Keywords: Anabranching River, Ungauged River, 1D Hydraulic Model, Variational Assimilation, Satellite Altimetry, SWOT, Hydraulic Visibility

Variational inference of eective channel and ungauged anabranching river 1 discharge from multi-satellite water heights of dierent spatial sparsity 2 P

Introduction

Fresh water is a crucial earth's resource and its journey from the clouds to the oceans passes through the hydrographic network. In order to characterize hydrological uxes, an essential physical variable is river discharge (cf. Global Climate Observing system [START_REF]The global observing system for climate: Implementation needs[END_REF]) representing an integration of upstream hydrological processes. In complement of in situ sensors networks which are declining in some regions (e.g. [START_REF] Fekete | The current status of global river discharge monitoring and potential new technologies complementing traditional discharge measurements[END_REF]), increasingly accurate measurements of hydrological and hydraulic variables, and especially river surface variabilities are now enabled by myriads of satellites for earth observation and new generations of sensors (e.g. [START_REF] Vorosmarty | Analyzing the discharge regime of a large tropical river through remote sensing, ground-based climatic data, and modeling[END_REF][START_REF] Alsdorf | Tracking fresh water from space[END_REF][START_REF] Calmant | Monitoring continental surface waters by satellite altimetry[END_REF][START_REF] Schumann | Exploiting the proliferation of current and future satellite observations of rivers[END_REF]).

The forthcoming Surface Water and Ocean Topography (SWOT) wide swath altimetric mission (CNES-NASA, planned to be launched in 2021) will provide a quasi global river surfaces mapping with an unprecedented spatial and temporal resolution on Water Surface (WS) height, width and slope -decimetric accuracy on WS height averaged over 1 km², 1 to 4 revisits every 21 days cycle [START_REF] Alsdorf | Measuring surface water 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] Rodriguez | SWOT Science requirements document[END_REF][START_REF] Biancamaria | The swot mission and its capabilities for land hydrology[END_REF][START_REF] Rodriguez | Wide-swath altimetry: A review[END_REF]. In addition to decades of nadir altimetry (e.g. [START_REF] Frappart | Preliminary results of envisat ra-2-derived water levels validation over the amazon basin[END_REF][START_REF] Charon | Contribution of the topex nasa radar altimeter to the global monitoring of large rivers and wetlands[END_REF][START_REF] Santos | Water level dynamics of amazon wetlands at the watershed scale by satellite altimetry[END_REF][START_REF] Calmant | 4 -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]) on inland waters, SWOT will enable an unprecedented hydraulic visibility, as dened from hydraulic analysis 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 | Physical basis for river segmentation[END_REF][START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF], of hydrological responses and hydraulic variabilities within river networks. Multi-satellite observations of water surfaces from the local to the hydrographic network scale indeed represent an unprecedented observability of hydrological responses through hydraulic processes signatures, especially on complex ow zones such as oodplains or anabranching rivers (see river morphology classication in [START_REF] Nanson | Anabranching rivers: Their cause, character and classication[END_REF]). This increased hydraulic visibility represents a great potential to learn hydrodynamic behaviors and infer hydrological uxes.

The estimation of river discharge from water surface observations (elevations, top width) remains an open and dicult question, especially in case of unknown or poorly known river bathymetry, friction or lateral uxes. Several open-channel inverse problems are studied in a relatively recent litterature in a satellite data context with more or less complex ow models and inverse methods (cf. [START_REF] Biancamaria | The swot mission and its capabilities for land hydrology[END_REF] for a review). Few studies started to highlight the benets of assimilating synthetic SWOT WS observations in simplied hydraulic models with sequential methods, for infering inow discharge assuming known river friction and bathymetry [START_REF] Konstantinos | Prospects for river discharge and depth estimation through assimilation of swath-altimetry into a raster-based hydrodynamics model[END_REF][START_REF] Biancamaria | Assimilation of virtual wide swath altimetry to improve arctic river modeling[END_REF] or infering 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, low-complexity methods have been proposed for estimating river discharge in case of unknown bathymetry and friction based on the kinematic wave assumption [START_REF] Durand | Estimating reach-averaged discharge for the river severn from measurements of river water surface elevation and slope[END_REF][START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF] or hydraulic geometries [START_REF] Gleason | Toward global mapping of river discharge using satellite images and at-many-stations hydraulic geometry[END_REF] or empirical ow models ( [START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF], see also [START_REF] Bjerklie | Satellite remote sensing estimation of river discharge: Application to the yukon river alaska[END_REF]). They are tested on 19 rivers with synthetic SWOT-like daily observations in [START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF] and their robustness and accuracy is found to uctuate, the importance of good prior guesses is highlighted.

The combined use of dynamic ow models and optimization methods enables to benet from WS observations for solving hydraulic inverse problems as shown for ood hydrograph inference in [START_REF] Roux | Use of parameter optimization to estimate a ood wave: Potential applications to remote sensing of rivers[END_REF] from WS width time series used to optimize a 1D hydraulic model or in [START_REF] Honnorat | Variational data assimilation for 2D uvial hydraulics simulation[END_REF][START_REF] Hostache | Assimilation of spatially distributed water levels into a shallowwater ood 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 ood model. Part I: mathematical method and test case[END_REF] by variational assimilation of ow depth time series in a 2D hydraulic model. The variational data assimilation (VDA) approach (see e.g. [START_REF] Dg Cacuci | Computational Methods for Data Evaluation and Assimilation[END_REF] and references therein) is well suited to solve the present inverse problem (see [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF][START_REF] Oubanas | River discharge estimation from synthetic swot-type observations using var-iational data assimilation and the full saint-venant hydraulic model[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] and references therein).

It consists in tting the hydraulic model response to the observed WS elevations by optimizing the input parameters in a variational framework. However, altimetry measurements of WS are relatively sparse in time compared to local ow dynamics. This important aspect of the inverse problem is investigated in [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF] with the introduction of the identiability maps . The latter consist to represent in space-time the available information: WS observables, hydraulic waves and an estimation of the mist with the local equilibrium. These maps enable to estimate if the sought upstream discharge information has been observed or not within the downstream river surface deformations; also they help to estimate inferable hydrograph frequencies [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF] or inferable hydrograph time windows [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF].

