Maricarmen Guerra
email: mnguerra@uc.cl

Rodrigo Cienfuegos

Cristian Escauriaza

Fabien Marche
email: fabien.marche@math.univ-montp2.fr

José Daniel

Galaz Mora

José Galaz
email: jdgalaz@uc.cl

Modeling rapid flood propagation over natural terrains using a well-balanced scheme

Keywords: Shallow water equations, shock capturing methods, well-balanced schemes, boundaryfitted curvilinear coordinates

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

In recent years, a number of catastrophic events have involved rapid flooding over complex topography, such as tsunamis and river floods. In Chile, for example, several Glacial-lake outburst floods (GLOF) have occurred in the Colonia river, a tributary of the Baker river (47 • 10' S; 73 • 20' W), as a consequence of the Cachet-II lake outburst. In fact, two major events in 2008 generated an increase of the Baker river free surface elevation of above 4.5 m and a peak discharge over 3,000 m 3 /s (DGA Satellite monitoring station at Baker River, DGA-MOP, Chile), flooding large parts of the Colonia and Baker river valleys and putting at risk the town of Caleta Tortel, located at the mouth of the Baker river [START_REF] Dussaillant | Repeated glacial-lake outburst floods in Patagonia: an increasing hazard?[END_REF]. This event has repeated two times a year since 2008, which is likely linked to significant increments of temperature that have been registered on the entire watershed in the last years.

Recently, on February 27 th 2010, an 8.8 Mw earthquake occurred off the coast of south-central Chile [START_REF] Fritz | Field survey of the 27 february 2010 Chile tsunami[END_REF][START_REF] Lay | Teleseismic inversion for rupture process of the 27 february 2010 chile (mw 8.8) earthquake[END_REF], generating a destructive tsunami that affected a significant portion of the coast, the Juan Fernández Archipiélago, and Easter Island, taking the lives of 124 people. As a consequence of these events, local authorities and the central government are currently developing new hazards and risk plans in different coastal communities along Chile, which consider the investigation of the inundation extent and maximum water depth estimations, peak discharge and velocities, among other hydrodynamic variables for riverine floods and tsunamis.

In order to study the large and costly consequences of these major rapid flooding events, it is necessary to develop instruments that can be used to predict accurately and efficiently the flow velocities and water depths, and assess their associated hazards and risks. A good estimation of flow features such as run-up, affected areas, and arrival time of the peak flood will lead to define better the mitigation plans, early warning systems, and improve the preparedness of people when facing such catastrophic situations, incorporating the hydrodynamic forces into the design of coastal and riverine infrastructure (Federal Emergency Management Agency 2011;American Society of Civil Engineers 2006;[START_REF] Yeh | Maximum fluid forces in the tsunami runup zone[END_REF].

The Nonlinear Shallow Water Equations (NSWE) are usually employed in these cases to describe the flow dynamics, and model fairly long-waves in a homogeneous and incompressible fluid.

They are obtained by vertically averaging the three-dimensional Navier-Stokes equations assuming a hydrostatic pressure distribution, resulting in a set of horizontal two-dimensional hyperbolic conservation laws that describe the evolution of the water depth and depth-averaged velocities [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF][START_REF] Stoker | Water Waves, The Mathematical Theory with Applications[END_REF]).

In the last decades, a significant number of numerical models have been developed to simulate these complex flows, employing finite-difference methods (e.g. [START_REF] Molls | Depth-averaged open-channel flow model[END_REF][START_REF] Molls | Depth-averaged simulation of supercritical flow in channel with wavy sidewall[END_REF], finite-element methods (e.g. [START_REF] Berger | Finite-element model for high-velocity channels[END_REF][START_REF] Tucciarelli | Finite-element modeling of floodplain flow[END_REF], or finite-volume methods (e.g. [START_REF] Valiani | Case study: Malpasset dam-break simulation using a two-dimensional finite volume method[END_REF][START_REF] Zhou | Numerical prediction of dam-break flows in general geometries with complex topography[END_REF][START_REF] Loose | Finite volume modeling of variable density shallow-water flow equations for a well-mixed estuary: application to the rio Maipo estuary in central Chile[END_REF]). In the framework of finite volume methods, Godunov-type formulations have become very useful to solve the NSWE, since they can reproduce complex discontinuities such as shock-waves or wetdry interfaces by solving a Riemann problem at each cell interface of the discretized domain [START_REF] Toro | Shock-Capturing Methods for Free-Surface Shallow Flows[END_REF][START_REF] Leveque | Finite Volumen Methods for Hyperbolic Problems[END_REF].

