T Malou

J Monnier

Two-scales diffusive wave equations dedicated to spatial rivers observations

Keywords: Diffusive Wave equations, two-scales asymptotic, river flow modeling, satellite observations, quantification of model terms

In river flow modeling, the Diffusive Wave equation is an interesting approximation of the Saint-Venant equations (Shallow-Water system) because it is simpler and still allows a rich numerical modeling. Moreover, the computed variable is the water surface elevation which can be directly measured using the altimetry satellites, unlike the discharge. A space-time Two-scales model that represents the physical scale (the small scale) and the observations scale (the large scale) is mathematically derived. At the (small) physical scale, the gradually varied assumption is retrieved. At the (large) observations scale, the spatio-temporal variations of the river width are not negligible anymore. The study results to a rigorously derived observations-scale diffusive wave equation which is consistent with spatial altimetry observations. The latter takes into account the spatio-temporal variations of the width through additional terms. This new observations-scale diffusive wave equation and the classical diffusive wave equation are numerically compared with each other and with the Saint-Venant system taken as reference, on numerically constructed channels with real river magnitudes. The observations-scale diffusive wave equation provides better estimates of the slope of the free surface and of the discharge than the classical ones. Finally, the importance of the additional terms specific to the observations-scale diffusive wave equation is quantified using a sparse identification method in a dictionary of differential model terms. This model learning based approach highlights that one of the newly introduced term is one of the dominant terms.

Introduction

The Diffusive Wave (DW) equation is a classical approximation of the Saint-Venant (SV) equations used to model gradually varied free surface of river flows, see e.g. [START_REF] Ponce | Shallow wave propagation in open channel flow[END_REF]. It is derived from the SV equations assuming the inertia term is negligible, and is therefore also called the non-inertial wave equation, see e.g. [START_REF] Yen | On noninertia wave versus diffusion wave in flood routing[END_REF]. The DW equation has been proven to be a reliable approximation of the SV system [Ponce et al., 1978, Santillana and[START_REF] Santillana | A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations[END_REF]. It is a rather simple approximation but it still is capable of modeling backwater effects, unlike the kinematic wave that also neglects the pressure term, see e.g. [START_REF] Ponce | Shallow wave propagation in open channel flow[END_REF], [START_REF] Yen | On noninertia wave versus diffusion wave in flood routing[END_REF]. The choice of a mathematical model can also be influenced by the data available. For example, solving the 1D SV system, which state variables are the wet area A and the discharge Q, requires imposing Q as upstream boundary condition which can be complex to measure or estimate, see e.g. [START_REF] Brisset | On the assimilation of altimetric data in 1d saint-venant river flow models[END_REF]. On the other hand, the DW equation, which state variable is the water surface elevation H, can be solved with H as Dirichlet boundary conditions, which is much easier to measure. This is especially true in a altimetry satellite context where the water surface elevation H is observed. Moreover, in this context, the assumption that the inertia terms are negligible (low Froude assumption) is rather well respected, see e.g. [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF], [START_REF] Larnier | Hybrid neural network -variational data assimilation algorithm to infer river discharges from swot-like data[END_REF]. In other respects, features such as the sensitivity with respect to the bathymetry may as well influence the choice of the mathematical model. For example, [START_REF] Guinot | Sensitivity analysis of 2d steady-state shallow water flow. application to free surface flow model calibration[END_REF] have analyzed the sensitivity of the SV system with respect to parameters such as the bathymetry. Based on this study, [START_REF] Aricò | Mast-2d diffusive model for flood prediction on domains with triangular delaunay unstructured meshes[END_REF] came to the conclusion that the DW equation might be less sensitive to variations in the bathymetry than the SV system.

The DW equation is a doubly nonlinear, potentially degenerated, equation. The nature of the equation makes the study of its mathematical properties rather difficult, see e.g. [START_REF] Alonso | On the diffusive wave approximation of the shallow water equations[END_REF] and [START_REF] Bögelein | Existence of solutions to a diffusive shallow medium equation[END_REF] for results on the existence and uniqueness of weak solutions. Still, it has been used in practical applications to model 1D flows (see e.g. [START_REF] Bates | A simple raster-based model for flood inundation simulation[END_REF]), 2D flows (see e.g. [START_REF] Hromadka | A two-dimensional dam-break flood plain model[END_REF], [START_REF] Santillana | A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations[END_REF]) or even 1D-2D coupling [START_REF] Aricò | The flo diffusive 1d-2d model for simulation of river flooding[END_REF].

In the present study, we introduce two scales which distinguishes the so-called physical scale which is a relatively small scale represented by the usual space-time variables (x, t), and the observations scale represented by the variables (y, s). These two scales are linked by a single small scaling ratio ε s : (y, s) = ε s × (x, t). In the present altimetry context, one typically has ε s ≈ 1/250, see e.g. SWOT mission [START_REF] Rodriguez | The surface water and ocean topography mission (swot): the ka-band radar interferometer (karin) for water level measurements at all scales[END_REF]. In the derivation of the DW equation, the flow is classically assumed to be gradually varied i.e. the water depth h varies slowly enough, see e.g. [START_REF] Chow | Handbook of applied hydrology[END_REF], [START_REF] Yen | On noninertia wave versus diffusion wave in flood routing[END_REF]. For sake of simplicity, the assumption that the geometry of the channel is a uniform rectangular is also quite common, see e.g. [START_REF] Ponce | Shallow wave propagation in open channel flow[END_REF], [START_REF] Ferrick | Analysis of river wave types[END_REF]. Both assumptions lead to neglecting the terms related to the width and its spatio-temporal variations in the derivation of the DW equation. These assumptions can be justified at the physical scale. Moreover, this assumption is natural if infering the river bathymetry from surface water elevation measurements (from satellite or not), see e.g. [START_REF] Tuozzolo | Estimating river discharge with swath altimetry: A proof of concept using airswot observations[END_REF].

In the present study, we investigate the derivation of a DW like equation in a context of spatial altimetry observations e.g. with data acquired by the spatial missions SWOT (NASA-CNES), Sentinel-3, Sentinel-6 (ESA) or IceSat2 (NASA). In this context and unlike the physical scale, the scale of the observations, width included in the case of SWOT, does not justify neglecting the width variation terms (two-scales assumption). The corresponding enriched DW equation in agreement with the observations scale, called the observations-scale DW equation, is rigorously derived.

The outline of the article is as follows.

In the first section, we present the two-scales assumption and the derivation of the two-scales DW equations from the SV system. These two-scales DW equations include the new observations-scale and the classical physical-scale one. In the second section, the observations-scale DW equation is compared with the physical-scale DW equation and with the SV system that is taken as reference. In the third section, we present a method to assess the importance of the different terms of the DW equation, based on the method to identify the terms of a model proposed in [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF].

Derivation of the two-scales DW model

In this section, the derivation of the two-scales DW equations from the SV system is presented. First of all, the SV system is recalled. Then, the additional hypothesis, especially the two-scales assumption, are presented. Finally, the two-scales DW equations are derived.

The reference SV equations

The most classical dynamical river flow model is the SV system, see e.g. [START_REF] Chow | Handbook of applied hydrology[END_REF]. In these equations, the natural state variables are the wet area A(x, t) [m 2 ] and the discharge Q

(x, t) [m 3 • s -1 ]:        ∂A ∂t + ∂Q ∂x = 0 (1a) ∂Q ∂t + ∂ ∂x Q 2 A + gA ∂H ∂x = -gAS f (1b)
with g the gravity magnitude (g ≈ 9.81m.s -2 ), H(x, t) [m] the free surface height and the friction term S f (x, t).

The bathymetry is denoted z b (x) [m], the average cross-section velocity u(x, t) [m] and the top width w(x, t) [m], see Fig. 3. The average width w(x, t) [m] is the river width averaged over a cross-section located at x, at time t, Fig. 3. The abscissa x is the curvilinear abscissa of the centreline of the river and

= Q(x,t) A(x,t) [m • s -1 ], the water depth h(x, t) = H(x, t) -z b (x)
x ∈ Ω = [0, L]. The modeling time window is t ∈ [0, T ].
The friction term is defined by the classical Manning-Strickler formula. Assuming Q > 0 and the hydraulics radius R h [m] satisfies R h ≈ h, the expression reads:

S f = Q 2 K 2 s A 2 h 4 3 (2) with K s the Strickler roughness coefficient [m 1/3 • s -1 ].
The SV system (1) is a conservative hyperbolic system. It is here solved using the classical Preissmann scheme [START_REF] Cunge | Practical aspects of computational river hydraulics[END_REF] implemented into DassFlow software [Larnier et al.].

(a) Cross-section geometry.

