Sylvain Biancamaria
email: sylvain.biancamaria@legos.obs-mip.fr

Thomas Schaedele
email: thomas.schaedele@legos.obs-mip.fr

Denis Blumstein
email: denis.blumstein@cnes.fr

Frédéric Frappart
email: frederic.frappart@get.obs-mip.fr

François Boy
email: francois.boy@cnes.fr

Jean-Damien Desjonquères
email: jean-damien.m.desjonqueres@jpl.nasa.gov

Claire Pottier
email: claire.pottier@cnes.fr

Fabien Blarel
email: fabien.blarel@legos.obs-mip.fr

Fernando Niño
email: fernando.nino@legos.obs-mip.fr

Validation of Jason-3 tracking modes over French rivers

Keywords: Jason-3, altimetry, rivers, Open-Loop/DEM tracking mode

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Introduction

Continental water plays an important role in the Earth"s water and energy cycles, and is at the interface between the atmosphere and the ocean. For climate studies, long-term (i.e. multidecadal) observations of the time variations in lake and river levels are necessary to estimate lake storage variability and river discharges. These data are also essential for water management, as water is a resource that is vital to human societies. This monitoring has traditionally been performed using in situ gage networks, which are usually operated at local or national level.

However, viewed from a global perspective, in situ networks are heterogeneous. The total number of gages has been decreasing over recent decades (IAHS Ad Hoc Group on Global Water Data Sets et al., 2001), and their measurements are not always shared publicly, especially in transboundary basins [START_REF] Gleason | Crossing the (watershed) divide: satellite data and the changing politics of international river basins[END_REF]). That"s why, satellite data, including water elevation measurements obtained from radar altimeters have been used since the early 90"s as an additional observation system (e.g. [START_REF] Birkett | The contribution of TOPEX/POSEIDON to the global monitoring of climatically sensitive lakes[END_REF], even if they cannot replace in situ gages, as noted by [START_REF] Fekete | Rationale for monitoring discharge on the ground[END_REF].

Since the launch in August 1992 of Topex/Poseidon, developed by NASA (National Aeronautics and Space Administration) and CNES (Centre National d"Études Spatiales), nadir radar altimeters have been widely used to measure ocean surface topography, which was the scientific objective of the mission (e.g. [START_REF] Fu | Satellite Altimetry and Earth Sciences[END_REF]. However, soon after its launch, the high potential of the altimeter to retrieve river and lake water elevations has been demonstrated by [START_REF] Koblinsky | Measurement of river level variations with satellite altimetry[END_REF]. In the present study, water elevation is defined as the distance from the water surface to a reference surface (an ellipsoid or a geoid). Pioneering studies demonstrated that 24 large lakes (with an area greater than 300 km 2 ) and 8 large rivers (more than 2 km wide) could be measured accurately with, at best, RMSE (Root Mean Square Error) of around 0.04 m and 0.11 m respectively [START_REF] Birkett | The contribution of TOPEX/POSEIDON to the global monitoring of climatically sensitive lakes[END_REF][START_REF] Birkett | Contribution of the TOPEX NASA radar altimeter to the global monitoring of large rivers and wetlands[END_REF]. Subsequently, radar altimetry has been increasingly used to monitor inland water bodies (for review see for example [START_REF] Crétaux | Hydrological applications of satellite altimetry: rivers, lakes, man-made reservoirs, inundated areas[END_REF]. The launch of Topex/Poseidon was followed by that of several nadir altimetry satellites: ERS-2 in 1995, GFO in 1998, Jason-1 in 2001, ENVISAT in 2002, Jason-2 in 2008, CryoSat-2 in 2010, SARAL in 2013, Jason-3 in 2016and Sentinel-3A in 2016. Online databases (G-REALM, https://www.pecad.fas.usda.gov/cropexplorer/global_reservoir/; Hydroweb, http://hydroweb.theia-land.fr; River and Lake database, http://tethys.eaprs.cse.dmu.ac.uk/RiverLake/shared/main; DAHITI, http://dahiti.dgfi.tum.de; HydroSat, http://hydrosat.gis.uni-stuttgart.de/php/index.php) using this series of altimeters have been developed over recent decades to provide freely accessible water elevation time series for selected rivers and lakes. Continuity of measurements is also guaranteed, as many nadir altimetry missions are already scheduled for launch in the near future. These include Sentinel-3B (early 2018), Sentinel-3C and -3D (launch dates currently unknown, as they will provide continuity of observation after Sentinel-3A and -3B), Sentinel-6/Jason-CS A (around 2020), and SWOT (around 2021; this satellite will carry not only a nadir altimeter, but also a wide swath altimeter).

Even where the instrument footprint on the ground is a few kilometres wide, enhancements made to the instruments, especially after ENVISAT and Jason-2, have improved the ability to observe smaller water bodies (width < 500 m) to an extent not previously thought possible (e.g. Santos da [START_REF] Da Silva | Water levels in the Amazon basin derived from the ERS 2 and ENVISAT radar altimetry missions[END_REF][START_REF] Michailovsky | River monitoring from satellite radar altimetry in the Zambezi River basin[END_REF][START_REF] Baup | Combining high-resolution satellite images and altimetry to estimate the volume of small lakes[END_REF]Frappart et al., 2015a;[START_REF] Sulistioadi | Satellite radar altimetry for monitoring small rivers and lakes in Indonesia[END_REF][START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF]. The ability of an altimeter to observe a river is dependent on river width, but is determined to an even greater extent by the "surrounding topography, the observation configuration, previous measurements and the instrument design" [START_REF] Baup | Combining high-resolution satellite images and altimetry to estimate the volume of small lakes[END_REF][START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF].

Nadir altimeters present some limitations. For instance, they provide elevation measurements only along or close to their track (in the nadir direction or in a quasi-nadir direction) and therefore miss many water bodies (the number of objects overflown depends on the satellite orbit). In addition, altimeters are mainly designed to observe oceans with the best accuracy. They are not optimized for the monitoring of continental surfaces, which could pose a problem with steep-sided rivers. For these rivers, the altimeter might measure the top of the surrounding topography, instead of the river valley. This problem has been extensively described and discussed by [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF]. However, in the future, more researchers and water resources managers will be interested in the water elevation of steep-sided rivers and smaller rivers. It is therefore important to address this issue as soon as possible, in order to obtain the longest possible time series in the coming decades, over regions previously unobserved by altimetry. To do so, CNES has developed an alternative instrument operating mode, usually referred to as "Open-Loop" (OL) or Digital Elevation Model tracking mode or DEM tracking mode. It requires the loading of an along-track DEM on board the altimeter. Very few studies have analysed measurements made for such rivers in this tracking mode, in comparison with the usual "Closed-Loop" (CL) tracking mode [START_REF] Birkett | Investigating the Performance of the Jason-2/OSTM Radar Altimeter over Lakes and Reservoirs[END_REF].

The purposes of this study are: [START_REF] Frappart | Preliminary results of ENVISAT RA-2 derived water levels validation over the Amazon basin[END_REF] to validate measurements of the newly launched Jason-3 satellite (17 January 2016) for small to medium-sized rivers (between 35 m and 300 m wide) over mainland France and (2) to study whether the Jason-3 alternative instrument operating mode is able to overcome the problem of the measurement of steep-sided rivers and is therefore advantageous. Section 2 of this paper discusses nadir altimetry and its potential issue with steepsided rivers. Section 3 introduces the study domain and the validation data used (sub-section 3.1), the computation of the Jason-3 on-board DEM over the study domain and the Jason-3 data validation methodology (section 3.2). Section 4 then presents and discusses the results obtained.

Finally, some conclusions and perspectives on the use of Jason-3 for the measurement of small to medium-sized rivers, in particular those that are steep-sided, are provided in section 5.

2. Satellite Nadir Radar Altimetry and the issue of steep-sided water bodies 2.1. Surface Elevation retrieval from nadir altimetry Nadir altimeters provide water elevation measurements at the intersections between satellite ground tracks and rivers (or lakes/reservoirs). These intersections are usually referred to as "virtual stations" (VS). A more detailed description of water surface elevation estimation from nadir altimetry over continents can be found, for example, in [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF] and in [START_REF] Crétaux | Hydrological applications of satellite altimetry: rivers, lakes, man-made reservoirs, inundated areas[END_REF]. For this reason, essential information only is mentioned in this section.