The inference of the hydraulic triplet (inow discharge Q(t), eective bathymetry b(x) and friction coecient K) from SWOT like WS observations is investigated in recent studies using 1D hydraulic and variational assimilation methods (e.g. [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF][START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full saint-venant hydraulic network model[END_REF][START_REF] Oubanas | River discharge estimation from synthetic swot-type observations using var-iational data assimilation and the full saint-venant hydraulic model[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]). However the inference of the triplet from WS observations remains a very challenging inverse problem because of the correlated inuence of temporal (discharge) and spatial (bathymetry-friction) controls on the simulated ow lines. This is especially true because of the bathymetry-friction equinality issue, see the discussions in [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]. Those recently developed VDA methods enable to infer accurately the inow discharge from water surface observables, considering unknown/uncertain channel bathymetry-friction, but from accurate prior information and synthetic WS observations. Note that a strong prior such as a known stage-discharge relationship (rating curve) downstream of a river domain as it is done in [START_REF] Oubanas | River discharge estimation from synthetic swot-type observations using var-iational data assimilation and the full saint-venant hydraulic model[END_REF] can control part of the simulated ow lines (uvial regime); as a consequence the VDA process may converge to the discharge hydrograph corresponding to the imposed (almost exact) rating curve. In the present study the downstream boundary condition (BC) is an unknown of the inverse problem.

A crucial point is the sensitivity of the triplet inference to the prior value from which the inference is started and it is studied in a SWOT observability context in [START_REF] Garambois | Inference of eective 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] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF][START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using airswot observations[END_REF]. The sensitivity of the estimated discharge (in the triplet) to the prior is highlighted by recent estimates performed from AirSWOT airborne measurements on the Willamette River [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using airswot observations[END_REF]. The temporal signal is well retrieved at observation times but using a biased prior hydrograph results in a biased hydrograph inference -see detailed investigations in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]. In view to infer worldwide river discharges from the future SWOT observations, especially for ungauged cases, a hierarchical modeling strategy HiVDI (Hierarchical Variational Discharge Inversion) is proposed in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]. The HiVDI approach includes low complexity ow relations (under the assumption of Low Froude and locally steady-state ows) which improves the robustness of the inferences in particular if an (unbiaised) average value of Q is provided. (It may be provided by a database or a regional hydrological model). Note that if introducing an a-priori information such as a single depth measurement, it enables to reconstruct an eective low-ow bathymetry see [START_REF] Gessese | Reconstruction of river bed topography from free surface data using a direct numerical approach in one-dimensional shallow water ow[END_REF][START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF].

All the studies mentioned above mostly address single channel natural rivers (∼ 100km in length) without lateral inows and using synthetic datasets (except in [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using airswot observations[END_REF] with AirSWOT data). Moreover very few studies address the modeling of eective 1D channels from real satellite data (e.g. [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]).

The present paper investigates the eective hydraulic modeling of anabranching river ows from real multisensor satellite observations of WS, the challenging inference of the hydraulic triplet (Q(t), b(x), K(x, h)) and its sensitivity to observation density in space. Anabranching rivers are characterized by complex hydraulic geometries relationships across ow regimes as shown in [START_REF] Schubert | Metric-resolution 2d river modeling at the macroscale: Computational methods and applications in a braided river[END_REF] through an analysis of a metric resolution 2D shallow water model of an anabranching portion of the Platte River, US. The key point here is to build up a suciently complex model to describe anabranching river ows and in coherence with the spatio-temporal scales of satellite altimetry measurements.

Based on the inverse method presented in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF][START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF], an eective hydraulic modeling strategy is adapted for tackling anabranching river ows using: (i) eective 1D cross sections based on real multi-satellite data from low to high ows (ii) a spatially distributed friction law depending on modeled water depth h. The inference of distributed hydraulic parameters patterns is investigated on a 71km long reach of the Xingu River (Amazone basin) from real altimetric observations gained on a single ENVISAT track or from synthetic SWOT observations, low identiability index (as introduced in [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF] and detailed in section 4). The inuence of the spatial density of WS observations on the identiability of spatial controls patterns (in the unknown triplet) is studied. A piecewise linear bathymetry representation is introduced along with a friction power law with piecewise constant parameters to put in coherence the observations and the ow model grids. Their constraining eect on the inversions is studied with spatially (and temporally) sparse satellite observations. Furthermore, numerical investigations are performed to test the sensitivity of hydraulic inferences to prior hydraulic values and also assess the correlated inuence of bathymetry and friction on the modeled ow lines (equinality) across ow regimes.

This study is organized as follows. Section 2 presents the 1D Saint-Venant ow model and the eective modeling approach for anabranching rivers including: (i) a spatially distributed friction law depending on the modeled ow depth, (ii) the construction of an eective channel geometry from multi-satellite observations, (iii) an inverse method based on variational data assimilation. Section 3 focuses on the calibration of the eective model on 8 years of WS observations gained from ENVISAT altimeter on a single track along this anabranching river. Using this model as a reference, section 4 proposes detailed investigations of the hydraulic inferences from real ENVISAT or synthetic SWOT observations considering this anabranching river as ungauged. The discussion in section 5 presents a numerical sensitivity analysis to the hydraulic prior and some investigations on the bathymetry friction equinality.

Modeling approach:

This section proposes an original 1D eective modeling approach of adequate complexity for modeling anabranching river ows across (uvial) regimes and in coherence with satellite observations. The approach is built on an eective channel cross-section derived from multi-satellite measurements and a spatially distributed friction law depending on the ow depth.

The ow model

River ows are classically modeled using the 1D Saint-Venant shallow water equations involving an integration of the ow variables over the cross section (see e.g. [START_REF] Chow | Handbook of applied hydrology[END_REF][START_REF] Guinot | Wave Propagation in Fluids: Models and Numerical Techniques[END_REF] for detailed assumptions). In (A, Q) variables, A the wetted-cross section m 2 , Q the discharge m 3 .s -1 , the equations read as follows [START_REF] Chow | Handbook of applied hydrology[END_REF]:

       ∂ t (A) + ∂ x (Q) = 0 ∂ t Q + ∂ x Q 2 A = -gA ∂ x Z -gAS f (1)
where g is the gravity magnitude m.s -2 , Z is the WS elevation [m], Z = (b+h) with b is the river bottom elevation

[m] and h is the water depth [m]. The friction slope S f is parameterized with the classical Manning-Strickler law

such that S f = |Q|Q /K 2 A 2 R 4/3 h with K the Strickler friction coecient m 1/3 .s -1 , R h = A /P h the hydraulic radius
[m] , P h the wetted perimeter. The discharge Q is related to the average cross-sectional velocity u m.s -1 such as Q = uA. A spatially distributed Strickler friction coecient is dened as a power law in the water depth h:

K(x, h(x, t)) = α(x)h(x, t) β(x) (2)
where α and β are two constants. Similar approaches based on hydraulic geometry or power law resistance equations are developed in the litterature for predicting mean ow velocity for example on a wide range of in situ river ow measurements in [START_REF] Bjerklie | Comparison of constitutive ow resistance equations based on the manning and chezy equations applied to natural rivers[END_REF] or else for gravel bed streams in [START_REF] Ferguson | Flow resistance equations for gravel-and boulder-bed streams[END_REF]. The friction depends on the ow depth through the proposed power law relation (Eq. 2) enabling a variation of the friction eect in function of the ow regime for complex ow zones for instance; this spatially distributed friction law is richer than a constant uniform value as it is often set in the literature from a-priori tables of frictions in function of river types for instance (e.g. [START_REF] Chow | Open-channel Hydraulics[END_REF]).