H u (t) H d (t) z = z b (x) z = H(x, t) h(x, t) u(x, t) x = 0 x = L

Derivation of the two-scales DW equations

To derive the DW equations from the SV system, some additional hypothesis must be made. In a first time, we present some additional but classical hypothesis such as a hypothesis on the geometry of the cross-sections and the so-called "low Froude assumption". Then, the hypothesis specific to our two-scales DW equations, namely the two-scales assumption, is presented. Finally, these hypothesis are injected in the SV system to derive the two-scales DW equations.

Basic assumptions

Geometry assumption While the state variables of the SV system (1) are naturally (A, Q), the natural state variable of the spatial observations is H. However, a bijective relation between A and H is naturally built up. Following the forthcoming SWOT type data, the effective geometry is chosen as trapezoid like in [START_REF] Brisset | On the assimilation of altimetric data in 1d saint-venant river flow models[END_REF], [START_REF] Larnier | Hybrid neural network -variational data assimilation algorithm to infer river discharges from swot-like data[END_REF], Fig. 1a:

A(H; x, t) = w(x, t) (H(x, t) -z b (x))
(3) with w(x, t) the average width of the wet area at a given time and location. Note that the width variable w depends on time and is an observable.

The low Froude assumption

We classically define the geometrical ratio ϵ of the flow as ϵ = h0 L0 , where h 0 is a characteristic depth and L 0 a characteristic wave length. For large rivers, the long-wave assumption ϵ ≪ 1 is easily verified with a reasonable minimal wave length L 0 . The SV model is an accurate flow model under this long wave assumption (shallow flows). Moreover, the Froude number F r = U 0 /c (with U 0 a characteristic velocity and c = gA/w the celerity) is in a great majority of large rivers strictly lower than 1 (fluvial regime). More precisely, the "low Froude assumption" introduced in [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF], [START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF], [START_REF] Frasson | Exploring the factors controlling the error characteristics of the surface water and ocean topography mission discharge estimates[END_REF] holds for the great majority of world rivers (in standard regimes). That is we can assume that:

F r 2 = O(ϵ) (4)
The two-scales assumption The two-scales assumption is motivated by differences between the usual flow scale (continuum mechanics derivation scale) and the large scale of the altimetry observations, see e.g. [START_REF] Rodriguez | The surface water and ocean topography mission (swot): the ka-band radar interferometer (karin) for water level measurements at all scales[END_REF] in the case of SWOT.

The time and distance characteristics of the "physical scale" are quite small: of the orders of magnitude of an hour and a meter. On the contrary, the time and distance characteristics of the altimetry observations are quite large: time between two satellite flybys at a virtual station (intersection of the satellite track with the river) is a dozen days (10 days for the Jason-3 satellite, 27 for Sentinel-3 and 10 -21 for SWOT). Also, the characteristic lengths are between a few hundred meters (scale of a satellite swath, ∼ 250 m for SWOT [START_REF] Rodríguez | Swot science requirements document[END_REF]) and several tens of kilometres (distance between two virtual stations) e.g. between 0.7 km and 100 km on the Rio Negro, see e.g. [START_REF] Malou | Generation and analysis of stage-fall-discharge laws from coupled hydrological-hydraulic river network model integrating sparse multi-satellite data[END_REF] and Fig. 2. Thus, the "physical scale" and the "observation scale" are distinct. Henceforth, the variables of the physical scale are denoted by (x, t) and the variables of the observation scale by (y, s). They are satisfy the relation:

(y, s) = ε s × (x, t) (5)
with ε s defining the ratio between the two scales. On the basis of the current satellite missions, it is remarkable that the same scaling factor ε s appears both in time and in space. This ratio currently satisfies: ε s ≈ 1/250.

River flows are classically considered to be "gradually varied" (in the sense that the cross-section geometry and the water depth does not vary significantly over a relatively short time and distance), see e.g. [START_REF] Chow | Handbook of applied hydrology[END_REF], [START_REF] Yen | On noninertia wave versus diffusion wave in flood routing[END_REF]. This "gradually varied" assumption implies that:

∂ x w ≪ 1 and ∂ t w ≪ 1 (6)
At physical scale, this assumption is justified since the relative width variations ∂xw w and ∂tw w are very small, in particular for large rivers. In the present examples Rio Negro and Garonne river, ∂xw w ≈ 10 -4 , see Fig. 2, Fig. 4 and Tab. 1. Moreover, in an altimetry context, the following statements hold, see e.g. [START_REF] Rodríguez | Swot science requirements document[END_REF], [START_REF] Malou | Generation and analysis of stage-fall-discharge laws from coupled hydrological-hydraulic river network model integrating sparse multi-satellite data[END_REF].

• At a given virtual station and flyby, the width data measured by the satellite are averaged over the swath to provide a single width value with better accuracy. The swath size is comparable to the characteristic length of the observation scale. Hence, spatial variations in the width at a scale smaller than the swath size are not observed (see Fig. 3a).

• The average width mentioned above is computed at a virtual station at each flyby. Thus, the time and space intervals between two width values are comparable to the observation scale (see Figs. 3b and2).

On the other hand, the width variations are not negligible at the observation scale, see Fig. 2. Indeed, we have:

∂ y w ∼ 1 and ∂ s w ∼ 1 (7)
These variations can be computed from satellite measurements of width, in particular the forthcoming SWOT data [START_REF] Rodríguez | Swot science requirements document[END_REF][START_REF] Rodríguez | Observing rivers with varying spatial scales[END_REF]. Width and its relative variation on the Rio Negro (Amazon basin). Fine scale values of w are computed from satellite images (static estimation derived from the optic landsat imagery with a 30-meter resolution, as in [START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF]). Large scale values of w are the altimetric-like derived values from the fine scale width (one point per virtual station with linear interpolation between). In this test case, ε s = 1/250.

Finally, the two-scales assumption states that the flow can be considered gradually varied at physical scale but not at the observation scale.

From now, the generic variables (ξ, τ ) denote either (x, t) for the physical scale or (y, s) for the observation scale.

The two-scales DW equations

First, the low Froude assumption (4) implies that the momentum equation (1b) reads :

gA∂ ξ H = -gS f A (8)
The Manning-Strickler formula (2) with the low Froude momentum equation ( 8) implies the following Manning-Strickler relation (assuming that ∂ ξ H < 0):

Q(ξ, τ ) = K ξ s |∂ ξ H(ξ, τ )| 1 2 h(ξ, τ ) γ-1 A(ξ, τ ) ⇐⇒ u(ξ, τ ) = K ξ s |∂ ξ H(ξ, τ )| 1 2 h(ξ, τ ) γ-1 (9)
The power law is set here to its classical value γ = 5 3 [START_REF] Chow | Handbook of applied hydrology[END_REF].

155

Note that the expression of u depends on the considered scale. However, the value of u does not depend on the scale since time and space are scaled with the same scaling factor ε s . In order to obtain the same value of u between the two scales, the Strickler coefficient for the observation scale K y s has to be scaled accordingly as:

K y s = ε 1 2 s K x s , with K x
s the Strickler coefficient for the physical scale (that is the classical Strickler coefficient). The Manning-Strickler law (9) or similar stage-fall-discharge laws have been used to provide an estimate of Q from 160 large scale measurements H and ∂ x H, see e.g. [START_REF] Durand | Estimating reach-averaged discharge for the river severn from measurements of river water surface elevation and slope[END_REF], [START_REF] Garambois | Inference of effective river properties from remotely sensed observations of water surface[END_REF], [START_REF] Malou | Generation and analysis of stage-fall-discharge laws from coupled hydrological-hydraulic river network model integrating sparse multi-satellite data[END_REF], [START_REF] Mansanarez | Bayesian analysis of stage-falldischarge rating curves and their uncertainties[END_REF].

A scalar "one-equation" model is obtained by injecting the Manning-Strickler relation (9) into the mass equation (1a) ("lubrication theory", see e.g. [START_REF] Chang | Complex wave dynamics on thin films[END_REF]): with u(H) defined by the parametrization (9). Using the geometry relation (3), the two-scales assumption ( 6) and ( 7), the derivatives of A read:

165 ∂ τ A(H) + u(H)∂ ξ A(H) + (γ -1)u(H) A(H) H -z b ∂ ξ (H -z b ) + 1 2 u(H) ∂ 2 ξξ H ∂ ξ H A(H) = 0 (10)
∂ t A(H) = w∂ t H and ∂ x A(H) = w (∂ x H -∂ x z b ) (11) ∂ s A(H) = (H -z b ) ∂ s w + w∂ s H and ∂ y A(H) = (H -z b ) ∂ y w + w (∂ y H -∂ y z b ) (12)
Next the assumptions (3), ( 11) and ( 12) are taken into account and the two-scales DW equations follows:

∀(x, t) ∈ [0, L] × [0; T ], ( ∂ t H -µ(H)H + v(H)∂ x H ) (x, t) = ( v(H)∂ x z b ) (x, t) ( ∂ s H -µ(H)H + v(H)∂ y H + ρ(H)H ) (y, s) = ( v(H)∂ y z b + ρ(H)z b ) (y, s) (13a) (13b) with the effective diffusion coefficient [m 2 • s -1 ]: µ(H; ξ, τ ) = 1 2 u(H; ξ, τ ) (H(ξ, τ ) -z b (ξ)) |∂ ξ H(ξ, τ )| (14) the wave velocity [m • s -1 ]: v(H; ξ, τ ) = γu(H; ξ, τ ) (15)
and the width variation coefficient [s -1 ]:

ρ(H; y, s) = 1 w(y, s) (∂ s w(y, s) + u(H; y, s)∂ y w(y, s)) (16)
Recall the DW equations are closed with the Manning-Strickler law (9).