Nadir altimeters emit a radar pulse in the nadir direction (or local vertical, Figure 1) and then record the radar echo using a pulse compression technique. This record, also known as the waveform, contains the value of the returned power as a function of time or, equivalently, of distance between the radar and the reflectors [START_REF] Chelton | Satellite altimetry[END_REF][START_REF] Frappart | Satellite altimetry: principle and major applications in Earth Sciences[END_REF]. The two-way travel time (T), given by the leading edge of the waveform signal, is estimated accurately using a "retracking" algorithm applied to the waveforms by the mission processing centre. This processing is based on the use of a theoretical model built for ocean conditions [START_REF] Brown | The average impulse response of a rough surface and its application[END_REF] and provides a high level of precision and accuracy over oceans. Over inland waters, estimation of the two-way travel time is more difficult since the waveform shape, affected by several complex backscattering phenomena, differs markedly from the theoretical model. For these cases, alternative retracking algorithms should be used (e.g. [START_REF] Frappart | Preliminary results of ENVISAT RA-2 derived water levels validation over the Amazon basin[END_REF]. The two-way travel time T is converted to the distance between the satellite centre of mass and the ground surface, called the altimeter "range" (R, Figure 1), assuming a constant electromagnetic wave propagation speed equal to the speed of light in vacuum. However, estimates of the range R need to be corrected for propagation delays of the radar signal within the atmosphere (these corrections are labelled ΔR propagation ) and for some known geophysical signals that affect measurement (these corrections are labelled ΔR geophysical ) (e.g. [START_REF] Chelton | Satellite altimetry[END_REF][START_REF] Frappart | Satellite altimetry: principle and major applications in Earth Sciences[END_REF]. In our study, ΔR propagation includes the ionosphere correction, the dry troposphere correction, and the wet troposphere correction (for more details, see [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF]. ΔR geophysical corresponds to corrections for crustal vertical motions due to the solid Earth and pole tides. Moreover, the satellite altitude (H, Figure 1) can be calculated with an accuracy better than 0.02 m (e.g. [START_REF] Couhert | Towards the 1 mm/y stability of the radial orbit error at regional scales[END_REF]. All these computations allow for the estimation of the ground elevation above the reference ellipsoid (h) from the nadir altimeter, as follows: [START_REF] Desjonquères | Poseidon-3 radar altimeter: new modes and in-flight performances[END_REF]. Jason-2 is on a 10-day repeat cycle orbit at an altitude of 1336 km, and a 66° inclination. Its ground-track spacing is around 315 km at the equator and 220 km over mainland France. Jason-3, developed by the same agencies as Jason-2 plus the European Union (through the Copernicus program), was launched on 17 January 2016 and was put on the same orbit as Jason-2. The two satellites were 80 seconds apart. Jason-3 has a similar payload to Jason-2, in particular a Poseidon-3 class altimeter (Poseidon-3B). After more than 6 months sharing a common orbit, Jason-2 was put on an "interleaved" orbit on 13 October 2016. This has the same characteristics as its previous orbit, but is shifted in longitude by half the ground-track spacing at the equator.

ℎ = 𝐻 -(𝑅 + ∆𝑅 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑖𝑜𝑛 + ∆𝑅 𝑔𝑒𝑜𝑝ℎ𝑦𝑠𝑖𝑐𝑎𝑙 ) (

Altimeter tracking modes

For technical reasons associated with the pulse compression technique used for emission and owing to the limited signal bandwidth [START_REF] Chelton | Satellite altimetry[END_REF], the altimeter records the power returned from the ground over only a limited window of ranges (called "tracking window" on Figure 1). The vertical size of the range window (see Figure 1) for Poseidon-3 class altimeters (i.e. for Jason-2 and Jason-3) is equal to 50 m. As it is orders of magnitude smaller than the altitude of the satellite (around 1336 km), this window needs to enclose the water surface elevation. Otherwise, the altimeter processing unit will not receive any signal backscattered by the river and it will not be possible to derive the actual river elevation. The position of this window (R track on Figure 1) is continuously updated on board to roughly follow the ground topography. This process is called "tracking". It is performed automatically by the Adaptive Tracking Unit (ATU), based on the analysis of waveforms received previously. More information on the ATU system can be found in [START_REF] Chelton | Satellite altimetry[END_REF] and [START_REF] Desjonquères | Poseidon-3 radar altimeter: new modes and in-flight performances[END_REF]. This tracking mode is generally referred to as the "autonomous" or "Closed-Loop" (CL) mode. The retracking processing determines the position of the water surface in the waveform (a distance known as epoch). Finally, the range R is estimated using equation (2).

𝑅 = 𝑅 𝑡𝑟𝑎𝑐𝑘 + 𝑒𝑝𝑜𝑐ℎ

(2)

As nadir altimeters are designed to observe oceans, the ATU is not optimized to follow continental surfaces, where elevations may vary by tens to hundreds of meters over an alongtrack distance of a few kilometres. This is an issue for steep-sided rivers enclosed in valleys that are a few kilometres wide (i.e. similar to the footprint of the instrument) and more than 50 m (i.e.

a distance greater than the tracking window size) lower than the surrounding hills. In this case, the altimeter can remain "locked" on the top of the surrounding hills (or even lose tracking data)

and never observe the river below. This problem could even occur with slightly shallower valleys. More detailed explanations and specific examples of this issue can be found, for instance, in [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF].

A new instrument mode, implemented on board Jason-2, SARAL, Jason-3, and Sentinel-3A/B, was designed by CNES and its contractors to solve this specific issue. It consists in setting the tracking window to an a priori value close to the river valley elevation, rather than letting the ATU estimate its position based solely on previous waveforms. This solution requires storing on board a priori elevation values for targets of interest (e.g. water bodies), as well as their location along the satellite track. Then, the on-board software uses these a priori elevations and the altitude of the satellite computed in real time by the DIODE (Doris Immediate Orbit on board Determination) software to force the tracking window to these positions when the satellite overflies them [START_REF] Desjonquères | Poseidon-3 radar altimeter: new modes and in-flight performances[END_REF]. To build the on board tables used by the OL tracking mode, an a priori list of targets of interest (therefore a water mask) and an elevation associated with each target is required. Accuracies of both target geolocation and associated elevation are crucial to the observation of these water bodies. Therefore, the OL tracking mode has been used on a very small number of cycles on 5,7,34,209,and 220) and on SARAL (passes 601 to 800 during cycle 1). The elevations used by Jason-2 over France were taken from the 1 km Altimetry Corrected Elevation global DEM (ACE; [START_REF] Berry | ACE: a new GDEM incorporating satellite altimeter derived heights[END_REF] and the water mask from the Generic Mapping Tools (GMT, http://gmt.soest.hawaii.edu/) [START_REF] Desjonquères | POSEIDON3 DEM/Diode Coupling Mode[END_REF]. The a priori global DEM used for water body elevations for SARAL was ACE2 [START_REF] Berry | ACE2: the new global digital elevation model[END_REF] and the land/water mask was derived from Globcover (http://due.esrin.esa.int/page_globcover.php), supplemented by some water body elevations available on the Hydroweb database.

In OL mode, the leading edge of the waveform (therefore the tracking window, see Figure 1)

should ideally be centred on the first third of the tracking window. As this window size is around 50 m for Jason-2 and Jason-3, CNES estimated that the accuracy for water body a priori elevations should be around ±10 m.

The OL tracking mode has not been studied extensively. Birkett and Beckley (2010) evaluated both the CL and the OL tracking modes for Jason-2 over 28 lakes and reservoirs. For these targets, they found that CL and OL modes provide similar results when the water body is observed in both modes, but that the CL mode observed more targets than the OL mode. Birkett and Beckley (2010) assumed that ""inadequate resolution and/or data in the DEM" might be the reason for OL mode underachievement in comparison with CL mode. They concluded that the on-board DEM might not be ""optimized for all regions". [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF] showed that some locations in the steep-sided Garonne River valley (within mainland France) could not be observed by nadir altimeters in CL mode. During the first SARAL cycle, two VS were observed in OL mode. The on-board DEM value for one VS was close to the river valley elevation and therefore the river was observed (which was not the case during subsequent cycles in CL mode).

However it was not the case for the second VS. The issue was an incorrect water mask and not a lack of accuracy in the a priori DEM used to compute the on-board DEM. For this reason [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF] concluded that the on-board DEM is highly dependent on both input water mask and the a priori DEM used, which should be internally consistent.

The launch of Jason-3 and the need to validate its measurements during 2016 represented a good opportunity to construct a finely tuned on-board DEM over a specific zone, where precise local DEMs are available. The methodology used to achieve this end and the results obtained are presented in the following sections.

3. Methods

Study domain and a priori/validation data

The study domain chosen to validate Jason-3 measurements over rivers corresponded to mainland France (Figure 2). The four main rivers (ranked by their mean annual discharge at their outlet indicated in brackets) in this study domain are: the Rhône (1700 m 3 .s -1 ), the Loire (900 m 3 .s -1 ), the Garonne (600 m 3 .s -1 ) and the Seine (520 m 3 .s -1 ). This region was selected because of the large number of in situ gages operated by regional public agencies, i.e. DREALs (Directions Régionales de l"Environnement, de l"Aménagement et du Logement). These observations are collected by the SCHAPI (Service Central d"Hydrométéorologie et d"Appui à la Prévision des Inondations), which releases them publicly via the online "Banque Hydro" national database (http://www.hydro.eaufrance.fr) within a period of a few weeks to a few months. Gages used in this study are shown in Figure 2 (green stars). Their records were downloaded from the Banque Hydro database or obtained directly from the operator (DREALs). For all these gages, water elevations measured were referenced to NGF-IGN69, the official vertical reference system for mainland France. Their measurements were available for the period of interest: the tandem phase when Jason-2 and Jason-3 shared the same orbit and ground track. The time series obtained from 13 these gages have a non-uniform time step, which is dependent on the water elevation stage.

