Violaine Piton
email: violaine.piton@legos.obs-mip.fr

Sylvain Ouillon

Duy Vu

Gael Vinh

Marine Many

Patrick Herrmann

Marsaleix

Vinh

Gaël Many

Marine Herrmann

Patrick Marsaleix

Seasonal and tidal variability of the hydrology and suspended particulate matter in the Van Uc estuary, Red River, Vietnam

Keywords: suspended sediments, estuary, estuarine turbidity maximum, neap/spring cycle, Red River, Vietnam

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Estuaries are links between land and ocean where freshwater coming from the rivers mixes with saline oceanic water [START_REF] Wolanski | Estuarine Hydrology[END_REF]. They are typically characterized by strong gradients of salinity, temperature, water surface level and suspended sediment concentrations (Priya et al., effort each year, with 6.6 million US $ spent on dredging activities in 2013 (Vietnam maritime administration 2017). Recent studies have arisen from this issue, and the tidal pumping has been pointed out as one of the physical processes responsible for the siltation, by creating a net upstream transport of sediments in the dry season, which settled in a near bed layer at low turbulent energy [START_REF] Mari | Seasonal aggregation dynamics along a salinity gradient in the Bach-Dang estuary, North Vietnam[END_REF]. In the Cam-Nam Trieu estuary, the Estuarine Turbidity Maxima (ETM, mass of highly concentrated suspended sediments) was shown to start developing at salinities as low as 0.1 PSU [START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF] and bio-aggregation was shown to be enhanced at salinity around 10-15 PSU due to increasing stickiness properties of bio-exopolymeric substances [START_REF] Mari | Seasonal aggregation dynamics along a salinity gradient in the Bach-Dang estuary, North Vietnam[END_REF]. In the coastal zone, the decrease of the suspended sediment loads caused alternative patterns of erosion and accretion to develop. For example, the Ba Lat River mouth (Fig. 1) has expended seaward due to a combined effect of sediment resuspension by wave actions and river floods [START_REF] Van Maren | Water and sediment dynamics in the Red River mouth and adjacent coastal zone[END_REF], and increased turbidity was observed in the touristic region of Do Son, while Hai Hau coasts have experienced a severe erosion of several meters per year. In this context, investigating the fine scale physical processes in estuaries and in their regions of influence is of particular interest for better understanding the Red River system.

The Van Uc River (Fig. 1) is a typical distributary of the Red River, with rice crops all along the banks that require water and construction of soil dykes for flooding protection, and with constant river bed dredging for both navigable waterways and selling purposes. It is ranked at the third position of the Red River system in terms of both water and sediment discharges. [START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF]. Above all, the Van Uc estuary is an interesting site to study estuarine dynamics because this meso-to macrotidal estuary (maximum tidal range: 4m) is both influenced by a strong seasonal river signal and a monsoon regime. Using the MIKE11 numerical model, [START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF] estimated that 70 % of the annual water discharges were issued in the wet season (June-October) while 16 % were issued in the dry season (December-April), the 14 % remaining being issued in intermediate periods (May and November). For the sediments discharges, 86 % of the total annual discharges were issued in the wet season, against 4 % in the dry season (10% were issued in the intermediate seasons). If seasonal budgets of water and sediments are globally known, forced by the river inputs [START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF], if seasonal variations of hydrology (water flux, salinity, stratification) and suspended matter was described at one station in the Ba Lat river mouth (Fig. 1) (van Maren, 2004) and if transects along the estuaries of the Red River distributaries were described at different seasons at spring tides [START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF], no effort has been initiated so far to document the hydrology and the suspended matter dynamics of Red River estuaries in terms of tidal variability. This question represents the central objective of the present study, which aims at documenting the lack of knowledge regarding the hydro-sedimentary processes in the Red River.

Based on new in situ data, this paper focuses on the analysis of the hydrology and on the characteristics of the suspended particulate matter at different seasons and tidal cycles, along the Van Uc estuary. It is the first step of a more comprehensive modeling study that aims at further analyze the transport and fate of sediments from the Red River to the Gulf of Tonkin, using the hydrodynamical-sediment transport model SYMPHONIE-MUSTANG [START_REF] Marsaleix | Energy conservation issues in sigma-coordinate free-surface ocean models[END_REF][START_REF] Hir | Dynamics of sand and mud mixtures : A multiprocess-based modelling strategy[END_REF] that is configured for the region. Hence, the choice of the parameters studied was made in a modelling perspective.

In Section 2, the estuary of interest is presented and in Section 3, field campaigns and methodologies for data sampling, instruments calibrations and data analyses are described. In Section 4, findings on water and sediment fluxes, water column stratification and particles properties, which are largely impacted by the lunar tidal cycle and related circulation are detailed. The dynamical mechanisms that can explain these results are discussed in Section 5: in particular the distinct effects of gravitational circulation and of the along-estuary tidal distortion on sediment fluxes and deposition.

Regional settings -The Van Uc estuary

The Van Uc estuary is located next to Hai Phong city (the third largest city of Vietnam), northeast of Vietnam (Fig. 1). The estuary is fed with fresh water and sediments by the Van Uc river, one of the nine distributaries of the Red River, which receives water and sediments from both the Red River, through the Duong River, and from the Thai Binh River through a complex network of connections within the Red River Delta (see [START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF]. The total river discharge through the Van Uc estuary was estimated of about 17.7 x 10 9 m 3 y -1 (for the period 1989-2010), which corresponds to 14.5 % of the total water discharge from the Red-Thai Binh system to the Gulf of Tonkin [START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF]. These authors also evaluated the sediment fluxes and found a drastic drop in the annual fluxes due to the Hoa Binh dam impoundment in the 1980s (from 11.5 to 5.1 Mt y -1 ). The Van Uc sediment flux represents 14.4 % of the total sediment flux of the Red-Thai Binh River to the Red River coastal area and is therefore the third most important distributary of the Red River Delta in terms of river and sediment discharges (after the Day and the Ba Lat rivers).

The Van Uc estuary is subjected to the South-East Asian sub-tropical monsoon climate [START_REF] Wyrtki | Physical oceanography of the Southeast Asian waters[END_REF], that is characterized by an alternation of a southwest summer monsoon and a northeast winter monsoon. The wet summer season (June to September) accounts for 85-95% of the total annual rainfall in the region (~1161 mm) and is dominated at 72% by winds blowing from the South-West. In the dry winter season, the winds blow at 92% from the North-East and bring cold and dry conditions over the region [START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF]. The seasonal variation of the Red River hydrological regime is therefore strong and largely influences the Van Uc estuary. [START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF] showed that 71-79% of the annual total water discharge of the Van Uc River was brought to the coastal zone during the wet season and only 9-18% in the dry season (December to March).

The South China Sea and the Gulf of Tonkin are one of the few regions in the world where diurnal tides dominate the semidiurnal tides (Fang, 1986). [START_REF] Wyrtki | Physical oceanography of the Southeast Asian waters[END_REF] was the first to identify the main tidal constituents in the South China Sea: O1, K1, M2, S2 (classified by their importance in amplitudes). The diurnal tide dominates due to a double resonance : in the South China Sea because of Helmholtz resonance [START_REF] Zu | Numerical study of the tide and tidal dynamics in the South China Sea[END_REF], then in the Gulf of Tonkin, higher for O1 than K1 due to a resonant peak period of 29h (Nguyen et al., 2014). The tide gauge measurements next to the Van Uc River mouth (at Hon Dau, Fig. 1) show a tidal amplitude of about 0.5-1.0 m in neap tides and of about 2.6-3.6 m in spring tides.

Material & Methods

Field data and sampling procedure

Four field surveys were performed in the Van Uc estuary during neap and spring tides in the wet and dry seasons. Surveys are noted WN for wet season at neap tide, WS for wet season at spring tide, DN and DS for surveys in the dry season at neap tide and spring tide, respectively.