Many environmental flows such as bore propagation [START_REF] Hibberd | Surf and run-up on a beach: a uniform bore[END_REF], tsunami 3 Guerra et al., February 11, 2014 inundations [START_REF] Yeh | Tsunami bore runup[END_REF], or glacial lake outburst floods [START_REF] Cenderelli | Peak discharge estimates of glacial-lake outburst floods and normal climatic floods in the Mount Everest region, Nepal[END_REF], fall within this category of extreme flood events that can be represented by the NSWE. They are characterized by rapid wetting and drying over highly variable topographies, giving rise to complex unsteady free surface dynamics which pose a significant challenge for the numerical models. Numerical strategies for integrating the governing equations in these shallow extreme flows must also deal with complicated geometries and the highly complex dynamics of wave breaking and run-up. Similarly, the discretization of the boundaries of the physical domain may have a strong influence in the development of the flow dynamics, introducing errors or numerical instabilities if not carefully performed [START_REF] Baghlani | A shock-capturing model based on fluxvector splitting method in boundary-fitted curvilinear coordinates[END_REF].

Motivated by these applications, in this investigation we develop an efficient numerical model to solve the two-dimensional dynamics of extreme flows over natural terrains. We extend the method of [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF], which has shown to resolve complex features of free-surface flows by implementing a well-balanced approach. Well-balanced schemes are specifically conceived to preserve local and global mass conservation to machine accuracy, maintaining also the steady and motionless states. To achieve this requirement, it is necessary to discretize carefully the friction and bed-slope source terms (see [START_REF] Greenberg | A well-balanced scheme for the numerical processing of 23 Guerra et al[END_REF][START_REF] Leveque | Balancing source terms and flux gradients in high-resolution godunov methods: the quasi-steady wave-propagation algorithm[END_REF][START_REF] Gallouet | Some approximate Godunov scheme to compute shallow-water equations with topography[END_REF][START_REF] Audusse | A fast and stable wellbalance scheme with hysdrostatic reconstruction for shallow water flows[END_REF][START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF], for more details). We formulate and solve the governing equations in a non-orthogonal generalized curvilinear coordinate framework to model extreme flows propagation over natural terrains. We use a finite volume well-balanced approach based on a robust VFRoe-relaxation Riemann solver [START_REF] Gallouet | Some approximate Godunov scheme to compute shallow-water equations with topography[END_REF][START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: A class of relaxation schemes[END_REF], calculating mass and momentum fluxes at cell interfaces and performing the hydrostatic reconstruction method proposed by [START_REF] Audusse | A fast and stable wellbalance scheme with hysdrostatic reconstruction for shallow water flows[END_REF]. The source term that accounts for friction effects is treated with the semi-implicit fractional-step approach of [START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF]. Validation of the new model is presented through the comparison with benchmark tests, which are specifically chosen to assess its ability to deal with wet-dry interfaces, complex geometries, shocks, friction and bathymetric source terms.