Since Q is assumed positive, u is also positive. Since h = (H -z b ) is positive by construction, µ defined by ( 14) is strictly positive. Note that |∂ ξ H| appears as the denominator in µ. For large rivers under the low Froude assumption (4), |∂ ξ H| is likely to be low (see e.g. Fig. 5). Therefore, µ is likely to be large (even extremely large in some locations).

If (y, s) → w(y, s) is sufficiently smooth, the coefficient ρ, defined by ( 16), is bounded in finite time.

In the end, the rigorously derived observations-scale DW equation (13b) actually corresponds to the classical gradually varied physical-scale DW equation (13a) (see e.g. [START_REF] Yen | On noninertia wave versus diffusion wave in flood routing[END_REF], [START_REF] Weill | Accuracy and efficiency of time integration methods for 1d diffusive wave equation[END_REF]) plus the 0-th (reaction) term ρ(H)(H -z b ). This additional term is simply related to the width variations.

Numerical results

The objective of this section is to study numerically the solutions of the DW equations ( 13). The solutions of the two-scales DW equations are compared to each other in order to highlight the impact of the width variation coefficient ( 16) specific to the observations-scale DW equation (13b). The solutions of the DW equations are also compared to a reference solution: a fine grid solution of the (complete) SV system (1). The two-scales DW equations ( 13) are solved using a implicit finite difference scheme with a Newton-Raphson algorithm. This scheme relies on centered second order approximations of the space derivatives. This solver have been implemented into DassFlow software [Larnier et al.].

Test cases description

The numerical experiments rely on channels with real river magnitudes. The different solutions are compared in an altimetry-like context. Steady-state flows are computed. Assuming that ∂x w w ≈ 0 (the witdh variations are negligible at physical scale), the DW equations ( 13) simplifies as:

- 1 2 (H(ξ) -z b (ξ)) |∂ ξ H(ξ)| ∂ 2 ξξ H(ξ) + γ∂ ξ H(ξ) + ∂ ξ w(ξ) w(ξ) H(ξ) = γ∂ ξ z b (ξ) + ∂ ξ w(ξ) w(ξ) z b (ξ) (17)
Note that the velocity u does not appear anymore in the steady-state equation ( 17).

The reference solution is the SV system one, see Eq. ( 1), in variables (A, Q). and the same discharge Q in on the DW and SV equations is not straightforward (especially in unsteady cases). In the steady case, the calibration of K s only rescales and shifts the resulting discharge so Q = Q in is satisfied upstream. This shift depends on the geometry of the channel. Hence, the estimation of Q and the resulting comparison of the two-scales DW equations are biased with respect to the geometry of the channel.

In order to study the behavior of the DW solutions only, such calibrations are not performed. In all the numerical results below, we have:

K s = K x s = 20 and K y s = ε 1 2 s K x s .
The steady-state DW solutions are compared through their differences with the reference solution (the SV system solution). For a state variable η computed by a DW equation, the following relative difference is computed:

210 ϵ η (x) = |η(x)-η SV (x)| |η SV (x)|
, with η SV the solution obtained by the SV system.

The presented results are obtained on two different numerically built channels with regular mesh. The geometry and boundary conditions Q A fine-scale width is constructed from the mean width (denoted w m ), the standard deviation of the width (denoted σ w ) and a characteristic wavelength (denoted λ w ) of the real river width, see Figs. 2 and4. The large-scale width is constructed from the values of the fine-scale width at virtual stations which are separated by a distance depending on the characteristic length of the observation scale denoted by L vs . Between the virtual stations, the large-scale width is linearly interpolated, see the lower part of Figs. 5 and6.

The first channel is based on a 900 km section of the Rio Negro data (Amazon basin); the width and its variations are plotted on Fig. 2. The lenght L vs corresponds to the average distance between two virtual stations on this river considering the Sentinel-3A, -3B and Jason-3 satellites, L vs ≈ 22 km.

The second channel is based on the Garonne River data. Its width is plotted on Fig. 4. On this channel, the distance L vs = 300 m corresponds to the magnitude of a swath size [START_REF] Rodríguez | Swot science requirements document[END_REF]. The Froude number of this channel is higher than the one of the Rio Negro-like channel, see Tab. 1. This river portion presents one of the highest Froude number among the 46 rivers portions analysed by the SWOT Science Team community [START_REF] Frasson | Exploring the factors controlling the error characteristics of the surface water and ocean topography mission discharge estimates[END_REF][START_REF] Durand | An intercomparison of remote sensing river discharge estimation algorithms from measurements of river height, width, and slope[END_REF]. River

w m (km) λ w (km) σ w (m) L vs (km) ∂xw w F r Q in (m 3 .s -1 ) z d b + h ref d (m)

Results and comparisons

In the first part of this section, the results obtained on the two academic but real-like channels (Rio Negro-like and Garonne-like) are presented. The two-scales DW solutions, see Eq. ( 13), are compared with each other and with the reference solution (solution of the SV system (1)).

Numerical results

In the steady-state case, the mass equation in the SV system, see Eq. (1a), simply reads: ∂ x Q = 0. This implies that Q is constant over the whole domain. After solving the SV system (1), the surface elevation H is then deduced from the geometry relation A = wh with h = (H -z b ).

On the contrary to the SV system, the DW equations ( 13) are non-conservative diffusive equations, which state variable is H. The DW equations are closed by the algebraic Manning-Strickler relation (9) (modeling locally uniform steady-state flows) between the three state variables H and (A, Q). In the present context of rectangular cross-sections this relation reads:

Q(ξ) = K ξ s |∂ ξ H(ξ)| 1 2 w(ξ)h(ξ) 5/3 (18)
Given a constant friction coefficient K ξ s (like in the present test case), the Manning-Strickler relation ( 9) does not enable to preserve simultaneously Q and A (and hence H) everywhere at every instant.

Moreover, due to the boundary conditions imposed on the DW equations and to the Strickler coefficient K s , A is conserved upstream but Q is not, see Fig. 5 and Fig. 6. On both the Rio Negro-like channel (see Fig. 5) and the Garonne-like channel (see Fig. 6), the solution of the observations-scale DW equation (green curve) provides an estimate of H which has similar variations and oscillations to those of w (but out of phase). The estimate of Q computed by the observations-scale DW equation also has very small variations similar to those of w.

Since z b is defined as a constant slope and there is no width variation term in the physical-scale DW equation, which is equivalent to considering a rectangular channel of constant width, the depth h = (H -z b ) estimated by the physical-scale DW equation (blue curve) is almost constant, except at the outflow boundary where the boundary condition must be satisfied. This implies that the slope ∂ x H is almost constant. Thus, the state variables A and Q estimated by flow law (18) present similar variations and oscillations to those of the width as well.

Comparisons

= w(H -z b ) = wh, max(ϵ A ) = max(ϵ h ).

Comparison of the classical physical-scale DW equation with the reference SV system

The classical physical-scale DW equation provides an estimate of A that is rather close to the reference solution computed by the SV equations on the Rio Negro-like channel but not satisfactory on the Garonne-like channel, see Tab. 2 and Figs. 5 and 6. Note that, since the cross-sections are rectangular, w is constant and the estimates of h have the same accuracy than the estimates of A.

However, the estimate of Q computed by the physical-scale DW equation is not so close to the reference solution. The reason is the following. Recall that to obtain an estimate of Q from the DW equations, the Manning-Strickler closure law (18) must be evaluated. Therefore, the accuracy of Q depends on the accuracy of the slope ∂ x H. Even though H computed by the physical-scale DW equation approximates rather accurately the reference solution on the Rio Negro-like channel, the variations of the reference surface elevation H (induced by w) are not well approximated by the physical-scale DW equation (whose solution (H -z b ) is nearly constant). As a result, the physical-scale DW equation estimates the slope ∂ x H quite poorly; it consequently also estimates Q rather poorly.

On the Garonne-like channel, the poor approximations of both A and ∂ x H explain the poor approximation of Q.