The dates of each survey with corresponding tidal cycles and seasons are listed in Table 1.

Three stations (ST1, ST2 and ST3) spaced by 10 km were selected for sampling (Fig. 1). ST1 is located next to the river mouth while ST3 is the further upstream. Each station was sampled during 24 hours with cross-river transects measurements every two hours to cover a full diurnal cycle, starting at ST1 and finishing at ST3. Each series of measurements consisted of four different steps. Firstly, cross river velocity profiles were measured with a hull-mounted, downward-looking, 1200 kHz Acoustic Doppler Current Profiler (ADCP RDI Workhorse in bottom tracking mode). Current magnitude and direction were measured in depth cells of 0.25 m with a range of 15 m using a frequency of 0.2 Hz. Secondly, vertical profiles of conductivity and temperature were measured for each transect, in the deep channel at the middle of the river, using a profiling CTD Compact-CTD (ASTD687, Alec Electronics Co., Nishinomiya, Japan, now released by JFE Advantech Co., Nishinomiya, Japan, as Rinko-Profiler). Thirdly and parallel to CTD measurements, depth profiles of floc size distribution and concentration were measured using an in situ laser scattering and transmissometry instrument with a 90% path reduction module (LISST-100X, type B, Sequoia Scientific Inc., Bellevue, WA, USA). The LISST-100X determines the volumetric particulate concentration in 32 logarithmically spaced size classes from 1.25 to 250 µm and uses a red laser (660 nm) through a path of 5 cm with a sampling rate of 1 Hz. In order to study the relationship between turbidity and suspended particulate matter concentration, water samples (of ~1 l) near subsurface (at ~1.5 m depth) were also collected every four hours (i.e. every two CTD deployments) with a Niskin bottle. Turbidity of these samples were measured onboard using a Hach 2100Q Turbidimeter.

Data at the hydrographic station Trung Trang (Fig. 1), provided by the National Hydro-Meteorological Service (NHMS), were also used in this study. They include hourly water discharges and water elevations. Table 1 : Van Uc surveys characteristics and dates of sampling. Each sampling started at 8 am and finished at 6 am the following day.

Hydrological parameters

Salinity and density were derived from CTD data (temperature and conductivity) following general UNESCO polynomial relationship (UNESCO, 1983).

The Brunt-Väisälä frequency N² (in s -2 ), also known as the buoyancy frequency, was computed from the density depth-profiles and used as an indicator of the water column stratification:

(1) ² N = ρ -g δz δρ where g is the acceleration of gravity (m 2 s -1 ) and ρ the potential water density (kg m -3 ). A stable stratification is considered when N 2 > 0, while a stratification is considered unstable when N 2 ~0 [START_REF] Turner | Buoyancy effects in fluids[END_REF]. (

2) i R = N ² ( )² δz δV
When Ri ≥ 0.25, the flow is considered stable and if Ri < 0.25, the turbulence overcomes the density stratification and generates vertical mixing [START_REF] Miranda | Princípios de oceanografia física de estuários[END_REF]Kirinus et al., 2002 ).

Furthermore, the Simpson parameter gives the potential energy (in J m -3 ) that represents the ϕ mechanical energy required to completely mix the water column. This parameter was used to quantitatively analyze the physical mechanisms contributing to mixing and stratification in an estuary [START_REF] Simpson | Fronts on the continental shelf[END_REF] and was computed as follows:

(3)

z z ϕ = 1 H ∫ 0 -H g (ρ ) -ρ δ
with H, the water column depth.

Water currents and derived water discharges

The ADCP transmits high frequency acoustic signals (pings), that are backscattered from water constituents of different origins: suspended sediment, phyto/zooplankton and bubbles.

These constituents are assumed to travel with the speed of water and therefore, the current velocity (horizontal and vertical) and the current direction as a function of depth are estimated through the Doppler effect. Following the recommendations of [START_REF] Mueller | Measuring discharge with the Acoustic Doppler Current Profilers from a moving boat (ver 2.0[END_REF], water discharges (noted Q Measured ) were computed using the measured water velocities (noted V w ) and the boat velocities (noted V b ):

(4.a) tdz Q Bin = (V V V ) wu bv -V wv bu d (4.b) Q M easured = ∑ Ensembles j=1 ∑ Bins i=1 Q Bin
where V wu and V wv correspond to the meridional and zonal water velocities, respectively, and V bu and V bv correspond to the meridional and zonal boat velocities, respectively. Q Bin (in m 3 s -1 ) was computed for each ensemble (i.e. time between two measurements) dt (5 s) and for each vertical bin dz (0.25 m). As recommended by [START_REF] Mueller | Measuring discharge with the Acoustic Doppler Current Profilers from a moving boat (ver 2.0[END_REF], the total water discharge Q ADCP of the river is the sum of Q Measured and the water discharges estimated in the unmeasured zones by the ADCP (i.e. the top and bottom layers and the right and left edges of the river) as:

(5)

Q ADCP = Q Lef tEdge + Q T op + Q M easured + Q Bottom + Q RightEdge
Detailed descriptions and formulations of Q LeftEdge , Q RightEdge , Q Top and Q Bottom are available in [START_REF] Mueller | Measuring discharge with the Acoustic Doppler Current Profilers from a moving boat (ver 2.0[END_REF] and were computed accordingly.

Furthermore, the flux throughout the water column can flow in opposite directions (downstream or upstream), whether it is dominated by the river flow or by flood tides. Therefore, we chose to compute Q ADCP + as the component of the flux flowing downstream and Q ADCP -as the component of the flux flowing upstream. Note that the sum of Q ADCP + and Q ADCP -is equal to Q ADCP .

3.4 Suspended particulate matter parameters 3.4.1 SPM concentration from water samples Following the process proposed by [START_REF] Aminot | Hydrologie des écosystèmes marins: paramètres et analyses[END_REF], the suspended particulate matter (SPM) concentrations were determined by filtering onboard a known volume of water (about 100-150 ml) per sample through two kinds of pre-weighed filters: polycarbonate Nuclepore filters (porosity 0.45 µm) and Whatman GF/F glass microfiber filters (porosity 0.7 µm). After filtration, filters were rinsed three times with 5.0 mL of milli-Q water to remove salt.

Then, they were dried for 24 hours at 70°C in an oven and weighed on a high-precision electrobalance. The sediment concentration obtained after drying the filters is the total SPM concentration, called SPM Filters hereafter, including organic and inorganic matter.

The particulate inorganic matter concentration was measured after burning the GF/F filter at 450 °C during 4 hours: this process removes all the organic matter from the filter. The difference between the total SPM Filters and particulate inorganic matter provided the particulate organic matter concentration and thus the ratio of organic to inorganic matter (particulate organic matter/particulate inorganic matter) within the SPM Filters .

Optical backscattering sensors (OBS)

Turbidity is an optical proxy of SPM concentration which is measured by nephelometers or optical backscattering sensors (OBS). The CTD was equipped with an OBS measuring in the infrared. The sensor outputs (in Formazin Turbidity Unit -FTU) were calibrated against gravimetric SPM concentrations measurements. The SPM concentrations measured by the CTD are called hereafter SPM OBS .

ADCP's backscatter index

For the past twenty years, acoustic instruments such as ADCP, originally designed to measure velocity profiles, have been used to quantify SPM concentrations in ocean and coastal environments (Thorne and Hanes, 2002;Tessier et al., 2008;Le Coz et al., 2009). The measurement principle is based on the fact that the acoustic signal is emitted with an initial intensity, that progressively decreases as it travels through the water column, and which is backscattered by suspended particles. Therefore, the intensity of the backscattered acoustic signal is a function of the quantity of the particles present in suspension in the environment. The acoustic method has the advantage of being non-intrusive.

