A G Filippini

S Bellec

M Colin

M Ricchiuto

On the nonlinear behavior of Boussinesq type models : amplitude-velocity vs amplitude-flux forms

Keywords: Wave propagation, nonlinear regime, wave breaking, wave shoaling, Boussinesq equations, amplitude velocity form, amplitude flux form Propagation de vagues, regime non-linéaire, déferlement, shoaling, modéles de Boussinesq, fomulaiton en vitesse, formulation en flux

In this paper we consider the modeling of nonlinear wave transformation by means of weakly nonlinear Boussinesq models. We show that for a given linear dispersion relation and linear shoaling parameter one can derive, within the same asymptotic truncation, two system of PDEs differing only in the form of the linear dispersive operators. In particular, these can either be formulated in terms of derivatives of the velocity, or in terms of derivatives of the flux. In the first case we speak of amplitude-velocity form of the model, in the second case of amplitude flux form.

We show examples of these couples for several linear relations, including a new amplitude-flux variant of the model of Nwogu. We then show, both analytically and by numerical nonlinear shoaling tests, that while for small amplitude waves it is important to have accurate linear dispersion and shoaling characteristics, when approaching breaking conditions it is only the amplitude-velocity or amplitude-flux form of the equations which determines the behavior of the model, and in particular the shape and the height of the waves. This knowledge has tremendous importance when considering the use of these models in conjunction with wave breaking detection and dissipation mechanisms.

Introduction

The modeling of wave transformation in the near shore region requires a physically correct description of both dispersive and nonlinear effects. The use of asymptotic depth averaged Boussinesq Type (BT) models for this task is quite common [START_REF] Brocchini | A reasoned overview on boussinesq-type models: the interplay between physics, mathematics and numerics[END_REF]. These models have to be used with much care. Quite often, weakly-nonlinear variants of these models, such as those proposed in [START_REF] Abbott | Computations of short waves in shallow water[END_REF][START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF][START_REF] Nwogu | An alternative form of the boussinesq equations for near-shore wave propagation[END_REF][START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF][START_REF] Peregnine | Long waves on a beach[END_REF], are used outside of their range of applicability, e.g. when reaching breaking conditions. In these cases fully-nonlinear models should be used instead [START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF]. Moreover, to actually include the energy dissipation effects associated to wave breaking, either ad-hoc viscosity terms are included, or a coupling with the Shallow Water equations is introduced [START_REF] Brocchini | A reasoned overview on boussinesq-type models: the interplay between physics, mathematics and numerics[END_REF][START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq type equations[END_REF][START_REF] Tissier | A new approach to handle wave breaking in fully nonlinear boussinesq models[END_REF][START_REF] Tonelli | Simulation of wave breaking over complex bathymetries by a boussinesq model[END_REF]. Despite of the fact that they are theoretically well adapted only for small amplitude waves, in practice these models provide accurate results also when used outside they domain of validity [START_REF] Brocchini | A reasoned overview on boussinesq-type models: the interplay between physics, mathematics and numerics[END_REF][START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq type equations[END_REF][START_REF] Cheung | Boussinesq-type model for energetic breaking waves in fringing reef environments[END_REF].

The key to this success is actually the use of a properly designed wave breaking model, which includes a breaking detection criterion, and a dissipation mechanism. The challenge for a correct capturing of these fronts, is the understanding of the genuinely nonlinear physics underlying breaking, as well as the behavior of the underlying dispersive wave propagation model, and in particular the wave shoaling when approaching the nonlinear regime. Accounting for genuinely nonlinear effects is thus a research topic of high priority [START_REF] Brocchini | A reasoned overview on boussinesq-type models: the interplay between physics, mathematics and numerics[END_REF]. While the linear properties of the models can be thoroughly studied analytically [START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF], in the nonlinear case some properties, such as e.g. the shoaling behavior, must be studied numerically.

There exist several types of weakly nonlinear BT models. These all provide different approximations of the nonlinear wave (or Euler) equations. The design properties of these models are often the linear dispersion relation and shoaling coefficients which should be as close as possible to those of the linear wave theory for in the range of wave numbers relevant for the applications sought. Given a linear dispersion relation and linear shoaling coefficient, it is known that two nonlinear set of Partial Differential Equations (PDEs) can be formulated, both degenerating to the same linearized system. Denoting by a the wave amplitude, d the mean water level, and λ the wavelength, these two models are alternate forms within the same asymptotics in terms of the nonlinearity ε = a/d and dispersion σ = d/λ parameters. The main difference is in the nature of the higher order derivatives, which can either be applied to the velocity u, or to the flux q = hu, h denoting the depth. These formulations are referred here to as amplitude-velocity, and amplitude-volume flux forms. Examples of such couples for some dispersion relations are given for example in [START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF].

To gain better understanding in the properties of weakly nonlinear BT models, this paper presents a thorough analytical and numerical characterization of their nonlinear behavior. For a given couple linear dispersion relation-linear shoaling parameter, we start by recalling how to construct, within the same asymptotic accuracy, two nonlinear set of PDEs : one in amplitudevelocity form, the other in amplitude-volume flux form. The theory is applied to four linear relations corresponding to the models of Peregrine [START_REF] Peregnine | Long waves on a beach[END_REF], and to the enhanced models of Beji and Nadaoka [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF], Madsen and Sørensen [START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF], and Nwogu [START_REF] Nwogu | An alternative form of the boussinesq equations for near-shore wave propagation[END_REF]. For each of these models, we show the corresponding alternate formulation. We will show that the amplitude-volume flux form of the Peregrine system leads to the model used by Abbott in [START_REF] Abbott | Computations of short waves in shallow water[END_REF], that the amplitude-volume flux form of the Beji and Nadaoka model actually correspond to a slight modification of the Madsen and Sørensen model, and vice versa that the amplitude-velocity form of the latter can be obtained by a small modification of the model of [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF]. Finally, for the equations of Nwogu, we derive a new BT system which is the corresponding amplitude-volume flux form. We then study these Inria models, and the main result of the paper can be summarized as follows : while in the linearized case four types of behaviors are observed, corresponding to the given four linear relations, when approaching the nonlinear regime, only two type of behaviors are observed, which are practically independent on the linear dispersion relations and shoaling parameters, and only depend on whether the model is in amplitude-velocity or amplitude-volume flux form. This observation is confirmed by both theoretical arguments, and numerical results.

The present study gives important insight in the behavior of BT models, especially in view of the applications of breaking detection criteria. In particular, our result shows that these criteria must not only take into account the type of breaking expected in the flow, but also the underlying form of the propagation model. For simplicity, we only consider here models well suited for the near shore range (reduced wave numers kd ≤ π), however very similar arguments can be used to study deep water variants [START_REF] Madsen | Higher-order boussinesq-type equations for surface gravity waves: Derivation and analysis[END_REF].

The structure of the paper is the following. In section §2. we present the derivation of weakly nonlinear Boussinesq equations, and we discuss the construction of models in amplitude-velocity, and amplitude-flux forms for different particular cases. Section §3 presents the theoretical analysis of the systems of PDEs obtained, and in particular the analysis of the propagation of higher order harmonics, which gives an indication of the non-linear behavior of the models. Finally, numerical tests in both the linear and nonlinear regime are discussed in section §4. The paper is ended by conclusive remarks and by an overview of future works.

Weakly nonlinear Boussinesq type models

We review a certain number of weakly nonlinear Boussinesq type (BT) models. We recall that these models are obtained as depth averaged asymptotic approximations of the incompressible Euler equations. In particular, if a denotes a reference wave amplitude, d 0 a reference water depth, and λ a typical wavelength, we consider the nonlinearity parameter ǫ and the dispersion parameter σ defined by

ǫ = a d 0 , σ = d 0 λ .
Weakly dispersive BT models are obtained under the small amplitude hypothesis ǫ = O(σ 2 ), as asymptotic approximations of the order O(ǫσ 2 , σ 4 ). We refer to [START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF][START_REF] Lannes | The water waves problem. Mathematical analysis and asymptotics[END_REF] for details concerning the basic derivation of the models. Our objective is to recall some of the models most commonly encountered in literature, and to discuss the construction of amplitude-velocity and amplitudeflux forms having the same linearized behavior.

In the following sections we use the notation d(x) for the variable mean/still water level, η for the wave amplitude w.r.t. a reference zero level, and h = η + d for the depth (cf. figure 1), Figure 1: Sketch of the free surface flow problem, main parameters description.

