and downstream boundary conditions are imposed on flow state variables depending on flow regimes and detailed in the following case studies. For the fluvial flows of interest, inflow hydrographs Q in (t) and q l,d (t) at d ∈ [1..N d ] are classically imposed respectively upstream of the river domain and at known injection cells along the river domain using the main channel flow speed. Let us define the Froude number Fr = U /c = Q 2 w / √ gA 3

comparing the average flow velocity U to wave celerity c. The initial conditions are set as the steady state backwater curve profile

Z 0 (x) = Z Q in (t 0 ) , q l,1..N d (t 0 )
Signification of the momentum conservation equation terms (see e.g. [START_REF] Ancey | Hydraulique à surface libre[END_REF]):

(1) Local acceleration term: describes velocity change over time.

(2) Convective acceleration term: describes velocity change over space and accounts for heterogeneous velocity speeds. It allows to model progressive hydrograph skewness as the flood wave is propagated in the network and hysteresis behavior in h-Q relations. Nullifying this term transforms the SWE into the diffusive wave equations.

Terms ( 1) and (2) constitute the "inertial" term and are omitted in some simplified hydraulic models.

(3) Pressure gradient term: accounts for heterogeneous water depths and allows to model wave attenuation as the flood wave is propagated in the network. Nullifying this term transforms the SWE into a kinematic wave equation.

(4) Friction source term: accounts for momentum loss due to all sources of momentum dissipation (bank and bottom friction, eddies, obstacles, ...). It can be complemented by additional terms (e.g. a drag force source term).

(5) Injected flow momentum term: this additional term accounts for the momentum added by the injection of a lateral flow.

2D Shallow Water equations

We consider a 2D river domain Ω 2D ⊂ R 2 and denote by t ∈ ]0, T] the physical time. The 2D SWE including the Manning-Strickler friction term in their conservative form write as follows (see e.g. [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF]; [START_REF] Ancey | Hydraulique à surface libre[END_REF] and Appendix A):

∂ ∂t U + ∂ ∂x F (U) + ∂ ∂y G (U) = S g (U) + S f (U) U =   h hu hv   , F (U) =     hu hu 2 + gh 2 2 huv     , G (U) =     hv huv hv 2 + gh 2 2     , S g (U) = 0 -gh∇b , S f (U) =   0 -g n 2 u h 1/3 u   (1.6)
with h the water depth [m] and u = (u, v) T the depth-averaged velocity [m/s] being the flow state variables (see Fig. 1.12(b)). The flow model parameters are g m/s 2 , the gravity magnitude, b [m], the bed elevation, and n s/m 1/3 , the Manning-Strickler friction coefficient. F (U) is the flux of the variable U, S g (U) is the gravitational source term, S f (U) is the mass and friction source term.

Resolution methods

1D SW model resolution with DassFlow1D

As a matter of facts, the analytical resolution of the set of non linear partial differential equation (PDE) Eq. (1.2) or Eq. (1.5) with initial and boundary conditions is not possible. The approximation of solutions is classically performed using a numerical resolution method based on spatio-temporal discretization of the river and temporal domain. Classical initial and boundary conditions adapted to real cases are considered in numerical resolution methods in DassFlow platform (see [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]; [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]). In this work, the numerical resolution of the 1D SW equations (Eq. 1.2) is performed with a classical semi-implicit finite difference scheme (a Preissmann scheme, see e.g. [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF]). The 1D SW equations (Eq. 1.5) are solved with an explicit finite volume scheme (a HLL scheme, for Harten-Lax-van Leer, see [START_REF] Harten | On upstream differencing and godunov-type schemes for hyperbolic conservation laws[END_REF]; [START_REF] Toro | The HLL and HLLC Riemann solvers[END_REF]). Solvers are used on a regular grid of spacing ∆x and with a time step ∆t, either fixed or adaptative. Both are implemented into the computational software DassFlow [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]). Source terms corresponding to lateral flows have been implemented in the present work in DassFlow1D, for HLL and Preissmann schemes. The detailed Preissmann scheme discretization of 1D SW model with lateral flows, is given in Appendix B. The addition of a lateral inflow source term is not documented here.

2D SW model resolution with DassFlow2D

In DassFlow2D (see [START_REF] Couderc | User and Developer Guide[END_REF]), finite volume Godunov-type schemes [START_REF] Godunov | Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics[END_REF]) with well-balancing properties are used to solve the SWE. A classical first order Euler explicit time-stepping and an Implicit-Explicit (IMEX) Runge-Kutta time-stepping scheme with global second order accuracy are used with a HLLC Riemann solver. The solver is an improvement over the HLL scheme from [START_REF] Harten | On upstream differencing and godunov-type schemes for hyperbolic conservation laws[END_REF] using a three-wave model for flux calculation, see [START_REF] Toro | The HLL and HLLC Riemann solvers[END_REF]. The second order scheme is notably accurate and robust in wet/dry front propagation scenarios (see [START_REF] Couderc | User and Developer Guide[END_REF] for more details).

Remark that given a discretization of the computational domain Ω 2D ⊂ R 2 , finite difference schemes solve for variables at a spatial point, while finite volume schemes solve for averaged variables over a cell. Explicit schemes depend on variables at the previous time-step, whereas implicit schemes depend on current and previous time-steps.

Over a given cell K of area m K , consider the piece-wise constant approximation U K = 1 m K K UdK. Recall that the finite volume approach applied to the homogeneous part of the hyperbolic system of equations 1.6 -that is without the source terms S g and S f -writes as follows:

U n+1 K = U n K - ∆t n m K ∑ e∈∂K m e F e U n K,i , U n K e , n e i ,K (1.7)
where U n K and U n+1 K are the piece-wise constant approximations of U at time t n and t n+1 (with t n+1 = t n + ∆t n ), F e stands for Riemann fluxes through each edge e of the border ∂K of the cell K, with each adjacent cell K e . The length of edge e is m e and n e,K is the unit normal to e oriented from K to K e .

Source term treatment During a computational time step, the full conservative SW equations are solved first, independently from non-conservative source terms in S = S g + S f . Then, source terms are handled by a two-step splitting algorithm. This can be written as:

     Ūn+1 i = U n i -∆t n F n i+1/2 -F n i-1/2 ∆x i U n+1 i = Ūn+1 i + ∆t n S f Ūn+1 i (1.8)
with Ūn+1

i an intermediate value of the computed variables and ∆t the computational time step. This time-splitting method is implemented in first and second order schemes in Dass-Flow2D (see [START_REF] Couderc | User and Developer Guide[END_REF]). Details on the 2D solver and on friction source term implicitation are provided in Subsection 3.A.

Computational hydraulic models

Computational hydraulic models are common tools, that include numerical methods required to solve the flow models detailed in Subsection 1.2.1. The following paragraphs presents some widely-used hydraulic models, distinguished between 1D and 2D approaches, numerical schemes and resolution methods, solved hydraulic equations and applied data assimilation methods.

1D numerical hydraulic models are widely used in river engineering applications but also in water level forecasting since they require relatively few data and are not computationally costly. 2D models generally require more information, like fine 2D bathymetry, and their computational cost is higher. They are used for local and detailed hydraulic studies where estimating dynamics 2D flow patterns is part of the modeling goals, e.g. dynamic flood mapping in urban areas.

Simplified 2D hydraulic models, that are less computationally costly but also have more limited applicability, are also used for larger scale applications such as regional flood modeling. Some existing 1D and 2D hydraulic models are presented in Table 3.1 along with a description of their main features in terms of model, numerical scheme, and in some cases optimization and assimilation methods. Multi-physics features such as sediment transport and erosion-deposition in river channels (e.g. [START_REF] Langendoen | Improved numerical modeling of morphodynamics of rivers with steep banks[END_REF]; [START_REF] Gibson | A physically-based channelmodeling framework integrating HEC-RAS sediment transport capabilities and the USDA-ARS bank-stability and toe-erosion model (BSTEM)[END_REF]) are out of scope here.

1D hydraulic models enable to simulate meaningful spatio-temporal variabilities of flow sections/lines and discharge, from the reach scale to the river network scale, see e.g. [START_REF] Vozinaki | Comparing 1D and combined 1D/2D hydraulic simulations using high-resolution topographic data: A case study of the Koiliaris basin, Greece[END_REF] with HEC-RAS, [START_REF] Barthélémy | On operational flood forecasting system involving 1D/2D coupled hydraulic model and data assimilation[END_REF] with Mascaret or [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF]; Malou and Monnier [2021] with DassFlow1D. Moreover, effective parameterizations enable to account for the effect of various hydraulic structures (such as frontal or lateral weirs, culverts, ...) or for lateral exchanges of mass with floodplains via "casiers" (see [START_REF] Brunner | HEC-RAS River Analysis System[END_REF] and references therein) or even mass and momentum exchange with effective floodplain flows (e.g. ISM model, [START_REF] Proust | Nonuniform flow in compound channel: A 1-D method for assessing water level and discharge distribution[END_REF] and references therein, see also [START_REF] Bousmar | Momentum transfer for practical flow computation in compound channels[END_REF]). Among reference 1D models, one can cite the widely used HEC-RAS [START_REF] Brunner | HEC-RAS River Analysis System[END_REF]) or Mike11 [START_REF] Havnø | Mike 11-a generalized river modelling package[END_REF]) models. Note that the unsteady solver of HEC-RAS uses a locally inertial simplification (LPI, see [START_REF] Fread | An LPI numerical implicit solution for unsteady mixedflow simulation[END_REF]) and a Preissmann numerical scheme (see [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF] and references therein). A full 1D hydraulic model is solved in MAGE (Preissmann scheme, [START_REF] Guertault | One-dimensional modeling of suspended sediment dynamics in dam reservoirs[END_REF]), Mascaret (HLL scheme, see [START_REF] Goutal | A finite volume solver for 1D shallow-water equations applied to an actual river[END_REF]) and DassFlow1D (Preissmann and HLL schemes, see [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]). Note that in DassFlow1D, an accurate finite volume scheme is implemented using a "pressure term" pre-balancing in the continuous model (Eq. 1.5) as proposed by Vila [1986b], next used by [START_REF] Goutal | A finite volume solver for 1D shallow-water equations applied to an actual river[END_REF].

Full 2D SW numerical models enable to simulate high resolution dynamic flooding of floodplains (e.g. [START_REF] Finaud-Guyot | Macroscopic flood modelling: Taking into account directional flows and main channelfloodplain transfer[END_REF]; [START_REF] Galland | Telemac: A new numerical model for solving shallow water equations[END_REF] and references therein), including complex urban flooding (SW2D, see [START_REF] Guinot | Dual integral porosity shallow water model for urban flood modelling[END_REF]; [START_REF] Steinstraesser | SW2D-LEMON: a new software for upscaled shallow water modeling[END_REF]), but also recently at the scale of a river network of a medium sized catchment of 808 km 2 in [START_REF] Nguyen | A high resolution coupled hydrologic-hydraulic model (HiResFlood-UCI) for flash flood modeling[END_REF]. In [START_REF] Fleischmann | Trade-offs between 1-D and 2-D regional river hydrodynamic models[END_REF] with the MGB model or in [START_REF] Grimaldi | Effective representation of river geometry in hydraulic flood forecast models[END_REF]; [START_REF] Uhe | Model cascade from meteorological drivers to river flood hazard: flood-cascade v1.0[END_REF] with LISFLOOD-FP, raster based inundation modeling over very large domains is performed with a simple 2D storage cell inundation model obtained from 1D inertial model [START_REF] Bates | A simple inertial formulation of the shallow water equations for efficient two-dimensional flood inundation modelling[END_REF] following Hunter et al. [2008], recently adapted to unstructured meshes in [START_REF] Sridharan | Local-inertial shallow water model on unstructured triangular grids[END_REF]). In [START_REF] Hocini | Performance of automated flood inundation mapping methods in a context of flash floods: a comparison of three methods based either on the Height Above Nearest Drainage (HAND) concept, or on 1D/2D shallow water equations[END_REF], an original 2D hydraulic modeling approach proposed by [START_REF] Davy | A precipiton method to calculate river hydrodynamics, with applications to flood prediction, landscape evolution models, and braiding instabilities[END_REF] and based on "precipiton" is used to compute steady inundation maps of various return periods at high resolution (5 m) for river networks and floodplains at catchment scale (several thousands of square kilometers, up to 5050 km 2 ). In [START_REF] Nguyen | A high resolution coupled hydrologic-hydraulic model (HiResFlood-UCI) for flash flood modeling[END_REF], a full unsteady 2D hydraulic model (BreZo, Begnudelli and Sanders [2006]) is applied at relatively high resolution (10 or 30 m) in the river network and floodplains on a 808 km 2 catchment. One can also cite the interesting macroscopic approaches inspired by flows in porous media (see [START_REF] Guinot | Dual integral porosity shallow water model for urban flood modelling[END_REF] and references therein), with a reduced computational cost compared to high resolution full 2D models, though it is still faced with the issue of sub-grid parameters estimation to approximate flows physics (either fine 2D or real flows).

In order to combine local accuracy and computational efficiency, a combination of full 1D and 2D hydraulic models is appropriate for simulating a basin-scale network in a way that is both practical and adequately accurate. Methods for coupling models of different dimensions have been developed (Miglio et al. [2005a,b]; [START_REF] Amara | Hydrodynamical modelling and multidimensional approximation of estuarian river flows[END_REF]), classically using domain de-composition [START_REF] Gervasio | Heterogeneous coupling by virtual control methods[END_REF]), or more recently using 2D 'zooms' overlapping with the 1D domain, in a variational data assimilation framework [START_REF] Gejadze | On a 2D zoom for the 1D shallow water model: Coupling and data assimilation[END_REF]; [START_REF] Marin | Superposition of local zoom models and simultaneous calibration for 1D-2D shallow water flows[END_REF]). An iterative coupling strategy is applied in [START_REF] Barthélémy | On operational flood forecasting system involving 1D/2D coupled hydraulic model and data assimilation[END_REF] between a 1D operational model and a 2D operational model with a sequential assimilation technique for water level forecasting.

Reaching high resolution accuracy and computational hydraulics efficiency for large scale applications remains a difficult challenge given multi-scale hydrodynamics and multiple uncertainty sources. The accuracy of high resolution computations may still be affected by complex dynamics with wet-dry front propagations, uncertain quantities at open boundaries (upstream inflows but also lateral ones, downstream controls and backwater effects), multi-scale and uncertain topography-structures and flow model parameters (e.g. friction effects), internal in/outflows in urban areas, and large computational domains (e.g. [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]).

In the river hydraulics community, the most employed data assimilation methods are based on sequential algorithms, using Kalman filters and variants. Let us cite for example [START_REF] Roux | Estimation de paramètres en hydraulique fluviale, à partir de données caractéristiques de l'imagerie aérienne[END_REF]Dartus [2005, 2006] who estimate flood hydrographs in the 1D Saint-Venant model from dense water surface width measurements; the bathymetry and friction are given. Variational data assimilation algorithms based on the adjoint method have been proposed for 2D hydraulic modeling [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF] and references therein) or 1D hydraulic modeling [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]) and implemented in the DassFlow platform. Using VDA, one can infer boundary conditions like upstream hydrographs, as in [START_REF] Honnorat | Variational data assimilation for 2D fluvial hydraulics simulation[END_REF][START_REF] Hostache | Assimilation of spatially distributed water levels into a shallow-water flood model. Part II: Use of a remote sensing image of Mosel River[END_REF] in 2D or as in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] in 1D, and distributed bathymetry-friction parameters, from multi-source data. VDA is particularly adapted to address high dimensional inverse problems with non-linear dynamic hydraulic models. In the hydraulic community, the calibration of model friction, usually set as spatially uniform by lack of constraining data, is often performed manually or with simple optimization methods. Note that Monte Carlo simulations have been applied with full 2D hydraulic models in [START_REF] Aronica | Probabilistic evaluation of flood hazard in urban areas using monte carlo simulation[END_REF] for flood hazard evaluation or in [START_REF] Chen | Variance based sensitivity analysis of 1D and 2D hydraulic models: An experimental urban flood case[END_REF] for computing Sobol sensitivity indices of a 2D SW model (Telemac2D) applied for flood modeling over urban street networks, but remain computationally expensive.

The DassFlow5 computational platform is used in this work. This platform allows for the resolution of 1D and 2D full shallow water equations with finite difference schemes and well balanced finite volume schemes respectively (more details in Subsection 1.2.2), as well as the use of variational data assimilation using adjoint code obtained via automatic differentiation (Tapenade tool, [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF]). To our knowledge, DassFlow is the first numerical tool proposing large scale multi-D river network modeling with VDA capabilities. 1.5: Some established freeware hydraulic models. The equations resolved are either formulated in (A, Q) (flow section m 2 and at-a-section discharge m 3 /s ) or in (h, u, v) (water depth [m] and 2D depth-integrated flow velocities [m/s]). "Max order" refers to the maximum demonstrated scheme order.

Hydrological models

As discussed in Subsection 1.1.4, the measurement of hydraulic variables at the catchment scale is difficult and data are generally lacking, especially for larger domains. To estimate inflows to be injected into basin-scale river network hydraulic models, hydrological models can be used and coupled to them. Hydrological models used in this work were "weakly" coupled, i.e. the hydrological model generates inputs for the hydraulic model without being influenced by its parameters or states. Hydrological models aim to predict how atmospheric forcings translate into catchment response, i.e. stream flow at the model outlet [START_REF] Beven | Uniqueness of place and process representations in hydrological modelling[END_REF]), especially for forecasting of extremes (floods and droughts). They generally take into account meteorological inputs in the form of rain, snow, evaporation and transpiration, as well as some catchment characteristics appropriate to the modeling approach. Modeling scales can range from upstream catchments (e.g. [START_REF] Roux | A physically-based parsimonious hydrological model for flash floods in mediterranean catchment[END_REF]; [START_REF] Perrin | Modèles hydrologiques du génie rural (GR). Cemagref[END_REF]) to large basins [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]; Paiva et al. [2013a]).

Modeling approaches

As a matter of facts, hydrological processes are difficult to observe and contrarily to hydraulics, "first principle" hydrological laws do not exist. Therefore, approaches with a certain degree of empirism have been proposed in the literature to represent catchment scale hydrological processes. So-called downward approaches consider the inputs and outputs first in model design, while upward approaches start from physically-based relationships.

• Empirical relations can be established from time series of catchments inputs and outflows, e.g. through frequency analysis [START_REF] Xu | Black-Box Hydrological Models[END_REF]). Pure data driven models have recently been proposed using deep learning techniques such as "long short term memory" (LSTM) neural networks (see [START_REF] Kratzert | Towards learning universal, regional, and local hydrological behaviors via machine learning applied to largesample datasets[END_REF], study of hydrological regimes in France in [START_REF] Hashemi | How can regime characteristics of catchments help in training of local and regional LSTM-based runoff models?[END_REF]). Although these models can be accurate, they work as a "black box" and carry no information on the physical catchment itself. Moreover, their extrapolation capability is still limited outside of the learning set but their "interpolation power", especially for tough problems as regionalization (prediction at ungauged locations) is interesting as shown in [START_REF] Kratzert | Towards learning universal, regional, and local hydrological behaviors via machine learning applied to largesample datasets[END_REF]; [START_REF] Hashemi | How can regime characteristics of catchments help in training of local and regional LSTM-based runoff models?[END_REF].

• Conceptual models separate identified hydrological processes into a series of conceptual relations. Bucket-type models divide catchment processes into water stores linked by drainage laws such as in GR4 model [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]; [START_REF] Mathevet | Quels modèles pluie-débit globaux au pas de temps horaire? Développements empiriques et comparaison de modèles sur un large échantillon de bassins versants[END_REF]). The latter consists in a parsimonious and robust model structure, lumped in space, established on large catchment sets to model their outlet discharges. The conceptual basis of those models makes internal state variables and parameters difficult to relate to real catchment states (such as soil moisture) and physiographic characteristics.

• Physical models include physically-based relations for hydrological fluxes (e.g. the Green-Ampt infiltration model [START_REF] Green | Studies on soil phyics[END_REF])) and for flow calculation (e.g. the kinematic wave approximation of SWE in [START_REF] Singh | Kinematic wave modeling in water resources: Environmental hydrology[END_REF]). Parameters of those models can be inferred from physiographic data but one is generally faced with scale issues, overparameterization and data requirements.

Another distinction can be made between spatially lumped or distributed approaches. The former handle in-catchment variabilities through sub-catchment discretization and modeling, for example using lumped models linked by transfer laws. The latter spatially discretize incatchment parameters and variables. For example, a version of the GR hydrological model [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF]) is adapted in a semi-distributed version called GRSD, to account for the effect of rainfall spatial variability [START_REF] Lobligeois | Mieux connaître la distribution spatiale des pluies améliore-t-il la modélisation des crues ? Diagnostic sur 181 bassins versants français[END_REF], recently adapted to account for rainfall intensity in [START_REF] Peredo | Investigating hydrological model versatility to simulate extreme flood events[END_REF].

Recall that modeling is generally a trade-off between modeling goal(s), data availability and model complexity as discussed for hydrological models in [START_REF] Grayson | Spatial patterns in catchment hydrology: observations and modelling[END_REF] (see Fig. Given a 2D-spatial domain Ω rr ⊂ R 2 (a basin/catchment/watershed), a hydrological model can be viewed as a dynamic operator M rr relating state variables h rr (x, t) , ∀x ∈ Ω rr (e.g. internal water storage and fluxes) with several observable input fields I (x, t) (precipitation, evaporation, ...), observable outputs Y rr (x, t) (e.g. discharge) and "unobservable" parameters θ rr (x). Note that the number of unobservable parameters and states coupled to their correlated influence on observable signals renders the inverse problem for hydrological model ill-posed (see Subsection 1.4.1). This means its solution is non-unique since several parameter-and-states set can lead the same observable signal (so called equifinality problem [START_REF] Bertalanffy | General System Theory[END_REF]; [START_REF] Beven | A manifesto for the equifinality thesis[END_REF]). The calibration of hydrological models is complex and often involves feeding a priori information to the calibration method such as, for example, proximity constraints in a calibration algorithm applied to the GRSD hydrological model on a river network De Lavenne et al. [2019]).

In this work, hydraulic models are coupled to spatially distributed conceptual hydrological models in the context of river network modeling and parameter inference from multi-source data. First, the large scale MGB model [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]; [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]) is used for inflowing a large scale 1D hydraulic modeling. Next, the coupling of the GR4H model [START_REF] Perrin | Modèles hydrologiques du génie rural (GR). Cemagref[END_REF]), used in a semi-distributed setup, to a 2D hydraulic model is investigated. More precisely, in view to investigate inverse problems with a hydraulic-hydrological assimilation chain, the source code of GR4H, in its state-space version [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF]), is included in the DassFlow2D source code and differentiated to obtain its adjoint. The GR4H and MGB models used in this work are detailed after.

MGB: a large scale hydrological model

MGB is a large scale semi-distributed rainfall-runoff model designed to simulate discharge in large river basins [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]; [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]) schematized in Fig. 1.14. The model discretizes basins into "unit catchments", themselves divided into hydrological response units (HRUs). It takes into account precipitations, interception, evapotranspiration and infiltration in order to model surface, sub-surface and groundwater flows, and finally inflows to the river network.

The hydrological module applied to each unit catchments consists in a soil reservoir where the partition between surface runoff and infiltration is described with a variable contributing area (VIC) model (concept from [START_REF] Liang | A simple hydrologically based model of land surface water and energy fluxes for general circulation models[END_REF]; [START_REF] Zhao | The Xinanjiang model[END_REF]; [START_REF] Todini | The ARNO rainfall-runoff model[END_REF]; Ludwig and Bremicker). Evaporation from soil and vegetation is described with a Penmann-Monteith model (e.g. [START_REF] Wigmosta | A distributed hydrology-vegetation model for complex terrain[END_REF]). Subsurface flow is taken from the soil reservoir and described with a function similar to the Brooks and Corey non-saturated hydraulic conductivity equation [START_REF] Rawls | Infiltration and soil water movement[END_REF]). A linear relation is used to model percolation from the soil reservoir to the groundwater reservoir. The three lateral fluxes, arising above or from soil reservoir for each HRU of a unit basin, are simply collected and delayed with one linear reservoir for each. The river flow routing scheme, a Muskingum Cunge based method (see Paiva et al. [2013b] and references therein), is expected to represent wave delay and attenuation along the network, but not backwater controls or loss of mass and momentum in the main channel due to floodplain exchanges.

MGB exists in 1D and 2D versions that are compared in [START_REF] Fleischmann | Trade-offs between 1-D and 2-D regional river hydrodynamic models[END_REF]. In the 1D version [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]), large numbers of hydrological catchments are linked by a routing model based on the inertial flood wave routing method (solving simplified SWEs). In the hydraulic part of the model, these catchments are considered as storage units. The routing equations are simplified and aimed at modeling stream flow at a large scale. They allow taking advantage of altimetric WS observations (e.g. through sequential DA methods in Paiva et al. [2013c]). In the 2D version, the basin is discretized into a regular Cartesian grid in which simplified SW equations (locally inertial diffusive formulation) are solved. Floodplain flows are computed between each orthogonal neighbor based on the LISFLOOF-FP sub-grid model [START_REF] Neal | A subgrid channel model for simulating river hydraulics and floodplain inundation over large and data sparse areas[END_REF]), while channel flows may be further informed by sub-grid DEM and computed between cells connected by identified channels [START_REF] Fleischmann | Trade-offs between 1-D and 2-D regional river hydrodynamic models[END_REF]).

In this work, MGB was used in weak coupling to DassFlow1D to compute spatially distributed hydrological inflows into the river network. This assumption is based on the good performance at flow gauge of the spatially distributed and regionalized MGB model [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]).

GR4H: a bucket-type rainfall-runoff model

The GR4H model [START_REF] Mathevet | Quels modèles pluie-débit globaux au pas de temps horaire? Développements empiriques et comparaison de modèles sur un large échantillon de bassins versants[END_REF]) is a lumped continuous model that runs at the hourly time step. It is based on the GR4J model formulation of [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF] and uses a robust and parsimonious production function proposed in [START_REF] Michel | Un modèle pluie-débit journalier à trois paramètres[END_REF]). This model is widely used and has been compared to other hydrological models (see e.g. [START_REF] Tian | Assessment of climate change impacts on river high flows through comparative use of GR4J, HBV and Xinanjiang models[END_REF]; [START_REF] Nepal | Spatial GR4J conceptualization of the Tamor glaciated alpine catchment in Eastern Nepal: evaluation of GR4JSG against streamflow and MODIS snow extent[END_REF]). It is used in this work because it is a well established, parsimonious and robust model, based on Ordinary Differential Equations (ODEs, state-space version only [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF])) hence differentiable. This property is interesting in the present work as it is a necessary condition to generate an adjoint code for variational data assimilation (see Section 1.4).

Several GR models exist, featuring varying amounts of parameters, stores and time steps (see [START_REF] Perrin | Modèles hydrologiques du génie rural (GR). Cemagref[END_REF]). The "state-space" version of the GR4H model (see Fig. 1.15) is described by the following set of ordinary differential equations:

          Ṡ Ṡh,1 Ṡh,2 . . . Ṡh,nres Ṙ           =           P s -E s -Perc P r -Q Sh,1 Q Sh,1 -Q Sh,2 . . . Q Sh,nres-1 -Q uh Q 9 -Q r + F           (1.9)
It consists in a production store S, a routing store R and a series of 11 Nash cascade stores S h,i , ∀i ∈ [1..11], that replace a unit hydrograph function from the classical GR4H version. The precipitation input P n is divided between the production store S and the upper Nash cascade store S h,1 . The production store R loses water to an evapotranspiration relation and is linked to the upper Nash cascade store S h,1 by a discharge law (Perc). The reservoirs of the Nash cascade are identical and linked in a chain by discharge relations (Q Sh,i , ∀i ∈ [1..10]). The last cascade reservoir outlet S h,11 is linked to the routing store R and the model output through a 9-1 tenth repartition. The routing store S and direct routing branch can both exchange water through flux F which represents an exchange for example with aquifers.

The internal model variables are the 13 store levels. Four parameters ((x 1 , x 2 , x 3 , x 4 ), see Subsections 3.3.3 and 3.C), rule the draining of the production store, of the routing store, of the 11 identical cascading stores and the non-conservative exchange law. They are given per catchment, i.e. spatially distributed. Initial store states are given using a 1 year warm-up period. A series of parameters used in flux calculations are fixed (values and fluxes are given in Subsection 3.C). A R-based global optimization method for the classical GR4 formulation can be found here6 and was used for background value estimation in this work.

In this work, the state-space" version of GR4H was integrated to the DassFlow2D assimilation tool-chain. The weakly-coupled hydrological module was used to model semi-distributed catchments linked to a 2D full SWE hydraulic model. Calibration of hydrological parameters using WS elevation and discharge observations were carried out using the gradient-based descent algorithm from the DassFlow framework to reach a local optimum.

Inverse problems and data assimilation

Given the rising availability, accuracy and resolution of river network monitoring products, the development of hydraulic-hydrological models should strive to account for their exploitation. The synergy between models and data can be achieved through data assimilation methods that enable to solve inverse problems, i.e. methods that use observations of reality to correct model states and parameters so that they better fit that reality. This section first introduces inverse problems in hydraulic models, the solving of which is a step toward synergy between hydraulic models (presented in Section 1.2) and river network observability (presented in Subsection 1.1.4). Then, data assimilation methods are described. They are linked to a unifying Bayesian approach, given Gaussian priors and linear models. Finally, the iterative variational data assimilation algorithm used in the following chapters is described summarily (more details can be found in the following chapters).

Direct and inverse problems

Definitions

Determining the cause of a phenomenon occurring in a physical system through its observation is a key question in various scientific domains. Solving a direct problem means using a cause to determine an effect, while solving an inverse problem means observing the effect to determine its cause (see [START_REF] Tarantola | Inverse problem theory and methods for model parameter estimation[END_REF] for theory of inverse problems).

Modeling a physical system generally consists in determining state variables that verify a physical model given parameters and boundary conditions. In the context of hydrodynamic modeling, the cause is a set of unknown/estimated parameters in the form of spatial fields and temporal forcings and the effect is composed of heterogeneous water surface observables. This problem and model are naturally the direct problem, as the hydraulic models describing the variability of flow mass and momentum in terms of state variables (A, Q) in 1D (Eq. 1.2) or (h, u, v) in 2D (Eq. 1.6). Nevertheless, in practice, boundary conditions, initial conditions and parameters can be uncertain or unknown, because of imperfect modeling and observability of the physical system of interest (see river network observability in Section 1.1.4). For example, for river models, bathymetry-friction or lateral exchanges of mass and momentum in SW models (see Section 1.2) can be poorly-known, which can lead to insufficient accuracy on the simulated state variables. The inverse problem consists in finding unknown or uncertain model parameters that enable to fit observations of physical system response (e.g. [START_REF] Tarantola | Inverse problem theory and methods for model parameter estimation[END_REF]). Consequently, the observed quantities have to be calculable from model variables.

One of the main difficulties encountered in inverse problems resolution is ill-posedness. As defined by the mathematician J. S. Hadamard, a mathematical modeling problem of a physical phenomenon is a well-posed problem if the solution exists, is unique and depends continuously on model boundary and initial conditions, and parameters.

Inverse problems

This work aims to infer uncertain or unknown parameters of hydraulic and hydrological models using multi-source river flow observations provided by various in situ or satellite measurements. In particular, this work focuses on the optimization of control vectors of coupled hydraulichydrological models. The considered datasets are heterogeneous in nature with observations among WS deformations (WS elevation, width, slope), flow velocity (at the free surface or not), flow depth or discharge estimates. The unknown or uncertain parameters of the direct hydraulic models (Eq. 1.2 or Eq. 1.6) weakly coupled to a hydrological model (Eq. 1.9) are among bathymetry, friction and lateral exchanges, including inflows from hydrological models. The unknown or uncertain parameters of the conceptual hydrological model (Eq. 1.9) considered in this work are among reservoir and sub-catchment exchange parameters. An underlying hypothesis of hydrological parameter inference in this work is that the conceptual representation of hydrological catchment introduced in GR4 (i.e. reservoir number and connections) is pertinent for the considered catchments and thus these parameters are not inferred. The correction of model forcings and reservoir states also represent interesting problems.

In hydraulic models, the inference of at least two type of unknowns among river bathymetryfriction-inflow discharge from WS signature observations is an ill-posed problem (see [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]). Indeed, these flow controls have a correlated influence on water surface deformations. For example, as remarked in [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF], an infinity of bathymetry-friction couples can lead to the same local WS elevation.

A supplementary difficulty arises from the spatio-temporal heterogeneity of data and the relatively sparse sampling they provide of the physical processes of interest. For example, inferrable flood frequency patterns, using sparse altimetric sampling of a relatively short river reach by a single satellite (SWOT like data), are studied at the river reach scale in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]. Regarding the spatialization of controls, a higher spatial density of controls than of observation leads to under-determined inverse problems, as discussed for bathymetry and friction inference of 1D hydraulic models from satellite data in [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]. See also bathymetry inferences in [START_REF] Lai | Assimilation of spatially distributed water levels into a shallow-water flood model. Part I: Mathematical method and test case[END_REF] or friction inferences on a 2D hydraulic model from in situ and Synthetic Aperture Radar (SAR) data in [START_REF] Hostache | Assimilation of spatially distributed water levels into a shallow-water flood model. Part II: Use of a remote sensing image of Mosel River[END_REF]. Those studies suggest the combination of sparse satellite data with a priori physical knowledge and in situ data to better constrain the inferences.