Instruments automatically adjust recording time steps to water elevation variations. However, the median time step for all gages is, on average, around 30 minutes. ), where Jason-3 on-board DEM values were computed. Red diamonds are virtual stations where Jason-2 and Jason-3 river elevation time series were compared to in situ gage (green stars) measurements. The background image is derived from the NASA MODIS "Blue Marble Next Generation" image [START_REF] Stöckli | The Blue Marble Next Generation -A true color earth dataset including seasonal dynamics from MODIS[END_REF] 3.2. Jason-3 on-board DEM and validation methodology 3.2.1. On-board DEM computation over France

The IGN BD Carthage database with rivers wider than 50 m corresponds to the water mask used in this study. These river reaches were intersected with the Jason-2/-3 orbit obtained from the Aviso+ website (http://www.aviso.altimetry.fr), leading to the creation of 89 VS. Three intersections with the river network and orbit tracks in the most southern part of the Garonne basin (see Figure 2) were not selected. For these locations the on-board DEM has been set by CNES to the altitude of transponders used for calibrating/validating the altimeter.

Next, it was necessary to estimate the DEM values for the 86 selected VS (white and red diamonds on Figure 2). This was achieved using the 25 m resolution DEM from IGN"s BD Alti database (http://professionnels.ign.fr/bdalti). The absolute vertical accuracy of this DEM is assessed by IGN at between 2 to 8 m, depending on the local data source used. Elevations are provided as integers and are referenced to the NGF-IGN69 vertical reference system. Rather than simply extracting DEM values at the position of each VS, to take into account the facts that [START_REF] Frappart | Preliminary results of ENVISAT RA-2 derived water levels validation over the Amazon basin[END_REF] the altimeter ground footprint is a few kilometres wide, (2) the satellite ground track is controlled to within ±1 km around its nominal position [START_REF] Dumont | Jason-3 product handbook[END_REF] and (3) rivers are in the lowest part of the valley, we computed the minimum elevation over a 2 km radius centred on each VS.

Different values of this radius have been investigated (1 km to 5 km). A 2 km radius is a good compromise between having sufficient variation to capture the river elevation (when compared to a small number of VS close to a gage) and not being excessively impacted by elevations outside the river valley targeted. As a quality check, the DEM was inspected visually and comparisons made with Google Earth images for the 22 VS where the difference between the minimum elevation computed on this circle and the elevation of the closest Banque Hydro database in situ gage was greater than 10 m. Different cases were observed as follows: the difference was due to an obvious issue with the gage elevation in the Banque Hydro, the closest gage was on a different river to the VS, or the VS was on a reservoir and the gage was not (or the reverse). For each case, the final elevation value was selected manually to take into account the specificity of each case and the local topography in the vicinity of the VS. For all other VS, the minimum elevation on the 2 km circle was selected. This radius might change a little for regions other than France, depending, for example, on the accuracy of the DEM and the water mask. But the methodology should be fairly robust, easily adapted to any other location, and should not be specific to French rivers. Finally, these elevations were sent to CNES, which uploaded them on board Jason-3 on 2 May 2016.

Validation of Jason-3 measurements

Between 17 February and 2 October 2016, Jason-2 (cycles 281 to 303) and Jason-3 (cycles 1 to 23) flew on the same orbit, 80 seconds apart. Therefore, they observed the same locations almost simultaneously, which allowed the direct comparison of measurements from both satellites.

Jason-2 was in CL tracking mode throughout the study period, whereas thirteen Jason-3 cycles were in OL tracking mode, for the most part after May 2016 (see Table 1 for cycle numbers and start dates). Therefore, it is possible to evaluate the benefits of the OL tracking mode for small to medium-sized steep-sided rivers by comparing Jason-3 cycles in OL mode to Jason-2 data and in situ measurements. To validate Jason-3 measurements (section 4), an initial qualitative test was performed to compare Jason-2 and Jason-3 measurements over the 86 VS. For each VS, the absolute difference (noted dR on Figure 1) between R track (the tracked distance between the satellite and the tracking window, see Figure 1 and section 2.2) and R DEM (the distance between the satellite and the position of the surface given by the DEM, see Figure 1) was computed. dR allows to quantitatively estimate how the position of the tracking window in CL mode is far from the on board DEM value (which should be close to the river elevation). Figure 1, illustrates a case when Jason-3 in CL mode successfully observes the river. In this case, dR is smaller than the size of the tracking window. Therefore, dR should provide useful information to assess if a SV in CL mode observe the river (dR much smaller than 50 m) or the surrounding topography (dR higher than 50 m). That"s why, in section 4, its temporal mean value for Jason-3 cycles in CL mode is analysed. Similar averages were computed for corresponding Jason-2 data during the same cycles. It allowed us to estimate, on average, how the tracked elevation differs between Jason-2 and Jason-3, when their altimeters are both in CL mode and when Jason-3 is in OL mode and Jason-2 is in CL mode. In addition, the backscatter coefficient, which takes a high value in the event that water contributes to the power recorded by the altimeter (e.g. Frappart et al., 2015c), was analysed.

For quantitative validation, all VS close to a "Banque Hydro" in situ gage were selected. For these VS, Jason-2 and Jason-3 measurements were processed to extract elevation time series (see section 2.1), without using any a priori information from the gage time series. Jason-3 time series were computed using all available measurements, without differentiating between cycles in CL mode and in OL mode. For both Jason-2 and Jason-3, some cycles had valid measurements of river elevations for many VS, whereas other cycles measured the elevations of the surrounding topography. Measurements from the latter cycles were easily excluded, as they were much higher than those of valid cycles. For this reason, some VS time series present observation gaps.

A thorough inspection led to the selection of 24 in situ gages (Figure 2, green stars) which were compared to 21 VS (Figure 2, red diamonds). The number of VS selected was different to the number of gages selected, as some VS had two in situ gages sited nearby (one upstream and one downstream), or conversely, some gages were sited close to two VS. The selection criteria were that [START_REF] Frappart | Preliminary results of ENVISAT RA-2 derived water levels validation over the Amazon basin[END_REF] the distance between gages and VS should be a few kilometres, (2) they should be located on the same river with no large intervening tributaries, and (3) the in situ gage time series should be available from the Banque Hydro or directly from regional DREAL agencies (see section 3.1). More information (WGS84 longitude/latitude coordinates, river width, and the intervening distance) for each pair of gage and VS compared is presented in the supplementary material.

To carry out this quantitative assessment for validation purposes, Jason-2 and Jason-3 time series were both compared to the in situ gage time series, using the same statistical tests as [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF]: correlation coefficient, mean bias, and Root Mean Square Error (RMSE) between the two time series for absolute water elevations referenced to NGF-IGN69. In situ and altimetry water elevation anomalies were also computed, by removing their respective temporal mean (calculated for the same common dates). The RMSE and the Nash-Sutcliffe (NS) coefficient [START_REF] Nash | River flow forecasting through conceptual models Part I -A discussion of principles[END_REF] were computed for anomalies. The NS coefficient (which varies between -∞ and 1) was used in this study for its ability to assess the accuracy of the match between two time series, both in time and amplitude. It is more informative than the correlation coefficient, which only provides information about the linear relationship between two time series, without taking into account amplitude agreement [START_REF] Moriasi | Model evaluation guidelines for systematic quantification of accuracy in watershed simulations[END_REF]. The NS coefficient for absolute elevation is not provided, as a bias is expected between altimetry measurements and the gage time series, caused at least partly by the slope of the river between the VS and the gage.

The Jason-2 and Jason-3 data used in this study were taken from the Geophysical Data Records (GDR), version D, provided by the CNES/NASA processing centres and formatted by the Centre de Topographie de l"Océan et de l"Hydrosphère (CTOH, http://ctoh.legos.obs-mip.fr) in a consistent NetCDF format with geophysical corrections coherent with previous altimetry missions. The retracking algorithm used in this study is Ice-1/ICE [START_REF] Wingham | New techniques in satellite altimeter tracking systems[END_REF], as recommended by [START_REF] Frappart | Preliminary results of ENVISAT RA-2 derived water levels validation over the Amazon basin[END_REF] for continental surfaces. Water elevation time series were computed using Multi-mission Altimetry Processing Software (MAPS) that allows a refined selection of valid altimetry data to build VS (Frappart et al., 2015b). MAPS is a graphical user interface that allows the loading of altimetry data formatted by CTOH that are located within the boundaries of a VS polygon, the computation of water elevations using Eq. 1 and the manual selection/removal of some measurements. These water elevations provided by MAPS correspond to ellipsoid heights. They have then been referenced to NGF-IGN69, for comparison to in situ water elevation time series.

Results

Jason-3 statistics over the study domain

Figure 3 shows maps of Jason-2 and Jason-3 absolute difference dR (see section 3.2.2 and Figure 1), averaged over Jason-3 CL mode cycles (Figure 3 Table 2 summarizes these results by showing some basic information for the 86 VS. Jason-2 behaves consistently over the whole time period. Therefore, the differences between the Jason-3 measurements in CL and OL modes can only be explained by the differences in tracking mode.