In addition, the future studies of these flows will require new experiments, representing the 4 Guerra et al., February 11, 2014 complex features of rapid flooding events over realistic arbitrary geometries, to test and improve the numerical models. In this investigation we also perform a dam-break experiment on a scaled physical model, representing the bed and banks of a mountain river. We register time-series of water depth, and then compare the results with simulations carried out with the new well-balanced numerical model, showing that it can capture the most relevant characteristics of the flow. Furthermore, in order to compare our model performance with previously validated numerical models, we simulated this experience using two well-known shock-capturing NSWE models, AnuGA [START_REF] Mungkasi | Validation of anuga hydraulic model using exact solutions to shallow water wave problems[END_REF] and GeoClaw [START_REF] Berger | The geoclaw software for depthaveraged flows with adaptive refinement[END_REF]. From this comparison we can establish the improvement achieved by our numerical approach in terms of the overall agreement of the free surface variations in time, the estimation of the maximum amplitude of the propagated bore and its arrival time to different locations.

The paper is organized as follows: In section 2 we present the non-dimensional governing equations and the partial transformation to generalized non-orthogonal curvilinear coordinates that are employed in the model. The numerical scheme and the different algorithms used to integrate the NSWE are briefly described in section 3. Validation tests and comparisons of numerical simulations with benchmark cases and previously published experimental data are presented in section 4. In section 5 we describe new dam-break experiments conducted in the Hydraulic Laboratory of the Pontificia Universidad Católica de Chile, intended to further validate the model over a realistic and highly variable topographic configuration. Conclusions and future perspectives of this work are discussed in section 6.

GOVERNING EQUATIONS

The two-dimensional NSWE are a system of nonlinear partial differential equations representing the mass and momentum conservation laws, which were originally derived by Saint-Venant (1871). The fluid is assumed as incompressible and homogeneous, with hydrostatic pressure distribution. The shallow water or long-wave hypothesis considers negligible vertical velocities and depth-uniform horizontal velocities. Hence, the NSWE are often applied to river or nearshore flows where the characteristic horizontal wave-length is much longer that the characteristic water 5 Guerra et al., February 11, 2014 depth (see Cunge et al. 1980, for more details).

In what follows, we will work with a non-dimensionalized set of NSWE by choosing characteristic horizontal and vertical length-scales and a velocity scale (L, H, and U respectively). By defining the length and velocity scales of the flow, the time-scale is represented by T = L/U, and the dimensionless Froude number by F r = U/ √ gH, which quantifies the relative importance of inertial effects over gravity (g). The dimensional variables, noted with a hat ( ), are hereafter defined as x = Lx, y = Ly, z = Hz, h = Hh, u = Uu, v = Uv, and t = T t, where x and y represent the Cartesian directions, z defines the bed elevation, h is the water depth, u and v are the depth-averaged flow velocities in each Cartesian direction, and t is the time. In order to reproduce better complex arbitrary geometries, we introduce a boundary-fitted curvilinear coordinate system in two dimensions denoted by the system (ξ, η). Generalized curvilinear coordinates are chosen to follow the boundaries of the physical domain, adapting the grid to the geometrical details of the terrain. With this transformation we can have a better resolution in zones of interest and an accurate representation of the boundaries, resulting in an efficient discretization of the flow domain [START_REF] Lackey | Role of artificial dissipation scaling and multigrid acceleration in numerical solution of the depth-averaged free-surface flow equations[END_REF][START_REF] Liang | A boundary-fitted numerical model for flood routing with shock-capturing capability[END_REF][START_REF] Baghlani | A shock-capturing model based on fluxvector splitting method in boundary-fitted curvilinear coordinates[END_REF]).

The Cartesian NSWE can be partially transformed to this new coordinate system maintaining the hydrodynamic variables referenced to the Cartesian frame. This procedure is known as partial transformation and only modifies the mass and momentum fluxes of the governing equations. The full transformation would change the hydrodynamic variables vector, Q = [h, hu, hv] T , to the velocity components in the ξ and η directions, using the so-called contravariant velocity components, and the derivatives in the convective terms would yield the well-known Christoffel symbols of the second kind [START_REF] Ahn | Depth-Averaged model of open-channel flows over an arbitrary 3D surface and its applications to analysis of water surface profile[END_REF]. Therefore, considering only bed-slope and friction source terms, the non-dimensional NSWE can be written in curvilinear coordinates in the following form [START_REF] Lackey | Role of artificial dissipation scaling and multigrid acceleration in numerical solution of the depth-averaged free-surface flow equations[END_REF],