Note that satellite altimetry mostly observes the downstream parts of river networks (top width W > 100m for SWOT), mainly in subcritical and mostly low Froude ows at the observation scales (cf. [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF][START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF]). The discharge Q in (t) is classically imposed upstream of the river channel with a discharge hydrograph. At downstream a normal depth is imposed using the Manning-Strickler equation depending on the unknowns (A, Q; K) out (it is classically integrated in the Preissmann scheme equations). The initial condition is set as the steady state backwater curve prole Z 0 (x) = Z(Q in (t 0 )); also depending on the unknowns. Note that these boundary and initial conditions are updated during the iterative inverse method presented in what follows. This 1D Saint-Venant model (Eq. 1) is discretized using the classical implicit Preissmann scheme (see e.g. 50) on a regular grid of spacing ∆x. It is implemented into the computational software DassFlow (DassFlow [START_REF] Dassflow | Data assimilation for free surface ows[END_REF]).

Eective anabranching river model from multisatellite data

A L = 71km long portion of the Rio Xingu containing anabranching reaches is considered (Fig. 1, cf. [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF],). WS observations are available at 6 virtual stations along a single ENVISAT track (#263) representing 77 samples of WS proles between mid 2002 and mid 2010 (cf. [START_REF] Santos | Water level dynamics of amazon wetlands at the watershed scale by satellite altimetry[END_REF]); that is Z obs s,p env S,P with S = 6 corresponding to the locations of the virtual stations simultaneously observed at P = 77 times (see Tab. 1).

An eective hydraulic modeling strategy of this anabranching river is proposed based on:

Cross-sectional water surface widths {W } jers S,2 obtained from JERS mosaics (Courtesy of GRFM, NASDA/MITI) in low and high ows. The eective water surface width is the sum of the width of all individual river channels for anabranching reaches. Note that the cross section geometry of this (ungauged) anabranching river might be changing over a hydrological year, from disconnected channels in low-ows to a mono-channel with forested oodplains during the ood season. The available satellite images resulted in an estimation of a larger eective top width in high-ow.

An a priori river bottom {b} r V S obtained from altimetric rating curves from [START_REF] Paris | Stage-discharge rating curves based on satellite altimetry and modeled discharge in the amazon basin[END_REF]. The authors determined eective bottom elevations by adjusting the scalar parameters γ and δ of a classical stage discharge relationship Eective cross-sections geometries are dened at the 6 virtual stations with the bathymetry b given by altimetric rating curves and from eective widths such that low ow width (resp. high ow) is reached for the rst (resp. ninth) decile of observed WS elevations for each cross section. The nal model geometry is obtained by linear interpolation between these 6 eective cross sections on the model grid with ∆x = 50m. It is shown in Fig. 1 along with ENVISAT and SWOT spatial samplings. The friction law (Eq. 2) introduced above and depending on the ow depth h is distributed using patches with constant values for each reach between two successive virtual stations.

Q = γ(Z -b) δ I 1/2 ,

The computational inverse method

This paper investigates the estimation of the hydraulic triplet (Q(t), b(x), K(x, h)) from observations of WS variabilities only on an anabranching river. The employed inverse method is those presented in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] (see also [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF]) with an augmented composite control vector c; it is detailed in Appendix 7. c contains a spatially distributed friction coecient enabling to model complex ow zones (while it is an uniform friction law K(h) in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]). This denition of K(x, h) enables to consider more heterogeneous bathymetry controls.

7

The principle is to estimate (discrete) ow controls minimizing the discrepancy between Z obs the observed ow line and Z the modeled one; the latter depending on the unknown parameters vector c through the hydrodynamic model (Eq. 1). This discrepancy is quantied through the cost function term:

j obs (c) = 1 2 Z obs -Z(c) 2 2 (3)
see Appendix 7 for details. The control vector c contains the unknown input parameters of the 1D Saint-Venant shallow water ow model (Eq. 1) considering eective cross sections (see Fig. 1). In the present study, c reads as:

c = (Q in,0 , ..., Q in,P ; b 1 , ..., b R ; α 1 , ..., α N , β 1 , ..., β N ) T (4)
where temporally and spatially distributed controls are the upstream discharge Q in,p , the river bed elevation b r and the distributed friction parameters α n and β n .

The subscript p denotes the observation time p ∈ [0..P ] and r denotes the reach number, r ∈ [1..R].

α n and β n are the parameters of the friction law depending on the model state h (Eq. 2) for each patch n ∈ [1.

.N ]

with N ≤ R.

The inversion consists to solve the following minimization problem: c * = argmin j(c) (Eq. 9).

This minimization, optimization problem is solved using a rst order gradient-based algorithm, more precisely the classical L-BFGS quasi-Newton algorithm. The main steps of the method are illustrated in Fig. 2.

Model Calibration

This section presents the calibration of the eective hydraulic model based on the reference eective geometry dened above (cf. section 2.2). The observed water elevation time series Z obs s,p env S,P at S = 5 ENVISAT virtual stations are used to calibrate the friction law of the 1D Saint-Venant ow model (Eq. 1). Since friction has a local and upstream inuence on a ow line (low Froude uvial ows, Fig. 10) the remaining ENVISAT time series at VS#6 downstream of the river domain will be used for infering the full control vector c in next section -recall that a normal depth is used as downstream BC (cf. section 2.1).