Analysis and contribution of the new observations-scale DW equation

The observations-scale DW equation provides an estimate of A that is close to the reference solution on both the Rio Negro-like and the Garonne-like channels. The observations-scale DW equation provides an estimate of Q that is very close to the reference (especially on the Rio Negro-like case, where the low Froude assumption is clearly satisfied). Indeed, on Figs. 5 and6, the estimate of h computed by the observations-scale DW equation has very similar variations and oscillations (that are induced by w) than the one computed by the SV system. Therefore, the estimate of the slope ∂ x H computed by the observations-scale DW equation is close to the one computed by the reference solution. Hence, the observations-scale DW equation provides better accuracy for estimating A and significantly better ac-curacy for estimating of Q than the physical-scale DW equation.

Let us note that the solutions (especially A) of both the physical-scale and observations-scale DW equations are less accurate on the Garonne-like channel than on the Rio Negro-like channel, see Fig. 6. Indeed, on the channel Garonne-like, the difference between the DW equations and the SV system are mainly due to the low Froude assumption which is less respected here. Still and unlike the physical-scale DW equation, the observations-scale DW equation provides an estimate of A that is rather close to the reference estimate on this Garonne-like channel.

One can note that A estimated by the three models are a bit out-of-phase between each other, see e.g. the Garonne-like channel on Fig. 6. The oscillations of A computed by the physical-scale DW equation are in phase with w since h is (nearly) constant, see the blue curve on Fig. 6. The oscillations of A computed by the SV system (red curves) are out-of-phase with w because of the inertia term and the kinetic induced by the variations of w. The out-of-phase between the A estimated by observations-scale DW equation and by the physical-scale DW equation (and hence w) is the result of the kinetic induced by the variations of w.

The comparison of the physical-scale DW and observations-scale DW equations (13) can be summarized as follows:

• On channels with very low Froude number, the observations-scale DW equation provides more accurate estimates of A than the physical-scale DW equation, although both estimates are relatively accurate. Both equations preserve the mass rather well on such channels. On the channels with higher Froude number (F r ≈ 0.2) however, the estimates of A provided by the observations-scale DW equation remains accurate while the physical-scale DW equation estimates A poorly.

• On the one hand, the estimate of H computed by the physical-scale DW equation ( 13a) is controlled only by z b (and the boundary conditions). On the other hand, the estimate of H computed by the observations-scale DW equation is also controlled by w through the width variation terms ρ(H)H and ρ(H)z b in the equation (13b). Thus, the estimate of the slope ∂ x H computed by the observations-scale DW equation takes into account the variations of w. The same is true for the estimate of Q computed by (9). Hence, Q is better estimated by the observations-scale DW equation than by the physical-scale DW equation, in particular in reaches where w has a non-negligible influence.

• Both the physical-scale DW and the observations-scale DW provide less accurate solutions on rivers with higher Froude number (see case on the Garonne river with F r ≈ 0.22). Still, the observations-scale DW equation provides much better results than the physical-scale DW equation whose results are not satisfactory on such channel.

• Same experiments have been done on channels with more complex w and z b (e.g. the real Rio Negro channel).

The conclusions of the comparison of the observations-scale DW and physical-scale DW equations are similar to the one mentioned above.

Quantifying the importance of each term

The purpose of this section is to quantify the importance of each term of the flow model equations ( 13), and in particular the width-variation terms ρ(H)H and ρ(H)z b specific to the observations-scale DW equation (13b).

Method

Brunton et al. [2016] proposes a sparse regression method named SINDy (Sparse Identification of Nonlinear Dynamics). This method aims at determining the most important terms of an ODE (contained in a dictionary), thus discovering the governing equation from observations. The method was then applied to PDEs in [START_REF] Rudy | Data-driven discovery of partial differential equations[END_REF].

This approach is here adopted to order each term importancy of Eqn (17) at physical scale. Eqn ( 17) can be re-written as:

1 2 (H(x) -z b (x)) |∂ x H(x)| ∂ 2 xx H(x) = D(x; H, z b , w)Θ T (19)
with Θ the vector containing the coefficient associated to the (potential) terms of the equation. These terms are contained in the dictionary D, that is in the present case:

D(x; H, z b , w) = ∂ x H , ∂ x w w H , ∂ x z b , ∂ x w w z b , ∂ 2 xx H , H∂ x H , ∂ xx z b (20)
The method aforementioned aims to solve the following multi-objective optimization problem:

Θ * Θ * = ΘJ res (Θ; H, z b , w) + κ∥Θ∥ 0 (21)
with the norm of the residual:

J res (Θ; H, z b , w) = ∥ 1 2 (H -z b ) |∂ x H| ∂ 2 xx H -D(x; H, z b , w)Θ T ∥ 2 2
and the number of non-zero coefficients ∥Θ∥ 0 = #{Θ i ; Θ i ̸ = 0}. Since this optimization problem can not be resolved, the sparsity-objective term ∥•∥ 0 is approached by a L 1 term like in the classical LASSO method [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]. Then, the following convex but non-differentiable optimization problem is solved:

Θ * Θ * = ΘJ res (Θ; H, z b , w) + λ∥Θ∥ 1 (22)
The choice of λ or κ (weights between the different objectives of the optimization) is not straightforward but crucial to adjust the number of non-zeros coefficients.

In the present study, the optimal coefficients are computed for several values of λ (LASSO path) in order to investigate the order of importance of each mathematical terms. This order is defined as the order in which the coefficients of the terms activate (i.e. are non-zero) as λ decreases. Indeed, when λ is large enough, the optimization will not keep any non-zeros coefficients. Then, when λ decreases the terms will progressively activate. The purpose of this order is to determine whether or not the terms ρ(H)H and ρ(H)z b are important terms in the two-scales DW equations ( 13) compared to the other terms.

To do so, the considered observations of H are the steady state solution values of the SV system over the entire domain: H = H SV . The derivatives of H SV must also be observed or estimated. Here, they are computed using the standard centered finite differences.

Recall from the definition of the dictionary (20) that all the terms of the observations-scale DW equations (13b),

except 1 2 (H SV -z b ) |∂xH SV | ∂ 2 xx H SV ,
are in the dictionary D(H SV , z b , w). Several terms are also added, such as an additional diffusion term. Note that this numerical experiment aims at investigating what are the most important term in a DW equations to approximate H SV computed by the SV system under the low Froude assumption (4). This order of importance is the order of importance to counterbalance the term in the left hand side of (19). By expertise, the left hand side, which is the diffusion term, is a dominant term.

The optimization problem ( 22) with H = H SV is solved in two steps using a home-developed Python routine derived from the routine released by [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF] 1 . In a first step, the optimization problem is solved using the FISTA algorithm with the Nesterov step [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF]. This step helps at determining the non-zeros coefficients, denoted Θ + . In the second step, J res is minimized with respect of the non-zero coefficients in order to avoid the effect of the L 1 norm: Θ * = ΘJ res (Θ + ; H SV , z b , w).

Results

The following numerical investigation is performed over the real Rio Negro set-up, see Fig. 2 and a few information in Tab. 1. Note that the low Froude assumption is here well respected. In this experiment, the coefficient of D 4 = ∂x w w z b is activated in the last position in the order of importance. Hence, it is likely to be negligible.

In sum, the ρ(H)H term, specific to the new observations-scale DW equation (13b), is important to accurately approximate the solution of the SV system (1) by a DW equation. On the contrary, the ρ(H)z b term is negligible.

The term D 5 = ∂ 2 xx H activates very early when λ is rather high. The coefficient Θ 5 is rather large, Θ 5 ≈ 6 × 10 3 while the nonlinear diffusion coefficient satisfies 3×10 Recall that this analysis is done in the steady case of the Rio Negro channel constructed from satellite data, and thus on a channel at the scale of the satellite observations. The 5 th and 6 th terms in the order of importance,

* 1 ≈ γ = 5/3, Θ * 2 ≈ 1, Θ * 3 ≈ -γ = -5/3, Θ * 4 ≈ -1.

Conclusion

This article presents a rigorous derivation of two-scales DW equations dedicated to spatial altimetry observations. The derivation of these equations are based on a small parameter ε s measuring the ratio between the usual ("physical") space-time scale and the observations one, see Eq. ( 5). The width variations are negligible at the physical scale (this corresponds to the classically made assumption) but not at the observations scale. The obtained physical-scale DW equation is then the classical DW equation, see e.g. [START_REF] Yen | On noninertia wave versus diffusion wave in flood routing[END_REF], [START_REF] Weill | Accuracy and efficiency of time integration methods for 1d diffusive wave equation[END_REF]. The obtained observations-scale DW equation is an enriched version in which the time and space derivatives of the width lead to an additional term ρ(H)H and an additional source term ρ(H)z b , see Eqs. (13b) and ( 16).