∂Q ∂t + J ∂F ∂ξ + J ∂G ∂η = S b (Q) + S f (Q) (1)
where Q is the vector of hydrodynamic variables, F and G are the flux vectors expressed in terms of the new spatial coordinate system ξ and η, respectively, and S b (Q) and S f (Q) are the source terms vectors. These vectors are given by the following expressions:

Q =       h hu hv       , F = 1 J       hU 1 uhU 1 + 1 2F r 2 h 2 ξ x vhU 1 + 1 2F r 2 h 2 ξ y       , G = 1 J       hU 2 uhU 2 + 1 2F r 2 h 2 η x vhU 2 + 1 2F r 2 h 2 η y       , (2)
S b (Q) =       0 -h F r 2 (z ξ ξ x + z η η x ) -h F r 2 (z ξ ξ y + z η η y )       , S f (Q) =       0 -S f x -S f y      
where h represents the water depth, u and v are the non-dimensional depth-averaged flow velocities in each Cartesian directions; z defines the bed elevation, z ξ and z η define the local bed slope with respect to the transformed coordinate system (ξ, η) and S f represents the friction source term.

The additional terms that appear in the fluxes, ξ x , ξ y , η x , and η y are the resulting metrics associated to the coordinate change, and J = ξ x η y -ξ y η x is the Jacobian of the transformation, which will remain constant for a fixed grid. U 1 and U 2 are the contravariant velocity components, expressed as U j = uǫ x + vǫ y with (j, ǫ) ∈ {(1, ξ), (2, η)}. The transformed system of equations is discretized on a rectangular and uniform grid in the transformed space (ξ, η) using the finite volume method that is described in the next section.

NUMERICAL SCHEME

The curvilinear NSWE system given in Eq. ( 1) is integrated using a finite volume well-balanced scheme, coupled with a splitting strategy for the treatment of source terms [START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF]. We decompose the solution associated to the system of equations ( 1) at each time step by solving two systems, one associated to the NSWE with topography source terms and a second associated to the remaining friction terms. In the following subsections we describe the different steps of the algorithm, including the implementation of the boundary conditions, and the stability criterion of the numerical solution.

Solution of the NSWE with Topography Source Terms

In this step we solve the following system associated to NSWE with topography source terms,

∂Q ∂t + J ∂F ∂ξ + J ∂G ∂η = S b (Q) (3)
We seek a numerical strategy that provides stable shock-capturing integration of system [START_REF] Cunge | Practical Aspects of Computational River Hydraulics[END_REF] with a precise control of the spurious oscillations induced by numerical dispersion. In addition, the scheme should be able to handle the complex interactions between flow and topography, including the preservation of motionless and steady states. We choose to adapt the robust second-order finite volume scheme introduced by [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF] to the non-orthogonal boundary-fitted coordinate framework.

Taking into account the new system of coordinates, the spatial discretization of Equation ( 3) can be recast under the following semi-discrete finite-volume formalism,

d dt Q i,j + J i,j ∆ξ (F * i+ 1 2 ,j -F * i-1 2 ,j ) + J i,j ∆η (G * i,j+ 1 2 -G * i,j-1 2 ) = S b(i,j) (4)
where Q i,j is the vector of cell-centered hydrodynamic variables, J i,j is the cell-centered Jacobian of the coordinate transformation, F * i±1/2,j and G * i,j±1/2 correspond to the numerical flux functions through the (i, j) cell interfaces, and S b(i,j) to the centered discretization of the bed-slope source term. We denote by ∆ξ and ∆η the cell sizes, and the interface between the (i, j) th cell and the (i + 1, j) th by (i + 1 2 , j), as depicted in Fig. 1.