For Jason-2, 25 VS (29% of all VS) are always locked on the top of the topography (first line in Table 2). This number increases to 38 VS (44%) for Jason-3 in CL mode. Around half of the VS had some cycles affected by this behaviour, as 34 (40 %) VS (Jason-2) and 49 (57 %) VS (Jason-3 in CL mode) have an average dR of over 50 m. Table 2 also indicates the mean and the 20 maximum values of the backscatter coefficient σ 0 (in dB) averaged over all VS. The average value of σ 0 over all VS is equal to 29 dB and 25 dB for Jason-2 and Jason-3 in CL mode, respectively. σ 0 tends to be lower over land than over water (smoother and more reflective surface than land, e.g. Frappart et al., 2015c). Therefore, its lower value for Jason-3 measurements in CL mode confirms its tendency to be locked on the surrounding topography more often than Jason-2. Nevertheless the σ 0 for both altimeters remain close. However, the average σ 0 for Jason-3 in OL mode increased to 40 dB, confirming that Jason-3 in OL mode observed more rivers than Jason-2. However, it is not sufficient to show that the signal backscattered by the river is measured by the altimeter. There remains the task of checking that these measurements can be converted to useful water elevation time series. For this reason, the next section compares the Jason-2 and Jason-3 time series to in situ water elevation time series.

21 404 Figure 3. Mean absolute difference (dR) between R track and R DEM (see Figure 1) for Jason-2 (a. and b.) and for cycles when Jason-3 is in closed-loop mode (approximately between February and May, 2016; a. and c.) and when Jason-3 is in open-loop mode (approximately between May and September, 2016, see Table 1 for the list of cycles; b. and d.)

Table 2. Statistics computed using the 86 VS shown on Figure 2 (red and white diamonds) indicating the number of VS for which the min, max, and mean difference between the altimeter tracked height and on-board DEM values (dR) is greater than the size of the tracking window (50 m), values of the mean and max dR (in m ) over all VS and the mean and max backscatter coefficient σ 0 (in dB) over all VS

Jason-2 (J-3 CL cycles)

Jason-3 (CL cycles)

Jason-2 (J-3 OL cycles) This section focuses on the 21 VS (Figure 2, red diamonds) that are close to in situ gages (Figure 2, green stars) which were selected for a quantitative assessment of Jason-2 and Jason-3 errors (see section 3.2.2).

Table 3 and Table 4 show the comparative statistics for in situ, and Jason-2 and Jason-3 satellite time series, respectively, for all VS/gage pairs. Data shown in these two tables suggest that altimeters are capable of accurately monitoring some small rivers, such as the JA-002 VS on the Marne River, which is 40 m wide. On this river both Jason-2 and Jason-3 display an RMSE of 0.20 m for water elevation anomalies, whereas the amplitude of the in situ time series at the satellite observation times is much larger (2.72 m). However, this instance is very favourable for altimetry, as the topography is quite flat, with little surrounding vegetation. For Jason-2, correlation coefficients greater than 0.9 are associated mainly with rivers wider than 130 m, whilst for Jason-3, correlation coefficients greater than 0.9 are mainly associated with rivers wider than 100 m. The Jason-2 RMSE for elevation anomalies for these rivers ranged from 0.20 to 0.63 m, whereas the Jason-3 RMSE was between 0.11 and 0.28 m. Eight Jason-2 VS (38% of VS selected) were significantly impacted by the surrounding topography, with fewer than 5 cycles (out of 23) observing the river valley or with a bias compared to in situ measurements around 50 m or more. All of these VS were observed correctly by Jason-3 only when operating in OL mode (except for JA3-014 for which 7 cycles in CL mode observed the river valley) with a small bias for absolute elevation compatible with what might be expected from the river slope (a couple of metres over a distance of a few kilometres). The same statistics as those shown in Table 3 and4, but computed only for Jason-3 CL mode cycles and Jason-3 OL mode cycles for both Jason-2 and Jason-3, are presented in the supplementary material associated with this article. They show that the accuracy of Jason-3 measurements is fairly similar for those VS that work well in both CL and OL modes. This result indicates that the OL tracking mode is operating as expected and allows the observation of steep-sided rivers that previously were practically never observed, resolving the issue described in [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF]. It should also be noted that Jason-3 observed the river valley for all VS during the 13 OL mode cycles. In Table 4, for VS with only 13 cycles in their time series, these cycles correspond solely to OL mode cycles (there are no observations for CL cycles). This is shown clearly in the tables presented in the supplementary material. There are three Jason-3 VS (JA3-075, JA3-017, and JA3-010) that are always locked on the surrounding topography in CL mode (0 cycle used in Table 4 in the supplementary material), whereas the corresponding Jason-2 VS provide some measurements of the river elevation (i.e. two or more cycles could be used, their bias with in situ measurements is below 50 m, and their correlation coefficient with in situ measurements is above 0.50)..

However, it should be noted that, for JA-075 and JA-017 VS, unlike JA-010 VS, even for Jason-3 OL mode cycles or Jason-2 cycles with valid measurements, the RMSEs for water elevation anomalies are still large, because of the small river widths (see supplementary material for these RMSEs). For other Jason-3 VS with 0 cycle used in CL mode, Jason-2 does not provide useful measurements of the river elevation (see supplementary material). For some other VS, the Jason-3 time series had fewer observation dates than the corresponding Jason-2 time series, owing to there being more Jason-3 cycles in CL tracking mode locked on the surrounding topography (see 448 table in supplementary material). This is consistent with the results obtained by the qualitative 449 assessment (section 4.1). 450 Table 3. Correlation coefficient, mean bias, RMSE (m), RMSE for anomalies (m), Nash-Sutcliffe (NS) coefficient for anomalies between Jason-2 and in situ time series. River width at the VS, amplitude (Ampl.) of the in situ time series for dates in common with Jason-2, and the number of these common dates are also provided. Rows shaded red correspond to VS/gage pairs for which the mean bias is around 50 m or more and/or the correlation coefficient is below 0.5 and the NS coefficient for anomalies is negative. Rows shaded orange correspond to VS for which half or more cycles are locked on the surrounding topography ( 11 ,5, and 6 show the comparison of elevation time series derived from Jason-2 and Jason-453 3 against in situ data for three pairs of VS/gage: JA-082/L8700020, JA-030/O9000010, and JA-454 064/K6830020, respectively. Blue lines and dots correspond to gage time series, whereas red 455 lines and dots correspond to nadir altimetry measurements. For the altimetry time series, the grey 456 vertical bar for each observation time corresponds to the mean absolute deviations of all selected 457 altimetry measurements for that time within the VS boundaries computed by MAPS software 458 (see section 2.1 and for more details Frappart et al., 2015b and[START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF]. 459

In the case shown in Figure 4, Jason-3 cycles perform well in both CL and OL tracking modes, similarly to Jason-2. It shows that the OL mode, when the on-board DEM is accurate, performs as well as the CL mode with valid cycles. Both modes perform well probably because there is an elevation difference of less than 50 m between the surrounding hills and the river valley. The high mean absolute deviation for one Jason-3 cycle (early April 2016) shown in Figures 4.b and 4.d is due to an outlier in the selected points within the VS, which may correspond to a measurement of the surrounding topography. In the case shown in Figure 5 all Jason-2 cycles (i.e. in CL tracking mode) observe the river, whereas only 3 (out of 10) Jason-3 cycles in CL (and all cycles in OL) observe the river. Finally, Figure 6 illustrates a Jason-2 VS that never observes the river and remains locked on the surrounding hills, whereas only Jason-3 cycles in OL mode are able to measure the river elevation. All these figures, and in particular their bottom panels show that a Jason-2/-3 sampling period of 10 days is coarse for this type of river.

Combining measurements from multiple altimetry missions might be beneficial here. The study time period (mid-February to late September 2016) covered a period of high flows (February/March), a flooding event on the Loire River lasting several days (in early June 2016) and a low flow period (July/September). The altimeter time series allowed sampling of the seasonal cycle (transition from high to low flow periods), but missed local maxima, especially the flood peak on the Loire at VS JA3-064. As previously stated by [START_REF] Biancamaria | Satellite radar altimetry water elevations performance over a 200 m wide river: Evaluation over the Garonne River[END_REF], Jason-2/-3 time sampling allows the study of the seasonal cycles of small to medium-sized rivers, but will miss high frequency dynamics (periods of hours to a few days). Having measurements of this kind will not assist flood forecasting for the type of rivers presented in this study, but they are important for studying the seasonal cycle, the annual, interannual, and even decadal dynamics of small and medium-sized rivers, especially in areas where no gaging network is present or maintained. The OL tracking mode will help to obtain more observations from around the world for these types of river.

Figure 4. a.,c. and e.) and b.,d. and f.) water elevation time series (red lines and red dots) on the Loire River at La Menitré (JA-082) compared to in situ time series (blue lines and blue dots) at Gennes. Top panels (a. and b.) correspond to absolute elevations with reference to (wrt) NGF-IGN69 at satellite observation times. Middle panels (c. and d.) correspond to elevation anomalies at satellite observation times. Bottom panels (e. and f.) correspond to times series with all available times for both satellite and in situ time series. Pale rose zones on panels b., d., and f. correspond to time periods when Jason-3 altimeter was in CL tracking mode Figure 5. a.,c. and e.) and b.,d. and f.) Results from both qualitative (section 4.1) and quantitative (section 4.2) Jason-3 validation assessments showed that Jason-3 in CL tracking mode was more frequently locked on the surrounding topography than Jason-2. Jason-3 and Jason-2 altimeters use the same CL tracking algorithm, called the median tracker. The aim of the median tracker is to "track" as much backscattered energy from the ground as possible, in the hope that the tracked ground corresponds to water bodies. This is a statistical approach, which almost always works, but which does not discriminate between surface types. In other words, in CL tracking mode we cannot give preference to water bodies over other ground targets. The fact that Jason-2 tracks more hydrological targets than Jason-3 in CL mode may appear to be counterintuitive.