Models of Peregrine and Abbott

The most common BT model is the one introduced by Peregrine in [START_REF] Peregnine | Long waves on a beach[END_REF]:

     η t + [(d + η)u] x = 0 u t + uu x + gη x - d 2 [du] xxt - d 2 6 u xxt = 0 . ( 1
)
Here u denotes the depth-average velocity. If we multiplying the first equation of (1) by u, the second one by h and summing up the resulting expressions, we can rewrite the Peregrine equations in terms of the conservative variables (h, q), with q = hu the volume flux. Denoting the Non-Linear Shallow Water (NLSW) flux by

F SW = q 2 h + g h 2 2 , (2)
we obtain:

   h t + q x = 0 q t + F SW x -ghd x -hP t (u) = 0 (3)
System (3) allows to underline the structure of the Peregrine model, which is obtained by adding to the NLSW equations a term given by the depth times the time derivative of an elliptic linear differential operator applied to the depth averaged velocity u. This elliptic operator, denoted by P (•) in (3), is defined by :

P (•) = d 2 [(•)d] xx - d 2 6 (•) xx . (4)
To proceed further, we consider now the non-dimensional form of (1). To this end, we recall that the reference depth, amplitude, and wavelength are denoted by d 0 , a, and λ respectively. Introducing a typical horizontal length scale L, we then introduce the set of non-dimensional variables :

x = x L , t = √ gd 0 L t, η = η a , d = d d 0 , ũ = d 0 a √ gd 0 u .
With this notation, one can easily show that (1) is equivalent to :

         ηt + ( d + εη)ũ x = 0 ũt + εũũ x + ηx -σ 2 d 2 [ dũ] xx - d2 6 ũxx t = 0 . ( 5
)
We recall that the truncation w.r.t. the Euler equations, which should appear on the right hand side, is of order O(εσ 2 , σ 4 ). We now repeat the same algebraic steps done previously : we multiply the first equation by εũ, the second one by h = d + εη, and we add the expressions thus obtained, resulting in

( d + εη)ũ t + ǫũ( d + εη)ũ x + ǫη ηx + dη x -( d + εη)σ 2 d 2 [ dũ] xx - d2 6 ũxx t = 0 .
Since dũ = q + O(ε), we deduce that:

( d + εη)σ 2 d 2 [ dũ] xx - d2 6 ũxx = σ 2 d2 2 qxx - d3 6 ( q d ) xx + O(εσ 2 , σ 4 ) . ( 6
)
As a consequence, an asymptotically equivalent system is obtained by replacing the second equation in ( 5) by

( d + εη)ũ t + ǫũ( d + εη)ũ x + ǫη ηx + dη x -σ 2 d2 2 qxx - d3 6 ( q d ) xx t = 0 . (7)
Coming back to dimensional quantities, we obtain the following BT model presented by Abbott in [START_REF] Abbott | Computations of short waves in shallow water[END_REF] :

   h t + q x = 0 Q t (q) + F SW x -ghd x = 0 (8)
In the linearized case, the models of Peregrine and of Abbott are identical, so the two systems have the same linear dispersive and linear shoaling characteristics. In the nonlinear case, the two systems are both approximations of the Euler equations of order O(εσ 2 , σ 4 ), however they do differ as the dispersive terms are expressed in terms of the derivatives of u in the Peregrine one and in terms of the derivatives of q in (8). In the latter, we see once more the appearance of the time derivative of a linear elliptic operator, denoted by Q(•) and defined by :

Q(•) = (•) -d d 2 (•) xx - d 2 6 (•) d xx (9)
Compared to (3), system (8) has a more compact and seemingly conservative structure, as it does not involve any additional non-conservative product w.r.t the NLSW equations.

The limited accuracy systems (1) and ( 8), e.g. in terms of linear dispersion relations, has pushed the development of the so-called enhanced models. These are improved approximations which, while still remaining of order O(εσ 2 , σ 4 ) w.r.t the Euler equations, provide substantially improved approximations of the linearized dispersion relations and shoaling coefficients of the original three dimensional equations. These models are considered in the following sections.

Beji-Nadaoka and BNA models

Following [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF], an improved system can be obtained as follows : starting from the non-dimensional system of Peregrine (5), we add and subtract the term σ 2 α B P (ũ t) ; we then use the fact that in the non-dimensional form ũt + ηx = O(ε, σ 2 ) to replace +σ 2 α B P (ũ t) by -σ 2 α B P (η x) plus orders of O(εσ 2 , σ 4 ) which can be neglected. The final result is the enhanced system proposed by Beji and Nadaoka [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF] (model BN), in dimensional variables reading :

η t + q x = 0 u t + uu x + gη x -(1 + α B )P t (u) -gα B P (η x ) = 0 (10)
with P (•) the elliptic operator [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF]. The model of Beji and Nadaoka can be also written in terms of the conservative variables (h, q) as :

   h t + q x = 0 q t + F SW x -ghd x -(1 + α B )hP t (u) -ghα B P (η x ) = 0 (11)
The additional free parameter α B allows to improve the dispersion relation of the system w.r.t. the one of Peregrine's model.

System [START_REF] Kazolea | Mathematical and computational modeling for the generation and propagation of waves in marine and coastal environments[END_REF] is in amplitude-velocity form, as the dispersive terms involve derivatives of the velocity and not of the flux q. We can derive an asymptotically equivalent system in amplitudeflux form proceeding exactly as done for the Peregrine equations in non-dimensional form (details are omitted for brevity). The final result is the Beji-Nadaoka-Abbott model (BNA) reading :

   h t + q x = 0 Qt (q) + F SW x -ghd x -gdα B P (η x ) = 0 (12)
where the operator Q(•) is very close to Q(•) in [START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF], except for the presence of the tuning parameter α B :

Q(•) = (•) -(1 + α B )d d 2 (•) xx - d 2 6 (•) d xx (13)
While for α B = 0 system (11) reduces to the model of Peregrine, model [START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq type equations[END_REF] reduces in this limit to the equations of Abbott. When α B = 0 the BNA model has a less compact form, compared to [START_REF] Kirby | A time-dependent numerical code for extended boussinesq equations[END_REF], due to the presence of the additional term -gα B dP (η x ).

Madsen-Sørensen and MSP models

The enhanced model proposed by Madsen and Sørensen in [START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF] is obtained with a procedure very similar to the one discussed in the previous paragraph : starting from the non-dimensional form of the second equation in Abbott's model [START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF], we add and subtract the quantity σ 2 β qxx t ; we then use the facts that qt + hη x = O(ε, σ 2 ) and that h = d + O(ε) to replace +σ 2 α M qxx t with -σ 2 β( dη x) xx . The system analyzed in [START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF] is finally obtained by neglecting all terms of order O(εσ 2 , σ 4 ), and in the mild slope hypothesis by neglecting terms containing d 2

x and d xx . The Inria resulting equations read :

   h t + q x = 0 Qt (q) + F SW x -ghd x -gβd P (η x ) = 0 , (14)
where P (•) and Q(•) are defined by :

P (•) = d 2 (•) xx + 2dd x (•) x , (15)
and

Q(•) = (•) - 1 3 + β d 2 (•) xx - d 3 d x (•) x , (16)
The system reduces exactly to the one of Abbott for the choice of β = 0, but it has a more complex structure due to the term gβd P (η x ). The free parameter β, however, allows to substantially improve the linear dispersion and shoaling properties w.r.t. [START_REF] Kirby | A time-dependent numerical code for extended boussinesq equations[END_REF].

The model is clearly in amplitude-flux form, and an asymptotically equivalent in wave amplitude velocity form can be obtained by manipulations of the non-dimensional form of the equations very similar to those done for the other models (details omitted for brevity). The final form of the model in the mild slope hypothesis is

   h t + q x = 0 q t + F SW x -ghd x + h Pt (u) -βgh P (η x ) = 0 , (17)
with

P (•) = - 1 3 + β d 2 (•) xx -(1 + 2β)dd x (•) x , (18)
In this case the limit β = 0 gives back the model of Peregrine (3), however [START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF] has substantially improved linear characteristics. In the following we will refer to this model as to the Madsen-Sørensen-Peregrine system (MSP).

Note that simple manipulations show that in the case of d = const, the BN and the MSP systems as well as the BNA and the MS systems collapse onto one another provided that β = α B /3. All these models indeed share the same linear dispersion relation. For non-constant bathymetries, these models define two families of linear shoaling parameters, as we will see in section 3.1.

Nwogu and NA models

The study of enhanced Boussinesq equations would not be complete without taking into account the model of extended model of Nwogu [START_REF] Nwogu | An alternative form of the boussinesq equations for near-shore wave propagation[END_REF]. This model is obtained by enhancing the system of Peregrine by replacing the depth averaged velocity by the velocity at an arbitrary elevation z θ . For simplicity, in the following developments we still denote by u the quantity u(t, x, z = z θ ), but it should be clear that the physical meaning of this quantity is different from the depth averaged horizontal speed present in all the previous models. Setting z θ = θd, for some θ ∈ (-1, 0), the model of Nwogu can be written as [START_REF] Nwogu | An alternative form of the boussinesq equations for near-shore wave propagation[END_REF] :

   η t + q + dP b (u) x = 0 u t + uu x + gη x + P a t (u) = 0 (19)
This model has a more complex structure, with additional operators appearing in both equations.