In order to infer the particle load from the signal recorded by the ADCP, the sonar equation (in dB) was used. The sonar equation represents an energy balance that expresses a signal-to-noise ratio, taking into account the technical characteristics of the instrument. The received acoustic level (RL), can be expressed as follows [START_REF] Deines | Backscatter estimation using broadband ADCP[END_REF]Lurton, 2002):

(6) I RL 2T L SL B = + -
with SL, the source level emitted by the ADCP (in dB), TL the transmission loss due to attenuation of the acoustic wave within the ambient environment (in dB), and BI the backscatter index, which depends on the SPM concentration.

In our case and following the recommendation of [START_REF] Mullison | Backscatter estimation using broadband Acoustic Doppler Current Profilers -Updated[END_REF], the RL was computed as follows:

(7) L 10log (10

) Br R = Kc (E-Er)/10 -1 +
where Kc (in dB/counts) is the factor used to convert the amplitude reported by the ADCP, in counts, to decibels. E the backscattered intensity of the acoustic wave, which was measured by the instrument (in counts), Er the backscattered intensity signal, which is seen by the instrument in the absence of any signal (the noise) and Br, the associated pressure level furnished by the manufacturer (dB). In our case, since we will be looking at relative SPM concentrations, neglecting Br will not affect our results.

Two types of transmission losses can be estimated. One is associated to the spherical spreading, which corresponds to the decrease of the signal intensity as the signal travels away from the emission source. The other type of loss corresponds to the acoustic signal attenuation due to the ambient environment, i.e. the attenuation due to the water (𝝰 w ) and the attenuation due to the suspended particles characteristics (𝝰 s ). In the presence of particles, the total transmission loss, with a distance R from the emission source and the beam angle (20°), is expressed as follows: ϕ

(8) L 20log ( ) r T = 10 R cos(ϕ) + ∫ R 0 (α (r) (r) ) w + α s d
The attenuation coefficient 𝝰 w (dB m -1 ) was estimated from the model of [START_REF] François | Sound absorption based on ocean measurements: Part I: Pure water and magnesium sulfate contributions[END_REF]Garrison (1982a, 1982b). According to [START_REF] Tessier | Caractérisation et dynamique des turbidités en zone côtière: l'exemple de la région marine Bretagne Sud[END_REF], the particle attenuation 𝝰 s becomes significant when the SPM concentration is of the order of 200 mg l -1 for a 1200 kHz acoustic signal and fine particles. The concentrations of SPM measured during the surveys (estimated from CTD OBS profiling) were rather low in the dry season, and never exceeded 170 mg l -1 in the wet season.

Therefore, the choice was made to neglect 𝝰 s in the calculation of the backscatter index (BI).

An accurate estimate of the acoustic source level (SL) emitted by the ADCP is difficult. It can either be obtained by laboratory calibrations of the instrument, or by using the average reference values provided by the manufacturer. Laboratory experiments showed that for a 1200 kHz ADCP, the values range from 216 to 218 dB [START_REF] Tessier | Caractérisation et dynamique des turbidités en zone côtière: l'exemple de la région marine Bretagne Sud[END_REF]. Furthermore, the SL can vary with the battery voltage. This effect could infer a bias in the measurement, when using the manufacturer reference. In our case, the ADCP was not calibrated prior to the survey, and therefore SL is neglected. Again, since we only look at relative SPM concentrations, neglecting SL does not alter our results.

Lastly, the best regression relationship between SPM concentrations measured by optical backscattering (SPM OBS ) and BI is found to be linear between 10 log 10 (SPM) and BI. It is used to estimate SPM concentrations (in mg l -1 ) from acoustic measurements (in dB), called SPM AC hereafter, following:

(9) 0log (SP M ) aBI b 1 10 AC = +
with a and b the slope and the intercept of the linear regression.

However, the uncertainties associated with the different terms of the sonar equation remain important, and the calibration of the acoustic signal is done in practice by adjusting the slope.

Sediment discharges from ADCP

For each ensemble dt (5 s) and vertical bin dz (0.25 m), the sediment discharge (noted Q BinSED ) was computed as the product of instantaneous Q ADCP and SPM AC as:

(10.a) Q P M )dtdz Q BinSED = ( ADCP × S AC
and the total sediment discharge is then :

(10.b) Q SED = ∑ Ensembles j=1 ∑ Bins i=1 Q BinSED
Furthermore, Q SED + was computed as the component of the sediment flux flowing downstream and Q SED -as the component of the sediment flux flowing upstream. Note that the sum of Q SED + and Q SED -is equal to Q SED .

Particles properties measurements from LISST

The laser diffraction spectrum, discretized over 32 classes, was first converted into a particulate volume concentration (in µl l -1 ) using the factory volume calibration constant. The sum of the contributions provided the SPM volume concentration (SPMVC). However, following the recommendation of [START_REF] Mikkelsen | In situ particle size distributions and volume concentrations from a LISST-100 laser particle sizer and a digital floc camera[END_REF], the extreme size classes #1, #2, #31 and #32 were removed before calculating the SPMVC, the particle size distribution (PSD) and the mean apparent diameter (D 50 ). Indeed, in stratified waters, water masses have different densities and therefore different refractive indexes which can cause significant light scattering.

This phenomenon is known and called the schlieren effect. The LISST probe detects the schlieren effect as an accumulation of large or small particles (i.e., the extreme classes of the spectrum) in the pycnocline [START_REF] Styles | Laboratory evaluation of the LISST in a stratified fluid[END_REF][START_REF] Mikkelsen | The influence of schlieren on in situ optical measurements used for particle characterization[END_REF]). In addition, the measurements for which peaks of particles in extreme classes occurred have been removed, as we found that schlieren effect happened to also contaminate classes as high as #4 and as low as #26. Based on data between classes #3 (1.74 µm) and #30 (179.53 µm), the particulate volume concentration was normalized by the width of each logarithmically-spaced class and gave the PSD (in µl l -1 µm). Then, D 50 (in µm) was calculated as the diameter corresponding to 50% of the cumulative volume concentration of particles.

Results

The effects of seasonal, tidal and spatial forcings on various water column parameters were investigated. Water and sediment fluxes are presented in the first part. The second part is dedicated to the hydrology and turbidity of the water column. Lastly, the suspended particulate matter variability is discussed. The best-fit regression relationship between SPM OBS and BI was used at each survey to derive SPM concentrations SPM AC from the acoustic signal (Fig. 2). The correlation coefficient R² reflects the different sensitivity of optical and acoustic sensors to changes in SPM size and characteristics/properties. Indeed, backscattering is influenced by sediment type, size, color and composition, with optical and acoustical backscattering showing different sensitivities and responses to these parameters [START_REF] Thorne | Measuring suspended sediment concentrations using acoustic backscatter devices[END_REF][START_REF] Fugate | Determining concentration and fall velocity of estuarine particle populations using ADV, OBS and LISST[END_REF][START_REF] Voulgaris | Temporal variability of hydrodynamics, sediment concentration and sediment settling velocity in a tidal creek[END_REF][START_REF] Downing | Twenty-five years with OBS sensors: The good, the bad, and the ugly[END_REF]. [START_REF] Gartner | The promises and pitfalls of estimating total suspended solids based on backscatter intensity from acoustic Doppler current profilers[END_REF] and [START_REF] Gartner | Estimation of suspended solids concentrations based on acoustic backscatter intensity: theoretical background, Turbidity and other sediment surrogates workshop[END_REF] showed that optical sensors are more sensitive to small particles while acoustic sensors are more sensitive to large particles. Furthermore, [START_REF] Kim | Estimation of suspended sediment concentration in estuarine environments using acoustic backscatter from ADCP[END_REF] found a slight bias in the SPM AC estimates when the sediment is silt or finer. [START_REF] Rouhnia | Do changes in the size mud flocs affect the acoustic backscatter values recorded by a Vector ADV?[END_REF] showed that the intensity of the backscattered acoustic signal from flocs is mainly affected by the flocculi (i.e the microflocs).

