Verley(4)(5), A.-S. Montazem(1)(7)(8), S. Calmant (7)(8)

Keywords: Multichannel River, Ungauged River, 1D Hydraulic Model, Data Assimilation, Satellite Altimetry, SWOT, Hydraulic Visibility

Eective channel and ungauged braided river discharge estimation by assimilation of multi-satellite water heights of dierent spatial sparsity 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 et al. [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. Fekete and Vorosmarty [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 observations and new generation of sensors (e.g. Vorosmarty et al. [START_REF] Vorosmarty | Analyzing the discharge regime of a large tropical river through remote sensing, ground-based climatic data, and modeling[END_REF], Alsdorf and Lettenmaier [START_REF] Alsdorf | Tracking fresh water from space[END_REF], Calmant et al. [START_REF] Calmant | Monitoring continental surface waters by satellite altimetry[END_REF], Schumann and Domeneghetti [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 50, 5. In addition to decades of nadir altimetry (e.g. Frappart et al. [START_REF] Frappart | Preliminary results of envisat ra-2derived water levels validation over the amazon basin[END_REF], Birkett [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. [START_REF] Da Silva | Water level dynamics of amazon wetlands at the watershed scale by satellite altimetry[END_REF], Calmant et al. [START_REF] Calmant | 4 -principles of radar satellite altimetry for application on inland waters[END_REF]) and imagery (e.g. Allen and Pavelsky [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 Garambois et al. [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF], Montazem et al. [START_REF] Montazem | Physical basis for river segmentation[END_REF], Montazem et al. [START_REF] Montazem | Physical basis for river segmentation from water surface observables[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 braided rivers. 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. Biancamaria et al. [START_REF] Biancamaria | The swot mission and its capabilities for land hydrology[END_REF] for a review). Few studies started to highlight the benet of assimilating synthetic SWOT WS observations in simplied hydraulic models with sequential methods, for infering inow discharge assuming known river friction and bathymetry (Andreadis et al. [START_REF] Andreadis | Prospects for river discharge and depth estimation through assimilation of swath-altimetry into a raster-based hydrodynamics model[END_REF], Biancamaria et al. [START_REF] Biancamaria | Assimilation of virtual wide swath altimetry to improve arctic river modeling[END_REF]) or infering bathymetry assuming known friction (Durand et al. [START_REF] Durand | Estimation of bathymetric depth and slope from data assimilation of swath altimetry into a hydrodynamic model[END_REF], Yoon et al. [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 Manning-Strickler's law (Durand et al. [START_REF] Durand | Estimating reachaveraged discharge for the river severn from measurements of river water surface elevation and slope[END_REF], Garambois and Monnier [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF]) or hydraulic geometries (Gleason and Smith [START_REF] Gleason | Toward global mapping of river discharge using satellite images and atmany-stations hydraulic geometry[END_REF]) or empirical ow models (Durand et al. [START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF], see also Bjerklie et al. [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 20 and their robustness and accuracy is found to uctuate, the importance of good priors is highlighted; none of the tested river is braided.