In particular, the operators P a (•) and P b (•) are linear elliptic operators with identical structure to P (•) appearing in the Peregrine system (cf. equation ( 4)), but with different coefficients:

P a (•) = a 1 d 2 6 (•) xx -a 2 d 2 [(•)d] xx , (20)
P b (•) = b 1 d 2 6 (•) xx -b 2 d 2 [(•)d] xx , (21)
The values of such coefficients are expressed as function of the tuning parameter θ and read :

a 1 = 3θ 2 ; a 2 = -2θ; b 1 = 3θ 2 -1; b 2 = -2θ -1 .
The equations of Peregrine are recovered for the choice

a 1 = a 2 = 1, b 1 = b 2 = 0.
Note that it is not possible to find a unique value θ ∈ (-1, 0), such that this condition is verified. With simple manipulations, we can recast model [START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF] in terms of time evolution of the flux q as :

   h t + q + dP b (u) x = 0 q t + F SW x -ghd x + hP a t (u) + u dP b (u) x = 0 (22)
Last equations show that, among the amplitude-velocity enhanced models, the model of Nwogu is the one with the most complex structure when written for the couple (η, q)

We now aim to find an asymptotically equivalent system in amplitude-flux form which degenerates to the same linearized equations as the model of Nwogu. Retracing steps already made in the previous subsections, we first consider the non-dimensional form of [START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF] :

             ηt + ( d + εη)ũ + σ 2 b 1 d2 6 ũxx -b 2 d 2 [ũ d] xx x = 0 ũt + εũũ x + g ηx + σ 2 a 1 d2 6 ũxx -a 2 d 2 [ũ d] xx t = 0 (23)
We multiply the first equation by εũ, and the second by ( d+εη), we sum the expressions obtained, and once more invoke the relations

( d + εη)σ 2 a 1 d2 6 ũxx -a 2 d 2 [ dũ] xx = σ 2 a 1 d3 6 ( q d ) xx -a 2 d2 2 qxx + O(εσ 2 , σ 4 ) (24) Inria and σ 2 d b 1 d2 6 ũxx -b 2 d 2 [ dũ] xx = σ 2 d b 1 d2 6 ( q d ) xx -b 2 d 2 qxx + O(εσ 2 , σ 4 ) (25)
to modify the linear dispersive terms. Neglecting the terms of order O(εσ 2 , σ 4 ), and coming back to the physical variables, we finally obtain the enhanced Boussinesq model :

   h t + Q b x (q) = 0 Q a t (q) + F SW x -ghd x = 0 (26)
where the operators Q a (•) and Q b (•) are defined by

Q a (•) = (•) + d a 1 d 2 6 (•) d xx -a 2 d 2 (•) xx (27) Q b (•) = (•) + d b 1 d 2 6 (•) d xx -b 2 d 2 (•) xx (28)
The system obtained has a similar compact structure as [START_REF] Kirby | A time-dependent numerical code for extended boussinesq equations[END_REF], but it has the exact same linear characteristics of the equations of Nwogu. In this case, for a 1 = a 2 = 1, b 1 = b 2 = 0 we recover the model of Abbott. For this reason in the following we refer to these equations as the Nwogu-Abbott model (NA).

Summary

In this section we have considered the most popular BT models of the literature. We have shown that for each model, viz for a given pair of linear dispersion relation-linear shoaling coefficient, we can derive asymptotically equivalent amplitude-velocity and amplitude-flux forms. Some examples of these couples can be already found in [START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF]. We have given a thorough list of these couples, including a new form of Nwogu's model with a compact PDE structure similar to those of the equations of Abbott. We summarize here the models which will be studied in the rest of the paper listing them couples having the same linearized system : The objective of the following sections is to study the properties of these systems. First, we will consider the aspect of wave propagation, in both the linear and nonlinear case. Then we will provide an analytical and numerical study of the linear and nonlinear shoaling characteristics of these models.

Theoretical analysis of the models

This section is devoted to the analysis of the theoretical properties of the models. In the linear case, the eight systems of PDEs introduced in section §2 reduce by construction to four families of linearized models. On constant bathymetries, the models of Beji and Nadaoka and of Madsen and Sørensen also degenerate to the same system of PDEs. In this case, we will briefly recall the linear dispersion analysis, and then discuss the behavior of second order harmonics. The analysis of the higher order harmonics will show that the nonlinear form of the system has a dramatic effect on the amplitudes of the harmonics, which basically depends on whether the system is written in amplitude-velocity or amplitude-flux form.

For non-constant bathymetries, a linear ODE can be derived to compute the shoaling coefficient in the linearized case. We will briefly recall this procedure, and present are compare the shoaling coefficients for the models considered. The nonlinear shoaling properties will instead be investigated numerically in section §4.

Dispersion properties

The dispersion properties can be investigated by means of a Fourier analysis on a horizontal bottom. Although the derivation of the equations has been based on the assumption of σ 2 ≪ 1 and ε = O(σ 2 ), here we work making the hypothesis of ε ≪ 1 and arbitrary σ 2 . The procedure follows closely the work of Madsen and Schaffer [START_REF] Madsen | Higher-order boussinesq-type equations for surface gravity waves: Derivation and analysis[END_REF]. Taking the non-dimensional form of the models, we introduce a solution of the form:

η = a 1 cos(ξ) + εa 2 cos(2ξ); ũ = u 1 cos(ξ) + εu 2 cos(2ξ) (29)
with ξ = ωtkx and being ω the non-dimensional angular frequency and k the corresponding non-dimensional wave number:

ω = L √ gd 0 ω; k = Lk
We give some details for the derivation for Peregrine's model, the analysis of the other models is similar with small variations discussed in a final remark.

First-order solution. The linear dispersion properties of the models emerge looking at the first-order solution. Substituting (29) in system (5), for instance, and collecting terms of O(1) leads to the system :

   -ωa 1 + k du 1 = 0 -ωu 1 + ka 1 + σ 2 ω k2 d2 3 u 1 = 0 , (30)
which can be easily solved, giving

         u 1 = ω k d a 1 ω2 k2 d = 1 1 + φ2 3 , (31)
with φ = σ k d. The second equation of (31) represents the non-dimensional phase velocity c of the Peregrine model which, expressed in terms of dimensional variables becomes :

c 2 = ω 2 k 2 = gd 1 + φ 2 3 , (32)
Inria with φ = σkd. This represents the rate at which the phase of the wave propagates in space (so the velocity at which every frequency of the wave propagates) ; in this sense it represents the linear dispersion relation for the Peregrine model. The reference to which this value must be compared is the linear dispersion relation of Stokes :

c 2 Stokes = gd tanh(φ) φ (33)
Figure 2 summarizes the linear dispersion characteristics of the models considered here. The figure reports the ratio c/c Stokes for all the models considered. As anticipated, the eight systems of PDEs introduced reduce in the linearized case and for constant bathymetry to only three families of linear dispersion relations, which explains the presence of only four curves in the figure, including the exact solution.

The linear dispersion relations depend on the choice of the values of the tuning parameters α B , β and θ. For all the models one can match the Padé approximation of the dispersion relations of the Stokes wave theory. This can be quickly shown to lead to the following relations between the parameters of the different enhanced models [START_REF] Lee | A note on linear dispersion and shoaling properties in extended boussinesq equations[END_REF] :

β = α B /3; θ = 1 -2α B 3 -1 (34)
The comparison can thus be done by using α B as the only tuning parameter. As shown on figure 2, in fact, the Beji-Nadaoka and Madsen-Sørensen curves are superimposed, and correspond to the classical choice of α B = 0.2 [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF][START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF]. Setting α B = 0.17, yields the optimum value for model of Nwogu [START_REF] Nwogu | An alternative form of the boussinesq equations for near-shore wave propagation[END_REF]. 3)), A for Abbott (eq. ( 8)), BN and BNA for Beji-Nadaoka and Beji-Nadaoka-Abbott (eq.s [START_REF] Kazolea | Mathematical and computational modeling for the generation and propagation of waves in marine and coastal environments[END_REF] and (12) resp.), MS and MSP for Madsen-Sørensen and Madsen-Sørensen-Peregrine (eq.s ( 14) and (17) resp.), and N and NA for Nwogu and Nwogu-Abbott (eq.s ( 22) and (26) resp.).

Second-order solution. After (29) has been introduced into the system (5), and collecting terms of O(ε), one obtains :

   -2ωa 2 + 2 k du 2 + ka 1 u 1 = 0 -2ωu 2 + 2 ka 2 + k 2 u 2 1 - 8 3 σ 2 k2 ωu 2 = 0 . ( 35
)
Using the first equation of system (31), we can obtain:

m11 m12 m21 m22 a 2 u 2 = a 2 1 d F1 F2 (36)
with:

m11 = 2ω; m12 = -2 k d; m21 = -2 k; m22 = 2ω(1 + 4/3 φ); F1 = ω; F2 = ω2
2 dk Solving the linear system, using the dispersion relation to simplify ω, and passing to dimensional variables, leads to the expression :

a 2 = 3 4 a 2 1 d 1 φ 2 1 + 8 9 φ 2 (37)
with φ = kd. this value can be compared to the one found for the Stokes theory [START_REF] Madsen | Higher-order boussinesq-type equations for surface gravity waves: Derivation and analysis[END_REF] :

a Stokes 2 = 1 4 a 2 1 d φ coth(φ)(3 coth 2 (φ) -1) (38)
Remark 3.1 In adapting this procedure to the several BT models of section 2, a particular attention must be paid for the A, MS and NA systems. For these models, the relevant ansatz is q = q 1 cos(ξ) + εq 2 cos(2ξ). Subsequently,

ε q2 d + εη = ε q2 d + O(ε 2 ).
which can be used to show

ε q2 d + εη x = ε2 k q 2 1 d sin(ξ) cos(ξ) + O(ε 2 ).
The analysis of the first and second harmonics can then be performed assembling the proper order linear systems as shown for the Peregrine equations.

The results obtained for the different models are compared on figure 3 in terms of the ratio a 2 /a Stokes 2 . The first obvious remark is that the second harmonic, taking into account the nonlinear behavior of the PDEs, reveals six different families of models. This is expected since, as already remarked, for flat bathymetry the MS and BNA models and the MSP and BN coincide.

The most striking result, however, is that only two different trends are observed depending uniquely on whether the model is in amplitude-velocity or amplitude-flux form. In particular, all the models in amplitude-flux form underestimate, the error monotonically increasing as the reduced wavenumber increases. On the contrary, all the models in amplitude-velocity form give a non-monotone trend, with an initial overestimation of the amplitude, and a peak close to kd = 1 Inria for the enhanced models. Also, the error obtained with this class of models is smaller, the ration being closer to one.