However, flocs can have a smaller influence on the backscatter signal than sand grains of the same size, due to their lower density [START_REF] Ha | Estimation of high-resolution sediment concentration profiles in bottom boundary layer using pulse-coherent acoustic doppler current profilers[END_REF]. Therefore, the SPM AC should be considered with caution and as estimates only, rather than exact SPM measurements.

Good correlations between SPM OBS and BI were found for surveys WN, WS and DS with R 2 of 0.83, 0.84 and 0.70, respectively (Fig. 2 a,b,c). Only survey DN showed a poor correlation (0.04) between the datasets (Fig. 2 d). Following the circulation and thermohaline conditions of the water column in survey DN (Fig. 3 and see section 4.2), the water column was, at some specific tidal stages, divided in two water masses flowing in opposite direction. The choice was made to compute different calibrations for the top and bottom layers, and the boundary between the two layers was defined by the reversal in current direction. A good correlation was found for the bottom layer (0.85) (Fig 2 f), but a poor correlation remained in the top layer (0.13) (Fig. 2 e). Therefore, for this particular case (top layer of the water column, survey DN), SPM AC could not be estimated. As a first guess and instead of using Eq. 10 b., the contribution of this layer to the sediment discharge Q SED was computed as the product of the mean SPM OBS with the mean Q ADCP of the top layer of the water column. Results of these estimates are presented in the following section. Seasonal variations in Q ADCP were high, with wet season water discharges (2425 m 3 s -1 in average over the survey WN and 1757 m 3 s -1 for survey WS, Table 2) 4 to 11 times larger than the discharges in dry season (478 and 222 m 3 s -1 for survey DS and DN, respectively). Suspended sediment discharges Q SED varied accordingly, with 3 to 170 times larger values in wet season than in the dry season (0.34 t s -1 in WN and 0.11 t s -1 in WS, as compared to 0.04 and 0.002 t s -1 in surveys DS and DN, respectively).

During the wet season, the total water and sediment discharges Q ADCP and Q SED were dominantly directed down-estuary (Q ADCP + and Q SED +) and only very limited fluxes directed up-estuary (Q ADCP -and Q SED -) were observed (Fig. 4 a,b ; Table 2). In neap tide, a mean Q ADCP + of 2432 m 3 s -1 was measured against a mean Q ADCP -of 7 m 3 s -1 , the associated mean Q SED + being of 0.34 t s -1 against a mean Q SED -of 10 -4 t s -1 . In spring tide, up-estuary fluxes were slightly higher but less than 4% of the total water and sediment discharges (Table 2).

As expected, the up-estuary component of the water discharge (Q ADCP -) was much higher in the dry season than in the wet season (Table 2). During the dry season, both down-estuary and up-estuary flows contributed to the total water and sediment fluxes. Globally, the down-estuary water and suspended sediment discharges were respectively 61 and 66% higher than the up-estuary water and suspended sediment flows in the dry season (Table 2). The range of Q ADCP was around 3 times higher in spring tide (from -3976 to 3178 m 3 s -1 ) than in neap tide (from -1252 to 1826 m 3 s -1 , Fig 4 . c,d). The difference was highly enhanced in the range of Q SED (from -0.55 to 0.45 t s -1 in spring tide, while they ranged from -0.015 to 0.022 t s -1 in neap tide, Fig 4.

c,d).

Fluxes were very sensitive to tidal variations in dry season, and in wet season during spring tide (Fig. 4).

In the dry season and in the wet season at spring tide, Q ADCP was influenced by the tidal diurnal cycle with high discharges at low tides, decreasing discharges at rising tides, low discharges at high tides and increasing discharges at falling tides (Fig. 4 b,c,d). Q SED followed the same patterns.

In This particular case resulted from the strong stratification at neap tide and dry season, with two distinct water masses as explained above in 4.1.1 : the upper flowing down-estuary, and the lower flowing up-estuary.

Q SED followed Q ADCP patterns with mean net down-estuary sediment fluxes Q SED 20 times larger in spring tide (0.04 t s -1 , Table 2) than in neap tide (0.002 t s -1 ) in dry season, and with negligible up-estuary sediment fluxes (Q SED -) in wet season lower than 4% of the down-estuary fluxes (Q SED +), both for spring and neap tides. However, the sediment sources are different: while Q SED + refers to the fluxes of fluvial suspended sediment, Q SED -refers to the upward transport of sediment from the coastal zone or resuspended in the estuary. A main information of the measurements is to quantify this very high up-estuary flow of coastal sediment Q SED -in dry season and spring tide (0.08 t s -1 , Table 2) which is 2/3 of the down-estuary discharge of suspended matter Q SED + from the river basin (0.12 t s -1 ). In other words, the estuary is silting up and the delta is consolidating mainly in spring tide during the dry season.

At last, the time variations of tidal amplitude and discharge at Trung Trang during spring tides (both in dry and wet seasons) clearly show that flood tides are shorter than ebb tides in the Van Uc estuary. Such a non linear tidal distortion may be explained either by a high tidal energy dissipation by friction (reflected as a/h= tidal amplitude/water depth) or by a low intertidal storage (measured by V s /V c =volume of intertidal storage/volume of channels at mean sea level) (Friedrichs and Aubrey, 1998;[START_REF] Kang | Flood and ebb dominance in estuaries in Korea[END_REF].

25 2). However, in the dry season, Q ADCP is 27% higher than the mean discharge at TT (Q TT ) in spring tide, and is 41% lower than Q TT in neap tide. Due to its location roughly 15 km upstream of ST3 (Fig. 1), TT station is less affected than our transect stations by the variation of discharges due to the tides, especially during the dry season, and therefore, discrepancies in measured discharges at our stations and discharges at TT may appear. Overall, time series of Q ADCP and Q TT were highly correlated in surveys WS, DS and DN (R²=0.80, 0.81 and 0.81 respectively) and only survey WN showed a weaker (but still significant at 99%) average correlation (R²=0.46) that may be explained by the small amplitudes of variations in Q ADCP in this season and tidal range. Better correlations were obtained with a lag between Q TT and Q ADCP in surveys WS, DN and DN. The best R² values (R²=0.97, 0.98 and 0.97 for survey WS, DS and DN, respectively) were found at each station and for each survey with a lag of two hours, which indicated that Q TT are leading Q ADCP by two hours all along the estuary, in average, as can be seen in Fig 4 . b,c,d.

Survey Q T T (m 3 s -1 ) (m 3 s -1 ) Q ADCP R² (Q ADCP / Q TT ) Q ADCP + Q ADCP - Q SED (t s -1 ) R² (Q ADCP / Q SED ) + Q SED - Q SED WN
Increases in Q ADCP of about 52%, 33% and 42 % were observed between ST3 and ST2 during surveys WN, WS and DS, respectively (Table 2). These increases are likely due to water inputs from the Luoc River which flows into the Van Uc River, approximately 2 km downstream of ST3 (see Figure 1). The 40% drop in water discharge that was observed during the survey DN between ST3 and ST2 coincided with decreasing water discharges measured at Trung Trang station (Q TT ) during the 3-day period of measurements (Table 2 ; Fig. 4 c), and can thus be explained by a decrease of discharge from the day of ST3 measurements to the day of ST2 measurements. Decreases in water and sediment discharges amplitudes were observed from the river mouth upwards (from ST1 to ST3) during surveys WS, DS and DN (Fig. 4 b,c,d). These decreases could be explained by the variations in the tidal elevation range along the estuary that impact the capability of currents to transport water and sediments. The tidal range increases towards the mouth before decreasing in the river section, when the convergence effects of the estuary sides exceeds the frictional effect of the bed [START_REF] Dyer | Chapter 14 Sediment transport processes in estuaries[END_REF].

