Denys Dutykh
email: denys.dutykh@univ-savoie.fr

Didier Clamond
email: diderc@unice.fr

Marx Chhay
email: marx.chhay@univ-savoie.fr

Serre-type equations in deep water

Keywords: deep water approximation, Serre-Green-Naghdi equations, variational principle, free surface impermeability MSC: [2010] 76B15 (primary), 76B07, 76M30 (secondary) PACS: [2010] 47.35.Bb (primary), 47.35.-i, 45.20.Jj (secondary) deep water approximation, Serre-Green-Naghdi equations, variational principle, free surface impermeability

This manuscript is devoted to the modelling of water waves in the deep water regime with some emphasis on the underlying variational structures. The present article should be considered as a review of some existing models and modelling approaches even if new results are presented as well. Namely, we derive the deep water analogue of the celebrated Serre-Green-Naghdi equations which have become the standard model in shallow water environments. The relation to existing models is discussed. Moreover, the multi-symplectic structure of these equations is reported as well. The results of this work can be used to develop various types of robust structure-preserving variational integrators in deep water. The methodology of constructing approximate models presented in this study can be naturally extrapolated to other physical flow regimes as well.

Introduction

Water waves is a special case of mechanical waves propagating at the interface of water and air. They play a central rôle in the interactions taking place between the ocean and atmosphere [START_REF] Komen | Dynamics and Modelling of Ocean Waves[END_REF][START_REF] Thorpe | The Turbulent Ocean[END_REF]. R. Feynman described the complexity of water waves using the following words [START_REF] Feynman | Mainly Mechanics, Radiation, and Heat[END_REF]:

[ . . . ] the next waves of interest, that are easily seen by everyone and which are usually used as an example of waves in elementary courses, are water waves. As we shall soon see, they are the worst possible example, because they are in no respects like sound and light; they have all the complications that waves can have [ . . . ] The complete mathematical formulation describing the propagation of water waves is quite complex to deal with. It cannot be solved analytically (unless in some asymptotic sense) and even efficient numerical algorithms are developed since 1970's [10] and nowadays this problem is far from being fully understood. That is why the water wave theory has always been developing by constructing sophisticated approximations [START_REF] Craik | The origins of water wave theory[END_REF]. Traditionally we assume that the fluid is homogeneous (i.e. its density ρ = const) and ideal and the flow is incompressible. Additionally it is quite common to assume that the flow is also irrotational [START_REF] Stoker | Water Waves: The mathematical theory with applications[END_REF], i.e. the flow vorticity vanishes identically.

In order to describe mathematically this problem let us introduce a Cartesian coordinate system O x 1 x 2 y , where the horizontal plane O x 1 x 2 coincides with the still water level y = 0 . The free surface is given by the function y = η (x, t) . The axis O y points vertically upwards. The only force acting on the fluid is the gravity g = (0, 0, -g) . The sketch of the fluid domain is shown in Figure 1.

To have a compact notation we introduce the vector of horizontal independent variables x def := (x 1 , x 2 ) along with the associated horizontal gradient operator

∇ def := ∂ x 1 , ∂ x 2 .
The three-dimensional gradient will be denoted by

∇ def := ∂ x 1 , ∂ x 2 , ∂ y .
Similarly, the horizontal u = (u 1 , u 2 ) and vertical v components of the velocity can be separated for the sake of convenience. The three-dimensional velocity vector will be denoted by u = (u, v) . Throughout this study two-dimensional vectors are denoted by bold symbols and three-dimensional vectors have additionally an over bar (or an arrow).

From the irrotationality assumption it follows that there exists a function φ(x, y, t) called the velocity potential such that

u = ∇ φ , v = φ y .
By taking into account also the flow incompressibility we obtain that the velocity potential φ is necessary a harmonic function, i.e.

∇ 2 φ + ∂ 2 y φ = 0 , (x, y) ∈ R 2 × -∞, η (x, t) . (1.1)
On the free surface we have the kinematic condition (also known as the free surface impermeability):

η t + ∇ η • ∇ φ = ∂ y φ , y = η (x, t) . (1.
2) The free surface being an isobar, thus we have also the dynamic boundary condition:

φ t + 1 2 | ∇φ | 2 + 1 2 φ 2 y + g η = 0 , y = η (x, t) . (1.3)
The last equation is known as the Cauchy-Lagrange integral in which we chose the gauge where the Bernoulli constant vanishes. Let us assume that the wave field consists mainly of waves with a characteristic wavelength λ , which corresponds to the characteristic wavenumber κ ≡ 2 π λ . The average fluid depth is d . Thus, we can form a dimensionless number κ • d . If this parameter κ • d ≫ 1 then we can further simplify our problem by applying the so-called deep water approximation d → +∞ , which allows to 'evacuate' the solid impermeable bottom from the consideration. Basically, it allows to reduce the number of boundary conditions to satisfy by one. More precisely, we require that the fluid tends to the state of rest as we dive into it, i.e.

| ∇ φ | → 0 , φ y → 0 , as y → -∞ . In practice, even for values κ • d 2 the deep water approximation can be already successfully applied in some situations [52]. In general, we refer to [52] as an excellent review on deep water waves. Equations (1.1) -(3.5) will be referred to as the full water wave problem in deep water approximation.

The shallow water limit is much better understood nowadays. In shallow water there is a well-established hierarchy of hydrodynamic models:

• nonlinear shallow water (Saint-Venant or Airy (especially in UK) or fully nonlinear non-dispersive) equations [START_REF] Airy | On the laws of the tides on the coasts of Ireland, as inferred from an extensive series of observations made in connexion with the Ordnance Survey of Ireland[END_REF][START_REF] De Saint-Venant | Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit[END_REF] The deep water case is much less organized. The main difference between these two regimes comes from the dimensionless numbers which characterize the flow. When the depth d is finite, we have one parameter α 0 d which characterizes the wave nonlinearity and another parameter d 2 λ 2 to describe flow 'shallowness'. By applying asymptotic expansions in one or even two parameters we can obtain various approximate models. Now, if we take the limit d → +∞ both these parameters collapse in the deep water making this case somehow special. Traditionally, deep water waves have been described as perturbations of a certain carrier wave. In this way, the wave field has been conveniently described using wave envelopes [5]. Then, the envelope function is shown to satisfy the nonlinear Schrödinger [53] or Dysthe-type [START_REF] Dysthe | Note on a modification to the nonlinear Schrödinger equation for application to deep water[END_REF] equations depending on the desired asymptotic order of accuracy. These equations can be also recast into the Hamiltonian framework [START_REF] Gramstad | Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth[END_REF]. In the present study we take an alternative route without appealing to wave envelope techniques.

The present article is organized as follows. In Section 2 we present three variational formulations of the full water wave problem (1.1) -(3.5). Then, in Section 3 we briefly review the state of the art in deep water wave modelling and in Section 4 we present the variational derivations of Saint-Venant and Serre equations analogues in deep water. Finally, the main conclusions and perspectives of this study are discussed in Section 5.

Variational structures

In this Section we briefly describe the main variational structures of the deep water wave problem in the chronological order of their appearance. Of course, this list is not being exhaustive.

Lagrangian and Hamiltonian formulations

In perfect agreement with the themes of this special issue, the water wave problem is known since A. [START_REF] Petrov | Variational statement of the problem of liquid motion in a container of finite dimensions[END_REF] [START_REF] Petrov | Variational statement of the problem of liquid motion in a container of finite dimensions[END_REF] and V. Zakharov [53] to have the Hamiltonian structure. Below we present the classical Lagrangian and Hamiltonian formulations together since they are naturally related by Legendre transformation. A more advanced Luke's Lagrangian functional along with its generalizations will be presented below (see Sections 2.2 & 2.3).