Shoaling properties

To characterize the representation of wave shoaling in the linear case, one introduces the shoaling coefficient s which relates the rate of change in wave amplitude to the rate of change in depth :

A x A = -s d x d (39)
The procedure to compute the explicit expressions for the coefficient s is described in many papers and textbooks, see e.g. [START_REF] Beji | A formal derivation and numerical modelling of the improved boussinesq equations for varying depth[END_REF][START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF][START_REF] Kazolea | Mathematical and computational modeling for the generation and propagation of waves in marine and coastal environments[END_REF][START_REF] Lee | A note on linear dispersion and shoaling properties in extended boussinesq equations[END_REF][START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF]. To be able to actually compare the results of the analysis to the results obtained by numerically solving the models, there is a small catch, not often underlined in literature. To explain this, consider the main ansatz of the linear shoaling analysis, which is the assumption of a "moving shoaling" signal

η = A(x) sin(k(x)x -ω(x)t) .
This is injected into the model equations to derive ODEs for all the quantities involved, and allows to derive an analytical expression of the type s = (d, k). We refer to the references given for the details of this procedure. We underline, however, that for a given variation (usually linear) of the bathymetry, the local wavenumber k(x) must also be obtained by solving an ODE of the type

k x k = -γ d x d (40)
where d(x) is defined by the test set and the expression for γ can be found in literature ( [4,14,17] etc). The values of the shoaling parameter, can be thus represented in two ways. The one which is most classically reported is s 0 = s(d(x), k 0 ), representing its variation w.r.t. k 0 d, where k 0 is the wave number of the incoming wave. This leads to the classical result reported on the left on figure 4 for the models considered here1 . However, the variation shown in the picture is not the actual one obtained in practice. To obtain this variation, one must integrate (40) to obtain k(x), and use it to modify s(d(x), k(x)) when integrating (39) to compute the actual wave amplitudes. The shoaling coefficient can then still be plotted against k 0 d, by computing for a given x the corresponding values of k 0 d(x), and the local value of s(d(x), k(x)) = -A x (x)d(x)/(A(x)d x (x)). This results in the right picture on figure 4, similar to the one reported without any explaination in [START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF]. Note that only when including the variation of the wavenumber in the integration of 39 the correct amplitudes are obtained. 39). On the left is sketched s(h(x), k 0 ), thus computing the values of s using the initial wavenumber of the signal k 0 . On the right, instead, the variation of k along the domain is considered in the computation of the shoaling coefficient, thus: s(h(x), k(x)).

For completeness we discuss the comparison of the s coefficient for the models considered in this paper. Figure 4 shows classical results : the Peregrine (and Abbott) model shows a large error already for small wavenumbers with considerable low values of s ; the model of Nwogu gives a better approximation yet still underestimating the wave amplitude already for moderate wavenumbers ; the model of Madsen and Sørensen gives the best approximation among the models considered here ; the model of Beji and Nadaoka overestimates the wave amplitudes, with errors similar to those of the model of Nwogu.

Numerical experiments

Numerical discretization and implementation

The main aim of the numerical tests is to further study the properties of the models in the nonlinear range. In particular, we want to characterize the shoaling characteristics in regions close to breaking conditions. To achieve this, one should require the numerics to be well validated, and the results to be as much as possible independent on the numerical methods. For this reason, we have proceeded as follows. Two different numerical discretizations have been used for each test, which has been performed on several meshes, to guarantee as much as possible scheme and grid independence. In particular, we have used both a finite difference scheme, and a finite element one. The finite difference scheme, is based on the approach proposed by Wei and Kirby, which discretizes the shallow water terms of the equations using the fourth-order finite difference formula, while the dispersive terms to second order accuracy [START_REF] Kirby | A time-dependent numerical code for extended boussinesq equations[END_REF]. The finite element method used is instead a P1 continuous finite element discretization based on a standard Galerkin finite element solution of the elliptic sub-problems which define the auxiliary variables P (•), Q(•), Q(•), Q(•), etc, plus a standard Galerkin projection for the first order time dependent PDEs of the several models taken into account: (3), ( 8), ( 11), ( 12), ( 14), ( 17), ( 22), (26). This procedure, introduced in [START_REF] Walkley | A finite element method for the two-dimensional extended boussinesq equations[END_REF] to discretize the model of Nwogu, has been also recently used in [START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF] to solve the MS equations, and shown both analytically and numerically to have accuracy close to a fourth order finite difference scheme, and to that of the scheme of [START_REF] Kirby | A time-dependent numerical code for extended boussinesq equations[END_REF]. In all the tests the two different space discretizations have led to virtually indistinguishable results. In the following we will not show this comparison, but only report the main findings.

We will discuss two type of tests : initial verification benchmarks, in which the results are compared to analytical solutions for both flat and variable bathymetry ; physical experiments, in which we have investigated the behavior of the models in conditions close to wave breaking.

Initial validation

We will discuss two verification tests : the propagation of solitary waves over a flat bathymetry, and a linear shoaling test. These allow to verify our implementation of the models. Additionally, the linear shoaling tests allow to show that in the linear regime all the numerical models actually do recover the behavior predicted by the linear shoaling analysis.

Soliton propagation

In this section we consider the approximation of exact solitary wave solutions of the models presented. Quite often in literature approximate solitary waves are used. Our purpose here is to be able to quantitatively certify our implementation. For this reason we have chosen to use semi-analytical solutions. These solutions are obtained by generalizing the technique described in [START_REF] Taylor | From the paddle to the beach -a bousinesq shallow water numerical wave tank based on Madsen and Sørensen's equations[END_REF], and applied to the MS system of equations in [START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF], consisting in deriving a nonlinear ODE describing the shape of the solitary wave w.r.t the reduced parameter ξ = x -C t, with C a propagation speed which is also an output of the analysis, hence model dependent.

Details on the computation of these solutions, and the physical and mathematical conditions for their existence are given in [START_REF] Bellec | On the existence of solitary waves for boussinesq type equations and a new conservative model[END_REF]. The resulting solitary wave shapes are reported in figure 5 for a wave of amplitude A/d 0 = 0.2. Surprisingly, two families of solitaries seem to emerge, models in amplitude-velocity form giving steeper and more peaky profile. To verify our implementation of the models , we have performed a grid convergence analysis on the solitary wave of figure 5, characterized by and amplitude A/d 0 = 0.2, with d 0 = 1 [m]. The numerical results have been compared to the analytical (initial) profile after the wave has travelled for a length of 100 [m]. The meshes used for the tets contain 1000, 2000, 4000 and 8000 equally spaced cells. The results are summarized in terms of convergence of the L 2 error on the save amplitude in figure 6. The slopes obtained for the error show a convergence with third order of accuracy for all the models with the exception of the Nwogu-Abbott one (26). The accuracy obtained for the other models is similar to the one shown for the same schemes in [START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF] in the case of the Madsen and Sørensen equations, and confirms a proper implementation. The lack of convergence observed for the Nwogu-Abbott equations was at first quite surprising since the solution looks excellent at a simple visual inspection as shown on the left picture on figure 7. A closer inspection has revealed that actually a relatively large error at the foot of the solitary is produced by the semi-analytical technique of [START_REF] Bellec | On the existence of solitary waves for boussinesq type equations and a new conservative model[END_REF], especially when compared to the accurate and smooth profiles obtained for the other models. This error seems to be related to the nature of the nonlinear ODEs solved in [START_REF] Bellec | On the existence of solitary waves for boussinesq type equations and a new conservative model[END_REF] to compute the exact solitary wave for the Nwogu-Abbott model, and is currently still an unsolved issue.

Linear shoaling test

To further verify our implementation we have performed the shoaling test proposed by Madsen and Sørensen in [START_REF] Sø Rensen | A new form of the boussinesq equations with improved linear dispersion characteristics. part 2: a slowing varying bathymetry[END_REF]. A periodic signal of amplitude a = 0.05 m and period T = 4 s propagates over an initial constant water depth of h 0 = 13 m. The periodic signal has been generated using a source method discussed in [START_REF] Kirby | A time-dependent numerical code for extended boussinesq equations[END_REF][START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF]. The source of the periodic wave have been set at x 0 = 100 m of the domain [0, 850] m. The bottom is flat for the first 50 m from the position of the periodic wave generator, it has a constant up-slope of 1 : 50 from x = 150 m to x = 800 m and it then returns flat until the end of the domain. In this way the local values of the parameter ε = a 0 /h, which represents the effects of nonlinearity, vary between the values 0.0038 and 0.25, attesting we are working in the linear regime and the nonlinear effects can be actually considered negligible. Two 60 m wide absorbing sponge layers have been used at the two boundaries of the domain restricting the region of study of the shoaling to x = 790, but avoiding undesirable reflections due to the upper bottom shape discontinuity. The finest mesh used has a uniform grid size of ∆X = 0.333.

For this test case a theoretical envelope of the signal amplitude can be obtained from the linear shoaling analysis by integrating (39). This curve is used as a reference to compare and validate the numerical results. For this particular case, the theoretical envelopes of the models are sketched on the left picture on figure 8, while the right picture shows the typical wave profiles obtained2 . The comparison of the envelopes obtained by numerically solving the models with the theoretical ones is summarized on figure 9. We can observe that the theoretical results are very well-reproduced by all the models and that, as expected, the MS and MSP models produce almost superimposed results, as well as the BN and BNA models, and the N and NA ones. Very close to the end of the slope, nonlinear effects start begin non-negligible, and we can clearly see a difference appearing between the linear theory and the numerical solution of the nonlinear equations.