The observations-scale DW equation is numerically studied over channels build from real-river characteristic lengths in the altimetry observations context, in steady-state cases.

The results of the observations-scale DW and of the physical-scale DW equations, which the state variable is the water surface elevation H, are compared with each other and with a reference solution that is computed by the SV system, which state variables are the wet area A and of the discharge Q.

Overall, the new observations-scale DW equation provides much better estimations of Q than the classical physicalscale DW. Eventhough the physical-scale DW equation provides a good estimate on channels where the low Froude assumption is well satisfied, the new observations-scale DW equation also provides better estimates of A than the classical physical-scale DW, especially on channels where the low Froude assumption is less (but still) respected.

Overall, the mass is overall well preserved by the DW equations at both observations-scale and physical-scale on channels with very low Froude number but only by the DW equation at observations-scale on channels with higher

Froude number (F r ≈ 0.2).

Since the estimate of Q provided by the DW equations are computed using the Manning-Strickler closure law (9), the accuracy of the estimate of Q is related to the accuracy of the slope estimations ∂ x H too. Unlike the slope estimated by the physical-scale DW equation, the slope estimation computed by the observations-scale DW is influenced by the width and its variations. As a consequence, the observations-scale DW equation provides a much more accurate estimation of ∂ x H, and therefore of Q, than the physical-scale one. Of course, this is true especially on portions of rivers where the width affects in a significant way the flow.

Moreover, the contribution of the new width variations terms has been highlighted by a method to quantify the importance of the terms of a model. Following the sparse identification method presented in [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF], [START_REF] Rudy | Data-driven discovery of partial differential equations[END_REF], a method to order the importance of terms is developed. The algorithm orders the importance of the observations-scale DW equation terms plus additional terms present in a dictionary, to reproduce the water surface elevation H obtained by the SV system. This investigation of the importance of the terms shows that the newly introduced term ρ(H)H, specific to the observations-scale DW equation, is a rather important one. It also highlights the sensitivity of the DW equations to the (nonlinear) diffusion coefficient µ (14) and hence to the slope

∂ x H.
In the present study, the two-scales DW equations have been thoroughly analysed over numerically constructed real-like channels. They will be next tested on large datasets of real rivers such as the SWOT Science Team mission ones [START_REF] Frasson | Exploring the factors controlling the error characteristics of the surface water and ocean topography mission discharge estimates[END_REF]. Estimations of slopes ∂ x H are known to be a crucial point to estimate Q from Stage-Fall-Discharge (SFD) laws, see e.g. [START_REF] Mansanarez | Bayesian analysis of stage-falldischarge rating curves and their uncertainties[END_REF], [START_REF] Malou | Generation and analysis of stage-fall-discharge laws from coupled hydrological-hydraulic river network model integrating sparse multi-satellite data[END_REF]. As a consequence, the present enriched two-scales DW equation could be combined with such laws to provide complimentary estimations of discharge Q from altimetry measurements.

Figure 1 :

 1 Figure 1: Overall assumed geometry of the river.

  Figure2: Width and its relative variation on the Rio Negro (Amazon basin). Fine scale values of w are computed from satellite images (static estimation derived from the optic landsat imagery with a 30-meter resolution, as in[START_REF] Pujol | Estimation of multiple inflows and effective channel by assimilation of multi-satellite hydraulic signatures: The ungauged anabranching negro river[END_REF]). Large scale values of w are the altimetric-like derived values from the fine scale width (one point per virtual station with linear interpolation between). In this test case, ε s = 1/250.

  150

  ∆y r ∼ 10km (b) Illustration of the physical and observation scales.

Figure 3 :

 3 Figure3: Illustration of the two-scales assumption of the river width: real value and value averaged over a swath (providing an effective width). The index r is for the virtual station; the index p is for the pass of the satellite above the virtual station.

  in and H ref d are constructed with orders of magnitude of real rivers, see Tab. 1. The cross-sections are assumed to be rectangular: w = w. The channel bathymetry z b is the linear interpolation of the upstream and downstream bathymetry of the real river values: ∂ x z b is constant.

Figure 4 :

 4 Figure4: Width of the Garonne river (in red, from in situ data) and Garonne-like width (in blue, at fine-scale in solid line, at large-scale in dots, the mean width in dashed line).

Figure 5 :

 5 Figure 5: Rio Negro-like channel, steady state. Outputs of the two-scales DW equation (13) (Physical-Scale and Observations-Scale) and the reference SV system (1) on the four top figures: H (state variable of the DW equations), ∂ x H and (A, Q) (state variables of the SV system) versus x. On the lowest figure: the physical (fine) scale and observations (large) scale width versus x, large scale width deduced from the fine scale width at the virtual stations.

Figure 6 :

 6 Figure 6: Garonne-like channel, steady state. Outputs of the two-scales DW equation (13) (physical-scale and observations-scale) and the reference SV system (1) on the four top figures: H (state variable of the DW equations), ∂ x H and (A, Q) (state variables of the SV system) versus x. On the lowest figure: the physical (fine) scale and observations (large) scale width versus x, large scale width deduced from the fine scale width at the virtual stations.

Figure 7 :

 7 Figure7: From top to bottom: optimal coefficients Θ * i (i ∈ {1, • • • , 7}, top 7 figures, in red when = 0 and in blue otherwise), the norm of the residual J res (Θ * ) and the number of non-zero coefficients ∥Θ * ∥ 0 vs the weight coefficient λ. The vertical green lines are located at the largest value of λ such that Θ * i ̸ = 0 (on the i th figure: dashed line for the considered Θ * i and dotted lines otherwise). The order of importance is the order in which the Θ * i activate (see vertical green lines) as λ decreases.

  z b ) |∂xH| < 3×10 5 , when only the coefficients of D 3 = ∂ x z b and D 5 = ∂ 2 xx H activate. However, Θ 5 drops significantly, Θ 5 ≈ 90, when λ decreases and the coefficients of D 1 = ∂ x H and D 2 = ∂x w w H activate. The diffusion term D 5 = ∂ 2 xx H enables to compensate the non-modelled phenomena. The non-modelled phenomena include, at λ given, the phenomena from terms not yet activated or the phenomena specific to the SV system that are not classically present in the DW equation. Since more terms active when λ decreases and therefore less phenomena are non-modelled, the coefficient Θ 5 naturally becomes much less significant. The compensation of the non-modelled phenomena by a diffusion term highlights the sensitivity of the DW equations, and more specifically of the capability of the DW equations to reproduce the estimate of H computed by the SV system, with respect to the nonlinear diffusion coefficient 1 2 (H-z b )|∂xH| and more specifically to the slope ∂ x H.

  resp. D 7 = z b and D 6 = H∂ x H, are likely related to overfitting and non-physical compensations that are induced by the considered framework. Moreover, let us note that, for low values of λ, the optimal coefficients Θ * i related to the terms of the observationsscale DW equation (i.e. D 1 = ∂ x H, D 2 = ∂x w w H, D 3 = ∂ x z b and D 4 = ∂x w w z b ) are close to the coefficients given by the observations-scale DW equation (17): Θ

  The SV system is closed by imposing a discharge value Q in at inflow and a water surface elevation H out at outflow. The Strickler coefficient K s is assumed constant, K s = 20. The state variable of the DW equations is H only. H is the measured variable in an altimetry context. The upstream and downstream water surface elevations of the reference solution, resp. denoted by H ref

u and H ref d = H out , are imposed as boundary conditions of the DW equations. Calibrating K s to impose simultaneously at inflow the same elevation H ref u

Table 1 :

 1 Characteristics numbers of the two river-like channels.

	Rio Negro	4.5	22	1126	22	∼ 10 -4 < 0.15	8.2 × 10 3	0 + 4.86
	Garonne	0.149	4	43.46	0.3	∼ 10 -4 < 0.22	99	61.41 + 1.23

Table 2 :

 2 Maximum relative error of h, S = ∂ x H and Q for both Physical-Scale and Observations-Scale DW equations (resp. PS and OS DW), for the Rio Negro-like and Garonne-like channels. Since A

		Rio Negro-like	Garonne-like
		PS DW OS DW PS DW OS DW
	max(ϵ A )	2.84%	0.15%	21.69%	1.06%
	max(ϵ Q )	28.15%	0.42%	37.03%	1.96%
	max(ϵ S )	63.65%	1.5%	57.14%	2.93%

Routine available at: https://github.com/snagcliffs/PDE-FIND

Acknowledgments

This work is part of the PhD thesis work of TM (CIFRE PhD 2018/0725, funded by ANRT).


