A Minghelli
email: minghelli@univ-tln.fr

S Vadakke-Chanat

Malik Chami

M Guillaume

M Peirache

Benefit of the potential future hyperspectral satellite sensor (BIODIVERSITY) for improving the determination of water column and seabed features in coastal zones

Keywords: Hyperspectral, radiative transfer, ocean color, bathymetry, seabed mapping

Benefit of the potential future hyperspectral satellite sensor (BIODIVERSITY) for improving the determination of water column and seabed features in coastal zones

I. INTRODUCTION

There is a growing interest in the use of hyperspectral imagery, in the field of ocean optics along with the deployment of new hyperspectral satellites [START_REF] Emberton | Sensor Capability and Atmospheric Correction in Ocean Colour Remote Sensing[END_REF]. Hyperspectral imagery offers numerous advantages including the spectroscopic techniques for analyzing the remote signal which allows a better understanding of water optical properties even in turbid waters [START_REF] Louchard | Derivative analysis of absorption features in hyperspectral remote sensing data of carbonate sediments[END_REF]. A hyperspectral resolution allows more accurate determinations of water bio-optical properties because it enables the individual spectral signatures of the hydrosols and seabed features to be detected through the measurements at narrow bands [START_REF] Emberton | Sensor Capability and Atmospheric Correction in Ocean Colour Remote Sensing[END_REF], [START_REF] Gitelson | Review of constituent retrieval in optically deep and complex waters from satellite imagery[END_REF], [START_REF] Matthews | A current review of empirical procedures of remote sensing in inland and near-coastal transitional waters[END_REF]. Furthermore , the recent interest in hyperspectral imagery is amplified by the fact that it allows estimations of geophysical products over different geographic sites, in open as in coastal waters, without needing to have previously chosen the location of bands [START_REF] Hestir | Measuring freshwater aquatic ecosystems: The need for a hyperspectral global mapping satellite mission[END_REF]. This is in conjugation with the growing scientific interest in differentiating certain phytoplankton species/groups and characterizing seabed composition. Consequently, further studies in the field of hyperspectral imagery are necessary [START_REF] Bracher | Obtaining phytoplankton diversity from ocean color: A scientific roadmap for future development[END_REF].

For research on coastal waters and inland freshwater benthic habitats, this technique has been extensively exploited. For shallow waters, the determination of bathymetry as well as the composition of the benthic vegetation can be carried out based on ocean color technology. A water body is called optically shallow when the above water reflectance can be determined by the optical properties of the water column, seafloor, and the depth [START_REF] Maritorena | Diffuse reflectance of oceanic shallow waters: Influence of water depth and bottom albedo[END_REF]. Knowledge of the benthic vegetation is important for the characterization of marine habitats [START_REF] Nieto | Classifying benthic habitats and deriving bathymetry at the Caribbean Netherlands using multispectral imagery[END_REF]. Habitats are distinct bio-physio-geo-chemical environment regions in contrast to neighboring areas [START_REF] Valentine | Classification of Marine Sublittoral Habitats, with Application to the Northeastern North America Region[END_REF]. Studies using remote sensing performed to date in the optically shallow waters have generally attempted to map the spatial distribution of single benthic habitats or habitats with a single dominant species or multiple heterogeneous habitats (seagrasses, macroalgae, or coral reefs). Studies have also been conducted on the temporal dynamics of the benthic habitats which aid in the understanding of variations in environmental conditions [START_REF] Vahtmäe | How much benthic information can be retrieved with hyperspectral sensor from the optically complex coastal waters?[END_REF]. Bathymetry estimation is one of the research topics where remote sensing applications have been most frequently used. Thus, the remote sensing technique plays an important and expanding role in ocean applications [START_REF] Vahtmäe | How much benthic information can be retrieved with hyperspectral sensor from the optically complex coastal waters?[END_REF].

The vast majority of the signal received at the satellite level is caused by the atmospheric scattering [START_REF] Kirk | Light and photosynthesis in aquatic ecosystems[END_REF], that contributes to about 80-90% [START_REF] Singh | A novel method for estimation of aerosol radiance and its extrapolation in the atmospheric correction of satellite data over optically complex oceanic waters[END_REF] of the total signal in oceanic waters. Furthermore, part of the radiation reaching the sensor comes from the skylight and sunlight reflections onto the sea surface, which is often called the glint signal. Both the atmospheric and glint radiances need to be removed from the satellite data, which is the so-called atmospheric correction procedure. The part of incident light entering into the water column that is absorbed and scattered away does not reach the remote sensing detector. The radiation that interacts with water and constituents and scatters back to the atmosphere contains useful information about the water column but not from the seabed. Only the part of the radiation that reaches the seabed and reflects up to the sensor contains information about the bottom.

For nearly two decades, there has been an increase in the use of hyperspectral airborne images combined with the use of exact radiative transfer models, such as Hydrolight, [START_REF] Mobley | A NUMERICAL-MODEL FOR THE COMPUTATION OF RADIANCE DISTRIBUTIONS IN NATURAL-WATERS WITH WIND-ROUGHENED SURFACES[END_REF] or semi-analytical models such as Lee's [START_REF] Lee | Hyperspectral remote sensing for shallow waters. I. A semianalytical model[END_REF] to determine the water bio-optical properties, namely the Chlorophyll-a concentration (Chl), the Suspended Particulate Matter (SPM) concentration and the absorption coefficient of the Colored Dissolved Organic Matter (CDOM), and the seabed features, such as bathymetry and composition. The analysis of the propagation of light through the water column is carried out using radiative transfer models that link the remote sensing reflectance to the properties of the optically active water constituents. Different methods have been developed for the seabed characterization. Some methods use the correction of the water column attenuation [START_REF] Dekker | Intercomparison of shallow water bathymetry, hydro-optics, and benthos mapping techniques in Australian and Caribbean coastal environments[END_REF] or LUT (Look-up Table ) of seabed reflectance [START_REF] Mobley | Interpretation of hyperspectral remote-sensing imagery by spectrum matching and lookup tables[END_REF]. The inversion of a semi-analytical model can also be used to estimate the water column parameters, the abundance of bottom classes, and the seabed depth [START_REF] Lee | Hyperspectral remote sensing for shallow waters: 2. Deriving bottom depths and water properties by optimization[END_REF].

The abundance within each pixel using remote sensing in the optically shallow waters was rarely estimated for practical considerations due to the extensive field calibrations that are required by the estimation models [START_REF] Dierssen | Ocean color remote sensing of seagrass and bathymetry in the Bahamas Banks by highresolution airborne imagery[END_REF]- [START_REF] Hill | Evaluating Light Availability, Seagrass Biomass, and Productivity Using Hyperspectral Airborne Remote Sensing in Saint Joseph's Bay, Florida[END_REF]. More recently, the significance of benthic vegetation in the global carbon budget of the oceans has been getting wider recognition while it has been largely ignored in the past [START_REF] Duarte | Reviews and syntheses: Hidden forests, the role of vegetated coastal habitats in the ocean carbon budget[END_REF]. Therefore, more studies were undertaken with the goal of quantifying the productivity of benthic habitats using remote sensing technology as well as the mapping and abundance estimation ( [START_REF] Lesser | Bathymetry, water optical properties, and benthic classification of coral reefs using hyperspectral remote sensing imagery[END_REF], [START_REF] Dierssen | Benthic ecology from space: optics and net primary production in seagrass and benthic algae across the Great Bahama Bank[END_REF]).

The potential of remote sensing techniques in obtaining information from the seabed for shallow waters is promising though it is more difficult as compared with terrestrial targets. The exploitation of the seabed spectral information is made more challenging because of the influence of the water column on the light propagation from the sea surface to the bottom. As a consequence, the radiation that reaches the remote sensor is attenuated by both the water column and the atmospheric layers.

Various hyperspectral sensors are available but most of them are mounted on an airborne platform, such as CASI, AVIRIS, HYSPEX, which are characterized by a good spatial resolution and a good signal to noise ratio. Only a very small number of sensors are mounted on a satellite platform. As examples, HICO (90 m, 87 bands) was onboard the International Space Station, operated by NASA , between 2009 and 2014, PRISMA (30 m, 250 bands) [START_REF] Loizzo | Prisma: The Italian Hyperspectral Mission[END_REF] was launched in March 2019 by ASI, ENMAP (30 m, 88 bands) is planned to be launched in 2020 by DLR, and a BIODIVERSITY-like sensor (8 m, 53 bands) is planned to be launched in the future by CNES (French space agency) [START_REF] Rast | Earth Observation Imaging Spectroscopy for Terrestrial Systems: An Overview of Its History, Techniques, and Applications of Its Missions[END_REF]. PACE-OCI (1 km, 118 bands) is a moderate spatial resolution hyperspectral sensor that is planned to be launched in 2022. Three parameters are decisive for the analysis of these data: the spatial and spectral resolutions, and the signal to noise ratio. Note that a BIODIVERSITY-like sensor is more sophisticated in terms of spatial resolutions (8 m) and SNR than the existing hyperspectral sensors. The objective of this paper is to evaluate the benefit of a BIODIVERSITY-like sensor for determining the water column parameters and for characterizing the seabed depth and composition in coastal zones. The results obtained are compared with ENMAP and HICO sensor performances. The paper is organized as follows: the study area in Porquerolles Island, the in situ and hyperspectral data and the methodology of simulation are described in Section 2. The results of inversion of BIODIVERSITY simulation are presented in Section 3 and discussed in Section 4 in comparison with other hyperspectral satellite sensors performances.

II. DATA AND METHOD

A. Study area

The study area is the coastal water of Porquerolles Island, south-eastern France (Fig. 1.). The island is a sanctuary and a part of the National Park of Port-Cros. The seafloor is characterized by seagrass, sand, and algae covers. The seagrass in the region belongs to the endemic species Posidonia oceania while the algal species is Caulerpa taxifolia, a tropical invasive algae carried to the Mediterranean Sea by the Atlantic Ocean currents. Note that the first appearance of Caulerpa taxifolia occurred 3 decades ago [START_REF] Chisholm | A century of warfare shoots holes in anti-Caulerpa campaign[END_REF], [START_REF] Jaubert | Reevaluation of the extent of Caulerpa taxifolia development in the northern Mediterranean using airborne spectrographic sensing[END_REF]. The photophilic algae of the species Cystoseira Amentacea are also observed in the seafloor rocks [START_REF] Blanfuné | The CARLIT method for the assessment of the ecological quality of European Mediterranean waters: Relevance, robustness and possible improvements[END_REF].

B. Data

The airborne data acquisition campaign was conducted on 13 September 2017. The date was selected taking into account that summer is the best season to observe Caulerpa taxifolia due to its seasonal cycle. The airborne HYSPEX sensor, operated by Hytech Imaging, was used to obtain hyperspectral images (Fig. 2). At an altitude of 2666 m, the spatial resolution of the sensor is 1 m [START_REF] Köhler | Airborne Imaging Spectrometer HySpex[END_REF]. It provides 160 spectral bands between 400 nm and 1000 nm at 3.6 nm intervals. The atmospheric correction procedure was performed using the ATCOR algorithm [START_REF] Richter | Atmospheric / Topographic Correction for Satellite Imagery (ATCOR-2/3 User Guide, Version 8.2 BETA[END_REF].