Given the increasing amount of multi-source observations of river surface deformations, in combination with ancillary data and physical knowledge, arises the question of information feedback in integrated hydraulic-hydrological models, from local observation scale to upstream hydrological modules. Can we "do backward hydrology" [START_REF] Kirchner | Getting the right answers for the right reasons: Linking measurements, analyses, and models to advance the science of hydrology[END_REF])? With which type and spatio-temporal density of model controls and data? More precisely, the inverse calibration problem of a distributed hydrological model, from multi-source observations in the river network, will be studied in this work.

Adapting the complexity of direct models and inverse methods, to both modeling goals, features and available observations, is necessary in view to progress in the integration of multisource information and is studied in this PhD.

Data assimilation

Data assimilation is a mathematical discipline that aims to optimally combine a model with observations of a physical system in order to exploit the physical informative content of data (see e.g. [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Carrassi | Data assimilation in the geosciences: An overview of methods, issues, and perspectives[END_REF]; [START_REF] Monnier | Variational Data Assimilation and Model Learning[END_REF]). It has been used for decades in meteorology or oceanography (see e.g. [START_REF] Ghil | Data assimilation in meteorology and oceanography[END_REF]) or more recently in hydrology (see e.g. [START_REF] Park | Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications[END_REF]; [START_REF] Seo | Automatic state updating for operational streamflow forecasting via variational data assimilation[END_REF]).

The prediction of a flow model is generally fitted to observations by adjusting a parameter. In meteorology, this sought parameter is usually the initial condition of numerical models, which represents an important challenge for weather forecasting given the chaotic nature and relatively short relaxation scales of atmospheric flows [START_REF] Carrassi | Data assimilation for chaotic dynamics[END_REF]). In this work, the parameter, called control, and denoted c, is composed of boundary conditions and flow models parameters (see e.g. [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] and references therein). Such composite controls, composed of different types of unknowns, lead to multivariate data assimilation problems. Moreover, the controls of non-linear hydrodynamic models considered in this work are distributed in space and time and are of a relatively high dimension.

Schematically, data assimilation has been developing around two families of methods: the variational approach that stems from the optimal control theory (see e.g. the work of J. L. Lions on the optimal control of PDE) and the statistical estimation with sequential methods. In sequential methods, model states are corrected recursively at discrete observation times, only using previous system state and observations [START_REF] Carrassi | Data assimilation in the geosciences: An overview of methods, issues, and perspectives[END_REF]). Sequential methods notably include Kalman filters [START_REF] Swerling | First order error propagation in a stagewise differential smoothing procedure for satellite observations[END_REF]; [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]) and smoothers [START_REF] Carrassi | Data assimilation in the geosciences: An overview of methods, issues, and perspectives[END_REF]).

In the former, model states are corrected at observation times only. In the latter, model states are constrained/smoothed at unobserved times. In variational methods, all observations are accounted for simultaneously and guide the model trajectory. The Bayesian framework offers a unifying framework for the two approaches.

In this work, the variational approach is used. It is an adapted approach to deal with nonlinear dynamic models and with heterogeneous observations in space and time, but also to deal with multivariate data assimilation problems and large control vectors. It is an efficient approach that is well suited to solve the inverse problems of interest (defined above in Section 1.4.1), as experienced for hydraulic inferences in satellite observability context (e.g. [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]) or for spatially distributed hydrological calibration problem [START_REF] Castaings | Sensitivity analysis and parameter estimation for distributed hydrological modeling: potential of variational methods[END_REF]; [START_REF] Jay-Allemand | On the potential of variational calibration for a fully distributed hydrological model: application on a mediterranean catchment[END_REF]). Moreover, note that global optimization algorithms are not adapted for the present inverse problems and local descent algorithms will be used in the variational approach.

Problem statement

The mathematical framework used in data assimilation is briefly recalled here (see DA course, e.g. [START_REF] Carrassi | Data assimilation in the geosciences: An overview of methods, issues, and perspectives[END_REF]; [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Monnier | Variational Data Assimilation and Model Learning[END_REF]; [START_REF] Asch | Data assimilation: methods, algorithms, and applications[END_REF]). Let U be the space of the input variables (controls) of flow models (Eq. 1.2) or (1.6) or (1.6 weakly coupled to 1.9). Considering c ∈ U , a finite dimension real-valued control vector, Let c b denote the background control vector, called "prior" in statistics, and c a the analysis that is expected to best approximate, in a sense to be defined, the true control value c t . Note that because of uncertainties on the components of c, the prior is the best available guess and writes:

c b = c + b (1.10)
Let us assume that model state variables are observed in a set Y = H (c) ∈ Y with Y the observation space and H : X → Y the observation operator from state space X to observation space Y. The actual observation is Y o = Y + o with o the observation uncertainty arising for example from measurement noise.

In the Bayesian framework, the posterior probability density of c conditioned on observations Y o can be written with the Bayes formula:

p (c|Y o ) = p (Y o |c) p (c) p (Y o ) (1.11)
Searching for the mode of the posterior density p (c|Y o ), that is maximizing p (c|Y o ), is the essence of data assimilation (see [START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model[END_REF]). Assuming Gaussian prior and observations uncertainties, that is o ∼ N (0, O) and b ∼ N (0, B) with B and O respectively the background and observation covariances, maximizing p (c|Y o ) is equivalent to minimizing the cost function:

j (c) = 1 2 O -1 (H (c) -Y o ) 2 + 1 2 B -1 Y -Y b 2 (1.12)
This is a classical definition of the cost function j (c) = j obs + j reg . The first term j obs is the observation term measuring the misfit between model prediction and observations. The second term is a Tikhonov regularization term [START_REF] Tikhonov | Méthode de résolution de problémes mal posés[END_REF]; [START_REF] Kaltenbacher | Iterative regularization methods for nonlinear illposed problems[END_REF]). The latter helps to add a priori constraint on the sought solution and "convexifying" the original ill-posed problem with observation term only. As matter of fact, background covariance is generally not known.

The optimization algorithm

The variational data assimilation problem consists in the following optimization problem [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]; [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF], see 1.16):

c a = argminj (c) (1.13)
The first order optimality condition of this optimization problem reads: ∇j (c) = 0. The optimization problem (Eq. 1.13) is numerically solved with a Quasi-Newton descent algorithm: here the classical L-BFGS algorithm or its bounded version L-BFGS-B (see [START_REF] Zhu | Algorithm 778: L-BFGS-B: Fortran subroutines for largescale bound-constrained optimization[END_REF]). The iterative process requires, at each iteration, the gradient of the cost function with regards to control vector components.

The gradient of the cost function is computed with the adjoint state method enabling to consider large control vectors (see e.g. [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]; Oubanas et al. [2018a]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]). The adjoint code is obtained by applying the automatic differentiation tool Tapenade to the forward numerical codes: DassFlow1D [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]) or Dass-Flow2D [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]) in the following. The automatic differentiation tool Tapenade [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF]), based on the derivation rule of composite functions, provides the tangent and adjoint codes of a forward code. Given a function F :

X ∈ R n → Y ∈ R n evaluated over [0, T]
in the form of a computer program P, Tapenade provides an adjoint code P by com- puting the analytical derivative of each elementary mathematical operation of P. This adjoint code evaluates X = Ȳ × F (X) over [T, 0] (Hascoet and Pascual [2013]). Detailed know-hows and historical references can be found for example in courses [START_REF] Carrassi | Data assimilation in the geosciences: An overview of methods, issues, and perspectives[END_REF]; [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Monnier | Variational Data Assimilation and Model Learning[END_REF]).

Trajectory change during the iterative process can be represented within the parameter space, that is a N-dimension space (with N the number of components of the control c) within which the misfit to observations (or cost) is evaluated. In the present VDA method, a single local minimum is sought, using a gradient-based descent algorithm, background parameters and initial conditions.

Mono-objective DA refers to approaches that use a single kind of observable in their cost function (e.g. WS elevation), while multi-criteria cost functions and multi-objective methods use multiple (e.g. WS elevation and at-a-section discharge). See [START_REF] Efstratiadis | One decade of multi-objective calibration approaches in hydrological modelling: a review[END_REF] for a review of multi-objective problems in hydrology. Mono-variate DA refers to methods inferring a single parameter type (potentially discretized in space and time), while multi-variate methods aim to simultaneously infer composite control vectors (e.g. the upstream inflow and bathymetry-friction couple in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]).

Python Wrapping of DassFlow

Python has become the most used computer language in 2021, overtaking C and Java, according to the Tiobe index7 . Python packages and libraries are freely available and cover uses such as data representation, scientific computing, global optimization, statistical analysis, Geographic Information System (GIS) tools and Machine Learning (ML) libraries. Python is easy to use and is an interpreted language that does not need to be compiled. The DassFlow computational platform is written in Fortran, a low-level compiled language. Fortran is capable of efficient matrix computations and high performance computing, but does not have access to numerous libraries like those for Python and is, in general, less practical to use.

The "wrapping" of a code consists in creating an interface between two programming languages. Python has emerged as a key language to interface older codes to, thanks to its ease of use and numerous libraries and toolboxes. The Python wrapping of DassFlow consists of the interfacing of its Fortran efficient computational core to Python. This allows preserving the core code and handling the pre-and post-processing, as well as some of the other less computationally costly parts of the code.

The wrapping of the DassFlow Fortran code (see [START_REF] Huynh | Wrapping Python-Fortran du code DassFlow2D[END_REF]; Ma [2021]) has seen the interfacing of most variables and parameters structures, the launch of both direct and inverse models through a Python script, and the adding of a Python-based minimizer instead of the previous Fortran one. Note that the gradient calculation is a computationally costly step, while the minimization of the cost function, given the gradient, is not.

The wrapping tool is f90wrap, a Python-Fortran interface generator developed by James Kermode [START_REF] Kermode | f90wrap: an automated tool for constructing deep Python interfaces to modern Fortran codes[END_REF] and that builds on the f2py Python package. The code is open-source and available at https://github.com/jameskermode/f90wrap. The tool generates modules (.mod files) and .py and .f90 interfacing files. These files are then used to generate a importable Python module with f2py. It allows to read and write chosen Fortran variables and routines from .py files. Multi-thread parallel computation is handled as well. More details on the wrapping methodology and development process are available in [START_REF] Huynh | Wrapping Python-Fortran du code DassFlow2D[END_REF]; Ma [2021], technical details can be found in [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF].

Context summary and thesis objectives

Strong societal issues drive the need for the cartographic representation of hydrological cycle phenomenons. The corresponding scientific challenges are the improvement of state and fluxes representation in hydrographic network models and the synergy of these models with new data sources and data types. Depending on modeling objectives (e.g. river discharge, water depth or flood extent) and data availability, multiple model scales with different degrees of complexity have been developed. 1D approaches are generally used to model near mono-dimensional hydraulic phenomenons at local, regional and global scales. 2D models are used to model bidimensional phenomenons at local scale but, due to their higher computational cost, are generally used with simplified hydraulic equations and rough spatial resolution at regional or global scales. However, the use of full SW models and adapted numerical schemes is needed for accurate modeling of wet/dry front propagation at the local scale. Over a river network, both 1D and 2D approaches can have their advantages. Coupled 1D-2D modeling approaches theoretically allow to use the best of both approaches (e.g. by limiting costly 2D computation to pertinent regions), but add a layer of difficulty regarding their coupling/interfacing. Hydraulic network models are generally inflowed by hydrological models to feature complete basin-scale state-fluxes numerical representations. At such scale the accurate estimation of both hydraulic and hydrological parameters remains an important difficulty.

Observations of hydrological phenomenons are given by a growing array of sensors, with increasing accuracy and spatial and temporal resolution. This leads to an increasing number of multi-scale, multi-sourced heterogeneous data products that contain an often under-exploited wealth of information, especially for ungauged river network portions. Single data products are usually integrated in models through data assimilation methods, classically with sequential DA methods. Variational methods are less common, especially because of the difficulties related to the obtention of an adjoint model. Yet, they are very pertinent for the inference of large multi-variate control vectors of hydrodynamic models. Inferences, in SW models, of channel parameters (the bathymetry-friction couple), initial and boundary conditions and temporal forcings (inflows) using multi-sourced altimetric measurements of the WS and other flow measurements have been carried out in recent works using data assimilation methods. The resolution of these ill-posed problems is a current scientific issue in hydraulic modeling at the reach and catchment scale. In hydraulic-hydrological models, the inference of hydrological parameters and states from upstream feedback of hydraulic signatures is also a current scientific issue. Given the growing observation data resolutions and the consequent potential to infer spatially dense parameters, this work uses variational data assimilation in the context of remotely-sensed and multi-sourced observations. It builds on a powerful VDA framework from the DassFlow platform and investigates complex inverse problems with hydrodynamic models from multi-source flow observations. This work focuses on the following scientific problems: i) the effective representation of river networks through multi-D basin-scale hydraulic-hydrological models, ii) the integration of multisource multi-scale heterogeneous observations in such models, iii) from this data, the inference of spatio-temporally distributed river channel parameters and inflows through a variational data assimilation method and iv) the upstream informational feedback of river network observations to integrated hydrological models. To answer these questions, the variational data assimilation platform DassFlow was expanded upon, notably by implementing spatio-temporally distributed temporal forcings and integrating a conceptual hydrological model to the variational tool-chain. A new method for 1D-2D modeling of catchment scale river networks was developed using a single finite volume solver. Numerous inferences of large composite control vectors using multisourced heterogeneous data were carried out on 1D and 1D-2D models coupled with hydrology with the VDA method.

49

This chapter is based on [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF]. The article's content is featured with minor layout and notation adjustments.

Extended abstract

With the upcoming SWOT satellite mission, which should provide spatially dense river surface elevation, width and slope observations globally, comes the opportunity to assimilate such data into hydrodynamic models, from the reach scale to the hydrographic network scale. Based on the HiVDI (Hierarchical Variational Discharge Inversion) modeling strategy (from [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]), this study tackles the forward and inverse (see runs in Subsection 1.4.1) modeling capabilities of distributed channel parameters and multiple inflows from multi-satellite observations of river surface. The considered model solves the full 1D (A, Q) SW equations with a lateral inflow source term (see Subsection 1.2.1). The lateral inflow source term was implemented in the DassFlow platform in a Preissmann scheme (see Appendix B) and a HLL scheme, for direct and inverse modeling.

It is shown on synthetic cases that the estimation of both lateral inflows q l (x, t) and the bathymetry-friction couple (b (x) , K (x)) is achievable with a minimum spatial observability be- tween inflows as long as their hydraulic signature is sampled. Next, a real case is studied: 871 km of the Negro river (Amazon basin) including complex multichannel reaches, 21 tributaries and backwater controls from major confluences. An effective modeling approach is proposed using (i) WS elevations from Envisat data and dense in situ GPS flow lines [START_REF] Moreira | Apport des données de géodésie spatial pour l'étude du bassin hydrologique amazonien[END_REF]), (ii) average river top widths from optical imagery [START_REF] Pekel | High-resolution mapping of global surface water and its long-term changes[END_REF]), (iii) upstream and lateral flows from the MGB large-scale hydrological model (Paiva et al. [2013a]). The calibrated effective hydraulic model closely fits satellite altimetry observations and presents real like spatial variabilities; flood wave propagation and water surface observation frequential features are analyzed with identifiability maps following [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] (see Fig.2.1). Synthetic SWOT observations are generated from the simulated flow lines and allow to infer model parameters (436 effective bathymetry points, 17 friction patches and 22 upstream and lateral hydrographs) given hydraulically coherent prior parameter values. The unprecedented spatial coverage provided by the SWOT wide-swath altimeter enables to infer relatively densely discretized parameters at a cartographic scale. Inferences of channel parameters carried out on this fine hydraulic model applied at a large scale give satisfying results using noisy SWOT-like data at reach scale (see Fig. 2.2). Inferences of spatially distributed temporal parameters (lateral inflows) give satisfying results as well, with even relatively small scale hydrograph variations being inferred accurately on this long reach. This study brings insights in: (i) the hydraulic visibility of multiple inflows hydrographs signature at large scale with SWOT; (ii) the simultaneous identifiability of spatially distributed channel parameters and inflows by assimilation of satellite altimetry data; (iii) the need for prior information; (iv) the need to further tailor and scale network hydrodynamic models and assimilation methods to improve the fusion of multi-source information and potential information feedback to hydrological modules in integrated chains.

Section 2.3 presents the modeling approach with the 1D Saint-Venant flow model and the inverse computational method. Section 2.4 investigates the capabilities of the inverse method for identifying spatially distributed inflows with and without unknown channel parameters given observation patterns of WS signatures including overlapping backwater effects. Section 2.5 presents the effective modeling approach from multi-satellite data applied to 871 km of the Negro river (Amazon basin) and the analysis of flow propagation features against SWOT observability. Section 2.6 proposes inference tests for spatially distributed inflows with and without unknown parameters on the Negro case in the presence of strong backwater effects. The original article conclusion is featured in Section 2.7. The red-blue heatmap gives the WS elevation anomaly with regards to the average local WS elevation Z ano (x, t) = Z (x, t) -Z (x), where Z (x) is the average local WS elevation over the seasonal flood peak (may-june). The blue line represents the travel of an intumescence traveling through the reach at the modeled kinematic speed ( 5 /3u) and starting at the upstream BC at the time of the flood peak at this point, in a base model calibrated with satellite altimetry. Dotted colored lines show the sensitivity of this travel time estimation to model parameters and forcings: variations of +30% and -30% from the base model are represented for the Manning friction parameter K (in green), the bathymetry b (in cyan) and all model inflows (in red). The vertical bars indicated lateral inflows q l (x, t) for a total of 21 lateral mass and momentum contributions (bold lines are denote the major tributaries). Horizontal bars represent observability given by the SWOT large swath altimeter.

(a) (b) Figure 2.2: Simultaneous inferences of lateral inflows q l (x, t), bathymetry b (x) and friction patches K (x) in a twin experiment setup on the Negro river 1D model. The control vec- tor is composed of bathymetry points, friction patches and select lateral inflow hydrographs and writes

c ext = (Q l,1 (t) , Q l,2 (t) , Q l,3 (t) , Q l,4 (t) , b (x) , K (x)
). Considered observations are noisy SWOT observations from the SWOT simulator (inferences in orange) and spatially and temporally dense observations (inferences in green).

Introduction

Hydrographic networks represent major flow paths for freshwater in the water cycle and an interface with the space of human societies. It is of prior importance in a context of climate change to improve the knowledge and representation of continental water fluxes, including river discharge, defined as an essential physical variable (see Global Climate Observing System Organization, 2011]). However, modeling flows structure in the different compartments of a catchment remains a hard task (see [START_REF] Schuite | Improving the spectral analysis of hydrological signals to efficiently constrain watershed properties[END_REF] and references therein) especially at poorly gauged locations. In complement of in situ sensors networks, which are declining in several regions (e.g. [START_REF] Fekete | The current status of global river discharge monitoring and potential new technologies complementing traditional discharge measurements[END_REF]), new generations of earth observation satellites and sensors provide increasingly accurate and dense measurements of water surface variabilities.

The Surface Water and Ocean Topography (SWOT) satellite, to be launched in 2021, will bring observations of water surface (WS) with an unprecedented spatio-temporal coverage [START_REF] Alsdorf | Spatial and temporal complexity of the Amazon flood measured from space[END_REF]; [START_REF] Durand | The Surface Water and Ocean Topography Mission: Observing Terrestrial Surface Water and Oceanic Submesoscale Eddies[END_REF]; [START_REF] Rodríguez | SWOT Surface Water and Ocean Topography Mission (SWOT)[END_REF]; [START_REF] Biancamaria | The SWOT Mission and Its Capabilities for Land Hydrology[END_REF]; [START_REF] Rodríguez | Wide-Swath Altimetry: A Review[END_REF]. This will yield greater hydraulic visibility (see definition in [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF]; [START_REF] Montazem | Wavelet-based river segmentation using hydraulic controlpreserving water surface elevation profile properties[END_REF]; [START_REF] Rodríguez | Observing rivers with varying spatial scales[END_REF]) of hydrological responses through WS signatures from the local scale to the hydrographic network scale, hence an opportunity to better characterize hydrological fluxes and potentially constrain local to integrated hydrodynamic models and inverse problems. However, estimating river discharge Q from "geometric" observables of flow surface (elevation Z, width W and slope S) remains a difficult inverse problem particularly in case of poor knowledge on river bathymetry and friction (see [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] and references therein).

Hydraulic inverse problems with various model complexities, data-unknowns types and amounts are investigated by recent studies in a satellite data context (see [START_REF] Biancamaria | The SWOT Mission and Its Capabilities for Land Hydrology[END_REF] for a review). A few studies started to test the benefit of assimilating (synthetic) SWOT WS observations with sequential methods in simplified hydraulic models, for estimating inflow discharge assuming known river friction and bathymetry Andreadis et al.;BIA [2011] or inferring bathymetry assuming known friction [START_REF] Durand | Estimation of bathymetric depth and slope from data assimilation of swath altimetry into a hydrodynamic model[END_REF]; [START_REF] Yoon | Estimating river bathymetry from data assimilation of synthetic SWOT measurements[END_REF]. Next, methods based on low-complexity models have been proposed for estimating river discharge from WS observables in case of unknown bathymetry b and friction K, based on the low Froude model [START_REF] Durand | Estimating reachaveraged discharge for the river Severn from measurements of river water surface elevation and slope[END_REF]; [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF], hydraulic geometries [START_REF] Gleason | Toward global mapping of river discharge using satellite images and at-many-stations hydraulic geometry[END_REF] or empirical algebraic flow models [START_REF] Bjerklie | Satellite remote sensing estimation of river discharge: Application to the Yukon river Alaska[END_REF]. The inter-comparison of low complexity methods in [START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF] highlights the difficulty of estimating the so-called unknown triplet (Q, K, b) from WS observables as well as the importance of good prior guesses on the sought parameters.

The combined use of dynamic flow models of river systems and optimization methods enables to solve hydraulic inverse problems, as shown for upstream flood hydrograph(s) estimation by [START_REF] Roux | Use of parameter optimization to estimate a flood wave: Potential applications to remote sensing of rivers[END_REF] from WS width time series and a 1D Saint-Venant model or by [START_REF] Honnorat | Variational data assimilation for 2D fluvial hydraulics simulation[END_REF][START_REF] Hostache | Assimilation of spatially distributed water levels into a shallow-water flood model. Part II: Use of a remote sensing image of Mosel River[END_REF]; [START_REF] Lai | Assimilation of spatially distributed water levels into a shallow-water flood model. Part I: Mathematical method and test case[END_REF] using variational assimilation of flow depth time series in a 2D shallow-water model. The variational data assimilation (VDA) approach (see e.g. [START_REF] Cacuci | Computational Methods for Data Evaluation and Assimilation[END_REF] and references therein) is suitable to address the present hydraulic inverse problem from WS observations (see [START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model[END_REF]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; Oubanas et al. [2018b]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] and references therein -single upstream hydrographs in all studies except multiple "step-wise" off-takes on synthetic and densely observed irrigation-like cases in [START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model[END_REF]). It consists in fitting the modeled flow features to observations through the optimization of control parameters in a variational framework. To be solved efficiently, such an ill-posed inverse problem needs to be regularized: see [START_REF] Kaltenbacher | Iterative regularization methods for nonlinear illposed problems[END_REF] for the theory of regularization of such inverse problems and [START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] for the present inverse flow problem.

Crucial aspects of this difficult inverse problem are (i) the spatio-temporal sparsity of altimetric observations regarding flow controls -as analyzed in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] for inferrable hydrographs frequencies with the introduction of the identifiability maps and in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF] for inferrable channel parameters patterns; (ii) the sensitivity of the triplet inference to good prior guesses on the sought parameters as highlighted in a SWOT context by [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]; [START_REF] Yoon | Improved error estimates of a discharge algorithm for remotely sensed river measurements: Test cases on Sacramento and Garonne rivers[END_REF]; [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using AirSWOT observations[END_REF]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]. The latest is highlighted by recent discharge estimates (in a triplet setup) from synthetic SWOT data on the Pô, Garonne and Sacramento Rivers in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] (see also Oubanas et al. [2018a]), from AirSWOT airborne measurements on the Willamette River in [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using AirSWOT observations[END_REF] or from Envisat altimetric data on an anabranching portion of the Xingu River [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]. Using a biased prior hydrograph results in a biased estimate of inflow hydrograph despite a correct temporal variability at observation times -see [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] for detailed analysis. A hierarchical modeling strategy HiVDI (Hierarchical Variational Discharge Inversion) is proposed in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] including low complexity flow relations (Low Froude and locally steady-state) for providing robust prior guesses to the VDA process by taking advantage of databases or regional hydrological models.

Most studies mentioned above tackle the estimation of a single upstream inflow discharge hydrograph from WS observations on relatively short river reaches regarding the spatio-temporal sparsity of (satellite) observations sampling and without complex flow zones -confluences, multichannel portions (except [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]), floodplains. Moreover, few recent studies address the effective modeling of (ungauged) river channels using multi-satellite data [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF]; [START_REF] Schneider | Application of CryoSat-2 altimetry data for river analysis and modelling[END_REF]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; O'Loughlin et al. [2019].

The present study investigates the challenging inference of multiple inflows and channel parameters patterns from hydraulic signatures in a SWOT context. Particular attention is paid to the difficult inference of HC with correlated effects on WS signatures including overlapping backwater effects. Moreover, we present an effective hydraulic modeling approach based on multi-satellite observations of WS and accounting for hydrological model inputs. It is applied to a long river reach including confluences with tributaries and strong backwater effects in the Amazon basin. The computational inverse method, based on the full 1D Saint-Venant equations, is that presented in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] with a spatially distributed friction power law in water depth and a simple piece-wise linear channel bathymetry [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]. It is adapted here to account for lateral inflows/off-takes and is weakly coupled to the large scale MGB hydrological model [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]; [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]; Paiva et al. [2013a]. Numerical investigations of the resulting WS signatures and identifiability tests are presented along with sensitivity analysis to the parameters of both the (forward) hydraulic model and the inverse method. The challenging inference of multiple inflows and channel parameters patterns is investigated with various observations densities including the assimilation of synthetic SWOT ones.

Modeling Approach

The flow model

The Saint-Venant equations (Barré de Saint-Venant [1871]) consist in the unidirectional form of the shallow water equations and are commonly used to describe open channel flows (see e.g. [START_REF] Chow | Handbook of applied hydrology[END_REF]; [START_REF] Ancey | Hydraulique à surface libre[END_REF]; [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF] for detailed assumptions including the long wave one). In what follows, x denotes the curvilinear abscissa from upstream to downstream along a reach of length L (usual simplifying hypothesis are used) and t ∈ [0, T] denotes the time. In this representation, let A (x, t) m 2 be the flow cross-sectional area and Q (x, t) m 3 .s -1 the discharge such that U = Q/A m.s -1 represents the longitudinal XS-averaged velocity . The Saint-Venant equations in (A, Q) variables at a flow XS read as follows:

   ∂A ∂t + ∂Q ∂x = k l q l ∂Q ∂t + ∂ ∂x Q 2 A + gA ∂Z ∂x = -gAS f + k l Uq l (2.1)
where t) [m] the wetted perimeter, g is the gravity magnitude m.s -2 , q l (x, t) is the lineic lateral discharge m 2 .s -1 , and k l is a lateral discharge coefficient chosen equal to one here since we consider inflows only. In DassFlow, the friction term S f is classically parameterized with the empirical Manning-Strickler law established for uniform flows:

Z (x, t) [m] is the WS elevation and Z = (b + h) [m] with b (x) [m] the river bed level and h (x, t) [m] the water depth, R h (x, t) = A /P h [m] the hydraulic radius , P h (x,
S f = |Q|Q K 2 S 2 R 4/3 h (2.2)
The Strickler friction coefficient K m 1/3 .s -1 is defined as a power law in h:

K (x, h (x, t)) = α (x) h (x, t) β(x) (2.3)
where α and β are spatially distributed parameters. This spatially distributed friction law enables a variation of friction effects in function of the flow state (see effective modeling of multichannel flows in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]). Inflow hydrographs Q in (t) and q l,d (t) at d ∈ [1..D] are classically imposed respectively up- stream of the river domain and at known injection cells, that is in-between two computational XSs along the river channel. Let us recall the Froude number definition Fr = U /c comparing the average flow velocity U to pressure wave celerity c = √ gA /W where W is the flow top width [m]. Considering subcritical flows (Fr < 1) in a satellite observability context (see [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]), a boundary condition is imposed at the downstream end of the model using the Manning-Strickler equation depending on the unknowns (A, Q, K) out . The initial condition is set as the steady state backwater curve profile Z 0 (x) = Z (Q in (t 0 ) , q l,1..D (t 0 )) for hot-start. This 1D Saint-Venant model (Eq. (2.1)) is discretized using the classical implicit Preissmann scheme (see e.g. [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF]) on a regular grid of spacing ∆x using a double sweep method enabling to deal with flow regimes changes, ∆t is precised in numerical cases. This is implemented into the computational software DassFlow [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF].

The computational inverse method

The paper studies the estimation of spatially and temporally distributed flow controls from WS observables using the inverse method presented in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] (see also [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]) with an augmented composite control vector c; the method is detailed in 2.A. The principle of the inverse method is to estimate (discrete) flow controls by minimizing the discrepancy between observed and simulated flow lines, Z obs and Z (c) respectively, the latter depending on the unknown parameters vector c through the hydrodynamic model (Eq. (2.1)). This discrepancy is classically evaluated with the observation cost function term

j obs (c) = 1 2 (Z (c) -Z obs ) 2
O computed on the observation spatial and temporal grids, see details in 2.A.

The control vector c contains temporally and spatially distributed unknown "input parameters" of the 1D Saint-Venant model: a friction law [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]) and lateral inflows, unlike in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], where there is a spatially uniform friction law K (h) without lateral flows. It reads:

c = Q 0 in , ..., Q P in ; Q 0 l,1 , ..., Q P l,1 , Q 0 l,2 , ..., Q P l,D ; b 1 , ..., b I ; α 1 , ..., α N ; β 1 , ..., β N T (2.4)
where

Q p in is the upstream discharge (the superscript p ∈ [1..P] denotes the observation time), Q p l,d is the lateral discharges injected in the inflow cell d ∈ [1..D] (note that Q l,d = q l,d ∆x), b i the river bed elevation (i ∈ [1.
.I] denotes the computational XS index in space) and, for each patch n ∈ [1..N] with N ≤ I, the spatially distributed parameters α n and β n of the friction law (Eq. ( 2.3)) depending on the flow depth.

The inversion consists in solving the following minimization problem: c * = argmin c j(c) starting from the so-called prior c prior in the parameter space. This minimization problem is solved using a first order gradient-based algorithm, more precisely the classical L-BFGS quasi-Newton algorithm (see 2.A). Note that the sought parameters have a correlated influence on the modeled flow lines, therefore leading to an ill-posed inverse problem. In order to be solved efficiently, the optimization problem is "regularized" as detailed in 2.A. The main steps of the method are illustrated in Fig. 2.3.

Inference capabilities from WS signatures: synthetic test cases

In order to calibrate the parameters of a hydraulic model (Eq. (2.1)) from WS observables, one has to identify and understand the influence of these parameters on the observable(s): in our case the WS profile. Fluvial flows are studied here in the context of satellite altimetry (see [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]). Following [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF], the influence of the parameters on the modeled flow lines is referred to as their "hydraulic signature" (HC) and a reach is defined inbetween two fluvial HCs. Fluvial HCs can be defined in steady state (see [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF]) as "local maximal deviations of the flow depth from the normal depth h n (equilibrium), imposing the upstream variation of the water depth profile h (x) over the so-called control length Samuels [1989]". They can stem from a change in either the hydraulic resistance, XS shape, bottom slope or total flow variation through lateral exchanges.

This section studies the influence of inflows on hydraulic signatures, the capabilities of the inverse method described above to infer multiple inflows and channel parameters (either spatially constant or not), with a focus on the influence of the spatial observability of those hydraulic signatures.