The computation of the numerical fluxes F * i±1/2,j and G * i,j±1/2 is achieved using a robust VFRoerelaxation scheme proposed by [START_REF] Gallouet | Some approximate Godunov scheme to compute shallow-water equations with topography[END_REF]. To achieve a second order accurate scheme, we straightforwardly apply the MUSCL extrapolation proposed by [START_REF] Van Leer | Toward the ultimate conservative difference scheme V. A second order sequel to Godunov's method[END_REF]. This technique considers that numerical fluxes are computed by linearly reconstructing the hydrodynamic variables, leading to more accurate reconstructed states at each side of the interface of every cell as shown in Fig. 1 (see Bouchut 2004, for details). In order to handle topographic variations and the requirement for the preservation of static flows, we also adapted the well-balancing discretiza-tion for the bed-slope term proposed by [START_REF] Audusse | A fast and stable wellbalance scheme with hysdrostatic reconstruction for shallow water flows[END_REF] to the boundary-fitted curvilinear coordinate system of equations. In this step of the algorithm we built a linear reconstruction of the topography considering the MUSCL reconstructed hydrodynamic variables as shown in [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF]. Finally, the source term S b (Q) is estimated through the new reconstructed values of the fluxes. For a detailed description of the computation of the fluxes and the hydrodynamic variables the reader is referred to the work of [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF] and [START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: A class of relaxation schemes[END_REF].

Solution of the Friction Source Terms

The friction source term is incorporated using the splitting semi-implicit method proposed by [START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF]. The corresponding ordinary differential equation of the splitting operation is defined as follows,

dQ dt = S f (5)
where

S f = (0, -τ f x , -τ f y ) T (6)
In this equation, the terms τ f x and τ f y are the non-dimensionalized bed shear-stresses for each cartesian direction. The magnitude of the bed shear stresses in each direction can be calculated as follows:

τ f x = C f u √ u 2 + v 2 (7) τ f y = C f v √ u 2 + v 2 (8)
where C f is a non-dimensionalized bed friction coefficient, which can be expressed using one of the standard existing approaches developed for uniform flows such as Manning or Chézy.

Following the algorithm developed by [START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF], equation ( 5) is integrated using an implicit scheme and a second-order Taylor series expansion. Note that an additional friction limitation may be locally added to prevent from unphysical flow reversing, due to large 9 Guerra et al., February 11, 2014 drag forces in vanishing depth areas, as mentioned by [START_REF] Burguete | Friction term discretization and limitation to preserve stability and conservation in the 1D shallow-water model: Application to unsteady 22[END_REF].

As far as time-discretization is concerned, we use a classical second-order Runge-Kutta scheme for each time-step in this splitting approach. In order to solve system shown in Eq. ( 1

VALIDATION

As previously explained, the numerical model in non-orthogonal generalized coordinates is based on the method of [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF], incorporating bed-friction with the splitting semiimplicit method of [START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF]. Initially, we validated the model using various benchmark cases that are not shown herein, for rectangular domains employing discretizations in Cartesian coordinates. These first tests involved shock-capturing and moving shoreline problems, obtaining quantitatively accurate results in comparison with analytical solutions and laboratory data, which are the same cases previously studied by [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF]. The following series of benchmark tests are intended to illustrate the improvements obtained when a boundary-fitted curvilinear discretization is used, and to prove the ability of the model to deal with complex geometries, bed-slope, and friction source terms.

Dam-Break in a Convergent-Divergent Flume

We test the numerical model with a dam-break induced flow in a convergent-divergent channel, performing simulations of two-dimensional flood waves studied experimentally by [START_REF] Bellos | Experimental investigation of two-dimensional dambreak induced flows[END_REF]. The channel is 21.2 m long and has a rectangular cross-section of variable width. At a 10 Guerra et al., February 11, 2014 distance of 5 m downstream from the beginning of the flume, there is a smooth curved contraction and expansion of 0.6 m of minimum width, the flume has a constant bed-slope that can be changed and ranges between ±1%. For a detailed description of the domain the reader is referred to the work of [START_REF] Bellos | Experimental investigation of two-dimensional dambreak induced flows[END_REF].