Targets were fixed on the ground in the field of view of the sensor to adjust the atmospheric correction process. Ground targets consisted of black, grey, and cream fabric, each covering an area of 5 m 2 . The cream color was chosen to avoid saturation of the radiance in the image. Their hyperspectral reflectance was measured in the laboratory prior to their deployment in the field. The weather conditions during the field experiment were windy with wind speed values between 10 and 15 m s -1 . The resulting sea surface roughness caused the sunlight reflection on the sea surface (i.e., so-called sun glint) to affect the airborne data. A correction procedure for the sunglint radiance was thus performed using the signal measured in the near-infrared bands (one band between 800 nm and 900 nm) where the water reflectance is considered to be null [START_REF] Hedley | Simple and robust removal of sun glint for mapping shallow-water benthos[END_REF]. The Litto3D data [START_REF] Louvart | The Litto3D project[END_REF], which were obtained from a LIDAR (light detection and ranging) campaign carried by the SHOM (hydrographic and oceanographic department of the marine office) and the IGN (national geographic institute) French institutes in 2015, provide a bathymetric model with a 1 m resolution and a precision of 95% (Fig. 3). These data were used for bathymetry validation in this study. A portable ASD Handheld2 sensor was used to collect the reflectance spectra for observing the abundance of the bottom classes. The spectral resolution of the sensor is 1 nm and the spectral range of measurements is from 350 nm to 1000 nm. The individual reflectance spectra of the seabed composition classes, namely Algae, Seagrass, and Sand, were measured on the boat using samples taken from the seabed by a diver. The resulting reflectance spectra showed that algae and seagrass spectra exhibit a reflectance increase around 700 nm, which is typical of vegetation targets (Fig. 4). The sand reflectance gradually increases with wavelength. Caulerpa taxifolia shows a pronounced peak at 550 nm because of its light green color. Posidonia has a seasonal cycle. On spring, new dark green leaves grow. During the summer, some leaves are colonized by epiphytes algae and the color can become lighter. Underwater Red-Green-Blue images were obtained by the Vortex Remotely Operated Vehicle (ROV) of IFREMER (French research institute for the exploitation of the sea) for a given transect between stations 1 and 4 over a distance of 1500 m. The resolution of the Nikon D5200 camera is 6000x4000 pixels, the spatial resolution of each pixel is 2.3 cm. All the images making up the mosaic were classified into 3 classes (Sand, Posidonia and Caulerpa taxifolia). The photophilic algae, located on the rocks are absent on this transect. The proportion of each class within 1-meter square along the transect was calculated. A profile can then be obtained with the proportion of each class on the transect from station 1 to 4 (Fig. 5) and can be used to validate the seabed mapping obtained with the simulated images. The water column inherent optical properties (IOP) and the concentration of water constituents were also measured for 6 selected stations (Fig. 1). The backscattering coefficients at 3 wavelengths (440 nm, 532 nm, 650 nm) were measured using the ECO-BB3 instrument from WETLabs/SeaBird (USA). The absorption and attenuation coefficients were measured using the ac-s meter from WETLabs/SeaBird (USA), which measures the spectra at 4 nm intervals. The chl concentration was measured along the water column using the multiparameter data probe HYDROLAB DS5 equipped with a fluorometric probe. The values were averaged over the water column for each station. Suspended Particulate Matter (SPM) and Colored Dissolved Organic Matter (CDOM) were also measured by collecting field samples 1 m beneath the sea surface. For SPM measurements, water samples (2l) were filtered and weighed. For CDOM measurements, 60 ml water samples were collected and the absorbance was measured in the laboratory using a spectrometer equipped with a double beam monochromator; the excitation wavelengths are between 220 nm and 600 nm with 0.2 nm resolution (UV 1800 Shimadzu).Depth was measured with the sounder of the boat (CLIPPER NASA Marine) and corrected for the sea level [START_REF] Rey | On the use of long-term observation of water level and temperature along the shore for a better understanding of the dynamics: example of Toulon area[END_REF]. Chlorophyll concentration, Suspended Particulate Matter, Colored Dissolved Organic Matter and depth for each station are given in Table 1. The in situ measurements are only used for validation purpose.

C. Methodology

The first part of the study is dedicated to the analysis of radiative transfer synthetic data to investigate the sensitivity of the various tested hyperspectral satellite sensors on the estimation of the water and seabed optical properties. The second part of study is dedicated to the simulation of a satellite image from aerial hyperspectral images.

Simulations using synthetic data

Lee's model [START_REF] Lee | Hyperspectral remote sensing for shallow waters. I. A semianalytical model[END_REF] is used to simulate remote sensing reflectance for three water types (clear, moderately turbid, and turbid), with a depth (Z) varying between 1 m and 20 m, and for 45 configurations of the seabed composition that consist of various combinations of 1, 2 or 3 materials taken from among 5 classes, namely sand, fresh Posidonia, colonized Posidonia, Taxifolia and photophilic algae. 𝑎 𝑖 corresponds to the abundance of the bottom i. Clear waters correspond to Chl=0.3 mg.m -3 , SPM=1 g.m -3 , CDOM=0.01 m -1 , moderately turbid waters to Chl=1 mg.m -3 , SPM=10 g.m -3 , CDOM=0.07 m -1 and turbid waters to Chl=5 mg.m -3 , SPM=30 g.m -3 , CDOM=0.2 m - 1 ). The satellite sensor data are also simulated (spatial resolution, bands and noise) and an error due to the atmospheric correction is taken into account. Then, the inversion of the simulated satellite radiances is performed to determine the error of estimation of both water column parameters the seabed abundance (𝑎 𝑖 ′ ) and the seabed abundance (Chl', SPM', CDOM' and Z'). The flowchart showing the method used for performing the simulations is presented in Fig. 6. Fig. 6. Flowchart of the method used for performing the theoretical simulations.

Simulation with airborne hyperspectral imagery

The second part of the study consists of simulating a BIODIVERSITY-like image to compare the retrieved estimations of the bio-optical parameters (e.g., SPM and Chl, depth) and of the seabed abundance with various types of data such as the in situ measurements, the airborne HYSPEX data and the data that will be retrieved from satellite hyperspectral sensors such as ENMAP and HICO (Fig. 7).

The input dataset is the hyperspectral aerial image with 1m and 160 spectral bands between 400 and 1000 nm acquired with the HYSPEX sensor. This image was corrected for the atmospheric effect to obtain water reflectance. The atmospheric contribution was added to simulate the atmospheric effect from the surface level to the satellite level to obtain a top of atmosphere radiance. The spatial resolution and the spectral configuration of the airborne hyperspectral image are degraded to match with the satellite sensors specifications (BIODIVERSITY, ENMAP and HICO); the sensor noise is also added. The simulated satellite image is then corrected for the atmospheric effects to derive the water reflectance, taking into account uncertainties due to the atmospheric corrections. The result of the inversion of the satellite radiance can then be compared to in situ measurements at the sea surface level as shown in Fig. 7. The details of the different steps are given in the following paragraphs. Fig. 7. Flowchart scheme of the method used for determining the water and seabed properties from airborne data.

Lee's Model

The semi-analytic model used in this study is the model developed by Lee et al. [START_REF] Lee | Hyperspectral remote sensing for shallow waters. I. A semianalytical model[END_REF] . This model provides the remote sensing reflectance (denoted Rrs) as a function of the water composition (chlorophyll concentration, SPM concentration, CDOM absorption coefficient at 440 nm, the depth and the abundance of the material composing the seabed). Inherent optical properties are calculated as the non-water absorption coefficient (𝑎 𝑛𝑤 , 𝑒𝑞. , are the constituent specific absorption or backscattering coefficients, and 𝑠 𝐶𝐷𝑂𝑀 , 𝑠 𝑁𝐴𝑃 , 𝑦 Φ and 𝑦 𝑁𝐴𝑃 are the spectral slopes of the optical properties of these constituents respectively. Φ stands for phytoplankton pigments and NAP stands for non-algal particle. The standard values of specific inherent optical properties given in [START_REF] Brando | A physics based retrieval and quality assessment of bathymetry from suboptimal hyperspectral data[END_REF] were used in this study because this study site consists of clear water type. The measurement of backscattering coefficients, absorption and attenuation coefficients were used to check that the backscattering and absorption coefficients calculated by the model were consistent with the standard constant values used. The reflectance of the water column and seabed are 𝑟 𝑟𝑠 𝐶 and 𝑟 𝑟𝑠 𝐵 respectively (eq. 4). The seabed reflectance (𝜌 𝑏 ), as given in equation ( 3), is a linear relationship between abundance and reflectance of each seabed class within the pixel. 𝑎 𝑠 , 𝑎 𝑝 , 𝑎 𝑐𝑝 , 𝑎 𝑡 , 𝑎 𝑎 are respectively the abundance of sand, Posidonia, colonized Posidonia, Caulerpa taxifolia, photophilic algae and 𝜌 𝑠 , 𝜌 𝑝 , 𝜌 𝑡 , and 𝜌 𝑎 are respectively the reflectance of each class are considered known. The sum of abundance is considered to be 1.

rrs(λ) = rrsC + rrsB (4) 𝑅 𝑟𝑠 (λ) = 0.52 𝑟 𝑟𝑠 (λ) 1-1.56 𝑟 𝑟𝑠 (λ) (3)
The rrsC and rrsB depends on the depth parameter (Z). The higher the depth, the greater rrsC and the lower rrsB and viceversa. The reader is referred to Lee et al. (1999) [START_REF] Lee | Hyperspectral remote sensing for shallow waters. I. A semianalytical model[END_REF] for more details about the model and the equations.

Simulation of the atmosphere

The atmospheric radiative transfer model MODTRAN is used to simulate the radiance at the top of the atmosphere. The Mid-latitude (45°N) summer model is selected as the atmospheric model and the marine aerosol is selected as the aerosol model with a visibility of 40 km. For the geometric configuration, the solar zenithal angle (𝜃 𝑠 ) was set to 40°, the viewing zenithal angle (𝜃 𝑣 ) at 0° and the difference of azimuthal angles Δ𝜙 at 167°, which is the relative azimuth value of the airborne measurements acquired in this study. The outputs of the MODTRAN simulation are the atmospheric reflectance, the spherical reflectance, and the direct and diffuse transmittance. The TOA reflectance (𝜌 𝑇𝑂𝐴 ) and the TOA radiance (𝐿 𝑇𝑂𝐴 ) can then be calculated using eq. ( 5) and ( 6):

𝜌 𝑇𝑂𝐴 = 𝜌 𝑎𝑡𝑚 + 𝑡 ↓ 1-𝜌 𝑠𝑝ℎ ⋅𝜌 𝑒 (𝑡 𝑑𝑖𝑟 ↑ ⋅ 𝜌 𝑤 + 𝑡 𝑑𝑖𝑓 ↑ ⋅ 𝜌 𝑒 ) (6)
where 𝜌 𝑎𝑡𝑚 is the atmospherical reflectance, 𝜌 𝑠𝑝ℎ is the spherical atmospheric reflectance, 𝑡 ↓ is the downwelling transmittance, 𝑡 𝑑𝑖𝑟 ↑ , 𝑡 𝑑𝑖𝑓 ↑ are the upwelling direct and diffuse transmittance and 𝜌 𝑒 is the environmental reflectance.