Test case design

Three test cases configurations representing typical hydraulic-observations setup of increasing complexities involving lateral inflows are presented (see Fig. 2.4). Cases Ch1 and Ch2 are designed to study the effect on the inference of the overlapping signatures triggered by the propagations of, respectively, one or two lateral hydrographs, concomitantly with the one of the upstream inflow hydrograph. Case Ch3 is a complexification of Ch2 through the introduction of a non-flat bottom and a variable friction pattern K = α (x) as needed in a real river case in the next sections (β = 0 in Eq. ((2.3)) -see investigations on spatialized friction laws with multi-scale bathymetry controls in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]). For all three channels the boundary conditions (fluvial) consist in: (i) a normal depth (equilibrium) imposed downstream and (ii) sinusoidal hydrographs (see Tab. 2.1) imposed upstream and at lateral injection cells. The simulation time step is set to ∆t = 20 s for all cases. They are set up as twin experiments, where a forward run of the flow model (Eq. (2.1)) is used to generate perfect WS elevation observations which are then used to infer an unknown parameter vector c (Eq. (2.4)) with the inverse method described in section 2.3.2 and 2.A. The inferences are started from erroneous prior guesses c (0) that verify Manning-Strickler law for hydraulic consis- defined by linear interpolation between the 4 bathymetry points (diamonds, b = {2, 1.88, 1.28, 1.12} m) -and friction defined by constant values on 3 patches (α = {30, 12.5, 30} m 1/3 /s). Upstream inflow at x = 0; for Ch1, 2, 3 respectively lateral injections at abscissae (in m): x = 300, x = {300, 700}, x = {350, 700}, and observations at {x S1 , ..., x S5 } = {150, 500, 850, 450, 550}, {x Sb1 , ..., x Sb5 } = {150, 325, 450, 600, 800}, {x Sc1 , ..., x Sc4 } = {0, 300, 600, 1000}. Bottom: Sample waterlines with visible upstream and downstream controls and signatures. For the sake of clarity here, upstream and injected flow are set at 100 m 3 /s (Fr ∼ 0.12 -0.

Case

Q in Q l,1 Q l,2 Froude range Q 0 a Q T Q 0 a Q T Q 0 a Q T Ch1
100 0 6300 100 20 6300 -0.

(x, t) = Q 0 (x) + a Q (x) sin 2π
T t used in synthetic channels and resulting modeled Froude ranges. Flows in m 3 /s, time T in [s].

tency, that is unbiased priors (see investigations in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]); hydrograph priors are constant values equal to the average value of the target hydrographs.

Increasingly challenging inverse problems are considered, with increasing number of unknowns sought simultaneously and various observations densities. Cases Ch1 and Ch2 are used to infer temporal parameters only, given a channel of constant slope and friction. Case Ch3 is the most challenging case with all inflows and non constant channel parameters sought simultaneously.

Informative content of hydraulic signatures: inflow inferences

The fluvial signature from a single lateral inflow is divided in two parts (see Ch1 on Fig. 2.4, bottom): (i) in the reach downstream of the injection point, the cumulative flow

(Q = Q in + Q lat,1 )
is uniform with a water depth corresponding to the normal depth imposed downstream, (ii) in the reach upstream of the injection point an M1 backwater curve profile (see [START_REF] Chow | Open-channel Hydraulics[END_REF], [START_REF] Montazem | Représentation et segmentation hydraulique effective de rivières pour le calcul de débit par altimetrie SWOT à l'échelle globale[END_REF]; [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF] in the present "altimetry context") is obtained given the upstream flow Q in and the water depth imposed downstream of this reach as the normal depth corresponding to the cumulative flow. In the case of two distinct lateral injections (Ch2), WS signatures overlap in the most upstream reach because of the stronger backwater effect created by two downstream inflows, which represent a more challenging inference problem.

Inference trials in case Ch1 with control vector

c 1 = Q 0 l,1 , ..., Q P l,1 T , assuming a known constant Q in (x = 0, t),
show that a single observation point in space with a dense sampling in time, placed either upstream (S1, in Fig. 2.4, top) or downstream (S2) from the lateral inflow, is sufficient to infer one lateral inflow hydrograph perfectly -noiseless twin experiments -(not presented). Indeed, the hydraulic signature of a lateral inflow is visible and fully informative either upstream from it because of its downstream control on the upstream flow line or downstream from it, in the signature of the cumulative flow. This means that as long the river is well temporally-observed regarding its response time (see [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] without lateral inflows) and that the temporal variations of the observed system stem from a single control, only one spatial point is needed to infer this parameter.

In the case of two distinct lateral injections (Ch2), WS signatures overlap in the most upstream reach because of the stronger backwater effect created downstream by the two inflows, which represents a more challenging inference problem considering the unknown control vec-

tor c 2 = Q 0 in , ..., Q P in ; Q 0 l,1 , ..., Q P l,1 , Q 0 l,2 , ..., Q P l,2 T
. Several variants of Ch2 are considered to study the possible misattribution of flow controls (locations, amplitudes and frequencies) in case of identical inflow hydrographs (Ch2a), the backwater influence of inflow hydrographs on Q in downstream signature observed at S1 given 4 times larger inflow amplitude (Ch2b) or 10 times higher frequency (Ch2c), different observations samplings "mixed" inflows signatures (see Fig. 2.4 and Tab. 2.1).

Inference of multiple inflows

For all cases, using perfect and dense observations in space (1 every 10 m) and also in time leads to quasi perfect inferences. The influence of a sparser sampling and of the observability patterns of overlapping WS signatures on the identifiability of multiple inflows with the present inverse method is studied here -without a priori weighting of the parameters in the inverse method, that is equal and unadjusted σ values (see 2.A). The inferred hydrographs are summed up in Fig. 2.5. Scores are given in Tab. 2.2, including cost function values and iterations number at convergence.

Variant Ch2a

Given only one observation station by reach (S1, S2, S3) very satisfying inferences of the 3 inflows are obtained(Fig. 2.5, red line). Hence sufficient information is provided by those three stations observing distinct signatures in each reach from upstream to downstream: (S1) propagation of the inflow Q in (x = 0, t) in presence of the overlapping backwater effects due to

Q l,1 (x = 300, t) and Q l,2 (x = 700, t); (S2) propagation of Q in (x = 0, t) + Q l,1 (x = 300, t)
in presence of the overlapping backwater effect due to Q l,2 (x = 700, t); (S3) the propagation of the total discharge without downstream control.

Variant Ch2b

Assimilation is more difficult than in Ch2a but inferred hydrographs (Fig. 2.5, red line) are still accurate (Tab. 2.2). This testifies to the ability to discriminate multiple sources of various amplitudes given observations of hydraulic signatures at higher frequency and at pertinent locations (S1, S2 and S3).

Interestingly, this case highlights the expected misattribution behavior between inflow sources as shown by an intermediate iteration (Fig. 2.5, orange line) and remaining to a lesser extent at convergence (red line): Q in and Q l,1 are respectively over-and underestimated). This may be due to the relatively higher contribution of Q in to the observed signature (it impacts WS elevation at S1, S2 and S3) and consequently its contribution in the cost function (observation part).

Note that the final overestimation of Q in in Ch2b is slightly greater than in Ch2a. This is likely due to greater WS elevation variation at S1 caused by backwater from Q l,1 , which is first attributed to Q in since it has more impact on the cost function. Remember that, with perfect observations of WS signatures, at the end of the optimization process, nearly perfect hydrographs are inferred. However, the small flow misattributions during this optimization shows the difficulty of inferring multiple controls using an observation located in a strong backwater signal.

Variant Ch2c Perfect inferences are obtained. An intermediate iteration (Fig. 2.5, orange line) shows that the expected misattribution of frequencies for all 3 inflows is present, though it disappears at convergence (Fig. 2.5, red line). This testifies to the ability to discriminate multiple sources of various frequencies given observations of hydraulic signatures at higher frequency and at pertinent locations (S1, S2 and S3). Variant Ch2d Convergence is achieved but the flow upstream of S4 is misattributed between Q in and Q l,1 . Signatures of Q in and Q l,1 are only observed mixed, downstream of Q l,1 (at S4 and S5) and downstream from both Q l,1 and Q l,2 (at S3). Given that all stations are located in the downstream influence of both inflows, the distribution of flow between them makes little difference on the observed WS dynamics. This confirms the need to have at least one observation station between each sought inflow in order to be able to "separate" them.

Synthesis

These first tests showed that for inferring multiple inflows, i.e. spatially distributed temporal controls, a minimal spatial observability of their WS signature is required with one observation point between each inflow here. In case of observation stations affected by backwater influence, the potential difficulty of separating multiple inflows from their "mixed signature" is highlighted; using a higher spatial density of (simultaneous) observations leads to improved inferences in the present configuration. Moreover, using observations with high temporal density (with regards to the response time in the considered river system) and low spatial density,

Case Q in Q l,1 Q l,2 Cost N ite RMSE rRMSE (×10 -6 ) RMSE rRMSE (×10 -6 ) RMSE rRMSE (×10 -6 ) Ch2a 0.
RMSE = 1 n n ∑ i=1 Q i target -Q i in f ered 2 , rRMSE = RMSE/ n ∑ i=1 Q i target
different frequencies can be correctly attributed to multiple inflows (as highlighted for a single upstream inflow in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]). Furthermore, note that if a supercritical regime occurs in a reach between inflows, their hydraulic signatures are disconnected (not shown), effectively reducing the assimilation problem to that of case Ch1.

Multiple and composite controls inference

In this section multiple inflows are sought simultaneously with channel parameters on case Ch3. Three friction patches are consistently applied to sub-reaches in-between the 4 sought bathymetry points. The control vector is:

c 3 = Q 0 in , ..., Q P in ; Q 0 l,1 , ..., Q P l,1 , Q 0 l,2 , ..., Q P l,2 ; b 1 , b 2 , b 2 , b 4 ; α 1 , α 2 , α 3 T (2.5)
. Searching both inflows and channel parameters creates a configuration (intently) prone to equifinality problems on the sought parameters having correlated influence in the water surface signal. Three observation configurations (see Fig. 2.4) are studied: one with a high station density (Ch3a: 100 stations, 1 every 10 m), another with fewer stations (Ch3b: 9 stations, Sb 1..5 and Sc 1..4 ) and a third one with even fewer stations (Ch3c: 4 stations, Sc 1..4 ). Priors for inflows are those defined for case Ch2 (subsection 2.4.2.1), priors for channel parameter are hydraulically consistent with flow priors and initial flow line. For this equifinality prone configuration, the σ values used in the inverse problem regularization, related to the sought parameters (see section 2.A) and denoted as weights, are given in Tab. 2.3.

Inference results are presented in Fig. 2.6. In red, the final estimate of c 3 for Ch3a with the "default" weights set (see Tab. 2.3). In green, final inferences for variant-specific parameter weights adjusted through trial and error. In orange, intermediate inferences with the "default" set of parameter weights. Equal values of 1, corresponding to "no weighting", were also tested: they lead to inaccurate inferences (not shown) and thus the "default" weights producing more interpretable results are preferred. In further iterations, after the ones plotted in orange, behav-iors similar to the Ch3a "default" weights inferences (Fig. 2.6, in red) appear (not shown), i.e. a shift of inferred hydrographs and Strickler coefficients away from the target. Also note that the inferred flow oscillation in the first time step stems from the influence of the initialization scheme (see section 2.3.1) in the optimization on this quickly responding channel.

Variant Ch3a Q in is underestimated while the local friction is overestimated, denoting a local tendency to equifinality. This is linked to a strong backwater influence, created by both Q l,1 and the increase in friction at x = 300 m. This local inflow error leads to compensation in downstream hydrographs. By adjusting parameter weights through trial and error, accurate inferences are obtained (Fig. 2.6, in green). This means that dense observations of the WS elevation are not sufficient for inferring all flow controls contained in c 3 and that spatially distributed regularization parameters, acting as weights in the parameter search, are required.

Variant Ch3b and Ch3c With sparse observations, the "default" weight set leads to worse inferences. However, the existence of a set of adjusted weights that lead to good inferences (Fig. 2.6, in green) is enough to show that the minimum observation spatial density of 1 station between each inflow can be sufficient to infer the extended control vector c 3 . Note that adjusted weight for Ch3b and Ch3c are different from adjusted weights for Ch3a (see Tab. 2.3).

Using less observation points in space, the influence of spatial parameters decreases without loss of meaningful information and thus the relative influence of inflows increases. This simple test highlights the weighting influence of the σ parameters in the regularization method in the present flow configuration. The spatial distribution and density of WS observations along with the weights change the hydraulic representativity of spatially distributed parameters in the optimization process.

The main difficulty uncovered with these academic cases is the challenge presented by simultaneous inferences of multiple inflows and channel parameters from their potentially overlapping hydraulic signatures. However, in the case of unbiased prior parameters and dense WS observation patterns relatively to those of spatio-temporal controls, satisfying inferences are obtained with the present inverse method. A real and complex river case is considered hereafter. variants. In red, final inference with "default" parameter weights (see Tab. 2.3). In green, final inference with adjusted parameter weights. In orange, intermediate inferences with "default" parameter weights.

Case σ Q in m 3 s -1 σ Q l,1 m 3 s -1 σ Q l,2 m 3 s -1 σ α m 1/3 s -1 σ b (m) Cost N ite "Default" Ch3a 50

Effective hydraulic model of the Negro river

After addressing increasingly challenging hydraulic inverse problems on synthetic test cases in the previous section, a real complex river flow case is now considered. It consists in 871 km of the Negro river, including several confluences with tributaries and anabranching flow zones.

The reach is located upstream of the Solimões-Negro confluence which is responsible for significant backwater effects (see e.g. [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF]). This section presents the elaboration of effective flow models in view of performing forward and inverse flow modeling from WS observations of varying sparsity in the next section.

The modeling approach consists in (i) a 1D hydraulic model (full Saint-Venant equations, see subsection 2.3.1) (ii) based on effective XSs defined from multi-satellite and in situ data and (iii) weakly coupled to the large scale hydrological model MGB Collischonn et al. [2007]; [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]; Paiva et al. [2013a]. The idea is to build an effective river flow model both in coherence with the main hydrological signals (inflows) propagations along with observable flow surface signatures and HCs (see [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF]). As shown in what follows, this 1D approach allows for a fair representation of flow propagation and longitudinal signatures, which are the core focus of this paper.

Study zone

The study domain corresponds to the main stream of the Negro river, a major "left-bank" Amazon tributary draining the north part of the basin, with an average discharge of 28 400 m 3 /s Agência Nacional de Águas e Saneamento Básico (ANA). The reach covers the 871 km upstream of its confluence with the Solimões and presents singular channel morphologies such as multichannel flow zones mainly located in two large grabens [START_REF] Latrubesse | The late quaternary evolution of the negro river, amazon, brazil: Implications for island and floodplain formation in large anabranching tropical systems[END_REF]). Part of the reach is strongly influenced by the control imposed by the Solimões river at its confluence (average discharge of 100 819 m 3 /s according to ORE HYBAM gauge data [START_REF] Cochonneau | L'Observatoire de Recherche en Environnement[END_REF], their confluence gives birth to the Amazon river). This HC is due to higher discharge and a consequently lower slope of the Negro river in its lower reach when compared to the Solimões river near to the confluence [START_REF] Filizola | The Rio Negro and Rio Solimões confluence point-hydrometric observations during the 2006/2007 cycle[END_REF]; [START_REF] Callède | Détermination de l'altitude du zéro des stations hydrométriques en Amazonie brésilienne. Application aux lignes d'eau des Rios Negro, Solimões et Amazone[END_REF]. The reach of interest has been crosscut by 18 Envisat ground tracks every 35 days from 2003 to 2010 (see Da [START_REF] Da Silva | Water level dynamics of Amazon wetlands at the watershed scale by satellite altimetry[END_REF]), representing 68 to 79 measurements of WS elevations at each of the 18 Virtual Stations (VS). Note that the measurements are not simultaneous for each station.

Effective models construction

This section presents the elaboration of effective flow models from multi-satellite data. First, a G1 "sparse" channel geometry is built from effective bottom elevations at Envisat VS resolution. Next, in view to test the additional constraints brought by spatially dense satellite data (synthetic SWOT), a more spatially detailed effective channel geometry G2 is built using a high resolution water mask and an in situ flow line as explained below.

Effective geometry G1 from altimetry and optical data

An effective 1D channel with effective rectangular XSs is set up from available multi-satellite data (altimetry, optical) and a large scale hydrological model following Garambois et al. [2017, Figure 2.7: Study zone on the Negro river. Envisat Virtual Stations (red points) are numbered from 1 to 18 starting from upstream.

The boundaries of the studied reach are defined by stations 1 and 18. The 21 tributaries (green points) are divided between actual rivers (light green) and inflow points from the hydrological model (dark green). SWOT swaths give an almost full spatial coverage of the reach. The average water extent map (from [START_REF] Pekel | High-resolution mapping of global surface water and its long-term changes[END_REF], in gray) was used to extract width values. 2020]. According to [START_REF] Latrubesse | The late quaternary evolution of the negro river, amazon, brazil: Implications for island and floodplain formation in large anabranching tropical systems[END_REF], high width to depth ratios make the rectangular channel a pertinent effective modeling approach of the true geometry, even in highly anastomosed reaches -where an error on the actual hydraulic perimeter P h hence R h (see Subsection 2.3.1) is expected. This is supported by a qualitative analysis of some additional ADCP measurements of river flow and cross-sectional bathymetry.

• The river centerline from [START_REF] Allen | Patterns of river width and surface area revealed by the satellitederived North American River Width data set[END_REF], formed by 30×30 m pixels, is used to calculate the river length and to project all spatial objects, such as VS, widths and inflow points, on a single one-dimensional reference.

• A longitudinal profile of cross-sectional WS width W is calculated from the average river extent map derived from 31 years of optic Landsat imagery by [START_REF] Pekel | High-resolution mapping of global surface water and its long-term changes[END_REF]. A single width value per centerline point is extracted in order to build a 1D rectangular geometry. For multi-channel reaches, the effective width is the sum of the widths of all channels. This underestimates the actual hydraulic perimeter. Specific hand-filtering based on hydraulic expertise was necessary in some anabranching parts of the model where the water extent may include inactive flow zones not accounted for in the present 1D effective model. Note that [START_REF] Park | The hydro-geomorphologic complexity of the lower amazon river floodplain and hydrological connectivity assessed by remote sensing and field control[END_REF] concurs to the necessity of reach-scale flow zone evaluation in the Amazon river catchment.

• An effective channel bottom elevation b env is obtained at each VS (Fig. 2.8, in red) from altimetric rating curves (RC) from [START_REF] Paris | Stage-discharge rating curves based on satellite altimetry and modeled discharge in the Amazon basin[END_REF]. Its slopes range from -7.1×10 -5 to 2.0×10 -4 m/m with an average of 7.0×10 -5 m/m. RCs were obtained by adjusting the parameters (γ, δ) of a stage discharge relationship Q = γ (Z satb) δ S 0.5 sat using WS elevations Z sat and slopes S sat gained by satellite altimetry and discharge Q simulated with the large scale hydrological model MGB [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]; [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]; Paiva et al. [2013a]) on the temporal window of interest.

Effective rectangular XS geometries are defined at the R = 18 VS using the above defined effective bottom elevations {b env } r∈ [1..R] and river widths {W 1 } r∈ [1..R] . The final model geometry (G1 = {b env , W 1 } r∈ [1..R] ) is obtained by linear interpolation between those 18 effective XSs on the model grid with ∆x = 200 m.

Effective geometry G2 at increased spatial resolution

Spatially dense WS elevation data is introduced in the form of an in situ GPS flow line with G = 579 spatial points. It was collected by survey ship along the whole studied reach over 7 days during the low-flow period in december 2010 [START_REF] Moreira | Apport des données de géodésie spatial pour l'étude du bassin hydrologique amazonien[END_REF]); it provides local WS elevations Z every 1.4 km on average and WS slopes S for every 25 km reach (ranging between 2.0×10 -5 and 8.11×10 -5 m/m, averaging at 3.4×10 -5 m/m). Under the hypothesis of a wide rectangular XS and a steady uniform flow, the Manning equation writes:

Q = K (Wh) 5/3 (W + 2h) -2/3 √ S (2.6)
The water depth writes h = (Zb) and the bottom elevation is sought using (i) the fixed WS width pattern W 2 from imagery, (ii) the WS elevation Z GPS and slope S GPS given by the GPS profile and (iii) the discharge Q from the hydrological model (see Subsection 2.5.1) on the river domain at the corresponding time t * . We invert an effective bathymetry b GPS using equation 2.6 by minimizing the square sum of misfits to b env at Envisat stations through the modification of M = 14 friction values ((α m , β m = 0) , m ∈ [1..M], friction law Eq. ((2.3))). They are simply spa- tialized into M "hydraulic" patches consistent with large scale morphological features classified as follows: single channels, multiple channels (from 2 to 3), lightly anastomosed and heavily anastomosed (Fig. 2.8, in purple). The friction coefficient values are coherent with the physical properties of the classified reaches.

The new bathymetry b GPS is coherent with the best available reference data and its corresponding set of physically distributed Strickler patches. The final model geometry is

G2 = {b GPS , W 2 } G∈[1..G] .
In the following, using either geometry G1 or G2, the hydraulic model is inflowed with time series at a daily time step upstream of the river domain and at 21 tributaries (both river tributaries and runoff inflows) corresponding to the 21 catchment cells feeding into the Negro river cells in the large scale hydrological model MGB [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]; [START_REF] Collischonn | The MGB-IPH model for largescale rainfall-runoff modelling[END_REF]). The largest of these tributaries is the Branco river at 657 km.

Effective Models calibration against altimetry

The friction of the hydraulic model (Eq. (2.1)) is calibrated against altimetric WS elevation time series following [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF], i.e. Z obs s,p env S=16, P∈[68..79] at Envisat VS, the most downstream VS being used as BC (see Subsection 2.3.1). The friction law is distributed using N = 17 "Envisat" patches with constant (α n , β n = 0) , n ∈ [1.

.N] values for each reach between two successive VS. This choice is made to avoid spatial "overparameterization" in the calibration process regarding the spatial sparsity of Envisat observations of WS signatures. The aim of parameter calibration is to obtain a "real-like" model as close as possible of the sparse observation set. Three models are considered, to assess the impact of the bathymetry refinement and of the downstream BC on the modeled hydraulic signatures and on inverse problems: a "sparse" model (M1) using channel geometry G1 and the WS elevation time series from VS 18 as BC, a refined model (M2a) with channel geometry G2 including all the spatial variability from multi-source data described above while keeping the same BC and a further changed refined model (M2b) where the BC is changed to an altimetric RC which is of interest for "operational-like" applications in other rivers and basins.

The inverse method presented in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] and described in Subsection 2.3.2 and 2.A is used here, without regularization terms, for friction calibration. Effective Strickler patches, starting from priors corresponding to average values of the "hydraulic" patches used above (Fig. 2.8, in purple), are calibrated following [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF] who use observations of the same nature. Friction patterns c * G1 , c * G2a and c * G2b found with the inverse method are shown in Fig. 2.9. Most differences in calibrated friction from M1 (Fig. 2.9, in red) to M2a (Fig. 2.9, in blue) correspond to their difference in bathymetry at the virtual station point (Fig. 2.9, gray line), i.e. a lower slope in M2a leads to a higher inferred Strickler parameter in order to match WS observations (e.g. in patch 2 and 6. Inferred parameters for M2b roughly match those of M2a, with some discrepancies in patch 2, 15 and 17. Using a different BC influences WS sensitivity to parameters and the relative contribution to the cost function of local WS misfits, which explains differences in patch 15 and 17 ; the one in patch 2 stems for the high friction values, hence lower WS sensitivity as analyzed after.

Water levels analysis

The following presents a detailed analysis of the effective hydraulic model for configuration M1, along with an analysis of changes obtained for configurations M2a and M2b.

The simulated WS elevation are compared to observed WS elevation at each Envisat virtual station in Fig. 2.10 -other time series are available in 2.B. For the 3 models calibrated above, the modeled WS are fairly close to observed WS given the limited modeling complexity and data uncertainties. More precisely, the fit to the altimetric WS elevation time series is fairly good, as shown for M1 in Fig. 2.10, and nearly unbiased as shown in Fig. 2.11(left). The WS elevation global RMSE is at 0.936 m for M1 ; similar results are found with M2a (see Tab. 2.4). Errors are greater in low and high flows, with consistent underestimations of flow amplitude upstream (VS 1 -4) which turns into overestimation downstream (VS 9 -13), before disappearing closer to the BC (VS 14 -18). VS 5 to 8 are particularly accurate. Error metrics are coherent with those from current state of the art models using satellite data (see e. Concerning the hydraulics, from downstream to upstream, relative errors are lower in anabranching reaches outside of the backwater influence starting at the Branco tributary (x = 657 km up to around x = 350 km) and in the backwater influence of the (known "perfect") downstream BC. Overall, relative errors are higher upstream, in single channeled, low water height reaches and in the Branco backwater influences. Note that 2D complex lateral flows in floodplains or retention behaviors from "igarape" rivers may happen in high flow periods (see [START_REF] Fleischmann | Modeling the role of reservoirs versus floodplains on large-scale river hydrodynamics[END_REF]; Fassoni-Andrade et al. [2020]). These unaccounted phenomenons may decrease flood wave velocities and cause hydrograph skewness [START_REF] Collischonn | Hydraulic causes for basin hydrograph skewness[END_REF]; [START_REF] Alsdorf | Spatial and temporal complexity of the Amazon flood measured from space[END_REF]; [START_REF] Fleischmann | On riverfloodplain interaction and hydrograph skewness[END_REF]).

The 1D modeling of water levels compared to altimetry observations (Fig. 2.10) can first be analyzed as follows:

• Stations 14 to 18 are located in reaches with different morphological properties. Stations 14 and 15 are located in a densely anastomosed reach upstream of the Branco river confluence, a major tributary. Stations 16 and 17 are in single channel reaches, upstream from the confluence with the Solimões river. Station 18 is in a densely anastomosed reach at the location of the BC forcing on WS elevation. Their low relative misfits do not testify to the absence of complex hydraulic behaviors in this area but rather to the dominating influence of the BC.

• Stations 5 to 13 are located in mostly homogeneous anastomosed reaches, with stations 5 to 8 in a less densely anastomosed region than stations 9 to 13. This spatial division corresponds to two trends in relative misfit, where lower misfit is seen in the less anastomosed reaches. This testifies to the difficulty of modeling potentially 2D hydrodynamics using 1D approach. Indeed, the more channels there are, the further away the simulated wetted perimeter is from the true wetted perimeter (and so the hydraulic radius). Note that parameterizing the Strickler coefficient as described in Eq. ((2.3)) and including β (x) in the control vector during the calibration process, instead of the simpler β (x) = 0 used here, does not yield a better fit in this complex case modeled with a single rectangular channel.

• Stations 1 to 4 are located on single channel reaches. Although the area seems the most suitable to be modeled in 1D, it still has the highest relative misfit to Envisat observations. For stations 1 and 2, this is partly due to effective width estimation errors being more prevalent in the relatively narrow channel (around 2 km in width). Furthermore, note that effective channel bottom elevation for these stations are respectively 37.3 m and 36.3 m while the lowest Envisat WS elevation observation are respectively 36.6 m and 35.8 m. This corresponds to low-flow water heights of 0.7 m and 0.5 m which do not fit field measurements. Consequently, relatively high friction coefficients are inferred between station pairs 1-2 and 2-3 to fit low water depth. This misfit might be due to data error, including effective width errors for stations 3 and 4 located in areas of sharp width variations. Note that the higher the friction values, the less sensitivity of the modeled WS elevation, which explains the highest spread of Strickler coefficient (K = 40 to 55 m 1/3 /s) in reach 2 found during calibration for the 3 models (Fig. 2.9).

The introduction of the refined geometry G2 in M2a, recall for generating spatially distributed SWOT data and to perform inference tests hereafter, has low impact on WS elevation bias and errors at Envisat VS (see Fig. 2.11), with only stations 1, 2 and 3 showing significant change. Using a rating curve as downstream BC in M2b mostly impacts the downstream part of the model where some misfit to altimetry data appears. Indeed, it is more difficult, using a simple power law depending on the local flow variables, to capture the influence of the confluence with the Solimões river -not modeled. The latest having strong discharge variations out of phase with the one of the Negro river itself (e.g. [START_REF] Montazem | Wavelet-based river segmentation using hydraulic controlpreserving water surface elevation profile properties[END_REF]).

Global RMSE (m) Global Average bias (m) Upstream RMSE (m)

Effective model analysis

As a preliminary to hydraulic parameters inference from WS observables, this subsection studies the spatio-temporal features of the simulated hydraulic signatures, their sensitivity to model parameters and their observability given a SWOT sampling. First, an analysis of a flood wave propagation, resulting from multiple inflows, and its hydraulic signature visibility is performed using identifiability maps following [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]. The latter consist in a space-time representation of the WS signal and flow propagation features against the observability pattern. These maps, inspired by the theory of characteristics (see [START_REF]Hydrodynamique de l'environnement[END_REF]; [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF]), enable to read how the sought upstream discharge information is sampled in the downstream WS deformations and help to estimate inferrable hydrograph frequencies. Next, a numerical sensitivity evaluation of the flow model is carried out.

In the context of regional hydrological modeling including river networks representation, the sensitivity of the present flow model is studied by using erroneous inputs. These inputs are also used in Section 2.6 as erroneous priors for various assimilation setups.

• Inflow: two hydrograph sets (containing lateral inflows and the upstream BC inflow) corresponding respectively to 70% and 130% of the true hydrographs are used as erroneous values for sensitivity trials and are referred to as Q -30 FG and Q +30 FG respectively.

• Friction: two Strickler repartitions, with coefficient values worth respectively 70% and 130% of the truth are used as erroneous values for sensitivity and are referred to as K -30 FG and K +30 FG respectively.

• Bathymetry: the inflow sets Q -30 FG and Q +30 FG and the true Strickler values are used to dig two bathymetries as described in Section 2.5.3. The bathymetry derived from underestimated flows is referred to as b -30 FG (it overestimates the true bathymetry), and the other is referred to as b +30

FG .

Identifiability map SWOT will provide spatially distributed observations with interesting revisit frequencies at the scales of the current river domain and hydrological signal propagations. Fig. 2.12 shows the evolution of the simulated WS elevation anomaly during the yearly peak flow (red-blue heatmap) as well as its multiple SWOT observability (in black). Based on the modeled flow, accounting for several inflows, the propagation of an intumescence corresponding to the annual flood wave signature is represented along the river through the maximum WS elevation in time (following [START_REF] Montazem | Représentation et segmentation hydraulique effective de rivières pour le calcul de débit par altimetrie SWOT à l'échelle globale[END_REF]) (Fig. 2.12, top, blue points). This intumescence propagation is visible on the upstream 400 km of the river from day 164 to day 173. It is detected by a SWOT swath at t = 166 d and another one at t = 170 d. It is more difficult to detect this signature in the downstream part of the river (x > 400 km) affected by the strong downstream control imposed by high water depths at the Negro-Solimões confluence; a downstream control due to the Branco tributary also overlaps from x = 657 km to around x = 400 km. This control can be seen through the tracked WS elevation maximum (Fig. 2.12, top, in gray), where an early rise in WS elevation originates from x = 657 km, and through the extreme waterlines (Fig. 2.12, bottom, in blue), which highlights the change in length of this influence in low and high flows. As a consequence, WS observations on the downstream part may contain combined information due to the upstream hydrographs propagation but also to the expression of downstream controls.

The maximum WS elevation is tracked for simulations with erroneous parameters as defined above (Fig. 2.12, top, in red, green and cyan). They are not plotted where the flow displays "pool behavior" (gray points). They highlight the sensitivity of propagation to model parameters which is also an important point when they are varied during an optimization process as featured in Section 2.6. The propagation time from 0 to 400 km can be evaluated to around 10 days, and is estimated as follows for the rest of the river domain.

The conservative part of the Saint-Venant equations (i.e. without source terms) is hyperbolic: some quantities depending on the water depth and velocity (known as the Riemann invariants) are transported by waves at speeds different from the flow speed (see e.g. [START_REF]Hydrodynamique de l'environnement[END_REF]; [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF]). The wave celerities are U + c and Uc with c = gh for rectangular XSs (see analysis of propagation features in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]). For the fluvial regime of interest here (Fr = U/c < 1), information propagates both downstream and upstream and the Riemann invariants are modified along the wave due to the source term effects. The wave celerities obtained on the Negro river model are given by reach in Table 2.5, relatively high wave speeds are obtained hence propagation of information both upstream and downstream, with spatio-temporal variability. The WS signature (and the discharge) thus reflects the nonlinear combination of information coming from both upstream (due to inflows variations) and downstream (due to local HCs or downstream BC -see the method of characteristic in [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF]). This highlights the difficulty of inferring multiple inflows from sparse observations of WS signature, especially given uncertain channel parameters and backwater effects.

Nevertheless interesting frequential information can be gained from the identifiability map as introduced in the case of a single inflow. Using the kinematic wave speed 5 3 U (Fig. 2.12, top, dashed blue line) which compares fairly well to the intumescence speed on the upstream part of the reach (Fig. 2.12, top, x < 400 km). This gives an approximate propagation time T wave = 26 d on the whole domain, greater than the SWOT observation cycle period of 21 days. This brings the reach identifiability index to I ident = 1.23 (defined as I ident = T wave /∆t obs , i.e. the average number of time a wave is observed, see [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]). However, in the present case, the notion defined by [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] accounts for a single upstream inflow, not spatially distributed lateral inflows with potential upstream backwater controls. Actual identifiability indices for reaches in between each lateral inflow would be much lower (estimated identifiabilities in between each inflow pair are given in Tab. 2.5 considering a fictitious ∆t obs = 1 d full domain observability). Furthermore, SWOT swaths observations consist in WS snapshots on different parts of the river domain at given times, hence containing various and mixed signatures (in the sense introduced in Section 2.4) of both several inflows and channel parameters -the more downstream, the more aggregated is the inflow information. Inferences of multiple inflows and frequential analysis are presented in the next section given known or uncertain channel parameters, spatio-temporally dense or sparse (SWOT) observations.