The curves corresponding to the P and A models are not present in figure 9 due to the fact that for such low amplitude waves we where unable to obtain a stable periodic signal by the internal wave generator strategy, at least not in the form described in [START_REF] Ricchiuto | Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries[END_REF] and implemented here.

Physical experiments

In this section finally we investigate the nonlinear properties of the models. We consider two experiments. The first is the nonlinear shoaling tests of [START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF]. The second involves the propagation of monochromatic waves over a submerged bar in two configurations : a non-breaking, and a breaking one. For the second configuration no breaking model is included. However, the results will allow to study and compare the shape of the wave obtained with the different models in a more complex flow.

Nonlinear shoaling test

This test allows to study the wave shoaling characteristics obtained with the models considered in conditions close to wave breaking. The test has been initially proposed by Grilli et al. in [START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF], and it consists of a solitary wave of amplitude A/d 0 = 0.2 m propagating on a water depth d 0 = 0.44 m, and shoaling onto a constant slope of 1 : 35. Note that in this test the local values of the nonlinearity parameter are roughly ε = a 0 /h ∈ [0.2; 2.2], which is clearly in the nonlinear range ε ≥ 1. A representation of this test is given below in figure 10. The numerical results obtained with the models studied in the paper are compared to the data of the laboratory experiments of [START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF]. The data available consist in the values of the free-surface elevations measured in 10 gauges positioned at stations from 0 to 9 (figure 10), with gauge 0 positioned just before the toe of the slope and gauge 9 located close to the wave breaking point. We refer to [START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF] for the precise description of the setup.

In our test, gauge 0 is used to calibrate the phase of the solutions obtained, the semi-anaytical solitary waves traveling at a celerity depending on the form of the model. The resulting shoaling wave profiles are compared to the experiments in figures 11 and 12, while the spatial evolution of the peak height is compared to the experiments in figure 13. The results show two distinct behaviors. All the models in amplitude-velocity form provide waves with considerably higher peaks and fronts with larger slope compared to the data, while all the models in amplitude-flux form give shorter waves with smaller slopes. This independently on the quality of the linearized system. This result is extremely important if one is to use these models in conjunction with a breaking detection plus dissipation mechanism. In particular, whether based on the slope of the front, on wave curvature, or on wave height, two distinct parametrizations of the detection criterion are necessary for these two family of models. In particular, looking at gauge 9, which is roughly where wave breaking should be detectable, we can see that while amplitude-velocity models have front slopes and wave heights larger than those of the data, velocity-flux models give lower amplitudes and, more importantly, much smaller front slopes. For criteria based on the shape of the profiles, such as those discussed e.g. in [START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq type equations[END_REF][START_REF] Tissier | A new approach to handle wave breaking in fully nonlinear boussinesq models[END_REF], this might mean that for a given parametrization of the constants involved in the breaking criterion, amplitude-velocity models might give an early breaking, while amplitude-flux models will most likely give a late breaking, or not break at all. We remark once more, that this result is independent on the quality of the model in terms of linear phase accuracy, the models of Peregrine and Abbott giving results considerably close to their enhanced versions.

Periodic wave propagation over a submerged bar

The last test is meant to verify the behavior of the models in a more complex situation. We consider the classical tests of monochromatic waves propagating on a submerged bar, for which extensive experimental data exist [START_REF] Beji | Numerical simulations of nonlinear-wave propagation over a bar[END_REF]. The geometry of the tests considered is sketched on figure 14. Monochromatic waves are generated and propagate on a depth of d 0 = 0.4 m before reaching the submerged bar. The periodic wave shoal over the 1:20 front slope, developing higher harmonics which are then released from the carrier frequency on the 1:10 slope of the lee side of the obstacle. We consider two configurations. Case (a) : wave amplitude A/d 0 = 0.025 and period T = 2.02s. Case (b) : wave amplitude A/d 0 = 0.0675 and period T = 2.525s. The first case is often used to validate dispersive wave propagation models. In the second case, considerably nonlinear conditions are obtained toward the end of the slope, where breaking is known to occur.

The periodic wave is generated by means of a periodic wave generator placed at x = 10 m of the domain [0, 35] m and two sponge layers with 3 m of thickness are used at the two boundaries of the domain in order to absorb any wave reaching the boundaries. In both the cases, the finest grid used has a grid size of ∆X = 0.04 m. The numerical results are compared with the data of Beji and Battjes [START_REF] Beji | Numerical simulations of nonlinear-wave propagation over a bar[END_REF]. In particular, in both cases the first gauge is used to phase calibrate the signal of the simulations with the experiments, while the other probes are then used to compare and validate the models. For further details on the tests, and for the exact location of the gauges the interested reader can consult [START_REF] Beji | Numerical simulations of nonlinear-wave propagation over a bar[END_REF][START_REF] Dingemans | Water Wave Propagation Over Uneven Bottoms: Linear wave propagation[END_REF][START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq type equations[END_REF]. The following remarks can be made. On the plateau, after shoaling has occurred, models in amplitude-velocity form give waves which have slightly higher peaks and steeper slopes, however the differences are small (roughly 5%d or below). The main remark that can be perhaps made is that the models of Peregrine and Abbott seem to have a larger phase advance in gauge 7 when compared to the other models. This is confirmed by the data in section 10, reported for completeness in figure 20, where the models of Paregrine and Abbott are compared with the respective Nwogu enhanced variant, giving results very similar to those of the other models. The picture shows that clearly for this case having a good approximation of the linear dispersion relations is more important than the form of the models. Case (b). In this case, wave breaking is expected to occur around gauge 2, at the end of the slope. As already remarked, no breaking criteria are included in our results so that the nonlinear behavior of the Boussinesq models is observed. For this case, the numerical results are phase calibrated w.r.t. experimental data measured at gauge 1, as shown in figure 21.

We then compare the computed wave heights with the data in gauge 2, at the end of the slope, and gauge 4, at the end of the plateau. The results are reported on figures 22, 23, 24 and 25. As before, in all the figures continuous lines are used for models in amplitude-flux form, and dashed lines for models in amplitude-velocity form.

From the figures we can see that already in gauge 2 there is a substantial difference in the height of the peak, with models in amplitude-velocity form giving peaks with taller wave heights, with differences of more than 20%d, and consequently, steeper wave fronts. This difference is even more visible in gauge 4, where the difference gous up to more that 35%d. models in amplitudeflux form clearly give gentler slopes, and shorter waves. As for the nonlinear shoaling test section §4.3.1, the results are practically independent on the linear characteristics of the systems, and only depend on the fact the the model is in amplitude-velocity, or amplitude-flux form. Once more, from the point of view of wave breaking detection, we see that the same criterion cannot be applied to models of the two families, or anyways not with the same parametrization.

Conclusions and perspectives

In this paper we have considered the impact of the form of weakly nonlinear Boussinesq models on their behavior in situations in which nonlinearity is not negligible. We have recalled how, within the same asymptotic truncation, a given linearized form, hence a give dispersion relation and shoaling coefficient, allow two derive two sets of PDEs : one, referred to as amplitude-velocity form, in which dispersive terms contain differential operators applied to the velocity ; the other, referred to as amplitude-flux form, in which these operators are applied to the flux. We have give four examples of these couples, including a new formulation of the model of Nwogu in terms of the flux.

The analytical and numerical study of these models has shown that : as soon as nonlinear effects start being relevant, the main factor influencing the behavior of the model is its amplitude-velocity or amplitude-flux form, enhanced models giving the same results as Peregrine's or Abbott's equations. This fact has been demonstrated analytically by the study of the propagation of higher harmonics, following [START_REF] Madsen | Higher-order boussinesq-type equations for surface gravity waves: Derivation and analysis[END_REF], and numerically on tests involving shoaling in genuinely nonlinear regimes. So, while in the linear case we have as many families of models as the number of linear dispersion relations times the linear shoaling coefficient, in the nonlinear case, only two type of behavior are observed.

This work has important consequences on the way in which wave breaking conditions are applied to these models, as well as on the way in which wave breaking dissipation is included. Clearly, breaking criteria should not be the same for the two family of models, and perhaps even the amount of dissipation in breaking regions necessary in the two cases should be different. Future development of this work will definitely involve the study of the coupling of the models considered in this paper with physical breaking criteria as those used in e.g. [START_REF] Bacigaluppi | A 1d stabilized finite element model for non-hydrostatic wave breaking and run-up[END_REF][START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq type equations[END_REF][START_REF] Tissier | A new approach to handle wave breaking in fully nonlinear boussinesq models[END_REF][START_REF] Tonelli | Simulation of wave breaking over complex bathymetries by a boussinesq model[END_REF] and references therein. Other future developments will involve the study of deep water models, as well as the one-to-one comparison of different forms of weakly-nonlinear with fully nonlinear models coupled with breaking detection and dissipation mechanisms.

•

  Models of Peregrine (3) (amplitude-velocity) and of Abbott (8) (amplitude-flux) ; • Model of Beji and Nadaoka (11) (amplitude-velocity) and BNA model (12) (amplitudeflux) ; • Model of Madsen-Sørensen (14) (amplitude-flux) and MSP model (17) (amplitude-velocity) ; • Model of Nwogu (19) (amplitude-velocity) and NA model (26) (amplitude-flux) .

Figure 2 :

 2 Figure2: Phase velocity ratio for all the models : P stands for Peregrine (eq. (3)), A for Abbott (eq. (8)), BN and BNA for Beji-Nadaoka and Beji-Nadaoka-Abbott (eq.s[START_REF] Kazolea | Mathematical and computational modeling for the generation and propagation of waves in marine and coastal environments[END_REF] and (12) resp.), MS and MSP for Madsen-Sørensen and Madsen-Sørensen-Peregrine (eq.s (14) and (17) resp.), and N and NA for Nwogu and Nwogu-Abbott (eq.s (22) and (26) resp.).