𝐿 𝑇𝑂𝐴 = 1 𝜋 ⋅ cos 𝜃 𝑠 ⋅ 𝐹 0 ⋅ 𝜌 𝑇𝑂𝐴 (7)
where 𝜃 𝑠 is the solar zenithal angle (40°), 𝐹 0 the is the solar irradiance given by MODTRAN [START_REF] Berk | MODTRAN4 user's manual[END_REF].

Consideration of the sensor radiometric and spectral specifications

The monochromatic simulated reflectances are averaged over the spectral bands of the satellite sensor (table 2). The Signal-to-Noise Ratio (SNR) simulation is operated by adding to the top of atmosphere radiance a Gaussian noise with a null mean value and a standard deviation provided for the sensor at the given band (Eq. ( 7)).

The BIODIVERSITY-like noise was calculated based on Eq. 8. The values of 𝛼() and 𝛽() were provided by the CNES for all the 88 spectral bands (CNES, personal communication).

𝜎 𝑠𝑒𝑛𝑠𝑜𝑟 (𝜆) = √𝛼(𝜆) 2 + 𝛽(𝜆) ⋅ 𝐿 𝑇𝑂𝐴 [START_REF] Nieto | Classifying benthic habitats and deriving bathymetry at the Caribbean Netherlands using multispectral imagery[END_REF] The SNR of the BIODIVERSITY-like sensor, ENMAP and HICO are shown in Fig. 8. The SNR of ENMAP and HICO sensors were obtained from the literature [START_REF] Lucke | Hyperspectral Imager for the Coastal Ocean: instrument description and first images[END_REF], [START_REF] Cerra | Restoration of Simulated EnMAP Data through Sparse Spectral Unmixing[END_REF] for a given spatial resolution of the sensor. Note that the BIODIVERSITY SNR values are consistent with the recent recommendation of the scientific community requirement for future hyperspectral sensors through the CEOS report [START_REF] Dekker | Feasibility Study for an Aquatic Ecosystem Earth Observing System Version 1[END_REF]. They cannot be compared to each other without taking into account the resolution information. The standard deviation of the noise is obtained by dividing the radiance LTOA by the SNR. For the simulation with airborne hyperspectral imagery, the spatial resolution is obtained by spatially averaging and down sampling the image to obtain the required spatial resolution.

Fig. 8. Signal-to-Noise Ratio (SNR) values used for the 3 sensors

Atmospheric correction simulation

To take into account an error due to the atmospheric correction, two other noises are added to the simulated reflectance. One is due to the error made on the surface reflectance retrieval and one is due to the error made on the aerosol model. The details of simulation are given in [START_REF] Gross | Artificial neural networks for modeling the transfer function between marine reflectance and phytoplankton pigment concentration[END_REF].

Estimation of seabed and water optical properties

The inversion of the water reflectance is achieved by minimizing the Euclidian distance between the model and the measured reflectance of each pixel the image through optimization. The minimization is operated by a curve-fitting in least-squares sense using the "lsqcurvefit" Matlab function with bounds for each parameter. The outputs of the inversion are the optimized values of Chl, SPM, CDOM, depth, and the seabed abundances of sand, Posidonia, colonized Posidonia, Caulerpa taxifolia, and photophilic algae for each pixel. At the end of the inversion process, the spatial distribution of Chl, SPM, CDOM, depth, and seabed abundances of sand, Posidonia, Taxifolia, and photophilic algae are then obtained.

D. Validation

All the estimated seabed and water parameters are compared with validation data. The metrics of the Root Mean Square Error (RMSE) and the Relative Error (RE) (Eq. 9) are used to quantify the performance of the retrieval.

𝑅𝐸( %) = 𝑎𝑏𝑠(𝑃 ̂-𝑃 𝑟𝑒𝑓 ) 𝑃 𝑟𝑒𝑓) * 100 (3)
where 𝑃 ̂ is the estimated parameters and 𝑃 𝑟𝑒𝑓 the desired values.

III. RESULTS

A. Theoretical simulations

Firstly, the influence of the seabed depth on the retrieval performance of each parameter is analyzed. About 100 simulation runs were carried out for each case to provide relevant statistics. Fig. 9 shows the relative error on the retrieval of SPM concentration when using a BIODIVERSITY-like sensor configuration for three water turbidities, namely clear, moderately turbid, and turbid waters. The relative error decreases with the seabed depth because of the stronger influence of the water column relative to the influence of the seabed, which leads to an improvement in the performance of the SPM retrieval. The relative error also decreases with turbidity. This is because the influence of the SPM on the subsurface reflectance is greater. The relative error on SPM could reach 100% in very shallow waters. It is lower than 40% for moderately turbid waters and lower than 20% for turbid waters.

Fig. 9. Variation of the relative errors of the estimated SPM concentration with the seabed depth and water turbidity when a BIODIVERSITY-like sensor configuration is used.

The variations of the relative error on the chlorophyll and CDOM retrieval performances with bathymetry and water turbidity (not shown here) are similar to those observed for SPM. Typically, the relative error on the chlorophyll estimation is greater than 70% in clear waters, around 30% in moderately turbid waters, and less than 10% in turbid waters. The relative error on CDOM absorption coefficient at 443 nm is higher than 50% in clear waters, lower than 30% for moderately turbid water and around 10% in turbid waters.

The relative error that is obtained for the retrieval of the seabed depth regularly increases with bathymetry from 10% to 30% and from 10% to 60% for clear and moderately turbid waters respectively for depths ranging between 1m and 20 m. (Fig. 10). Interestingly, the relative error remains lower than 30% for clear waters including for a seabed depth of 20 m for which the retrieval is often considered as challenging. For moderately turbid waters, higher values of the relative error than the clear water case are expected because the hydrosols mostly contribute to the sea surface reflectance for deeper waters. The relative errors found for the turbid water case sharply increase from 10% to 80% up to 6 m depth as a result of the increase of the influence of the hydrosols on the light scattering reaching the sea surface. Beyond 6 m depth, the seabed reflectance no longer influences the surface reflectance and the absolute error reaches an asymptotic value. Note that since the relative error is obtained by dividing the absolute error by the actual depth, the relative error shows a decrease beyond 6 m depth.

Fig. 10. Variation of the relative errors of the estimated seabed depth with bathymetry and water turbidity when a BIODIVERSITYlike sensor configuration.

The relative error is also analyzed with respect to the seabed composition (Fig. 11) in the clear water case. The seabed depth is much better retrieved for bright targets such as sand and colonized Posidonia than for dark targets such as the fresh Posidonia.

Fig. 11. Variation of the relative errors of the estimated seabed depth with bathymetry for various seabed compositions and for clear

water, when a BIODIVERSITY-like sensor configuration is used.

The relative errors obtained for the retrieval of the seabed material abundance are shown in Fig. 12 for the case of a seabed composed of a pure material and Fig. 13 for the case of a seabed composed of mixed materials. The error increases with depth as a result of the increasing influence of light scattering by hydrosols of the water column (Fig. 12). The material abundance is also better estimated on bright targets (Sand) than on dark target (Posidonia).

Fig. 12. Variation of the relative errors of estimated seabed material abundance (in %) with bathymetry for seabed composed of pure materials in clear water, when a BIODIVERSITY-like sensor configuration is used.

A pixel composed of mixed material (e.g. 1/3 Sand, 1/3 fresh Posidonia and 1/3, also demonstrates that the seabed material abundance is better estimated on target (sand and Caulerpa taxifolia) than on dark target (fresh Posidonia) (Fig. 13). In particular, in clear water the errors are lower than 5% when the bathymetry is lower than 10 m. Note that the performance of the retrieval significantly deteriorates for Fresh Posidonia where the error rapidly increases up to 100% when the bathymetry is higher than m. Thus, a dark target is not well determined for deep waters.

Fig. 13. Variation of the relative errors of the estimated seabed material abundance (in %) with bathymetry for seabed composed of mixed materials, namely 1/3 of Sand, 1/3 of fresh Posidonia, 1/3 of

Caulerpa taxifolia species for BIODIVERSITY-like sensor configuration.

Table 3 presents the RMSE on the estimation of water column parameters and seabed material abundances in clear waters for a BIODIVERSITY-like sensor and other sensors including ENMAP, and HICO. If the noise level (SNR) is given for a resolution noted R, the noise level at a coarser resolution (noted R') is given by SNR'=SNR/(R'/R). Then R'>R and SNR'>SNR.

The band features of the sensors are simulated by integration between 400 and 700 nm. It is observed that the RMSE decreases with the spatial resolution of the sensor (see first 2 columns of Table 3, which is likely to be due to the averaging effect on the noise. For the given spatial resolution of 30 m, lower values of RMSE are obtained for the BIODIVERSITY-like sensor relative to the ENMAP sensor because of its SNR. The RMSE obtained for ENMAP is lower than HICO for a spatial resolution of 90 m while the BIODIVERSITY like sensor performs better than both ENMAP and HICO for all the parameters for the same spatial resolution. Finally, when the resolution and the SNR are fixed, the RMSE increases when the number of bands decreases. When the number of bands and the SNR are fixed, the RMSE decreases when the spatial resolution decreases.

When the spatial and the spectral resolution are fixed, the RMSE increases when the SNR decreases. But as the SNR of the BIODIVERSITY-like sensor is better than ENMAP's and HICO's at the same spatial resolution, the RMSE is lower with BIODIVERSITY's 26 bands than with ENMAP'43 bands or HICO's 50 bands. This means that a high SNR can balance with a low number of bands and inversely.

B. Satellite simulation with airborne data

The BIODIVERSITY-like simulated image is shown in Fig 14 without and with noise. Even low, noise is visible because the land has been removed from the images and a linear histogram stretch has been applied on the water area. The retrieved maps of water quality parameters derived from the airborne data acquired using the HYSPEX sensor for which the radiance has been propagated to the satellite level by considering both the atmospheric radiance and the BIODIVERSITY-like sensor noise are shown in Fig. 15.

Fig. 15. Retrieved maps of water column parameters and bathymetry derived from the inversion of the airborne data once they have been propagated to the satellite level: (top) for noise-free data, (bottom) for noisy data.

When the noise is ignored in the satellite data, the concentration of Chl is low (<0.2 mg.m -3 ) except near the beach (>0.2 mg.m -

3

). The SPM is quite high (>1 g.m -3 ) at intermediate seabed depths, typically between 5 m and 10 m. CDOM absorption coefficient is weak (<0.05 m -1 ) over the whole area. Bathymetry logically increases with the distance from the beach. The retrieved parameters can be compared with in-situ measurements. For that purpose, the error and the relative error obtained for each parameter and each station are reported in Table 4.