Let us compute the kinetic K and potential energies of a deep fluid moving under the force of gravity g :

K def := 1 2 ˆR2 ˆη -∞ | u | 2 dy dx, = 1 2 g ˆR2 η 2 dx .
According to Hamilton's principle [2], the fluid motion has to provide a stationary value to the following action functional

S = ˆt1 t 0 ρ L dt , (2.1)
where L is the Lagrangian density classically defined as

L def := K - .
Below in Section 2.2 we shall give another Lagrangian density. We have to keep in mind that the flow is incompressible, i.e. ∇ • u = 0 , and on the free surface we also have the kinematic boundary condition that we shall write as

η t = 1 + | ∇η | 2 • u n ,
where u n def := u • n is the normal velocity at the free surface and n is the outer unitary normal vector

n = 1 1 + | ∇η | 2 -∇η 1 .
We have to incorporate these conditions into Hamilton principle using two Lagrange multipliers φ = φ(x, y, t) and ϕ = ϕ(x, t) :

L = K - + ˆR2 ˆη -∞ φ ∇ • u dy dx + ˆR2 η t - 1 + | ∇η | 2 • u n ϕ dx .
By taking the variation of this functional with respect to u and requiring that it vanishes in the fluid bulk we obtain δ u : u -∇φ = 0 .

(2.2) Consequently, the flow is necessarily irrotational. It is a direct consequence of assumptions made above and the Lagrange multiplier φ is a velocity potential. From Kelvin's circulation theorem we know that the flow initially irrotational will remain irrotational forever [3]. The variational description of flows with vorticity is out of scope of the present study.

Taking into account (2.2), from now on we can substitute u = ∇φ into the Lagrangian density L . By applying the Gauß-Ostrogradsky theorem to the Lagrangian density we obtain

L = ˆR2 ϕ η t + u n 1 + | ∇η | 2 • ϕ -φ y = η dx -K - .
By taking the variation with respect to the normal velocity u n we obtain

δu n : ϕ -φ y = η = 0 .
Thus, the other Lagrange multiplier ϕ is simply the trace of the velocity potential at the free surface, i.e. ϕ(x, t) ≡ φ x, y = η(x, t), t .

Finally, the Lagrangian density L becomes

L = ˆR2 ϕ η t dx -H
where H def := K + is the total fluid energy being also the Hamiltonian of the water wave problem:

H = 1 2 ˆR2 ˆη -∞ | ∇φ | 2 dx + 1 2 g ˆR2 η 2 dx .
The last Hamiltonian functional was independently rediscovered by Broer (1974) [START_REF] Broer | On the Hamiltonian theory of surface waves[END_REF], then by Miles (1977) [START_REF] Miles | On Hamilton's principle for water waves[END_REF] and probably several other researchers.

The evolution equations for canonical variables are

η t = δ H δ ϕ , ϕ t = - δ H δ η .
By δ H we denote the variational (Gâteaux's) derivative. In order to compute the Hamiltonian H one has to solve the Laplace equation

∇ 2 φ = 0 ,
with corresponding boundary conditions:

φ y = η = ϕ , | ∇φ | → 0 , as y → -∞ .
In general, it is not possible to solve this problem analytically. Consequently, in deep water one uses in practice asymptotic expansions with respect to the small parameter ε ∼ ∇η ∼ α 0 • κ .

Luke's Lagrangian formulation

In 1967 J.C. Luke proposed to use the following functional [START_REF] Luke | A variational principle for a fluid with a free surface[END_REF] (in finite depth case):

L = ˆη -d φ t + 1 2 | ∇φ | 2 + 1 2 φ 2 y + g y dy ,
where d is the constant water depth. The action integral is defined in (2.1) as above.

Without free surface effects this functional was proposed in 1929 by H. Bateman [4].

One can easily recognize that the expression under the integral sign is the well-known Cauchy-Lagrange integral. In his seminal paper [START_REF] Luke | A variational principle for a fluid with a free surface[END_REF] Luke justified the advantages of this functional over the classical Lagrangian L = K -described above. In order to apply the deep water approximation we have to take the limit d → +∞ . The term g y is not integrable, so before taking this limit we integrate it over the depth and remove the constant term -g d 2 2 which disappears under the Gâteaux derivative operation. As a result, we obtain the following Lagrangian density:

L = ˆη -∞ φ t + 1 2 | ∇φ | 2 + 1 2 φ 2 y dy dx + 1 2 g η 2 .
(2.3)

In order to recover the water wave problem equations (1.1) -(3.5) in deep water, we write down the Euler-Lagrange equations corresponding to the functional (2.3):

δφ : ∇ 2 φ + φ yy = 0 , δφ y = η : η t + ∇φ • ∇η -φ y = 0 , δη : φ t + 1 2 | ∇φ | 2 + 1 2 φ 2 y + g η = 0 .
Luke's variational principle has at least one important advantage over the Hamiltonian principle: the flow incompressibility (1.1) is incorporated into the variational principle and it does not have to be additionally assumed as a constraint. It appears as one of Euler-Lagrange equations.

Relaxed Lagrangian formulation

Recently, two authors of this manuscript proposed a generalization to Luke's Lagrangian [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF]. The so-called 'relaxed variational principle' will be extensively used in this study and we proceed to a brief description of the main ideas behind this generalization. Earlier in the literature this this method was introduced also under the name of a "motivated Legendre transform" (see e.g. [START_REF] Henyey | Hamiltonian description of stratified fluid dynamics[END_REF] for more details). In this Section we follow closely the broad lines of our previous publication [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF].

We would like to introduce more variables into the Luke Lagrangian (2.3) which has the velocity potential φ(x, y, t) and free surface elevation η(x, t) in its original form. Let us introduce also explicitly the components of the velocity field u = ∇φ and v = φ y by using two Lagrange multipliers µ and ν :

L = -ϕ η t + 1 2 g η 2 + ˆη -∞ 1 2 ( u 2 + v 2 ) + µ • (∇φ -u) + ν (φ y -v)
dy , where we took also the term φ t out of the integral sign for the sake of convenience. By applying the Gauß-Ostrogradsky theorem we can rewrite the Lagrangian L in the following equivalent form:

L = -η t + μ • ∇η -ν ϕ + 1 2 g η 2 + ˆη -∞ 1 2 (u 2 + v 2 ) -µ • u -ν • v -(∇ • µ + ν y ) φ dy .
The tildes denote the quantities evaluated at the free surface, i.e. ν(x, t) def := ν x, y = η(x, t), t . The last functional L is the so-called relaxed variational principle. Let us count the degrees of freedom:

(1) η (x, t) is the free surface elevation (2) φ(x, y, t) is the velocity potential (3) u(x, y, t) is the horizontal velocities vector (4) v(x, y, t) is the vertical velocity (5) µ(x, y, t) is the Lagrange multiplier associated to the horizontal velocities (6) ν(x, y, t) is the Lagrange multiplier associated to the vertical velocity So, instead of having two degrees of freedom in the original Luke Lagrangian, the relaxed Lagrangian has six. This extra freedom can be used to derive various approximations which was illustrated in [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF].

Lagrange multipliers

For the purposes of the present study we may content with four degrees of freedom by eliminating the Lagrange multipliers µ and ν. Indeed, let us compute the variations of the relaxed Lagrangian with respect to u and v:

δu : u -µ = 0 , δv : v -ν = 0 .
This computation gives us also the physical sense of Lagrange multipliers -they are pseudo-velocities, which coincide with physical velocities u and v at least in the unconstrained case. Thus, we can substitute µ = u and ν = v into L to obtain

L = -η t + ũ•∇η -ṽ ϕ + 1 2 g η 2 - ˆη -∞ 1 2 (u 2 + v 2 ) -(∇•u + v y ) φ dy .
(2.4) We shall use extensively this Lagrangian in the rest of this manuscript.