Inferences from satellite observables

This section studies the challenging inference of ungauged channel parameters and multiple inflows on the Negro river case, which represents a real and complex large scale problem. Typical inverse problems in hydraulic-hydrological modeling are studied here considering SWOT WS observations. The inference of channel parameters or/and inflows in the 1D Saint-Venant model is addressed using the inverse method presented in Subsection 2.3.2 (see also Section 2.A). The downstream BC is set as a known altimetric rating curve. Three observation sets are generated: spatially and temporally dense (∆t = 600 s) observations (DenseSet), SWOT observations from the hydraulic model outputs masked by SWOT swaths (SWOTSet) and noisy SWOT observations using the large scale simulator CNES to add realistic measurement noise (SWOTNoiseSet). We first present inferences of inflows only, then of channel parameters, and finally of all those spatiotemporal controls simultaneously.

Multiple hydrographs inferences

Depicting flow structure within a river network and a catchment is a key issue in hydrological modeling, especially in ungauged basins. Seeking to infer, from distributed WS observations, prior guess c (0) consisting in true hydrographs affected by uncertainties of ±30%, that is Q +30 FG and Q -30 FG as defined above. Note that the inference is started from a hydraulically consistent initial state using an unbiased prior in the first time steps (see investigations in [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]); the prior values of regularization parameters σ Q l correspond to inflows magnitudes.

The inferred hydrographs from inflow prior Q -30 FG are presented in Fig. 2.13 for DenseSet (green lines) and SWOTNoiseSet observations (orange lines). Results from prior Q +30

FG are available in 2.C. SWOTSet and SWOTNoiseSet give almost identical inferences, therefore only the SWOTNoiseSet inferences are presented. For under-and overestimated priors, the assimilation of dense and SWOT observations enables to infer the true hydrographs fairly well. RMSE ranges from 8.86 m 3 s -1 at x = 465 km up to 578.31 m 3 s -1 for the Branco tributary at x = 657 km. RMSE for all inferences presented in Fig. 2.13 can be found in Section 2.C. Some inferences show global under-or overestimations (e.g. x = 216 , 388 , 789 km). These biases are linked to the prior bias. Strong and numerous overlapping backwater signals may also influence flow misattribution, as discussed in the academic cases (Section 2.4) for a small scale model. As tested in numerical experiments (not shown), increasing a scalar value σ Q l,d can give more effective weight to an hydrograph Q l,d in the inference and it can be found further away from its prior guess, which highlights the role of the covariance matrix used for regularization.

Note that temporal oscillations appear on the inferred hydrographs when using SWOTNois-eSet which is "temporally sparse" observation patterns compared to flow propagation, which is not the case of DenseSet. These oscillations are especially present in downstream inflows, which may link them to particular hydraulic responses in the BC influence zone, although they can be seen in upstream inflows as well. They tend to be prevalent in declining limbs of hydrographs (e.g. in Fig. 2.13, at 789 km, from day 120 to 300).

Note that, regardless of oscillations, inferences tend to be further from the truth in decreasing hydrographs. These oscillations are not the effect of signal misattribution, as they are present with any number of inferred hydrographs (not shown), nor are they caused by the prior's shape, as filtered priors also lead to oscillations (not shown). Instead, the oscillations seems to stem from the combination of the low observation frequency compared to the spatially distributed inflow hydrographs and the nonlinear hydraulic response. Keep in mind that we track flow information through WS elevation deformations caused by the nonlinear propagation of parameter signatures (see Subsection 2.5.3.2).

A sample illustration of those oscillations on the simulated WS elevation is presented in Fig. 2.14, at 870 km, downstream for three oscillating inferred hydrographs (at x = 738, 754 and 789 km). The inferred waterline from SWOTNoiseSet is compared to the truth at all simulated times and at SWOT pass times only. The misfit is logically lowest at SWOT pass times (goal of the optimization), while unobserved periods exhibit a slightly oscillating (unconstrained) misfit. Higher frequency observations, such as DenseSet, prevent this behaviors through a more com-plete spatio-temporal observability of the WS signatures, hence constrain the spatio-temporal parameters inference further. Some model configurations where temporal parameters are discretized at a greater time step than observation one do no exhibit such behaviors (e.g. results with DenseSet, [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]). As already shown in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] for the identification of a single hydrograph, the identification is possible only in time windows representing the wave propagation time T wave ∼ 26 days in the present case, yet with multiple inflows and observation sample (see Subsection 2.5.3.2). As a consequence, outside the "identifiable time windows", the inferred values are not necessarily representing reality (see related WS misfit in Fig. 2.14). They are the optimal solution corresponding to the considered priors of the optimization problem. In practice, this means that introducing an additional regularization term j reg , like t 0

∂ 2 h
∂t 2 dt at observation points, would smooth (as following a spline curve) between the identifiable windows instead of the obtained oscillations. This smooth discharge curve would not be more physical than the present oscillations and we made the choice to not hide this well understood phenomena. It is a logical consequence of the disparity between the samplings of observations and parameters and does not impede interpretations of hydraulic signatures and identifiability.

Seeking to infer a control that is both temporally and spatially distributed represents a challenging assimilation problem. In the present case: (i) the observation frequency now plays a role in identifying the hydraulic signature, on top of its spatial density and resulting flow propagation: (ii) varying nonlinear flow propagation, and so WS signatures, can result in different inferences depending whether they are performed from observations of rising/declining hydrographs propagations (local Q (Z) hysteresis) and (iii) indirect contributions to parameter weight in the inverse method appear, as successive hydrographs influence the whole downstream water line (established in Subsection 2.4.2), which gives greater "effective weight" to upstream hydrographs as the cause of the misfit is observed in more stations and thus accounted for multiple times in the cost function. The inferred flow variations may be compensating for errors made at upstream stations with different SWOT pass times, impacting their WS elevation either through backwater control or through the modification of the BC and its own backwater effect.

Spatial parameters inference

The inference of effective channel parameters is studied here considering a control vector composed of all friction coefficient values and bathymetry points. The bathymetry is composed of I = 436 free points (1 every 2000 m) between each of which it is obtained by linear interpolation. SWOTSet is used, with a spatial observations sampling (1 point every 200 m), i.e. 10 times greater than the sought bathymetry for observed reaches. Two inferences from hydraulically consistent priors are presented, one with the refined bathymetry b -30 FG introduced in Subsection 2.5.3.2 (Fig. 2.15, in green) and another one with b +2.6m env , a shifted bathymetry from the M1 model in Subsection 2.5.2.1, in red. The 2.6 m correspond to the spatially averaged shift of b -30 FG compared to b env . b +2.6m env does not contain a priori information on target bathymetry shape -such as a coarse DEM prior. The friction prior is K -30 FG for both inferences. Using b -30

FG and b +2.6m env as priors, the inference of channel parameters (friction and bathymetry) respectively reach a RMSE of 4.362 m 1/3 s -1 , 1.231 m and 4.762 m 1/3 s -1 , 1.913 m . Upstream (x = 0 -110 km), irregularities appear in both inferred bathymetries and correspond to un- FG . This testifies to the role of the a priori bathymetry shape knowledge contained in the prior when the WS elevation sensitivity to bathymetry is low. Strong backwater effects make the inference of channel parameters more difficult and parameter compensation appears due to the lessened sensitivity of the simulated WS (e.g. in green, at the last friction patch). The refined bathymetry prior b -30 FG leads to inferred bathymetry and friction that are closer to the truth. It will be used in the extended control vector inferences below.

Inference of channel parameters and inflows

This section investigates the simultaneous inference of both unknown inflows and channel parameters on the large scale Negro river case; it combines all previously mentioned difficulties and corresponds to an ungauged configuration. In the following, the aim is to determine: 1) if SWOT data are sufficient to infer the extended control vector given unbiased prior parameters; 2) how the added spatial complexity from lateral inflows impacts spatial parameter inference. In addition, further investigations on the impact of lateral inflow prior shape, representing for example hydrological modeling uncertainty in a simple manner, are presented. The considered extended control vector is the following:

c ext = (Q l,x=65 (t) , Q l,x=502 (t) , Q l,x=657 (t) , Q l,x=754 (t) ; b 1 , ..., b H ; α 1 , ..., α N ) (2.7)
The inferences are performed from DenseSet and SWOTNoiseSet. The bathymetry and friction priors are b -30

FG and K -30 FG respectively. Four major lateral inflows located at x = 65, 502, 657 and 754 km (Fig. 2.12, bold dashed bars) are considered. Their reduced number facilitates the analysis of their spatial impacts. The other inflows are set to their target values. Two inflows prior types are used: Q f lat FG , that gives no a priori on hydrograph shapes and Q f ilter FG , hydrographs obtained by applying a 80 days moving average filter to the true hydrographs. Prior flow values in Q f lat FG are set to the target flow values from the first time step up to 120 days for the sake of initial hydraulic consistency. Inferences of all parameters for these inflow priors are presented in Fig. 2.16 and Fig. 2.17. The inferred control vectors are referred to as c * f lat and c * f ilter . Inferred parameter scores can be found in Tab. 2.6.

Inferred spatial parameters patterns are similar to those obtained previously without unknown inflows in Subsection 2.6.2. c * f lat features a fair bathymetry fit downstream (x = 600 -870 km) while c * f ilter stays close to the prior value. This may be due to the different range of the simulated hydraulic responses in the first iterations: using Q f lat FG leads to an increase in WS elevation sensibility to bathymetry. Upstream (x = 0 -110 km), increased bathymetry irregularities in c * f lat are linked to the erroneous prior Q f lat FG leading to bathymetry errors in the first iterations, coupled with lower inferred Strickler coefficients, hence a lessened impact of bathymetry on the water surface and the inability to correct the "initial" errors.

In terms of temporal behaviors, both priors give fair estimates of HCs for DenseSet. Inferences from SWOTNoiseSet are close to those from DenseSet with the presence of oscillations and the rising part of hydrographs are better fitted than decreasing ones, as observed in Subsection 2.6.1. In both c * f ilter and c * f lat and for both observation sets, a correlation between sought inflows at x = 502; 657 and 754 km appears. The Branco river flow, at x = 657 km, is better inferred and its well fitted peaks are also found in the two smaller rivers (e.g. at 520 days), which are in its upstream and downstream influences zones (see Fig. 2.12, left). In all inferences, the total flow at the downstream BC closely matches that of the truth, which means that only hydraulic signature is misattributed across the 4 inferred temporal parameters, not on the total flow. In c * f ilter , more accurate inferences are obtained, with a smaller influence of the Branco river on other inflows in its influence zone and more accurate inference of small scale behaviors. The filtered prior Q f ilter FG introduces information on low frequency behaviors of the sought inflows, helping the assimilation process to converge to correct the target inflows. This configuration allows for a better fit of small scale variation in the controls.

Conclusion

This paper investigated the inference of river channel parameters and multiple inflows from water surface signatures in the context of satellite altimetry with the forthcoming SWOT mission and using water extents from optical data as well. The HiVDI inverse method presented in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], based on the 1D Saint-Venant equations and a variational assimilation scheme adapted to account for lateral inflows (mass and momentum injections). Given hydraulically consistent prior guesses and regularization weights, it is successfully applied to synthetic test cases and a long reach of the anabranching Negro river in the Amazon basin using multisatellite data. Through low Froude synthetic cases, it is shown that the signature of a lateral inflow is visible downstream from the inflow point through the total flow signature and can be visible upstream in case of downstream control at the injection. Following this analysis and using the HiVDI variational assimilation method (global in time and space), a study of the minimum spatial density of water surface observations necessary to infer lateral inflows from their hydraulic signatures is carried out. Synthetic twin experiments yield the following results: (i) given high observation temporal frequency relative to model hydraulic response, perfect inflows inferences can be obtained; (ii) to correctly attribute signatures between multiple lateral inflows, a minimum of 1 observation point between each injection cell is necessary; (iii) when simultaneously inferring inflows and/or channel parameters, a sensitivity to parameter weights (see Section 2.A) appears; (iv) given a priori parameter weights, accurate inferences of inflows and channel parameters is achievable even with the minimum spatial observability.

A method for building effective river models in coherence with multi-satellite data and including realistic spatial variations is introduced based on multi-source data of water surface elevation, width and slopes. This method makes use of (i) multi-mission altimetric rating curves (see [START_REF] Paris | Stage-discharge rating curves based on satellite altimetry and modeled discharge in the Amazon basin[END_REF]) or equivalently a distributed hydrological model and altimetry data and (ii) water surface width like those from current databases (see [START_REF] Allen | Global extent of rivers and streams[END_REF]); it should be applicable to rivers from the future SWOT database. It is applied here to build a simple effective 1D model of the Negro river upstream from its confluence with the Solimões river. It fits currently available satellite water surfaces signatures and contains real-like spatial variabilities and flood wave propagation features.

The inference capabilities of spatially distributed channel parameters and inflows from synthetic SWOT observations are highlighted on the Negro river case given hydraulically coherent priors. The inference of temporal parameters in the form of 21 spatially distributed lateral inflow hydrographs leads to accurate estimates and low water surface misfit at observation times. High frequency observations give good inferences, with an expected sensitivity to both prior bias (see [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]) and prior shape.

SWOT-like observations lead to comparable inferences, with slight oscillations due to the frequential disparity between observations temporal controls combined to their spatial distribution and the resulting nonlinear flow propagation on the domain, as analyzed with identifiability maps. Those oscillations related to model-observations time scales could be overcome by introducing additional regularizations -not done here for the sake of hydraulic analysis. Inference of purely spatial parameters (bathymetry/friction) were carried out as well, leading to some complementary remarks: (i) channel parameters equifinality is most present in the downstream part of the model, where the waterline is influenced by the strong backwater effect of the boundary condition (Solimões river) which diminishes water surface sensitivity to other parameters; (ii) bathymetry prior shape influences the inferred bathymetry. Finally, simultaneous inference of channel parameters and spatially distributed inflows was achieved with satisfying accuracy. We show that, with the present method, large scale temporal parameter variations can be found from synthetic SWOT observations even without a priori knowledge of the shape of the hydrological response, but that small scale variations can be better inferred with a priori hydrograph shape knowledge.

Recall that the estimation of discharges and channel parameters from (SWOT) WS observations is a difficult inverse problem because of the correlated influence of flow controls on the observable water surface signatures -non uniqueness/equifinality issues. It is therefore necessary to use hydraulically consistent priors as investigated in [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using AirSWOT observations[END_REF]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] with HiVDI method that contains low complexity flow relations for deriving robust prior guesses from databases and hydrological models, or even in situ depth/discharge data when available (see [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]) -not the scope of this paper. As already discussed in [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], the VDA solution depends on the priors which are the first guess value and the covariances matrices. Ongoing research efforts in the SWOT community, in view of global discharge estimates, focus on the determination of priors through the construction of a SWOT a priori database based on [START_REF] Andreadis | A simple global river bankfull width and depth database[END_REF]; [START_REF] Allen | Global extent of rivers and streams[END_REF] and global/regional model outputs (see [START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]), constrained with available in situ gauge measurements. Note that a priori estimations/databases could be enriched or reprocessed during or after the SWOT mission lifetime and HiVDI would enable to refining discharge estimates (see [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]). Moreover, as shown in [START_REF] Larnier | Hybrid neural network-variational data assimilation algorithm to infer river discharges from swot-like data[END_REF], priors obtained by deep learning can greatly improve global estimation.

More generally, reaching unbiased estimates of discharge, from downstream to upstream of river networks with varying densities of in situ discharge data hence ungauged river portions/basins, is a crucial challenge in hydrology that could benefit from the fusion of complementary in situ and remotely sensed data in integrated models. The present study brings insight in lateral inflows inference from hydraulic signatures and paves the way for further research on integrated hydraulic-hydrological assimilation chains for river networks and in coherence with multi-satellites observables (of local hydrodynamic signatures) to benefit from them in a regionalization perspective.

Searching for distributed channel parameters and inflows given temporally sparse SWOT data and a global assimilation method brings the issue of signal attribution to the forefront, especially at the scale of a river network. Further research should focus on tributaries that could be amenable to the use of SWOT and multi-satellite observations to better constrain estimates of lateral inflows and next distributed fluxes on network models considering hydraulic-hydrological couplings. The introduction of pertinent signatures, scales and constrains in the forward-inverse models (e.g. forward operators, covariance matrices, cost function terms) should help maximizing the use of various information sources and enable smooth discharge estimates and better signal attribution, given uneven and heterogeneous satellite data in combination with other complementary databases/knowledge. This could help leveraging better inferences of hydrological responses and flow structure within a river basin and eventually enable information feedback to rainfall-runoff modules and ultimately regionalization issues.

2.A The computational inverse method

The computational inverse method is based on Variational Data Assimilation (VDA) applied to the 1D Saint-Venant model (Eq. (2.1)). The computational inverse method is the one presented in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] with an augmented composite control vector c (Eq. (2.4)): c contains a spatially distributed friction coefficient enabling to model complex flow zones (while it is an uniform friction law K (h) in Larnier et al. [2021]). This definition of K (x, h) enables to consider more heterogeneous bathymetry controls. c also contains lateral flow hydrographs Q l,d (t) , d ∈ [1..N d ] to deal with in/off-takes due to tributaries or underground flows. It is important to point out that the imposed downstream BC is an unknown of the inverse problem. It is constrained with the observed water elevations and inferred river bottom slope using a locally uniform flow hypothesis (i.e. Manning equation, see Subsection 2.3.1) -except in the last real case above. The cost function j (c) is defined as:

j (c) = j obs (c) + γ j reg (c) (2.8)
where γ > 0 is a weighting coefficient of the so-called "regularization term" j reg (c). The term j obs (c) measures the misfit between observed and modeled WS elevations such that:

j obs (c) = 1 2 (Z (c) -Z obs ) 2 O
(2.9)

The norm This optimal control problem is solved using a Quasi-Newton descent algorithm: the L-BFGS algorithm version presented in [START_REF] Gilbert | Some numerical experiments with variable-storage quasi-Newton algorithms[END_REF]. The cost gradient ∇j (c) is com- puted by solving the adjoint model; the latter is obtained by automatic differentiation using Tapenade software [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF]. Detailed know-hows on VDA may be found e.g. in the online courses [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Monnier | Variational Data Assimilation and Model Learning[END_REF].

• O = O 1/2 • 2 is
To be solved efficiently this optimization problem needs to be "regularized". Indeed the friction and the bathymetry may trigger indiscernible surface signatures therefore leading to an ill-posed inverse problem; we refer e.g. to [START_REF] Kaltenbacher | Iterative regularization methods for nonlinear illposed problems[END_REF] for the theory of regularization of such inverse problems and to [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] for a discussion focused on the present inverse flow problem.

Following [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], the optimization problem (Eq. ( 2.10)) is regularized as follows. First the regularization term j reg is added to the cost function (Eq. ( 2.8)). We simply set:

j reg (c) = 1 2 b" (x) 2
2 . Therefore this term imposes (as weak constraints) the inferred bathymetry profile b (x) to be an elastic interpolating the values of b at the control points (i.e. a cubic spline).

A specificity of the present context is the large inconsistency between the large observation grid (altimetry points) and the finer model grid. Between the sparse observations points (equivalently the control points), the bathymetry profile b (x) is reconstructed as a piece-wise linear function. It is worth to point out that the resulting reconstruction is consistent with the physical analysis presented in [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF]; [START_REF] Montazem | Représentation et segmentation hydraulique effective de rivières pour le calcul de débit par altimetrie SWOT à l'échelle globale[END_REF]. This study analyses the adequacy between the SW model (Eq. (2.3.1)) behavior and the WS signature.

Next and following [START_REF] Lorenc | The Met. Office global three-dimensional variational data assimilation scheme[END_REF]; [START_REF] Weaver | Correlation modelling on the sphere using a generalized diffusion equation[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], the following change of control variable is made:

k = B -1/2 c -c prior (2.11)
where c is the original control vector, c prior is a prior value of c and B is a covariance matrix. The choice of B is crucial in the VDA formulation; its expression is detailed below. After this change of variable the new optimization problem reads:

min k J (k) with J (k) = j (c) (2.
12)

It is easy to show that this leads to the following new optimality condition: B 1/2 ∇j (c) = 0; somehow a preconditioned optimality condition. For more details and explanations we refer to Haben et al. [2011a,b] and [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] in the present inversion context.

Assuming uncorrelated controls the matrix B is defined as block diagonal such that B = diag (B Q , B Ql,1 , . . . , B Ql,D , B b , B ff , B fi ). Still following Larnier et al. [2021], the covariance matrices B Q , B Q l and B b are set as the classical second order auto-regressive correlation matrices:

(B Q ) i,j = (σ Q ) 2 exp - t j -t i ∆t Q , B Q l,d i,j = (σ Q l ) 2 exp - |tj-ti| ∆t Q and (B b ) i,j = (σ b ) 2 exp - x j -x i L b (2.13)
The VDA parameters ∆t Q and L b represent prior hydraulic scales and act as correlation lengths. We refer to [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF] for a thorough analysis of the discharge inference in terms of frequencies and wave lengths and [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] in the present river-observation context. In the present study, the friction parameters applied to deca-kilometric patches are assumed to be uncorrelated thus the matrices B α and B β are diagonal:

(B α ) i,i = (σ α ) 2 , B β i,i = σ β 2 (2.14)
The scalar values σ may be viewed as variances ; their values are given in the numerical results section.

Finally, in a noised observation context and to avoid over-fitting noisy data, we denote by δ the noise level such that Z obs -Z true 2 ≤ δ with Z obs the observed and Z true the true WS elevation profiles. A common technique to avoid over-fitting noisy data, in the context of Tykhonov's regularization of ill-posed problems, is Morozov's discrepancy principle, (see e.g. [START_REF] Kaltenbacher | Iterative regularization methods for nonlinear illposed problems[END_REF] and references therein): the regularization parameter γ (see Eq. ( 2.8)) is chosen a posteriori such that j does not decrease below the noise level.

2.B Extended friction calibration results for the Negro river

101

This chapter is based on [START_REF] Pujol | Multi-dimensional hydrological-hydraulic coupling with multi-variate variational data assimilation for river networks and floodplains: forward and inverse cases with DassFlow[END_REF]. The article's content is featured with minor layout adjustments.

Extended abstract

This chapter presents a novel multi-dimensional (multi-D) hydraulic-hydrological model intended to allow modeling of large and complex river network at the appropriate scales and at a low computational cost, see Fig. 3.1. In turn, this would allow exploring synergies between spatially distributed flow models and datasets over large domains, combining in situ observations with high-resolution hydro-meteorology and satellite data.

The multi-D hydraulic model consists in the 2D shallow water equations (Eq. 1.6) solved with a single finite volume solver (see Subsection 1.2.2). 1Dlike reaches are built through meshing methods that cause the 2D solver to generate into 1D (see Fig. 3.2). They are connected to 2D portions that act as local zooms, for modeling complex flow zones such as floodplains and confluences, via 1Dlike-2D interfaces (see Fig. 3. 3). An existing parsimonious hydrological model, GR4H, is implemented and coupled to the hydraulic model. These developments are carried out within the DassFlow variational data assimilation framework. The adjoint of the whole tool chain is obtained by automatic code differentiation. The forward-inverse multi-D computational model is successfully validated on academic and real cases of increasing complexity. Assimilating multiple observations of flow signatures leads to accurate inferences of multi-variate and spatially distributed parameters among bathymetry-friction, upstream/lateral hydrographs, and hydrological model parameters. This notably demonstrates the possibility for information feedback towards upstream hydrological catchments, that is backward hydrology. A 1Dlike model of part of the Garonne river is built and accurately reproduces flow lines and propagations of a 2D reference model. A multi-D model of the complex Adour basin network, inflowed by the semi-distributed hydrological model, is built. High resolution flow simulations are obtained on a large domain, including fine zooms on floodplains, with a relatively low computational cost since the network contains mostly 1Dlike reaches. The current work constitutes an upgrade of the Dassflow computational platform. The source code and synthetic cases are open source and available upon request1 . Section 3.3 presents the multi-D modeling approach, the flow model and integrated hydrological module and the inverse computational method. Section 3.4 presents the validation of the 1Dlike modeling strategy using academic and real case and investigation of the VDA capabilities of the integrated hydraulic-hydrological tool chain. Inferences of hydrological parameters are carried out on an academic confluence case. Inferences of channel parameters are carried out on the 1Dlike model of the Garonne river. Inferences of upstream hydrographs are carried out on the 1Dlike model of the Adour river network. The initial article conclusion is featured in Section 3.5. Right: Example of multi-D interfaces. At the top, a "frontal" interfacing of a 1Dlike network to the channel part of a 2D mesh, featured in the Adour network model (see Subsection 3.4.3.2). At the bottom, an academic "lateral" interfacing of a 1Dlike river channel to a 2D floodplain. Both interface are numerically identical, but may be used for different purposes.

Introduction

The accurate estimation of storage and fluxes in surface hydrology is an essential scientific question linked to major socio-economic issues in floods and droughts forecasting, particularly with regards to the ongoing climate change and potential intensification of the water cycle and hydrological hazard [START_REF] Masson-Delmotte | Global warming of 1.5°c. An IPCC Special Report on the impacts of global warming[END_REF]; [START_REF] Iturbide | An update of IPCC climate reference regions for subcontinental analysis of climate model data: definition and aggregated datasets[END_REF]). In this context, advanced numerical modeling tools are crucially needed to both perform meaningful and detailed representations of basin-scale hydrological processes and provide sensible local forecasts. The quantities of interest range from discharge hydrographs on upstream ungauged parts of the drainage network to their translation into flow depth, velocities and submersion times on downstream floodplains. This information is difficult to access, especially for floods over large territories. Indeed, given the complexity of physical processes involved, their limited observability and the resulting hydrological responses, hydrological modeling remains a hard task and internal state-fluxes are generally tinged with uncertainties [START_REF] Beven | Prophecy, reality and uncertainty in distributed hydrological modelling[END_REF]; [START_REF] Schuite | Improving the spectral analysis of hydrological signals to efficiently constrain watershed properties[END_REF]; [START_REF] Milly | Climate, soil water storage, and the average annual water balance[END_REF]).

Moreover, the accuracy of high resolution hydraulic computations may still be affected by complex dynamics with wet-dry fronts, multi-scale and uncertain topography-structures and flow model parameters (e.g. friction), uncertain quantities at open boundaries (upstream inflows but also lateral ones due to sudden local runoff, downstream controls and backwater effects), internal in/outflows in urban areas, and large computational domains [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]). Thus, integrated hydrological-hydraulic approaches are required (e.g. [START_REF] Nguyen | A high resolution coupled hydrologic-hydraulic model (HiResFlood-UCI) for flash flood modeling[END_REF]; [START_REF] Hocini | Performance of automated flood inundation mapping methods in a context of flash floods: a comparison of three methods based either on the Height Above Nearest Drainage (HAND) concept, or on 1D/2D shallow water equations[END_REF]). Such approaches are now enabled by the increasing informative richness of multi-source datasets provided by high resolution hydrometeorology and satellite remote sensing in complement to in situ measurements. Nevertheless, reaching high resolution accuracy and computational efficiency for large scale applications remains a difficult challenge because of multi-scale non-linear hydrodynamic processes over large computational domains and multiple uncertainty sources. These uncertainties could be reduced by the optimal combination of models and multi-source datasets, including high resolution maps, spatially sparse in situ flow measurements but also the growing amount of earth observation data provided by new generations of satellites, drones and sensors (e.g. [START_REF] Biancamaria | The SWOT Mission and Its Capabilities for Land Hydrology[END_REF][START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200m wide river: Evaluation over the garonne river[END_REF]; [START_REF] Schumann | Exploiting the proliferation of current and future satellite observations of rivers[END_REF] among others). Indeed, remote sensing provides very interesting cartographic observations of the variabilities of worldwide catchments characteristics (topography, soil occupation, surface moisture, snow cover, ...), as well as an unprecedented and increasing hydraulic visibility over river networks [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF]; [START_REF] Montazem | Wavelet-based river segmentation using hydraulic controlpreserving water surface elevation profile properties[END_REF]; [START_REF] Rodríguez | Observing rivers with varying spatial scales[END_REF]). This growing wealth of multi-sensed information is key to the design and improvement of basin-scale models, as shown for accurate river network 1D hydraulic modeling enabled by recent multi-source altimetric and optical satellite data in [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF] and in Malou et al. [2021] (see also references therein) or accurate 2D local floodplain models with radar sensed flooding extent [START_REF] Hostache | Assimilation of spatially distributed water levels into a shallow-water flood model. Part II: Use of a remote sensing image of Mosel River[END_REF]). In order to exploit this wealth of hydrological and hydraulic information, the complexity of integrated models and assimilation methods has to be adapted to these data that are both heterogeneous in nature and of varied spatio-temporal resolutions. Moreover, an integrated flood modeling approach should also be computationally efficient in order to be applicable over entire catchments that is large computational domains. This study proposes a new integrated hydrological and multi-dimensional hydraulic modeling approach, based on the accurate and robust 2D hydraulic solver presented in [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]. It is capable of multi-variate optimization problems of high dimension using multi-source data.

Cascades of 2D hydrological-hydraulic models have been proposed in recent literature, for inundation mapping at large scales using worldwide DEM (e.g. [START_REF] Grimaldi | Effective representation of river geometry in hydraulic flood forecast models[END_REF]; [START_REF] Fleischmann | Trade-offs between 1-D and 2-D regional river hydrodynamic models[END_REF]; [START_REF] Uhe | Model cascade from meteorological drivers to river flood hazard: flood-cascade v1.0[END_REF]) with simplified hydraulic modeling) or at finer scale, e.g. at catchment scale for flash floods in [START_REF] Nguyen | A high resolution coupled hydrologic-hydraulic model (HiResFlood-UCI) for flash flood modeling[END_REF]; [START_REF] Hocini | Performance of automated flood inundation mapping methods in a context of flash floods: a comparison of three methods based either on the Height Above Nearest Drainage (HAND) concept, or on 1D/2D shallow water equations[END_REF]. In those studies, conceptual hydrological models of upstream-lateral sub-catchments are used to inflow hydraulic models of river network and floodplains in a weak coupling approach, mostly performed via external coupling of numerical models. In [START_REF] Grimaldi | Effective representation of river geometry in hydraulic flood forecast models[END_REF]; [START_REF] Fleischmann | Trade-offs between 1-D and 2-D regional river hydrodynamic models[END_REF]; [START_REF] Uhe | Model cascade from meteorological drivers to river flood hazard: flood-cascade v1.0[END_REF], a simple 2D storage cell inundation model obtained from 1D non inertial model [START_REF] Bates | A simple inertial formulation of the shallow water equations for efficient two-dimensional flood inundation modelling[END_REF] following Hunter et al. [2008], implemented in LISFLOOD-FP model), enables raster based inundation modeling over very large domains at relatively low computational cost (see also [START_REF] Fleischmann | Trade-offs between 1-D and 2-D regional river hydrodynamic models[END_REF] for coupling of this non inertial model with the large scale hydrological model MGB Collischonn et al. [2007]; [START_REF] Pontes | MGB-IPH model for hydrological and hydraulic simulation of large floodplain river systems coupled with open source GIS[END_REF]). In [START_REF] Hocini | Performance of automated flood inundation mapping methods in a context of flash floods: a comparison of three methods based either on the Height Above Nearest Drainage (HAND) concept, or on 1D/2D shallow water equations[END_REF], an original 2D hydraulic modeling approach, using "precipiton" for the resolution of the full shallow water model, proposed by [START_REF] Davy | A precipiton method to calculate river hydrodynamics, with applications to flood prediction, landscape evolution models, and braiding instabilities[END_REF], is used to compute steady inundation maps of various return periods at high resolution (5 m) for river networks and floodplains at catchment scale of several thousands of square kilometers (up to 5050 km 2 ). In [START_REF] Nguyen | A high resolution coupled hydrologic-hydraulic model (HiResFlood-UCI) for flash flood modeling[END_REF], an unsteady full 2D shallow water model (see [START_REF] Sanders | ParBreZo: A parallel, unstructured grid, Godunovtype, shallow-water code for high-resolution flood inundation modeling at the regional scale[END_REF]) is applied at relatively high resolution (10 or 30 m) in the river network and floodplains on a 808 km 2 catchment. Note that sequential data assimilation methods based on the Kalmann filter have been carried out extensively for mono-variate data assimilation with such models (see e.g. [START_REF] Brêda | Assimilation of satellite altimetry data for effective river bathymetry[END_REF] with simplified hydraulics in a satellite observability context references therein and Table 3.1) at varying spatio-temporal resolutions. Current model development strives to propose combinations of high resolution accuracy and fast computation times over large domains and to incorporate multi-source data assimilation methods for large spatially and temporally distributed controls. This paper aims at providing an innovative and effective way to achieve these goals.