A reduced control vector c cal = (α 1 , ..., α N , β 1 , ..., β N ) consisting in spatially distributed friction parameters only is considered here. In order to avoid a spatial overparameterization regarding the 5 water height timeseries available at VS, the choice is made to spatialize friction on N = 5 patches, on each reach downstream an altimetric VS. The inverse method presented in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] and described in appendix (section 7) is used here with no regularization nor variable change for this simple calibration problem.

An optimal friction distribution c * cal is found with the inverse method and the calibrated values of α n=1... for each virtual station (cf. Fig. 3). The spatially distributed friction law (Eq. 2) enables a fairly good reproduction of the observed water level variations on this anabranching river, across a wide range of ows, even with an eective 1D model built on multi-satellite data (Fig. 3).

A constant friction in time would lead to systematical errors for a large range of ows as shown by the grey curves on Fig. 3. The calibrated friction exponents β n range between 0.482 and 1.133 except for the second reach (SV2-3) where a small β n is found, that is a barely constant friction across ow regimes for this short reach (cf. Fig. 3). The spatial pattern of α n values calibrated here corresponds to signicant friction eects, varying across ow regimes, and necessary to eectively represent anabranching reaches using a 1D eective cross section. Indeed the lattest leads to an underestimation of the hydraulic radius R h = A /P h hence of the friction slope 10

S f = |Q|Q /K 2 A 2 R 4/
asssessed in what follows regarding the spatial sparsity of observations. First is presented the numerical experiment framework, then the inferences with relatively sparse ENVISAT measurements and nally those with SWOT synthetic observations.

Design of the numerical experiments

The eective hydraulic model described in section 2.2 and calibrated in section 3 is used as a reference ( target ) in the following numerical experiments. The control vector (Eq. 4) containing discharge, bathymetry and friction is sought with the inverse method decsribed in section 2.3 (see also appendix, section 7). It is tested rst with real ENVISAT time series repesenting a relatively sparse spatial sampling of WS signatures with 6 VS on this 71km

long river, and next with synthetic SWOT observations sampling the ow line at ∆x = 200m (RiverObs product, see [START_REF] Prata De Moraes Frasson | Automated river reach denition strategies: Applications for the surface water and ocean topography mission[END_REF]).

The Xingu River is observed either by a single along-stream ENVISAT track at 6 observation points (virtual stations) of ow lines every 35 days, or two SWOT tracks providing dense WS observations in space twice per 21 days repeat cycle (5 days delay, cf. section 2.2). Note that the temporal sparsity of observations (35 days for ENVISAT or 5 days between the two SWOT passes every 21 days) only enables to identify low hydrograph frequencies, at observation times (see [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF] for a detailled analysis and the identiability maps). Indeed the hydraulic wave propagation time is around T wave ∼ 9h which is much smaller than the lowest satellite revisit time of 5 days.

This propagation time is calculated using the kinematic wave velocity for rectangular channels c k = 5 /3U and maximal high ow velocity U = 2.17m/s from calibrated model outputs c k = 2.2m/s (second hydrograph peak at t = 490 days, see ow variables on Fig. 10). Let I indent = T wave /∆t obs be the identiability index dened in [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF] as the ratio between ood wave propagation time and observation time step. This leads to a very low temporal identiability index for this 71km river: I ident = 7.5 × 10 -2 for SWOT and I ident = 10 -2 for ENVISAT.

Consequently, only low temporal dynamics and discharge at observation times are inferable as shown in [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF]; SWOT and ENVISAT observations are thus considered separately in the present study.

The starting point of the VDA process in the parameter space, the so-called prior c prior (cf. section 7), consists in a rough hydrological prior: Q (0) = Q M GB the mean discharge estimated from the MGB hydrological model, a spatially constant α (0) friction dened a priori from classical hydraulic ranges (e.g. [START_REF] Chow | Open-channel Hydraulics[END_REF]) and β (0) = 1, the bathymetry b (0) is dened as a simple straight line over the whole domain for hydraulic analysis rst. Note that the sensitivity of the inference to the prior denition is investigated in section 5.

In a noised observation context, we denote by δ the noise level such that Z obs -Z true 2 ≤ δ for all spatial locations r with Z obs r the observed and Z true r the true WS elevation. A common technique to avoid overtting noisy data, in the context of Tykhonov's regularization of ill-posed problems, is Morozov's discrepancy principle, (see e.g.

[59] and references therein): the regularization parameter γ (see Eq. 7) is chosen a-posteriori such that j does not decrease below the noise level. In the present numerical experiments, the convergence is stopped if j obs (c) ≤ 10 -1

or if j obs is not decreased anymore for higher discrepencies.

Inference from spatially sparse ENVISAT snapshots

In this section the assimilation is based on WS elevations Z env s,p S,P at S = 6 virtual stations observed simultaneously by ENVISAT during 8 years every 35 days, i.e. P = 77. In this spatially sparse observation context, the impact of spatial controls density is investigated.

First, we consider a full control vector c (cf. Eq. 4) including P = 77 inow discharges, all 1D model bathymetry points R = 1420 and N = 5 friction patches between ENVISAT virtual stations (cf. section 2.2).

The infered inow discharge, bathymetry and friction are presented in Fig. 4 (case Env.a). Despite the satisfying value of the hydraulic controls reached at iteration 35, the descent is still possible as shown by j obs decreasing of about 20% at iteration 96. Allthough it enables to t the observations according to the a priori convergence criteria dened in section 4.1, the solution found after the VDA process is not very accurate nor realistic as shown by peak ow underestimations and signicant oscillations of the identied friction and bathymetry. The spatial sparsity of observations prevents to infer these relatively dense bathymetry controls; in this case the considered inverse problem is underconstrained.

In order to better constrain the inverse problem in case of sparse spatial observability, a bathymetry representation is consistently introduced at the scale of the observation grid and applied to the ner ow modeling grid.

Based on the physical analysis of the SW model (Eq. 1) behaviour and the WS signature of bathymetry/friction controls (see [START_REF] Montazem | Physical basis for river segmentation[END_REF][START_REF] Montazem | Representation et segmentation hydraulique eective de rivieres pour le calcul de debit par altimetrie SWOT Ã l'echelle globale[END_REF][START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF]), a linear bathymetry interpolation is used between the successive couples of bathymetry controls dened at observation points only. The resulting bathymetry b(x) ∈ C 0 (R), ∀x ∈ [0, L] is piecewise linear and strongly constrains the bathymetry prole between the sought bathymetry points -instead of using only a weak constraint j reg (c) = 1 2 b"(x) 2 2 in the optimization process (cf. appendix 7) as done in the next section 4.3 with spatially dense SWOT observations. Using this bathymetry constraint with R = 6 bathymetry controls dened at each ENVISAT virtual station results in 5 reaches and N = 5 friction patches are consistently applied to each. This leads to a more robust and accurate inference as shown in Fig. 5 (case Env.b). The discharge infered for 8 years is fairly correct (RMSE = 520 m 3 /s, Nash = 0.95) and relatively realistic bathymetry/friction patterns are found, with some compensations between spatial controls locally in space, which is further analyzed in what follows.