Pierre Larmande
email: pierre.larmande@ird.fr

Konstantin Todorov
email: todorov@lirmm.fr

AgroLD: a Knowledge Graph for the Plant Sciences

Keywords: Knowledge Graph, Linked Data, Plant Sciences

Recent advances in sequencing technologies and high-throughput phenotyping have revolutionized the analysis in the field of the plant sciences. However, there is an urgent need to effectively integrate and assimilate complementary information to understand the biological system in its entirety. We have developed AgroLD, a knowledge graph that exploits Semantic Web technologies to integrate information on plant species and in this way facilitate the formulation and validation of new scientific hypotheses. AgroLD contains around 900M triples created by annotating and integrating more than 100 datasets coming from 15 data sources. Our objective is to offer a domain specific knowledge platform to answer complex biological and plant sciences questions related to the implication of genes in, for instance, plant disease resistance or adaptative responses to climate change. In this paper, we present results of the project, which focused on genomics, proteomics and phenomics. We present the AgroLD pipeline for lifting the data, the open source tools developed for these purposes, as well as the web application allowing to explore the data.

Introduction

The understanding of genotype-phenotype interactions, which stand for the regulation of gene expression conferring a phenotype, is one of the most critical research areas in agronomy. However, these interactions are complex to identify because they are expressed at different molecular levels in the plant and are strongly influenced by environmental factors. The new challenges consist in identifying these interactions between various molecular entities involved in the expression of the phenotype, which, we believe, can only be addressed by integrating information from different levels in a global model using a systemic approach in order to understand the real functioning of the biological system.

Recent high-throughput technologies such as Next Generation Sequencing, which allow DNA to be sequenced much more rapidly than previous methods, can only partially capture these dynamics of interactions [START_REF] Kemble | Recent insights into the genotype-phenotype relationship from massively parallel genetic assays[END_REF]. Similarly, high-throughput phenotyping, which allows to produce a large amount of experimental data in various environmental conditions, lack in filling the gap with genomics data because of missing links in the data. Even if these new technologies allow to go further and further in obtaining new data, the current limitations and challenges are mainly at the level of data integration and data analysis. Moreover, a methodology to standardize and share data according to the FAIR (Findable, Accessible, Interoperable, Reusable) principles should allow to group data efficiently, and thus contribute to the improvement of the biological knowledge [START_REF] Wilkinson | The FAIR Guiding Principles for scientific data management and stewardship[END_REF]. Indeed, it appears that this knowledge is still fragmented and this fragmentation hinders the elucidation of the molecular mechanisms that govern the expression of complex phenotypes [START_REF] Weighill | Multi-Phenotype Association Decomposition: Unraveling Complex Gene-Phenotype Relationships[END_REF]. The question is, therefore, how to structure and manage the complexity of biological data in order to extract knowledge that can be used to identify the molecular mechanisms controlling the expression of plant phenotypes. Our hypothesis is that weaving these data and disparate information into a knowledge graph (KG) would enable the formulation and validation of research hypotheses that would link genotype to phenotype, hence unlocking the potential of the currently available decentralized scientific data. We have developed AgroLD (for Agronomy Linked Data) [START_REF] Venkatesan | Agronomic Linked Data (AgroLD): a Knowledge-based System to Enable Integrative Biology in Agronomy[END_REF], 4 a FAIR knowledge graph powered by Semantic Web technologies as a structure to integrate data, to enable knowledge sharing and to allow information retrieval at scale. It is designed to integrate available information on various plant model species in the agronomic domain such as rice, arabidopsis and wheat, to name a few. The online documentation 5 shows the complete list of species with the total number of related protein entities. Among the contributions of the project is the development of the AgroLD schema, which combines newly created concepts/properties with concepts/properties imported from various ontologies from the biology field. Because life sciences and bioinformatics produce a large plethora of specific data formats, specific open source tools for data conversion to the Resource Description Framework (RDF) following the AgroLD schema have been developed and discusses in this paper. These different steps have led to the construction of several graphs on plant molecular interactions, which have been interlinked to form the AgroLD KG. We present these tools, together with a data fusion approach that allows for the construction of the pivotal AgroLD graph. Finally, we introduce an exploratory search engine that allows to browse the knowledge graph.

Related Work

In the last decade, many initiatives emerged in the biomedical and bioinformatics fields aiming at providing integrated environments to formulate scientific hypotheses about the role of genes in the expression of phenotypes or the emergence of diseases. Among them, we cite Bio2RDF [START_REF] Belleau | Bio2RDF: towards a mashup to build bioinformatics knowledge systems[END_REF], EBI RDF [START_REF] Jupp | The EBI RDF platform: linked open data for the life sciences[END_REF], Uniprot RDF [START_REF] Redaschi | UniProt in RDF: Tackling Data Integration and Distributed Annotation with the Semantic Web[END_REF], WikiPathways [START_REF] Waagmeester | Using the semantic web for rapid integration of WikiPathways with other biological online data resources[END_REF], OpenPhacts [START_REF] Chichester | Drug discovery FAQs: workflows for answering multidomain drug discovery questions[END_REF] and PubChemRDF [START_REF] Fu | Pub-ChemRDF: towards the semantic annotation of PubChem compound and substance databases[END_REF]. Moreover, we can mention the BioHackathon6 [START_REF] Aoki-Kinoshita K | Implementation of linked data in the life sciences at Bio-Hackathon 2011[END_REF] which gather multidisciplinary scientists to solve biomedical and bioinformatics issues in data integration and knowledge representation. Since 15 years, the BioHackathon produces tools, ontologies [START_REF] Bolleman | FALDO: a semantic standard for describing the location of nucleotide and protein feature annotation[END_REF] and guidelines [START_REF]DBCLS guidelines for RDFizing databases[END_REF] for RDF modelling and conversion. Recently, the DisGeNET [START_REF] Piñero | The DisGeNET knowledge platform for disease genomics: 2019 update[END_REF] RDF platform and the Monarch Initiative [START_REF] Mungall | The Monarch Initiative: An integrative data and analytic platform connecting phenotypes to genotypes across species[END_REF] were created for human biology data. OntoForce7 developed a new tool named DISQOVER for data discovery in life sciences. However, to the best of our knowledge, there was no equivalent in the plant sciences field before the AgroLD platform [START_REF] Venkatesan | Agronomic Linked Data (AgroLD): a Knowledge-based System to Enable Integrative Biology in Agronomy[END_REF] was launched in 2015. In a related topic, KNETMINER [START_REF] Hassani-Pak | KnetMiner: a comprehensive approach for supporting evidence-based gene discovery and complex trait analysis across species[END_REF] is a graph database for plant molecular network that has been developed with Neo4J and provides also a subset of its datasets through a SPARQL endpoint. Both KNETMINER and AgroLD have the same purpose and target the same scientific community. However, KNETMINER offers limited access to its features in its free version, while AgroLD has the advantage of being open and FAIR.

3 The AgroLD Knowledge Graph

Overview

AgroLD is built in phases spanning vast aspects of plant molecular interactions. The current phase (second phase) covers information on genes, proteins, predictions of homologous genes, metabolic pathways, plant phenotype and genetic studies. At this stage, we have integrated data from several resources such as Ensembl plants [START_REF] Bolser | Ensembl Plants: Integrating Tools for Visualizing, Mining, and Analyzing Plant Genomics Data[END_REF], UniProtKB [START_REF]UniProt: a worldwide hub of protein knowledge[END_REF], Gene Ontology Annotation [START_REF] Huntley | The GOA database: gene Ontology annotation updates for 2015[END_REF]. The choice of these sources has been guided by the biological community, as they are widely used and have a strong impact on the user's confidence. We have also integrated resources developed by the local SouthGreen platform [START_REF] South Green Collaborators | The South Green portal: a comprehensive resource for tropical and Mediterranean crop genomics South Green collaborators[END_REF] such as Trop-GeneDB [START_REF] Hamelin | TropGeneDB, the multi-tropical crop information system updated and extended[END_REF], a tropical plant genetics database, OryGenesDB [START_REF] Droc | OryGenesDB 2008 update: database interoperability for functional genomics of rice[END_REF], a rice genomics database, GreenPhylDB [START_REF] Valentin | Green-PhylDB v5: a comparative pangenomic database for plant genomes[END_REF], a comparative genomics database for tropical plants, OryzaTagLine [START_REF] Larmande | Oryza Tag Line, a phenotypic mutant database for the Genoplante rice insertion line library[END_REF], a rice phenotype database and SniPlay [START_REF] Dimou | RML: A generic language for integrated RDF mappings of heterogeneous data[END_REF], a rice genomic variation database. These resources bring together experimental data produced by researcher groups in Montpellier and the South of France. The online documentation provides an overview of the integrated data sources 8 . The conceptual framework of AgroLD is based on well-established ontologies in the plant field such as Gene Ontology [START_REF] Ontology | The Gene Ontology Resource: 20 years and still GOing strong[END_REF], Plant Ontology [15] or Plant Trait Ontology [START_REF] Cooper | The Planteome database: An integrated resource for reference ontologies, plant genomics and phenomics[END_REF]. Furthermore, we developed a dedicated schema9 that creates links between the imported ontologies and introduces new classes and properties. The online documentation 10 shows the complete list of the used ontologies. The majority of these ontologies are hosted by the OBO Foundry project [START_REF] Smith | The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration[END_REF].