Figure 3 :

 3 Figure 3: Ratio of the second harmonic a 2 /a Stokes 2 for the models considered. Continuous line : amplitude-velocity models. Dashed lines : amplitude-velocity models. Refer to figure 2 for the legend.

Figure 4 :

 4 Figure4: Linear shoaling: representation of the linear shoaling coefficient s of the several BT models defined by (39). On the left is sketched s(h(x), k 0 ), thus computing the values of s using the initial wavenumber of the signal k 0 . On the right, instead, the variation of k along the domain is considered in the computation of the shoaling coefficient, thus: s(h(x), k(x)).

Figure 5 :

 5 Figure 5: Semi-analytical exact solitary waves for the Boussines models of the paper.

Figure 6 :

 6 Figure 6: Grid convergence results for the Galerkin finite element (left) and Wei and Kirby finite difference (right) scheme.

Inria

Figure 7 :

 7 Figure 7: Solitary wave propagation. Left : comparison between numerical and analytical solution for the NA model. Right : error at the foot of the semi-analytical wave for the NA model

Figure 8 :

 8 Figure 8: Linear shoaling. Left : theoretical envelope of the maximum wave amplitude. Right: line plots of the surface elevation computed using the MS model.

Figure 9 :

 9 Figure 9: Linear shoaling: maximum elevation

Figure 10 :

 10 Figure 10: Shoaling of a solitary wave; computational configuration and gauges position.

Figure 11 :

 11 Figure 11: Nonlinear shoaling. Comparison between computed wave heights at gauges 1, 3, 5, 7 and 9 and data of[START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF] ; models in amplitude-flux form.

Figure 12 :

 12 Figure12: Nonlinear shoaling. Comparison between computed wave heights at gauges 1, 3, 5, 7 and 9 and data of[START_REF] Grilli | Shoaling of solitary waves on plane beaches[END_REF] ; models in amplitude-velocity form.

Inria

Figure 13 :

 13 Figure 13: Nonlinear shoaling. Comparison between computed wave peak evolution and data from [9].

Figure 14 :

 14 Figure 14: Periodic wave propagation over a submerged bar: sketch of the computational configuration and of the gauges position; case (a) on the left, case (b) on the right.

Figure 15 :

 15 Figure 15: Propagation over a submerged bar, case (a); data in gauge 4 are used for the signal syncronization.

Figure 16 :

 16 Figure 16: Propagation over a submerged bar, case (a); data in gauge 5 (left) and gauge 10 (right): experimental data [3] (• • •), A model (---), P model (------).

Figure 17 :

 17 Figure 17: Propagation over a submerged bar, case (a); data in gauge 5 (left) and gauge 10 (right): experimental data [3] (• • •), BNA model (---), BN model (------).

Figure 18 :

 18 Figure 18: Propagation over a submerged bar, case (a); data in gauge 5 (left) and gauge 10 (right): experimental data [3] (• • •), MS model (---), MSP model (------).

Figure 19 :

 19 Figure 19: Propagation over a submerged bar, case (a); data in gauge 5 (left) and gauge 10 (right): experimental data [3] (• • •), NA model (---), N model (------).

Figure 20 :

 20 Figure 20: Propagation over a submerged bar, case (a); data in gauge 10 for the models of Peregrine and Abbott (left), and for the enahnced variants of Nwogu ad Nwogu-Abbott (right). Experimental data [3] (• • •). Left figure : A model (---), P model (------). Right figure : NA model (---), N model (------).

Figure 21 :

 21 Figure 21: Propagation over a submerged bar, case (b); data in gauge 1 are used for the signal syncronization.

Figure 22 :Figure 25 :

 2225 Figure 22: Propagation over a submerged bar, case (b); data in gauge 2 (left) and gauge 4 (right): experimental data [3] (• • •), A model (---), P model (------).

RR n°8573

Inria

The optimal values of the free parameters give in section §3.1 are used for the plot Inria

Result of the Madsen and Sørensen model. RR n°8573

351, Cours de la Libération Bâtiment A 29

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399


Karen Fort
email: karen.fort@loria.fr

Gilles Adda
email: gilles.adda@limsi.fr

Benoît Sagot
email: benoit.sagot@inria.fr

Joseph Mariani
email: joseph.mariani@limsi.fr

Alain Couillault
email: alain.couillault@univ-lr.fr

Karën Fort

Crowdsourcing for Language Resource Development: Criticisms About Amazon Mechanical Turk Overpowering Use

Keywords: Amazon Mechanical Turk, language resources, ethics

come   L'archive ouverte pluridisciplinaire

Introduction

Developing annotated corpora, as well as other language resources, involves such high costs that many researchers are looking for alternative, cost-reducing solutions. Among others, crowdsourcing, microworking 6 systems which enable elementary tasks to be performed by a huge number of on-line people, are possible alternatives. Nowadays, Amazon Mechanical Turk (mturk) is the most popular of these systems, especially in the Speech & Language community. Since its introduction in 2005, there has been a steady growth of mturk use in building or validating language resources [START_REF] Fort | Amazon mechanical turk: Gold mine or coal mine?[END_REF].

Costs are drastically reduced due to available sparse time of human language experts on-line. But mturk raises, among others, ethical and quality issues which have been minimized until now, and we will investigate them in this paper. However, because we are aware that the development costs of corpora often stand in the way of language research and technologies, especially for less-resourced languages, we are also sensible of some visible advantages of crowdsourcing. Developing a crowdsourcing system which retains some of the main qualities of mturk (rapidity, diversity, access to non-expert judgment) while avoiding the ethical and labor laws issues is (theoretically) possible, but this solution will require some delay (in the best case scenario) and the help of our scientific associations (ISCA, ACL, ELRA) and of the national and international funding agencies. Therefore, we will propose existing alternatives aiming at producing high quality resources at a reduced cost, while deliberately keeping ethics above cost savings. In parallel, we created the French Ethics and Big Data Charter that will allow funding agencies to select the projects they want to finance according to ethical criteria.

2 mturk: legends and truth 2.1 mturk, a hobby for the Turkers?

In order to evaluate the ethics of mturk, we need to qualify the activity of Turkers while they are participating in mturk. Is it a voluntary work, as the one in Wikipedia? Looking at the mturk site or at Turker blogs, where the monetary retribution is a major issue, the answer is clearly no.

Studies in social sciences [START_REF] Ross | Who are the crowdworkers?: shifting demographics in mechanical turk[END_REF][START_REF] Ipeirotis | Demographics of mechanical turk[END_REF], using surveys submitted within mturk, give some insight 7 into Turkers' socio-economic facts (country, age. . . ) or the way they use mturk (number of tasks per week, total income in mturk . . . ), and how they qualify their activity. 91% of the Turkers mentioned their desire to make money [START_REF] Silberman | Sellers' problems in human computation markets[END_REF], even if the observed wage is very low; 8 when 60% of the Turkers think that mturk is a fairly profitable way of spending free time and getting some cash, only 30% mentioned their interest for the tasks, and 20% (5% of the Indian Turkers) say that they are using mturk to kill time. Finally, 20% (30% of the Indian Turkers) declare that they are using mturk to make basic ends meet, and about the same proportion that mturk is their primary source of income.

Looking at the tasks which are performed within mturk is another way to qualify the Turkers' activity. Innovative kinds of tasks can be found which can be seen as creative hobby activities. However, many tasks correspond to activities which used to be performed by salaried employees, and therefore are working activities; for these tasks, mturk could be assimilated to off-shoring on the Web to decrease production costs. For years, speech corpora transcription (and translation) tasks were being performed by employees of agencies like ldc or elda: these are jobs. The 20% of the most active Turkers who spend more than 15 hours per week in mturk [START_REF] Adda | Language resources and amazon mechanical turk: legal, ethical and other issues[END_REF], and produce 80% of the activity, can be called laborers when performing these tasks.

It is difficult to be conclusive about the nature of the Turkers' activity. Many different types of tasks are proposed within mturk and the Turkers' motivations are heterogeneous. Nevertheless, those 20% of the Turkers for whom mturk is a primary income, and those Turkers who perform tasks which are actually performed by employees, produce an activity in mturk corresponding to a real labor.

Qualifying the mturk activity as labor raises issues about the setup of mturk. The very low wages (below $2 an hour [START_REF] Ross | Who are the turkers? worker demographics in amazon mechanical turk[END_REF][START_REF] Ipeirotis | Demographics of mechanical turk[END_REF][START_REF] Chilton | Task search in a human computation market[END_REF]) are a first point. A further point concerns Amazon's choice of hiding any explicit relationship between Turkers and Requesters, even the basic workplace right of unionization is denied and Turkers have no recourse to any channels for redress against employers' wrongdoing, including the fact that they have no official guarantee of payment for properly performed work. Some regulation between Requesters and Turkers exists through Turkers' blogs or forums 9 , or the use of Turkopticon10 , a tool designed to help Turkers reporting bad Requesters. However, all these solutions are unofficial and nothing explicitly protects the Turkers, especially the new ones who are mostly unaware of these tools.

mturk drastically reduces costs?