Although Jason-2 and Jason-3 share a common algorithm, the hardware is not exactly the same.

Algorithm parameters values must be adapted to each instrument. They are defined during altimeter ground tuning and testing. These are critical values and are generally not updated in flight unless strictly necessary. Owing to some unexpected in-flight behaviour of Jason-2 ("ghost echoes") the post-launch modification of the original tuning of the altimeter was unavoidable. A side-effect of this procedure was a slight degradation of radar sensitivity in CL tracking mode.

When the altimeter operates in CL mode, it first searches for a backscattered radar signal. For this purpose, it scans from higher to lower altitudes. Then when the energy received by the altimeter is sufficiently strong with respect to its sensitivity, it stops scanning altitudes and locks its tracking loop to the current target altitude.

In the case of areas with relief, generally the higher the altitude, the weaker the backscattered signal. As the sensitivity of the Jason-2 altimeter has been degraded, the result is that, in principle, for such regions, it will lock its tracker at a lower altitude than the Jason-3 altimeter.

As rivers are particularly located in valleys, Jason-2 will have a better chance of tracking a hydrological target than Jason-3.

The drawback of the median and more generally the CL tracker is the lack of control and of target prioritization. For this reason, the OL tracking mode was designed to overcome this issue.

It has the capability to designate a target and therefore to give highest priority to water bodies.

Prior to the Jason-3 launch, it was not anticipated that Jason-2 altimeter tuning was improving water elevation observation over continents. As this modification was not applied to Jason-3 (or to any other past, present or planned altimeter), it should be expected that Jason-3 in CL tracking mode will be locked over surrounding topography for steep-sided rivers more frequently than Jason-2, as shown in this study. However, the yield of Jason-3 in OL tracking mode will be dramatically better than that of Jason-2 if the behaviour observed over mainland France, can be obtained for all global locations.

Conclusions and Perspectives

The 6 month period during which the recently launched Jason-3 altimetry mission shared the same orbit as Jason-2 was the perfect time to validate the Jason-3 Closed-Loop (CL) tracking mode and to assess the benefit of the Jason-3 Open-Loop (OL) tracking mode. A high-quality DEM and water mask available from the IGN were used to compute the on-board DEM for the OL mode over mainland France which was loaded onto Jason-3 on 2 May 2016 (the first Jason-3 cycles being in CL tracking mode). Accurate water elevations from in situ gages were used to validate both Jason-2 and Jason-3 measurements for river widths between 35 and 300 m.

This study has shown that altimeters such as Jason-2 and Jason-3 are able to accurately measure rivers with a width of 100 m or greater with an RMSE for elevation anomalies of around 0.20 to 0.30 m. The 10 day sampling period is adapted for observing the seasonal cycle, but misses some local maxima. Smaller rivers could also be observed, but with a higher RMSE and time sampling could be more critical for this type of river.

This study demonstrates that Jason-2 and Jason-3 in CL mode provide similar results, although

Jason-3 has a tendency to remain locked on the surrounding topography more frequently than Jason-2. This issue is due to a Jason-2 altimeter post-launch tuning, which solved a problem but led to a slight degradation in radar sensitivity. It was not anticipated that this modification would improve observations over continents. As the problem had been resolved on the Jason-3 altimeter, this modification was not applied to the Jason-3 CL tracking mode. Of the Jason-2 VS, 38% measured the top of the surrounding hills almost all of the time. On the contrary, water elevation measurements were always measured at the corresponding Jason-3 VS in OL mode.

Even for Jason-2 VS that were only partially affected by this issue (a small number of cycles locked on the surrounding topography), Jason-3 was able to observe the river in OL mode for all cycles, increasing the number of observation times. This shows the significant advantages provided by the OL tracking mode for steep-sided rivers that were unobserved or only partially observed by previous altimetry missions, thus potentially significantly increasing the number of river reaches that could be monitored worldwide by satellite radar altimeters. Even much wider rivers could benefit from the OL tracking mode if they are located in a narrow river valley (a few kilometres across), or are very close to surrounding hills that are higher than the river elevation by 50 m or more. Jason-3 has greater flexibility than previous missions (i.e. Jason-2 and SARAL) had in OL mode, because small portions (a few kilometres) of the track can be in OL tracking mode (for locations where the water mask and the a priori DEM are coherent and sufficiently accurate), while the rest of the track remains in the classical CL tracking mode.

Demonstrating the potential of the OL tracking mode is important in the early stages of the Jason-3 mission and also in the context of recently launched (Sentinel-3A), or future altimetry missions (Sentinel-3B planned for launch in early 2018, Sentinel-3C/D whose launch dates are currently unknown, as they will ensure continuity of observation after Sentinel-3A/B, Jason-CS/Sentinel-6 in 2020, SWOT in 2021, and follow-ons), that operate or will operate using this tracking mode. The use of the OL tracking mode will substantially increase the number of river basins that can be monitored using satellite radar altimetry.

The OL mode over mainland France successfully provided observations of river stages, because an accurate on-board DEM could be computed, based on accurate local database. Unfortunately, such a database is not available globally and other sources will have to be used for the rest of the world. For some locations where the global DEMs (GDEM) are sufficiently accurate, these could be used to compute the on-board DEM for OL mode. This could be the case for GDEMs such as the latest version of the Shuttle Radar Topography Mission GDEM released in September 2014

(see https://www2.jpl.nasa.gov/srtm/), the ASTER GDEM [START_REF] Tachikawa | Characteristics of ASTER GDEM version 2[END_REF], or the upcoming DLR TanDEM-X GDEM [START_REF] Zink | TanDEM-X: the new global DEM takes shape[END_REF], whose absolute elevation in some regions has been assessed to be within ±10 m. They will also need to be coupled with a coherent global water mask more precise than that of Globcover. For example, the global water mask established from satellite imagery computed by [START_REF] Pekel | High-resolution mapping of global surface water and its long-term changes[END_REF] could represent a good alternative. Some local information provided by the broader scientific community (based on field trip measurements, local maps…) could also be used. For locations where an accurate on-board DEM over water bodies cannot be computed, the altimeter would be left in classical CL tracking mode, to avoid setting the tracking window to an incorrect position for all cycles. In the coming years, more extensive work will be needed to improve the Jason-3 on-board DEM, but also for incoming altimetry missions that are not on the same orbit (for example the Sentinel-3 series of satellites).

The benefits of the OL tracking mode for lakes and reservoirs need to be carefully investigated.

Similar results are anticipated to those obtained for rivers where an accurate on-board DEM can be computed. However computation of the on-board DEM may be even harder than for rivers, as many reservoirs (and lakes) are in mountainous regions where the DEMs usually available are not accurate to within ±10 m. Further, for large lakes and reservoirs, the OL tracking mode might not be suitable, as they can display seasonal and/or interannual variations much larger than those of rivers. For such variations, setting the tracking window to one constant value might not be appropriate for all observation times. A potential solution would be to update the on-board DEM on a seasonal basis. Specific studies are needed to investigate these cases. CNES, NASA, EUMETSAT, NOAA, and Copernicus are acknowledged for providing free access for the scientific community to measurements from Jason-2 and Jason-3 altimeters.

The CTOH observation service at LEGOS (http://ctoh.legos.obs-mip.fr/) is also acknowledged for processing and providing altimetry data in a uniform format.

Constructive comments from Nicolas Picot (CNES) helped to improve this article.

This study was funded by the ""Réseau Thématique de Recherche Avancée -Sciences et Technologies pour l"Aéronautique et l"Espace" (RTRA-STAE, Toulouse, France), through a grant awarded to the ""Ressources en Eau sur le bassin de la GARonne: interaction entre les composantes naturelles et anthropiques et apport de la téléDétection" (REGARD) project.

This work was supported by the CNES. It is based on observations with Poseidon-3 and Poseidon-3B embarked on Jason-2 and Jason-3, respectively. Especially, funding from the CNES Terre-Océan-Surfaces Continentales-Athmosphère (TOSCA) committee to the "AltiWaveforms" and CTOH projects, and from the Ocean Surface Topography Science Team (OSTST) FOAM project is acknowledged.

Finally, comments and suggestions from three anonymous reviewers improved this article.

Figure 3. Mean absolute difference (dR) between Rtrack and RDEM (see Figure 1) for Jason-2 (a. and b.) and for cycles when Jason-3 is in closed-loop mode (approximately between February and May, 2016; a. and c.) and when Jason-3 is in open-loop mode (approximately between May and September, 2016, see Table 1 for the list of cycles; b. and d.)

1 )

 1 Figure 1. Conceptual view of nadir altimeter measurements and notations used in this study (the copyright of the Jason-3 satellite image used in this sketch belongs to CNES/Mira Production)

Figure 2

 2 Figure2shows Jason-2/-3 tandem phase orbit tracks (red lines) superimposed on the study

Figure 2 .