In the following, we describe the components of the knowledge graph and the process of its construction.

Statistics

As of today, AgroLD contains more than 900 Millions triples resulting of the integration of roughly 100 datasets gathered in 33 named graphs. Table 1 gives a summary of all number of features. Table 3 gives an overview of available resources and tools. All datasets are available in Zenodo under the Creative Commons Attribution 4.0 International license (CC-BY 4.0). Each resource can contain several datasets, for instances, one dataset per species or per data type. Combining all ontologies and datasets imported, AgroLD graph gather 383 classes and 793 properties. Among the pipelines developed to lift up the datasets, we focused also on connecting our datasets with others. The property rdfs:seeAlso reach the total number of almost 80 millions of outbound links making the AgroLD graph correctly linked with other datasets in the LOD. Besides, we paid attention to increasing the number of semantic annotations with imported ontologies, which increased the number of links between datasets making the overall graph denser. We created more than 14 million semantic links linking entities to ontological classes. Finally, our data linking strategy (see next section) allowed us to create around 160,000 owl:sameAs links between entities. exist within the Semantic Web community, including Datalift [START_REF] Scharffe | Enabling linked data publication with the Datalift platform[END_REF], Tarql [START_REF] Tarql | SPARQL for Tables[END_REF], RML.io [START_REF] Dimou | RML: A generic language for integrated RDF mappings of heterogeneous data[END_REF], none of them were adapted to take into account the complexity of data formats in the biological domain (e.g. VCF format [18]) or even the complexity of the information they could contain. A simple example illustrates this complexity through the GFF (Generic Feature Format) [START_REF]The formal specification of GFF3[END_REF], which represents genomic data in a TSV type format (file with tabs as separators). It contains a column with key = value type information, of variable length and having different information depending on the data source. In this case, the transformation needs to be adapted according to the data source. Furthermore, the large volume of data was a limiting factor for the above-mentioned tools. In this context, we developed RDF conversion tools adapted to a large range of genomics data standards such as GFF [START_REF]The formal specification of GFF3[END_REF], Gene Ontology Annotation File (GAF) [START_REF]Gene Annotation File (GAF) specification[END_REF], Variant Call Format [18] and we are currently working on packaging these ETL tools in an API 11 . These data standards represent a first step, as they are indeed the most widely used in the community. However, we plan to develop more tools as we will integrate new data standards.

The AgroLD Schema

In order to match the different data types and properties, we developed a schema that associates the classes and properties identified in AgroLD with corresponding ontologies. Fig. 1 shows an overview of the AgroLD ontology including these mappings. For instance, the Protein class 12 is associated with the SO polypeptide class 13 with the owl:equivalentClass property. Similar mappings have been done for the properties. For example, the has function property is linked with properties from the RO ontology, 14 with owl:equivalentProperty. When an equivalent property did not exist, we associated it with the higher level property with rdfs:subPropertyOf. For example, the property has trait 15 , linking entities with TO terms is associated with a more generic property from RO: causally related to 16 . So far, 55 mappings have been manually identified.

URI design

In the transformation pipelines, RDF graphs share a common namespace and are named according to the corresponding data sources. Entities in RDF graphs are linked by the common URI principle. In general, we build URIs by referring to Identifiers.org [START_REF]The formal specification of GFF3[END_REF] which provides design patterns for each registered source. For instance, genes integrated from Ensembl Plants are identified by the base URI. 17 When they are not provided by Identifiers.org, new URIs are constructed and in this case URIs take the form. 18 In addition, the properties linking the entities are constructed as form. 19In order to link identical entities from different data sources, we used the approach based on URI pattern matching. Its principle is to scan the URIs in order to look for similar patterns in the terminal part of the URI (i.e. Entity ID). In addition, we also followed the common URI approach which recommends to use the same URI pattern for two identical entities. Therefore, for the same entity, this allowed us to aggregate information from different RDF graphs. In addition, we used cross-reference links by transforming them to URIs and linking the resource to the rdfs predicate seeAlso. This significantly increases the number of outbound links by reaching almost 80 million links, making AgroLD better integrated with other data sources. In the future, we plan to implement a similarity entity profile approach to identify matches between entities with different URIs.

Challenges in Creating the AgroLD Graph

The process of creating a knowledge graph is complex and challenging. In this section, we will present some of the challenges we had to address and in particular those related to managing the heterogeneity of the datasets and their sizes, aligning the entities and assessing the data quality.

Concerning data heterogeneity, the main problem was the variety of the data formats which we solved by having RDF as unified format. We proposed several pipelines that were able to handle this variety and manage the size of the datasets. Indeed, as discussed in Section 3.3, in the majority of cases, we preferred developing our own solutions instead of using generic tools to manage better the complexity or the size of the datasets. Another problem was the heterogeneity of the genomic coordinates (i.e. different naming of the chromosome identifier, missing information, etc.). We solved it by choosing a unique representation and transforming all coordinates in URIs patterns following the FALDO ontology representation [START_REF] Bolleman | FALDO: a semantic standard for describing the location of nucleotide and protein feature annotation[END_REF].

Concerning the entity linking problem (i.e. same entities having different names or identifiers), we managed to only partially solve this problem, by using pattern matching in URIs, or database cross-links to identify mappings between entities. Indeed, in the case where entities have a different namespace URI (e.g. namespace1:identifier1 and namespace2:identifier1), we search patterns matching in the URIs and create a new URI doing the mapping between them. In the case when entities have different URIs with no matching patterns but having synonym properties (i.e. skos:altLabel, skos:prefLabel, skos:synonym or specific ones), we search matches with these properties and the URIs patterns. For entities that do not contain the above information, we adopt a more global approach based on properties and values analysis. However it is an open challenge that we are currently working on.

Concerning the processes followed for data quality assessment, pre-processing quality assessments such as input file format, raw line and missing value check were developed for the resources used by the ETL pipeline. Then, the produced triples were validated for syntax with built-in libraries (e.g. with RDFlib). Further assessments include counting the number of entities (e.g. genes, proteins, chromosomes, etc) and checking the presence/absence of properties with sets of SPARQL queries. More complex quality assessment such as type restriction on properties is planned in the future.

Data Access and Applications

The AgroLD KG is available for access via a SPARQL enpoint 20 . However, although the SPARQL language is efficient to build queries, regarding access to 20 http://agrold.southgreen.fr/sparql RDF data, it remains difficult to handle for our main users, which are mainly biologists with little or no background in formal query languages. Therefore, we propose a web application implementing various elements of a semantic search systems, such as pattern-based querying, graphical visualization, information retrieval tools. Hence, the AgroLD platform provides three entry points, as described in [START_REF] Venkatesan | Agronomic Linked Data (AgroLD): a Knowledge-based System to Enable Integrative Biology in Agronomy[END_REF]:

-Quick Search is a faceted search plugin made available by Virtuoso that allows users to search by keywords and browse AgroLD content by navigating through links. Fig. 2A shows the result of a keyword search. In this example a user submitted the GRP2 keyword which stands for a gene name. Results are ranked according to the number of occurrences found in various fields of the entities. -Advanced Search is an interface allowing specific searches by class of entities such as filtering by Gene, Protein, Pathway and having an aggregation engine for external resources (Fig. 2C). The Advanced Search form is based on a RESTFul API. The purpose of this interface is to provide a tool to query the knowledge graph while hiding the technical aspects of SPARQL querying. The interest of coupling the AgroLD RESTFul API and this interface is to be able to interactively combine searches with external services API such as for instance Pubmed or EMBL APIs. Moreover, this is made possible through the user interface and also programmatically. As shown in Fig. 2C, the users begin by entering one or several keywords and select which type of entity they want to retrieve. In this example, we selected the type Gene and the keyword TBP1. The results are presented in the form of a table which can be sorted and explored. Moreover, results could be visualised as a graph as shown in Fig. 2B. This tool was adapted from KnetMaps [39]. -The SPARQL Editor is a query editor that provides an interactive environment for formulating SPARQL queries. We developed the editor based on the YASQE and YASR [START_REF] Rietveld | The YASGUI Family of SPARQL Clients[END_REF] tools and adapted them for our system. In addition, we proposed a list of modular and customizable query patterns according to the users' needs that can be automatically executed through the editor. Fig. 3 shows the SPARQL Editor. The editor is divided in three areas. The main area on the top left corresponds to the query area. Thanks to the YASQE tool, proposing several code editing functions, the code syntax is highlighted and checked for errors. Furthermore, users can load their own queries stored in a file in order to run batch queries and save their results in various file formats. It is also possible to build up queries by using query patterns. In this case, the users can select one of the dozen query patterns shown on the right. The query appears in the left-hand area within the query text box. Users can read the code of the query and see differently colored pieces of code according to the type of variable, string fields and SPARQL syntax. Hence, users can directly modify the code. There is also a text box above the query box that allows users to modify the value of the string parameter and by clicking on the apply button, it modifies the string value of the query. Finally, the results are displayed at the bottom of the editor as a table (by default), but they can also be displayed in JSON or graph-based formats. Each column can be sorted and text filters can be applied to search among the results. Data can be downloaded as a CSV file.