The impact, on the infered parameters, of searching a spatially uniform friction law is tested with the piecewise linear bathymetry representation used above. The resulting discharge inference is fairly correct (RMSE = 608 m 3 /s, Nash = 0.93) and interestingly the bathymetry spatial pattern is well retrieved but shifted above the reference one (cf. Fig. 6) (case Env.c). The infered friction coecients are α = 22.621, β = 0.217, which represents a lower friction eect on most ow regimes regarding the calibrated ones (cf. Tab. 1). These infered eective friction law and bathymetry patterns, leading to somehow eective stage-discharge relationships locally given the infered hydrograph and its propagation, enable to approximate the observed WS variations (j obs = 1.269 ) but with a less accurate t than with spatially distributed friction (j obs = 0.118). Note that in this case of lower model complexity an underestimation of the low ow discharges occurs. Recall that the observations consist in real measurements of WS elevations gained by nadir altimetry on anabranching reaches of the Xingu River. The complexity of the forward-inverse modeling approach, in coherence with the spatial sparsity of the observation grid, enables to approximate satisfactorily the one of the observed anabranching ow. The additionnal constraint provided by spatially dense ow lines observations is investigated in the next section with SWOT synthetic data. In this section the full hydraulic control c (cf. Eq. 4) is infered by assimilating SWOT-like observations. Those noisy data are computed using the SWOT hydrology simulator applied to ow lines from the eective hydraulic model calibrated above (cf. section 3). The SWOT spatio-temporal pattern over the studied river is obtained by overlapping the river centerline and the expected SWOT orbit and swaths (cf. enable to constrain the inference of bathymetry controls at a ne spatial resolution (model grid). The inverse method includes: (i) a regularization term j reg in the cost function (Eq. 7); (ii) covariance matrices acting as spatial or temporal smoothers/regularizations (cf. Eq. 12 in appendix). The infered discharge and spatially distributed controls are slightly more accurate than previously in a comparable inversion scenario with sparse ENVISAT observations in space and piecewise linear bathymetry constrain (case Env.b, cf. Tab. 2 and Fig. 5).

Note that the friction is sought by reaches which enables to consider more dense bathymetry controls. Again, the compensation between spatial controls appears locally in space but enables the best t to the distributed measurements of WS elevations given the infered discharge (j obs = 0.099).

Discussion and numerical investigation of the bathymetry-friction equinality

This section discusses the challenging inference of spatially and temporally distributed river ow controls from water surface observations through numerical investigations. Indeed, the considered ow controls (Q(t), b(x), K(x, h)) have a correlated inuence and can produce undiscernable signatures in the modeled ow lines therefore leading to an ill-posed inverse problem (cf. [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] for investigations on this bathymetry-friction equinality in a comparable data-inversion context). The hydrograph is responsible for ow variability in time, hence enabling to retrieve the temporal dynamics of the observed ow lines [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF].

Given altimetric measurements of WS variabilities and the rst guess c prior1 , the regularized inverse method enables to infer a complex control vector composed of temporally and spatially distributed controls of the 1D SW model (Eq. 1). In the numerical experiments above, the discharge hydrograph Q(t) is accurately infered at observation times but because of the ill-posedness of the inverse problem, compensations can occur between the sought parameters and especially between the spatial controls -the bathymetry b(x) and the distributed friction parameters α(x) and β(x). These infered friction laws and bathymetry patterns -simultaneously infered with the discharge hydrograph -correspond to eective rivers enabling to t the observed variability of ow lines.

Note that the spatial density of SWOT data enables to constrain ow controls that are relatively dense in space, here on a complex anabranching ow case using the eective 1D river representation and a friction law pattern depending on water depth. Improving the physical segmentation, parameterization and sparse representation of river networks and ow signatures (e.g. [START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF]) seems of great importance to take advantage of the forthcoming SWOT observations along with other data.

Importantly, as already pointed out in the VDA inferences performed with the DassFlow model using SWOT like data in [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] and AirSWOT data in [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using airswot observations[END_REF], the accuracy of the inferred discharge depends on the quality of the prior.

In other words spatially distributed WS observations enable to depict spatio-temporal signatures and eventually propagation dynamics but a quantitative biais remains regarding uxes, from the river reach to the network scale.

In the following subsection the inuence of the prior value on the quality of the inferences with spatially distributed controls is investigated rst. Next, is proposed a numerical analysis of the sensitivity of the friction slope (source term) S f in the Saint-Venant equations (Eq. 1) to the ow controls (triplet) that are embeded in it (Manning-Strickler parameterization).

Sensitivity to the prior guess

The sensitivity of the inference to the quality of the prior guess of the control vector c prior is investigated here for the most challenging inverse problem with spatially and temporally distributed controls and sparse ENVISAT data. First the inow prior is varied of ±30% around the mean true discharge; the river bottom elevation and friction priors are set as previously in c prior1 . The infered hydraulic controls are presented in Fig. 8 and various inference scores are summed up in Tab. 2. For each inow prior, the temporal variations of the inow hydrograph are very well retrieved as shown on Fig. 8 -runs Env.b2 and Env.b3. However a biased inow prior results in a biased hydrograph estimate (with correct temporal variations at observation times) which is coherent with results of [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF][START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using airswot observations[END_REF]). Next, the sensitivity to the prior bathymetry and friction is tested. The prior bathymetry is infered with the low-complexity system proposed in the hierarchichal HiVDI model chain ( [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]) for ungauged rivers. It consists in estimating an eective prior bathymetry from WS observables using the low Froude model and prior discharge from a hydrological model (Q M GB here) and prior friction (α (0) , β (0) ). Two prior guesses c man1 and c man2 are considered with prior friction under/over-estimations compared to calibrated ones (cf. Fig. 9). As shown on Fig. 9, the inference in case Env.b31 (blue) results in an accurate estimation of discharge, very similar to Env.b (purple). It is started from a prior guess c man1 that underestimates river bottom elevation and overestimates the spatially averaged friction eect compared to calibrated values (cf. Fig. 9, bottom). In that case, tting WS elevations enables to infer an eective river channel (bathymetry and friction) but also to infer a fairly realistic upstream temporal control (discharge hydrograph). Using the prior guess c man2 that overestimates both river bottom elevation and spatially averaged friction eect results in a comparable t to the observed WS elevations. However this correct t stems from the compensation between an infered eective channel of reduced conveyance capacity (comparable friction eects but overestimated bed levels) and consequently an infered hydrograph with underestimated low-ow discharges (in yellow).