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 Roux and Dartus [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 Honnorat et al. [START_REF] Honnorat | Variational data assimilation for 2D uvial hydraulics simulation[END_REF], Hostache et al. [START_REF] Hostache | Assimilation of spatially distributed water levels into a shallow-water ood model. Part II: Use of a remote sensing image of Mosel River[END_REF], Lai and Monnier [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. Cacuci et al. [START_REF] Cacuci | Computational Methods for Data Evaluation and Assimilation[END_REF] and references therein) is well suited to solve the present inverse problem (see Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF], Oubanas et al. [START_REF] Oubanas | River discharge estimation from synthetic swottype observations using var-iational data assimilation and the full saint-venant hydraulic model[END_REF], Larnier et al. [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 Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF] with the introduction of identiability maps. The latter consist to represent in space-time the available information: WS observables, hydraulic waves and an estimation of the mist with 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 Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF] or inferable hydrograph time windows Larnier et al. [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. Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF], Gejadze and Malaterre [START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full saint-venant hydraulic network model[END_REF], Oubanas et al. [START_REF] Oubanas | River discharge estimation from synthetic swottype observations using var-iational data assimilation and the full saint-venant hydraulic model[END_REF], Larnier et al. [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 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]. 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 swottype observations using var-iational data assimilation and the full saint-venant hydraulic model[END_REF] highly controls the simulated ow lines (uvial regime); as a consequence the VDA process converge to the discharge hydrograph corresponding to the imposed (almost exact) rating curve. In the present study the downstream boundary condition 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 only studied in a SWOT data context in Garambois and Monnier [START_REF] Garambois | Inference of eective river properties from remotely sensed observations of water surface[END_REF], Yoon et al. [START_REF] Yoon | Improved error estimates of a discharge algorithm for remotely sensed river measurements: Test cases on Sacramento and Garonne Rivers[END_REF], Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], Tuozzolo et al. [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 (Tuozzolo et al. [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 Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]. In view to infer worldwide river discharge from the future SWOT observations, especially for ungauged cases, a hierarchical modeling strategy HiVDI (Hierarchical Variational Discharge Inversion) is proposed in Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]. HiVDI approach includes low complexity ow relations (under the assumption of Low Froude and locally steady-state) which improve the robustness of the inferences in particular if an average value of Q is provided. (It may be provided by a database or a large scale 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 30, 28, 42. All the studies mentioned above address single thread natural rivers (∼ 100km in length) without lateral inows and using synthetic datasets (except in Tuozzolo et al. [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. Garambois et al. [START_REF] Garambois | Hydraulic visibility: Using satellite altimetry to parameterize a hydraulic model of an ungauged reach of a braided river[END_REF], Schneider et al. [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 braided river ows from real multi-sensor 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. Multichannel rivers are characterized by complex hydraulic geometries relationships across ow regimes as shown in Schubert et al. [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 a braided portion of the Platte River, US. The key point is to build up a suciently complex model to describe multichannel river ows and in coherence with satellite altimetry measurements spatio-temporal scales.

Based on the inverse method presented in Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF], an eective hydraulic modeling strategy is adapted for tackling multichannel river ows using: (i) eective 1D cross sections based on real multisatellite 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 along a single ENVISAT track or from synthetic SWOT observations, low identiability index (as introduced in 10 and detailed in section 4). The inuence of the spatial density of WS observations on the identiability of spatial controls patterns (in the 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 sparse 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 multichannel rivers including: (i) a spatially distributed friction law depending on 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 braided river. Using this model as a reference, section 4 proposes detailed investigations of hydraulic inferences from real ENVISAT or synthetic SWOT observations considering this braided river as ungauged. Section 5 presents numerical sensitivity analysis to the hydraulic prior and investigations on the bathymetry friction equinality.

Eective hydraulic modeling approach:

This section proposes an original 1D modeling approach of adequate complexity for modeling multichannel river ows across 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 ow is classically modeled using the 1D Saint-Venant shallow water equations involving an integration of the ow variables over the cross section (see e.g. Chow [START_REF] Chow | Handbook of applied hydrology[END_REF], Guinot [33] for detailed assumptions). In their non-conservative form 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 term 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 literature for predicting mean ow velocity for example on a wide range of in situ river ow measurements in Bjerklie et al. [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 Ferguson [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 (2) enabling a variation of friction eect in function of 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 table of frictions in function of river types for instance (e.g. [START_REF] Chow | Open-channel Hydraulics[END_REF]).

The discharge Q in (t) is classically imposed upstream of the river channel. At downstream the Manning-Strickler equation depending on the unknowns (A, Q; K) out is imposed (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 )). This 1D Saint-Venant model ( 1) is discretized using the classical implicit Preissmann scheme (see e.g. 16) on a regular grid of spacing ∆x. It is implemented into the computational software DassFlow DassFlow.

Eective braided river model from long altimetric time series, satellite images and a hydrological model

A L = 71km long portion of the Rio Xingu containing braided reaches is considered (gure 1, cf. Garambois et al. [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. Da Silva et al. [START_REF] Da Silva | 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 table 1).

An eective hydraulic modeling strategy of this braided 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 braided reaches.

An a priori river bottom {b} r V S obtained from altimetric rating curves from Paris et al. [START_REF] Paris | Stage-discharge rating curves based on satellite altimetry and modeled discharge in the amazon basin[END_REF]. They are determined by adjusting the parameters of a classical stage discharge relationship on WS elevations gained by satellite altimetry and discharge simulated with the large scale hydrological model MGB (de Paiva et al. [START_REF] De Paiva | Largeâscale hydrologic and hydrodynamic modeling of the amazon river basin[END_REF]) on the temporal window of interest -called true discharge in what follows.