Given the scarcity of water discharge data and of their tidal and seasonal variability in the Van Uc river, those values will provide a precious basis for assessing the role of the neap-spring cycle on estuarine dynamics, but will also provide a new benchmark for the calibration/validation of an estuarine hydrodynamic and sediment model. which illustrated an oscillation between fresh and unstratified to salty and stratified conditions. The PSDs alternated between bimodal and trimodal distributions from a survey to another, with varying contributions of each particle class to the total particle volume (Fig. 7 a).

As mentioned in the literature, PSDs of observed suspensions are more often bimodal or multi-modal rather than unimodal (Ludwig and Hane 1990;[START_REF] Green | The measurement of constituent concentrations in non-homogeneous sediment suspensions using optical backscatter sensors[END_REF].

In wet season, trimodal PSDs were measured with a first peak for particle diameters < 4 µm, a second peak for particles of diameter 5-10 µm, and a third peak for aggregates of larger diameters (of 30-70 µm for survey WN and of ~30 µm for survey WS). From hereafter, we consider the particles < 4 µm as primary particles (PP), the particles of 4-20 µm as flocculi, and the particles > 20 µm as microflocs. In the wet season, spring tides likely induced stronger turbulent stresses than neap tides (see Fig. 4b as compared to 4a) that may cause a breakage of the largest particles, hence a decrease in populations of microflocs > 30 µm in spring tides. This is consistent with the results from [START_REF] Mikkelsen | Single-grain, microfloc and macrofloc volume variations observed with a LISST-100 and a digital floc camera[END_REF], who demonstrated that when the water column stress increases, the volume occupied by large particles (in their case : macroflocs) decreases, while the volume occupied by finer particles (in their case : microflocs) increases. As seen in section 4.2.2 and Fig. 6 a,b, salinity maxima oscillated between 0.8 and 27 PSU during the dry season, and [START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF] showed that flocculation happened for salinity as low as 0.1 PSU in the Cam Nam estuary (Fig. 1). Besides, [START_REF] Eisma | Suspended Matter in the Aquatic Environment[END_REF] indicated that salinity flocculation is complete at 8 PSU or less, and [START_REF] Mari | Seasonal aggregation dynamics along a salinity gradient in the Bach-Dang estuary, North Vietnam[END_REF] demonstrated that the stickiness properties of organic exopolymeric substances increase between 10 and 15 PSU, enhancing the bio-induced flocculation in the Cam Nam estuary.

It is important to note that the high concentrations of PP observed in almost all surveys and which sometimes corresponded to 5-15 % of the SPMVC, are to be considered with caution.

Rather than providing an exact number, these concentrations should be interpreted as an indication of the presence of very fine particles (Fettweis and Baeye, 2015). Indeed, the LISST-100X (type B) detectors show some limitations and uncertainties related to the characteristics of the particles. Significant concentrations of particles smaller than the lowest LISST detection limit (<1.25 µm) can influence the entire PSD with increase of concentrations of the two smallest size classes, a decrease of the next classes and an increase on the largest size classes [START_REF] Andrew | Using laser diffraction data to obtain accurate particle size distributions: The role of particle composition[END_REF][START_REF] Graham | Interpreting LISST-100X measurements of particles with complex shape using digital in-line holography[END_REF].

Spatial variations

We further investigated the transfer of particles from a class to another, and the relative contribution of each class to the total SPMVC (Fig. 7 c). During spring tides (surveys WS and DS), the proportion of each class remained quite unchanged along the estuary, while longitudinal variations were observed in neap tides (surveys WN and DN), with decreasing proportion of primary particles and increasing proportion of microflocs from upstream (ST3) to the river mouth (ST1). In survey WN, for example, microflocs made up 23% of the total volume at ST3, 35% at ST2 and 39% at ST1, while PP contribution to SPMVC decreased from ST3 (34%) to ST2 (22%) and ST1 (18%). Similar longitudinal transfers from PP to microflocs were observed in survey DN, even though the PP contributions to SPMVC were about twice larger in survey WN. These discrepancies in PP concentrations between dry and wet seasons could be due to differences in sediments discharges (Q SED ) between the seasons (section 4.1.2.1), with mean Q SED approximately 170 times bigger in survey WN than in DN (Table 2). In survey DN, salinity-induced flocculation may explain the transfers of PP to microflocs (see previous developing an estuarine turbidity maximum near the upstream boundary of the gravitational circulation [START_REF] Dronkers | Tide-induced residual transport of fine-sediment. In Physics of shallow estuaries and bays[END_REF]. This is particularly observable in neap tides when a sediment flux is oriented seaward in a surface layer at the same time than a landward sediment flux in the underlying salted layer. The ETM (estuarine turbidity maximum) was described by [START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF] in another distributary of the Red River, close to the Van Uc River, and our measurements indicates that an ETM is likely located between stations ST1 and ST2 in dry season. In wet season, salinity remained very low in the riverine part of the estuary (with maximum value of 0.09 PSU at the river mouth), indicating that mixing between riverine and marine waters develops in the bay and in coastal waters, like in the Cam-Nam Trieu estuary [START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF]. Since stratification remains low during the tidal cycle, another process than gravitational circulation should explain the landward flux of suspended sediment in the wet season, which was observed at spring tides. Important asymmetry (or distortion) can occur in the tidal curve and velocities, in estuaries characterized by larger tidal ranges than water depths and by cross-river sections rapidly decreasing going up-estuary [START_REF] Dyer | Coastal and Estuarine Sediment Dynamics[END_REF]. In these cases, flood tides become shorter and stronger than ebb tides. The distortion of the tide wave in spring tide may be the dominant process to explain the up-estuary sediment mobility in wet season at spring tide during flood. A combination of the two main estuarine processes could thus explain the estuarine behaviour between two extreme conditions, at neap tides in dry season when gravitational circulation dominates, and at spring tides in wet season, when tidal distortion dominates. The effects of both processes may be further quantified and compared from numerical simulations. corresponds to the so-called wash load transport, which keeps particles in motion by turbulence at all current speeds and can therefore occur at any season and any tidal stages. The coarser fraction (microflocs) is likely in motion and transported by suspension, re-suspension and bedload [START_REF] Dyer | Chapter 14 Sediment transport processes in estuaries[END_REF]. While suspension can occur at any season, bedload occurs mostly during the high-flow season with highest current velocities (high water discharges).

This study took place in the framework of a more comprehensive modeling project which aims at representing the transport and fate of sediment from the Red River to the Gulf of Tonkin and analyzing the relevant processes. As tides have a major effect on the sediment dynamics within the estuaries and in the plume area [START_REF] Pritchard | A study of the salt balance in a coastal plain estuary[END_REF][START_REF] Pritchard | The dynamic structure of a coastal plain estuary[END_REF]Allen et al., 1980;[START_REF] Fontes | Numerical study of circulation on the inner Amazon Shelf[END_REF][START_REF] Vinh | Estuarine Turbidity Maxima and Variations of Aggregate Parameters in the Cam-Nam Trieu Estuary, North Vietnam, in Early Wet Season[END_REF], and as monsoonal estuaries are highly affected by seasonal water discharges [START_REF] Wolanski | Estuarine Hydrology[END_REF][START_REF] Vinh | Impact of the Hoa Binh dam (Vietnam) on water and sediment budgets in the Red River basin and delta[END_REF]Nowacki et al., 2015), it is necessary to document and understand the estuarine hydrology and suspended matter dynamics, before setting up a set of numerical models and investigating the fine scale sediment physics. This study complemented the understanding of the Van Uc estuary, which, to our knowledge, has never been sampled before. Further studies are also encouraged to focus on the mixing processes and estuary configuration in wet season along the Red River delta.