Table 4. Absolute error (AE) and relative error (RE) of the retrieved water column and bathymetry parameters calculated using the in-situ measurements collected over the 6 sampling stations for the cases where the noise is accounted for in the data. Note that the absolute error is low for Chl, SPM and CDOM while the relative error is high because their concentrations are weak thus not significantly influencing the reflectance; the retrieval performance of these variables with a high degree of accuracy remains challenging using semi-analytical model inversion. Considering the bathymetry, the relative error is lower than the other parameters while the absolute error is high because of the considerable depth of the stations: four of them are depths of over 10 m (Table 1). Maps of the retrieved seabed material (or classes) are shown in Fig. 16. The materials that are derived from the data are Sand, Posidonia, Taxifolia and Photophilic algae. The Sand component is identified close to the beach. Posidonia covers the largest area while the occurrence of Taxifolia is restricted to a small area. Photophilic algae are located on the rocks which is consistent with in situ observations. Fig. 16. Retrieved abundance of seabed materials, namely Sand, Posidonia, Caulerpa taxifolia and photophilic algae, from the inversion of satellite data simulated for the BIODIVERSITY-like sensor configuration.

The retrieval of the seabed material abundance using the BIODIVERSITY-like simulated image can be compared with the in situ profile (Fig. 5) acquired using the IFREMER ROV along a transect between the sampling station and 4 (station 1, 2 3 and 4) in Fig. 1. The four stations (1 to 4) are area of clear water. Fig. 17 shows the measured material abundance (top), the measured bathymetric profile and the retrieved material abundance from the inversion of the simulated satellite data for the cases where the noise is ignored and taken into account. For the case where the noise is ignored, the sand component is correctly mapped in shallow waters (<10 m depth), but not in deep waters (>10 m). This because the sand is mixed with Posidonia and Taxifolia in deeper waters. Posidonia is appropriately classified for all depths because even though Posidonia is a dark seabed, pixels are composed of 100% of Posidonia, making its detection easier (Fig. 12). Taxifolia is detected even at 10 m depth but its abundance is underestimated. When the noise is taken into account, the seabed components retrieval is not significantly different from the noise-free case.

To compare the BIODIVERSITY-like sensor performances for the estimation of water column parameters with those of other hyperspectral sensors (e.g. ENMAP, HICO), its resolution was downgraded from 8 m to 30 m and 90 m. The RMSE values obtained for different spatial and spectral resolutions are calculated (Table 5). At 30 m resolution, all the parameters are better estimated using the BIODIVERSITY-like sensor configuration than for the ENMAP sensor. Similar conclusions are observed at 90 m when compared with both HICO and ENMAP.

IV. DISCUSSION

Despite multiple limitations, the hyperspectral remote sensing has developed as a potentially promising technology to study marine and freshwater benthic maps as well as bathymetry [START_REF] Vahtmäe | How much benthic information can be retrieved with hyperspectral sensor from the optically complex coastal waters?[END_REF]. The detectable information is restricted within a maximum depth depending on the sensor characteristics and environmental factors.

The results of the synthetic data simulations performed in this study, for which the noise affecting the top of atmosphere data was rigorously taken into account, provide an insight into the maximum possible accuracy and associated errors with the given configuration of the seabed and the sensor characteristics. The simulations with in situ data provide a realistic understanding of the accuracy and associated errors for the same configuration of the seabed and the sensor characteristics. An important factor that affects the accuracy of the estimated water column or seabed parameters is the amount of information that the signal contains from the seabed or water column. The stronger the signal from the water column, the more accurate the estimation of the water column parameters; a similar tendency is observed for the seabed abundances. The radiation from the seabed is higher for clearer waters while a higher proportion of the signal from the water column is observed for turbid waters, which explains the RMSE trends in the estimations from the theoretical simulations. Moreover, the amount of individual spectral information contained in the signal also influences the accuracy of estimation. Depth estimation from remote sensing data depends on the physical changes in the spectral radiance that is reflected from the seabed which is correlated with the water depth variation. However, beyond a threshold depth, the increase in depth affecting the benthic reflectance is too small to be resolved by the sensor and inversion models [START_REF] Vahtmäe | How much benthic information can be retrieved with hyperspectral sensor from the optically complex coastal waters?[END_REF], [START_REF] Legleiter | Defining the limits of spectrally based bathymetric mapping on a large river[END_REF].

Nevertheless, the focus of this work was on the simulation of the BIODIVERSITY-like sensor which provided an understanding of its suitability for the estimation of the water column and seabed parameters with the given characteristics such as the position and number of the measurement bands in the spectrum, the spatial resolution and the SNR of the sensor. The goal of this work to evaluate the potential of a BIODIVERSITY-like sensor in comparison with ENMAP and HICO sensors was accomplished as demonstrated in the results. Remarkably, the error in the estimation of the parameters with the BIODIVERSITY-like sensor is generally lower despite having a fewer number of bands as compared to the HICO and ENMAP sensors in the same spectral resolution as can be observed from the results of the theoretical simulations and the satellite simulation using airborne data (Table 3 andTable 5). In this study the information provided by each sensor (spatial and spectral resolutions and SNR) was analyzed and it has been shown that a high SNR can balance with a low number of bands and vice-versa. Note that these results could be degraded if a lower SNR value concerning? BIODIVERSITY sensor was used.

The difficulty observed in the estimation of the proportion of bottom classes in seabed mapping when the seabed is composed of multiple classes is associated with the difficulty in separating individual spectral signatures from an already weak signal modulated by the water column attenuation. As the depth increases, the attenuation increases, and the signal is further weakened and thereby larger errors occur in the estimations, which explain the difficulty in estimating the bathymetry for depths higher than 10 m.

The results in this work are remarkable particularly when compared with a recent study that estimated the detectability limits for seabed estimation for the CASI hyperspectral sensor in the complex waters of the Baltic sea [START_REF] Vahtmäe | How much benthic information can be retrieved with hyperspectral sensor from the optically complex coastal waters?[END_REF]. Vahtmäe et al. observed that the detectability threshold for the brightest seabed composed of sand was 7.5 m, 5 m for brighter benthic flora of green macroalgae, and 3 m for darker benthic flora composed of higher plants and brown macroalgae. Moreover, they observed that the bathymetric mapping was possible in the coastal waters of the Baltic Sea up to a maximum of 4 m depth whereas the BIODIVERSITY sensor can estimate the depth up to10 m with an error lower than 40% even in moderately turbid waters (fig. 10). This can be due to the lower sensitivity of CASI to the bottom reflectance. Despite the fact that the BIODIVERSITY-like sensor is a satellite sensor that could observe through a thicker layer of atmosphere and that is designed with more spectral bands than the airborne CASI sensors, the BIODIVERSITY-like configuration is likely to perform better than the CASI sensor in a similar environment.". Louchard et al. [START_REF] Louchard | Optical remote sensing of benthic habitats and bathymetry in coastal environments at Lee Stocking Island, Bahamas: A comparative spectral classification approach[END_REF] and Mobley et al, [START_REF] Mobley | Interpretation of hyperspectral remote-sensing imagery by spectrum matching and lookup tables[END_REF] used spectral classification and a lookup-table based approach to map the benthic habitats and to bathymetry from portable hyperspectral imager (PHILLS) airborne sensors for clear and shallow coastal waters. The current study focused on the viability of a potential satellite attached BIODIVERSITY-like sensor for different water types (by simulation). Louchard et al., [START_REF] Louchard | Optical remote sensing of benthic habitats and bathymetry in coastal environments at Lee Stocking Island, Bahamas: A comparative spectral classification approach[END_REF] were able to estimate depth with a mean accuracy of 83% with the airborne sensor while Mobley et al. [START_REF] Mobley | Interpretation of hyperspectral remote-sensing imagery by spectrum matching and lookup tables[END_REF] demonstrated an accuracy of 95.3% on average for constrained inversion and up to 87.4% for unconstrained inversion. Villa et al., [START_REF] Villa | A rule-based approach for mapping macrophyte communities using multi-temporal aquatic vegetation indices[END_REF] implemented a rule-based approach for mapping functional types of macrophytes in the seabed and obtained 90.1% accuracy with up to 20% misclassified mixed sands. Kotta et al., [START_REF] Kotta | In-air spectral signatures of the Baltic Sea macrophytes and their statistical separability[END_REF] investigated the measured reflectance spectra of major macrophyte species in the Baltic sea by using hyperspectral datasets and adopted a methodology to allow statistically significant discrimination of spectral signatures of various macrophyte species.

Most studies on seabed mapping to date were performed for shallow waters with a depth of often less than 5 m while the present study is performed for waters up to 20 m depth. The relatively low errors in depth estimation (<20%, except for Posidonia in shallow waters) for depths up to 20 m using the sensor characteristics in this study in conjugation with the inversion model can be a promising approach for bathymetry estimation which is critically important in the fields of navigation safety, planning of marine farming, delimitation of protected areas, engineering of nearshore infrastructures (e.g. Ports and wind energy structures). A synoptic and rapid method such as using hyperspectral remote sensing sensors for the bathymetry chart preparation can aid in situations where the hydrographic survey is difficult. Moreover, it can also help in cost reduction and time efficiency as compared to a shipborne bathymetric survey even when the latter is feasible [START_REF] Vahtmäe | How much benthic information can be retrieved with hyperspectral sensor from the optically complex coastal waters?[END_REF].

Additionally, the ability of the BIODIVERSITY sensor in the benthic habitats mapping as revealed in this study can be expected to have substantial impacts in the ecological studies after a potential setting up of the studied sensor in a satellite platform. The results and the conclusions in this study can be used as advisory with the implications for the new launch of satellite missions taking into consideration the sensor characteristics studied.

V. CONCLUSION

The present study was conducted using both synthetic data and data from aerial hyperspectral images of the coastal zone of Porquerolles Island (France). The results showed that the BIODIVERSITY-like sensor configuration characterized by 28 spectral bands on the range 400-700 nm (53 bands on the range 400-1000 nm), 8 m resolution and a SNR of 200 at 400 nm will be able to better estimate the water column parameters including the chlorophyll concentration, the SPM, the CDOM and the seabed depth than the retrieval obtained for the ENMAP and HICO spectral and radiometric configurations. The BIODIVERSITY-like sensor was able to provide estimates of the Chl, SPM, CDOM and depth with low RMSE of 0.41 mg.m -3 , 0.43 g.m -3 , 0.06 m -1 and 3.09 m corresponding respectively to relative errors of 101%, 81%, 640% and 28%. The absolute error is low for Chl, SPM and CDOM while the relative error is high because their concentrations are weak thus not significantly influencing the reflectance. Considering the bathymetry, the relative error is lower than the other parameters while the absolute error is high because of the considerable depth of the stations: four of them are located at a depth of over 10 m. The BIODIVERSITY-like sensor will allow the seabed to be mapped for the pixels composed of only one material (e.g., sand or Posidonia). The false detections can be reduced, especially for depths of less than 10 m, and remarkably for Posidonia cover. Moreover, the abundance estimations are also fairly accurate where less than 25, 35, and 45 % errors respectively for sand, fresh Posidonia, and Caulerpa taxifolia were observed (Fig. 12). On the other hand, there were difficulties in retrieving the pixels composed by several materials (e.g., Sand and Posidonia). Further work could consist in taking into account the environmental effect of surrounding seabed in the inversion.

Fig. 1 .