The simulation of the experiment was carried out using a non-uniform boundary-fitted mesh of 241 × 41 cells and the channel was assumed to have a mild bed-slope, S 0 = 0.002, and a Manning friction coefficient n = 0.012 consistent with the recommendations of [START_REF] Bellos | Experimental investigation of two-dimensional dambreak induced flows[END_REF]. The propagating back and perturbing the numerical solution. The simulations were carried out for 70 s using a CFL number equal to 0.9 in order to ensure numerical stability during the computations.

The geometry of the channel, dimensions and location of measurement points studied by [START_REF] Bellos | Experimental investigation of two-dimensional dambreak induced flows[END_REF] along with the computational mesh are shown in Fig. 2.

The results show that dam-break phenomena is correctly captured by the numerical model as observed in Fig. 3. At the breaking, the shock wave spreads through the expansion and propagates downstream inundating the dry bottom. A rarefaction wave propagates upstream decreasing the water depth, which is then reflected at the upstream closed boundary of the flume. Then, the water depth in the flume starts decreasing, reaching a minimum of 0.014 m after 70 s. From the images depicted in Fig. 3, it can be seen that computed water depths and arrival times of the front are in excellent agreement with experimental data for all the studied measurement points, confirming the abilities of the model to capture shocks and deal with wet-dry cells and source terms over a curvilinear geometry. Guerra et al., February 11, 2014 Dam-Break over a closed basin with steep topography and friction

An important characteristic of the model is its capability of handling frictional source terms in the numerical solution of the NSWE, especially in situations where the flow is shallow or when the process of wave run-up/run-down is important. Here we test the numerical model with the flood produced by a dam-break over a closed channel with three conical obstacles. This test was first proposed by [START_REF] Kawahara | Finite element method for moving boundary problems in river flow[END_REF] and subsequently used by many researchers (e.g. [START_REF] Brufau | A numerical model for the flooding and drying of irregular domain[END_REF][START_REF] Brufau | Unsteady free surface flow simulation over complex topography with multidimensional upwind technique[END_REF][START_REF] Gallardo | On a well-balanced higher-order finite volume scheme for shallow water equations with topography and dry areas[END_REF][START_REF] Nikolos | An unstructures node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography[END_REF] to assess the ability of numerical models to deal with steep bed-slopes and friction source terms, wetting/drying proceses, and test local and global mass conservation.

The basin is 70 m long in the streamwise x direction and 30 m wide in the cross-stream or y direction. The bed topography is defined by the following equation,

z(x, y) = max            0 1 -0.1 • (x -30) 2 + (y -22.5) 2 1 -0.1 • (x -30) 2 + (y -7.5) 2 2.8 -0.28 • (x -47.5) 2 + (y -15) 2            (9)
Initial conditions consist of a motionless free surface elevation upstream of the dam location (at x = 16 m) equal to 1.85 m, and a dry bottom bed downstream. The simulation was performed employing a 101 × 101 uniform mesh using the suggested Manning coefficient of n = 0.018, and no-flow boundary conditions were applied to all the sidewalls of the basin. Computations were carried out for 400 s in order to achieve steady state as shown in previous investigations [START_REF] Nikolos | An unstructures node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography[END_REF], using a CFL number equal to 0.9 to ensure the stability of the model during the computations.

Evolution of the free surface is shown in Fig. 4. After the dam-break, the flood wave wets the small obstacles and a reflected wave is propagated back to the upstream boundary. At the same time, the front passes through the small obstacles and runs up and down over the larger obstacle, 12 Guerra et al., February 11, 2014 which is partially dry. Later, the wetting front separates and symmetrically goes around the larger mound as it crosses back at the middle of the channel downstream from the larger conical obstacle.