Fig. 1

 1 Fig. 1 Location of stations ST1, ST2 and ST3 (in red) in the Van Uc River and of Trung Trang

  of the optical and acoustical SPM, SPM OBS and SPM AC Calibrations of optical instruments must be carried out in environments where flocculation of fine grains/particles occurs and where hydrodynamics and grain types/particles properties are highly variable(Bunt et al., 1999). A linear regression was used between the OBS signal (i.e. the turbidity, in FTU) and the filtered SPM Filters concentration (in mg l -1 ), for the wet season surveys and for the dry seasons surveys. The slope (a) and the intercept (b) of the fitted regression lines were a=0.90 and b=-9.84 for the wet season (R²=0.96), and a=1.08 and b=-3.86 for the dry season (R²=0.88). Based on these relationships, SPM OBS concentrations were determined from the turbidity profiles, for each season.

Fig. 2 Fig. 3

 23 Fig. 2 10log 10 of suspended matter concentration estimated from CTD OBS (SPM OBS in mg l -1 )

  the wet season, only spring tides have induced large enough amplitudes (of ~2 m) to balance the riverine discharge and induce up-estuary flow (Q ADCP -) at stations ST1 and ST2 (Fig. 4 b) ; tidal currents at neap tides (maximum amplitudes of 0.5 m) were not strong enough to impact the riverine discharges (Fig. 4 a). The rest of the diurnal cycle (i.e. low, ebb and flood tides) at each station was dominated by a down-estuary flow (Q ADCP +), which increased with ebb tides and was the highest around low tides. A noticeable difference can be seen on the flow regime between spring and neap tides in dry season. During spring tides, the flow was alternatively strictly down-estuary and strictly up-estuary with positive Q ADCP values and no Q ADCP -component during 12 hours then the reverse during the next 12 hours from the end of flood until halfway through ebb (Fig. 4 d). At neap tides (survey DN, Fig 4 c), the reversion of the total flow also occurred, but was more nuanced with both Q ADCP -and Q ADCP + components contributing at the same time to the flow at ST1 and ST2.

Fig. 4

 4 Fig. 4 Tidal elevation (in m, black line) and Q TT (m 3 s -1 , blue line), measured Q ADCP (m 3 s -1 , red dots)

Fig. 5 :Fig. 6

 56 Fig. 5 : Simpson parameter values (J m -3 ) at ST3 (left), ST2 (center) and ST1 (right) for survey

  In dry season, bimodal PSDs were measured with peaks moving in the spring-neap tide cycle. Peaks of PP and flocculi were observed in spring tides (survey DS, Fig. 7 a), while peaks of flocculi and microflocs appeared in neap tides (survey DN). However, the tidal average PSDs in spring tides masked a diurnal alternation of PSDs. At low tide (Fig. 7 b), the instant PSD was similar to the average PSD of survey DS (Fig. 7 a) but at high tide, the instant PSD (Fig. 7 b) appeared to be similar to the PSDs of survey DN (Fig. 7 a). These observations highlight the fact that riverine waters, which dominated the water column around low tides in spring tide (see section 4.1.2.2, Fig. 4 d and section 4.2.2, Fig. 6 a), carried PP and flocculi. On the other side, marine salty waters, which dominated the water column around high tide in spring tides and all along neap tides, carried flocculi and microflocs. We suggest that the microflocs were formed by salinity-induced flocculation of PP or covarying aggregation by bio-processes, either within the estuary, or in the coastal area and they were brought back into the estuary by the tidal advection.

  section), intensifying closer to the river mouth (ST2 and ST1). For spring tide surveys (DS and WS), the proportion of each class remained spatially almost unchanged because the along-estuary salinity gradient was non-existent, limiting the flocculation and therefore the transfer of PP to microflocs.Lastly, the SPMVC progressively decreased along the estuary with higher volume concentrations recorded at ST3 than at ST2 and ST1, in each case (see the values on Fig.7 a). The volume concentration ratio between ST3 and ST1 decreased from 12 for survey WN, to ~3 for surveys WS, DS and DN. Although SPMVCs should not be considered as exact values but only as indicators, such a volume loss may be explained by particles deposition, with zones of accumulation spanning from ST3 to ST1. Here, hydrodynamical-sediment transport modelling would be a useful tool to detail the transport and deposition of SPM, and could highlight zones of accumulation / accretion in the estuary and along the shorelines.

Fig. 7

 7 Fig. 7 a) PSD at each station for all surveys and their associated SPMVC. b) PSD at ST2 in

4-

  Seasonal and tidal flow dynamics are also reflected in grain-size characteristics. All types of particles (PP, flocculi and microflocs) are transported during the high-flow season at any tidal stages, while mainly flocculi and microflocs are transported in the low-flow season, at high tides in spring tides and all along neap tides. The presence of the finest fraction (PP and flocculi)

Table 2 :

 2 Mean water discharges at Trung Trang (Q TT ), mean water (Q ADCP ) and mean sediment

	(Q SED ) discharges derived from ADCP measurements and corresponding correlation coefficients
	(R²), for the four surveys (for our 36 values time series, the 99% significativity threshold is
	reached for a correlation of 0.42). The means are computed over the 3-days period of each
	survey (36 values), and computed over 24h when showed for each station (12 values).
	4.1.2.2 Spatial variations
	Q

ADCP measured in the estuary exhibited very small differences compared to the mean discharges from the hydrological station Trung Trang (noted TT from hereafter) in the wet season (below 1% for WN and around 7% for WS, see Table

ST3:0.0019

Acknowledgments

This paper is a contribution to the LOTUS International Joint Laboratory ( lotus.usth.edu.vn ). The authors warmly thank Gaël Orban de Xivry and Pablo Lipchitz for their precious work and help during the field campaigns, as well as the boat crew.

Authors Contribution

VP, SO and MH conceived the study. VP, VDV, SO and MH organized the field trip, and VP and VDV performed the measurements. VP, SO and GM analyzed the data. VP and SO prepared the manuscript with contributions of GM, MH and PM.

Conflicts of Interest

The authors declare no conflict of interest.


Lorenzo Riparbelli
email: lorenzo.riparbelli@unifi.it

Fabrice Brémand

Paolo Dionisi-Vici

Jean-Christophe Dupré

Giacomo Goli

Delphine Jullien

Paola Mazzanti

Marco Togni

Elisabeth Ravaud

Luca Uzielli

Riparbelli Lorenzo

Brémand Fabrice

Dionisi-Vici Paolo

Jean-Christophe Dupre

Goli Giacomo

Hesser Franck

Jullien Delphine

Mazzanti Paola

Togni Marco

Ravaud Elisabeth

Uzielli Luca

Gril Joseph

Learning from Objects: the use of advanced numerical methods to exploit a complete set of information from experimental data, for the Mona Lisa's Digital-Twin

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Abstract. The approach to wooden artefacts of historical importance, and panel paintings in particular, is a task that requires a multidisciplinary approach based on experimental observation of the artwork and advanced techniques to make these data actually useful for the knowledge and preservation of the object. This study illustrates how a series of scientific observations and instrumental analyses can be used to construct a numerical simulation that allows a deeper understanding of the physical structure and behaviour of the object itself, namely to construct a hygro-mechanical predictive model (a "Digital-Twin") of Leonardo da Vinci's Mona Lisa panel. Based on specific request from the Louvre Museum, a group of experts with different and complementary skills cooperated and are still cooperating to construct a complete set of experimental observation and non-invasive tests; so, the integration of the collected data made the construction possible of the panel's Digital-Twin. This paper also specifically examines how the Digital-Twin can be used to compare two framing conditions of the panel; although the two experimental configurations are not inherently comparable, the comparison is made possible by the introduction of a technique of projection of the fields obtained as results of the two analyses, named the Projected Model Comparison (PMC), which has been developed specifically for this research.