Intermediate conclusions. The variational structure in general (such as Hamiltonian or Lagrangian functionals) is important in many respects. First of all, since the full equations (1.1) -(3.5) enjoy this variational structure, we should seek for approximate models which enjoy the same structure and, thus, preserve some sub-set of qualitative properties of the base model. For instance, the Hamiltonian formalism [START_REF] Zakharov | Hamiltonian formalism for nonlinear waves[END_REF] allows to simplify asymptotic developments in powers of the nonlinearity parameter ε def := α 0 λ , which is the wave steepness in the deep water regime. Finally, the Hamiltonian formulation allows also to put the problem of hydrodynamic waves in a unified framework of nonlinear waves in various media [54,[START_REF] Zakharov | Hamiltonian formalism for nonlinear waves[END_REF]. Thus, methods developed in other fields might be directly transposed to water waves.

State of the art

The golden standard in deep water wave modelling is incontestably the cubic Zakharov model used, recently e.g. in [START_REF] Dyachenko | Weak turbulence of gravity waves[END_REF]32] to study wave (weak) turbulence [START_REF] Zakharov | Kolmogorov Spectra of Turbulence I[END_REF]. These equations are obtained by expanding the Hamiltonian in the wave steepness parameter

H = H 0 + H 1 + H 2 + . . .
The cubic Zakharov equations are obtained by truncating this expansion after quartic terms (thus, the governing equations are effectively cubic after taking the variations). This model is weakly nonlinear but it is valid for the whole spectrum of gravity waves. Cubic Zakharov equations are well understood nowadays. Consequently, in the present study we focus on models which do the opposite: on one hand, there are a priori no assumptions on the nonlinearity parameter, on the other hand, we describe waves around certain wavenumber κ . Let us review the state of the art by following the main steps of [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF]. However, below we generalize their computations to the three-dimensional case.

Consider the incompressible Euler equations:

∇ • u + v y = 0 , (3.1)
u + ∇p = 0 , (3.2) v + p y + g = 0 , (3.3)
where p is the fluid pressure and the over dot denotes the total material derivative, i.e.

( •) def := (•) t + u • ∇(•) + v (•) y .
The governing equations are completed with the following boundary conditions:

η t + u • ∇η = v , y = η (x, t) , (3.4) p = p a , y = η (x, t) , (3.5)
| u |, | v | → 0 , y → -∞ , (3.6)
where p a is the constant atmospheric pressure.

In order to derive an approximate model, Kraenkel et al. [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF] propose to take the following solution ansatz for the Euler equations (3.1) -(3.3):

u(x, y, t) = u 0 (x, t) e κ y , v(x, y, t) = - 1 κ (∇ • u 0 ) e κ y . (3.7)
From above ansatz it is straightforward to understand the physical sense of the variable u 0 -it is simply the value of the 3D horizontal velocity u on the surface y = 0 . Other ansätze will be considered below. Here, κ = const is the wavenumber around which we model water waves in the spectral domain. The vertical velocity ansatz is chosen to satisfy identically the free surface incompressibility (3.1). The velocity fields under a linear periodic wave predicted by this ansatz is represented in Figure 2. Substituting ansatz (3.7) into the kinematic boundary condition (3.4) we readily obtain the mass conservation equation:

κ η t + ∇ • u 0 e κ η = 0 . (3.8)
In order to derive momentum balance equations, we compute first the material derivatives using ansatz (3.7):

u = u 0 t e κ y + e 2 κ y , -κ v = A e κ y + B e 2 κ y ,
where A , B and = C 1 , C 2 are defined as

A = ∇ • u 0 t , B = ∇(∇ • u 0 ) • u 0 -(∇ • u 0 ) 2 , (a) u 1 (x 1 , y, t) (b) v (x 1 , y, t)
C 1 = u 2 ∂u 1 ∂x 2 -u 1 ∂u 2 ∂x 2 , C 2 = u 1 ∂u 2 ∂x 1 -u 2 ∂u 1 ∂x 1 .
By substituting v into (3.3) and taking into account the boundary condition (3.5), we obtain the pressure distribution in the fluid bulk:

p -p a = g (η -y) + 1
κ 2 e κ ye κ η A + 1 2 e 2 κ ye 2 κ η B .

(3.9)

Notice that the pressure field p diverges when y → -∞ due to the hydrostatic effects (in agreement with the Archimedes law). The pressure distribution under a linear periodic wave is shown in Figure 3.

The problem now is to satisfy in some sense equation (3.2) with available expressions for u and p . Kraenkel et al. proposed the following weak formulation:

ˆη -∞ u + ∇p e m y dy = 0 ,
where m > 0 is a modelling parameter to be chosen later. So, the Newton law is satisfied in an average sense. The exponential weight function allows to overcome the problem of Archimedean pressure divergence. In order to derive the momentum equation in a conservative form, we shall use the equivalent form:

ˆη -∞ u e m y dy + ∇ ˆη -∞ p e m y dyp a e m η • ∇η = 0 . After performing all computations we obtain the desired horizontal momentum equation:

e (m + κ) η m + κ u 0 t + m + κ m + 2 κ e κ η + ∇ g e m η m 2 -A e (m + κ) η κ m (m + κ) - 1 2 B e (m + 2 κ) η κ m (m + 2 κ) = 0 . (3.10)
Just derived equations (3.8) and (3.10) constitute a closed system which describes the evolution of water waves in deep water.

Remark 1. In order to make a check of the derivation made above, we shall restrict our attention to the two-dimensional case. Here we can set u 0 1 u , u 0 2 0 and dependent coefficients become:

A u x t , B u u x x -u 2 x , 0 .
Thus, equations (3.8) and (3.10) in 2D become:

κ η t + u e κ η x = 0 , (3.11) e (m + κ) η m + κ u t + g e m η m 2 -u x t e (m + κ) η κ m (m + κ) -u u x x -u 2 x e (m + 2 κ) η κ m (m + 2 κ) x = 0 .
(3.12)

By switching to dimensionless variables (g 1 , κ 1) we recover equations (2.18) and (2.20) from [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF].

As we saw above, the water wave problem possesses Hamiltonian and Lagrangian variational structures. The question we can ask is whether just derived equations (3.8), (3.10) (which are supposed to approximate the full Euler equations) possess at least one of these structures? By looking at equations (3.8), (3.10) the answer is not clear. Even in 'Conclusion and comments' Section in [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF] the Authors admit that they did not succeed in finding a Hamiltonian formulation even for the short wave limit of these equations. We shall propose some fixes to this problem below in Section 3.2.

Choice of the modelling parameter

In order to derive a physically sound value for the modelling parameter m, we consider the governing equations in 2D and we linearize equations (3.11), (3.12):

κ η t + u x = 0 , 1 m + κ u t + g m η - 1 κ m (m + κ) u x t x = 0 .
It is easy to eliminate the variable η from the above equations to obtain the linear version of the so-called improved Boussinesq equation:

m κ u t t -g (m + κ) u x x -u t t x x = 0 .
Then, we look for plane wave solutions of the form:

u(x, t) = α 0 e i (k x -ω t) .
Such solutions exist only if wave frequency ω and wavenumber k are related by the following relation, which is called the dispersion relation:

c p (k) = ω(k) k = g m + κ m κ + k 2 .
The ratio c p (k) is called the phase velocity. Notice that in the limit k → κ we obtain

lim k → κ ω(k) k = g κ ,
which coincides with the exact phase velocity of the full Euler equations in deep water. Finally, in order to determine the modelling parameter m we consider the group velocity:

c g (k) def := ∂ω(k) ∂k .
In the limit k → κ we obtain

lim k → κ c g (k) = m m + κ • g κ .
Now it is straightforward to notice that we recover the exact expression c g (κ) = 1 2 • g κ for deep water Euler equations only if m ≡ κ . This is the desired value of the modelling parameter m .

Variational derivations

In this Section we propose alternative derivations of the model equations similar to (3.8), (3.10), but based on the relaxed variational principle (2.4). Of course, the resulting models obtained through variational derivation might (and actually will) be different from (3.8), (3.10). However, the advantage here is that the Lagrangian structure is preserved by construction.