In order to combine local accuracy and computational efficiency, the association of full 1D and 2D hydraulic models is an appropriate approach for simulating a basin-scale network in a way that is both practical and adequately accurate. Methods for coupling models of different dimensions have been developed Miglio et al. [2005a,b]; [START_REF] Amara | Hydrodynamical modelling and multidimensional approximation of estuarian river flows[END_REF], classically using domain decomposition [START_REF] Gervasio | Heterogeneous coupling by virtual control methods[END_REF], or more recently using local 2D 'zooms' overlapping with the 1D domain, in a variational data assimilation framework [START_REF] Gejadze | On a 2D zoom for the 1D shallow water model: Coupling and data assimilation[END_REF]; [START_REF] Marin | Superposition of local zoom models and simultaneous calibration for 1D-2D shallow water flows[END_REF]. An iterative coupling strategy is applied in [START_REF] Barthélémy | On operational flood forecasting system involving 1D/2D coupled hydraulic model and data assimilation[END_REF] between a 1D Mascaret and a 2D Telemac operational model and a sequential data assimilation technique is performed for correcting water levels forecasting. A summary of some established 1D and 2D numerical hydraulic models, external coupling methods and optimization-assimilation methods is presented in Table 3.1. One can spot DassFlow 2D [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF] as the only 2D hydraulic model with a second order solver with accurate wet-dry front treatment, parallel computation and adjoint based variational data assimilation capabilities.

The present study details upgrades to the DassFlow variational data assimilation framework [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]) in the form of a new multi-D hydraulic computational code and an integrated hydrological module. The proposed multi-D hydraulic code consists in a single finite volume solver applied to a 2D river network. The network is discretized into "1Dlike" reaches connected to high resolution 2D meshes in a single formulation of the SWE. The resulting product allows building large 1Dlike river networks, connected to fine local zooms. The method can lead to low computational costs over large networks and local fine scale accuracy at zooms where pertinent. The hydraulic model is coupled with a well-established conceptual hydrological model (GR4H state-space [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF]) in a semi-distributed setup. The variational platform can solve high-dimensional optimization problems with descent algorithms and using gradients computed with the adjoint model obtained via the automatic differentiation tool Tapenade [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF]). It enables tackling multi-variate, i.e. with composite control vectors (bathymetry, friction, boundary conditions, hydrological parameters), data assimilation problems given multi-source dataset, heterogeneous in nature and spatio-temporal resolutions (see e.g. [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF]). This integrated tool chain enables information feedback within the whole computational domain (basin) and especially from downstream to

The computational hydrological-hydraulic chain

This section presents the integrated and multi-dimensional hydrological-hydraulic model and the data assimilation approach. The model is designed for simulating spatio-temporal flow variabilities over an entire river network, from upstream hydrological responses to complex flow zones (confluences, multi-channel portions, floodplains, ...).

The modeling approach which is detailed below, is based on the following ingredients:

• An integrated multi-D hydraulic model: the 2D shallow water equations (SWE) with finite volume solvers from [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF] are applied to "1Dlike"-2D composite meshes of river networks using a numerical flux splitting method and an effective friction power-law depending on flow depth.

• A numerically coupled hydrological model, the widely used GR4 model from [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF] in its state-space version [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF], for the sake of model differentiability.

• A computational inverse method based on VDA algorithms from [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF] enabling spatially distributed calibration and variational data assimilation with the whole chain.

Multi-D hydraulic-hydrological modeling principle

The flow model consists in a spatially-distributed modeling of hydrological responses coupled to a seamless multi-scale "1Dlike"-2D hydraulic model. The core idea of this work is to apply the 2D SW hydraulic model (Eq. 3.1) on a multi-D discretization D hy of the computational domain Ω. The discretization (mesh) D hy is composed of N mixed unstructured triangular/quadrangular cells with interfaces between 1Dlike and 2D zones (Fig. 3.6). Ω is composed of a hydrological domain Ω rr connected to a hydraulic domain Ω hy (Fig. 3.4). Γ hy-rr is the border of Ω hy containing interfaces with Ω rr . The unstructured lattice covering Ω consists in hydrological units for describing upstream/lateral sub-catchments in Ω rr and mixed unstructured triangular/quadrangular elements in Ω hy .

Hydraulic module

Numerical hydraulic models describing open channel flows generally rely on the resolution of cross-sectionally or depth integrated flow equations, respectively the 1D Saint-Venant or 2D SWE (see e.g. [START_REF] Guinot | Wave propagation in fluids: models and numerical techniques[END_REF]). While 1D hydraulic models enable a physically sound representation of river flows variabilities in terms of wetted section A and discharge Q for instance, 2D hydraulic models in flow depth h and depth integrated velocity u = (u, v) T enable to tackle more complex flow zones such as confluences/diffluences and floodplain flows. The 2D shallow water model used in the proposed approach is presented here with the adaptation of the finite volume solver from [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF] for multi-D modeling. Note that 1D Saint-Venant equations are presented in appendix (3.B) along with their resolution method in DassFlow 1D [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]) that is used for comparison in this study. Next, this section presents the hydrological module and the inverse algorithm.

Mathematical flow model

On the hydraulic computational domain Ω hy ⊂ R 2 and for a time interval ]0, T], the 2D SWE with the Manning-Strickler friction term, in their conservative form, write as follows:

∂ t U + ∂ x F(U) + ∂ y G(U) = S g (U) + S f (U) U =   h hu hv   , F(U) =     hu hu 2 + gh 2 2 huv     , G(U) =     hv huv hv 2 + gh 2 2     , S g (U) = 0 -gh∇b , S f (U) =   0 -g n 2 u h 1/3 u   (3.1)
with h the water depth [m] and u = (u, v) T the depth-averaged velocity [m/s] being the flow state variables. g is the gravity magnitude m/s 2 , b the bed elevation [m] and n the Manning- Strickler friction coefficient s/m 1/3 being the flow model parameters. Classical initial and boundary conditions adapted to real cases are considered (see [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]; [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF] for details).

An effective friction law consisting in a simple power-law n = αh β is introduced, as previously done for 1D SWE for effective modeling with simplified multi-channel river geometry in [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF].

Building-up equivalencies between 2D and 1D flow states

1Dlike reaches refer to river reaches where the following meshing strategy has been applied: quadrangular cells are built such that their interfaces are perpendicular to the main flow direction and span the whole river (bankfull) width as a traditional 1D XS would. Examples can be found in Fig. 3.9 and Fig. 3.16. This leads to a series of quadrangular cells, each linked to a single upstream and downstream cell. The 1Dlike approach implicitly assumes a rectangular XS shape which potentially impacts the representation of: (i) at-a-section hydraulic geometry [START_REF] Leopold | The hydraulic geometry of stream channels and some physiographic implications[END_REF]), (ii) longitudinal hydraulic controls and flow variabilities.

In view to put the multi-D model in coherence with real flow physics, a continuity condition between 1D and 1Dlike models states and parameters is required. This continuity condition is enforced at-a-section, in a prismatic channel such that the uniform permanent flows, that is equilibrium, are preserved.

Let us consider a reference 1D model in (A, Q) variables with the bankfull width value W 1D . The friction term reads:

S f ,1D = n 2 Q|Q| A 2 R 4/3 h (Appendix 3.B).
In the corresponding 1Dlike model in (h, u, v) variables, see Section 3.3.2.1, the friction term reads: S f ,1Dlike = n 2 ||u|| h 1/3 u Considering 1D flow states over an idealized river section (Fig. 3.5, left) the hypothesis of local flow equilibrium (uniform, steady-state) with identical wetted areas A and WS widths W, the continuity condition implies that: effective friction and bathymetry. Left: Equivalency between a 1D idealized XS (purple) and a 2D single-cell rectangular XS (green), with the same flow section A and WS elevation. Right: Variation of the hydraulic radius R h (h) for 3 XS shapes (of similar dimensions). This showcases the potential over-and under-estimation of state variables using "1Dlike equivalent friction" from Eq. (3.2).

n 1Dlike = n 1D A W h 1/3 1Dlike R 4/3 h,1D (3.2)
where n 1Dlike (resp. h 1Dlike ) is the Manning-Strickler friction coefficient (resp. flow depth) in the 1Dlike model i.e. the coefficient in the 2D SWE (3.1).

With the additional assumption of a rectangular XS (as it will be assumed in some test cases),

we have h 1Dlike = A /W 1D , which leads to

n 1Dlike = n 1D h 1D R h,1D 2/3
. This "1Dlike equivalent friction" leads to a perfect fit in WS elevation of 1Dlike model and 1D model in a straight prismatic channel at the given uniform regime (results not shown here for brevity). Fig. 3.5, right, shows the evolution of the ratio ( h /R h ) vs h. For rectangular and parabolic XS, the ratio n 1Dlike/n 1D is expected to increase with h. Thus, one can naturally expect an overestimation (resp. underestimation) of the actual friction coefficient by n 1Dlike at lower flows (resp. greater flows). This would lead to an overestimation (resp. underestimation) of the 1D WS elevation by the "equivalent" 1Dlike model. However, later it will be considered a power-law in h to model the friction coefficient (Subsection 3.3.2.1), which provides a better fit to the 1D WS elevations outside of the considered permanent flow.

Note that longitudinal controls and flow variabilities in 1Dlike models are assessed using synthetic cases in Subsection 3.4.2.2.

Multi-dimensional hydraulic model

Over a given cell K ∈ Ω hy of area m K , the piece-wise constant values U K = 1 m K K UdK are approximated. Recall that the finite volume approach applied to the homogeneous part of the hyperbolic system of Eq. 3.1 (that is without the friction source term S f but including a consistent discretization of the gravitational source term S g ) writes as follows:

Ūn+1 K = U n K - ∆t n m K ∑ e∈∂K m e F e U n K,i , U n K e , n e i ,K (3.3)
1.7 where U n K and U n+1 K are the piece-wise constant approximations of U = (h, hu, hv) T at time t n and t n+1 (with t n+1 = t n + ∆t n ), F e stands for Riemann fluxes through each edge e of the border ∂K of the cell K, with each adjacent cell Ke. The length of edge e is m e and n e,K is the unit normal to e oriented from K to Ke.

The finite volume schemes are those developed in [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]; [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF].The discretization of the friction source term is described in Appendix 3.A.

Based on the first and second order finite volume solvers of [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]; [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF], a 1D2D coupling technique is introduced following a similar concept to Finaud-Guyot et al. [2018] (urban geometries and porosity context) to compute numerical fluxes on each interface between a 1Dlike quadrangular mesh cell connected to several 2D cells as schematized in Fig. 3.6.

At the multi-D interfaces, that is in the case of n > 1 cells x K,i i ∈ [1.

.n] adjacent to the same interface of another cell Ke (see Fig. 3.6 for notations and e.g. Fig. 3.19 for a real-like example), a special treatment is applied. It consists in the Riemann fluxes being calculated for each cell K i using the state from the same corresponding Ke cell over an interface of length m e i . In the end, the flux crossing the interface e = ∪e i is equal to the sum of the fluxes crossing the e i interfaces:

F e = ∑ i=1..n m e i F e i U n K i , U n Ke , n e i ,K .
This type of internal interface has been implemented in the numerical solvers from [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF] in the DassFlow platform which includes a solver with second order accuracy in space. This solver, developed in [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]; [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF], is accurate and robust for wet-dry front propagations and fully applies in the present context. Note that the lateral distribution of variables across the 1D2D interface is not constrained. The source code and synthetic cases are available upon simple request.

Hydrological module

In order to simulate the hydrological response of sub-catchments within a river basin, a hydrological module is coupled to the 2D SW flow model. The widely used, parsimonious and robust conceptual hydrological model GR4 [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]), in its "state-space" version from Santos et al. [2018], was chosen. The original lumped hydrological model has been deployed in a semi-distributed manner in the DassFlow framework.

The model is composed of two non-linear stores for production (soil moisture accounting) and routing, and a Nash cascade composed of a series of linear stores replacing the unit hydrograph from [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]. Being a set of ODE with explicit dependency to parameters, this hydrological model is differentiable. Moreover, the Fortran code is differentiable with the automatic differentiation tool Tapenade [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF].

Let us consider a sub-catchment bv i among sub-catchments {bv 1 , ..., bv C } in the discretized river basin hydrological domain Ω rr . The hydrological model can be seen as a dynamic operator Figure 3.6: Internal multi-D domains interface, general case. At each cell center x, the state variables U = (h, hu, hv) T and the bathymetry b are defined. The total numerical flux is conserved:

F e = ∑ i=1..n m e i F e i U n K,i , U n Ke , n e i ,K .
M relating sub-catchment state variables vector h (x i , t) with observable input of spatially aver- aged (on sub-catchment x i ) rainfall P (x i , t) and potential evapotranspiration E (x i , t), observable outputs Y (x, t) and "unobservable" parameters θ (x).

Omitting the sub-catchment index i for readability, the hydrological model consists in the following set of ordinary differential equations:

dh dt =                      ḣp = P s -E s -P erc ḣ1 = P r -Q Sh,1 ḣ2 = Q Sh,1 -Q Sh,2 ... ... ḣnres = Q Sh,nres-1 -Q Sh,nres ḣr = Q 9 + F -Q r (3.4)
where

P s , E s , P r , Q Sh,i , i ∈ [1.
.nres], Q 9 , F, Q r are model internal fluxes given in appendix 3.C along with their internal parameters. The evolution of reservoir states and model outputs and inputs is presented for a sample rain event in Fig. 3.7.

The input of the hydrological model are the evapotranspiration and precipitation E n and P n , the output is discharge q (t) = Q r + Q d [mm/h]. E n and P n are classically imposed from data time series as piece-wise constant on fixed temporal resolution (e.g. hourly). The numerical resolution is achieved with an implicit Euler algorithm with an adaptative sub-step algorithm enabling to reduce numerical errors especially for high flows [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF]). The initial states of the stores is given by a 1 year warm-up run. The discharge q is injected into Ω hy at a sub-interface of Γ hy-rr , either as an upstream or lateral flow.

The calibrated parameters will be the classical 4 parameters (c i ) i∈1..4 of GR4 (they will con- stitute the control vector considered in the forthcoming VDA experiments). Other parameters, such as several drainage law exponents or the number of Nash cascade stores are not optimized in this study. They are set at values from [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF].

Inverse algorithm: Variational Data Assimilation

Given spatio-temporal flow observables, provided by in situ and airborne sensors for instance, the inverse algorithm consisting in Variational Data Assimilation (VDA) aims at estimating the unknown or uncertain "input parameters" of the hydrological-hydraulic chain composed of a hydraulic model, presented in Section 3.3.2, and a hydrological model, presented in Section 3.3.3.

We consider the following set of spatio-temporal observations of water surface and discharge over the river domain Ω ⊂ R 2 :

Z o,k (t) , ∀k ∈ [1..N o,Z ] , Q o,k (t) , ∀k ∈ [1..N o,Q ] (3.5)
with Z o the observed WS elevation [m] above reference elevation, Q o the observed discharge m 3 /s , N o,Z the number of altimetric observations points and N o,Q the number of observed discharges over Ω.

Note that observed discharge Q o may be a value of a hydraulic discharge at a flow XS in Ω hy , or within the hydrological domain Ω rr and especially at the outlet of a sub-catchment here. Also note that other water surface observables could be considered, such as water surface velocity observations or dynamic water masks -not the scope of this study.

Given river stage and/or discharge observations, the aim is here to estimate unknown or uncertain quantities of the hydrological-hydraulic model among: discharge hydrographs Q i (t) , i ∈ [1, N] on the border of the hydraulic domain, spatially distributed hydraulic parameters (bathymetry elevation b or friction n) and hydrological models parameter sets (c i ) i∈1..4 . The control vector containing the sought quantities is denoted θ in what follows:

θ = θ hy , θ rr = Q 0 1 , ..., Q T 1 , ..., Q 1 N , ..., Q T N , n 1 , ..., n M , b 1 , ..., b M , (c 1 , ..., c P ) (3.6)
with θ hy and θ rr the control vectors of respectively the hydraulic and the hydrological modules, N the number of inflows points and T the number of inflow values in time, M the number of modeled cells and P the number of hydrological units.

We consider a cost function j obs aiming at measuring the discrepancy between simulated and observed flow quantities on the computational domain Ω. This cost function is defined as:

j obs (θ) = j Q (θ) or j obs (θ) = j Z (θ) (3.7)
This cost function contains either misfit to WS elevation,

j Z (θ) = 1 2 Z o (t) -Z (θ, t) 2 O Z , or misfit to discharge, j Q (θ) = 1 2 Q o (t) -Q (θ, t) 2 O Q .
The metrics (symmetric positive definite matrices) O Z and O Q are based on the inverse of the observation error covariances. This enables to regularize the inverse problem, see e.g. [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Asch | Data assimilation: methods, algorithms, and applications[END_REF]; [START_REF] Monnier | Variational Data Assimilation and Model Learning[END_REF] and references therein for related discussions.

Moreover, we classically enrich the cost function with a regularization term: j (θ) = j obs (θ) + γj reg (θ) with j reg a Tikhonov type regularization term. Here we consider a regularization on the bathymetry only:

j reg (θ) = 1 2 M ∑ i=1 (∂ x b i ) 2 + ∂ y b i
2 with θ defined by (3.6).

The regularization term adds convexity to the cost function. Moreover, it here dampens the bathymetry highest frequencies. Recall that low Froude flows i.e. subcritical flows naturally act as a low pass filtering of the bathymetry shape, see [START_REF] Martin | Inverse rheometry and basal properties inference for pseudoplastic geophysical flows[END_REF]; [START_REF] Gudmundsson | Transmission of basal variability to a glacier surface[END_REF].

The total cost function j is minimized starting from a background value θ (0) . Following [START_REF] Lorenc | The Met. Office global three-dimensional variational data assimilation scheme[END_REF], see also [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], the following change of variables is applied:

k = B -1/2 θ -θ (0) (3.8)
with B the covariance matrix of the background error.

Then by setting J (k) = j (θ), the optimization problem which is solved is actually the follow- ing:

min k J (k) (3.9)
The first order optimality condition of this optimization problem (3.9) reads: B 1/2 ∇j (θ) = 0. The change of variables based on the covariance matrix B acts as a preconditioning of the optimization problem. This optimization problem is solved using a first order gradient-based algorithm, more precisely the classical L-BFGS quasi-Newton algorithm (limited-memory Broyden-Fletcher-Goldfarb-Shanno bound-constrained, [START_REF] Zhu | Algorithm 778: L-BFGS-B: Fortran subroutines for largescale bound-constrained optimization[END_REF]) or, in some cases in this study, its bounded version L-BFGS-B [START_REF] Zhu | Algorithm 778: L-BFGS-B: Fortran subroutines for largescale bound-constrained optimization[END_REF]) but without variable change 3.8. Detailed know-hows on VDA may be found e.g. in online courses (see e.g. [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF]; [START_REF] Monnier | Variational Data Assimilation and Model Learning[END_REF]). The gradient is computed with the help of the adjoint model. The latter is obtained by automatic differentiation, using Tapenade [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF]. The choice of the covariance matrix B, represents an important a priori and greatly influences the computed solution of the inverse problem. Assuming the unknown parameters are independent variables, the matrix B is defined as a block diagonal matrix:

B = blockdiag B Ω hy , B Ω rr , with B Ω hy = blockdiag (B Q , B n , B z b ) (3.10)
Each block matrix of B Ω hy is defined as a covariance matrix (positive definite matrix) using 2D kernels for spatial controls. Here following [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF], we consider:

(B Q ) i,j = (σ Q ) 2 exp - t j -t i L Q (3.11) (B b ) i,j = (σ b ) 2 exp - x j -x i + y j -y i L b , (B n ) i,j = (σ n ) 2 exp - x j -x i + y j -y i L n (3.12)
The parameters L Q and (L b , L n ) act as correlation lengths. These parameters are usually empirically defined. However the expression of B b and the correlation lengths can be derived from physically-based estimations following Malou and Monnier [2021].

The following stopping criterion are used to stop the iterative optimization process if: i) the cost function does not decrease over a set number of iterations or ii) the current cost gradient, normalized by the initial cost gradient, goes under a set objective value. The multi-D hydrological-hydraulic tool chain presented above has been implemented in the latest version of [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF].

Results and discussion

Numerical experiments design

Both synthetic and real cases are considered to test the forward and inverse modeling capabilities of the proposed computational chain for river networks and floodplains simulation. First, the multi-D hydraulic model (Section 3.3.2.3) is validated against reference hydraulic models on synthetic cases corresponding to typical hydraulic complexities: (i) simple straight channel, (ii) confluence and (iii) straight, rectangular and parabolic, channels with effective parameterization of friction and bathymetry. For fluvial regimes in the context of altimetry, these hydraulic complexities generate hydraulic controls. Following [START_REF] Montazem | Wavelet-based river segmentation using hydraulic controlpreserving water surface elevation profile properties[END_REF], we define hydraulic controls as characterized by a maximal deviation of the water depth from the normal depth (see e.g. [START_REF] Chow | Open-channel Hydraulics[END_REF]; [START_REF] Dingman | Fluvial hydraulics[END_REF] for definition) at the reach scale. Next, a series of inference cases are considered in twin experiment setups. In a twin experiment, a reference model acts as a synthetic truth and is used to generate observations of model variables (e.g. b + h in Ω hy or Q in Ω rr ). No observation noise is considered in this study. The VDA method is then applied with these observations on an altered version of the reference model. Inferences of temporal forcings (inflow hydrographs Q i (t) , i ∈ [1..N]) are presented on a 1D2D confluence case. Inferences of channels parameters are presented on a straight 1Dlike case. Next, inferences of hydrological parameters (c i ) i∈1..4 are presented using hydraulic observables. Finally, the model is tested on two real cases: (i) the capability of the 1Dlike model to reproduce real flow lines and propagations, through effective bathymetry-friction (b (x) , n (x)) inference, is assessed against a reference 2D model built on fine bathymetry of 75 km of the Garonne river and (ii) the whole multi-D hydraulic hydrological tool chain is tested at basin scale on the real complex case of the Adour River network.

Synthetic cases

First, the proposed multi-D hydraulic solver (Subsection 3.3.2.3) is evaluated and compared to a fine 2D reference model on two simple configurations, a straight channel and a confluence, that feature contain frontal interfaces between 1Dlike and 2D meshes. Next, to investigate the reproductibility of hydraulic controls using a 1Dlike meshing approach and effective modeling, this approach is compared to a 1D reference model in three typical channel configurations: (i) a rectangular prismatic channel, (ii) a rectangular channel with a slope break and (iii) a parabolic prismatic channel. Finally, inferences of inflow hydrographs Q i (t) , i ∈ [1..N] and of a friction power-law n = αh β are carried out using WS observables.

Forward multi-D hydraulic cases

Straight channel case

A prismatic rectangular channel and a multi-D mesh are considered (1Dlike to 2D to 1Dlike, see Fig. 3.9). The channel width is 300 m and its length is 2300 m. A rating curve is imposed downstream. This multi-D model is compared to a reference 2D model with refined mesh (mesh not shown, 2400 cells, average edge length 25 m). The multi-D waterline is validated against the 2D model at permanent flow Q = 100 m 3 /s and the modeled downstream discharges are compared for a flood hydrograph (Fig. 3.9(b)). Both first (not shown) and second order (Fig. 3.9) numerical solver allow a close fit to the target water line (Fig. 3.9, bottom).

At permanent flow, a slight misfit is observed between the 2D and multi-D WS elevations with the second order scheme (Fig. 3.9, top, relative misfit < 0.15% at 1000 m). This is due to approximation error in the multi-D model caused by large spatial steps in 1Dlike reaches (dx = 200 m). Indeed, this misfit is reduced by reducing the 1Dlike cells length (not shown). This is confirmed and showcased in the next subsection.

At the interface between 1Dlike and 2D meshes, a slight jump in WS elevations can be observed at all 2D cells (Fig. 3.9, top). This is due to the second order scheme, which is currently not designed for multi-D interfaces. Recall that no constraint is imposed on the lateral distribution of computed variables. The technical implementation of this reconstruction for 1D2D interfaces will be done in next version of DassFlow.

During a varied flow event, the outflow of both models is close to identical (Fig. 3.9, middle). The flow is correctly transmitted at multi-D interfaces and, at this scale, the 1Dlike meshes are adequate to model a flood wave propagation.

Simple confluence case

A simple symmetrical confluence is modeled using a multi-D mesh. The channel width is 300 m in the downstream reach and 150 m in the upstream reaches. Two inflow hydrographs are imposed at the two upstream interfaces. The maximum abscissa of the mesh points are 0 and 2075 m. The 2D part (average edge length 5 m) contains a confluence flow zone (Fig. 3.10) while the 1Dlike mesh (dx = 100 m) covers the upstream and downstream reaches. At permanent flow, the multi-D model compares well with the reference 2D fine model (mesh not shown, 15 000 cells, average edge length 5 m) and leads to similar conclusions to that of the above paragraphs. Using a shorter spatial step in the 1Dlike reaches (10 m) reduces the difference between reference and multi-D model and allows for a nearly perfect fit to the reference WS elevation at permanent flow (Fig. 3.10, in green). During a varied flow event, the outflows modeled with the multi-D model are very close to reference ones -slight differences during rising and falling limbs, same peak time, NSE = 0.996. (a) Top: Waterlines for a permanent flow of 100 m 3 /s at both upstream boundaries. In red: reference 2D waterline. In blue: waterline for the mesh (left). In green: waterline for a denser 1D2D mesh (not shown). Middle: In black: total upstream discharge during sample varied flow event (evenly distributed between upstream boundaries). In red/green/blue: downstream simulated flow for the considered meshes. Bottom: WS elevation observations for the varied flow event.

(b) Station locations, noted as black dots on the mesh. Figure 3.10: Multi-D confluence case results with second order scheme

Hydraulic controls and effective friction

To investigate the reproductibility of hydraulic controls for fluvial flows, which are characterized by a maximal deviation of the water depth from the normal depth at the reach scale (see definition in [START_REF] Montazem | Wavelet-based river segmentation using hydraulic controlpreserving water surface elevation profile properties[END_REF]), using a 1Dlike approach, a set of typical channel variabilities and hydraulic controls are considered. Let us compare 1D and 1Dlike waterlines in a series of synthetic cases: (i) a straight rectangular channel, (ii) a straight rectangular channel with a mid-channel slope break and (iii) a straight parabolic channel.

Direct calibration

Recall the equivalent friction formulation (Eq. 3.2) designed to make match the water line at-a-section and at equilibrium. For each 1Dlike case, waterlines with 1D friction (n = 0.05 s/m 1/3 ) and effective equivalent friction are generated. Effective friction values are calculated using (Eq. 3.2) and results from the corresponding 1D case. Reference 1D flow lines are computed with DassFlow solving 1D Saint-Venant equations in (A, Q) state variables with a Preissmann scheme (see 3.B and also [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]).

In a straight rectangular prismatic channel, with the 1D normal water depth imposed downstream (Fig. 3.11(a)), a backwater curve is computed by the DassFlow1D model (in red). With a 1D homogeneous friction (n = 0.05 s/m 1/3 ), the 1Dlike approach yields an underestimated waterline (in blue). An homogeneous effective friction allows to correct the underestimation and to better match the 1D normal water depth over the whole domain (in green). The remaining misfit can be attributed to numerical errors, especially numerical diffusion from the Preissmann scheme (1D model spatial step is dx 1D = 100 m).

A first complexification of this case consists in the introduction of a local hydraulic control point in the form of a slope break at x = 10 km (Fig. 3.11(b)). In this setup, both 1D and 1Dlike models generate M2 backwater curves, see e.g. [START_REF] Dingman | Fluvial hydraulics[END_REF]. The hydraulic control generated at the slope break is well represented with a 1Dlike approach, given the aforementioned numerical errors due to the coarse grid.

Another complexification of the first case consists in changing from a rectangular XS to a parabolic one (Fig. 3.5). In this case, both equivalent friction -a single homogeneous patch -and effective bathymetry, in the form of an homogeneous shift δ b of the reference bathymetry, are needed to match the 1D WS elevation. Equivalent friction only allows to model identical wetted sections at permanent bankfull flow for a given channel width (Section 3.3.2.2). In this case, matching the 1D wetted section does not equate to matching its WS elevation, thus an effective bathymetry is used (Fig. 3.5(c)).

According to the above model comparisons, effective parameterization of channel parameters is sufficient to reproduce 1D behaviors with a 1Dlike approach for permanent bankfull flows in simple geometries. Outside of the permanent bankfull flow, a friction power-law n = αh β can be used (Subsection 3.3.2.1). This friction aims at better 1Dlike representativity when modeled wetted surfaces (and other XS variables) are different in 1D and 1Dlike models for the same WS elevation.

Inverse calibration

In the following comparison, power-law parameters α and β are obtained using VDA in a twin experiment setup. Observations are taken at two stations (Fig. 3.12). Prior values are α = 0.05 s/m 1/3 and β = 0.

The classic friction law and a power-law with calibrated parameters are compared in Fig. 3.12 during a varied flow event. A range of water depth higher than the "bankfull" depth used to calculate equivalent friction parameters is simulated. This results in an increase in misfit to the 1D WS elevation at high flow (Fig. 3.12, in blue and red). The calibrated friction power-law (in orange) somewhat reduces the misfit during high flows in this simple synthetic channel and within the simulated water depth range.

Multi-D hydraulic-hydrological data assimilation

Temporal forcings inference

Simple inference tests are carried on the confluence case from Subsection 3.4.2.1, following [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF], in a twin experiment setup. A control vector c = (Q 1 (t) , Q 2 (t)) is considered, where Q 1 and Q 2 are sinusoidal inflow hydrographs injected at the upstream cell of the two upstream reaches. [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF] shows that the minimal requirement to infer multiple spatially distributed temporal forcings simultaneously is to observe either (a) both of their unmixed signatures, at 1 point each, or (b) one of their unmixed signatures at 1 point and their mixed signatures at a second point. We verify this for configuration (b). Unmixed signatures observations are generated at a 1Dlike cell of the upstream reaches, mixed signatures observations are generated at a cell in the 2D part (Fig. 3.13, left, red crosses). Prior values for the inflow hydrographs are constant and set to their average value Q 1 (t) = Q 2 (t) = 100 m 3 /s . Using sets of two observations points, we are able to reproduce the results of [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF] but with the proposed multi-D hydraulic model: the VDA algorithm enables to infer Q 1 and Q 2 close to perfectly from configurations (a) and (b) (Fig. 3.13, right). In this second twin experiment, the issue of spatially distributed calibration of a hydrological model is studied, from multi-source observations of the river network. The confluence case above is used and this time, the upstream inflows are generated by GR4H module applied on two upstream catchments inflowing the hydraulic module. For each catchment, synthetic rain and evaporation time series and a hydrological parameter set θ rr = (c 1 , ..., c 4 ) are used to generate a discharge time series over a 1-year period (not shown). Mixed observables are used: WS elevation is observed at a single downstream point and hydrological model discharge is observed at the outlet of one of the catchments. They provide the same signal observability -mixed and unmixed signals -as in the above paragraph but with observations of different nature. Since the observed variables are of different nature and amplitude, we introduce a normalization. The cost function is here j obs = j Z + j Q , with each term being normalized by the number of observations and by their range of variation such that:

Hydrological parameters inference

j Z = N o,Z T Z N o,Z T Z +N o,Q T Q 1 (Z o,max -Z o,min ) 2 N o,Z ∑ 1 T Z ∑ 1 (Z o (t) -Z (θ, t)) 2 and j Q = N o,Q T Q N o,Z T Z +N o,Q T Q 1 (Q o,max -Q o,min ) 2 N o,Q ∑ 1 T Q ∑ 1 (Q o (t) -Q (θ, t)) 2 .
Those correspond to two separate normalized squared RMSE with N o,Q and N o,Z denoting the number of observations station and T Q and T Z the number of observation time steps.

The control vector c = (θ rr1 , θ rr2 ) contains the two sets of 4 hydrological parameters each. For this synthetic case, an inequality constraint of the control parameters is imposed with the bounded L-BFGS-B algorithm [START_REF] Zhu | Algorithm 778: L-BFGS-B: Fortran subroutines for largescale bound-constrained optimization[END_REF]). Indeed, restricted research intervals are considered for the three first parameters of each catchment, namely a 5% bracket around their target values used as prior, while c 4 is sought in its expected variation range from an erroneous prior (Table 3.2). Expected ranges for GR4H parameters are provided in Le [START_REF] Lay | Modélisation hydrologique dans un contexte de variabilité hydro-climatique. Une approche comparative pour l'étude du cycle hydrologique à méso-échelle au Bénin[END_REF], for the classical GR4 formulation.

In this setup, both j Q and j Z are reduced during the iterative steps and the inferred value for Inferred value for all parameters including c 4 are very close to the target. The "/" separates values for the 2 distinct hydrological units.

both c 4 parameters are very close to the target value starting from an erroneous prior (Fig. 3.14, right). Bounded parameters c 1 and c 2 vary slightly between their bounds, while c 3 is locked at its lower bound from the first iteration.