Since 2004, the wooden panel on which Leonardo da Vinci painted his "Mona Lisa" has been studied by an international research Team, and several experimental campaigns have been carried out to understand its mechanical, hygroscopic and shape characteristics and behaviour, to evaluate its present state of conservation, and provide related suggestions in order to optimize its conservation. The artwork is painted on one face of a flat-sawn Poplar (Populus alba L.) panel, doubly curved (convexity toward the front face), and pressed against the rebate of the frame (in French châssis-cadre, an intermediate frame, see Fig. 2) by two crossbars screwed onto the frame itself. The châssis-cadre, with the panel inside, is placed and fixed by metal brackets in a sculptured and gilded external frame, which significantly contributes to the stiffness of the system. An ancient crack runs through the wood from the upper edge of the panel down to the top of the Lady's forehead. -studying the artwork by direct observation and measurements (once a year, during the few hours when it is removed from its climate-controlled exhibition case, for the routine inspection of its conservation conditions, or for the execution of exceptional measurements); -analysing the propagation risk of the fracture [START_REF] Gril | Mona Lisa saved by Griffith theory: assessing the crack propagation risk in the wooden support of a panel painting, International conference on integrated approach to wood structure, behaviour and application[END_REF]; -monitoring the behaviour of the panel during its permanence in the exhibition case, by special ad-hoc equipment, conceived and implemented specifically for this task [START_REF] Gril | The Mona Lisa Project: An update on the progress of measurement and monitoring activities, Euromech conference[END_REF]; -developing and calibrating a numerical model (a so called "Digital-Twin"), capable of reproducing accurately the original artwork's reactions to simulated external stresses (hygroscopic and mechanical) that might (actually or potentially) affect it.

The observations methods include: (i) scientific and technological analysis of the wooden panel, including wood anatomy, macrostructure and physical and mechanical properties;

(ii) optical measurements of its shape [START_REF] Brémand | Mechanical structure analysis by Digital Image Correlation and Fringe Pattern Profilometry[END_REF];

(iii) continuous monitoring of the forces acting on the panel and of its deformations;

(iv) detection and description of contact areas between the panel and the châssis-cadre using pressuresensitive film.

In the course of fifteen years, the Team's knowledge about the panel has progressively increased along with measurement and modelling techniques; based on the collected data, numerical models of the wooden panel have been developed to accurately simulate its hygro-mechanical behaviour and its constraints.

The aspect that the Authors want to emphasize in the present paper is that numerical modelling tools are also being used in order to enrich experimental investigations, thus allowing to reach deeper information levels.

The Digital-Twin and its meaning

The general problem that arises in the modelling of panel paintings is the need to correctly understand its hygro-mechanical characteristics and structure and based on that give information to the conservators to and then protect beforehand the artworks; which makes mandatory the use of noninvasive technologies for characterization. At the same time, the uniqueness of the artworks (inter alia due to the variability of wood, the interaction between the elements composing the system, the conservation history including restoration works executed in the past), makes it difficult to apply literature parameters in the definition of a predictive numerical model. In any case, understanding the stresses and strains acting on the actual art piece is of great interest for its conservation and for the study of potentially dangerous conditions. For these reasons the systematic series of non-invasive experimental observations made on the Mona Lisa panel aimed towards, and made it possible, to develop a numerical model totally calibrated on the artwork itself.

The model was calibrated by means of the following non-destructive measurements [START_REF] Riparbelli | Mona Lisa's digital twin: identifying the mechanical properties of the panel combining experimental data and advanced finite-element modelling[END_REF]:

-the shape of the panel was derived from optical surveys and reconstructed in a numerical threedimensional model; -the boundary conditions, i.e. contact zones and pressures between the panel and the framing components surrounding it, have been detected using ad hoc developed techniques, and then have been transferred within the three-dimensional model; -through visual anatomical observations and results from X-ray investigations, the anatomical directions of the wood at each point of the panel have been reconstructed, and this made it possible to construct a model consistent with the anatomy and the stiffness orientation of the actual wooden board; -from the load cells readings, it was possible, through an optimization process, to calculate the actual stiffness characteristics of the wooden panel (including the preparation and pictorial layers).

In-depth studies are available in the literature that examine the fundamental aspects of the above methods [START_REF] Allegretti | The influence of dovetailed cross beams on the dimensional stability of a panel painting from the Middle Ages[END_REF][START_REF] Remondino | Review of Geometric and Radiometric Analyses of Paintings[END_REF][START_REF] Pelagotti | Comparing two 3D measurement techniques for documenting painted wooden panels surface deformations on a real test case: "Mystical Marriage of Saint Catherine" by Renaissance artist Piero di Cosimo[END_REF][START_REF] Gauvin | Image correlation to evaluate influence of hygrothermal loading on wood[END_REF]. The present paper focuses on the study of contact areas in various configurations, necessary for the correct modelling of the panel, and on analytical techniques to compare the results of two numerical analyses representing two different mechanical conditions.

Materials and Methods

Experimental campaigns on contact areas

Two experimental campaigns were conducted to investigate the areas of contact between the panel and the châssis-cadre. In the first one (performed in 2012) the panel was directly in contact with the wood of the châssis-cadre [START_REF] Goli | Locating contact areas and estimating contact forces between the "Mona Lisa" wooden panel and its frame[END_REF]. In the second one (performed in 2016), the effects of the interposition of a closed cell polymer foam (Plastazote©) between the wood of the panel and the châssis-cadre were investigated [data still unpublished]. In order to identify and evaluate the contact zones, in both cases a pressure sensitive film (LLLW Ultra Super Low Pressure, two-sheet film, Fujifilm Prescale®) was used to perform some totally noninvasive tests: strips of the film were placed on the rebate surface of the châssis-cadre, and the panel was carefully inserted and pressed during the prescribed short time (few seconds), reproducing the actual assembly situation, and then disassembled. Red patches appear on the film at the contact areas, and the level of pressure is indicated by the density of the colour (Figure 3). The restraining forces are applied by the crossbeams on the back the panel, near its four corners, in order to partially contain its tendency to cup; four miniature load cells continuously measure these forces, and the data are fed into a continuous recording system [START_REF] Gril | The Mona Lisa Project: An update on the progress of measurement and monitoring activities, Euromech conference[END_REF]: this arrangement makes it possible to have available both the contact profiles and the four forces balancing the contact forces, in other words to have under control the whole system of the boundary conditions and the forces acting on the panel.

Numerical modeling

The interpretation of the above results made it necessary to create an ad hoc FEM model, since it was not possible to evaluate the stresses and strains resulting from the two constraint configurations from the mere observation of the patterns of stains . An elastic orthotropic model was therefore developed, respecting the constraints represented by the contact areas identified by the tests and the forces measured at the points on the back where the load cells are located. This model is only an approximation as far as viscoelastic behaviour is concerned, but it is efficient in representing ("photographing") the mechanical situations of the panel in both specific cases of the test by applying the experimentally detected conditions. The shape of the panel was determined by optical methods by the Pprime Institute and based on these measurements a three-dimensional model was built within the open source ecosystem Salome-Meca. The two obtained solid models, one for each test, of the panel were enriched with the contact areas detected during the Prescale campaigns, by partitioning the surface of the geometry concerned. The two different models were subsequently meshed with a Netgen algorithm, always within Salome-Meca, as shown in Figures 45, with mesh refinement in the zones of the contacts and of the crack. The FEM solution has been realized with the open source solver, code_aster [START_REF] De | Finite element code_aster[END_REF], developed by EDF, for both cases in orthotropic linear elasticity with the mechanical characteristics of the wood, derived from [START_REF] Mohen | Au Coeur de La Joconde[END_REF], and the pressure data derived from the Prescale tests.