 1 Fig. 1. Study area of Porquerolles Island (France) and in situ sampling stations measurements (red square).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Hyperspectral image acquired by the HYSPEX sensor abovethe study area on Sept, 13, 2017.

Fig. 4 .

 4 Fig. 4. In-situ measurements of the hyperspectral reflectance of the various components of the seabed, namely the seagrass, sand and algae.

Fig. 5 .

 5 Fig. 5. Proportion of each class composing the seabed, namely Sand, Posidonia and Caulerpa taxifolia within a 1-meter square resolution on the transect between the sampling stations 1 and 4.

  𝑎 𝑛𝑤 (λ) = 𝐶ℎ𝑙 * 𝑎 Φ * (λ) + 𝐶𝐷𝑂𝑀 * 𝑒 𝐶𝐷𝑂𝑀 (λ-440) (λ) + 𝑆𝑃𝑀 * 𝑎 𝑁𝐴𝑃 * (440)𝑒 -𝑆 𝑁𝐴𝑃 (λ-440) (1)

Fig. 14 .

 14 Fig. 14. Color composite of the simulated BIODIVERSITY-like image without and with noise (bands 440, 552, 680 nm)

Fig. 17 .

 17 Fig. 17. (top) in-situ measurements of the seabed material abundances acquired by the ROV (IFREMER) along the transect going from station 1 to station 4,(middle) bathymetry along the transect, (bottom) retrieved seabed composition for the cases where the noise is ignored and taken into account and for satellite data simulated for the BIODIVERSITY-like sensor configuration.The left side of Fig.17corresponds to shallow waters (~ 6 m) and the right side corresponds to deep waters (12 m). For the case where the noise is ignored, the sand component is correctly mapped in shallow waters (<10 m depth), but not in deep waters (>10 m). This because the sand is mixed with Posidonia and Taxifolia in deeper waters. Posidonia is appropriately classified for all depths because even though Posidonia is a dark seabed, pixels are composed of 100% of Posidonia, making its detection easier (Fig.12). Taxifolia is detected even at 10 m

  corresponds to shallow waters (~ 6 m) and the right side corresponds to deep waters (12 m).

Table 1 .

 1 In situ measurements of water column parameters

	Station	Chl	SPM	CDOM	Depth (m)
		(mg/m 3 )	(g/m 3 )	(m -1 )
	1	0.265	0.46	0.01	2.73
	2	0.481	0.44	0.01	10.15
	3	0.485	0.21	0.01	11.83
	4	0.433	0.65	0.06	9.74
	5	0.447	0.69	0.01	24.04
	6	0.413	0.47	0.01	15.33

Table 2 .

 2 Spectral and spatial resolutions of the hyperspectral sensors examined in this study. FWHM is the Full Width Half Maximum.

	Sensor	Total number	FWHM	Domain	Spatial
		of	(nm)	(nm)	resolution
		bands/number			(m)
		of bands in
		the range
		[400-700 nm]
	HYSPEX	160/79	4,5	404-990	1 m
	Biodiversity	53/26	10	413-990	8 m
	ENMAP	88/43	8.2	415-	30 m
				1000
	HICO	87/50	10	410-	90 m
				1000

Table 3 .

 3 of RMSE calculated for the estimated water column parameters and seabed material abundances in clear waters using various hyperspectral satellite sensors.

	Sensor	Biodiversity Biodiversity ENMAP Biodiveristy ENMAP	HICO
	Spat. Res.	8m	30m	30m	90m	90m	90m
	Spect. Res.	26 bands	26 bands	43 bands	26 bands	43 bands 50 bands
	Chl (mg.m -3 )	0.5687	0.2782	0.4361	0.1018	0.2159	0.4584
	SPM(g.m -3 )	0.5863	0.2796	0.3525	0.1477	0.1920	0.5925
	CDOM (m -1 )	0.0223	0.0056	0.0124	0.0023	0.0048	0.0106
	Z (m)	4.1775	1.7717	2.6241	0.9158	1.3277	2.9958
	Sand	0.2278	0.1269	0.1626	0.0913	0.1125	0.1536
	Fr. Posidonia	0.5037	0.4132	0.4537	0.3248	0.3699	0.4799
	Col. Posidonia	0.3812	0.2815	0.3374	0.2043	0.2525	0.3322
	Caulerpa taxifolia	0.3194	0.1970	0.2314	0.1251	0.1415	0.2427
	Phot. Algae	0.7732	0.4453	0.5505	0.2744	0.3386	0.5145

407 101% 0.446 81% 0.071 640% 3.545 28%

		AE	RE (%)	AE	RE (%)	AE	RE (%)	AE	RE
	1	0.392 148% -0.310 -67% 0.077 771% -0.790 -29%
	2	0.389 81% -0.333 -76% 0.088 879% 2.899 29%
	3	0.417 86% -0.108 -52% 0.075 745% 3.808 32%
	4	-0.413 -95% 0.451 69% 0.034 56% 3.015 31%
	5	-0.427 -96% 0.412 60% 0.075 752% -5.490 -23%
	6	-0.403 -98% 0.777 165% 0.064 636% -3.554 -23%
	Total	0.

Station Chl (mg.m -3 ) SPM (g.m -3 ) CDOM (m-1) Depth (m)

Table 5 .

 5 Comparison of the RMSE for the retrieval of water column parameters (Chl, SPM, CDOM, seabed depth) between the simulated datasets and the in-situ measurements collected in the sampling stations.

		Biodiversity	ENMAP	Biodiversity	ENMAP	HICO
		degraded to		degraded to	degraded to
	RMSE	30m	30m	90m	90m	90m
			43			50
		26 bands	bands	26 bands	43 bands	bands
	Chl (mg.m -3 )	0.3900	0.4880	0.4000	0.3200	0.4200
	SPM (g.m -3 )	0.3800	0.4100	0.3600	0.4900	0.5600
	CDOM (m -1 )	0.0700	0.0500	0.0900	0.0800	0.0900
	Z (m)	4.1900	6.8100	0.9800	4.5500	5.4300

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JSTARS.2020.3031729, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing

AKNOWLEDGEMENTS

The authors would like to thank the French Spatial National Agency (CNES) for the funding of the TOSCA/HYPCOLAC project. We are grateful to Manchun Lei (IGN) for the atmospheric simulation, Thierry Tormos (Onema) and Tristan Harmel (GET) for their help during the in situ campaign and Xavier Briottet (ONERA) for helpful discussions about the BIODIVERSITY mission. The data campaign was also funded by the ANR/DGA (French Research Agency/French Defense Agency), on the behalf of the HypFoM project ANR-15-ASTR-0019.


Sylvie Su

David Cébron

Henri-Claude Nataf

Philippe Cardin

Jérémie Vidal

Max Solazzo

Yann Do

Acoustic spectra of a gas-filled rotating spheroid

Keywords: acoustic resonances, spheroid, velocimetry, rotation Gas-filled rotating spheroid acoustic spectra 1

The acoustic spectrum of a gas-filled resonating cavity can be used to indirectly probe its internal velocity field. This unconventional velocimetry method is particularly interesting for opaque fluid or rapidly rotating flows, which cannot be imaged with standard methods. This requires to (i) identify a large enough number of acoustic modes, (ii) accurately measure their frequencies, and (iii) compare with theoretical synthetic spectra. Relying on a dedicated experiment, an air-filled rotating spheroid of moderate ellipticity, our study addresses these three challenges. To do so, we use a comprehensive theoretical framework, together with finite-element calculations, and consider symmetry arguments. We show that the effects of the Coriolis force can be successfully retrieved through our acoustic measurements, providing the first experimental measurements of the rotational splitting (or Ledoux) coefficients for a large collection of modes. Our results pave the way for the modal acoustic velocimetry to be a robust, versatile, and non-intrusive method for mapping large-scale flows.

INTRODUCTION

The acoustic response of a resonant cavity is directly related to its geometry and physical properties. In the past, seismology has used those free oscillations to probe the Earth's interior 1,2 . This method has successfully contributed to determine properties such as density, elasticity and anisotropy parameters taking into account the rotation and the shape of the Earth 3 . Similarly, eigenmodes measurements have also been developed in helio 4 and asteroseismology 5 , giving insights on the stellar interiors 6,[START_REF] Aerts | Asteroseismology[END_REF] .

Inspired by asteroseismology, a recently presented laboratory velocimetry method relies on acoustic resonances to invert flows within a spherical shell [START_REF] Triana | [END_REF] . By contrast with usual velocimetry techniques, for example particle tracking methods such as Particle Image Velocimea) david.cebron@univ-grenoble-alpes.fr try (PIV) 9 or methods based on Doppler effect such as laser Doppler [START_REF] Albrecht | Laser Doppler and Phase Doppler Measurement Techniques[END_REF] and ultrasound Doppler velocimetry [START_REF] Brito | [END_REF]12 , modal acoustic velocimetry does not require a seeded fluid as it measures the fluid acoustic response directly [START_REF] Triana | [END_REF] . This is of great interest for all experiments where seeding the fluid is not practically feasible, in particular for gases [START_REF] Melling | 12 Gas-filled rotating spheroid acoustic spectra 14 Beebe R[END_REF] . Another benefit is that modal acoustic velocimetry is a non-intrusive technique providing flow fields in the whole volume, even in opaque fluids.

Interested in zonal jets, their formation, geometry, and amplitude in geo-astrophysical flows [14][15][16] , we built a reduced model that obeys similar force balances. Our apparatus is a rotating oblate spheroid called ZoRo (Zonal flows in Rotating fluids). It has been shown that in some cases 17 , the zonal flows are geostrophic, i.e. invariant along the axis of rotation of the fluid, forming concentric cylindrical shells 18,19 . Such flows are expected to have significant influence on acoustic global modes [START_REF] Aerts | Asteroseismology[END_REF][START_REF] Triana | [END_REF] . Following previous experimental studies [START_REF] Triana | [END_REF]20 , we stress that three challenges must be met in order to effectively map these flows: (i) identify a large enough number of modes; (ii) accurately measure their frequencies; (iii) compare with theoretical synthetic spectra that take into account the effect of rotation. The present study addresses these three goals, and focuses on measuring and computing the effect of rotation on a large collection of acoustic modes in a spheroid.

The paper is organised as follows. Section I presents the ZoRo experimental apparatus. In Section II, we present the theoretical framework we use to predict the acoustic response of our experiment. In Section III, we analyse and interpret experimental acoustic spectra. To refine this interpretation, in Section IV we use a second-order geometrical perturbation theory, and test it against spectral and finite-element calculations. Then, in Section V we introduce rotation within the experiment, and compare the measured acoustic response to perturbation theory. Finally, Section VI summaries the different approaches and presents some perspectives.

I. THE ZORO EXPERIMENT

The experimental apparatus ZoRo is a 1 cm thick shell made with aluminium based alloy (Thyssenkrupp by Constellium cast, MecAlu+ 7000 series) enclosing an axisymmetric oblate spheroidal cavity of equatorial radius r eq = 20 cm and polar radius r pol = 19 cm filled with air (see Figure 1). The ellipticity is defined as e = (r eqr pol )/r eq = 0.05. Special care has been given during the fabrication process, especially to the junction plane between the two hemispheres and the coaxiality (both between the hemispheres and the shafts) which have been identified in the literature as common defaults 21 . This allows us to ensure the dimensions of the shell down to 0.1 mm. Chosen excerpts of technical drawings are provided in Supplementary material 22 than 0.2 g of unbalanced mass (for a total rotating spheroid mass of 32.5 kg).