Real cases

Garonne river: 1Dlike effective model of a 2D reference case

A 1Dlike model of a reach of the Garonne river, in southern France, is built from real data and calibrated here via assimilation of spatially distributed WS observations in a twin experiment setup. A full 2D model of 75 km of the Garonne river (not shown, 867 500 cells including floodplains, average edge length 25 m) is used to generate real-like observables on this well known case [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF]; [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]; [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]). First, river extent is derived from the 2D model output at bankfull flow Q = 400 m 3 /s with an homogeneous friction n = 0.05 s/m 1/3 . This extent is in turn used to build a 1Dlike mesh, over the whole reach: single quadrangular cells cover the whole river width and are linked sequentially along the river reach (Fig. 3.15). 1Dlike cell interfaces are perpendicular to the flow direction, as would be XS in a 1D model. Each cell is about 100 m in longitudinal length. This mesh was generated using the SMS meshing tools. Cell bathymetry is first set using the lowest bathymetry point at each corresponding 2D XS.

Permanent bankfull flow calibration

A first expectedly imperfect 1Dlike model (Model A1) is built using the 1Dlike coarse mesh and expectedly underestimated bathymetry elevation, and an homogeneous friction parameter of 0.05 s/m 1/3 . The simulated steady WS elevation at bankfull flow is lower than that of the 2D model (average misfit of 0.858 m), which is expected since the 1D bathymetry is that of the lowest point of the 2D XS (Fig. 3.5). Furthermore, the 1Dlike friction is underestimated and leads to an underestimated simulated water depth (and flow surface). The WS elevation misfit does not seem to follow a significant trend from upstream to downstream, although it varies sharply at points of width variation (Fig. 3.16, e.g. around cell 410). Recall that the goal of this effective modeling approach is to accurately reproduce water surface signatures, including WS elevation and, tangentially, flow section (Subsection 3.3.2.2).

Calibration by hand

We here propose to reduce the misfit using, on one hand, effective friction and, on another hand, bathymetry as follows. In Model A2, the introduction of an equivalent friction parameter (Eq. 3.2), calculated at each 1D cell using observations from 2D XSs, improves the WS elevation misfit (mean friction of n = 0.062 s/m 1/3 , average WS elevation misfit of 0.562 m). It reduces misfit overall, but has no significant local influence (Fig. 3.16). In Model A3, we use the average WS elevation misfit from Model A2 to create a simple bathymetry shift δ b that helps fit the observations. Equivalent friction parameters from Model A2 are kept and all bathymetry points are shifted by δ b = 0.562 m (the bottom slope is conserved). The average WS elevation misfit at bankfull flow reaches 0.003 m for Model A3, which is a very satisfying result.

Calibration by VDA Now, the same calibration problem is addressed with an inference based on the VDA method. It is applied to the same reference model permanent flow waterline, observed at each cell. The control vector contains a single homogeneous friction parameter, as before, and spatialized bathymetry b (x) for each cell. To constrain the parameter search, two VDA processes are performed separately: a bathymetry regularization and a change of variable. Inference with bathymetry regularization leads to Model B1, with γ = 1 (Subsection 3.3.4). Inference with change of variable (Eq. 3.8) leads to Model B2, with L b = 500 m (Eq. 3.12). Both models lead to average misfits close to that of Model A3: 0.0839 m and 0.0844 m respectively.

Two other inference setups based on Model B1 are considered. The number of observations is divided by 10: 72 stations, homogeneously distributed at 1 per each 1 km, are considered. In Model B1a, no regularization term is considered (γ = 0). In Model B1b, a regularization is added (γ = 1, chosen by trial and error). This weight can be optimally determined using iterative regularization [START_REF] Malou | Covariance kernels investigation from diffusive wave equations for data assimilation in hydrology[END_REF]). It is dependent on the spatial scales of observed signals and on the discretization of inferred parameters. As presented in Fig. 3.16 and in Table 3.3, both inferences lead to low misfits of the 2D permanent WS elevation. The regularization tends to reduce extreme bathymetric variations that tend to appear far from the observation points. The iterative minimization process can be followed through the cost function value and the parameter gradients (Fig. 3.17). Values are normalized over their initial (iteration 0) value. For Model B1a, the optimal control is reached after 40 iteration (for a normalized cost of 8.6 × 10 -10 ), while for Model B1b it is reached in 85 iterations (for a normalized cost of 5.5 × 10 -5 ). Both models follow the same trajectory up to around iteration 12, where the regularization term j reg reaches the same order of scale as the WS elevation misfit term j obs .

Variational calibration of channel parameter has allowed to fit a permanent regime water line. The following paragraphs study the reproduction of propagation in a calibrated models during a varied flow event.

Variational calibration for a flow event

Let us now consider a varied flow event, without flooding in the 2D model. This event last 10 days, with a peak discharge of 702 m 3 /s at the start of day 2, a minimum discharge of 160 m 3 /s and and average discharge of 382 m 3 /s. Given our previous inference attempts, we know that we can find a couple (n, b (x)) or (n, δ b ) that minimizes the average misfit. For the For models A1, A2 and A3 (in grays), the bathymetry and homogeneous friction are manually calibrated. For model B1 (yellow), B1a (magenta), B1b (red) and B2 (green), the bathymetry and friction are calibrated by VDA. For models B1 and B2, inferences are carried out from observations at each cell (720 total). For models B1a and B1b, inferences are carried out from observations every 10 cells (i.e. around every 1 km, 72 stations total). Vertical bars indicate these 72 stations. sake of simplicity, we consider the couple (n, δ b ). For a varied flow event, this couple would be an optimal parameterization for the average observed water line. A first inference trial is carried out for a densely observed (in space and time) varied flow. Using Model A1 as a prior for bathymetry and friction and observations from the 2D model, we find the following optimal parameters: n = 0.054 s/m 1/3 , δ b = 0.669 m . The resulting optimal model is Model C.

To allow a better fit during high and low flows, we introduce the friction power-law n = αh β (Subsection 3.3.2.1). Using observations from the 2D model during the varied flow event, we infer the couple (β, δ b ). The value of α is set to 0.054 s/m 1/3 , the inferred value from Model D. Three different β priors are tested: -0.5, 0, 0.5. δ b is included in the control vector to allow modeling of more varied water depths, which may be needed to reach the optimum. All three inferences lead to the very close optimal control. Their averaging, a slight bathymetric shift δ b = 0.099 m and β = -0.017, leads to Model D. The direct simulation results of Model D is compared to the reference model. They correspond to an average WS elevation misfit of 0.026 m at high flow and of 0.11 m at low flow. Compared to Model C (average misfit of 0.086 m and 0.20 m at resp. high and low flow), this is an improvement. Note that NSE values for both model (Table 3.3) are both extremely close to 1.

The variational calibration of the global bathymetric shift δ b and of an homogeneous friction value n in a 1D-like model has allowed to closely fit WS elevation observations of a 10 days flood wave over the reference Garonne model. The calibration of depth-dependent friction, in the form of the β parameter in n = αh β , has allowed an even closer fit to this reference WS elevation over the high and low flows, i.e. a better representation of the observed flood wave propagation.

Adour basin: multi-D hydrological-hydraulic model

The whole hydrological-multi-D-hydraulic tool chain is now tested in a real context both for forward and inverse problems resolution. A real and complex basin case is now considered: the Adour river basin in the South West of France, with a total drained area of 16 890 km 2 at the estuary. It is probably one of the most difficult basins to model in the country, because of contrasted hydrological regimes including nival effects in the south, flash floods on small ungauged catchments, complex river network morphology, anthropized floodplains and tidal effects from downstream.

In this section, a multi-D model, composed of 1Dlike meshes and 2D zooms over floodplain area is built from available data (Fig. 3.18, 2D area in green). Then, forward and inverse flow simulations on the river network are presented.

Model construction and rainfall to inundation simulation

The following model of the Adour river combines a multi-D hydraulic network model and several hydrological models of sub-catchments (Fig. 3.18). It encompasses around 180 km of river reaches and includes the Adour river from its tidal boundary downstream of Bayonne up to a gauging station around 70 km upstream, and part of its main tributaries: the Nive river (around 45 km), the Oloron and Pau rivers (around 65 km in total). The river networks contains mostly single-branch reaches, with some notable flood areas around the city of Bayonne. The WS is observed in situ at 10 points, 5 of which are used as boundary conditions (Dax, Orthez, Escos, Cambo and Convergent, red points in Fig. 3.18). Out of the 5 remaining stations kept for data assimilation, 3 are located on river reaches (Peyrehorade, Urt and Villefranque, blue points in Fig. 3.18) and 2 are located in the Bayonne area (Pont-Blanc and Lesseps, blue points in Fig.

3.18).

At the 4 upstream points of the modeled river network, 4 sub-catchments are modeled with 4 lumped hydrological models (Subsection 3.3.3). Their drainage areas are respectively about 7811, 842, 2480 and 2464 km 2 . The hourly discharge have been extracted from the HYDRO3 database while the rainfall data from the radar observation reanalysis ANTILOPE J+1, which merges radar and in situ gauge observations, is provided by Météo France. The interannual temperature data is provided by the SAFRAN reanalysis, Météo France, and then used to calculate the potential evapotranspiration using the Oudin formula [START_REF] Oudin | Which potential evapotranspiration input for a lumped rainfall-runoff model?: Part 2 Towards a simple and efficient potential evapotranspiration model for rainfall-runoff modelling[END_REF]. The rainfall and PET are at a spatial resolution of 1 km 2 square grid, and processed into hourly time step. Spatial averages of the rainfall and PET computed with SMASH distributed platform [START_REF] Jay-Allemand | On the potential of variational calibration for a fully distributed hydrological model: application on a mediterranean catchment[END_REF]; [START_REF] Colleoni | SMASH v1.0 platform for spatially distributed hydrological modeling and data assimilation: hypothesis testing and signatures analysis[END_REF]) over the 4 catchments are used as inputs for the lumped GR4H model. Lumped parameter sets for the 4 GR4H models are simply obtained here using for each the airGR global calibration algorithm [START_REF] Coron | The suite of lumped GR hydrological models in an R package[END_REF]). In this section, inferences are carried out only for upstream inflow hydrographs, not hydrological parameters. Indeed, the study of global calibration and regionalization issues of spatially distributed hydrological models is left for further research.

Our multi-D hydraulic modeling approach (Section 3.3) is applied to this complex case as follows. First, a "1Dlike-only" model of the whole network is built. Then, a multi-D model is built based on the "1Dlike-only" model, with the addition of a 2D mesh of a floodplain (Fig. 3.19, left).

On the "1Dlike-only" model, the goal is to analyze 1Dlike signal propagation representation at a low computational cost. The Adour 1Dlike model is built similarly to the Garonne 1Dlike model, using DEM data to determine minor bed bank line placement and build the quadrangular cells. Bathymetry comes from 25 × 25 m DEM data, aggregated from fine LiDAR data from public databases, extracted at each cell. This rough approximation is sufficient to show the potential of the DassFlow assimilation tool chain on a large scale river network. This "1Dlikeonly" model contains 1409 cells.

Then, a multi-D model is obtained: the existing 2D mesh from a Telemac model is coupled to a 1Dlike mesh, similarly to the reference Telemac-Mascaret model from the regional flood forecast center SPC-GAD. The 1Dlike parts of the mesh are kept identical to the "1Dlike-only" model, while the 2D part is the mesh extracted from the Telemac-Mascaret model (Fig. 3.19, left) provided by SPC-GAD. For hydraulic coherence, bathymetry at coupled 1Dlike cells are taken as the average bathymetry of the linked 2D cells. This "1D2D" model contains 66 982 cells in the 2D area and 1342 cells in 1Dlike reaches. Results for a flooding event in the Bayonne 2D area are presented in Fig. 3.19, on the right. Modeled variables appear coherent over the 1D2D area and a high resolution flood map in coherence with flow conditions in the whole river network is obtained.

A 1 day (physical time) flooding event is computed in around 20 min (computation time) in the "1D2D" model (with 68 324 total cells, 6 threads in parallel). The same event leads to a 7 s computation time for the "1Dlike-only" model (with 1409 cells, same computational setup). Remember that computation time may vary depending on the number of wetted cells and on the adaptative time step calculation. This relatively low computation time, and the potential to decrease it further by using more threads in parallel, indicate that this multi-D method is suited to operational use. The code version this work is based on was proven scalable [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]). Additions made to the current version should not change this, but no numerical testing has been done.

Assimilation of WS observables to infer 4 upstream hydrographs

To investigate the 1Dlike effective modeling on a river network, a twin experiment setup is designed to infer large control vector from a realistic observability on the "1Dlike-only" model.

The considered control vector composed of the 4 upstream hydrographs is:

c = (Q Dax (t) , Q Esc (t) , Q Ort (t) , Q Cam (t)) (3.13)
Observations of WS elevation are generated at Peyrehorade, Urt and Villefranque stations, and additionally at a virtual station directly downstream from Orthez. These 4 points give theoretically sufficient observability to identify the 4 upstream hydrographs. Indeed, they sample mixed and unmixed signal similarly to the academic setup in Subsection 3.4.2.1). The observation pattern also corresponds to a reasonable expectation of spatial observability in French river networks. Prior hydrograph values are classically set to a constant average discharge value.

Simultaneous inference of the 4 hydrographs is satisfying. As shown in Fig. 3.20, left, upstream hydrographs injected at Cambo and Orthez are inferred very accurately (NSE > 0.95). This is due to their WS elevation signature being observed "unmixed" in the respective downstream reaches. The Escos hydrograph WS signature is observed at Peyrehorade, mixed with the Orthez hydrograph WS elevation signature. This leads to partial error of signature attribution and a less accurate inference (NSE = 0.57). The Dax hydrograph WS elevation signature is observed only at the Urt station, where its signal is mixed with that of Escos and Orthez. The resulting inference is the less accurate with a NSE of 0.50 and closer in shape to the inferred Escos and Orthez hydrographs. This hints at correlated influences of these hydrographs at observation stations and insufficient observability of the Dax hydrograph signal given the model hydrodynamics. Observed signals at the 4 stations are however all very accurate (Fig. 3.20, right). For upstream stations (Orthez and Villefranque), this is due to the accurate inference of upstream hydrographs. For stations under the influence of the tidal BC (Peyrehorade and Urt), this is due to the backwater influence of the BC, which compensates for the error in inferred hydrographs (namely at Dax, and less so at Escos).

In conclusion, this experiment shows the capability of the VDA tool chain to infer various upstream hydrographs on 1Dlike network models. Note that multi-D models are identical to the investigated 1Dlike model in terms of VDA capabilities. It paves the way towards investigations on multi-variate inferability of spatially distributed hydraulic-hydrological parameters.

In the future, investigation on the influence of data sparsity, observation weights and ill-posed problem constraints should be carried out.

Conclusions and perspectives

This article presents a new approach and numerical chain for the multi-D hydrological-hydraulic modeling of complex river networks with variational data assimilation capabilities. It is based on the VDA algorithm and the finite volume solvers (including second order one and accurate treatment of wet/dry front propagation) from [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]. The resolution of the full 2D shallow water equations (3.1) is performed with a single finite volume solver applied on a multi-D discretization of a river network domain. This lattice consists in "1Dlike" reaches meshed with irregular quadrangular cells connected, via 1Dlike-2D interfaces, to 2D zooms consisting in higher resolution unstructured meshes -either triangular or quadrangular. This hydraulic model is inflowed with a hydrological model enabling to describe upstream/lateral catchments inflow hydrographs. In this work, the parsimonious GR4 model is integrated [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]), in its state space version [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF]) for the sake of differentiability (for the VDA computations). This approach is implemented in the platform DassFlow [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]).

The adjoint of the whole tool chain is generated with Tapenade [START_REF] Hascoet | The Tapenade automatic differentiation tool: principles, model, and specification[END_REF]) and validated. Forward-inverse capabilities with the new components is assessed on several cases of increasing complexity. The 1Dlike effective hydraulic modeling approach as well as its coupling with higher resolution 2D zooms was validated, against: 1) reference 1D and 2D hydraulic models, on an array of academic; 2) complex real cases featuring 1D and 2D flow variabilities in river networks with confluences and floodplains. Those cases are also employed to test the coupling with the hydrological model implemented in a semi-distributed setup.

Considering single (resp. multiple) type(s) of flow signatures observations in those river networks through a single-(resp. multi)-objective observation cost function, the capabilities of the VDA method for inferring mono/multi-variate control vectors of large dimensions was successfully tested.

From the obtained results, the following conclusions can be raised:

1. A complete integrated multi-D model coupled with an hydrological model has been implemented and validated in a parallel environment. Moreover, this complete tool chain includes VDA capabilities based on the adjoint code.

2. The 1Dlike modeling approach enables to simulate fine physical flow states compared to reference 1D or 2D SW models; hydrograph propagation remains very close also.

3. The inference, from heterogeneous observations in the river network, of multi-variate controls among multiple inflow discharge hydrographs, bathymetry, friction of the multi-D hydraulic model but also hydrological model parameters is demonstrated. Very accurate inferences are obtained when the available information contained in system observability and priors is sufficient regarding the nature and quantity of unknown parameters (see discussion in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | River discharge and bathymetry estimation from swot altimetry measurements[END_REF]; [START_REF] Garambois | Variational estimation of effective channel and ungauged anabranching river discharge from multi-satellite water heights of different spatial sparsity[END_REF] and references therein).

4. Information feedback from the river network to upstream hydrological models of subcatchments is shown.

5. Real flows on complex channel geometries can be accurately simulated with the 1Dlike model, despite its intrinsic rectangular XS, thanks to the calibration of effective geometryfriction patterns. The depth-dependent friction law helps to reduce misfits across flow regimes.

6. High resolution simulations of real flows can be obtained on complex river networks including floodplains and confluences with reduced simulation costs.

To our knowledge, the present numerical tool is the first one proposing, large scale multi-D river network modeling with VDA capabilities. Short term perspectives will aim to taylor the data assimilation algorithm to perform complex data assimilation experiments at basin scale using various multi-source datasets. To be actually operational, improvements pertaining to the construction of 1Dlike models from global public databases are needed to deploy the multi-D approach to a large number of river networks. Coupling is ongoing with the SMASH spatially distributed hydrological platform [START_REF] Jay-Allemand | On the potential of variational calibration for a fully distributed hydrological model: application on a mediterranean catchment[END_REF]; [START_REF] Colleoni | SMASH v1.0 platform for spatially distributed hydrological modeling and data assimilation: hypothesis testing and signatures analysis[END_REF]) on which is based the French flash flood warning system Vigi-Crues Flash [START_REF] Garandeau | Vigicrues Flash, un service automatique d'avertissement pour les crues rapides[END_REF]). Further work will also test the integrated chain on flash flood Mediterranean basins as well as on larger basins in a satellite observability context. The implementation of porosity models [START_REF] Guinot | Flux closures and source term models for shallow water models with depth-dependent integral porosity[END_REF] and references therein) represents a very interesting research direction regarding effective floodplains and 1Dlike reaches parameterizations. This is especially true with depth-dependent porosity (Özgen et al. [2017] and references therein) applied to complex channel geometries and with spatially distributed calibration.

3.A 2D

(h, u, v) Shallow Water scheme

3.A.1 2D solver

Recall the rotational invariance property of the SWE (Eq. 3.1) which simplifies the sum of 2D problems in Eq. (1.7) to 1D Riemann problems. The fluxes F e are computed using a Riemann solver. Each local Riemann problem depends on left and right states at the interface e.

The HLLC approximate Riemann solver is used. This gives the expressions:

         FHLLC e 1,2 = s K e F ÛK 1,2 -s K F ÛK e 1,2 + s K s K e ÛK e 1,2 -ÛK 1,2
s * = s K h K e ûK e -s K e h K ûK -s K s K e (h K e -h K ) h K e ( ûK e -s K e ) -h K ( ûK -s K ) (3.16)
A Courant-Friedrichs-Levy (CFL) condition for the time step ∆t n applies, see e.g. [START_REF] Vila | Convergence of an explicit finite volume scheme for first order symmetric systems[END_REF].

3.A.2 Well-balancing

The numerical scheme must preserve the fluid at rest property, that is the gradient of bathymetry ∇z b must not provide u n+1 = 0 if u n = 0. In the presence of topography gradients (in particular those perpendicular to the streamlines) the basic topography gradient ∇z b discretization in the Figure 3.21: Typical situation of desired well-balanced property in presence of a wet/dry front. The WS elevation is here denoted by η (from [START_REF] Couderc | User and Developer Guide[END_REF]). gravity source term S g (U) generates spurious velocities. There is no discrete balance between the hydrostatic pressure and the gravity source term anymore: ∇ gh 2 /2 = -gh∇z b .

Following [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF], this issue is solved by considering the following change of variable.

From now, we consider the water depth h * e,K defined from the "reconstructed" topography z e (Fig. 3.21) at edge e as:

h * e,K = max (0, h e,K + z e,Kz e ) with z e,K = η e,Kh e,K z e = max (z e,K , z e,K e ) (3.17)

The conservative variable vector U n e,K in the semi-discrete scheme (Eq. 1.7) is now considered with the new variable: 3.18) Note that this new variable h * e,K depends on the bathymetry values (z e,K , z e ). The resulting well-balanced first order scheme reads:

U * e,K = h * e,K h * e,K u (
U n+1 K = U n K - ∆t n m K ∑ e∈∂K m e F e (U *

3.A.3 Prediction-correction time scheme

The friction source term is taken into account in the complete SW system by deriving a predictioncorrection time scheme, see e.g. [START_REF] Toro | Shock-capturing methods for free-surface shallow flows[END_REF].

We denote here by Ūn+1 K the FV solution at time t n+1 of the (well-balanced) scheme, either first or second order, of the SW system with the gravitational term S g but without the friction term S f . Recall that we denote: U = (h, hu) T .

At each time step, from n to n + 1,

• Step 1: computation of Ūn+1 , solution of the conservative SW system, i.e. the SW system without S f , i.e. the FV solution of the following system:

∂ t U + ∂ x F (U) + ∂ y G (U) = S g (U) (3.21)
• Step 2: given the "predicted value" Ūn+1 , compute U n+1 solution of:

∂ t U = S f (U) (3.22)
General schemes (explicit, implicit or semi-implicit ones) including the friction source term S f in the discretization of the model (Eq. 3.1) can be written as: for all K,

U n+1 K = Ūn+1 K + ∆ n t S f Ūn+1 K , Ūn+1 K e (3.23)
Note that this splitting scheme is consistent at first order in ∆t with the complete SWE. Splitting scheme second order in time is possible; it is not detailed later.

In the case of the Manning-Strickler law, the friction term reads:

S f = -gn 2 0 | ū| h 1 3
ū .

Therefore the equation to be solved (Eq. 3.22) reads:

∂ t h h ū = -gn 2 0 | ū| h 1 3 ū (3.24)
Since the friction source term S f is zero in the mass conservation equation, we remark that h n+1 = hn+1 . As a consequence, we consider the non-zero momentum component only:

∂ t (h ū) = -gn 2 | ū| h 1 3 ū 3.A.3.

First order expression of U n+1

Let us consider the implicit scheme :

h n+1 u n+1 -h n+1 ūn+1 ∆t n = -gn 2 u n+1 u n+1 (h n+1 ) 1 3
( 3.25) This implies that:

u n+1 u n+1 + h n+1 4 3 ∆t n gn 2 u n+1 -ūn+1 = 0 (3.26) Let us set c = (h n+1 )
vanishing velocities, it exists α ∈ ]0, 1] such that: u n+1 = α ūn+1 . Adopting these notations, we obtain ūn+1 ūn+1 α 2 + c ūn+1 αc ūn+1 = 0. This simplifies to:

ūn+1 α 2 + c α -c = 0 (3.27)
Since α ≥ 0, the root of this quadratic equation reads:

α = -c + c 2 + 4c | ūn+1 | 2 | ūn+1 | (3.28) Let us define the function ε = 1 c ūn+1 = ∆t n gn 2 | ūn+1 | (h n+1 ) 4/3 . Observe that ε = O (∆t n ), also ε = O | ūn+1 | (h n+1 ) 4/3
.By adopting this notation, Eq. (3.28) reads:

α = √ 1+4 -1 2
. After some rearrangements, we obtain α = 2 1+ √ 1+4ε . At first order in , we get: α ∼

1 1 + /4 ∼ 1 -/4.
Finally, we obtain:

U n+1 K = h n+1 h n+1 u n+1 =     hn+1 h n+1 ūn+1   2 hn+1 2/3 hn+1 2/3 + hn+1 4 3 + 4∆t n gn 2 | ūn+1 |       (3.29)
with ūn+1 the solution of Eq. (3.21).

3.A.3.2 Second order scheme

In order to obtain a globally second order scheme, a higher-order time stepping scheme is needed. Let us briefly describe the ingredients of this second order well-balanced positive scheme that is strictly the same as the one proposed in [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF]; [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF]. Actual second order accuracy, considering source terms S g and S f , is achieved through the combination of a Monotonic Upwind Scheme for Conservation Laws (MUSCL) spatial reconstruction and an IMEX RK time scheme (see [START_REF] Monnier | Inverse algorithms for 2D shallow water equations in presence of wet dry fronts: Application to flood plain dynamics[END_REF] and references therein), as well as a spatial discretization of S g and the semi-implicitation friction source term S f given by in the subsection above.

A monoslope second order MUSCL scheme is adopted, see e.g. [START_REF] Chévrier | A Van Leer finite volume scheme for the Euler equations on unstructured meshes[END_REF]; [START_REF] Buffard | Monoslope and Multislope MUSCL Methods for unstructured meshes[END_REF]. It leads to new expressions of U n K and U n K e . With this linear reconstruction, one can expect a scheme with a second-order accuracy in space (for regular solutions only). In order to prevent large numerical dispersive instabilities, the computed vectorial slopes are limited by applying a maximum principle. Furthermore, to handle the presence of wet-dry fronts that can break the Finite Volume mass conservation property, a Barth limiter [START_REF] Barth | Numerical methods for conservative laws on structured and unstructured meshes[END_REF]) is employed.

3.B 1D (A, Q) Saint-Venant equations

A river network Ω 1D is described by connected closed line segments. For a flow XS orthogonal to the main (longitudinal) flow direction of curvilinear abscissa x ∈ Ω 1D (distance from downstream), at time t ∈ [0, T], let A (x, t) be the flow cross-sectional area m 2 and Q (x, t) the discharge m 3 /s such that Q = U A with U (x, t) defined as the longitudinal XS averaged velocity [m/s]. The 1D Saint-Venant equations in (A, Q) variables at a flow XS write as follows:

   ∂A ∂t + ∂Q ∂x = 0 ∂Q ∂t + ∂ ∂x Q 2 A + gA ∂Z ∂x = -gA |Q|Q K 2 A 2 R 4/3 h (3.30)
where Z (x, t) is the WS elevation [m] and Z = (b + h) with b (x) [m] the river bed level and h (x, t) [m] the water depth , R h (x, t) = A /P h [m] the hydraulic radius , P h (x, t) [m] the wetted perimeter, g m/s 2 is the gravity magnitude. Let us recall the Froude number definition Fr = U /c comparing the average flow velocity U to pressure wave celerity c = gA W where W [m] is the flow top width.

The friction term S f is classically parameterized with the empirical Manning-Strickler law established for uniform flows

|Q| Q K 2 A 2 R 4/3 h where K m 1/3 /s is the Strickler coefficient.
The Saint-Venant equations are solved on each segment of the river network and the continuity of the flow between segments is ensured by applying an equality constrain on water levels at the confluence between two segments.

Boundary conditions are classically imposed (subcritical flows here) at boundary nodes with inflow discharges Q (t) at upstream nodes and WS elevation Z (t) at the downstream node; lateral hydrographs q l (t) at in/outflow nodes. The initial condition is set as the steady state backwater curve profile Z 0 (x) = Z (Q in (t 0 ) , q l,1..L (t 0 )) for hot-start. This 1D Saint-Venant model is discretized using the classical implicit Preissmann scheme (see e.g. [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF]) on a regular grid of spacing ∆x using a double sweep method enabling to deal with flow regimes changes with a one-hour time step ∆t. This is implemented into the computational software DassFlow [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | Modélisation thermohydraulique d'un tronçon de Garonne en lien avec l'habitat piscicole : Approches statistique et déterministe[END_REF]).

The numerical scheme is a semi-implicit finite difference scheme (generalized Preissmann scheme) with a double sweep Local Partial Inertial method to minimize the inertial terms (see documentation in [START_REF] Brisset | On the assimilation of altimetric data in 1D Saint-Venant river flow models[END_REF]; [START_REF] Larnier | Modélisation thermohydraulique d'un tronçon de Garonne en lien avec l'habitat piscicole : Approches statistique et déterministe[END_REF]).

3.C GR4 hydrological model operators

The state-space version of the lumped conceptual hydrological model GR4 presented in [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF] consists in the set of ordinary differential equations (ODE) given in Eq. (3.4) and recalled here for clarity:

dh dt =                      ḣp = P s -E s -P erc ḣ1 = P r -Q Sh,1 ḣ2 = Q Sh,1 -Q Sh,2 ... ... ḣnres = Q Sh,nres-1 -Q Sh,nres ḣr = Q 9 + F -Q r (3.31)
They involve the following fluxes:

                                                             E s = E 2h S c 1 - h p c 1 α P S = P 1 - h p c 1 α P erc = ν c 1 β-1 1 β-1 h + p β P R = P h p c 1 α + ν c 1 β-1 1 β-1 h + p β Q N1 = n res -1 c 4 h + 1 ... ... Q N11 = n res -1 c 4 h + n res Q 9 = Φ n res -1 c 4 h + n res Q 1 = (1 -Φ) n res -1 c 4 h + n res F = c 2 c ω 3 (h + r ) ω Q D = (1 -Φ) n res -1 c 4 h n res +1 -c 2 c ω 3 (h + r ) ω Q R = 1 (h + R ) γ-1 (γ-1) (h + r ) γ (3.32)
The following parameter are set following [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF] and [START_REF] Santos | Continuous state-space representation of a bucket-type rainfall-runoff model: a case study with the GR4 model using state-space GR4 (version 1.0)[END_REF]: α = 2, β = 5, γ = 5, ω = 3.5, ν = 4/9, Φ = 0.9, n res = 11. Calibrated parameter for the Adour case were obtained using the airGR global calibration algorithm [START_REF] Coron | The suite of lumped GR hydrological models in an R package[END_REF]) from the freely available package

Work perspectives

The goal of this work was the hydrodynamic modeling of river basins in the context of their multi-source observability. In Chapter 2, hydraulic signatures were analyzed and used to infer parameters of a 1D model applied at large scale. In Chapter 3, a multi-D hydraulic-hydrological model was developed to allow for the representation of large scale hydrographic networks at adequate scales and at relatively low computational cost.

The developed code and methodology should be further confronted to operational-like scenarios, such as the inference of discharge hydrographs of ungauged tributaries using heterogeneous altimetric observations or the fine, space-time accurate, reproduction of flood-inundation events (e.g. using the unprecedented 2021 Adour floods recorded by the SHPC-GAD). Near future works will focus on the improvement of the current Adour network models through the refinement of the a priori bathymetry extracted from fine LiDAR DEM and the calibration of effective hydraulic and hydrological parameters. Initial investigations into the assimilation of 2D observation fields are presented in Subsection 4.1.2.

Further work should focus on the integration of more complex distributed or semi-distributed hydrological models into the DassFlow tool chain and on their optimal synergy with in situ and satellite data using VDA methods. Given the ill-posedness of the considered inverse problems, their resolution would benefit from the exploitation of all available data. Furthermore, the inference of channel parameters, temporal forcings and distributed hydrological parameters is dependent on the spatial and temporal scales of observations, but also on a priori information fed into the assimilation process through background values and regularizations. These issues would be studied further in multi-D catchment-scale hydrological-hydraulic models.

Another perspective would consist in investigating porosity based approaches in 2D SW solver [START_REF] Guinot | Dual integral porosity shallow water model for urban flood modelling[END_REF]) for subgrid parameterizations in 1Dlike channels to account for channel shape variability or in 2D zones with complex topography. Further works on lateral flood plain modeling (at an appropriate computational cost and scale), hydrological parameter estimation (see Subsection 3.4.2.3), and on depth-dependent effective porosities [START_REF] Guinot | Multiple porosity shallow water models for macroscopic modelling of urban floods[END_REF]), both in floodplains and 1Dlike reaches, would represent interesting directions.

Finally, future work should also focus on strategies for automatic generation of 1Dlike meshes on river networks from available data products. This would allow for an easier deployment of this new modeling approach proposed and facilitate the use of VDA methods in basin scale modeling with the DassFlow platform.

Inferences on a large scale hydrographic network

General perspectives and goals The current Adour models should be better fitted to available WS observations. Improved bathymetry extraction from data shared by the SHPC-GAD1 and overall model calibration is an immediate perspective. The new a priori bathymetry (and friction) could be further calibrated using available WS observations (see e.g. observabilities in Table 4. 1 and Fig. 4.1). Recent extreme events (e.g. the extreme 2021 floods in the Bayonne area) could be used for calibration of the 2D area of the multi-D model (see Subsection 4.1.2for early trials).

The 1D-like model of the Adour river and its tributaries should be further exploited to analyze parameter inferability on a regional-scale hydrographic network. This exploitation should be based on an analysis of signature propagations, linked to hydraulic parameters, to the upstream hydrographs and to the downstream tidal boundary, and an analysis of parameter sensitivity across the domain.