Weakly compressible ansatz

Consider an ansatz similar to (3.7):

φ(x, y, t) = φ 0 (x, t) e κ y , u(x, y, t) = u 0 (x, t) e κ y , v(x, y, t) = v 0 (x, t) e κ y .
(3.13) The main difference with (3.7) is that here the vertical velocity approximation v 0 is kept independent from u 0 and it will be chosen by the variational principle. Substituting (3.13) into Lagrangian (2.4) and performing exactly all the integrations over the depth, we obtain the following Lagrangian density:

L = -η t + u 0 • ∇η -v 0 e κ η φ 0 e κ η + 1 2 g η 2 - 1 2 κ 1 2 (| u 0 | 2 + v 2 0 ) -(∇ • u 0 + κ v 0 ) φ 0 e 2 κ η . (3.14)
The governing equations are then obtained by computing variations of this functional:

δφ 0 : η t + u 0 • ∇η -v 0 e κ η = 1 2 κ ∇ • u 0 + κ v 0 e κ η , δu 0 : u 0 + ∇φ 0 + 4 κ φ 0 ∇η = 0 , δv 0 : φ 0 - 1 2 κ v 0 -κ φ 0 = 0 , δη : φ 0 e κ η t + ∇ • φ 0 u 0 e 2 κ η + g η -1 2 (| u 0 | 2 + v 2 0 ) -(∇ • u 0 + κ v 0 ) φ 0 e 2 κ η -κ η t + 2 u 0 • ∇η -v 0 e κ η φ 0 e κ η = 0 .
The Euler-Lagrange equations turn out to be rather complicated. However, we can learn some lessons nevertheless. The variation δu 0 gives us the connection between the horizontal velocity u 0 and the velocity potential φ 0 (thus, u 0 can be in principle eliminated from the equations). In particular, one can see that the flow is not irrotational. The variation δv 0 gives us the expression of the vertical velocity v 0 = 3 κ φ 0 in terms of the velocity potential. The variation with respect to η gives the analogue of the Cauchy-Lagrange integral (i.e. an unsteady Bernoulli equation). Finally, the variation with respect to φ 0 gives us the mass conservation equation and in order to have a conservative form, it is better to reinforce the flow incompressibility, i.e.

∇ • u 0 + κ v 0 ≡ 0 .
It will be done in the following Section.

Exactly incompressible ansatz

Now we take the same ansatz (3.13), but the vertical velocity approximation v 0 is chosen in order to satisfy identically the incompressibility condition. It is not difficult to see that this goal can be achieved by taking v 0 ≡ -1 κ ∇ • u 0 . In this way we recover ansatz (3.7) proposed by Kraenkel et al. [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF]. Substituting this expression for v 0 into the Lagrangian density (3.14), we obtain the following slightly more compact density functional:

L = -η t + u 0 • ∇η + 1 κ ∇ • u 0 e κ η φ 0 e κ η + 1 2 g η 2 - 1 4 κ | u 0 | 2 + 1 κ 2 (∇ • u 0 ) 2 e 2 κ η .
The Euler-Lagrange equations yield the following system:

δφ 0 : κ η t + ∇ • u 0 e κ η = 0 , δu 0 : ∇ (∇ • u 0 ) e 2 κ η -κ 2 u 0 e 2 κ η = 2 κ 3 φ 0 ∇η e 2 κ η -2 κ 2 ∇ φ 0 e 2 κ η , δη : φ 0 e κ η t + ∇ • φ 0 u 0 e 2 κ η + g η -κ ∇ • u 0 e κ η φ 0 e 2 κ η - 1 2 | u 0 | 2 + 1 κ 2 (∇ • u 0 ) 2 = 0 .
Even if these equations are more compact (than the system we obtained in Section 3.2.1) and the mass conservation coincides exactly with (3.8), still this system seems to be quite complicated. This time, it follows from the variation δu 0 that in order to reconstruct the horizontal velocity u 0 from the velocity potential φ 0 , one has to solve an elliptic (vectorial) equation.

Intermediate conclusions. We saw above that the variational method can easily lead to complicated and unamenable equations, even if the latter inherits naturally the variational structure. Consequently, the choice of good ansatz is absolutely crucial for the derivation of an approximate model. In this respect a better ansatz will be proposed below.

Alternative deep water ansatz

Let us slightly modify the ansatz (3.13) in the following way: instead of defining all the quantities by their values at y = 0 , we shall define them at the free surface y = η (x, t) in accordance with Hamilton's principle: φ(x, y, t) = ϕ(x, t) e κ (y -η) , u(x, y, t) = ũ(x, t) e κ (y -η) , v(x, y, t) = ṽ(x, t) e κ (y -η) .

(4.1) The horizontal and vertical velocities predicted by this ansatz are represented in Figure 4. Notice qualitative similarities with Figure 2. However, the magnitudes of velocities are slightly different.

Substituting the last ansatz into the relaxed variational principle (2.4) and performing exactly all integrations over the vertical coordinate y yields the following Lagrangian density:

2 κ L = 2 κ ϕ η t -g κ η 2 + 1 2 | ũ | 2 + ṽ 2 -ũ • ∇ϕ -κ ϕ ∇η -κ ṽ ϕ . (4.2)
The direct comparison with Lagrangian (3.14) shows that just obtained Lagrangian density (4.2) is much simpler. Indeed, in (4.2) we have only cubic nonlinearities at most, while in (3.14) the nonlinearities are of infinite order. Below we shall derive some approximate models from this Lagrangian density in deep water.

Saint-Venant equations in deep water

The Euler-Lagrange equations for functional (4.2) can be easily obtained:

δ ũ : ũ -∇ϕ + κ ϕ ∇η = 0 , δṽ : ṽ -κ ϕ = 0 , δϕ : 2 κ η t + ∇ • ũ -κ ṽ + κ ũ • ∇η = 0 , δη : 2 κ g η + 2 κ ϕ t + κ ∇ • (ϕ ũ) = 0 .
The first two variations show that our approximation (4.1) is exactly irrotational in the sense that we have u ≡ ∇φ , v ≡ ∂ y φ . The substitution of the first two relations in the last two yield the following system:

η t + 1 2 κ -1 ∇ 2 ϕ -1 2 κ ϕ = 1 2 ϕ ∇ 2 η + κ | ∇η | 2 , ϕ t + g η = -1 2 ∇ • ϕ ∇ϕ -κ ϕ 2
∇η . The last system was called the generalized Klein-Gordon equations (gKG) since its linearization coincides with the classical Klein-Gordon equation. The reasons why these equations can be considered as the analogue of nonlinear shallow water (or Saint-Venant [START_REF] De Saint-Venant | Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit[END_REF]) equations in deep water are explained in [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF]. The gKG equations were extensively studied in [START_REF] Dutykh | Numerical study of the generalised Klein-Gordon equations[END_REF]. In particular, it was shown that these equations possess the canonical symplectic (Hamiltonian) and multi-symplectic structures additionally to the variational structure incorporated into the Lagrangian density (4.2). Moreover, it was shown numerically that gKG equations may develop waves with an angular point at the crest as it is known since G. Stokes in the case of periodic travelling waves [START_REF] Stokes | Supplement to a paper on the theory of oscillatory waves[END_REF]. We note also that Kraenkel et al. [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF] also obtained some indications for the existence of peaked travelling waves in their model in the limit of small aspect ratio waves.

Remark 2. If we look for periodic linear travelling waves of the form η(x , t) = α cos(k xω t) , then the dispersion relation will be

c p (k) ≡ ω(k) k = 1 2 g k 2 + κ 2 κ k 2 .
In particular, one can see that for k = κ the phase and group velocities coincide with the exact values given by linearized Euler equations.

Serre equations in deep water

The next model of interest can be obtained if we impose the free surface impermeability as a constrain, i.e. ṽ = η t + ũ • ∇η .