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 (res. 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 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.

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 a braided river. The employed inverse method is those presented in Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] (see also Brisset et al. [10]) 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 Larnier et al. [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.

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 ( 1). This discrepancy is quantied through the cost function term j obs (c) = 1 2 Z obs -Z(c) 2

2

, see Appendix ow model (eq. 1) considering eective cross sections (see gure 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 (3)
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 (2) for each patch n ∈ [1.

.N ]

with N ≤ R.

The inversion consists to solve the following minimization problem: c * = argmin j(c) (eq. 8).

This minimization, optimization problem is solved using a rst order gradient-based algorithm, more precisely the classical L-BFGS quasi-Newton algorithm.

Calibration of the eective hydraulic model on historical satellite altimetry

Thi 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 [START_REF] Allen | Global extent of rivers and streams[END_REF]. Since friction has a local and upstream inuence on the ow line (low Froude uvial ows, gure 9) 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 Larnier et al. [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... 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.

Inverse hydraulic modeling method with WS elevations gained from nadir altimetry and SWOT

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. 3) containing discharge, bathymetry and friction is sought with the inverse method decsribed in section 2.3 (see also appendix, section 7 [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF] for a detailled analysis and 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 gure 9). Let I indent = T wave /∆t obs be the identiability index dened in Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant 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 Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant 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. Chow [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 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 ( 1) behaviour and the WS signature of bathymetry/friction controls (see Montazem et al. [START_REF] Montazem | Physical basis for river segmentation[END_REF], Montazem et al. [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF], Montazem [START_REF] Montazem | Representation et segmentation hydraulique eective de rivieres pour le calcul de debit par altimetrie swot à l'echelle globale[END_REF]), a linear bathymetry interpolation is used between 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 constrains 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 constrain 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 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. gure 5) (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. table 1). This infered eective friction law and bathymetry pattern, leading to somehow eective stage-discharge relationships locally given the infered hydrograph, enable to approximate the observed WS variations (j obs = 1.269 ) but with a less accurate t than with spatially - These infered friction laws and bathymetry patterns -simultaneously infered with the discharge hydrographcorrespond to eective rivers enabling to t the observed variability of ow lines. Recall that the observations consist in real measurements of WS elevations gained by nadir altimetry on multichannel reaches of the Xingu River.

The complexity of the forward-inverse modeling approach, in coherence with the spatial sparsity of observation grid, enables to approximate satisfactorily the one of the observed multichannel ow. The additionnal constrain provided by spatially dense ow lines observations is investigated in the next section with SWOT synthetic data. The inow discharge, bathymetry and friction are infered by assimilating SWOT WS observations Z SW OT obs r,p on the same spatial grid as that of the numerical hydraulic model with c prior1 . The estimates are presented on gure [START_REF] Birkett | Contribution of the topex nasa radar altimeter to the global monitoring of large rivers and wetlands[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 roughness enable to constrain the inference of bathymetry controls at a ne spatial resolution (model grid); the inverse method including covariance matrices acting as spatial or temporal smoothers/regularizations (cf. eq. 11 in appendix). The infered discharge and the 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. table 2 and gure 4). 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 distributed measurements of WS elevations given the infered discharge (j obs = 0.099).

Numerical investigation of the bathymetry-friction equinality

The hydrograph is responsible for ow variability in time, hence enabling to retrieve the temporal dynamics of the observed ow lines (Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF], Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]). The friction and bathymetry controls have a correlated inuence on the modeled ow lines therefore leading to an ill-posed inverse problem (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] for investigations on this bathymetry-friction equinality in a comparable data-inversion context). In this section the inuence of the prior value on the quality of the inferences with spatially distributed controls is investigated. Next, is proposed a numerical analysis of the sensitivity of the friction source term S f in the Saint-Venant equations (1) to the ow controls (triplet) that are embeded in it.

Sensitivity to the prior of the hydraulic inference from altimetric observations of WS signature

Given altimetric measurements of WS variabilities and the rst guess c prior1 , the inverse method enables to infer a complex control vector composed of temporally and spatially distributed controls of the 1D SW model (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). As already pointed out in the VDA inferences performed with the DassFlow model using SWOT like data in (Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF], Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF]) and AirSWOT data (Tuozzolo et al. [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.