The newly developed wrapped version of the Fortran code (see Section 1.5) may ease the use of Python libraries, including machine/deep learning, which opens plenty of perspectives in hydrological modeling. In hydrological modeling, [START_REF] Kratzert | Towards learning universal, regional, and local hydrological behaviors via machine learning applied to largesample datasets[END_REF]; [START_REF] Hashemi | How can regime characteristics of catchments help in training of local and regional LSTM-based runoff models?[END_REF] use LSTM to learn time series and others, e.g. [START_REF] Gao | Short-term runoff prediction with GRU and LSTM networks without requiring time step optimization during sample generation[END_REF], uses in prediction and forecasting. In hydraulic modeling, [START_REF] Kabir | A deep convolutional neural network model for rapid prediction of fluvial flood inundation[END_REF] uses surrogate models (deep convolutional neural networks) for real time flood inundation forecasting and [START_REF] Raissi | Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[END_REF] uses physics informed deep learning aimed at the synergetic combination of mathematical models and data.

Adour assimilation case

In the following paragraph, we propose work perspectives centered around multi-sourced data exploitation, that can come from in situ, airborne of satellite sensors. Recall the stated importance of the spatio-temporal coverage contained given by this multisourced cocktail to infer spatio-temporally distributed parameters.

In the area of interest of the Adour basin (see Subsection 3.4.3.2), the majority of river reaches have a width of 50 m or above, with 100 m reached on most of the Adour river and downstream from the Pau-Oloron confluence. This means that SWOT wide-swath observations should cover almost all of the network and that nadir altimetry virtual stations should still cover a significant portion (see Table 4.1 and Fig. 4.1 for a sample of satellite observability). Given the capabilities of the variational method for inferring large control vectors, initial calibration of reach-scale parameters could be carried out from this synthetic remote-sensed data. The dense temporal sampling of in situ station data (Fig. 4.1) also opens the way toward inferences of temporally dense hydraulic model forcings, possibly for operational use. The inference of ungauged/unknown/uncertain inflows was achieved in Chapter 3 in a 1D model and constitutes an achievable and interesting perspective in multi-D large scale models. The study of lateral fluxes represents an interesting research direction. These fluxes include gains and losses along river networks [START_REF] Cholet | Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection-diffusion equation[END_REF][START_REF] Le Mesnil | Impact of karst areas on runoff generation, lateral flow and interbasin groundwater flow at the storm-event timescale[END_REF]), for example via two-ways couplings with hydrogeological models. At the regional scale, [START_REF] Baratelli | Estimation of stream-aquifer exchanges at regional scale using a distributed model: Sensitivity to in-stream water level fluctuations, riverbed elevation and roughness[END_REF] (and references therein) links 1D river channels and saturated and unsaturated zones through conceptual relations. At a finer scale, [START_REF] Jeannot | A low-dimensional integrated subsurface hydrological model coupled with 2-D overland flow: Application to a restored fluvial hydrosystem (Upper Rhine River-France)[END_REF] links overland flows (using 1D and 2D diffusive wave models) to subsurface flows (using depth-integrated Richards equations) with superposed meshes and a coupling source term.

A first step towards assimilating this in situ and remotely-sensed data would be the creation of a twin experiment setup and the generation a synthetic target model. Proposed experimental setups could include: (i) the inference of temporal forcings from in situ WS elevation data, (ii) the inference of friction patches from satellite observabilities and (iii) the inference of the bathymetry-friction couple from satellite observabilities including spatially dense patterns. The use of a priori information and regularizations is an important component of inverse problem resolution and should be studied in the multi-D Adour model.

Inferences from 2D surface observation fields

The integration of new observations, for example from surface velocity fields provided by LSPIV techniques (see e.g. [START_REF] Dramais | Advantages of a mobile lspiv method for measuring flood discharges and improving stage-discharge curves[END_REF]; [START_REF] Fujita | Application of aerial LSPIV to the 2002 flood of the Yodo River using a helicopter mounted high density video camera[END_REF]; Le [START_REF] Le Coz | Performance of image-based velocimetry (LSPIV) applied to flash-flood discharge measurements in Mediterranean rivers[END_REF]) to constrain river network models, should be further explored. Indeed, surface velocity fields may represent constraining information as shown on simple test cases briefly presented here. Inferences from velocity observations are carried out in tandem with similar setups using WS elevation observations as a mean to compare them to better studied inverse problems (see e.g. [START_REF] Lai | Assimilation of spatially distributed water levels into a shallow-water flood model. Part I: Mathematical method and test case[END_REF]).

A multi-D confluence test case has been used for preliminary tests (mesh first defined in Subsection 3.4.2.3) and is used here. The three 1Dlike reaches are given a flat bottom and the 2D confluence mesh is given two anti-symmetrical bathymetry irregularities (a bump (in red) and a pool (in blue), see Fig. 4.2, right), with the aim of generating complex WS deformations on par with real-like complexities. The model is inflowed by two known identical varied upstream hydrographs and a rating curve is imposed downstream.

Two spatial observability patterns are setup to correspond. They are i) a 2D spatial field in the form of 20 × 20 m observation station grid (Fig. 4.2, left, in orange) and ii) local cross-sectional observation, in the two reaches, upstream from the confluence (Fig. 4.2, left, in green). They are meant to represent simplified in situ survey results, respectively LSPIV-like and ADCP-like data products. Both WS elevation and modeled velocities are observed, although they will be considered separately in the following inferences.

The model considered is the 2D (h, u, v) SW presented in Subsection 3.3.2 and 3.A). The control vector θ is composed exclusively of bathymetry points at the 2D confluence, i.e. 1Dlike cells are left out of the control vector. The prior bathymetry is flat over the whole case, which leads to bathymetry over-estimation (resp. under-estimation) at the north pool (resp. south bump) and corresponds to the target in the 1Dlike reaches. The following cost functions are considered, respectively for the assimilation of WS elevations and of flow velocities: j h,obs (θ) =

2 Z o (t) -Z (θ, t) 2 O and j uv,obs (θ) = 1 2 u o (t) -u (θ, t) 2 O + v o (t) -v (θ, t) 2 O , with Z o [m] the observed WS elevation and u o = (u o , v o ) T [m/
s] the observed flow velocity. Note that obser- vation covariance matrices O might differ for altimetric and velocity observations, although this distinction is not explored in this work, where they are set equal to identity matrices. Also note that, in the considered twin experiments, no noise is considered and observation locations are cell centers, hence the observation space corresponds to the state space. The regularizations term

writes j reg = 1 2 M ∑ i=1 ∂b i ∂x 2 + ∂b i ∂y 2
, with M the number of control cells, and is considered in the second set of inferences (Fig. 4.4).

Although the below inferences assume that the modeled flow velocity, i.e. depth-averaged velocity, is directly observable, through twin experiment setups, only the surface flow velocity is observable in practice. To account for this, observation operators accounting for known velocity profiles could be used, so as to translate observations into reasonable estimates of modeled depth-averaged velocity. The development of methods for this is outside the scope of this perspectives part and could be further studied.

Inferences from WS elevation observations with a grid pattern (pattern A, Fig The introduction of a bathymetry regularization term in the cost function leads to better inferences. For inferences using WS elevation observations, it leads to a smoother inferred bathymetry in coherence with patterns inferred without regularization, as expected from similar experiments of inferences in the 1Dlike model (see Subsection 3.4.3.1). Given grid WS elevation observation (Fig. 4.4(a) and (b)), inference results are unilaterally improved, as inferred bathymetry values are closer to the target in both cases. For inferences using flow velocity observations, results are also improved. Although, given pattern B observability, erroneous estimations at the downstream 1D2D interface are still present and a significant contribution of the observed bathymetry backwater control. This could be explained by the incomplete bathymetry reconstruction method at these interfaces (see Chapter 4).

Overall, these preliminary experimental setups highlight the interest of using 2D field for local bathymetry estimations. Surface velocity fields could also be used to constrain model flux, following the idea of local discharge estimation by LSPIV [START_REF] Dramais | Advantages of a mobile lspiv method for measuring flood discharges and improving stage-discharge curves[END_REF]; Le [START_REF] Le Coz | Performance of image-based velocimetry (LSPIV) applied to flash-flood discharge measurements in Mediterranean rivers[END_REF]). As a matter of facts, a misfit exists between depth averaged modeled flow velocities and observations of real surface velocities of a 3D turbulent flow. This requires to study adequate observation operators linking actual surface velocity and depth-averaged modeled velocity, through a priori knowledge of velocity profile shapes and taking into account measurement errors. Thus, given known bathymetry-friction over a XS (e.g. over an ADCP transect) and this observation operator at a a virtual station/XS, upstream discharge estimations could be carried out from surface velocity fields.

General conclusion

In this thesis work, flow modeling in complex hydrographic networks is studied in context of their observability by satellites. Water surface elevations and extents, respectively provided by radar altimetry (classical nadir or wide-swath) and optical sensors, are considered in complement of in situ physiographic and flow data. In this context, the work deals with networks of relatively large rivers and with fluvial flows (Fr < 1). The main issues in fluvial networks modeling pertain to the challenging estimation of unobserved spatio-temporal river discharge and bathymetry-friction. This works aims to tackle this issue by making the most of observations of river surface deformations, including satellite and in situ observations. These goals are related to several issues studied in this work with the aim to put in coherence model and data, which are (i) the effective hydraulic-hydrological modeling of river networks, including multi-scale hydraulic controls, (ii) the resolution of ill-posed inverse problems given heterogeneous and relatively sparse observability, compared to physical flow scales, of WS deformations and flows.

First, as synthesized in Chapter 1, this work builds upon previous knowledge and forwardinverse algorithms aimed at the inference of the bathymetry-friction couple and of a single upstream inflow. The difficulty of such inverse problems lies (i) in the correlated influences of the sought parameters on the observed water surface deformations, (ii) in the relatively sparse observation sampling when compared to the physical scale of signatures of hydraulic controls and (iii) in the heterogeneousness, in space, time and nature of observations. This work builds upon the DassFlow hydraulic modeling framework and its VDA algorithm enabling to tackle high dimensional inverse problems. The hydraulic inverse problems of interest, given WS observables, are even more challenging when considering distributed lateral inflows, which is yet necessary for modeling large river networks.

The article [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF], presented in Chapter 2, studies the forward and inverse modeling capabilities of distributed channel parameters and multiple inflows in a 1D SW model, from multi satellite observations of the river surface. Lateral inflows are implemented into the 1D SW numerical model in the DassFlow framework. Inferences of geometric parameters and multiple inflow hydrographs of the hydraulic model are carried out on synthetic 1D cases. In particular, the inference of distributed lateral inflows are studied based on analyses of their hydraulic signature in WS which also contains the footprint of channel parameters. To better constrain the considered ill-posed problems, classical regularizations and a change of variables in the parameter space are used. The hydraulic model, built from optical and historical altimetric data (Envisat), is coupled to the large scale hydrological model MGB and applied to a long portion of the Negro River in the Amazon basin. The effective hydraulic model, calibrated using multi-satellite observations, closely fits altimetry data and presents real-like spatial variabilities (backwater effects from control sections and downstream boundaries, lateral inflow influences). Note that this model building methodology was further applied in the Amazon basin in Malou et al. [2021] with multi-mission altimetry data (Sentinel, Jason) and used to derive altimetric stage-fall discharge laws based on simulated WS slopes. The global availability of such data enables the application of the approach on worldwide rivers.

Signal propagation and WS deformations are analyzed both on academic cases and on the Negro river with identifiability maps. In a twin experiment setup using noisy SWOT-like observability, simultaneous inferences of distributed bathymetry-friction and distributed inflow hydrographs lead to low misfit to WS observations. The study brings new insight into: (i) the large scale visibility of the signatures of distributed parameters and temporal forcings, using SWOT observability, (ii) the identifiability of large hydraulic control vectors using a variational method and satellite altimetry, (iii) the need to constrain these inferences with prior knowledge and (iv) the need to further taylor and scale hydrodynamic network models and assimilation methods to better synergize with rich multi-source observations. The modeling of large river networks would benefit from (i) multi scale river network hydraulic modeling (ii) with integrated hydrology model. To model complex flow zones, such as floodplains and confluences, local fine scale representations are also needed. This can be achieved by integrating seamless 2D zooms into complete river network models. These considerations motivated the modeling methods developed in Chapter 3.

The article [START_REF] Pujol | Multi-dimensional hydrological-hydraulic coupling with multi-variate variational data assimilation for river networks and floodplains: forward and inverse cases with DassFlow[END_REF], featured in Chapter 3, presented a new method for the multi-D hydraulic-hydrological modeling of river networks. The new modeling approach implemented in DassFlow allows fine hydrodynamic modeling of flow propagation within large river networks with zooms over complex flow zones such as floodplains and confluences. It is based on an existing solver, in the parallel framework of the DassFlow platform. A so-called "1Dlike" mesh allows for an effective hydrodynamic representation of river reaches under 1D hydraulic assumptions, while still allowing for a low computational cost over large domains. Fine 2D meshes of complex flow areas, such as confluences and floodplains, are connected to the 1Dlike reaches through 1D-2D interfaces, for which the finite volume solver has been adapted. The river network model is inflowed by the parsimonious hydrological model GR4H, in its state-space version, which has been integrated to the assimilation tool chain. The forward-inverse multi-D computational model is successfully validated on academic and real cases of increasing complexity, featuring typical 1D and 2D flow variabilities such as respectively channel hydraulic controls and confluences and floodplains. In particular, two hydraulic models based on real-like cases are built from reference models containing fine scale variabilities: (i) a 1Dlike model of a reach of the Garonne river, (ii) a large scale multi-D model of the Adour hydrographic network The 1Dlike modeling approach enables to simulate fine physical flow states compared to reference 1D or 2D shallow water models, hydrograph propagation remains very close also. The coupling with the hydrological model implemented in a semi-distributed setup is also successfully tested. Inferences of multi-variate controls composed of hydraulic and hydrological parameters are carried out from multi-source observations. Effective bathymetry-friction fields of the multi-D hydraulic model are inferred from WS observations and lead to accurate flow simulations as shown on the Garonne River case with low misfit to observed WS elevations and accurate flood wave propagation. Inferences of spatially distributed hydrological parameters are successfully carried from WS and discharge observations in the river network, via multi-objective cost function. This shows the possibility of informational feedback from downstream hydraulic observations towards upstream hydrological modeling. Inferences of spatially distributed hydraulic temporal forcings are successfully carried out on the Adour river network model, based on observability of WS elevation at in situ permanent stations. They lead to the successful estimation of poorlyknown inflows. This work presents a novel multi-dimensional hydraulic-hydrological numerical model with variational data assimilation in DassFlow framework. It allows multi-scale modeling over large domains and combining in situ observations with high-resolution hydro-meteorology and satellite data.

Over the course of this work, the synergy between hydraulic models of river networks and available WS observations is studied. It involves (i) the adaptation of a 1D SW hydraulic model for coupling with a spatially distributed hydrological model, (ii) the development of a multi-D hydraulic model , (iii) the integration of a hydrological module into the DassFlow framework and (iv) the resolution of a series of complex inverse problems using VDA. The integration of multi-source observations into hydraulic-hydrological network models allows the inference of spatio-temporal parameters from heterogeneous sets of observed flow signatures. Development carried out during this PhD lead to better modeling of river networks from altimetric and optical satellite data through numerical upgrades of the DassFlow platform. Model building methodologies are developed for flow simulations over complex river networks. Further research should focus on coupling the multi-D model with a distributed hydrological model and on optimal model synergy with in situ, satellite and high resolution hydrometeorological data in view of operational flood forecasting. Numerical model and method improvements will rely on pertinent effective modeling of hydrodynamic signatures in the multi-D model. This appendix details how the 1D and 2D SWE (Eq. A.4 and A.5 respectively) are obtained from the Navier-Stokes equations for an incompressible free surface flow of a Newtonian and viscous fluid -that is liquid water here. Let us consider an infinitesimally small volume of fluid far from the boundary layer. We assume that momentum loss can be modeled by a friction source term with the classical Manning formulation. Applying mass and momentum conservation principles to this volume (see mass fluxes representation on Fig. where g m/s 2 is the gravitational constant, θ [m/m] the bottom slope (see Fig. A.2), ρ kg/m 3 is the water volumetric mass, n s/m 1/3 is the Manning friction coefficient, h [m] is the water depth , u = u x , u y , u z T [m/s] is the flow velocity and p (x, y, t) = P atm + ρghcos (θ) is a pressure with P atm kg/m/s 2 the atmospheric pressure at the free surface.

Assuming the SWE hypotheses (recalled in the below box from Section 1.2.1) the mass and momentum equations of the incompressible Navier-Stokes equations are integrated over depth in what follows. For the sake of brevity, we will note u i (z = a) , i ∈ [w, y, z] as u i,a .

The BC for a free surface flow are: BC1: At the river banks, assumed parallel to axis y, the boundary adherence condition gives u y = 0.

BC2:

At the river bottom, the boundary adherence condition gives u z = 0.

BC3:

The kinematic boundary condition dictates that fluid velocity directed perpendicular 157 to a solid boundary must be 0. Thus, at the free surface we have ∂h ∂t + u x,H ∂h ∂x + u y,H ∂h ∂y = u z,H . Shallow Water flow hypotheses, recalled here from Section 1.2:

• The fluid is of constant volumetric mass ρ, which implies it is incompressible and that no sediment transport is considered.

• The flow is gradually varied, which implies that the flow surface is close to a plane and varies slowly in space, and that there is no sharp change in the flow section. Thus, flow lines are parallel to each other and close to parallel to the river bottom, and vertical velocities are negligible. This is also called the long wave assumption, as it means that with regards to the water depth, signal wavelength is high [START_REF] Boussinesq | Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire[END_REF]).

• The bottom slope (and thus the water surface slope) is low, such that the water depth can be measured as perpendicular to the river bottom.

The following demonstrations makes use of the Leibniz formula which writes: .3) where f (x, t) and ∂ f ∂x (x, t) are continuous in t and x in a region of the x, t plane, including where a (x) ≤ t ≤ b (x) and x 0 ≤ x ≤ x 1 , and where a (x) and b (x) are both continuous and have continuous derivatives for x 0 ≤ x ≤ x 1 .

d dx b(x) a(x) f (x, t) dz = b(x) a(x) ∂ f ∂x (x, t) dz + db dx (x) f (x, b (x)) - da ds (x) f (x, a (x)) (A
The complete SWE formulations demonstrated in the following sections of this appendix write: .4) where u = 1 h(x,t) h(x,t) 0 u x (x, y, z, t) dz is the depth-averaged velocity, A = w 2 w 1 H b h (x, y, t) dzdy is the flow section and Q = w 2 w 1 H b u x (x, y, t) dy the mass flux through A and H (x, y, t) is the water surface elevation. S f is generally given as a nonlinear friction term modeled using the classical Manning-Strickler's law

1D full SWE      ∂A ∂t + ∂Q ∂x = 0 ∂Q ∂t + ∂ ∂x Q 2 A + gA ∂H ∂x = gA -∂b ∂x -S f (A
S f = |Q| Q K 2 A 2 R 4/3 h
, where K is an effective friction parameter (see Eq. 1.3). R h = A /P is the hydraulic radius, with P the wetted perimeter. [START_REF] Dingman | Fluvial hydraulics[END_REF]). h is the water depth measured perpendicularly to the water surface, P atm is the atmospheric pressure, θ is the bottom slope in the direction the flow. u y (x, y, z, t) dz are the depth- averaged velocity over axes x and y respectively and H (x, y, t) is the water surface elevation. S f is generally given as a nonlinear friction term modeled using the classical Manning-Strickler's law S f = S f ,x S f ,y = n 2 u h 1/3 u, where n is an effective friction parameter (see Eq. 1.6).

2D full SWE

A.1 Continuity equation integration over depth

In this subsection, the 2D formulation of the continuity equation for SW flows is demonstrated.

Let us integrate Eq. A. u z dz = 0 (A.9)

Let us define the depth-averaged flow velocity u (x, t) = 1 h(x,t) h(x,t) 0 u x (x, y, z, t) dz and v (x, t) = 1 h(x,t) h(x,t) 0 u y (x, y, z, t) dz. Eq. A.9 becomes the 2D formulation of the SWE: To model a change in mass (e.g. due to rainfall, seepage or lateral flows), one can introduce q l = ∂Q ∂x , see e.g. [START_REF] Dingman | Fluvial hydraulics[END_REF].

A.2 Momentum equation integration over depth

Let us consider the momentum conservation component of the x axis: Let us integrate between the river bottom b (x, y) (see Fig. 1.12(a), we noted b (x) = min y b (x, y) the river bottom used in 1D modeling) and the free surface elevation H (x, y, t) over the z-axis, and then between the channel banks at w 1 and w 2 over the y-axis. x,H

∂u
By using BC3 at the free surface, we know that u (z) = 0, we write: [START_REF] Couderc | DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides[END_REF], by projecting on the y-axis following the same method and omitting integration over the width, we get the 2D formulation:

Appendix B

Preissmann scheme with lateral flows

Let us consider the classical non-conservative form of the 1D Saint-Venant equations in (A, Q) variables, with added lateral inflow q l (x, t) source terms (see e.g. Roux [2004]; Dingman [2009]). :

   ∂A ∂t + ∂Q ∂x = q l ∂Q ∂t + ∂ ∂x Q 2 A + gA ∂Z ∂x = -gAS f + q l u l (B.1) with S f = |Q|Q K 2 A 2 R 4/3
h the Manning friction term, q l the lateral unit discharge over a spatial step and u l the inflow velocity.

This appendix details the discretization of the 1D SWE (Eq. B.1) including mass and momentum source terms due to lateral inflows with the Preissmann scheme. The formulation is implemented into the DassFlow platform and is adapted from [START_REF] Roux | Estimation de paramètres en hydraulique fluviale, à partir de données caractéristiques de l'imagerie aérienne[END_REF], based on [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF].

B.1 Preissmann discretization scheme

The semi-implicit Preissmann discretization scheme enables greater time steps than explicit schemes, which allows for shorter simulation times, and is useful for simulation of large spatiotemporal domains.

The generalized Preissmann scheme [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF] is based on the following discretizations of a real valued function f at a spatial and temporal point P (x, t), where x ∈ Ω denotes space in the considered discretization Ω and t ∈ [0, T] denotes time within a time interval T. The scheme writes:

f | P = θ ψ f n+1 i+1 + (1 -ψ) f n+1 i + (1 -θ) ψ f n i+1 + (1 -ψ) f n i (B.2) ∂ f ∂t P = ψ f n+1 i+1 -f n i+1 ∆t + (1 -ψ) f n+1 i -f n i ∆t (B.3) ∂ f ∂x P = θ f n+1 i+1 -f n+1 j ∆x + (1 -θ) f n i+1 -f n j ∆x (B.4)

165

where j ∈ [1, N] is the space index, with N the number of grid points, that locates the cross- section taken in consideration and n ∈ [1, T] the time index, with N the number of time steps. θ (0 ≤ θ ≤ 1) and ψ (0 ≤ ψ ≤ 1) are respectively space and time weight. DassFlow1D uses values advised by [START_REF] Sart | Adaptation of Preissmann's scheme for transcritical open channel flows[END_REF]: ψ = 0.5 and θ = 0.55.

The following notations are adopted:

∆Q i+1 = Q n+1 i+1 -Q n i+1 , ∆Q i = Q n+1 i -Q n i (B.5) ∆Z i+1 = Z n+1 i+1 -Z n i+1 , ∆Z i = Z n+1 i -Z n i (B.6)
The following details the discretization with the Preissmann scheme, of mass conservation (Eq. B.8) in Subsection B.3.1 and next momentum (Eq. B.10) in section B.3.2.

B.2 Discretization summaries

Below is a short summary of the discretization obtained for the 1D Saint-Venant equations with the Preissmann scheme.

Continuity equation

For a given cross-section we have ∂A ∂Z x = W, so Eq. B.

Momentum conservation equation

By writing D = K s AR 2/3 h , Eq. B.1.2 can also be written as:

Q |Q| + D 2 ∂Z

B.3 Detailed SWE discretization B.3.1 Mass conservation equation

The lateral discharge q l term is considered between the cells j and j + 1 and between the time steps n and n + 1. It is denoted q l | n+1/2 j+1/2 . For the sake of simplicity, its temporal discretization is chosen identical to that of other terms in the Preissmann scheme such that: q l | n+1/2 j+1/2 = θ q l | n+1 j+1/2 + (1θ) q l | n j+1/2 . Temporal chronicles of q l | j+1/2 are given by the user.

In a cross-section located at x, ∂A ∂Z x = W (h) where h is the water depth and W is the total width, so ∂A ∂t + ∂Q ∂x = q l gives:

W ∂Z ∂t + ∂Q ∂x = q l (B.
+ ∂ ∂x Q 2 A + gA ∂Z ∂x = -gA |Q|Q K 2 A 2 R 4/3 h + u l q l (B.20)
Multiplying by D 2 (D = K s AR 2/3 h ) and dividing by gA leads to:

Q |Q| + D 2 ∂Z ∂x + 1 gA ∂Q ∂t + ∂QU ∂x -u l q l = 0 (B.21)
This appendix focuses on the discretization of the lateral inflow source term. The other terms are discretized by successive linearizations in [START_REF] Roux | Estimation de paramètres en hydraulique fluviale, à partir de données caractéristiques de l'imagerie aérienne[END_REF].

Discretization of u l q l The term u l q l corresponds to the momentum injected between sections j and j + 1. We write its discretization u l | n+1/2 j+1/2 q l | n+1/2 j+1/2 , which means it intervenes between sections j and j + 1 and time steps n and n + 1. For the sake of simplicity, the temporal discretization of u l | n+1/2 j+1/2 q l | n+1/2 j+1/2 will be identical to that of other terms in the Preissmann scheme. We write:

u l | n+1/2 j+1/2 q l | n+1/2 j+1/2 = θ (u l q l )| n+1 j+1/2 + (1 -θ) (u l q l )| n j+1/2 (B.22)
To avoid the need to provide values for u l | j+1/2 , and seeing that the flow velocity on the tributary depends on the hydrodynamics of the main reach, especially in subcritical regime, the velocity terms is estimated from calculated velocities in the reach (U j and U j+1 ). This work is focused on fluvial flow modeling in a context of satellite observability and does not feature subcritical and critical regimes at lateral inflow injection points. where α is the spatial weight of u l . It is set to 0.5, so that the lateral injection velocity is the average of computed channel velocities at the upstream and downstream cross-sections.

By introducing (B.23) in the discretization of u l q l , we can write:

u l | n+1/2 j+1/2 q l | n+1/2 j+1/2 = θ (u l q l )| n+1 j+1/2 + (1 -θ) (u l q l )| n

Contents 1 . 1 1 . 2 1 . 3

 111213 Fluvial hydraulics and hydrological variabilities . . . . . . . . . . . . . . . . . . 1.1.1 Catchment hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 River discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Fluvial hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 River networks monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Mathematical flow models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Resolution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Computational hydraulic models . . . . . . . . . . . . . . . . . . . . . . . Hydrological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Modeling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 MGB: a large scale hydrological model . . . . . . . . . . . . . . . . . . . . 1.3.3 GR4H: a bucket-type rainfall-runoff model . . . . . . . . . . . . . . . . . . 1.4 Inverse problems and data assimilation . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Direct and inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Data assimilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  (a) Qualitative representation from Britannica [Accessed 28 October 2021]. Athmospheric, surface and sub-surface exchanges are included.

Figure 1 . 1 :

 11 Figure 1.1: Complete continental water cycle.

Figure 1 . 3 :

 13 Figure 1.3: Metropolitan France and Amazon in situ observation station networks.

Figure 1 . 4 :

 14 Figure 1.4: In situ observation station measurements

Figure 1 .

 1 Figure 1.5: Impermanent data acquisition mission results (1)

Figure 1 .

 1 Figure 1.5: Impermanent data acquisition mission results (2)

  (a) Ground tracks of multiple altimetric missions over the Mediterranean sea (from AVISO). In blue: Jason-2, in green: Saral, in red: CryoSat, in pink: HY-2A.(b) Schematic representation of SWOT wide-swath altimetry coverage and global map of SWOT revisit numbers during its 21 days cycle (from[START_REF] Biancamaria | The SWOT Mission and Its Capabilities for Land Hydrology[END_REF]).

Figure 1 . 7 :

 17 Figure 1.7: Satellite observability of the Earth from altimetric sensors.

Figure 1 .

 1 Figure 1.8: Sentinel mission imagery, featuring contrasted lateral river variabilities (accessed through https://apps.sentinel-hub.com/sentinel-playground).

Figure 1 .

 1 Figure 1.11: Hydraulic models: 2D hydraulic models used for river reach and floodplain modeling (1)

  (d) Left: 2D notations. Right: Example mesh intended for overbank flow modeling, with gradually varied edge lengths/cell sizes between river bed and floodplain.

Figure 1 .

 1 Figure 1.12: Notations in 1D (A, Q) and 2D (h, u, v) models. (a) Longitudinal and cross-sectional notations of parameters and variables used in Eq. (1.2). (b) Notations of cell variables used in Eq. (1.6) and sample mesh of river bed and the surrounding floodplain (from Christopher and Arturo [2013]).

  1.13).

Figure 1 . 13 :

 113 Figure 1.13: Conceptual trade-off between goals, data availability and model complexity in hydrological models (from Grayson and Blöschl [2001]).

Figure 1 .

 1 Figure 1.14: MGB model reservoirs and propagation modeling (adapted from Fleischmann et al. [2020]; Collischonn et al. [2007]). Three linear reservoirs (for surface flow Dsup, subsurface flow Dint and groundwater reservoir Dbas) link the temporal forcings (over-canopy precipitation PC, evapotranspiration from the soil reservoir ET and from the canopy and vegetation interception reservoir EI) to flow in a simplified channel representation (routing model). W and Wm are respectively the current and maximum soil water storage.

Figure 1 .

 1 Figure 1.16: Complete VDA hydraulic-hydrological tool-chain

Figure 1 . 1 2 . 4

 1124 Figure 1.15: GR4 state-space conceptual model (Santos et al. [2018])

Figure 2 . 1 :

 21 Figure 2.1: Identifiability map of a flood wave on a 871 km reach of the Negro river in a large scale 1D model.The red-blue heatmap gives the WS elevation anomaly with regards to the average local WS elevation Z ano (x, t) = Z (x, t) -Z (x), where Z (x) is the average local WS elevation over the seasonal flood peak (may-june). The blue line represents the travel of an intumescence traveling through the reach at the modeled kinematic speed ( 5 /3u) and starting at the upstream BC at the time of the flood peak at this point, in a base model calibrated with satellite altimetry. Dotted colored lines show the sensitivity of this travel time estimation to model parameters and forcings: variations of +30% and -30% from the base model are represented for the Manning friction parameter K (in green), the bathymetry b (in cyan) and all model inflows (in red). The vertical bars indicated lateral inflows q l (x, t) for a total of 21 lateral mass and momentum contributions (bold lines are denote the major tributaries). Horizontal bars represent observability given by the SWOT large swath altimeter.

Figure 2 . 3 :

 23 Figure 2.3: Flowchart of the method using the HiVDI inverse method Larnier et al. [2021] for variational calibration, adapted from Monnier et al. [2016]; Garambois et al. [2020]; Monnier [2021].

Figure 2 . 4 :

 24 Figure 2.4: Spatial hydraulic signature of lateral inflows on an academic test case. Top: Academic test cases configurations. Rectangular channels of length L = 1000 m and constant XS width W = 300 m, constant bottom slope of 10 -3 m/m for Ch1, 2 and varying between 10 -4 and 10 -2 m/m for Ch3 -the bottom b (x)is defined by linear interpolation between the 4 bathymetry points (diamonds, b = {2, 1.88, 1.28, 1.12} m) -and friction defined by constant values on 3 patches (α = {30, 12.5, 30} m 1/3 /s). Upstream inflow at x = 0; for Ch1, 2, 3 respectively lateral injections at abscissae (in m): x = 300, x = {300, 700}, x = {350, 700}, and observations at {x S1 , ..., x S5 } = {150, 500, 850, 450, 550}, {x Sb1 , ..., x Sb5 } = {150, 325, 450, 600, 800}, {x Sc1 , ..., x Sc4 } = {0, 300, 600, 1000}. Bottom: Sample waterlines with visible upstream and downstream controls and signatures. For the sake of clarity here, upstream and injected flow are set at 100 m 3 /s (Fr ∼ 0.12 -0.3). Using the identifiability index I ident = T wave /∆t obs introduced in Brisset et al. [2018] with T wave = L/c k and the kinematic wave velocity for a rectangular channel c k = 5 /3U (c k = 1.16 m/s considering average speed U = 0.69 m/s) and a high observation frequency (∆t obs = 20 s), gives a high identifiability index I ident = 43 for the present flow observation configuration.