Spatio-temporal sensitivity of the friction term

The considered ow controls (Q(t), K(x, h), b(x)) of the 1D Saint-Venant shallow water equations (Eq. 1) have a complex non linear inuence on the modeled ow lines and consequently on the t to the observed onesthe lattest being evaluated globally in space and time with the current inverse method given the observation cost function (Eq. 3). The variation of momentum expressed by the second ow equation is due to a pressure source term -gA ∂ x Z (including the longitudinal variation of uid-to-uid pressure, the longitudinal variation of lateral

Case

Control Prior and bottom wall-to-uid pressure) and a dissipation term -gAS f . The discharge and the bathymetry appear in the momentum and pressure terms while all the ow controls are embedded in the friction source term S f . Note that for a locally steady uniform ow S f = -∂ x Z and an innity of friction and bathymetry values can correspond to a single value of discharge (cf. Garambois and Monnier [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF], Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]).

RMSE Q (0) rRMSE Q (0) Nash Q (0) RMSE b (0) RMSE α (0) RMSE β (0) (m 3 /s) (%) (-) (m) (m 1/3-β /s) (
We propose a simple calculation in order to make appear the sensitivity of the friction term to a change on the controls; let us express the dierential of S f assuming Q > 0:

dS f = d 1 K 2 Q 2 A 2 R 4/3 h = - 2 K 3 Q² A²R 4/3 h dK - 2 A 3 Q² K 2 R 4/3 h dA - 4 3R 7/3 h Q² K 2 A² dR h + 1 K 2 2Q A²R 4/3 h dQ (5)
Since dR h = d(A/P ) = 1 P dA -A P 2 dP = 1 P (dA -R h dP ) = 1 P (dA 0 -R h dP 0 ) + df (h) with A 0 = W 0 h 0 and P 0 = W 0 + 2h 0 respectively the unobserved low-ow area and perimeter under our modeling hypothesis (cf. section 2.2 and Fig. 1, see also Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] for details on cross section representation). It follows that f (h) is a function depending on the modeled water depth h and of the observed cross-section variation δA above low-ow (h 0 ), W 0 being dened from observables. We get dR h = 1 P 1 -2R h W0 dA 0 + df (h) and nally:

dS f = 1 K 2 Q A²R 4/3 h -2 Q K dK - Q A 2 + 4 3 1 - 2R h W 0 dA 0 + 2dQ -dφ(h) (6)
with φ(h) = 4 3R 7/3 h Q² K 2 A² df (h) a function depending on the observed geometry of a cross section above low-ow and of the simulated ow (A, Q hence h (A) given a channel geometry). We rewrite Eq. 6 as dS f = ∂ K S f dK + ∂ A0 S f dA 0 + ∂ Q S f dQ -dφ(h) and under our modeling hypothesis we have

∂ K S f < 0, ∂ A0 S f < 0, ∂ Q S f > 0 ∀x, t,
i.e. opposite eects of local values of friction K, low ow area A 0 and simulated local discharge Q values on S f .

Those terms are plotted on Fig. 10 along the Xingu River, on the model grid, from hydraulic variables simulated (forward run) with calibrated parameters (cf. Tab. 1). Note that dφ(h) is not studied with this simple method.

Interestingly, |∂ K S f | is about 100 times greater than |∂ A0 S f | or |∂ Q S f | at high-ow and about 10 times greater at low-ow. This is consistent with the singular value of friction that is found 1000 times greater than the one of reach averaged discharges by Garambois and Monnier [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF] through a singular value decomposition of the normal equations of reach averaged Manning equations -applied to 70km of the Garonne River downstream of Toulouse (France). In other words, the friction term in the present 1D modeling context must be more sensitive to a change in friction than unknown low-ow bathymetry or discharge.

Remark that for low-ow, S f is more sensitive to discharge than unknown cross sectional area

(|∂ Q S f | > |∂ A0 S f |)
and conversely for high-ow. Moreover the spatial variability of the three sensitivities is more pronouced at low-ow. Abrupt changes are highlighgted at locations corresponding to changes in the bottom slope or the channel width.

The inuences of the bottom slope break at x = 30km is clearly visible at low-ow and the inuence of the width contraction at x = 17km at high-ow, which is fully consistent with the ndings of [START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF]. Further investigations on the sensitivity of the full Saint-Venant equations, and especially the dierent contributions to the friction slope, in space and time could be of interest to better taylor, scale and constrain methods for tackling hydraulic inverse problems.

Conclusion

This paper investigates the challenging inference of the hydraulic triplet (discharge, bathymetry, friction) from real or synthetic altimetric WS observations only on an ungauged anabranching river.

The HiVDI inverse method presented in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] is adapted for reproducing an anabranching ow by introducing a spatially distributed friction law depending on modeled water depth h and by using multi-satellite data.

The friction law coecients are spatialized by reach to be coherent with the observation grid and with the (rather large) meaningful scale of these parameters in the 1D Manning-Strickler equation (see e.g. [START_REF] Guinot | Sensitivity equations for the one-dimensional shallow water equations: Practical application to model calibration[END_REF]). This eective modeling approach enables a fairly accurate reproduction of the anabranching ows observed during 8 years by nadir altimetry (ENVISAT) on this 71km anabranching river.

The inference capabilities of hydraulic parameters patterns from real altimetric observations along a single ENVISAT track or from the future spatially dense SWOT observations are demonstrated. For the present observed anabranching river complexity, the inverse method enables to infer a fairly realistic upstream discharge hydrograph along with an eective river channel. The estimated bathymetry and friction patterns somehow result in local and eective stage-discharge relationships. In case of spatially sparse observations, the coherence between the sparse observation grid and the dense model grid is ensured using a piecewise linear bathymetry representation along with a friction power law with piecewise constant parameters. This constrain on the VDA process provided by the above dened eective bathymetry-friction representation by reach is highlighted with spatially sparse ENVISAT observations. Moreover the additional constrain provided by the forthcoming SWOT observations to infer a discharge hydrograph and densely distributed spatial controls is assessed on this eective anabranching river representation; the denition of friction by reaches enabling to consider more dense bathymetry controls.