Acoustic pressure is measured by electret microphones (ref.

Projects Unlimited TOM-1545P-R) connected to a mixing C and atmospheric pressure, the cavity's fundamental mode is at 578 Hz. Both source and data files are sampled at 44.1 kHz and written in 16-bits in the uncompressed audio specific Waveform Audio File Format (WAVE), corresponding to standard CD audio quality.

The instrumentation is in contact with the gas in through holes in the aluminium shell. Air-tightness is ensured by custom made plastic joints on each hole and at the equatorial seam of the shell. Additional holes (see ), provided for future temperature and pressure sensors, are plugged closed. The apparatus can accommodate up to four speakers and fourteen electrets, half on each hemisphere, at respectively ±45 • and ±32 • latitude (Fig. 1). All speakers are on the same meridional plane, electrets are evenly spaced on their latitude; lower and upper hemispheres are strictly symmetric (for technical drawings, see Supplementary material 22 ). Electric signals are transmitted from the rotating to the laboratory immobile frames through two slip rings (PSR-HSC-36 and PSRT-38H-24 from Panlink) with gold-gold contacts.

II. THEORETICAL FRAMEWORK

A. Governing equations and boundary conditions

We consider the small perturbations [ρ, p, u, T ] upon a homogeneous basic (background) state of density ρ 0 , pressure p 0 , velocity u 0 , and temperature T 0 . We thus linearize the governing flow equations for a Newtonian fluid (see Supplementary material 23 ), assuming u 0 = 0 in the reference frame that is rotating at the constant rate Ω = Ω 1 z (with 1 z the unit vector along the polar axis). For uniform viscous and thermal diffusivities, the equations are 24

∂ρ ∂t = -∇ • (ρ 0 u), (1a)
ρ 0 ∂u ∂t = -2ρ 0 Ω × u -∇p + µ∇ 2 u + µ B + µ 3 ∇(∇ • u), (1b)
ρ 0 C p ∂T ∂t = αT 0 ∂p ∂t + λ∇ 2 T, (1c)
ρ = ρ 0 (β T p -αT ), (1d)
using the usual underlying assumptions (which removes p 0 from the dynamical equations, see the Supplementary material 23 for details). In equations (1a)-(1d), t is the time, λ is the thermal conductivity, α is the isobaric coefficient of thermal expansion, C p is the specific heat at constant pressure, β T = γβ s is the isothermal compressibility, where γ is the specific-heat ratio and β s is the isentropic compressibility, µ is the dynamic (shear) viscosity, and with the bulk viscosity µ B , which is related to the second coefficient of viscosity µ B -2µ/3. Note that the sound speed c is then given by c = (ρ 0 β s ) -1/2 . To solve the system of equations (1a)-(1d) we supplement them with boundary conditions. Assuming that the sound speed is much larger in the container than in the cavity, we consider a rigid spheroidal boundary and impose the noslip boundary condition u = 0. We also assume that the container thermal conductivity is much larger than λ and impose an isothermal boundary condition.

Finally equations (1a)-(1d) can be solved in the frequency domain by looking for time periodic solutions for [ρ, p, u, T ]. With an eigensolver, we can then obtain the fluid eigenmodes such as the acoustic modes [START_REF] Vidal | [END_REF] . One can also rather calculate the spectral fluid response if we define excitation sources for [ρ, p, u, T ], e.g. by modelling the experimental speakers using (imposed) time periodic flows at the speaker locations.

B. Principles of perturbation theory

We seek the fluid normal eigenmodes with a harmonic time dependence exp(iωt), where ω is the (possibly complex) pulsation. The system of governing equations (1a)-(1d) can be cast into a symbolic eigenvalue equation as

Hv = -ω 2 v, (2)
where H is a complex vectorial operator linear in v, which can be the velocity u or the Lagrangian displacement for isentropic fluids 5 ; this equation being supplemented by appropriate boundary conditions. Analytical solutions of the complete problem are usually not available. A convenient and classical approach is to define a reference model, solution of equation ( 2) with H = H 0 and simple boundary conditions, for which analytical solutions are available. The missing terms of the original operator H and boundary conditions are then treated as perturbations [START_REF] Morse | Methods of Theoretical Physics[END_REF][START_REF] Dahlen | Theoretical Global Seismology[END_REF] .

After choosing our reference model, we successively consider perturbations due to (i) Coriolis force, (ii) elliptical shape of the rigid shell, and (iii) dissipation. First-order perturbations are often sufficient, with the advantage that the frequency perturbation can be computed without calculating the corresponding eigenvector. However, accurate calculation of the modes may require higher-order perturbations [START_REF] Vidal | [END_REF] , for instance for the Coriolis force 1 or the elliptical shape [START_REF] Mehl | [END_REF] .

C. Modes of the reference model

We aim to study theoretically the homogeneous motionless basic state, and its eigenmodes, defined in section II A. Without loss of generality, we tackle this problem by considering, for our reference model, a diffusionless gas enclosed in a non-rotating sphere of (arbitrary) radius a. The latter value remains unspecified for the moment (see below). Equation ( 2) becomes

c 2 ∇ (∇ • u) = -ω 2 u.
We introduce the velocity potential Ψ such that u = ∇Ψ, which relates to the acoustic pressure in the gas by p = -∂Ψ/∂t. Then, problem (2) reduces to the Helmholtz equation

c 2 ∇ 2 p = -ω 2 p. (3)
In spherical coordinates (r, θ, ϕ), the solutions of equation ( 3) that satisfies the boundary condition ∂p/∂r = 0 are

p(r, θ, ϕ) = p 0 j l (k nl r) Y m l (θ, ϕ), (4)
where k nl is the radial wavenumber, Y m l is the spherical harmonic of degree l and order m and yields the angular dependency, j l is the spherical Bessel function of the first kind of degree l and carries the radial dependency (we remind that |m| ≤ l). The boundary condition dj l (k nl r)/dr| r=a = 0 completes the quantification of the modes, yielding a radial mode number n, labelling the n th zero of dj l (k nl r)/dr.

One (n, l, m) triplet fully characterises a given acoustic mode of our spherical enclosure. We denote it n S m l , following the convention of seismologists [START_REF] Dahlen | Theoretical Global Seismology[END_REF] . The spherical symmetry of our reference model implies that the frequencies f nl of the modes are independent of the azimuthal mode number m. One gets f nl = c k nl /(2π). The perturbations we consider lift this degeneracy partially or totally. They also modify the corresponding eigenvectors. However, since we consider rotation of the spheroid around its symmetry axis, we can still label the perturbed modes as n S m l .

D. Perturbations

Coriolis splitting

The Coriolis term -2ρ 0 Ω × u in equation (1b) breaks the ±ϕ-symmetry. As a consequence, the degenerate spectral peak of a n S l multiplet splits into 2l + 1 peaks, corresponding to all n S m l singlets. According to first-order perturbation theory, the frequency shift δ Ω of a n S m l singlet with respect to the n S l degenerated frequency is 5

δ Ω = -m f Ω C nl , (5)
where 0 ≤ C nl < 1 are called the Ledoux coefficients, and noting f Ω = Ω/(2π). Their expression is 7

C nl = a 0 [2 ξ r ξ h + ξ 2 h ] r 2 dr a 0 [ξ 2 r + l(l + 1) ξ 2 h ] r 2 dr , (6)
with ξ r = dj l (k nl r)/dr and ξ h = j l (k nl r)/r. We list the values of Ledoux coefficients for a chosen collection of acoustic modes in Supplementary material 23 . One of the main goals of our study is to retrieve the frequency splittings due to the Coriolis force in the ZoRo experiment, and compare them to predictions 5 .

Note that the splitting of acoustic modes in a gas-filled rotating cylinder has been used to design gyrometers 20 , and that seismologists 1 developed the second-order perturbation theory for the Coriolis term, in order to match the observed splitting of the lowest normal modes of the Earth.

Geometrical splitting

Coriolis splitting is very small for acoustic overtones with large n. In order to measure the splitting of an individual n S ±m l doublet, it is desirable to separate it from the other doublets of its n S l multiplet. This was the reason for designing the ZoRo experiment as an axisymmetric oblate spheroid.

First-order perturbation theory for elliptical flattening show that the induced frequency splitting δ geom is quadratic in m. Introducing γ nl the ellipticity splitting coefficient, the frequency perturbation writes 3,29

δ geom = f nl - 1 3 l(l + 1) + m 2 γ nl e, (7)
where f nl is the frequency of the n S l mode in the spherical reference model (with the same volume as the spheroid), and e the ellipticity. We give the expression of γ nl , and its value for a collection of modes, in Supplementary material 23 . We obtain the same first-order results using formulas from either Dahlen 29 or Mehl [START_REF] Mehl | [END_REF] provided the radius of the reference sphere is the same. We will see later that we need to consider second-order geometrical perturbations, which have been derived by Mehl 28 using Morse and Feshbach's formalism 26 .

Dissipation

In order to construct realistic spectra that can be compared with observations, it is essential to take into account dissipative effects, which control the width and the amplitude of the resonance peaks 30,[START_REF] Trusler | Physical Acoustics and Metrology of Fluids[END_REF] . Dissipation of acoustic modes in air is dominated by viscous friction and heat diffusion at solid boundaries, with minor contributions from bulk viscosity 30 .

Considering the spherical reference model, perturbation methods [START_REF] Kirchhoff | [END_REF]33 provide estimates for the diffusion effect g nl = g bound + g bulk , where g bound and g bulk are respectively the boundary and bulk contributions. The mode complex frequencies can then be written as f nlg bound + ig nl , where g bound and g bulk take positive (real) values 30 . Note that the imaginary part (i.e. the damping of the mode) is due to g nl but only g bound affects the real part, reducing the (real) eigenfrequency. Assuming that the container is a much better heat conductor than the gas, dissipation depends only upon physical properties of the gas: kinematic viscosity ν = µ/ρ and bulk viscosity ν bulk = µ B /ρ 0 , thermal diffusivity κ = λ/(ρ 0 C p ), and the adiabatic index γ. The g bound contribution is then given in Moldover 30 , equation (42), by

g bound = f nl 2a (γ -1) d T + l(l + 1) d U /z 2 nl 1 -l(l + 1)/z 2 nl , (8)
which, in our case, is much larger than the bulk contribution, corrected by 34

g bulk = z 2 nl 4πa 2 (γ -1)κ + 4 3 ν + ν bulk , (9)
where z nl = k nl a is the dimensionless radial wave number, and d T = κ/(πf nl ) and d U = ν/(πf nl ) are the thicknesses of the thermal and viscous boundary layers respectively. We use air thermodynamic properties 35 , see also https://encyclopedia.airliquide.com/ fr/air, to calculate these contributions and list our values of g nl coefficients for a collection of acoustic modes in the Supplementary material 23 .