Most articles dealing with mturk and resource production indicate low costs as the primary motivation. Given the observed salaries (for instance $0.005 to transcribe a 5-second speech segment [START_REF] Novotney | Cheap, fast and good enough: automatic speech recognition with non-expert transcription[END_REF]), the cost may indeed be very low. However, the overall cost is not to be limited to the mere salary: the time needed to develop the interface, and to tackle the spammer problem is not negligible [START_REF] Callison-Burch | Creating speech and language data with amazon's mechanical turk[END_REF]; validation [START_REF] Kaisser | Creating a research collection of question answer sentence pairs with amazon's mechanical turk[END_REF] and correction costs [START_REF] Xu | Paragraph acquisition and selection for list question using amazon's mechanical turk[END_REF] to ensure minimal quality are also to be considered. Furthermore, some tasks may become more expensive than expected. This may occur for instance, if the required Turkers' competence is hard to find: to transcribe Korean [START_REF] Novotney | Cheap, fast and good enough: automatic speech recognition with non-expert transcription[END_REF], wages were increased from $5 to $35 per hour.

mturk allows for building resources of equivalent quality?

Many technical papers have reported that at least for translation and transcription, the quality is sufficient to train and evaluate statistical translation or transcription systems [START_REF] Callison-Burch | Creating speech and language data with amazon's mechanical turk[END_REF][START_REF] Marge | Using the amazon mechanical turk for transcription of spoken language[END_REF]. However, some of these papers bring to light quality problems. 11Limitations due to the lack of expertise Turkers being non-experts, the requester has to decompose complex tasks into simpler tasks (HITs, Human Intelligence Tasks), to help performing them. By doing so, s/he can be led to make choices that can bias the results. An example of this type of bias is analyzed in [START_REF] Cook | Automatically identifying changes in the semantic orientation of words[END_REF], where the authors acknowledge the fact that proposing only one sentence per lexical evolution type (amelioration and pejoration) influences the results.

Even more problematic is the fact that the quality produced with mturk on complex tasks is often not satisfactory. This is for example the case in [START_REF] Bhardwaj | Anveshan: A tool for analysis of multiple annotators' labeling behavior[END_REF], in which the authors demonstrate that, for their task of word-sense disambiguation, a small number of well-trained annotators produces much better results than a larger group (the number being supposed to counterbalance non-expertise) of Turkers. From this point of view, their results contradict those presented in [START_REF] Snow | Cheap and fast -but is it good? evaluating non-expert annotations for natural language tasks[END_REF] on a similar, though much simpler, task. The same difficulty arises in [START_REF] Gillick | Non-expert evaluation of summarization systems is risky[END_REF], in which it is demonstrated that non expert evaluation of summarization systems is "risky", as the Turkers are not able to obtain results comparable to that of experts. More generally, this quality issue can be found in numerous articles in which the authors had to validate Turkers' results using specialists (PhD students in [START_REF] Kaisser | Creating a research collection of question answer sentence pairs with amazon's mechanical turk[END_REF]) or use a rather complex post-processing [START_REF] Xu | Paragraph acquisition and selection for list question using amazon's mechanical turk[END_REF]. Finally, the quality of the work from non experts varies considerably [START_REF] Tratz | A taxonomy, dataset, and classifier for automatic noun compound interpretation[END_REF].

Moreover, there is currently a "snowball" effect going on, that leads to overestimate the resources quality mentioned in articles: some researchers praise mturk [START_REF] Xu | Paragraph acquisition and selection for list question using amazon's mechanical turk[END_REF], citing research that did use the system, but would not have given usable results without a more or less heavy post-processing [START_REF] Kaisser | Creating a research collection of question answer sentence pairs with amazon's mechanical turk[END_REF]. A simplistic conclusion could be that mturk should only be used for simple tasks, however, besides the fact that mturk itself induces important limitations (see next section), it is interesting to notice that, for some simple tasks, Natural Language Processing tools already provide better results than the Turkers [START_REF] Wais | Towards building a high-qualityworkforce with mechanical turk[END_REF].

Limitations due to mturk itself In [START_REF] Tratz | A taxonomy, dataset, and classifier for automatic noun compound interpretation[END_REF], the authors note that the limits of the user interface constitute the "first and most important drawback of mturk". The authors also regret that it is impossible to be 100% sure that the Turkers participating in the task are real native English speakers. If pre-tests can be designed to address, at least partly, this issue, they represent an added cost and it will still be very easy to cheat [START_REF] Callison-Burch | Creating speech and language data with amazon's mechanical turk[END_REF]. Of course, one can always organize various protections [START_REF] Callison-Burch | Creating speech and language data with amazon's mechanical turk[END_REF], but here again, this requires time and therefore represents an additional cost that only few requesters are ready to pay for. 12 For example, in [START_REF] Xu | Paragraph acquisition and selection for list question using amazon's mechanical turk[END_REF], the authors identified spammers but did not succeed in eliminating them.

Finally, the impact of task payment should not be neglected, as it induces as logical behavior to place the number of performed tasks above quality, regardless of payment. In [START_REF] Kochhar | The anatomy of a large-scale human computation engine[END_REF] the authors thus reached the conclusion that an hourly payment was better (with some verification and time justification procedures).

3 Existing or suggested alternatives mturk is not the only way to achieve fast development of high quality resources at a low cost. First, and despite the lack of systematic studies, existing automatic tools seem to perform as well as (non-expert) Turkers, if not better, on certain tasks [START_REF] Wais | Towards building a high-qualityworkforce with mechanical turk[END_REF]. Second, the cost of tasks like manual annotation can be drastically reduced using the appropriate techniques. Third, exploiting as much as possible existing resources can be an inexpensive alternative to mturk. Finally, mturk is not the only crowdsourcing and microworking platform.

Unsupervised and semi-supervised techniques for low-cost language resource development

Unsupervised machine learning techniques have been studied in the Speech & Language community for quite a long time, for numerous and sometimes complex tasks, including tokenization, POS tagging [START_REF] Goldwater | A fully bayesian approach to unsupervised part-ofspeech tagging[END_REF], parsing [START_REF] Hänig | Improvements in unsupervised co-occurrence based parsing[END_REF] or document classification. Although such techniques produce results that are below stateof-the-art supervised or symbolic techniques, which both require resources that are costly to develop, it is unclear whether they produce results that are below what can be expected from mturk, especially for complex tasks such as parsing. Moreover, unsupervised techniques can be improved at a reasonable cost by optimizing the construction and use of a limited amount of additional information (annotations, external resources). This constitutes the semi-supervised learning paradigm [START_REF] Abney | Semisupervised Learning for Computational Linguistics[END_REF]. Such approaches for developing language resources rely on two (complementary) principles:

• Training models on a limited amount of annotated data and using the result for producing more annotation. For example, using one model, one can select within the automatically annotated data those that have a high confidence level, and consider that as additional training data (self-training, [START_REF] Yarowsky | Unsupervised word sense disambiguation rivaling supervised methods[END_REF]). Using two different models allows to rely on the high-confidence annotations of one model for augmenting the training corpus for the other, thus decreasing systematic biases (co-training, [START_REF] Blum | Combining labeled and unlabeled data with co-training[END_REF]). If one accepts to produce a limited amount of manual annotations not only in advance but also while developing the tools, one can request the manual annotation of carefully chosen data, i.e., data for which knowing the expected output of the system improves as much as possible the accuracy of the system (active learning [START_REF] Cohn | Active learning with statistical models[END_REF]). • Using data containing annotations that are less informative, complete and/or disambiguated than the target annotations, like a morphological lexicon (i.e., an ambiguous POS-annotation) for POS tagging [START_REF] Smith | Contrastive estimation: Training log-linear models on unlabeled data[END_REF], a morphological description for morphological lexicon induction [START_REF] Sagot | Automatic acquisition of a Slovak lexicon from a raw corpus[END_REF] or a partly bracketed corpus for full parsers [START_REF] Watson | Semi-supervised training of a statistical parser from unlabeled partially-bracketed data[END_REF].

Optimizing the cost of manual annotation: pre-annotation and dedicated interfaces

When using approaches that rely on expert annotation, this annotation can be sped up and sometimes even improved by automatic annotation tools used as pre-annotators. For instance, [START_REF] Fort | Influence of Pre-annotation on POS-tagged Corpus Development[END_REF] have shown that for pos tagging, a lowquality and non-costly pre-annotation tool can drastically improve manual annotation speed; 50 manually pos-annotated sentences are enough for training a pre-annotation tool that reduces manual work as much as a state-of-the-art pos tagger, allowing to developing a 10,000-sentence standard-size corpus in ∼100 hours of expert work. On the other hand, on such a task, one could question the ability of anonymous Turkers to correctly follow detailed and complex annotation guidelines. Obviously, the above-mentioned remarks by [START_REF] Tratz | A taxonomy, dataset, and classifier for automatic noun compound interpretation[END_REF] about the limitations of mturk interfaces apply more generally. Past projects aiming at developing syntactically and semantically annotated corpora have shown that both the speed and quality of the annotation is strongly influenced by the annotation interface itself [START_REF] Erk | The salsa annotation tool[END_REF]. This provides another source of improvements for annotation efficiency and quality. Put together, it might well be the case that even costly expert work can be used in optimized ways that lead to high-quality resources at a reasonable cost, even compared with that of mturk.