 2 Figure 2. Study domain (Mainland France, delimited by the black line) and the river network (light blue lines) from the IGN BD Carthage database (only rivers wider than 50 m according to this database are shown). Red lines correspond to Jason-2 and Jason-3 tandem phase orbit tracks. White and red diamonds correspond to virtual stations (VS), where Jason-3 on-board DEM values were computed. Red diamonds are virtual stations where Jason-2 and Jason-3 river

  .a and 3.c) and those where Jason-3 is in OL mode(Figure 3.b and 3.d). As the size of the Jason-2 and Jason-3 tracking window is around 50 m, wherever and whenever this difference is greater than 50 m (Figure3, red diamonds), it can be assumed that the altimeter is locked on the surrounding hills. This figure clearly shows that Jason-2 behaved in a similar manner before (Figure3.a) and after (Figure 3.b) May 2016. It also shows that Jason-3 exhibited similar behaviour to that of Jason-2 when it was in CL tracking mode (Figure 3.c), with a slight tendency to have a few more VS locked on the surrounding hills (especially in the downstream part of the Garonne River).

  Figure5. a., c. and e.) and b., d. and f.) water elevation time series (red lines and red dots) on the Garonne River at Caumont-sur-Garonne (JA-030) compared to in situ time series (blue lines and blue dots) at Tonneins. Panels are

  CEREMA (Vanessya LABORIE), DREAL Auvergne Rhône-Alpes (Eric KERMAREC and Philippe PRAX), DREAL Bourgogne-Franche-Comté (Erwan LE BARBU), DREAL Centre-Val de Loire (Franck GILLOUX), DREAL Grand Est (Christophe MAGE, Johan HABERT and Marc KLIPFEL), DREAL Nouvelle-Aquitaine (Olivier DEBINSKI et Dominique LAGORCE), DREAL Occitanie (Arthur MARCHANDISE et Didier NARBAÏS-JAUREGUY), and DREAL Pays de la Loire (Stéphanie POLIGOT-PITSCH) are also gratefully thanked for providing some in situ measurements and gage elevations used in this study.

Figure 4 .

 4 Figure 4. Jason-2 (left panels; a., c. and e.) and Jason-3 (right panels; b., d. and f.) water

Table 1

 1

	Jason-3 cycles in OL
	tracking mode	Cycle start time

. Number and start times of Jason-3 cycles in OL tracking mode, during the shared orbit period with Jason-2

Table 4

 4 or less cycles used) Ampl.) of the in situ time series for dates in common with Jason-3, and the number of these common dates are also provided. Rows shaded red correspond to VS/gage pairs for which the correlation coefficient is below 0.5 and the NS coefficient for anomalies is negative

	Jason-2	VS Riv.	Gage ID	Corr. Bias	RMSE	RMSE	NS	Ampl.	Nb
	VS ID	Width				(m)	anom	anom	(m)	cycles
		(m)					(m)			used
	JA2-074		M1511610	0.62	3.33	3.35	0.30	0.26	1.46	23
	JA2-075		M3230930	0.47	26.07	26.07	0.25	-0.39	0.65	17
	JA2-002		H5201010	0.97	-2.23	2.24	0.2	0.95	2.72	23
	JA2-017		U1074020	0.43	12.55	12.57	0.64	0.15	2.64	12
	JA2-011		P7121510	-	0.25	0.25	-	-	-	1
	JA2-014		Q1420010	0.88	3.67	3.7	0.44	0.63	1.85	4
	JA2-077		X3310010	-0.16	-11.39	11.51	1.63	-3.5x10 4	0.05	23
	JA2-077		X3500010	-0.25	16.42	16.51	1.67	-76.79	0.83	20
	JA2-020		U0610010	0.87	1.44	1.49	0.39	0.75	3.14	22
	JA2-071		M0520610	0.62	-5.59	5.59	0.15	0.30	0.59	20
	JA2-071		M0630610	0.62	5.56	5.57	0.18	0.38	0.71	17
	JA2-042		K0910010	0.56	1.04	1.16	0.5	-2.05	0.78	5
	JA2-012		Q5421020	0.8	-34.32	34.32	0.32	0.53	1.56	19
	JA2-012		Q5501010	0.76	13.73	13.74	0.36	0.30	1.35	19
	JA2-094		V3130021	0.4	127.92	127.93	2.02	-5.43	1.97	11
	JA2-009	110	P5550010	-	3.96	3.96	-	-	-	1
	JA2-093	120	V3130021	0.1	126.9	126.96	3.74	-23.98	1.97	13
	JA2-010	130	O9190010	0.97	-0.15	0.43	0.4	0.93	4.36	8
	JA2-030	155	O9090010	0.91	3.58	3.64	0.63	0.75	4.37	23
	JA2-030	155	O9000010	0.91	-3.01	3.08	0.63	0.78	4.53	23
	JA2-080	165	H8100021	0.55	82.39	82.4	0.14	0.31	0.66	12
	JA2-032	200	O9090010	0.95	0.96	1.03	0.37	0.89	3.56	10
	JA2-036	245	P5770010	0.74	3.89	3.98	0.81	-51.61	0.38	20
	JA2-064	250	K6830020	0.47	49.18	49.28	3.19	-8.54	3.28	23
	JA2-082	300	L8700020	0.99	-0.09	0.37	0.36	0.93	5.5	23
	JA2-082	300	L8700030	0.99	1.74	1.75	0.19	0.97	4.73	23
	451

.

Correlation coefficient, mean bias, RMSE (m)

, RMSE for anomalies (m), Nash-Sutcliffe (NS) coefficient for anomalies between Jason-3 and in situ time series. River width at the VS, amplitude (

Acknowledgements:

Our warm thanks to IGN for freely providing numerous georeferenced information across France (DEM is available at http://professionnels.ign.fr/ and river network data at www.sandre.eaufrance.fr).

The authors are grateful to SCHAPI for making gage measurements from the many gages operated by different agencies in France publicly available, through the "Banque Hydro" database (http://www.hydro.eaufrance.fr).

similar to those in Figure 4. Pale rose zones on panels b., d., and f. correspond to time periods when Jason-3 altimeter was in CL tracking mode Figure 6. a.,c. and e.) and b.,d. and f.) White and red diamonds correspond to virtual stations (VS), where Jason-3 on-board DEM values were computed. Red diamonds are virtual stations where Jason-2 and Jason-3 river elevation time series were compared to in situ gage (green stars) measurements. The background image is derived from the NASA MODIS "Blue Marble Next Generation" image [START_REF] Stöckli | The Blue Marble Next Generation -A true color earth dataset including seasonal dynamics from MODIS[END_REF] those in Figure 4. As Jason-2 is locked on the surrounding topography, panels c. and e. are not relevant, but are kept for consistency with Figure 4 and Figure 5. Pale rose zones on panels b., d., and f. correspond to time periods when Jason-3 altimeter was in CL tracking mode


Asmaa Amehraye

Dominique Pastor

So£a Ben Jebara

On the Application of Recent Results in Statistical Decision and Estimation Theory to Perceptual Filtering of Noisy Speech Signals

This paper combines perceptual £ltering to recent results in statistical decision and estimation theory in order to denoise speech signals corrupted by additive and independent white Gaussian noise. The resulting technique requires no Voice

Activity Detector and its performance is signi£cantly close to that obtained when the noise standard deviation is known and the masking threshold computed on the basis of the clean speech signals.

I. INTRODUCTION

Speech enhancement has greatly progressed over the past decades. Traditional methods basically improve the Signal to Noise Ratio (SNR). However, they introduce unpleasant residual noise known as musical noise. Various algorithms have been proposed for reducing the effects of musical noise ( [START_REF] Berouti | Enhancement of speech corrupted by acoustic noise[END_REF], [START_REF] Boll | Suppression of acoustic noise in speech using spectral subtraction[END_REF], [START_REF] Ephraim | Speech enhancement using a minimum mean square error short-time spectral amplitude estimator[END_REF]). They admittedly succeed under some conditions but there is still a need for more performant methods that reduce the amount of residual noise at very low input SNRs without introducing speech distortion.

The current trend is to exploit the auditory masking properties, widely used in perceptual audio coding. By using these properties, it is expected to make residual noise inaudible ( [START_REF] Hu | Incorporating a psychoacoustic model in frequency domain speech enhancement[END_REF], [START_REF] Tsoukalas | Speech enhancement using psycho-acousticriteria[END_REF], [START_REF] Virag | Singlehannel speech enhancement based on masking properties of the human auditory system[END_REF]). In this respect, the masking threshold becomes a constraint to take into account because the human auditory system is not sensitive to any residual noise masked by coexistant speech signal. The most usual method to calculate the masking threshold is presented in [START_REF] Johnston | Transforming of audio signals using perceptual noise criteria[END_REF] and relies on a critical band analysis modelling the behaviour of the inner ear.