Other perturbations

Other effects might influence the spheroid's acoustic spectrum. We list and explore a few of them that are mentioned in the literature as potentially relevant. They include uneven shell surface finishing, presence of holes in the shell or a seam between the two hemispheres. All of those are expected to be negligible for our apparatus 30 .

The finite elasticity of the container may modify significantly the acoustic resonances when acoustic and elastic eigenfrequencies are close 30 . Influences of the shell elasticity of our apparatus have been estimated with analytical calculations 36,37 and cross-checked with finite-element simulations, as detailed in Supplementary material 23 . Both methods predict frequency shift of 0.1 -1 Hz far from the elastic resonances, and much larger when elastic and acoustic resonances are close. However, due to the high complexity of our apparatus (complex shell geometry, presence of screws, mount on an outer frame), accurate predictions of the apparatus elastic eigenfrequencies remain out of reach. Note that, because of its finite elasticity, the container oscillates, leading to sound scattering in the surrounding air. Based on our elastic calculations, we have verified that this sound scattering can also be neglected 23 .

E. Synthetic spectra

For air enclosed in a rigid container, effects influencing the complex frequencies can be linearly superposed 30 . Adding the different perturbations to f nl , we obtain the predicted frequency f nlm of each n S m l mode. Assuming a Lorentzian resonance 30,[START_REF] Trusler | Physical Acoustics and Metrology of Fluids[END_REF] , we compute its contri-bution A nlm (f ) to the complete frequency spectrum by

A nlm (f ) = A nlm (f -f nlm ) -i g nl , (10)
with

A nlm = A |ζj l (k nl a) cos [m(ϕ el -ϕ sp )]| , (11)
where ζ = P m l (cos θ sp ) P m l (cos θ el ), A is a constant, P m l is the associated Legendre polynomial of degree l and order m, θ sp , ϕ sp and θ el , ϕ el the co-latitude and longitude of the speaker (source), resp. electret (receptor), see Supplementary material 22 for exact positions of the instrumentation.

III. EXPERIMENTAL SPECTRA AND MODE IDENTIFI-CATION

Using the instrumentation described in Section I, we measure the acoustic response of the ZoRo apparatus at rest. Acoustic spectra obtained as such display unambiguous resonances (Fig. 2) which we aim to interpret.

A. Spectrum interpretation

ZoRo's ellipticity is moderate, so at lowest order we expect resonance frequencies to be near those of the sphere of same volume. Eigenfrequencies are proportional to the sound speed c, which varies with temperature [START_REF] Zuckerwar | Speed of sound in real gases[END_REF][START_REF] Koulakis | [END_REF] . In order to compare the measured and predicted frequencies, we choose a non-splitting reference ( 2 S 0 0 ) and deduce the apparent sound speed, which we then use for the theoretical predictions. In the run shown in figure 2, the reference frequency is f 20 = 2177 Hz and corresponds to c = 346.5 m.s -1 with a taken to be a = 3 r pol r 2 eq in the following (see Supplementary material 23 ).

In figure 2 we show the experimental spectrum obtained with a continuous linear chirp of duration 90 s with frequencies ranging from 400 to 5000 Hz (which corresponds to the working range of the loudspeakers). We first observe that all peaks are close to the resonances of the reference model. Around the f nl , there are groups of peaks, corresponding to degeneracy lifting of the resonances. The lowest frequency modes are clearly separated. We note that those groups contain l + 1 peaks confirming that the degeneracy is only partially lifted, as predicted by the geometrical perturbation theory II D 2.

FIG. 2. Experimental acoustic spectrum at rest averaged over all electrets of one hemisphere. The spectrum is continued from the top to the bottom frame. Groups of peaks can be labelled with n S l according to theoretical prediction of a sphere of same volume (dashed lines).

B. Using symmetries to improve mode identification

The symmetric disposition of the instrumentation allows separation of acoustic modes based on their spatial pressure distribution. Since the pressure field at the resonator's surface is mapped on the spherical harmonics Y m l , as shown in equation ( 4), it displays the same symmetries. Acoustic modes with even lm are symmetric with respect to the equator; those with odd lm are anti-symmetric [START_REF] Morse | Methods of Theoretical Physics[END_REF] . Acoustic sources impose both wave phase and positions of the anti-nodes, so playing several speakers at the same time strengthens modes with the same symmetries as the source pattern, whereas modes with incompatible symmetries are extinguished, or, in practice, weakened. To amplify this effect, the acoustic response of the resonator is systematically measured with at least a pair of equatorially symmetric electrets and we make use of the symmetry properties in the data analysis. Measured signals are summed, or subtracted, to match the source pattern and further attenuate the unwanted modes. In figure 3, two speakers symmetric with respect to the equator simultaneously play first in-phase then in antiphase (also called phase opposition). The acoustic response is measured with four pairs of equatorially symmetric electrets. Measured time series of the in-phase speakers are summed for each pair. We compute the frequency spectra which are then averaged, shown in figure 3 (purple). We do the equivalent for phase opposition sources signal with subtraction of each pair, shown in figure 3 (blue). The summed spectra of the in-phase source (purple) enhance the even |m| (0 and 2 here), subtracted spectra of phase opposition source (blue) enhance the odd |m| = 1.

By successively extinguishing even and odd lm modes, we able to identify, meaning uniquely attribute a (n, l, m) triplet, each peak of the experimental spectrum up to 3500 Hz (for complete experimental spectrum identification, see Supplementary material 23 ).

Mode identification allows us to observe several features that seem to be qualitatively consistent throughout the spectrum. For a given n S l multiplet, the flattening partly lifts the mode degeneracy and separates the |m| peaks, with higher |m| at lower frequency, which agree with the simple first-order perturbation analysis. This is consistent with the rule of thumb stating that higher |m| modes are more sensitive to the equatorial region, hence feel a larger radius, yielding a lower frequency.

IV. BEYOND FIRST-ORDER EFFECTS

Upon more rigorous observations, exceptions to the features mentioned before are visible. We can see on the 1 S 2 multiplet, around 2050 Hz, that the m = 0 and |m| = 1 peaks have same frequency, and on 2 S 2 around 2950 Hz, that the m = 0 peak has lower frequency than |m| = 1 (Fig. 3). To explain those deviations, it seems necessary to go beyond first-order effects.

A. Dominant second-order effects: due to the ellipticity

To estimate the dominant second-order effects in the experiment, one can perform order of magnitude calculations. As e = 0.05 ν/(cr eq ) ≈ 10 -7 the dominant second-order correction is expected to be in e 2 , originating only from the container geometry second-order perturbation. To check if this effect can explain our observations, we compute diffusionless acoustic resonances of the exact spheroid, as a function of the ellipticity e, and we compare them with the experimental observations. To do so, we implement two different methods. First, we use the new global polynomial (Galerkin) approximation of the acoustic modes in rigid ellipsoids [START_REF] Vidal | [END_REF] . Second, we solve the acoustic equation (3) using the built-in acoustic interface of the finite-element commercial software COMSOL Multiphysics. In figure 4, we plot the frequency evolution as a function of ellipticity for different m of the 2 S 2 multiplet. Both computations match very well and are in good agreement with the experiment as well (crosses at e = 0.05). They also show that the first-order perturbation theory, given by the tangents at e = 0, is not sufficient for ZoRo's ellipticity (e = 0.05). This is clearly illustrated by the crossing between the m = 0 and |m| = 1 branches around 0.025 (Fig. 4). Frequency (Hz)

|m | = 2 m = 0 | m | = 1 FIG. 4.
Frequency evolution of 2 S m 2 modes with e (r eq being kept constant) given by finiteelement calculations (circles) and global polynomial computations 25 (dashed lines). Both agree with experimental (crosses) at the apparatus ellipticity e = 0.05 (black line).

B. Second-order ellipticity perturbation

A second-order perturbation theory for a homogeneous fluid enclosed in a quasi-spherical container has been developed 21,[START_REF] Mehl | [END_REF]40 , calculating the eigenfrequencies of a slightly aspherical acoustic cavity resonator enclosed within a spherical cavity [START_REF] Morse | Methods of Theoretical Physics[END_REF] . Our implementation in a Matlab package of Mehl's method [START_REF] Mehl | [END_REF] is described in Supplementary material 23 with more details. We compared our results both with his original results and with finite-element calculations.

To do so, we use the built-in modules of COMSOL Multiphysics, which solves the acoustic equation for a given (arbitrary) shape. To single out the second-order correction and check the validity of the perturbation theory, we use Lagrange elements of degree 5 to compute accurately the eigenfrequencies for a series of oblate spheroids with increasing ellipticity 0 < e < 10 -3 . Then for each n S m l mode, the evolution of its frequency with ellipticity is fit by a secondorder polynomial as f nlm ( ) = f nl +γ (1) nlm +γ

(2) nlm 2 with = e r eq /r pol , in order to easily compare with Mehl's results [START_REF] Mehl | [END_REF] . We have checked that the results are unchanged when higher orders are taken into account in this fit. nlm corrections for diffusionless acoustic eigenmodes in our experiment. We reproduced Mehl's original results [START_REF] Mehl | [END_REF] for modes l = 0 (connected by dotted line) and l = 1 (connected by dashed lines) and compare them with our implementation of Mehl's theory [START_REF] Mehl | [END_REF] (symbols) and with finite element calculations in oblate spheroids (dots connected by solid lines for a given m). All results from the three different methods are consistent.

Figure 5 displays the evolution of γ

(2) nlm with f nl as computed with our various methods. The solid (red) line joining the m = 0 data displays a saw-tooth pattern, with the l = 1 and l = 0 points yielding increasingly large positive values, while the l = 2 points get more and more negative as f nl increases. In his article [START_REF] Mehl | [END_REF] , Mehl only reports on the l = 0 (dotted red line) and l = 1 (dashed lines) cases. The mode crossing we observe occurs for large enough negative value of γ

(2) nlm and was therefore not present in his results. Our implementation of perturbation theory is in very good agreement with both Mehl's original results [START_REF] Mehl | [END_REF] and the finite-element numerical calculations, showing that perturbation theory up to second-order is sufficient for mode identification in ZoRo within our frequency range.

8

Gas-filled rotating spheroid acoustic spectra

C. Comparison with experimental spectra

Focusing on the 2 S 2 multiplet, the secondorder correction for m = 0 is negative while the other m are positive (close to zero) (Fig. 5), yielding in fine a lower frequency for 2 S 0 2 than 2 S 1 2 .

In figure 6 we plot the acoustic spectra from experimental data, perturbation theory and finite-element calculations. The first row shows the synthetic spectrum obtained from the perturbation theory (with the second-order correction in geometry) for the sound speed c = 343.194 m.s -1 , which is the typical value for dry air at 20 • C. The second row shows the experimental data. For the whole experimental spectrum, we apply a multiplicative correcting factor on the frequency axis to align the frequency of the mode 2 S 0 0 (indicated by an arrow) with the one of the theory. The third row displays finiteelement computation of the spectrum. For the sake of comparison with experiments and theory, we consider shear and bulk viscosities and no-slip and constant temperature boundary conditions, which prevents the use of the built-in acoustic equation of COMSOL. Instead, we have modified the built-in aero-acoustic interface, which is limited to m = 0. This non-trivial extension to arbitrary m is fully detailed in Supplementary material 23 . In practice, we use Lagrange elements of order 3 for the pressure and order 4 for the velocity and temperature. We then obtain the acoustic response for each m, and the complete spectrum is obtained by summing contributions of all m (Fig. 6, bottom). Note that a typical calculation of a unique fluid response takes ∼ 12 s on a current desktop computer, and the accurate calculation of the high-resolution full on 500 -3000 Hz requires thus a full week.