Reusing existing resources

Even less costly is the use of existing data for creating new language resources. An example is the named-entity recognition (ner) task. mturk has been used for developing ner tools, in particular for specific domains such as medical corpora [START_REF] Yetisgen-Yildiz | Preliminary experience with amazon's mechanical turk for annotating medical named entities[END_REF], twitter [START_REF] Finin | Annotating named entities in twitter data with crowdsourcing[END_REF] or e-mails [START_REF] Lawson | Annotating large email datasets for named entity recognition with mechanical turk[END_REF]. However, converting Wikipedia into a large-scale named-entity-annotated resource leads to building high-quality ner tools [START_REF] Nothman | Transforming Wikipedia into Named Entity Training Data[END_REF], including when evaluated on other types of corpora [START_REF] Balasuriya | Named entity recognition in wikipedia[END_REF]. Apart from Wikipedia (and the related dbpedia), other wiki projects (e.g., wiktionaries) and freelyavailable resources (lexicons, corpora) are valuable sources of information.

Collaborative or crowdsourced development beyond mturk

All these alternatives require a fair amount of expert work. Other approaches do exist that reduce this requirement to a low level, and in particular collaborative and game-based techniques, as well as crowdsourcing platforms other than mturk, which try to avoid at least in part its pitfalls.

Collaborative approaches for language resource development rely on the strategy set up by the Wikipedia and other Wikimedia projects, as well as other wikis such as semantic wikis (Freebase, OntoWiki. . . ). Anyone can contribute linguistic information (annotation, lexical data. . . ), but usually contributors are motivated because they are to some extent experts themselves. The quality control is usually done mutually by contributors themselves, sometimes by means of on-line discussions, often leading to high quality results. One of the first collaborative platforms for language resource development was the semantic annotation tool Serengeti [START_REF] Stürenberg | Web-based annotation of anaphoric relations and lexical chains[END_REF], currently used within the AnaWiki project. 13However, such approaches remain more suitable for developing medium-scale high-quality resources. For the fast development of large-scale resources, another strategy is to attract a large number of non-experts thanks to online games, that fall in the family of so-called games with a purpose (gwap). This idea was initiated by the esp on-line game [START_REF] Von Ahn | Games with a purpose[END_REF] for image tagging. Its success led researchers to develop such games for various tasks, including language-related ones. A well-known example is PhraseDetective [START_REF] Chamberlain | Phrase Detectives: a Web-based Collaborative Annotation Game[END_REF] for annotating anaphoric links, a reputedly complex task, which lead the authors to include a training step before allowing players to actually provide new annotations. However, the boundary between gwaps and crowdsourcing is not clear-cut. It is not the case that mturk remunerates a work whereas other approaches are purely "for fun". Indeed, even contributing to Wikipedia is a job, though a voluntary unpaid job. gwap and mturk cannot be distinguished either by the fact that mturk gives a remuneration, as some gwaps do propose non-monetary rewards (e.g. Amazon vouchers for PhraseDetective). Finally, collaborative and gwap-based techniques are not the only "ethical alternatives", since ethical crowdsourcing platforms do exist.

For gathering language data, in particular for less-resourced languages, crowdsourcing platforms apart from mturk seem to be particularly appropriate, as shown for example by speech corpus acquisition experiments using dedicated applications run on mobile phones [START_REF] Hughes | Building transcribed speech corpora quickly and cheaply for many languages[END_REF]. An example of an ethical crowdsourcing platform is Samasource, an ngo that allows really poor people to be properly trained and paid for specific tasks (e.g. translating sms in Creole after the earthquake in Haiti for helping victims and international rescuers to communicate). 14

Towards traceability: the Ethics and Big Data Charter

To adopt an ethical behavior in developing, funding, using or promoting language resources is first and above all a matter of choice: for the provider, deciding which approach to adopt -crowdsourcing or not -, or which platform to request on, or the level of remuneration of the workers; for the funding agency, choosing which project to fund; for users, choosing which resource to use or acquire. These choices have to be learned ones. We designed the Ethics and Big Data Charter [START_REF] Couillault | Charte Éthique et Big Data : parce que mon corpus le vaut bien ! In: Linguistique, Langues et Parole : Statuts, Usages et Mésusages[END_REF] in collaboration with representatives of interest groups, private companies and academic organizations, including the French CNRS15 , ATALA 16 , AFCP 17 and APROGED 18 . The purpose of this charter is to provide resources developers with a framework to document their resources and ensure their traceability and transparency.

Why Big Data?

In the process of writing the Charter, it soon appeared that the issues raised for language resources apply to a larger range of data sets, which can be described as Big Data. Indeed, Big Data are characterized not only by their volume, but also by the complexity of the data, which is in no doubt the case even for small sets of language resources. Reversely, the reflexions conducted for language resources can be generalized to and benefit to Big Data sets.

Contents of the Charter

The Ethics and Big Data Charter is provided as a form to be filled in by the dataset provider. It is split into three major sections: traceability, intellectual property and specific legislation, preceded by a short identification section containing the names of the resource, the contact and responsible persons and a short description of the data set.

Traceability Traceability is key to our purpose of putting forward ethical issues. The traceability part of the charter allows to precise the relationship between the resource provider and the workers involved in developing the resource, including legal bounding, workers skills, selection criteria. Specific focus is put on personal data, i.e. data, like voice or video recording, which can provide a means to identify a person directly or indirectly. The Charter requires to precise if and how the data is de-identified, and if and how the individuals were informed of the purpose of the data collection. Quality assurance is another major aspect of traceability addressed by the charter, as it requires to document the quality assurance strategy, so that the user of the data set is fully informed on the level of quality s/he can expect: what QA procedure the data were passed through? what portion of the data has been evaluated? What are the actual metrics used and their values?

License and copyright Thanks to a great deal of effort accomplished in the definition of -mainly open source -license schemes, it has become common practice to attach a license to a data set. The License and Copyright section of the Charter goes beyond this and puts the focus on questions which may be disregarded, like ensuring that the legal or moral copyrights of the persons who worked on compiling, enriching or transforming the data are respected. As an example, we saw to it that all the writers of the Ethics and Big Data Charter are mentioned in the license citation. Also, the Charter reminds data collectors and distributors that they should check whether they comply with any third party data license they may use. Specific legal requirements A third section of the Ethics and Big Data Charter deals with legal requirements that may arise from certain properties of the data set. For example, a country may have issued specifics laws regarding the storing, use and/or dissemination of personal data. The Charter serves as a reminder for checking if such requirements exist.

Availability

The Ethics and Big Data Charter is available on-line. 19 The website is currently in French, and an English translation of the Ethics and Big Data Charter is available. 20Examples of charters are also provided, including one for a corpus of e-mail messages, and one for a medical dataset. Both corpus raise privacy issues that the Ethics and Big Data Charter allows to deal with.

Conclusion and perspectives

We have tried to demonstrate here that mturk is no panacea and that other solutions exist allowing to reduce the development costs of high-quality language resources, while respecting those working on the resources and their skills.

We would like, as a conclusion, to go beyond the present facts and insist on the longer term consequences of this trend. Under the pressure of this type of low-cost systems, funding agencies could become more reluctant to finance language resources development projects at "normal" costs. The mturk cost would then become a de facto standard and we would have no other choice as for the development method.

We saw, in section 3.3, that a microworking system can generate paid tasks while preserving ethics. This can even represent a chance for people who cannot participate in the usual labor market, due to their remoteness, their handicap, etc., but it requires a strict legal framework to ensure that the system does not violate their rights as workers. This is why we propose that the concerned associations, like the ACL 21 for natural language processing, the ISCA 22 for speech and the ELRA 23 for Language Resources take care of this problem and push to the development and dissemination of the needed tools to better qualify the quality and ethics of the language resources, such as the Ethics and Big Data Charter. For that purpose, we already engaged with funding agencies at the French level, some of which have adopted the charter as part of their projects selection process. This effort would need to be extended to international organizations.

Excellence "Multilingual Europe Technology Alliance (meta-net)", co-funded by the 7th Framework Programme of the European Commission through the contract t4me (grant agreement no.: 249119). We would like to thank the authors 24 of the Ethics and Big Data Charter for their dedicated time and effort.

Microworking refers to the fact that tasks are cut into small pieces and their execution is paid for. Crowdsourcing refers to the fact that the job is outsourced via the web and done by many people (paid or not).

For instance mechanicalturk.typepad.com or turkers.proboards.com

turkopticon.differenceengines.com

Some of the problems reported, such as the interface problems, are not specific to mturk, but are generic to many crowdsourcing systems.

Interestingly, it seems that mturk recently decided to no longer accept the non-US Turkers, for quality and fraud reasons: http://turkrequesters.blogspot.fr/ 2013/01/the-reasons-why-amazon-mechanical-turk.html.

http://www.anawiki.org

http://www.samasource.org/haiti/

Centre National de la Recherche Scientifique/National agency for scientific research

Association pour le Traitement Automatique des Langues/Natural Language Processing Association http://www.atala.org

[START_REF] Gillick | Non-expert evaluation of summarization systems is risky[END_REF] Association Française de Communication Parlée/French spoken communication association, http://www.afcp-

parole.org[START_REF] Tratz | A taxonomy, dataset, and classifier for automatic noun compound interpretation[END_REF] Association de la Maîtrise et de la Valorisation des contenus/Association for mastering and empowering content, http://www.aproged.org

http://wiki.ethique-big-data.org

http://wiki.ethique-big-data.org/chartes/charteethiqueenV2.pdf

http://www.aclweb.org/

http://www.isca-speech.org/

http://www.elra.info/

http://wiki.ethique-big-data.org/index.php?title=Ethique_Big_Data: Accueil#Les_auteurs

Acknowledgments. This work was partly realized as part of the Quaero Programme, funded by oseo, French State agency for innovation, as well as part of the French anr project edylex (anr-09-cord-008) and of the Network of