Use-case Scenarios

A better understanding of genotype-phenotype relationships requires the integration of biological information of various kinds. However, this information is often dispersed in several databases on the Internet each with different data models, scales or distinct means of access. For biologists, it is difficult to search relevant information in these databases as the mass of information can be incomplete and hard to manage. These problems are particularly relevant in the context of genetic association analyses or GWAS (Genome Wide Association Studies), which allow to associate large regions of the genome (locus) with a phenotypic trait (trait). GWAS loci often include several hundred genes that need to be analysed in order to identify only a fraction of the genes associated with the trait under study. At some point, each scientist will have to choose which genes to investigate further in the laboratory.

In order to show how AgroLD can help in this type of analysis, we took the results published in [START_REF] Crowell | Genome-wide association and highresolution phenotyping link Oryza sativa panicle traits to numerous trait-specific QTL clusters[END_REF] and tried to reproduce the experiments. The paper studies the key genes that are responsible of the panicle architecture in rice. The authors outlined, based on a manual literature review, a list of 319 candidate genes known to regulate the plant architecture. The aim of our use-case study is to reproduce these results automatically.

The authors of [START_REF] Crowell | Genome-wide association and highresolution phenotyping link Oryza sativa panicle traits to numerous trait-specific QTL clusters[END_REF] identified numerous GWAS loci combining several trait associations all along the chromosomes and studied chromosome 4, which was associated with ten panicle and yield traits. We found less associations with the query Q122 in AgroLD. Indeed, only five phenotypic traits loci associated to "panicle" trait name were retrieved.

Next, the authors identified 20 candidate genes distributed along chromosome 4. By building a second query Q2, which retrieve the genes available for chromosome 4 and using a filter on "panicle", we obtained 15 genes results.

Finally, the authors narrowed down the genomic region of chromosome 4 between 30 Megabases and 32 Megabases. They identified five candidates genes namely OsKS3, OsKS1, OsKS2, OsMADS31 and NAL1. On our side, querying the same genomic region with similar filters (Q3), we obtained only one gene: NAL1. However, using a less restrictive query (Q4), we obtained 81 genes results including the five candidates identified by the authors. By comparing our results to the ones in the paper, we can first argue that the authors have a larger GWAS/QTL datasets than AgroLD currently has integrated. Thus, they get more genomic regions associated with phenotypic trait than we get in Q1. Second, they have a better selection of candidate genes for a given genomic region. Even if AgroLD contains a large number of genes (81 genes) for the same genomic region, the final result is smaller when genes are filtered by the name of the trait. After checking, we observed that this value is absent in the majority of cases. The authors extracted this information from the manual review of scientific papers.

Conclusion and Future Work

Data in the agronomic domain are highly heterogeneous and dispersed. For plant scientists to make informed decisions in their daily work it is critical to integrate information at different scales. Semantic Web technologies play a pivotal role in data integration and knowledge management. The biomedical domain provides a good example to follow by capitalizing on previous experience and addressing lessons learned. To build on this line of research in the agronomy field, we have developed the AgroLD KG. The knowledge base exploits the power of seamless data integration offered by RDF. It contains more than 900 Millions triples resulting of the integration of roughly 100 datasets gathered in 33 named graphs. However, it coverage with respect to the species and the data sources are expected to grow with the subsequent releases. To our knowledge, AgroLD is one of the first initiatives taken to bring Semantic Web practices to the agronomic domain, playing a complimentary role in the integrative approaches adopted by the community.

AgroLD is being actively developed based on feedback from domain experts. It also benefits from the support of the SouthGreen Bioinformatics Platform since its beginning in 2015 by providing IT support and infrastructure to host data and web applications. SouthGreen is one of the core platforms of the French Elixir-EU node, thus will provide a long lasting support for AgroLD. AgroLD is strongly linked to several use-cases of the D2KAB project 23 (National Research Agency funded project) to demonstrate the benefits of linked data to discover gene-phenotype interactions. With the achievement of the second phase, user feedback reveals some limitations and challenges on the current version. Thus, a number of issues are a matter of ongoing or future work.

On the one hand, the KG coverage has to be extended to a larger number of biological entities (e.g. miRNA) and relations (e.g. co-expression, regulation and interaction networks) in order to capture a broader view of the molecular interactions. For instance, we need to integrate information on gene expression and gene regulatory networks. On the other hand, the ETL process for KG creation is mostly based on domain specific approaches thus limiting its re-usability. We will investigate approaches using declarative functions for its creation.

Methods for knowledge augmentation need to be applied and adapted to our data. Indeed, we have observed that certain information remains hidden in the RDF literal contents, such as biological entities or relationships between them, while a wealth of related knowledge is available in external sources. We currently developing methods to extract information embedded in unstructured data such as the text fields from the KG or from external web documents and scientific publications and bring this information under a structured form to the knowledge base. Finally, we are in the process of extending state-of-art data linking techniques by considering the specificities of the biological domain.

Fig. 1 .

 1 Fig. 1. Overview of the AgroLD schema

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 21

Fig. 2 .

 2 Fig. 2. Overview of AgroLD Web interfaces. (A) displays the Faceted search interface. (B) displays results from the KnetMaps tool. (C) displays results from the advanced search interface.

Fig. 3 .

 3 Fig. 3. The SPARQL query editor. The Query patterns frame allows to select a query from a natural language question. The Query text frame allows to visualize and modify the SPARQL query. The results frame displays results returned from the query.

Table 1 .

 1 Features of the AgroLD knowledge graph.

	Features	Number of features
	datasets	100
	graphs	20
	triples	933,663,219
	classes	383
	properties	793
	rdfs:seeAlso	79,696,972
	owl:sameAs	166,551
	semantic annotations	14,652,812
	3.3 AgroLD Integration Pipelines
	Our contributions focus, among other things, on the development of various RDF
	conversion workflows for large agronomic datasets. Although several generic tools

Table 2

 2

. Data sources integrated in AgroLD. Ontologies are referenced as GO = gene ontology, PO = plant ontology, TO = plant trait ontology, EO = plant environnment ontology, SO = sequence ontology, CO = crop ontology (plant specific traits)

Table 3 .

 3 Links to AgroLD resource and tools

	Name of resource or tool and description, URL
	Data
	AgroLD datasets, https://doi.org/10.5281/zenodo.4694518
	List of graphs, http://www.agrold.org/documentation.jsp
	List of ontologies, http://www.agrold.org/documentation.jsp
	AgroLD vocabulary,
	https://github.com/SouthGreenPlatform/AgroLD ETL/tree/master/model
	AgroLD SPARQL Endpoint, http://agrold.southgreen.fr/sparql
	Example queries, http://www.agrold.org/sparqleditor.jsp
	Use case queries, https://github.com/pierrelarmande/ISWC-use-case
	Tools
	Web application, https://github.com/SouthGreenPlatform/AgroLD webapp
	RDF conversion pipelines (GFF2RDF, GAF2RDF, VCF2RDF, Datasets),
	https://github.com/SouthGreenPlatform/AgroLD ETL

www.agrold.org

http://www.agrold.org/documentation.jsp

http://www.biohackathon.org

https://www.ontoforce.com

http://www.agrold.org/documentation.jsp

https://github.com/SouthGreenPlatform/AgroLD ETL/tree/master/model

http://www.agrold.org/documentation.jsp

https://github.com/SouthGreenPlatform/AgroLD ETL

http://www.southgreen.fr/agrold/vocabulary/Protein

http://purl.obolibrary.org/obo/SO 000010

http://purl.obolibrary.org/obo/RO 0000085

http://www.southgreen.fr/agrold/vocabulary/has trait

http://purl.obolibrary.org/obo/RO 0002410

http://identifiers.org/ensembl.plant/{Entity ID}

http://www.southgreen.fr/agrold/resource/{Entity ID}

http://www.southgreen.fr/agrold/vocabulary/{property}

http://www.agrold.org

Example Use case, available from: https://github.com/pierrelarmande/ISWC-usecase