SWOT observations would represent unprecedented measurements of hydrological and hydraulic processes signatures from the local to the hydrographic network scales, including complex ow zones such as anabranching ones.

On-going researches focus on the detection and use of various hydraulic signatures in WS as highlighted here for bottom slope (resp. channel width) breaks in low (resp. high) ows (see WS curvature analysis and SW model behavior in [START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF]), on the estimation of reliable prior guesses on the sought parameters, model scaling and inverse problems at the scale of larger river network portions including complex ow zones.

coecient enabling to model complex ow zones (while it is an uniform friction law K(h) in [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]). This denition of K(x, h) enables to consider more heterogeneous bathymetry controls.

It is important to point out that the imposed downstream boundary condition is an unknown of the inverse problem. It is constrained with the observed water elevations and infered river bottom slope using a locally uniform ow hypothesis (i.e. Manning equation, cf. section 2.1).

The cost function j(c) is dened as: j(c) = j obs (c) + γ j reg (c) [START_REF] Alsdorf | Tracking fresh water from space[END_REF] where γ > 0 is a weighting coecient of the so-called regularization term j reg (c). The term j obs (c) measures the mist between observed and modeled WS elevations such that:

j obs (c) = 1 2 (Z(c) -Z obs ) 2 O (8)
The norm

• O = O 1/2 • 2 is
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 computed by solving the adjoint model; the latter is obtained by automatic dierentiation using Tapenade software [START_REF] Hascoët | The Tapenade Automatic Dierentiation tool: Principles, Model, and Specication[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: from optimal control to large scale data assimilation[END_REF].

To be solved eciently 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 ill-posed problems[END_REF] for the theory of regularization of such inverse problems and to [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] for a discussion focused on the present inverse ow problem.

Following [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], the optimization problem ( 9) is regularized as follows. First the regularization term j reg is added to the cost function, see [START_REF] Alsdorf | Tracking fresh water from space[END_REF]. We simply set: j reg (c) = 1 2 b"(x) Next and following [START_REF] Lorenc | The met. oce global three-dimensional variational data assimilation scheme[END_REF][START_REF] Weaver | Correlation modelling on the sphere using a generalized diusion equation[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], the following change of control variable is made:

k = B -1/2 (c -c prior ) ( 10
)
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) (11)
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 [START_REF] Stephen A Haben | Conditioning and preconditioning of the variational data assimilation problem[END_REF][START_REF] Stephen A Haben | Conditioning of incremental variational data assimilation, with application to the met oce system[END_REF] and [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] in the present inversion context.

Assuming uncorrelated controls B is dened as a block-diagonal matrix:

B =          B Q 0 0 0 B b 0 0 0 B α 0 0 0 B β          (12)
Still following [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], the matrices B Q 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 and (B b ) i,j = (σ b ) 2 exp - |x j -x i | L b (13)
The VDA parameters ∆t Q and L b represent prior hydraulic scales and act as correlation lengths. Given the frequency (few days) and spatial resolution of observations (200m long pixels for SWOT), the low Froude anabranching river ows of interest, adequate values for those parameters are: ∆t Q = 24 h and L b = 3km km We refer to [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF] for a thorough analysis of the discharge inference in terms of frequencies and wave lengths and Section 4.1 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 (14)

  with I the water surface slope gained from altimetry at large scale. They used WS elevations gained by satellite altimetry and discharges simulated with the large scale hydrological model MGB ([53, 54, 55]) on the temporal window of interest -called true discharge in what follows.

Figure 1 :

 1 Figure 1: Study zone (top) with ENVISAT track #263 and virtual stations (orange dots); simulated SWOT tracks #133 and #258 on the 1 st and 6 th day every 21 days repeat cycle (transparent white). Eective river bathymetry derived from altimetric rating curves ([52]) and water surface width from satellite images.

  5 and β n=1...5 are summed up in Tab. 1. The resulting water height time series are compared to altimetric observations

Figure 2 :

 2 Figure 2: Flowchart of the method using the HiVDI inverse method (Larnier et al. [1]) for variationnal calibration, adapted from Monnier et al. [56], Monnier [57].

Figure 3 :

 3 Figure 3: Calibration of variable friction K(x, h) with 8 years of ENVISAT measurements at 6 VS using the variational method with c = (α 1 , ..., α 5 , β 1 , ..., β 5 ) ; j obs = 0.07. (Bottom right) Eective friction law in function of water depth for each VS.

  Figure 4: Identication of (Q(t), K(x, h), b(x)) with ENVISAT observations and overparameterized c = Q in,0 , ..., Q in,P ; b 1 , ..., b R ; α 1 , ..., α N , β 1 , ..., β N T with P = 77, R = 1420, N = 5, bathymetry regularization weight γ = 10 -3 ; j obs = 0.098 at iteration 35 (top) and j obs = 0.077 at iteration 96 (bottom) (Env.a)

Figure 6 :

 6 Figure 6: Inferrence of Q(t), b(x) and spatially uniform K(h) = αh β with ENVISAT WS observations and eective c = Q in,0 , ..., Q in,P ; b 1 , ..., b R ; α, β T , P = 77, R = 6, no bathy γ = 0; j obs = 1.269 at iteration 54. The identied friction coecients are α = 22.621, β = 0.217. (Env.c)

Fig. 1 )

 1 Fig.[START_REF] Alsdorf | Tracking fresh water from space[END_REF]. The infered discharge hydrograph is accurate (RMSE = 391 m 3 /s, Nash = 0.97) and bathymetry/friction patterns are relatively well retrieved. Using SWOT spatially distributed observations and piecewise constant friction

Figure 8 :

 8 Figure 8: Sensitivity test to the prior discharge Q M GB ± 30% ; identication (var change) of (Q(t), K(x, h), b(x)) with ENVISAT observations c = Q in,0 , ..., Q in,P ; b 1 , ..., b R ; α 1 , ..., α S , β 1 , ..., β S T with P = 77, R = 6, N = 5 and with a piecewise linear b(x) and S = R = 5. Estimate (case Env.b) j obs = 0.118 at iteration 51, Estimate2 (case Env.b21) j obs = 0.125 at iteration 41, Estimate3 (case Env.b21) j obs = 0.125 at iteration 25.