By substituting this expression for ṽ into Lagrangian density (4.2), we obtain the following functional:

2 κ L = (κ η t + ∇ • ũ) ϕ -κ g η 2 + 1 2 | ũ | 2 + 1 2 (η t + ũ • ∇η) 2 .
The Euler-Lagrange equations give us the following relations among dependent variables:

δ ũ : ũ + (η t + ũ • ∇η) -∇ϕ = 0 , (4.3) δϕ : κ η t + ∇ • ũ = 0 , (4.4) δη : η t t + κ ϕ t + 2 κ g η + ( ũ • ∇η) t + ∇ • (η t ũ) + ∇ • ( ũ • ∇η) ũ = 0 . (4.5)
The variation δϕ shows that our approximation turns out to be exactly incompressible

(since δ L δϕ = 0 is equivalent to ∇ • u + v y = 0)
. Notice that we did not require this condition at the level of ansatz (4.1). It is the Hamilton variational principle which gives this property automatically in this particular case. On the other hand, the approximation is not exactly irrotational. To linear approximation equations (4.3) -(4.5) yield an improved linear Boussinesq equation:

(∇ 2 -κ 2 ) η t t + 2 g κ ∇ 2 η = 0 .
This Boussinesq equation admits travelling plane wave solutions with the dispersion relation

c p (k) = 2 g κ k 2 + κ 2 .
Again, it is not difficult to see that for k = κ we obtain the exact values of the phase and group velocities given by linear Euler equations. Notice, that to linear approximation the model (4.3) -(4.5) derived in this Section coincides with the model proposed by Kraenkel et al. [START_REF] Kraenkel | Theory of small aspect ratio waves in deep water[END_REF] (see also Section 3 above) if the modelling parameter m is chosen in the optimal way, i.e. m = κ . In [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF] equations (4.3) -(4.5) were named the deep water Serre equations (and this name was properly motivated). The existence of singular (i.e. peaked) travelling waves was also shown in [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF].

Evolution equations

The governing equations (4.3) -(4.5) may appear complicated to the reader. The main problem with the formulation given above is that equations are not written in an evolutionary form of a system of PDEs with time derivatives separated from other terms. We can recast equations (4.3) -(4.5) in a more amenable form. In order to obtain a compact form of equations (4.3) -(4.5), first we are going to expand them:

ṽ ≡ η t + ũ • ∇η = η t + ∇ϕ • ∇η 1 + | ∇η | 2 ≡ ũ • ∇η -κ -1 ∇ • ũ , ũ ≡ ∇ϕ -ṽ ∇η = ∇ϕ -η t ∇η + | ∇η | 2 ∇ϕ -∇ϕ • ∇η ∇η 1 + | ∇η | 2 , 0 = κ η t + ∇ • ũ , 0 = 2 κ g η + κ ϕ t + ṽ t + ∇ • (ṽ ũ) .
The last equation gives us a hint that the right evolution variable is

q def := ∇ ϕ + κ -1 ṽ = ũ + ṽ ∇η + κ -1 ∇( ũ • ∇η) -κ -2 ∇(∇ • ũ) .
We can notice that ∇(∇ • ũ) can be seen as an application of the operator matrix ∇ ⊗ ∇ to the vector ũ, i.e.-

q = I -κ -2 ∇ ⊗ ∇ ≡ D -1 • ũ + ṽ ∇η + κ -1 ∇( ũ • ∇η) ,
where I denotes the identity operator. Finally, the system of evolution equations can be written as

κ η t + ∇ • D • q = ∇ • D • ṽ ∇η + κ -1 ∇( ũ • ∇η) , (4.6) q t + 2 g ∇η = -κ -1 (∇ ⊗ ∇) • (ṽ ũ) . (4.7)
These equations have to be supplemented by two algebro-differential relations:

ṽ = ũ • ∇η -κ -1 ∇ • ũ , ũ = D • q -ṽ ∇η -κ -1 ∇( ũ • ∇η) .
The pseudo-differential operator D = Iκ -2 ∇ ⊗ ∇ -1 can be easily computed in the Fourier space:

D = I -κ -2 k ⊗ k -1 ,
where k = (k 1 , k 2 ) is the vector of wavenumbers. Otherwise, one has to invert an elliptic operator numerically as it is custom in the numerical analysis of classical Serre equations [START_REF] Dutykh | Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations[END_REF].

Open problem. By analogy to the classical Serre-Green-Naghdi equations the canonical Hamiltonian structure for deep water Serre equations (4.6), (4.7) does not probably exist. The Authors of the present manuscript did not succeed to find even a non-canonical Hamiltonian formulation which should exist in principle. Consequently, it remains an open problem so far. However, we succeeded to find the multi-symplectic formulation for deep water Serre-type equations.

Multi-symplectic formulation

The history of multi-symplectic formulations can be traced back to V. Volterra (1890) who generalized Hamiltonian equations for variational problems involving several variables [START_REF] Volterra | Sopra una estensione della teoria Jacobi-Hamilton del calcolo delle variazioni[END_REF][START_REF] Volterra | Sulle equazioni differenziali che provengono da questiono di calcolo delle variazioni[END_REF]. Later these ideas were developed in 1930's [19,[START_REF] Lepage | Sur les champs géodésiques du calcul des variations[END_REF][START_REF] Weyl | Geodesic Fields in the Calculus of Variation for Multiple Integrals[END_REF]. Finally, in 1970's this theory was geometrized by several mathematical physicists [START_REF] Goldschmidt | The Hamilton-Cartan formalism in the calculus of variations[END_REF]30,[START_REF] Krupka | A geometric theory of ordinary first order variational problems in fibered manifolds. I. Critical sections[END_REF][START_REF] Krupka | A geometric theory of ordinary first order variational problems in fibered manifolds. II. Invariance[END_REF] similarly to the evolution of symplectic geometry from the ideas of J.-L. Lagrange [START_REF] Lagrange | Mécanique analytique[END_REF][START_REF] Souriau | Structure of Dynamical Systems: a Symplectic View of Physics[END_REF]. In our study we will be inspired by modern works on multi-symplectic PDEs [7,40]. Recently this theory has found many applications to the development of structure-preserving integrators [START_REF] Bridges | Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity[END_REF][START_REF] Chen | The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs[END_REF][START_REF] Dutykh | Geometric numerical schemes for the KdV equation[END_REF]42].

Here we give the multi-symplectic structure for deep water Serre equations in the case of one spatial horizontal dimension x 1 ≡ x (and, thus, ũ ≡ ũ 1 ) for the sake of notation compactness. The generalization to the case of two horizontal dimensions is straightforward. The general form of multi-symplectic equations (with one spatial variable) is

M • z t + K • z x = ∇ z S (z) , (4.8)
where z ∈ R d is the vector of state variables and M, K ∈ Mat d×d (R) are some skewsymmetric matrices. It is not difficult to check that for deep water Serre equations (4.3) -(4.5) (in 1D) it is sufficient to take

z = ⊤ ϕ, η, ũ, γ, β, ṽ , S (z) = -g κ η 2 + β (ũ ṽ -γ) + 1 2
ũ 2ṽ 2 , and the skew-symmetric matrices M and K are defined as

M def :=           0 -κ 0 0 0 0 κ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0           , K def :=           0 0 -1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0           .
Equation (4.8) can be rewritten in the component-wise form for the sake of clarity:

-κ η t -ũ x = 0 , κ ϕ t + ṽ t + γ x = -2 g κ η , ϕ x = ũ + β ṽ , -η x = -β , 0 = γ -ũ ṽ , -η t = -ṽ + β ũ .
Now it is straightforward to check by making substitutions that equation (4.8) is indeed equivalent to the deep Serre equations (4.3) -(4.5). The generalization to the 2D case is straightforward.