In this paper, to denoise speech signals corrupted by independent and additive white Gaussian noise (AWGN), we combine a perceptually motivated method with a non parametric estimator of the noise standard deviation. This estimator avoids the use of any Voice Activity Detector (VAD). Because of the novelty of this estimator, we do not address the case of coloured noise yet. The estimate performed by this estimator is used twice. First, to adjust a recursive Wiener £ltering whose outcome serves to estimate the masking threshold of the speech signals to denoise; second, to tune the perceptual £lter proposed by [START_REF] Lee | Speech enhancement by perceptual £lter with sequential noise parameter estimation[END_REF]. We evaluate the quality of the £ltered speech signals by means A. Amehraye is with the GSCM-Faculté des Science de Rabat, Rabat, Morocco and also with Ecole Nationale Supérieure des Télécommunications de Bretagne asmaa.amehraye@enst-bretagne.fr D. Pastor is with Ecole Nationale Supérieure des Télécommunications de Bretagne, Technopôle de Brest-Iroise, CS 83818, 28238 Brest, France dominique.pastor@enst-bretagne.fr S. Ben Jebara is with Research unit TECHTRA , Ecole Supérieure des Communications de Tunis, Route de Raoued 3.5 Km, Cité El Ghazala, 2083, Ariana, Tunisia, sofia.benjebara@supcom.rnu.tn of two objective criteria, namely the standard Segmental Signal to Noise Ratio (SSNR) (see [START_REF] Quackenbush | Objective Measures of Speech Quality[END_REF]) and the Modi£ed Bark Spectral Distortion (MBSD) (see [START_REF] Yang | Modi£ed bark spectral distortion measure which uses noise masking threshold[END_REF]). The latter is a perceptually motivated metric that employs the concept of masking threshold in order to introduce no bias due to possible inaudible distortions. It proves to be highly correlated with subjective speech quality assessment [START_REF] Yang | Modi£ed bark spectral distortion measure which uses noise masking threshold[END_REF].

Section II presents the perceptual £ltering we consider. In section III, we describe the theoretical tools, the noise standard deviation estimator derived from these results and the application of this estimator to speech processing. The performance of the speech enhancement method that combines this estimator with the perceptual £ltering of section II is addressed in section IV. Concluding remarks and perspectives are given in section V.

II. PERCEPTUAL SPEEH ENHANCEMENT

Let s(n) some speech signal corrupted by additive and independent stationary noise x(n). The observed signal is

y(n) = s(n) + x(n). (1)
Given a frame of N samples y(n), n = 1, . . . , N , let Y k , S k and X k , k = 0, . . . , N -1, the Discrete Fourier Transform (DFT) coef£cients of y(n), s(n) and x(n) respectively. Generally, speech enhancement techniques consist in estimating the frequency components S k by

Ŝk = H k Y k , k = 0, . . . , N -1, (2)
where H k is a linear estimator chosen according to a suitable criterion. The error signal generated by this estimator is

e k = Ŝk -S k = (H k -1)S k + H k X k . (3)
The values (H k -1)S k , k = 0, . . . , N -1, are the DFT coef£cients of the speech distorsion due to the £ltering and the frequency components H k X k , k = 0, . . . , N -1, are the DFT coef£cients of the residual noise. The quantity

|H k | 2 E[|X k | 2
] is then the residual noise spectral power for the kth frequency component.

In order to take into account the properties of the auditory system, the £lter H k can be designed so as to yield inaudible residual noise. This can be achieved by forcing the residual noise spectral power to be below the masking threshold in each frequency bin; indeed, the human auditory system does not perceive any noise with spectral power less than the masking threshold. This is the approach followed in [START_REF] Lee | Speech enhancement by perceptual £lter with sequential noise parameter estimation[END_REF]. The £lter is then constrained by the inequality |H k | ≤ T k γ k , k = 0, . . . , N -1, where T k and γ k are the value of the masking threshold and that of the noise power spectral density in the kth bin. We choose real values for H k such that 0 ≤ H k ≤ 1. This inequality is the same as that satis£ed by the standard Wiener £lter and guarantees that the spectral power of the residual noise is less than or equal to that of the input noise. The expression of the perceptual £lter is then

H k = min( T k γ k , 1) (4)
This solution can also be derived by considering the £ltering problem addressed in this section as a constrained optimization problem [START_REF] Hu | Incorporating a psychoacoustic model in frequency domain speech enhancement[END_REF]. The £lter de£ned by ( 4) reduces only the noise frequency components that are above the masking threshold because the others are not audible and can even mask residual noise. In practice, the two terms of the ratio in the right hand side of ( 4) must be estimated and the performance of the resulting £ltering strongly depends on the accuracy of these estimates.

The masking threshold can be estimated on the basis of the outcome of a spectral substraction ( [START_REF] Hu | Incorporating a psychoacoustic model in frequency domain speech enhancement[END_REF], [START_REF] Virag | Singlehannel speech enhancement based on masking properties of the human auditory system[END_REF]). This solution reduces the amount of additive noise but introduces musical noise. The tone-like nature of musical noise increases the energy per critical band. This can induce an overestimation of the masking threshold.

To avoid such a drawback, we estimate the speech signals by Wiener £ltering. This method introduces less musical noise in comparaison with spectral subtraction methods [START_REF] Ephraim | Speech enhancement using a minimum mean square error short-time spectral amplitude estimator[END_REF]. The Wiener £ltering version that we use is based on the a priori SNR ξ k (m) estimated by [START_REF] Ephraim | Speech enhancement using a minimum mean square error short-time spectral amplitude estimator[END_REF]

ξ k (m) = (1-α)h (χ k (m) -1)+α | S k (m -1)| 2 | X k (m -1)| 2 , 0 ≤ α < 1.
In this equation, | S k (m -1)| and | X k (m -1)| are the amplitude estimates of the kth spectral components of the clean speech signal and noise in the (m-1)th analysis frame; h(x) = x if x ≥ 0 and h(x) = 0 otherwise; α is some weighting factor, we chose α = 0.98 for our experiments;

χ k (m) = |Y k (m)| 2 /| X k (m)| 2 is the a posteriori SNR where |Y k (m)
| is the amplitude of the kth spectral component of the noisy speech signal in the mth analysis frame. The Wiener £lter is then

W = ξ k (m) 1 + ξ k (m) , k = 0, . . . , N -1. (5)
The estimate performed by the Wiener £ltering thus de£ned is then employed to calculate the masking threshold.

In order to estimate the input noise spectral density, standard solutions are based on the use of a VAD. Frames detected by this VAD as noise alone serve to estimate the values γ k , k = 0, . . . , N -1. The accuracy of the estimates depends on the performance of the VAD. When speech signals are corrupted by independent and AWGN with noise standard deviation σ 0 , we have γ k = σ 2 0 . In this case, what follows explains how to get an estimate of σ 0 without using any VAD.

III. ESTIMATION OF THE NOISE STANDARD DEVIATION A. Theoretical results

The random variables encountered below are assumed to be de£ned on the same probability space and we write (a-s) for almost surely. Given a positive real value σ 0 , a sequence X = (X k ) k∈N of random complex variables is said to be a complex white Gaussian noise (CWGN) with standard deviation σ 0 the random variables X k , k = 1, 2, . . ., are complex, mutually independent and identically Gaussian distributed with null mean and variance σ 2 0 (X k ∼ N c (0, σ 2 0 )). The minimum amplitude a(S) of a sequence S = (S k ) k∈N of random complex variables is de£ned by

a(S) = sup {α ∈ [0, ∞] : ∀k ∈ N, |S k | ≥ α (a-s)} . (6)
If f is some map of the set of all the sequences of complex random variables into R, we say that the limit of f is ℓ ∈ R when a(S) tends to ∞ and write that lim a(S)→∞ f (S) = ℓ if, for any positive real value η, there exists some α 0 ∈ (0, ∞) such that, for every α ≥ α 0 and every S such that a(S) ≥ α, |f (S) -ℓ| ≤ η.

Let L 2 C (Ω) stand for the set of those complex random variables

Y such that E[|Y | 2 ] < ∞. We denote by ℓ ∞ (N, L 2 C (Ω)) the set of those sequences S = (S k ) k∈N of complex random variables such that S k ∈ L 2 C (Ω) for every k ∈ N and sup k∈N E[|S k | 2 ] is £nite.
The following result is a corollary of the limit theorem established in [START_REF] Pastor | A limit theorem for sequences of independent random vectors with unknown distributions and its applications to non-parametric detection[END_REF] for n-dimensional real random vectors. Given any random vector Y any real number , I(|Y | ≤ τ ) stands for the indicator function of the event {|Y | ≤ τ }.

Proposition 3.1: Let Y = (Y k ) k∈N be some sequence of complex random variables such that, for every k ∈ N,

Y k = ε k S k + X k where S ∈ ℓ ∞ (N, L 2 C (Ω)), X = (X k ) k∈N is
∆ m (σ, T ) = m k=1 |Y k |I(|Y k | ≤ σT ) m k=1 I(|Y k | ≤ σT ) -2σ T 0 u 2 e -u 2 du 1 -e -T 2 .
Then, σ 0 is the unique positive real number σ such that, for every

β 0 ∈ (0, 1], lim a(Λ)→∞ lim m ∆ m (σ, βg(a(Λ)/σ)) ∞ = 0 (7)
uniformly in β ∈ [β 0 , 1] where, for every x ∈ R, g(x) = I -1 0 (e x 2 )/2x with g(0) = 1 and I 0 is the standard zerothorder modi£ed Bessel function of the £rst kind.

In this statement, Y models a sequence of observations; for every given k ∈ N, S k stands for some possible random signal, ε k is the possible occurrence of S k and the complex noise is modelled by X. The assumption that S ∈ ℓ ∞ (N, L 2 C (Ω)) corresponds to the practical case of interest where the energies of the signals are £nite and bounded.

B. The algorithm

With the same notations as above, suppose that we have m observations Y 1 , . . . , Y m . Let L ∈ N and set β ℓ = ℓ/L for every ℓ ∈ {1, . . . , L}. The result stated above suggests estimating σ 0 by a possibly local minimum of sup ℓ∈{1,...,L} ∆ m (σ, β ℓ g(a(S)/σ)).