The sensitivity of the inference to the quality of the prior control vector is investigated here for the most challenging inverse problem with spatially 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 7 and various inference scores are sumed up in table 2.

For each inow prior, the temporal variations of the inow hydrograph are very well retrieved as shown on gure 7 -runs Env.b2 and Env.b3. However a biased inow prior results in a biased hydrograph estimate (with correct temporal variations) which is coherent with results of Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], Tuozzolo et al. [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 (Larnier et al. [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 c man1 and c man2 are considered with prior friction under/over-estimations compared to calibrated ones (cf. 8). As shown on gure 8, 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 c man1 that underestimates river bottom elevation and overestimates the spatially averaged friction eect compared to calibrated values (cf. gure 8, 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 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 (1) have a complex non linear inuence on the modeled ow line and consequetly on the t to the observed ow lines. 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 and bottom wall-to-uid pressure) and a dissipation term -gAS f . Discharge and bathymetry appear in the momentum and pressure terms while all 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]).

We propose a simple calculation in order to make appear the sensitivity of the friction term to a change on 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 (4)

Case

Control Prior 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 gure 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:

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) (
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) (5)
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 equation 5 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 gure 9 along the Xingu River, on model grid, from hydraulic variables simulated (forward run) with calibrated parameters (cf. table 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 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 found at locations corresponding to bottom slope or channel width changes. 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 Montazem et al. [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF]. Further investigations on the sensitivity of the full Saint-Venant equations in space and time could be of interest to better taylor 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 multichannel river.

The HiVDI inverse method presented in Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] is adapted for reproducing a multichannel 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. Guinot and Cappelaere [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 multichannel ows observed during 8 years by nadir altimetry (ENVISAT) on this 71km braided 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 multichannel 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 multichannel river representation; the denition of friction by reaches enabling to consider more dense bathymetry controls.

SWOT observations would represent unprecedented measurements of hydraulic processes signatures from the local to the hydrographic network scales, including complex ow zones such as braided 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 Montazem et al. [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF]), on the estimation of reliable priors and inverse problems at the scale of larger river network portions including complex ow zones.

This optimal control problem is solved using a Quasi-Newton descent algorithm: the L-BFGS algorithm version presented in 31. 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 Bouttier and Courtier [START_REF] Bouttier | Data assimilation concepts and methods march[END_REF], Monnier [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 Kaltenbacher et al. [START_REF] Kaltenbacher | Iterative regularization methods for nonlinear ill-posed problems[END_REF] for the theory of regularization of such inverse problems and to Larnier et al. [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 Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF], the optimization problem ( 8) is regularized as follows. First the regularization term j reg is added to the cost function, see [START_REF] Birkett | Contribution of the topex nasa radar altimeter to the global monitoring of large rivers and wetlands[END_REF]. We simply set: j reg (c) = 1 2 b"(x) A specicity of the present context is the large inconsistency between the large observation grid (altimetry points) and the ner 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 Montazem et al. [START_REF] Montazem | Physical basis for river segmentation[END_REF], Montazem et al. [START_REF] Montazem | Physical basis for river segmentation from water surface observables[END_REF], Montazem [START_REF] Montazem | Representation et segmentation hydraulique eective de rivieres pour le calcul de debit par altimetrie swot à l'echelle globale[END_REF]. (This study analyses the adequation between the SW model (1) behavior and the WS signature).

Next and following Lorenc et al. [START_REF] Lorenc | The met. oce global three-dimensional variational data assimilation scheme[END_REF], Weaver and Courtier [START_REF] Weaver | Correlation modelling on the sphere using a generalized diusion equation[END_REF], Larnier et al. [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 ) ( 9
)
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) (10)
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 35, 36 and Larnier et al. [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          (11)
Still following Larnier et al. [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 (12)
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 braided river ows of interest, adequate values for those parameters are: ∆t Q = 24 h and L b = 3km km We refer to Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant 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 (13)
The scalar values σ may be viewed as variances ; their values are given in the numerical results section.

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 (Paris et al. [49]) and water surface width from satellite images.

  (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 (de Paiva et al.[START_REF] De Paiva | Largeâscale hydrologic and hydrodynamic modeling of the amazon river basin[END_REF]).

Figure 2 : 3 h 4 .