Remark 3. The multi-symplectic structure for classical ( i.e. shallow water) one layer Serre-Green-Naghdi equations was recently reported in [START_REF] Chhay | On the multi-symplectic structure of the Serre-Green-Naghdi equations[END_REF]. Later this structure was generalized to the case of interfacial waves between two layers (with rigid lid approximation) in [START_REF] Clamond | Multi-symplectic structure of fully nonlinear weakly dispersive internal gravity waves[END_REF].

Discussion

Above we presented several approximate models in deep water regime and now we outline the main conclusions and perspectives of the present study.

Conclusions

In the present article first we discussed the most popular variational structures for the water wave problem in the deep water regime. Then, we assumed that the flow has an exponential profile in the vertical coordinate, which allowed us to derive some known and some new approximate models in deep water. The governing equations possess κ-dependent coefficients, where κ is the dominant wave number that we try to describe in the physical domain. This property is quite usual for models describing the modulation of wave trains. However, in models we consider this property appears without introducing any envelopes. Whenever it was possible, the underlying variational structure of these equations was discussed. Of course, the best strategy to preserve this structure is to derive approximations using variational methods [START_REF] Clamond | Practical use of variational principles for modeling water waves[END_REF]. For instance, one can expand the Hamiltonian functional in powers of some order parameter and truncate this expansion after a few terms [START_REF] Zakharov | Kolmogorov Spectra of Turbulence I[END_REF] or one can choose an ansatz for the flow, impose additional constraints, substitute everything into the Lagrangian density and perform simplifications [START_REF] Clamond | Modeling water waves beyond perturbations[END_REF]. The governing equations will be given by taking the variations of this functional with respect to all dependent variables. No matter which method is chosen, it is absolutely crucial to work and simplify the Hamiltonian or Lagrangian functionals instead of working with individual PDEs. In this study we illustrated also the weaknesses of the variational method. Indeed, a 'bad' ansatz may lead to cumbersome and unamenable equations which have a variational structure, but nevertheless we do not really want to work with such model equations. Only the simultaneous application of the variational method to a 'good' ansatz can yield new and practically useful approximate models.

Perspectives

Above we discussed mainly the modelling issues and we emphasized on the importance of the preservation of variational structure while deriving approximate models. However, these equations can be solved analytically only in rare special situations. Consequently, in the rest of cases we apply numerical numerical methods to solve these equations and we did not cover this topic in the present publication. For variational integrators in general we can refer to [START_REF] Lew | An overview of variational integrators[END_REF]. However, these integrators are much better understood in the ODE (i.e. symplectic) case. Some of the models given in this publication possess also the multisymplectic formulation as well. The corresponding multi-symplectic numerical methods are available as well [START_REF] Bridges | Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity[END_REF]. The general problem which exists in structure preserving numerical methods is to understand how symmetry and structure preservation affect usual properties of numerical methods (such as consistency, accuracy and stability) [START_REF] Chhay | Intégrateurs géométriques: application à la mécanique des fluides[END_REF][START_REF] Chhay | On accuracy of invariant numerical schemes[END_REF]. Of course, the quest for new interesting ansätze in deep water together with physically sound constraints has to be pursued.

Contents 1

 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Variational structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lagrangian and Hamiltonian formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Luke's Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Relaxed Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Choice of the modelling parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Variational derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weakly compressible ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exactly incompressible ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Alternative deep water ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Saint-Venant equations in deep water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Serre equations in deep water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Multi-symplectic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 .

 1 Figure 1. Sketch of the physical fluid domain.

Figure 2 .

 2 Figure 2. Vertical structure of the chosen ansatz (3.7) under a simple linear travelling periodic wave: (a) the horizontal and (b) vertical components of the velocity field. The periodic wave amplitude is α = 0.2 .

Figure 3 .

 3 Figure 3. Fluid pressure distribution under a linear travelling wave as predicted by equation (3.9) (a). The right panel (b) shows the dynamic pressure p d def := pp ag (ηy) distribution ( i.e. without hydrostatic effects). The periodic wave amplitude is α = 0.2 .

(a) u 1 Figure 4 .

 14 Figure 4. Velocity field distribution predicted by ansatz (4.1) under a periodic travelling wave: (a) horizontal and (b) vertical velocities. The periodic wave amplitude is α = 0.2 .

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgments

The authors would like to thank Professors V. Volpert and V. Vougalter for their kind invitation to prepare and submit this manuscript.


Eric Kergosien
email: eric.kergosien@univ-lille3.fr

Hugo Alatrista-Salas

Mauro Gaio
email: mauro.gaio@univ-pau.fr

Fábio N Güttler
email: fabio.nor-guttler@icube.unistra.fr

Mathieu Roche
email: mathieu.roche@cirad.fr

Maguelonne Teisseire
email: maguelonne.teisseire@teledetection.fr

When textual information becomes spatial information compatible with satellite images

Keywords: Natural language processing, Information retrieval, Spatial information, Land-use planning

With the amount of textual data available on the web, new methodologies of knowledge extraction domain are provided. Some original methods allow the users to combine different types of data in order to extract relevant information. In this context, we present the cornerstone of manipulations on textual documents and their preparation for extracting compatible spatial information with those contained in satellite images. The term footprint is defined and its extraction is performed. In this paper, we describe the general process and some experiments conducted in the ANIMITEX project, which aims to match the information coming from texts with those of satellite images.

INTRODUCTION

Nowadays, impressive amounts of high spatial resolution satellite data are available. This raises the issue of fast and effective satellite images analysis as it still requires a costly human implication. In this context, automated and semi-automated remote sensing approaches attempt to tackle this challenge.

Such large amount of data is also related to increasing temporal repetitivity of Earth Observation satellites (i.e. one image every 5 days for the new satellite Sentinel-2 instead of 16 days for the current Landsat-8). At the same time, the web offers a large amount of textual data and many researcher communities are interested in the issue of knowledge extraction including spatial information.

The ANIMITEX project aims at processing massive and heterogeneous textual data (i.e. big data context) in order to provide relevant information to enrich the analysis of satellite images. The project has many application areas such as image annotation [START_REF] Forestier | Knowledge-based region labeling for remote sensing image interpretation[END_REF]. For instance, identifying the precise type of crop or the function of a building is not always possible using only remote sensing images. Nevertheless, textual data could contain this kind of information and give additional meaning to the images. The development of approaches based on image/text matching becomes crucial in order to complete image analysis tasks. It also enables a better classification of data. Moreover, image-text matching will enrich Information Retrieval (IR) methods and it will provide users a more global context of data [START_REF] Sallaberry | Fuzzying gis topological functions for gir needs[END_REF]. This can be useful for experts involved in land-use planning and management.

In this paper, we investigate one specific scenario: The construction of a bypass north of Villeveyrac (a small town close to Montpellier, south of France). The aim of this case study is to show how to enrich images with spatial information present in newspaper articles provided by Midi Libre (French newspaper). The main difficulty in extracting spatial information, e.g. Spatial Features (SF), is the ambiguity inherent in natural language. SF extraction methods generally exploit two complementary fonctionalities: toponym recognition (geoparsing) and toponym resolution (geocoding) [START_REF] Leidner | Detecting geographical references in the form of place names and associated spatial natural language[END_REF]. The problem of the recognition of toponyms (place names) in text, can be seen as a particular category of named entity recognition and classification (NERC). According to [START_REF] Smith | Bootstrapping toponym classifiers[END_REF] there are actually several types of ambiguity involved in toponym resolution (i.e. associate a toponym with its spatial representation). The use of contextual elements (other than toponyms), such as words that have a geographical denotation ("river", "town", "basin" , etc.), can be extremely important in a toponym disambiguation task [START_REF] Hollenstein | Exploring place through user-generated content: Using flickr tags to describe city cores[END_REF].