Complete comparison of the predicted second-order frequencies with experimental spectrum shows that all major features are reproduced (peak frequencies differ by less than 1 Hz). Each experimental peak can be identified. Both frequencies (real part), relative amplitude of peaks and their shape (imaginary part) are very similar, except in some narrow frequency range (e.g. around 1600 Hz) where some experimental peaks seem to be missing. The dissipation in the experiment is expected to be higher than from the values used in the theory, where we neglected air humidity and other diffusion mechanisms. This might cause some peaks to be hidden in the experimental spectrum by the neighbouring peaks (see Supplementary material 23 for whole frequency range comparison). Other minor discrepancies can be attributed to slight geometry perturbations due to the presence of the instrumentation (e.g. loudspeakers). The numerical and theoretical spectra are very similar, showing that we can rely on the perturbative approach for an accurate and robust description of the experimental spectrum.

V. SPLITTINGS DUE TO SOLID-BODY ROTATION

In this section, we consider solid-body rotating flows in ZoRo. To obtain these flows, we impose a constant rotation rate to the container and we wait for 2 minutes, a long time compared to the spin-up time (∼ 10 s for air rotating at 10 Hz), such that the fluid is uniformly rotating with the container [START_REF] Greenspan | The Theory of Rotating Fluids[END_REF] . Then, the fluid is at rest in the frame rotating with the container.

A. Effects of the rotation on the 0 S 2 multiplet Using the protocol developed in Section III to acquire and identify acoustic modes at rest, we measure the acoustic response in presence of rotation. The top panel of figure 7 shows typical splittings due to solid-body rotations for the 0 S 2 multiplet. Focusing first on the original peak (i.e. with no rotation) around 910 Hz, we observe that it splits into two peaks about half-height on both sides of the original peak. We note that the splitting increases with the rotation rate.

In order to compare with the rotation splitting predicted by the theory, certain mode identification is needed, especially in regions where several peaks are close together, around 930 Hz for example. In the bottom panel of figure 7 we separate the 0 S 2 multiplet by symmetry at f Ω = 25 Hz. We use the symmetry method detailed in Section III B to separate odd and even m modes. We can also identify the m = 0 mode around 940 Hz by the fact that it is the only mode not influenced by rotation at first-order, as shown by equation (5). The vertical lines show the frequencies given by the perturbative Gas-filled rotating spheroid acoustic spectra FIG. 6. Acoustic spectra for ZoRo configuration at rest obtained with perturbation theory (top), experimental data (middle) and finite-element calculations (bottom). Mode frequencies and labels from the theory are given across the three spectra for comparison (vertical lines, different line types are used for n, different colours for m). Experimental spectrum is averaged over all electrets of one hemisphere. For each spectrum, amplitude is normalised by its mean on the given frequency window.

approach taking into account the first-order rotation effects (Section II). This allows to identify the mode 0 S -2 2 at 930 Hz and the mode 0 S +1 2 at 926 Hz: the two peaks cross each other. This lifts the ambiguity that could have arisen from a naive reading of the bottom purple curve (f Ω = 25 Hz) in the top panel. At f Ω = 20 Hz (blue curve), these 2 peaks merge in a higher peak at 928 Hz (see the top panel).

B. First experimental determination of the Ledoux coefficients

For a given n S m l mode, perturbation theory predicts a linear increase of the rotational splitting with the rotation rate. To verify this prediction and its validity domain, we extract splittings for a collection of non-axisymmetric eigenmodes over our range of working rotation rates (from rest to 30 Hz). We fit the observed ±m pairs of spectral peaks with synthetic spectra, carrying a grid search on the four following parameters: the frequency splitting between the two peaks, their mean frequency, their width, and their amplitude. For each combination of parameters, we evaluate the misfit as the rootmean square (rms) difference between the observed and synthetic spectra (in log scale) in the frequency window of the mode. We obtain the best frequency splitting from the combination yielding the smallest misfit. The error is estimated from the minimum and maximum splittings for which some parameter combinations produce a misfit of typically 1.05 times the minimum misfit. We provide examples of the fits in Supplementary material 23 .

In figure 8, we show a selection of experimental splittings as a function of rotation rate (the experimental splittings being measured by the difference between the ±m peak frequencies). At first glance, we observe that they agree well with the theoretical linear predictions |2δ Ω | given by equation ( 5) and shown by the lines. The blue crosses show our finite calculations predictions which are, in many cases, closer to the experimental data than the theory (see e.g. 0 S 2 2 or 0 S 1 2 ). In our range of rotation rates, the numerical splitting also follow a linear trend. Note however, that the observed slopes can be slightly different from the one predicted using Ledoux coefficients in the sphere (see typically 0 S 1 2 ). This is due to the effect of ellipticity [START_REF] Vidal | [END_REF] , and this slightly different slope defines spheroidal Ledoux coefficients, which now depends on but also on m by contrast with the spherical one (which is independent of m, see equation 6). In our range of rotation rates, rotational splittings in the sphere are predicted to be proportional to f Ω , but also to m through the spherical Ledoux coefficients. By successively focusing on various non-axisymmetric multiplets, we retrieve the rotational splittings for a larger collection than before. We have extracted rotational splittings for 24 modes at f Ω = 20 Hz with the method detailed above and computed the associated spheroidal Ledoux coefficients (frequency splitting divided by 2mf Ω ). Figure 9 shows these Ledoux coefficients for different ±m pairs of peaks as a function of l. For each mode, the experimental Ledoux coefficients (symbols) agree roughly with the linear theory (lines) and the deviations may be explained by the ellipticity of the container (see l = 2, n = 0 in figures 8 and 9). Conversely, the determination of the rotation rate can be obtained by measuring any splitting if the spheroidal Ledoux coefficient is known.

VI. CONCLUSIONS AND PERSPECTIVES

For this study, we built an experimental setup made of a gas-filled spheroid cavity (ZoRo) rotating up to 30 Hz. We successfully model the experimental acoustic spectrum of the cavity with a perturbation theory and finite-element calculations. To identify the modes, we need to introduce a second-order geometry correction and use the equatorial symmetries of the modes, sources and receivers. We have successfully measured the Coriolis effects on the acoustic response, allowing the first experimental determination of the Ledoux coefficients 5 and their correction due to the ellipticity of the cavity [START_REF] Vidal | [END_REF] .

We aim to use this modal acoustic velocimetry method to image flow fields in our ZoRo apparatus. Having identified modes up to l = 10 and n = 3, we can expect a spatial resolution of 1/5 of the cavity radius 8 . This modal acoustic velocimetry is thus a robust, versatile and non-intrusive velocimetry method. As a longterm goal, we aim to study more complex largescale rotating flows, including non-uniform rotation rates, and use acoustic splitting data to retrieve the time-dependent three components of flow velocity in the whole volume. It could also be interesting to increase rotation rates to probe the limits of the perturbation theory, to investigate higher-order Coriolis effects [START_REF] Vidal | [END_REF] and centrifugal effects 20,[START_REF] Reese | [END_REF] . Another perspective is to use other internal gases with different diffusive behaviour, leading to a change of peaks width, potentially allowing better peak separation (SF 6 has thinner n = 0 modes than air for example). We plan to use this modal acoustic velocimetry to describe the physics of zonal jets. Future planned experiments include changing (increasing or decreasing) the pressure within the spheroid and differential heating of the working gas to reproduce the forcings and forces balance seen in astro-geophysical bodies.

  FIG.1. Photo (top) and sketch (bottom) of the ZoRo experimental apparatus. Four speakers (blue) are at ±45 • latitude. The acoustic response of the gas-filled resonator (yellow) is measured with electrets (green), at ±32 • (colour online). The instrumentation is installed on the rotating apparatus, which can rotate around its symmetry axis up to |Ω|/(2π) = f Ω = 30 Hz.

  e = (r eqr pol )/r eq

FIG. 5 .

 5 FIG. 5. Ellipticity second order γ(2)

2 FIG. 7 .

 27 FIG.7. Experimental spectra centered on 0 S 2 . Top: for solid body rotation at increasing rotation rates from rest (top, red) to f Ω = 25 Hz (bottom, violet) with 5 Hz increment. Spectra are filtered (low-pass) and vertically shifted for visualisation purposes. Bottom: for solid body rotation at f Ω = 25 Hz separated by symmetry, m = ±1 (blue) and m = 0, ±2 (purple) (colour online). The dashed lines shows the perturbation theory predictions.

FIG. 8 .

 8 FIG.8. Rotational splittings of chosen modes for increasing rotation rates. Experimental (circles) and finite element (blue crosses) splittings are measured by the difference between the ±m peak frequencies. Theoretical linear splittings (lines) are given by |2δ Ω |, see equation(5). For theoretical splittings, equatorially symmetric modes are shown in red solid lines and anti-symmetric modes in black dashed line; we use the same choice of colours for the experimental splittings (colour online).

5 FIG. 9 .

 59 FIG. 9. Comparison between the theoretical Ledoux coefficients multiplied by l + 1 (connected by lines for n = 0, 1, 2) and their experimental counterparts deduced from the mode splitting measured in ZoRo (rotating at 20 Hz) for several l-modes of various m, being either symmetric (ES, hot-coloured triangles) or anti-symmetric (EA, cold-coloured circles) with respect to the equator (colour online). When possible, several ±m pairs are included for one n S l multiplet.

  table (ref. TASCAM US-16x08). Acoustic waves are pro-duced by 36 mm-diameter audio speakers with working range 400-6000 Hz (ref. Multicomp MCKP3648SP1-4758) connected to a sound card (ref. Asus TeK, Xonar DGX), through 20 W amplifiers (LEPY LP-808). Acoustic resonances are excited by chirps sweeping over the frequencies of interest. At T = 20

Gas-filled rotating spheroid acoustic spectra

Gas-filled rotating spheroid acoustic spectra

Gas-filled rotating spheroid acoustic spectra

ACKNOWLEDGMENTS

This research was supported by ANR-13-BS06-0010 (TuDy). JV was partly funded by STFC Grant ST/R00059X/1. HCN wishes to thank M. Moldover, M. Bruneau and C. Guianvarc"h for their interest and encouragements. This work is dedicated to the memory of J. Mehl, who gave the authors his very last encouragements to pursue the path he pioneered. ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). The MATLAB package to calculate topographic and rotation influence on acoustic modes with perturbation theory is available at https://www.isterre.fr/annuaire/ member-web-pages/henri-claude-nataf/. The COMSOL models are available at https://www.isterre.fr/annuaire/ member-web-pages/david-cebron/.

Most figures were produced using matplotlib (http://matplotlib.org/).