 234 Figure 2: 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 (

 4 Figure 4 (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.

  Figure 3: 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 5 :

 5 Figure 5: 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)

Figure 7 :

 7 Figure 7: Sensitivity test to 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 8 :

 8 Figure 8: Sensitivity test to prior friction and bathymetry estimated using the Manning method from Larnier et al. [42] (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 9 :

 9 Figure 9: Evaluation of the partial derivatives of the friction source term S f ; forward run with the calibrated parameter set (cf. table 1) and true inow discharge.

The norm • O = O 1 / 2 • 2

 122 is 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 constrains) the infered bathymetry prole b(x) to be an elastic interpolating the values of b at the control points (i.e. a cubic spline).

Table 1 :

 1 Summary of the eective hydraulic model parameters including calibrated friction parameters α cal

	5 and

A constant friction in time would lead to systematical errors for a large range of ows as shown by grey curves on gure 2. 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 small reach (cf. g. 2). The W low (x) Total low ow width [m] (derived from JERS) W high (x) Total high ow width [m] (derived from JERS

  ). 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 Frasson et al.[START_REF] 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

Brisset et al.

  a noised observation context, we denote by δ the noise level such that Z obs -Z true 2 ≤ δ for all spatial 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. 3) 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 gure (3) (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.

	4.2. Inference of distributed hydraulic controls (Q(t), K(x, h), b(x)) with spatially sparse WS observations: real
	ENVISAT altimetric snapshots
	In this section the assimilation is based on WS elevations Z env s,p S,P
	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.

Kaltenbacher et al. 40

and references therein): the regularization parameter γ (see eq. 6) 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.

  -)

	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 (Larnier et al. [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

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] Allen | Global extent of rivers and streams[END_REF]. The computational inverse method is those presented in Brisset et al. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint venant river ow models[END_REF], Larnier et al. [START_REF] Larnier | River discharge and bathymetry estimations from swot altimetry measurements[END_REF] with an augmented composite control vector c, see (3): c contains a spatially distributed friction coecient enabling to model complex ow zones (while it is an uniform friction law K(h) in Larnier et al. [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] Birkett | Contribution of the topex nasa radar altimeter to the global monitoring of large rivers and wetlands[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:


Kevin Godineau
email: kevin.godineau@ens-paris-saclay.fr

Sylvain Lavernhe

Christophe Tournier

Joint Special Interest Group meeting between euspen and

Keywords: Optical, Laser, Model, Laser beam machining (LBM)

The additive manufacturing machines used in laser powder bed fusion processes are composed of an opto-mechanical chain whose purpose is to focus the laser in the work plane to locally fuse the material with the required energy. The opto-mechanical chain does not only position the laser spot, it also induces a modification of its shape and thus the distribution of energy brought to the material. As a result, the shape of the laser spot is not the same at every point on the machine, which can have an influence on the characteristics of the produced parts. This aspect related to opto-mechanical chain technology is very often ignored, particularly in thermal modelling. In this work, we study the impact of opto-mechanical chains composed of post-objective focusing systems on the energy distribution of the laser spot in the entire achievable space by the system. For this purpose, mathematical models are used to evaluate the position of the laser spot in the work plane as well as the orientation of the laser beam. An energy model of Gaussian laser beams is then used and applied to these geometric models. The coupling of these two models makes it possible to study the energy distribution of the laser spot at any point in the workspace of an additive manufacturing machine. The work is confirmed with experimental measurements made on a test bench representative of an industrial additive manufacturing machine.

Introduction

In additive manufacturing by Laser Powder Bed Fusion (LPBF), knowledge of the size of the laser spot and the melting pool are important elements, as they condition many parameters during the trajectory generation process. Thus, this work focuses on the influence of the opto-mechanical chain on the distribution of energy supplied to the material. These opto-mechanical chains consist of two main blocks identified in figure 1: the focusing system and the positioning system. In LPBF additive manufacturing, whatever the machines, the positioning system is always the same. These are two movable mirrors that allow the orientation of the outgoing laser beam to be controlled. On the other hand, to focus the laser beam in the work plane, two main types of technological solutions are distinguished. Ehrmann [START_REF] Ehrmann | Optics for vector scanning[END_REF] describes these two types of focusing systems called pre-objective and postobjective. Pre-objective systems consist of a fixed lens located after the positioning system that focuses the laser beam in a focal plane. The post-lens systems shown in figure 1 consist of a movable lens and a fixed lens. The combination of these two lenses allows to control the focusing distance and therefore at any time to focus the laser beam in the work plane.