This paper focuses on the use and adaptation of Natural Language Processing (NLP) techniques for recognition of the spatial representation throughout the document. To achieve this, we have collected a set of newspaper articles (corpus of 3809 textual documents) about the Thau basin territory from 2010 up to 2013. NLP methods based on lexico-syntactic patterns [START_REF] Gaio | Towards heterogeneous resources-based ambiguity reduction of sub-typed geographic named entities[END_REF] were then used to automatically annotate linguistic expressions conveying more or less complex spatial information. In the proposed approach, SF appearing in a text, are composed of at least one named-entity allowing a geolocation and one or more spatial indicators specifying its location [START_REF] Lesbegueries | Geographical information access for non-structured data[END_REF]. Once the SF extracted, the problem is to identify their spatial characteristics in order to define a spatial representation throughout the document.

The paper is structured as follows. In Section 2, an overview of SF extraction methods is presented. In Section 3, the method to identify SF representation is detailed. Section 4 gives a short description of the corpus, reports experiments and lists associated prospects. The paper ends with conclusions and some perspectives.

RELATED WORK

NERC methods automatically annotate different types of named entities: dates, people, organisations, themes, numeric values, as well as place names. There is a significant number of systems available, both proprietary and open source, such as OpenNLP1 from Apache, OpenCalais2 from Thomson Reuters, and CasEN [START_REF] Maurel | Casen: a transducer cascade to recognize french named entities[END_REF]. More specific methods that are solely concerned with geographical data are known as geoparsing [START_REF] Leidner | Detecting geographical references in the form of place names and associated spatial natural language[END_REF]. In our work, we focus on this category and a first issue is to precisely identify named-entities allowing a geolocation using the definition proposed in [START_REF] Lesbegueries | Geographical information access for non-structured data[END_REF].

In this model, SF can then be identified in two different ways:

• an Absolute Spatial Feature (A SF) one Named-Entity (NE) allowing a geolocation, such as:

< (spatialIndicator) * , NE of Location>
A spatialIndicator is a term contained within a geographic lexicon ("river", "town", "mountain", etc.). Two examples of this type of SF are the "The Thau basin" and "the town of Sète";

• a Relative Spatial Feature (R SF) one spatial relationship (topological or Euclidean) with at least one SF. An R SF, including one A SF at the end of a pattern, is defined as:

< (spatialRelation) 1.. * , A SF > or < (spatialRelation) 1.. * , R SF >.
Five spatial relation types are considered: orientation ("in the south of", etc.), distance ("20 kilometres from", etc.), adjacency ("near", etc.) , inclusion ("in", etc.) , and geometry which defines the union or intersection linking two SFs (between A and B, etc.

). An example of this type of SF is "in the area of Cuzco" according to the pattern

< (spatialRelation) 1.. * , A SF >.
A second issue is related to the identification of spatial representation for each SF. In this sub-domain, first research works in the 90s have been focused on the representation of complex qualitative relationships such as orientation [START_REF] Frank | Qualitative spatial reasoning with cardinal directions[END_REF]. The direction of a SF is defined taking as reference the position of a second SF. To achieve this, the author proposes the model with cones to represent the four cardinal points: north, east, south, west. A second proposal was to represent the orientation relationship using a 3x3 grid in which the central cell is called the "neutral zone". Cells around represent eight cardinal points, north, northeast, east on southeast, etc. In [START_REF] Hernández | Qualitative distances[END_REF], the authors focus on the study of the representation of qualitative distances (far, near, etc.) and propose a representation model of for flexible distances at different levels of granularity. In [START_REF] Cohn | Calculi for qualitative spatial reasoning[END_REF], a state-of-the-art of SF representation is drawn. At first, the author focuses on the use of an ontology of geographic objects and presents a study on the extension of basic physical objects (points, lines, etc..) to build more complex figures (regions , roads, etc..). In a second step, the author takes into account the topological relationships between SF and relations (orientation, distance, size and shape of related objects). These representations, enabling to take into account the underlying abstractions, are then used to create complex qualitative models.

More recently, in [START_REF] Davis | Qualitative spatial reasoning in interpreting text and narrative[END_REF], the authors rely on a corpus of citations to identify the spatial relationships between spatial objects. The authors describe several problems related to the identification of relationships between SF, such as ambiguity in the recognition of the spatial representation, continuity of geometrical objects and many others.

SPATIAL FOOTPRINT AND DOCUMENTS

3.1 General process to match satellite images and textual documents

In the context of ANIMITEX project, we propose a new approach which is designed to semiautomatically match satellite images and textual documents in land-use planning contexts. This general approach is divided into three stages (see Figure 1). In the first stage, the approach focuses on the semiautomatic extraction of image objects from satellite images, using temporal clustering and segmentation.

In the second stage, NLP methods have been applied in order to identify linguistic features concerning spatial information in the documents. The use of lexicons and dedicated rules [START_REF] Gaio | Towards heterogeneous resources-based ambiguity reduction of sub-typed geographic named entities[END_REF] allows us (1) to identify the absolute (e.g., "Madrid") and relative (e.g., "south of Madrid") Spatial Features (AS F and R SF) and (2) to calculate the footprint of each document. In the final stage, we propose to select textual documents in relation with the studied land-use planning project. In this paper, we focus on the second and third stages. The following section describes the proposed approach to construct footprints from textual documents, and then to match these documents with satellite images using spatial representation.

Footprint Identification Process

To calculate the footprint, we propose a process consisting of three steps (see Figure 2). SF marking: the text marking step is achieved through an automatic framework containing all specifications proposed in [START_REF] Gaio | Towards heterogeneous resources-based ambiguity reduction of sub-typed geographic named entities[END_REF]. This framework outputs a structure incorporating the SF definition proposed in the previous section. A list of A SF and R SF can be extracted from each document of the treated corpus.

SF validation: for each identified A SF, we check on external resources if there is a corresponding spatial representation. In particular, we have used layers provided by the IGN3 (municipalities, roads, railways, buildings, etc.). In addition, if an A SF does not present on IGN resources, we use gazetteers (Geonames and Open Street Maps) to complete the information. Concerning the representation of R SF, we use spatial indicators of topological order associates to A SF.

Following the scopes proposed in [START_REF] Sallaberry | Fuzzying gis topological functions for gir needs[END_REF], the spatial indicators of topological order have been grouped in five categories:

• Proximity: Different indicators can be used in relationship of proximity, such as "near", "around", "beside", "close to", "periphery", etc. To represent this relationship, we define a Tolerance Zone T Z. This area built around the studied A SF will be calculated as follows:

T Z(A SF) = sur f ace(A SF) + φ * sur f ace(A SF)
where sur f ace(A SF) is the surface of A SF on which the relationship is docked. The value of φ is fixed by taking into account the needs of experts and in particular the size of the studied area.

• Distance: The indicators used in this relationship are of the form: "x km", "x miles", etc. Two representations are then proposed in our approach: 1) calculus of distance from the centroid of the A SF and construction of a circular buffer of size x from the centroid; 2) regarding the shape of the A SF and building a buffer of size x from the edge of the processed A SF.

• Inclusion: This binary operation allows us to check if an A SF is inside another by taking into account indicators such as "center", "in the heart", "in", "inside", etc. Two types of inclusion are considered: 1) full inclusion, for instance, the expression "the town of Sete is in the Herault region"; 2) partial inclusion (i.e. intersection), for instance, "departmental road RD2 crosses (partly included) Poussan town".

• Orientation: This unary relationship has been broadly studied in the literature. Different approaches have been proposed to identify a cardinal points of an A SF. We have chosen to use the conical model proposed in [START_REF] Frank | Qualitative spatial reasoning with cardinal directions[END_REF]. For this, we use the centroid of A SF and we build a buffer around. The size of this buffer will be calculated by taking into account the surface of the studied A SF. Then we decompose the buffer into four equal areas (forming a "X") from the centroid. Each intersection between the buffer and cones thus formed represent the four cardinal points.