According to proposition 3.1, the larger the minimum amplitude, the better the estimate. However, good results can be obtained even when the minimum amplitude of the signals is not large [START_REF] Pastor | Un théorÁ eme limite et un test pour la détection non paramétrique de signaux dans un bruit blanc gaussien de variance inconnue[END_REF]. Therefore, we propose to perform an estimate of the noise standard deviation by taking a(S) = 0, a trivial bound for any signal norm. By so proceeding, we discard any assumption about the probability distributions of the signals and simply that these signals less present than absent. Since g(0) = 1, we compute a minimum σ0 of

sup ℓ=1,L            m k=1 |Y k |I(|Y k | ≤ β ℓ σ) m k=1 I(|Y k | ≤ β ℓ σ) -2σ β ℓ 0 u 2 e -u 2 du 1 -e -β 2 ℓ            (8)
by means of a minimization routine for scalar bounded non-linear functions. For instance, experimental results presented in the next section were obtained with the MATLAB routine fminbnd.m. The search interval [σ min , σ max ] is constructed as follows. Sort the complex values Y 1 , . . . , Y m , k = 1, . . . , m, by increasing modulus. Let Y [k] , k = 1, . . . , m, be the resulting sequence. The right endpoint of the search interval is

σ max = |Y [m] |/ √ 2. Now, choose a real number Q close to 1 but less than or equal to 1- m 4(m/2-1) 2 . A typical choice is Q = 0.95, provided that m ≥ 24. Set h = 1/ 4m(1 -Q) and k min = m/2-hm. The left endpoint is σ min = |Y [kmin] |/ √ 2.
The reader is asked to refer to [START_REF] Pastor | A limit theorem for sequences of independent random vectors with unknown distributions and its applications to non-parametric detection[END_REF] and [START_REF] Pastor | Un théorÁ eme limite et un test pour la détection non paramétrique de signaux dans un bruit blanc gaussien de variance inconnue[END_REF] for justi£cations of this construction.

Preliminary tests of the same type as those described in [START_REF] Pastor | Un théorÁ eme limite et un test pour la détection non paramétrique de signaux dans un bruit blanc gaussien de variance inconnue[END_REF] show that σ0 tends to overestimate the value of σ 0 and suggest to estimate this noise standard deviation by

σ 0 = m k=1 |Y k | 2 I(|Y k | ≤ σ0 )/ m k=1 I(|Y k | ≤ σ0 ). (9)
We work on theoretical justi£cations of this £nal estimate.

According to [START_REF] Pastor | A limit theorem for sequences of independent random vectors with unknown distributions and its applications to non-parametric detection[END_REF] and [START_REF] Pastor | Un théorÁ eme limite et un test pour la détection non paramétrique de signaux dans un bruit blanc gaussien de variance inconnue[END_REF] and as a good trade-off between computational load and accuracy, it is recommended to use the algorithm proposed above on a few hundred observations and to choose L = m. For instance, the experimental results of section IV were obtained on the basis of observation sets with 200 hundred samples each.

C. Application to speech processing

Consider K samples of speech signals corrupted by independent AWGN with standard deviation σ 0 . Let F s stand for the sampling frequency. Split this set of observations into M disjoint frames of N = 2 p samples each. We have K = M N . In practice, p will be chosen such that N F s ≈ 20ms. Apply an N -DFT on each frame. We obtain a matrix [U i,j ] i∈{1,...,M },j∈{0,...,N -1} of complex values where i is the frame index and j the DFT bin number. Because of the Hermitian symmetry of the DFT, we restrict our attention to the values U i,j , i ∈ {1, . . . , M }, j ∈ {0, . . . , N/2 -1}.

For each frame i and each bin j, we assume the random presence of a speech frequency component S i,j and that the probability of presence of S i,j is less than or equal to one half. This probability of presence may be larger than one half for low frequency components; however, for high frequency components, this probability of presence becomes less than or equal to one half and even relatively small. We thus assume that U i,j = ε i,j S i,j + X i,j . As above, ε i,j ∈ {0, 1} indicates whether the speech frequency component S i,j is present or absent in the jth bin of the ith frame. Since noise is white and Gaussian with standard deviation σ 0 , the complex random variables X i,j are mutually independent and identically distributed with X i,j ∼ N c (0, N C 2 σ 2 0 ) when the DFT coef£cients U k , k = 0, . . . , N -1, of a sequence of N samples u 0 , . . . , u N -1 are U k = C N -1 n=0 u n e -i2πnk/N . Instead of performing an estimate of N C 2 σ 2 0 on the basis of the M N/2 = K/2 values we have, we follow the recommendation of the previous section and split our set of observations into subsets of m = 200 observations each. Each subset is used to perform an estimate of N C 2 σ 2 0 . We then compute the average value of the M N/2m estimates thus obtained. It then suf£ces to divide this average by C √ N to get an estimate of σ 0 . In order to deal with m observations that can reasonably be considered as mutually independent, these observations can be chosen randomly amongst the M N/2 values we have. However, this randomization does not affect signi£cantly the results presented below.

IV. PERFORMANCE EVALUATION

The perceptual £ltering associated with the estimator proposed in section III was tested as follows. We considered speech signals from the TIMIT database. These speech signals are recorded at 16 kHz and we downsampled them to 8 kHz before adding white Gaussian noise.

To estimate the noise standard deviation, we proceeded as described above with frames of N = 256 samples each. This estimate serves to estimate the masking threshold and tune the perceptual £ltering. Both the estimation of the masking threshold and the perceptual £ltering are achieved on the basis of 32ms-duration frames (256 samples per frame) with a 50% overlap. The computation of the masking threshold is based on 18 critical bands.

We computed the SSNR and the MBSD of the proposed method. The SSNR and the MBSD are objective criteria for measuring the performance of a speech enhancement system. The SSNR is the average of the SNR values on short segments [START_REF] Quackenbush | Objective Measures of Speech Quality[END_REF]. However, the SSNR is not relevant enough to measure the distortion of denoised speech signals and to assess the quality of perceptually motivated speech enhancement approaches. Such approaches purposely keep noise components that are inaudible because suppressing them could introduce unpleasant speech distortion. Many other objective measures have been developped. They correlate well with subjective measures of speech quality. Amongst them, the MBSD [START_REF] Yang | Modi£ed bark spectral distortion measure which uses noise masking threshold[END_REF] is an improved version of the Bark Spectral Distortion (BSD) [START_REF] Wang | An objective measure for predicting subjective quality of speech coders[END_REF]. It extends BSD by incorporating masking threshold so as to differentiate audible distortions from inaudible ones. The MBSD proves to be even more correlated with speech quality than the BSD [START_REF] Yang | Modi£ed bark spectral distortion measure which uses noise masking threshold[END_REF].

We computed the SSNR and the MBSD when the exact value of the noise standard deviation is used and the masking threshold is computed on the basis of the clean speech signals (before adding noise). The SSNR and the MBSD thus obtained are hereafter called the "Theoretical limits".

The SSNR and the MBSD were also measured for the perceptual £lter of section II when the noise standard deviation is estimated on the basis of signal-free time frames provided by an ideal DAV ("perceptual £lter + DAV").

Consider the Wiener £ltering based, as in section II, on the recursive computation of the a priori SNR. We computed the SSNR and the MBSD when the estimate of the noise standard deviation is achieved via the estimator of section III ("Wiener + Noise Estimator"). We also calculated the SSNR and the MBSD when this noise standard deviation is estimated on the basis of signal-free periods of time ("Wiener + DAV").

Fig. 1 presents the performance measurements obtained. The proposed method yields performance signi£cantly close to the theoretical limits at various input SNRs. It outperforms the Wiener £ltering. Surprisingly enough, it also performs better than the perceptual £ltering tuned by the noise standard deviation estimate derived from an ideal DAV. We think that this relates to the fact that the number of observations used by our estimator is signi£cantly larger than that available on the basis of signal-free time frames. Note also that the SNR improvement achieved by the proposed method is particularly signi£cant for low input SNRs.

V. CONCLUSION AND FUTURE WORK

By combining a perceptually motivated approach for speech enhancement with a new estimator of the noise standard deviation, we denoise speech signals corrupted by independent AWGN without using any VAD. The proposed method reduces residual noise, limits speech distortions and outperforms Wiener £ltering. Above all, its performance measurements are very promising because they signi£cantly approach those obtained by using the exact value of the noise standard deviation and computing the masking threshold on the basis of clean speech signals. Forthcoming work will involve further comparison to other techniques such as Ephraim and Malah's [START_REF] Ephraim | Speech enhancement using a minimum mean square error short-time spectral amplitude estimator[END_REF] as well extension of the approach to non-white Gaussian noise.

  some CWGN with standard deviation σ 0 and ε = (ε k ) k∈N is a sequence of random variables valued in {0, 1} respectively. Assume that (A1) for every k ∈ N, S k , X k and ε k are mutually independent; (A2) the random variables Y k , k ∈ N, are mutually independent; (A3) the random variables ε k , k ∈ N, are mutually independent; (A4) the priors P ({ε k = 1}), k ∈ N, are less than or equal to one half. Given any natural number m and any pair (σ, T ) of positive real numbers, de£ne the random variable ∆ m (σ, T ) by

Fig. 1 .

 1 Fig. 1. MBSD and SSNR improvement for speech signals in independent AWGN with various SNRs.