Previous work has been done to qualify the shape of the laser spot and its distribution. Deshmukh et al. [START_REF] Deshmukh | Optomechanical scanning systems for microstereolithography (msl) : Analysis and experimental verification[END_REF] presents measurements made on three different types of focusing systems. However, the work on the post-objective system is not well documented. In order to standardize the shape of the laser spot and its energy distribution over the entire working area, the development carried out in [START_REF] Yang | Laser galvanometric scanning system for improved average power uniformity and larger scanning area[END_REF] consists in proposing a new hybrid opto-mechanical chain. This one is qualified, but is not compared to standard opto-mechanical chains. It is also possible to find purely geometric approaches [START_REF] Li | Beam deflection and scanning by two-mirror and twoaxis systems of different architectures: a unified approach[END_REF] based on the deformation of a pixel by the opto-mechanical chain. However, not all these approaches directly and mathematically address the influence of the opto-mechanical chain.

The work carried out in this article concerns more specifically the qualification of the energy deposited on the work plane in opto-mechanical chains composed of post-objective focusing systems. The numerical values presented are typical values used in additive manufacturing machines.

The article is organized as follows: section 2 describes the assumptions and parameters used in mathematical models. Section 3 presents the two phenomena that influence the energy supplied to the material: the increase of the spot size as a function of the focal length and the incidence angle of the laser beam with respect to the work plane. Experimental measurements are carried out in section 4 which confirm the theoretical results. Finally, a conclusion on the study is provided in section 5 in order to highlight some important points.

Assumptions and parameters

In order to characterize the geometry and the intensity profile of the laser spot in the work plane, wave optics hypotheses are used. The assumptions and parameters are defined in the following list and are presented in figure 2.

1. The laser beam is considered as Gaussian beam; 2. The laser beam wavelength is noted 𝜆 𝑙 and is equal to 1.06 𝜇𝑚; 3. The laser beam diameter at the entrance of the equivalent lens of the focusing system is noted 𝑑 𝑙 and is equal to 20 𝑚𝑚; 4. The focal point is named 𝑷 𝒇 ; 5. The laser beam radius is called 𝑤. This radius is minimal at the focus point 𝑷 𝒇 and is named "waist" with 𝑤(𝑧′ = 0) = 𝑤 0 ; 6. The defocusing parameter is noted ℎ 𝑓 and corresponds to the height of the focal point with respect to the work plane;

The focusing distance 𝐷 𝑓 depends on the position of the laser spot in the work plane and the defocusing height.

The laser beam radius 𝑤(𝑧′) as no immediate meaning, it is an energy threshold. This distance corresponds to the value from which the intensity of the laser spot is equal to 1/𝑒 2 times the maximum intensity 𝐼(𝑧′) of the laser spot (with 𝑒 = exp( 1)). This criterion is often used to characterize the diameter of a laser beam. Opto-mechanical chains composed of post-objective focusing systems allow the laser beam to be focused in space and not only in the work plane. Thus the notion of defocusing in relation to the work plane is introduced with the parameter ℎ 𝑓 , this adjustable parameter therefore allows to modify the laser spot size and consequently the energy distribution in the work plane by acting on the focusing of the beam.

Before going into details, it is also necessary to understand that the reflection of the laser beam on the mirrors has no impact on the shape of the laser spot. Indeed, this reflection does not change the distance travelled by each light beam.

Modification of the shape and the distribution of energy

Influence of focal length

The first parameter influencing the laser spot geometry and in particular the waist size 𝑤 0 is the focusing distance 𝐷 𝑓 . The laws of optics allow to establish equation [START_REF] Ehrmann | Optics for vector scanning[END_REF] with 𝜃 the angle of divergence of the laser beam:

𝜃 = 𝜆 𝑙 𝜋 ⋅ 𝑤 0 (1)
Using figure 2, it is possible to define 𝜃 as a function of the focusing distance using the triangle formed by the lens and the waist (cf. equation ( 2)).

𝜃 = arctan ( 𝑑 𝑙 2 ⋅ 𝐷 𝑓 ) (2)
The combination of equations ( 1) and ( 2) allows to express the waist as a function of the focal length. This relationship is written in equation [START_REF] Yang | Laser galvanometric scanning system for improved average power uniformity and larger scanning area[END_REF].