  Figure 2.4: Spatial hydraulic signature of lateral inflows on an academic test case. Top: Academic test cases configurations. Rectangular channels of length L = 1000 m and constant XS width W = 300 m, constant bottom slope of 10 -3 m/m for Ch1, 2 and varying between 10 -4 and 10 -2 m/m for Ch3 -the bottom b (x)is defined by linear interpolation between the 4 bathymetry points (diamonds, b = {2, 1.88, 1.28, 1.12} m) -and friction defined by constant values on 3 patches (α = {30, 12.5, 30} m 1/3 /s). Upstream inflow at x = 0; for Ch1, 2, 3 respectively lateral injections at abscissae (in m): x = 300, x = {300, 700}, x = {350, 700}, and observations at {x S1 , ..., x S5 } = {150, 500, 850, 450, 550}, {x Sb1 , ..., x Sb5 } = {150, 325, 450, 600, 800}, {x Sc1 , ..., x Sc4 } = {0, 300, 600, 1000}. Bottom: Sample waterlines with visible upstream and downstream controls and signatures. For the sake of clarity here, upstream and injected flow are set at 100 m 3 /s (Fr ∼ 0.12 -0.3). Using the identifiability index I ident = T wave /∆t obs introduced in Brisset et al. [2018] with T wave = L/c k and the kinematic wave velocity for a rectangular channel c k = 5 /3U (c k = 1.16 m/s considering average speed U = 0.69 m/s) and a high observation frequency (∆t obs = 20 s), gives a high identifiability index I ident = 43 for the present flow observation configuration.

Figure 2 .

 2 Figure 2.5: Inflows inferences from WS observations for all Ch2 variants. Intermediate iteration in the assimilation process are represented for Ch2b and Ch2c ; they are hand-picked to illustrate "intermediate" behaviors before convergence ("inferred").

Figure 2 .

 2 Figure 2.6: Inflow, bathymetry and friction patch inferences from WS observations for all Ch3variants. In red, final inference with "default" parameter weights (see Tab. 2.3). In green, final inference with adjusted parameter weights. In orange, intermediate inferences with "default" parameter weights.

Figure 2 .

 2 Figure 2.8: Effective river channel bottom and width for spatially sparse, G1 = {b env , W 1 } r∈[1..R] , and spatially dense, G2 = {b GPS , W 2 } G∈[1..G] , model geometries and a low-flow GPS waterline from[START_REF] Moreira | Apport des données de géodésie spatial pour l'étude du bassin hydrologique amazonien[END_REF].

  g. O'Loughlin et al. [2019] on the Congo river).

Figure 2 .

 2 Figure 2.9: Friction patches after calibration against Envisat WS elevation observations. Inferred Strickler coefficient values are very close for all configurations for patches 1, 3 to 5 , 9 to 14 and 16. Patches 2, 6 and 15 are especially sensitive to model variations.

Figure 2 .Figure 2 .

 22 Figure 2.10: Modeled and observed WS elevation at Envisat VS after friction calibration at all stations for M1 (1)

Figure 2 .

 2 Figure 2.12: Identifiability map, flood wave propagation estimation and spatial variabilities of the large scale Negro river 1D model. Top: Identifiability maps and flood wave propagation during the yearly peak flow (may-june) in the Negro river model. The WS anomaly (heatmap) is given by Z ano (x, t) = Z (x, t) -Z (x), where Z (x) is the average local WS elevation from day 160 to 190. Blue dotted line: tracking of maximum WS elevation value Z m (x) = max t∈[0,365] Z (x, t) , ∀x ∈ [0, L]. Gray points: tracking of maximum WS elevation in the downstream pool. Dashed blue lines: fictitious trajectory at kinematic speed (sparse dashes) c k = 5 /3U and at U + c (dotted dashes), starting at x = 0, at the time of the local WS elevation peak. The speeds are calculated from the simulated flow speed U and water height h and updated every ∆x = 200 m, such as t p+1 = t p + ∆x c p k . Bottom: Extreme flow forcings and flow model variables over a 2 year period. Blue lines: Extreme simulated waterlines. Red lines: corresponding extreme Froude values. Green lines: corresponding cumulative injected flows. Vertical black dashes are lateral inflow locations. Bold vertical dashes are inflows inferred in Subsection 2.6.3.

Figure 2 . 13 :

 213 Figure 2.13: Lateral hydrographs inferences from SWOTNoiseSet and DenseSet, using the Q -30FG

Figure 2 .

 2 Figure 2.15: Bathymetry and friction pattern inferences using SWOTSet observability

Figure 2 .Figure 2 .

 22 Figure 2.16: Inflow, bathymetry and friction patch inferences from SWOT synthetic data: c * f lat , inferred control vector without a priori hydraulic behavior

  defined from an a priori positive definite covariance matrix O. Assuming uncorrelated observations O = diag (σ Z ). The modeled WS elevations Z depend on c through the hydrodynamic model (Eq. (2.3.1)) and the inverse problem reads as c * = argmin c j (c)(2.10)

Figure 2 .Figure 2 .

 22 Figure 2.18: Envisat WS elevation misfit after friction calibration at all stations for M2a (1)

1 1

 11 Intel Corporation 8th Gen Core Processor Host Bridge/DRAM Registers PCI bridge: Intel Corporation Xeon E3-1200 v5/E3-1500 v5/6th Gen Core Processor PCIe Controller Memory: 2x16Gb SODIMM DDR4 Synchronous 2667 MHz (0.4 ns) Resolution mode: sequential Resolution method: implicit-explicit Preissmann scheme Sample run: inference of the full triplet on the Negro model (inferred control vector c * f ilter in Subsection 2.6.3) • Control vector components: 4 × 740 flow points, 436 bathymetry points, 17 friction patches (3413 total sought values) • Total run time (direct): under 15 min • Total run time (inverse): 20.Extended abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -D hydraulic-hydrological modeling principle . . . . . . . . . . . . . 3.3.2 Hydraulic module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Hydrological module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Inverse algorithm: Variational Data Assimilation . . . . . . . . . . . . . . Numerical experiments design . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Synthetic cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Real cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A 2D (h, u, v) Shallow Water scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A.1 2D solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A.2 Well-balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A.3 Prediction-correction time scheme . . . . . . . . . . . . . . . . . . . . . . .

Figure 3 . 1 :

 31 Figure 3.1: Conceptual meshing approach for integrated hydraulic-hydrological and multidimensional modeling of a river network. The computational domain Ω is composed of the hydrological domain Ω rr connected to the hydraulic domain Ω hy . Ω hy contains 1Dlike meshes and classical 2D meshes, interfaced frontally at the red points. Inflows injected in Ω hy (blue points) can come from classic inflowing methods or from the coupling to hydrological catchments from Ω rr .

Figure 3 . 2 :

 32 Figure 3.2: 1Dlike mesh (quadrangular cells linked in a chain, black) overlaid on top of a simulated water extent (water depth heatmap) from a 2D reference model of the Garonne river (see Subsection 3.4.3.1). The bankfull water extent is used to set lateral 1Dlike cell edges at the banks (DEM data was also used to determine bank placements, see Subsection 3.4.3.2). Out-of-bank flows are not modeled in the 1Dlike model.

Figure 3 . 3 :

 33 Figure 3.3: 1D2D interface implementation. Left: Interfacing of the cell x Ke , part of a 1Dlike mesh, to cells x K,i , i ∈ [1, n]. Flows are calculated using an adapted 2D solver at each e i , i ∈ [1, n] interface. The flux over the 1Dlike edge e is given by F e = ∑ i=1..n m e i F e i U n K,i , U n K e , n e i ,K .

Figure 3 . 4 :

 34 Figure 3.4: Conceptual meshing approach for integrated hydraulic-hydrological and multidimensional modeling of a river network. The computational domain Ω is composed of the hydrological domain Ω rr connected to the hydraulic domain Ω hy . Ω hy contains 1Dlike meshes and classical 2D meshes, interfaced frontally at Γ rh . Inflows injected in Ω hy (blue points) can come from classic inflowing methods or from the coupling to hydrological catchments from Ω rr .

Figure 3 .

 3 Figure 3.5: Equivalency of 1D and 2D flow states at equilibrium (permanent uniform flows):effective friction and bathymetry. Left: Equivalency between a 1D idealized XS (purple) and a 2D single-cell rectangular XS (green), with the same flow section A and WS elevation. Right: Variation of the hydraulic radius R h (h) for 3 XS shapes (of similar dimensions). This showcases the potential over-and under-estimation of state variables using "1Dlike equivalent friction" from Eq. (3.2).

Figure 3 .

 3 Figure 3.7: Evolution of GR4 inputs, output and reservoirs states during a sample rain event. Top: temporal forcings (rain and evaporation) and modeled output (discharge). Bottom: hydrological model states (reservoir levels).

Figure 3 .

 3 Figure 3.8: Complete VDA hydraulic-hydrological tool chain

  (a) Top: Waterlines for a permanent flow of 100 m 3 /s. In red: reference 2D waterline. In blue: waterline for the mesh (left). The total misfit at the 1D2D upstream interface (at 1000 m) is around 10 -3 m, for a relative misfit < 0.15% of the local depth. Middle: In black: upstream discharge during sample varied flow event. In red/blue (total overlap): downstream simulated flow for the considered meshes. Bottom: WS elevation observations for the varied flow event (total overlap). (b) Station locations are noted as black dots on the mesh (left).

Figure 3 .

 3 Figure 3.9: Multi-D straight channel case results with second order scheme. The flow is subcritical with a maximum Froude value of 0.06.

Figure 3 .

 3 Figure 3.11: Effective friction analysis for steady waterline in academic cases: (a) rectangular channel with constant slope, (b) rectangular channel with slope break, (c) parabolic channel with constant slope. Values of effective friction parameter α s/m 1/3 : (a): 5.06 × 10 -2 , (b): (5.06 × 10 -2 , 5.04 × 10 -2 ), (c): 5.46 × 10 -2 . Bathymetric shift δ b [m]: (a): 0 , (b): 0 , (c): 0.591. 1D reference model: fixed time step 5 s spatial step 100 m; average Courant number equals 0.26 1Dlike models: adaptative time step with mean value of 9 s; 1Dlike cell length 100 m; average Courant number equals 0.48.

Figure 3 .

 3 Figure 3.12: Effective friction analysis for an unsteady waterline in case (a). The power-law is given by n = αh β , with α = 0.0446 and β = 0.0634. The injected hydrograph is symmetrical. Observations stations are at x = 2.5 km and x = 7.5 km. Adaptative time step with average value of 13 s. 1Dlike cell length: 200 m. Average Courant number: 0.33.

Figure 3 . 13 :

 313 Figure 3.13: Inflow hydrographs inference on confluence synthetic case in observation setup (b) with one station on a reach upstream of the confluence and one observing mixed flows downstream in the 2D part (see mesh on the left). Given sufficient observability and an unbiased prior, inferred hydrographs (in blue and orange) are both at target (black, total overlap). The flow is subcritical with a maximum Froude value of 0.05.

Figure 3 .

 3 Figure 3.14: Simultaneous inference of GR4H hydrological parameters in two catchments from multi-source observations: WS elevation and flow observations. (a) Multi-D hydraulic mesh and modeled hydrological catchments. Hydrological discharge at the outlet of Ω rr is inflowed upstream in Ω hy . Observation stations (red cross and red line) are used for inferring hydrological parameters. (b) Inference of 2 sets of 4 hydrological parameters: normalized costs and parameter values over the course of the iterative optimum search. The x 4 background values are erroneous. x 1 , x 2 and x 4 background values are set to target values.

Figure 3 .

 3 Figure 3.15: Garonne 2D model extent and simulated water depths using the 2D model and the 1Dlike model for a non-flooding event

Figure 3 .

 3 Figure 3.16: Comparison of 7 1Dlike models of 72 km of the Garonne river to a reference 2D SW fine model for a permanent flow (reference WS elevation in blue lines, bottom). (Up) Left: WS comparison of 1Dlike models to 2D reference model at each spatial point. Right: Zoom on 3 km long reach of interest. (Down) Left: WS elevation and bathymetry for the reference and 1D like models. Right: Zoom on 3 km long reach of interest.For models A1, A2 and A3 (in grays), the bathymetry and homogeneous friction are manually calibrated. For model B1 (yellow), B1a (magenta), B1b (red) and B2 (green), the bathymetry and friction are calibrated by VDA. For models B1 and B2, inferences are carried out from observations at each cell (720 total). For models B1a and B1b, inferences are carried out from observations every 10 cells (i.e. around every 1 km, 72 stations total). Vertical bars indicate these 72 stations.

Figure 3 .

 3 Figure3.17: Normalized cost functions and gradients for inferences of distributed bathymetry and homogeneous friction, from 72 observation stations, in models B1a and B1b. Top: Total cost normalized by initial cost j (θ) /j θ (0) vs iterations. Bottom: Cost gradients normalized by initial gradient for the inferred parameters vs iterations.

Figure 3 .

 3 Figure 3.18: Schematic view of the complete Adour river network and observability. Dimensions are not respected: total river lengths equal ≈ 180 km, 2D floodplain area equals ≈ 5 × 3 km 2 . Tidal BC influence (from the downstream BC at Convergent) is observed up to Dax (and further upstream), Peyrehorade and Villefranque.

Figure 3 . 3 .

 33 Figure 3.19: Zoom on the Bayonne area (green area in Fig. 3.18) using the 1Dlike and 1D2D approaches. Manning values are homogeneous throughout the network and floodplains at 0.05 s/m 1/3 . Left: Bathymetry in 1Dlike and 2D meshes. The 2D area has 66 982 cells and the coupled 1Dlike reaches account for 1342 cells. The south, east and west interfaces respectively feature 6, 18 and 14 2D cells. The exclusively 1Dlike mesh contains 1409 cells. Right: Simulated water height for the 1D2D model for a sample flooding event (low tide, Q Adour = 650 m 3 /s and Q Nive = 58 m 3 /s

Figure 3 .

 3 Figure 3.20: Upstream hydrograph inferences from 4 observation stations in twin experiment setup. Left: In blue: upstream inflow hydrographs used to generate observations. In green: inferred hydrographs at 20 iterations. Dotted lines: background hydrograph value given for iteration 0 (mean target inflow over assimilation window). NSE values are given for their informative value but are not used in the assimilation process. Right: In blue: target WS elevation. In green: WS elevation generated by the inferred hydrographs. Dotted lines: WS elevation generated by the background hydrographs. Downstream tidal boundary influence is felt at the two most downstream stations (Urt and Peyrehorade). RMSE value given are used to calculate cost in the assimilation process (Section 3.3.4).

Figure 4 . 1 :

 41 Figure 4.1: Satellite nadir tracks (Jason in red, Envisat in orange) and local permanent stations (Banque HYDRO network) over the Adour river network

1

 1

  . 4.3(a)) lead to accurate inferences at the observation points, with a global overestimation of the required correction. Outside of the immediate surroundings of the observation points, in the 2D zone, bathymetry remains mostly unchanged by the assimilation process (this is even more apparent in cases with lower sensitivity to bathymetry, not shown). Inferences from flow velocity observations with a grid pattern (pattern A, Fig.4.3(c)) lead to global inferred bathymetry patterns in coherence with the target. Inferences are closer to the target at the bottom bathymetry bump, where flow velocity sensitivity to bathymetry is higher.Inferences from cross-sectional upstream WS elevation observations (pattern B, Fig.4.3(b)) lead to a fair inference of the downstream pattern and amplitude of the bathymetry irregularities. Local erroneous estimations appear, especially near 1Dlike-2D interfaces. This is explained by the correlated influences of local bathymetry backwater control over pattern B observability. Inferences from cross-sectional upstream flow velocity observations(pattern B, Fig. 4.3(d)) are even more subject to this behavior, with bathymetry overestimation at the southern upstream and the downstream 1D2D interfaces. Note that all inferences lead to low cost function values and modeled flow velocities close to reference ones.

Figure 4 Figure 4 . 3 :Figure 4 . 4 :

 44344 Figure 4.2: Confluence twin experiment setup

  A.1) leads to the 3D Navier-Stokes equations that write (see e.g.[START_REF] Ancey | Hydraulique à surface libre[END_REF]):divu = 0 ∂u ∂t + u∇u = -1 ρ ∇p + g + gS (u,n)

Figure A. 1 :

 1 Figure A.1: Microscopic fluid control volume within an open surface flow with mass fluxes (from Dingman [2009]) .

Figure A. 2 :

 2 Figure A.2: Element of fluid in a free surface flow, projected on the flow direction axis. (from[START_REF] Dingman | Fluvial hydraulics[END_REF]). h is the water depth measured perpendicularly to the water surface, P atm is the atmospheric pressure, θ is the bottom slope in the direction the flow.

  flow and further integrating by channel width, over a cross-section perpendicular to the 1D flow, the 1D SWE are obtained in (A, Q) variables:

  using BC2 (u x, b = 0), we write:

b

  At the channel banks at w 1 and w 2 the integral over the water depth is 0, thus : (u x, b = 0, u y, b = 0), we write:∂u x u z ∂z dz = [u x u z ] (H) -[u x u z ] b (A.28)By using BC2 (u x, b = 0), we write:

  dz and v = 1 h H b u y dz.

Table 1 .

 1 1: Worldwide pluviometric regimes (adapted from

	Subtropical humid	100-150	Inland and coasts (e.g. Yangtze river basin, eastern Mississipi basin)
	Subtropical dry	25	Inland and coasts (e.g. Sahara region)
	Intertropical coastal	150	Coasts (e.g. eastern central America)
	Temperate continental	10-50	Inland (e.g. American great plains)
	Temperate oceanic	100	Inland and coasts (e.g. western Europe, 80-200 cm/year in metropolitan France)
	Polar and arctic	30	North of 60 th parallel (e.g. northern Canada)

Table 1 .

 1 4: Main satellite missions providing altimetric observations of inland water surfaces.

		Start	End	Cycle period (days)
	Seasat	1978	1978	17
	Geosat	1986	1989	17
	ERS-1 (5 missions)	1991 to 1995	1992 to 1996	3 to 168
	Topex/Poseidon	1992	2005	10
	ERS-2	1995	2003	35
	Jason-1 and 2	2001 and 2008	2014 and 2019	10
	Envisat	2002 and 2010	2010 and 2012	35 and 27
	CryoSat-2	2010	-	91
	SARAL	2013	-	35
	Jason-3	2016	-	10
	Sentinel-3A and 3B	2016 and 2018	-	27
	GRACE	2002 and 2018	2017 and -	4
	SWOT	2022/2023	-	21 (1 to 4 obs per cycle)

  is

Table 2 .

 2 1: Parameter values for sinusoidal hydrographs Q

	13-0.29

Table 2 .

 2 2: Inferred parameters misfits to the truth for Ch2 variants. The RMSE m 3 /s and rRMSE represent the misfit of the inferred parameters, while the cost function used in the assimilation process represents the misfit of variables.

		08	2.5	0.15	4.7	0.05	1.6	8.0×10 -6	54
	Ch2b	0.72	22.9	1.34	10.6	0.08	2.7	9.4×10 -6	261
	Ch2c	0.06	1.9	0.27	8.7	0.04	1.3	7.7×10 -6	78
	Ch2d	2.21	70.0	3.31	105.0	0.03	0.9	7.9×10 -6	24

Table 2 . 3

 23

						0.54	180
		Ch3b	50	50	600	0.54	97
		Ch3c				0.63	54
					2
		Ch3a	30	30	400	0.23	156
	"Adjusted"	Ch3b				0.25	108
		1	0.9	1.2	300
		Ch3c				0.26	100

: Parameter weight sets in Ch3 variants

Table 2 .

 2 Upstream Average bias (m) 4: RMSE and bias over 8 years for the M1, M2a and M2b models. Upstream metrics are calculated for stations 1 to 9 only, which are outside of the BC's backwater influence. The high global RMSE for M2b comes from the known dephasing of the Solimões and Negro peak flow, which is not reproduced by the RC.

	M1	0.94	-0.02	0.88	-0.08
	M2a	0.94	-0.02	0.91	-0.01
	M2b	1.72	0.23	0.90	-0.05

Table 2

 2

	Obs set	Prior	RMSE m 3 /s (rRMSE [-] )			Nash-Sutcliffe
				65 km	502 km	657 km	754 km	65 km 502 km 657 km 754 km
	DenseSet NoiseSWOTSet	Q	f lat FG	189 (0.12) 329 (0.17) 1472 (0.22) 430 (0.45) 209 (0.14) 360 (0.18) 1719 (0.26) 421 (0.44)	0.84 0.81	0.90 0.88	0.87 0.82	0.10 0.10
	DenseSet NoiseSWOTSet	Q	f ilter FG	101 (0.07) 195 (0.10) 102 (0.07) 208 (0.11)	412 (0.06) 503 (0.07)	158 (0.17) 154 (0.16)	0.94 0.94	0.97 0.96	0.99 0.99	0.49 0.51
							(a) Hydrograph scores
		Obs set	Prior RMSE α m 1/3 /s	RMSE b [m]
		DenseSet NoiseSWOTSet	Q	f lat FG	5.35 5.12		1.89 1.64
		DenseSet NoiseSWOTSet	Q	f ilter FG	5.30 5.13		2.06 1.99
				(b) Channel parameters scores

.6: Inferred parameter scores for extended control inferences

Table 2 .

 2 7: Inferred lateral inflows parameter weights and RMSE

	(a) With prior Q -30 FG	789 2400 249.73	754 2000 140.89	739 2800 127.97	700 1500 263.07	657 5000 421.26	628 1200 119.85	623 1000 40.75	598 600 28.69	528 1000 66.85	502 2000 116.21	1 465 600 8.86	418 800 60.64	388 1000 98.30	261 800 56.57	247 1000 89.82	234 800 112.76	216 700 37.04	193 1000 49.06	163 1000 64.17	121 1500 97.74	65 2500 79.62	x [km] µ [d] σ m 3 /s DenseSet RMSE m 3 /s
		283.92	158.19	173.35	264.27	578.31	126.13	44.83	29.08	83.55	194.86	12.64	49.12	125.94	45.62	105.24	120.43	41.48	50.75	70.44	134.59	122.34	RMSE m 3 /s	SWOTNoiseSet
	(b) With prior Q +30	789 2400 293.07	754 2000 147.88	739 2800 157.54	700 1500 250.54	657 5000 581.03	628 1200 123.45	623 1000 39.51	598 600 28.62	528 1000 87.67	502 2000 212.87	1 465 600 13.53	418 800 33.81	388 1000 146.13	261 800 35.82	247 1000 122.21	234 800 128.18	216 700 45.31	193 1000 45.66	163 1000 69.46	121 1500 142.70	65 2500 134.66	x [km] µ [d] σ m 3 /s DenseSet RMSE m 3 /s
		297.37	154.01	194.14	256.81	775.15	126.71	43.31	28.85	91.96	235.58	15.37	32.22	149.39	33.85	125.72	130.98	47.28	43.47	66.83	138.31	156.34	RMSE m 3 /s	SWOTNoiseSet

FG

Table 3 .

 3 2: Hydrological parameter inference results. Prior values are taken identical to target values, except for parameter c 4 , where under-and over-estimated prior value are considered.

	Parameters	c 1	c 2	c 3	c 4
	Target Prior	520.01	-3.523	78.75	0.137 0.167 / 0.107
	Inferred	520.00 / 519.46 -3.654 / -3.820 80 / 80 0.136 / 0.137

Table 3 .

 3 3: Garonne models parameters and metrics. "PR" stand for Permanent Regime. "High" observation density corresponds to 720 stations, or 1 station per 100 m. "Low" observation density corresponds to 72 stations, or 1 station per 1 km.

	.858	0.881

  = min 0, ûKgh K , ûK e -2 gh K e + gh K s K e = max 0, ûK e + gh K e , ûK + 2 gh Kgh K e(3.15) It has been demonstrated that this insures L ∞ stability, positivity and consistency with entropy condition under a CFL condition.For the intermediate wave speed estimate, following[START_REF] Toro | Shock-capturing methods for free-surface shallow flows[END_REF] we set:

	FHLLC e	3 = FHLLC e	s K e -s K 1 v * with v * =	vK e if s * < 0 vK if s * ≥ 0	(3.14)
	where the wave speed expression is those proposed in Vila [1986a]:
	s K

Table 3 .

 3 4 . 4: Calibrated hydrological parameters of the 4 upstream hydrological catchments from the Adour multi-D hydrographic network model. *Note that the x 4 calibrated parameters corresponds to the non-"state-space" GR4H version (not presented, see[START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]), for which the calibration tool is provided. x 4 values in the present run with the "state-space" were set at 0.15.

	Parameter Adour	Oloron	Pau	Nive
	x 1	413.744 1844.567 1118.78 982.401
	x 2	0.148	1.363	1.134	0.696
	x 3	86.418	117.919 112.168 90.017
	x 4 *	55.466	12.739	11.059	6.980

Contents 4.1 Work perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.1.1 Inferences on a large scale hydrographic network . . . . . . . . . . . . . . 146 4.1.2 Inferences from 2D surface observation fields . . . . . . . . . . . . . . . . 149 4.2 General conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Table 4 .

 4 1: Observability of Adour network WS from satellite sources

	Mission	Observability	Revisit time
	Envisat	3 points on 2 tracks	27 days
	Jason	2 points on same track	10 days
	SWOT	3 swaths, each around 80% of whole network, all network seen	21 days

  2.1 over the water depth h:Using the boundary condition at the river bottom (BC2, u z,0 = 0) and Eq. A.8, Eq. A.7 writes:

					0	h(x,t)	∂u x ∂x	dz +		0	h(x,t)	∂u y ∂y	dy +	0	h(x,t)	∂u z ∂z	dz = 0	(A.6)
	By using the Leibniz formula (Eq. A.3), we get:
				∂ ∂x	0	h	u x dz -	dh dx	u x,h +	∂ ∂y		0	h	u y dz -	dh dy	u y,h + [u z ] h 0 = 0	(A.7)
	Evaluating Eq. A.7 at the free surface gives the BC3 relation:
										∂h ∂t	+ u x,H	∂h ∂x	+ u y,H	∂h ∂y	= u z,H	(A.8)
	∂ ∂x	0	h	u x dz -	dh dx	u x,h +	∂ ∂y	0	h	u y dz -	dh dy	u y,h +	∂h ∂x	u x,h + u y	∂h ∂y	(h) +	∂h ∂t	= 0
																			∂h ∂t	+	∂ ∂x	0	h	u x dz +	∂ ∂y	0	h

  At the channel banks at w 1 and w 2 , the integral over the water depth is 0, thus :∂H ∂y = u z , thus -∂H ∂t -∂H ∂y u y,H + u z,H -∂H ∂x u x,H = 0. The left term (A) + (B) + (C) + (D) of Eq. A.16 write: The right terms (E) + (F) + (G) of Eq. A.16 write:With the assumption the the bottom slope θ is low, we reach this 1D formulation of the momentum equations:

	(A) + (B) + (C) + (D) =	∂Q ∂t	-		w 2 w 1	∂H ∂t	u x,H dy -	w 2 w 1	∂H ∂y	u x,H u y,H dy +	w 2 w 1	u x,H u z,H dy (A.35)
									+	∂ ∂x		w 2 w 1	H b u 2 x dzdy -	w 2 w 1	∂H ∂x	u 2 x,H dy
									=	∂Q ∂t	+	∂ ∂x	w 2 w 1	H b u 2 x dzdy +	w 2 w 1	u x,H -	∂H ∂t	-	∂H ∂y	u y,H + u z,H -	∂H ∂x	u x,H dy
															(A.36)
	At the free surface, we have ∂H ∂t + u x	∂H ∂x + u y
									∂Q ∂t	+	∂ ∂x		w 2 w 1	H b u 2 x dzdy =	∂Q ∂t	+	∂ ∂x	Q 2 A	(A.37)
	w 2 w 1	b	H	-gsin (θ) -	1 ρ	∂p ∂x	-gS f ,x dzdy = -	w 2 w 1	H b 1 ρ	∂x	dzdy	(A.38)
															∂h ∂x	+ S f ,x	(A.39)
	w 2 w 1	H b ∂u 2 ∂x	dzdy = = -	w 2 w 1 ∂ ∂x w 2 w 1	w 2 w 1 ∂H ∂x ∂Q ∂t	∂ ∂x H b u 2 H b u 2 x dz -x dzdy -∂Q ∂t + ∂ ∂x u 2 x,H dy + ∂ ∂x Q 2 A + gA ∂H u 2 x,H ∂x ∂w 2 ∂x b u 2 dy H x dz (w 2 ) + Q 2 A = -gA ∂h ∂x + S f ∂H ∂x = gA S g -S f	∂w 1 ∂x	(A.32) x dz (w 1 ) (A.33) (A.40) b u 2 H (A.41)
	with Q = Following	w 2 w 1	H b u x dzdy, S g = -∂b ∂x and S f = ν g	∂ 2 u x ∂x 2 .
									D =	∂ ∂x	w 2 w 1	H b u 2 x dzdy -	w 2 w 1	∂H ∂x	u 2 x,H dy	(A.34)
	The left side of Eq. A.16 becomes :

∂ (P atm + ρghcos (θ)) -Agsin (θ) -AS f ,x = -gA sin (θ) + cos (θ)

  1.1 becomes: From Eq. B.7, the following discretization is obtained (see Subsection B.3.1): CG.∆Q i+1 + CH.∆Z i+1 = CI.∆Q i + CJ.∆Z i + CK (B.8) where CG, CH, CI, CJ and CK are coefficients which depend on hydraulic and geometrical parameters, see details below in Subsection B.3.1. Note that only the CK coefficient is impacted by the lateral inflow term.

	W	∂Z ∂t	+	∂Q ∂x	= q l	(B.7)

  CL.∆Q i+1 + CM.∆Z i+1 = CN.∆Q i + CO.∆Z i + CP (B.10) where CL, CM, CN, CO and CP are coefficients that depend on hydraulic and geometric parameters, see details below in Subsection B.3.2.

	∂x	+	1 gA	∂Q ∂t	+	∂QU ∂x	-u l q l	= 0	(B.9)
	From Eq. B.9, the following discretization is obtained (see Subsection B.3.2):

  11)Discretization of Eq. B.11 according to the Preissmann scheme:The discretized mass equation obtained is written as:CG.∆Q j+1 + CH.∆Z j+1 = CI.∆Q j + CJ.∆Z j + CK (B.14)Let us recall Eq. B.1.2:

	B.3.2 Momentum conservation equation
				∂Q
				∂t
	θ	W n+1 j+1 -W n+1 j 2	+ (1 -θ)	W n j+1 -W n j 2	Z n+1 j+1 -Z n j+1 2∆t	+	Z n+1 j 2∆t -Z n j	(B.12)
					+θ	Q n+1 j+1 -Q n+1 j ∆x	+ (1 -θ)	Q n j+1 -Q n j ∆x	= θq l	n+1 j+1/2 + (1 -θ) q l	n j+1/2
	θ.∆Q j+1 +	∆x 4∆t	W n+1 j+1 + W n j	∆H j+1 = θ∆Q j -	∆x 4∆t	W n j+1 + W n j	∆Z j -Q n j+1 -Q n j	(B.13)
									+∆x. θq l	n+1 j+1/2 + (1 -θ) q l	n j+1/2
	With:
			CG =					θ	(B.15)
			CH =					∆x 4∆t	W n j+1 + W n j	(B.16)
			CI =					θ	(B.17)
			CJ =					-	∆x 4∆t	W n j+1 + W n j	(B.18)
			CK =		-Q n j+1 -Q n j + ∆x θq l	n+1 j+1/2 + (1 -θ) q l	n j+1/2	(B.19)

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3∆t n gn 2 , c ≥ 0. Note that u n+1 u n+1 + cu n+1 = c ūn+1 . Therefore for non

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Service d'Hydrométrie et de Prévision des Crues Gironde Adour Dordogne

upstream. The source code and synthetic cases are available upon simple request 2 .

The remainder of this article is organized as follows. In Section 2, the modeling hypothesis, the computational resolution and inverse methods are detailed. In Section 3, the multi-D coupling scheme is validated on a series of academic cases and several academic and real-like inference setups are investigated. The study is concluded in Section 4, which also outlines potential applications and improvement perspectives that the proposed method and findings bring. (A, Q) and (h, u, v), both locally non-inertial SWE (A,Q) (flow section m 2 and at-a-section discharge m 3 /s ) or in (h, u, v) (water depth [m] and 2D depth-integrated flow velocities [m/s]).

"Max order" refers to the maximum demonstrated scheme order.

Rearrangement of αU|

Let us rearrange the lateral inflow source term discretization such that it is expressed by its dependency on ∆Z and ∆Q.

Flow velocity U at time n + 1 writes:

Thus, the discretization of u l | n+1/2 j+1/2 q l | n+1/2 j+1/2 in Eq. B.25 becomes:

This can be written in compact form as:

With:

(B.33)

The discretization of the terms from eq:Conservation of Momentum to Discretize other than u l q l is detailed in [START_REF] Roux | Estimation de paramètres en hydraulique fluviale, à partir de données caractéristiques de l'imagerie aérienne[END_REF]. The current work adapted the method for the lateral source term, with the CLat i , i ∈ [1, 5] terms. By regrouping the discretized terms of Eq. B.21 from Roux [2004] and the above, we obtain the following discretized momentum conservation equation:

Rearranging the above expression, we can rewrite the above formula as:

With:

List of Figures 2 Simultaneous inferences of lateral inflows q l (x, t), bathymetry b (x) and friction patches K (x) in a twin experiment setup on the Negro river 1D model. . . . . . . 2.3 Flowchart of the method using the HiVDI inverse method for variational calibration 2.4 Spatial hydraulic signature of lateral inflows on an academic test case . . . . . . .