Figure 9 :

 9 Figure 9: Sensitivity test to the prior friction and bathymetry estimated using the Manning method from [1] (c man1 (α (0) = 7.5; β (0) = 0.5) and c man2 (α (0) = 12.5; β (0) = 1)); identication (var change) of (Q(t), K(x, h), b(x)) with ENVISAT observations c = Q in,0 , ..., Q in,P ; b 1 , ..., b R ; α 1 , ..., α S , β 1 , ..., β S T with P = 77, R = 6, N = 5 and with a piecewise linear b(x) and S = R = 5. Estimate (case Env.b) j obs = 0.118 at iteration 51, Estimate2 (case Env.b31) j obs = 0.116 at iteration 46, Estimate3 (case Env.b32) j obs = 0.122 at iteration 41. (Bottom) prior eective friction laws and spatially averaged calibrated friciton law (α cal = 10.74 and β cal = 0.6, Cal bar).

Figure 10 :

 10 Figure 10: Evaluation of the partial derivatives of the friction source term S f ; forward run with the calibrated parameter set (cf. Tab. 1) and true inow discharge.

  dened from an a-priori positive denite covariance matrix O. Assuming uncorrelated observations O = diag(σ Z ) with σ Z the a-priori observation error on Z obs -σ Z = 15cm in this study. The modeled WS elevations Z depend on c through the hydrodynamic model (1) and the inverse problem reads as c * = argmin c j(c)

2 2 .

 2 Therefore this term imposes (as weak constraint) the infered bathymetry prole b(x) to be an elastic interpolating the values of b at the control points (i.e. a cubic spline).A specicity of the present context is the inconsistency between the large observation grid (altimetry points) and the ner model grid. Between the sparse observations points (equivalently the control points), the bathymetry prole b(x) is reconstructed as a piecewise 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[END_REF][START_REF] Montazem | Representation et segmentation hydraulique eective de rivieres pour le calcul de debit par altimetrie SWOT Ã l'echelle globale[END_REF][START_REF] Montazem | Wavelet-based river segmentation using hydraulic control-preserving water surface elevation prole properties[END_REF]. (This study analyses the adequation between the SW model (1) behavior and the WS signature).

Table 1 :

 1 Summary of the eective hydraulic model parameters including calibrated friction parameters α cal (x) and β cal (x) (recall K(x, h) = α(x)h β(x) ) using 8 years of WS elevation variations (ENVISAT data) given eective channel bathymetry and upstream discharge from the MGB hydrological model ([START_REF] Paiva | Largeâscale hydrologic and hydrodynamic modeling of the amazon river basin[END_REF]).

	3
	h

This section studies the challenging inference of the hydraulic triplet (discharge, bathymetry, friction) from multi-satellite WS observations. The anabranching Xingu River morphology represents a supplementary diculty for inversions regarding the variability of local hydraulic behaviors accross ow regimes as evidenced above by the calibrated friction laws (β cal = 0). The impact of spatial controls density and bathymetry representation is

  -)

	Env.a	Dense b(x)		cprior1	2254	194	-0.01	1.19	4.93	0.49
	Env.b	Piec. b(x)		cprior1	"	"	"	"	"	"
	Env.c	Piec. b(x), K(h)		cprior1	"	"	"	"	"	"
	SWOT.a	Dense b(x)		cprior1	"	"	"	"	"	"
	Env.b21	Piec. b(x)	Q	(0) prior1 -30%	2433	97	0.18	1.19	4.93	0.49
	Env.b22	Piec. b(x)	Q (0) prior1 + 30%	2626	297	-0.37	"	"	"
	Env.b31	Piec. b(x)	cman1 (α (0) = 7.5; β (0) = 0.5)	2254	194	-0.01	0.77	5.63	0.34
	Env.d32	Piec. b(x)	cman2 (α (0) = 12.5; β (0) = 1)	2254	194	-0.01	1.13	5.43	0.49
	Case	Control		Prior	RMSEQ (m 3 /s)	rRMSEQ (%)	NashQ (-)	RMSE b (m)	RMSEα (m 1/3-β /s)	RMSE β (-)
	Env.a	Dense b(x)		cprior1	830	57	0.86	1.97	10	0.46
	Env.b	Piec. b(x)		cprior1	520	61	0.95	1.07	4.8	0.37
	Env.c	Piec. b(x), K(h)		cprior1	608	58	0.93	1.05	-	-
	SWOT.a	Dense b(x)		cprior1	391	38	0.97	0.91	5.67	0.2
	Env.b2	Piec. b(x)	Q	(0) prior1 -30%	1229	39	0.7	0.48	7.83	0.28
	Env.b3	Piec. b(x)	Q (0) prior1 + 30%	1473	104	0.57	0.75	5.09	0.22
	Env.bm2	Piec. b(x)	cman1 (α (0) = 7.5; β (0) = 0.5)	550	61	0.94	1.22	4.64	0.32
	Env.bm3	Piec. b(x)	cman2 (α (0) = 12.5; β (0) = 1)	885	78	0.84	1.30	5.50	0.35

Table 2 :

 2 Scores of the inferences (bottom) performed with various priors (top), ENVISAT (Env) or SWOT (SWOT) observations.

Author contributions and acknowledgments

The contributions of the respective authors are as follows. Pierre-André Garambois designed the research plan and performed the numerical investigations and analysis. Pierre-André Garambois, Pascal Finaud-Guyot, Kevin Larnier and Amanda Montazem contributed to the hydraulic understanding and sensitivity analysis. Jérôme Monnier is the principal designer of the inverse computational method and its analysis. Jonas Verley has started the present study during the beginning of his PhD. This study is warmly dedicated to him. The computational software DassFlow1D and satellite data curation toolbox were adapted from their previous versions ( [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]) by Jonas Verley, Pierre-André Garambois and Kevin Larnier, this last generated the SWOT synthetic data using the large scale simulator and computational ressources of CNES (Centre National d'Etudes Spatiales, French space agency); Amanda Montazem processed and analyzed the SWOT data. Stéphane Calmant provided the multisatelite dataset and interesting discussions related to the concept of hydraulic visibility .

Appendix: the computational inverse method

As already briey summarized in Section 2.3, the computational inverse method is based on Variational Data Assimilation (VDA) applied to the Saint-Venant ow model [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]. The computational inverse method is those presented in [START_REF] Brisset | On the assimilation of altimetric data in 1d saintvenant river ow models[END_REF][START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] with an augmented composite control vector c, see (4): c contains a spatially distributed friction