𝑤 0 = 𝜆 𝑙 𝜋 ⋅ arctan ( 𝑑 𝑙 2 ⋅ 𝐷 𝑓 ) (3)
The study of this function shows that an increase of the focusing distance leads to an increase of the waist. If we consider only the variation of the laser spot radius due to the focusing distance then it is possible to obtain figure 3, which shows the evolution of the laser spot size over the entire area accessible by the opto-mechanical chain (full field). The calculation of the focusing distance 𝐷 𝑓 as a function of the position of the laser spot in the working plane (𝑥, 𝑦) is performed using the geometric models described in article [START_REF] Godineau | Opto-mechanical modelling of an additive manufacturing laser scanning head including assembly defects[END_REF]. Thus for the full field the waist size increases by nearly 10 % between points 𝑷𝟏 (0, 0) 𝑚𝑚 and 𝑷𝟐 (330, 291) 𝑚𝑚. The point 𝑷𝟑 (260, 260) 𝑚𝑚 is used in the rest of the paper to perform measurements.

Influence of laser beam orientation

Once the influence of the focal length is quantified, it remains to define the influence of the laser beam orientation on the laser spot.

First, to simulate the real shape of the laser spot, it is necessary to start from the Gaussian equation that modelsthe whole laser beam. For a Gaussian laser beam, the intensity of a point 𝑴 of coordinates (𝑥′, 𝑦′, 𝑧′) in the waist coordinate system 𝓡 𝒘 is expressed by equations ( 4), ( 5) and (6).

𝑤(𝑧′) = 𝑤 0 ⋅ √ 1 + ( 𝜆 𝑙 ⋅ 𝑧′ 𝜋 ⋅ 𝑤 0 ) 2 (4)
𝐼(𝑧 ′ ) = 𝐼 0 ⋅ ( 𝑤 0 𝑤(𝑧 ′ ) ) 2

(5)

𝐼(𝑥 ′ , 𝑦 ′ , 𝑧 ′ ) = 𝐼(𝑧 ′ ) ⋅ exp (-2 ⋅ ( 𝑥 ′ 2 + 𝑦 ′2 𝑤(𝑧 ′ ) )) (6)
To obtain the laser beam intensity in the work plane, it is necessary to establish the transfer matrix from the waist coordinate system 𝓡 𝒘 to the coordinate system linked to the point 𝑷 located in the work plane 𝓡 𝒑 (cf. figure 4). This transfer matrix is determined by using 𝒖 𝟔 ⃗⃗⃗⃗ the orientation of the laser beam relative to the work plane, and 𝑷 = (𝑥, 𝑦) the position of the laser beam relative to the work plane. These two elements are obtained using the geometric models described in [START_REF] Godineau | Opto-mechanical modelling of an additive manufacturing laser scanning head including assembly defects[END_REF]. Once the transfer matrix has been established, the position of any point 𝑴 is expressed in 𝓡 𝒑 . The coordinates of 𝑴 in this coordinate system are named 𝑥 1 , 𝑦 1 and 𝑧 1 . This allows the intensity field of the laser beam to be calculated in 𝓡 𝒑 , i. e. 𝐼(𝑥 1 , 𝑦 1 , 𝑧 1 ). To evaluate the intensity of the laser beam in the work plane, it is therefore sufficient to set 𝑧 1 = 0.

Figure 5 shows several intensity distribution results in the work plane. All these results are expressed with the same intensity scale. This scale is established in relation to the maximum of the laser beam which is located in the upper left case. Each column presents the laser spot shape at a specific position in the work plane. This shows that at the center of the work surface the laser spot shape is circular, while elsewhere the spot is elliptical. The direction of the ellipse depends on the orientation of the incident beam and therefore on the position of the point in the work plane. The first row shows the results when the dependence of the waist size 𝑤 0 on the focal length 𝐷 𝑓 is not considered. The second raw shows the case where the defocusing parameter is set to 2 𝑚𝑚. Defocusing leads to a significant increase in spot size and a decrease in the peak intensity at the center (Gaussian circular or elliptical). The last row includes the variation of the waist size according to the focal length presented above. This corresponds to substituting in equations ( 4) and ( 5) the value of 𝑤 0 defined by equation (3). The maximum variation in the laser spot shape corresponds to the difference between the laser spot at the center of the work plane and at one extremity. Taking into account the two aspects previously shown, a laser spot with a radius of 30.6 𝜇𝑚 is transformed into an ellipse with a semi-major axis of 40 𝜇𝑚 and a semi-minor axis of 34 𝜇𝑚. The change in shape is therefore quite significant when using the full field. The impact in the manufactured area is slightly less important, as the printing area is most of the time a restriction of the reachable space.