Analysis of results and further processing

The analyses, whose results are shown in Figures 6789, indicate that in both cases the stresses are in a safe range, considering the stresses in the wood are definitely smaller than the failure stresses reported in the literature [START_REF] Mazzanti | Strength and MOE of Poplar wood (Populus alba L.) across the grain: experimental data[END_REF]; however, further investigations are necessary to understand the differences between the two solutions, i.e. to compare two results based on two different mechanical models. These two models are not directly comparable because the sum of the reactions applied by the load cells in z direction (normal to the panel) for each model are different:

• Model A: Sum of reactions 51.92 N • Model B: Sum of reactions 74.27 N

In order to clearly interpret the differences between the tensional states of the two configurations, a technique of projection of the fields obtained as results of the two analyses, here named the Projected Model Comparison (PMC), has been developed specifically for this research. Here follows a concise description of PMC.

After having normalized the two fields with reference to the constraint reactions listed above, a new mesh of comparison was created, and the numerical field of the model A was projected on such new mesh after normalization, while the model B was projected without manipulation (Figure 10). The numerical technique used for the projection of a numeric field from the original mesh to the comparison mesh is based on the following steps for each node of the comparison mesh [START_REF]Projection d'un champ sur un maillage -R7.20.01 -Électricité De France -EDF. Finite element code_aster[END_REF]:

1. We determine the element of the original mesh that contains the node of the comparison mesh.

2. We determine the position of the comparison node within the element identified in step 1, taking into account the fact that the meshes are non-linear, varying the jacobian of the geometric transformation on the element.

3.We use the shape functions of the element of the original mesh in the point of step 2 to determine the field value at the point determined in step 2.

Figure 10. Field normalized for the model A and projected on the final mesh

In summary, two meshes are generated, geometrically identical, containing the values of displacements perpendicular to the panel, and of stresses perpendicular to the grain in the plane of the panel, at the same nodes, one for each case, i.e. with and without Plastazote between the panel and the châssis-cadre.

Another model was then created, for comparison, cross-checking the data fields of the previous models, creating a point-to-point function between the two states of the panel, i.e. with and without the foam.

Here a simple function, valid in this case where displacements and bending deformed shape have everywhere the same direction, has been used point by point, extended to the entire 3d volume of the panel:

𝑓(𝑝𝑜𝑖𝑛𝑡_𝑡𝑜_𝑝𝑜𝑖𝑛𝑡)=|𝑣𝑎𝑙𝑢𝑒 𝑚𝑜𝑑𝑒𝑙 𝐴|-|𝑣𝑎𝑙𝑢𝑒 𝑚𝑜𝑑𝑒𝑙 𝐵|

This procedure makes it possible to highlight the differences between the two models, with respect to the displacements and stresses they provide. In Figures 11 and12 the red-coloured zones show where and how much the displacement/stress provided by model A (without Plastazote) is greater than the one provided by model B (with Plastazote); the reverse is true for blue-coloured zones. In particular, Figure 11 shows that (possibly due to the increase in the contact area) the presence of Plastazote leads to a decrease of the displacements (perpendicular to the plane of the panel) in the upper part of the panel, where the fissure is located; while the opposite behaviour (displacements increase) occurs in the lower part, where no fissures are present. Figure 12 in turn shows that due to the presence of Plastazote the stresses (perpendicular to the grain, in the plane of the panel in the wood situated in the contact zone with the preparation layers ) tend to increase very slightly (one order of magnitude lower than the stresses of Figure 7, model A) throughout the whole panel, except in the following very restricted areas, identified by very small red spots, where instead the stresses decrease: (i) in the upper part of the panel, along the edge of the rebate of the châssis-cadre, where the contact pressures between châssis-cadre and panel are larger (see Figures 7 and9) and where some very small aligned red spots appear; (ii) at the tip of the fissure, where one very small red spot appears; and (iii) on the edge of the lower rebate of the châssis-cadre, where one small red spot appears.

As far as stresses are concerned, the above results suggest that the two configurations (i.e. without and with Plastazote) can be considered equivalent, except for the tip of the fissure, and for the small contact areas along the upper and lower rebates of the châssis-cadre, near which the barbe extends (barbe is the French term for the preparation and colour crest located along the contact angle between a no longer existing frame and the panel surface, formed when the work was painted [START_REF]Projection d'un champ sur un maillage -R7.20.01 -Électricité De France -EDF. Finite element code_aster[END_REF]): here the presence of Plastazote produces a local decrease of the stresses in the wood. It should however be emphasized that these areas must be considered particularly significant for the integrity of both the panel and the pictorial layers.

Conclusions

The numerous and varied non-invasive experimental observations made on the Mona Lisa panel aimed towards, and made it possible, to develop the panel's Digital-Twin, i.e. a numerical model totally calibrated on the artwork itself. Also, the Projected Model Comparison (PMC) was developed, which is a general method of comparison between the fields of two different FEM analyses, by generating 3d fields extended to the entire volume of the panel, and making it possible to combine, subtract, or generate complex functions to compare the numerical fields of the two different solutions. PMC made it possible to compare two framing conditions of the panel analysed by means of the Digital-Twin, although the two experimental configurations are not inherently comparable This paper shows how the 3d comparison field is a point by point correlation between two different solutions, that can provide important information making it possible to compare different boundary conditions, mechanical properties, mechanical behaviour, loads (both mechanical and hygroscopic). With this method it was possible to evaluate how the possible insertion of a protective layer of polymer foam might reduce stresses in some significant areas of the panel, namely at the tip of the fissure and near the barbe. As far as the general state of stresses outside this area is concerned, there are no significant differences between the two configurations. The Authors emphasize that the present work expresses only the evaluation of the stresses in the wooden panel and is not in itself capable of providing indication of the possible fragility of the work of art as a whole.

Figure 1 .Figure 2 .

 12 Figure 1. Painting Figure 2. Framing system

Figure 3 .

 3 Figure 3. Contact zone in the upper and lower zone of the panel

Figure 4 .Figure 5 .

 45 Figure 4. Tetrahedral mesh of the contact configuration of the 2012 campaign (Model A)

Figure 6 .

 6 Figure 6. Model A: results in terms of displacements in the direction perpendicular to the panel [mm]

Figure 7 .

 7 Figure 7. Model A: results in terms of stresses perpendicular to the grain in the plane of the panel at the level of the preparation layers [MPa]

Figure 8 .

 8 Figure 8. Model B: results in terms of displacements in the direction perpendicular to the panel [mm]

Figure 9 .

 9 Figure 9. Model B: results in terms of stresses perpendicular to the grain in the plane of the panel at the level of the preparation layers [MPa]

Figure 11 .

 11 Figure 11. Difference between the displacements perpendicular to the plane of the panel; such difference is obtained by subtracting point-by-point the displacements provided by model B from those provided by model A [mm]

Figure 12 .

 12 Figure 12. Difference between the stresses perpendicular to the grain, in the plane of the panel; such difference is obtained by subtracting point-bypoint the stresses provided by model B from those provided by model A at the level of the preparation layers [MPa]

Acknowledgments

The Authors wish to thank:

-the Louvre Museum, and in particular the concerned Conservators (Vincent Delieuvin, Sebastien Allard, Vincent Pomarède, Cécile Scallierez) and Technicians (including Eric Bourgart and the late Antoine Carnino) for providing the access to the artwork, and for making it possible during several years to carry out monitoring and measurement activities, -Daniel Jaunard and Patrick Mandron, and recently also Jonathan Graindorge Lamour and Gilles Tournillon, Restorers of wooden panels, for their invaluable support in planning and performing the measurements, -Marco Fioravanti, Linda Cocchi, Bertrand Marcon, Cecilia Gauvin, J.P. Mohen, M. Menu and B.

Mottin and several other Colleagues, mostly co-Authors of the papers cited in the Bibliography, who in various ways supported and/or participated in the activities to which this paper refers.