• Geometry: Geometry relations are built from at least two A SF. These relationships are, for example, the union, the adjacency, the difference or a position of an A SF with respect to other A SF, for example, C between A and B (where A,B and C are A SF), etc.

Representation of the footprint: After the extraction step and spatial representation of the A SF and R SF, the footprint associated with the treated document can be mapped. In this process, two main problems have been identified. The first one is the persistent ambiguity of some named entity contained in SF because some named entities correspond to several places, e.g. "Montagnac". To address this issue, a configurable spatial filter based on predefined scenarios has been developed. For example, to identify events related to a specific land-use planning project occurred in a part of the area of the Thau lagoon, only the SF contained in this area will be explored. The second issue is related to the use of external resources and the identification of the spatial representation appropriate to each A SF. Taking into account the spatial indicator (e.g. "town", "road", etc.) preceding by the toponymic name is a first answer because it allows us to specify the type of the SF and thus take into account the appropriate spatial representation. As cited before, in this article we propose a process to identify and construct footprints from textual documents to help the remote sensing analysts to explain changes. To achieve this, we need two inputs.

EXPERIMENTS

1) In one hand, we have to identify a spatial representation of a spatial object related to the land-use planning project that we want to address. This spatial representation is obtained thanks to a remote sensing process applied over a time series of satellite images. The goal is to find abrupt and unexplained changes in this time series. The three Pleiades images used in this work were acquired on 10 July 2012, 14 September 2012, and 15 March 2013 (see Figure 3).

The process to match images and textual documents

Once the textual corpus collected, the process, described in Section 3.2, has been applied to extract SF (see Table 1). There are two outputs files. The first one is A XML file, in which SF are identified and marked by a tag < SF >. A SF is validated if it is present in the external geographical resources used (IGN and gazetteers).

In the XML file, one of the tags expresses a spatial relationship between one or two SF. Indeed, tag indirection represents one of the five categories of spatial relations described in Section 3. We can take advantage of indirections to build R SF. To do this, all components in < indirection > tag are explored in order to extract information to extend A SF to R SF. In this manner, some SF associated to spatial relationships can be extended to approximate SF towards the spatial representation of the event identified in the use-case.

The second file (structured in JSON format) represents the geographical characteristics of confirmed SF. In fact, once SF is validated, geographical information describing the SF is collected from resources. This information is, for example, the Lambert coordinates, the resource name, times that SF appears in resources, etc. This information is presented following a JSON structure.

Figure 5 represents validated spatial entities extracted from 3809 documents. As presented in Table 1 (see line 1), 252575 spatial entities are identified and validated.

In this context, an important issue has to be tackled: a validated SF could represent different locations. For example, the SF "Croissiles" is a municipality located in 3 different areas in France: Pas-de-Calais, Calvados and Orne. In the same way, "Calais" is a town located in United States and as well in the north of France.

To address this problem and reduce the noise associated to the ambiguity of spatial entities, we have filtered spatial entities located in the studied area (using Lambert coordinates). The treated experimental land-use planning project is the construction of the Villeveyrac bypass, which represents a small part of the Thau basin region. Thanks to this filtering step, we reduce by 95% potentially SF candidates (from 252575 points to 2090 SF, see Table 1, lines 1 and 2).

Nevertheless, our goal is to be as close as possible to an event that appeared in a specific area located thanks to our scenario. The representation using a simple point is not enough. In this work, we propose to take into account representation of A SF and R SF extracted by our process (see Section 3.2). For instance, the A SF "the town of Villeveyrac" can be represented by a polygon (corresponding to territory of the municipality), the A SF "the road R2" can be represented by lines (it is a departmental road), etc. These representations are obtained using external resources as IGN maps, GeoNames, etc. We have proposed a top-down geographic granularity level, from departments to buildings (in France). However, three types of representations have been considered for our experiments: municipalities, roads, and buildings because of the studied scenario that covers a relatively small area. Figure 6 shows 4 footprints, generated by our approach, which are in intersection relation with the spatial object representing the Villeveyrac bypass. As presented in Section 3.2, a footprint is obtained by the union of all validated SF extracted from a document.

We note that footprints are spatially closer to event previously identified in the representation of the treated spatial object (extracted from satellite images and related to the Villeveyrac bypass construction) thanks to Absolute and Relative Spatial Features (cf. A SF and R SF). Our method enables to identify a footprint for 754 textual documents, e.g. these doc- In this work the footprints are identified, the next step is to analyze the document content to identify the ones referring to the event defined in the scenario.

CONCLUSION AND FUTURE WORK

This paper presents the first step of a methodology for extracting spatial footprint associated with textual documents. The main issue is to identify textual documents containing spatial footprint associated to satellite images, and this work is part of the ANIMITEX project. We have used the spatial relation of intersection to identify textual documents related to the same area of the spatial object representing the scenario. The second step will be performed via several spatial operators (i.e. distance, orientation, covering, etc.) according to a protocol which has to be fixed and in adequacy with the needs of the experts.

Finally, we propose to evaluate the reproducibility and relevance of experiments on other corpora and associated to a distinct land-use planning context.

Figure 1 :

 1 Figure 1: General process of the ANIMITEX project

Figure 2 :

 2 Figure 2: From document to footprint definition: the threesteps process

4. 1

 1 Data preprocessing A set of newspaper articles (i.e. 3809 documents) concerning the Thau basin from 2010 up to 2013 has been acquired. A second part of the data set is composed of three Pleiades satellite images (2 m pixel size and 4 spectral bands) covering the entire Thau basin region. Satellite images are available via the GEO-SUD Equipex 4 .

Figure 3 :Figure 4 :

 34 Figure 3: Visible color composite of the three Pleiades images showing the construction of a bypass north of Villeveyrac (France). First, a temporal clustering is performed over the Villeveyrac area (K-means algorithm with 12 classes). One of the obtained clusters (see Figure 4.a. -blue color) is related to the atypical change representing the bypass construction. Then, a segmentation (MeanShift algorithm) is applied over the temporal clustering results. In total, 1711 objects are obtained from the segmentation (see Figure 4.b.) and 8 of them correspond to the bypass construction. These 8 objects are exported as a single multi-polygon vectorial layer (see Figure 4.c.) and represent the footprint of the Villeveyrac bypass extracted from the Pleiades time series.2) In the other hand, we have at our disposal an extract of our corpus containing 3809 newspapers articles provided by the French press Midi Libre. This corpus is related to the Thau basin region from 2010 up to 2013.

Figure 5 :

 5 Figure 5: Spatial entities identified from corpus over the French territory.

Table 1 :

 1 Experiments on a corpus of 3809 articles

		Steps		Nb Documents	Nb Points	Nb SF	Nb A SF	Nb R SF
		1. Initial corpus	3809	252575	-	-	-
		2. Filtering by the spatial area	758	2090	-	-	-
		3. Intersection with points	0	0	-	-	-
		4. Documents with footprints	758	-	2090	1992	98
		5. Intersection with footprints	279	-	763	677	96
				Map Key
				SF of the Villeveyrac bypass
				SF of the "road R2"
				SF of the "city of Villeveyrac"
				(present in all documents)
				Footprint of document id=89
				Footprint of document id=393
	0	1	2 km	Footprint of document id=2701 Footprint of document id=1453

Mediterranean Sea

Figure

6

: Representation of footprint related to 4 selected documents uments contain at least one SF located in the studied area (see Table 1, line 4). From this selection, 279 documents have got a footprint in intersection with the spatial object representing the Villeveyrac bypass (see Table 1, line 5).

https://opennlp.apache.org/

http://www.opencalais.com/

National Institute of Geographic and Forestry Information -www.ign.fr

http://www.equipex-geosud.fr/

Acknowledgments: The authors thank Midi Libre (French newspaper) for its expertise on the corpus, and all of the partners of ANIMITEX. This work is partially funded by Mastodons CNRS grant, and GEOSUD Equipex.