Experimental validation of the energy profile

In order to experimentally verify the previous simulation results, a test bench representative of an industrial manufacturing machine is used. This test bench is instrumented with a CCD matrix positioned in the work plane to visualize the laser spot trace. To avoid saturation of the measurement, optical filters are used, moreover the exposure time is also adjusted. For safety reasons and in order to be able to carry out a measurement, the laser beam used at the input of the optomechanical chain is at a very low power (1 𝑚𝑊).

Due to the small size of the focused laser spot (≈ 70 𝜇𝑚 ), the pixel size of the CCD matrix (5.3 𝜇𝑚) and the small spatial variation of the energy distribution to be measured, the measurement of the deposited energy is performed with a low spatial resolution. Finally, the experimental measurement is also limited by the incidence angle of the laser beam. Due to the size of the camera and optical filters, it is not possible to measure the energy deposited at point 𝑷𝟐 because the orientation of the incident laser beam is too important and the laser spot exits the CCD matrix. Thus the measurement is carried out at a point 𝑷𝟑 previously presented.

Figure 6 shows the experimental results from the laser spot measurement. The first row of the table shows that the laser beam used is not quite Gaussian, there is a slight deformation. The comparison of row 1 and row 2 of the table shows the influence of defocusing on the shape of the laser spot. The experimental results are not as obvious as the theoretical results, but they confirm the increase in diameter and the decrease in the maximum intensity of the laser spot. For larger defocusing, this effect is more pronounced. Finally, the measurement presented in the third row clearly shows the deformation of the laser beam and the increase in diameter. However, the energy intensity of this third measurement cannot be directly compared to previous measurements. Indeed, given the angle of incidence of the laser beam, it travels a greater distance in the optical filter and is therefore more attenuated. This third measure is therefore normalized separately.

Conclusion and perspectives

In opto-mechanical chains with post-lens focusing systems, the laser spot shape and the energy distribution are influenced by two elements. On one side the focusing distance which is variable in order for the waist to belong to the work plane and on the other side the orientation of the incident laser beam on the work plane which generates an elliptical shaped laser spot. These two aspects lead to significant deformations of the laser spot, particularly on the ends of the laser accessible areas.

In most machines currently in use, the effective printing area is a significant restriction of the laser spot accessible area. This aspect allows, among other things, contain the deformations of the laser spot and the variations in energy However, more productivity and to be able to manufacture larger parts, the build platform size must be larger, so manufacturers are looking to use the full area covered by the galvanometric heads as much as possible. In this perspective, it becomes necessary to know the energy distribution of the laser spot to consider it and finely control the process.

The work performed can also be applied to the measurement of the deposited energy during a trajectory and no longer only in static way. This would make it possible to quantify the influence of energy distribution on a complete trajectory and show the impact of the position of the part in the work plane. In dynamics, however, the energy interpretations are more complex due to the influence of the dynamics of the actuators and the numerical control on the realized trajectories. The model presented can also be an additional brick for the nesting of the part on the build platform before starting the manufacturing process.

Figure 1 .

 1 Figure 1. Opto-mechanical chain of an additive manufacturing machine by laser powder bed fusion. Post-objective focusing system.

Figure 2 .

 2 Figure 2. Parameterization and representation of the waist.

Figure 3 .

 3 Figure 3. Variation of the waist size in the work plane.

Figure 4 .

 4 Figure 4. Intersection between the Gaussian laser beam and the work plane.

Figure 5 .

 5 Figure 5. Energy distribution of the laser spot. The columns correspond to two positions of the laser spot P1 and P2. The rows correspond to three configurations (size of the laser spot fixed or depending on the focusing distance and defocusing equal to 0 or 2 mm).

Figure 6 .

 6 Figure 6. Experimental energy distribution of the laser spot. The three rows represent three different configurations. The left and right columns respectively represent the raw measurement and the post-processed measurement results.

Acknowledgement

The authors thank the AddUp company, which through the SOFIA project has provided us the means to experimentally validate the work carried out.