Finally, the wave hits the downstream solid boundary and is reflected back to the obstacles. The motion decays in time as a consequence of the friction force and after approximately 400 s, the steady state is reached leaving the three obstacles partially dry. These results agree well with those obtained by [START_REF] Gallardo | On a well-balanced higher-order finite volume scheme for shallow water equations with topography and dry areas[END_REF] and [START_REF] Nikolos | An unstructures node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography[END_REF], and illustrate the ability of the model to represent the interaction between dry and wet cells with high accuracy over a steep and frictional topography. Results also demonstrate that global mass conservation was achieved during the entire computation. Therefore, the robustness and stability of the friction scheme in conjunction with the well-balancing properties of the solution of the hyperbolic system are validated.

EXPERIMENTAL DAM-BREAK OVER COMPLEX TERRAIN AND NUMERICAL

SIMULATION

With the purpose of testing the numerical model, we carry out an experiment for the propagation of a dam-break wave generated by the rapid emptying of a reservoir over a scaled physical model of a river. The experiments were conducted in the Hydraulic Laboratory of the Pontificia Universidad Católica de Chile. In this section we present the details of the experiment, and the comparison between experimental results and numerical solutions obtained with our model and with two recognized NSWE models: AnuGA [START_REF] Roberts | AnuGA User Manual[END_REF][START_REF] Mungkasi | Validation of anuga hydraulic model using exact solutions to shallow water wave problems[END_REF] and GeoClaw (Clawpack Development Team 2013;[START_REF] Berger | The geoclaw software for depthaveraged flows with adaptive refinement[END_REF], showing the capacity of the models to handle highly demanding natural conditions analogous to the propagation of a tsunami wave over varying topography.

Experimental Set-up

Dam-break experiments were conducted in a physical model of a river consisting of a narrow and steep valley with complex topography presented in Fig. 5(a). The model was built using a geometrical scale equal to 1:60 and Froude similarity. The entire river reach is 14.6 m long and has a maximum width of 4.5 m. It starts with a narrow and curved zone with an average adverse bed-slope of nearly -4.5%, then it becomes wider towards its downstream end. The average bed-13 Guerra et al., February 11, 2014 slope over the river reach considered in the experiments is -1.5%. A longitudinal profile of the river reach along with the location of the measurements points is shown in Fig. 5(b). The river bed was built with a uniform concrete mix with fine gravel, which yields a roughness characterized by a Manning coefficient of n = 0.014.

Upstream of the river reach, there is a reservoir and a wooden gate that holds a fixed volume of water equal to 2.17 m 3 . The experiment we carry out consists of a sudden lift of the reservoir gate to release the water into a quiescent free surface downstream. A bore wave is then produced and propagated to the end of the river reach. The free surface evolution within the scaled-model is recorded during 60 s. At the reservoir, free surface elevation is set at 0.85 m, while at the river it is set at 0.56 m as shown in Fig. 5(b). Free surface variations are recorded at nineteen points in the river reach and at the reservoir as depicted in Fig. 5(b).

At the reservoir, free surface variations were measured using a KPSI brand pressure transducer recording voltage at 100 Hz. The accuracy of this instrument is ±1%. It was calibrated such that 1 V equals 1 m of water column. At downstream cross-sections, free surface variations over the mean water level were measured using wave DHI resistive gauges, which were located at the thalweg of the studied cross sections. Each gauge records voltage data at 100 Hz, the accuracy of these gauges is ±1.5 mm and the zero drift is ±5%, depending on the water temperature.

For the experiments, we use four resistive gauges, i.e. only four points could be measured at each run. Thus, five set of experiments with different gauge positions, are performed in order to cover the nineteen considered sections. As a verification of the repetability of the data, three repetitions were performed at each gauge location, maintaining the position of the pressure sensor in the reservoir in order to use it as a reference to synchronize the time series.

Numerical Simulation

The digitized bathymetry of the physical model is constructed from 39 measured cross sections within the river reach. Then, the bathymetric data together with the reservoir geometry is interpolated using a cubic spline method to create a boundary-fitted computational domain, which correctly represents the bathymetric features of the river reach and its geometry. The physical 14 Guerra et al., February 11, 2014 model is discretized into 130 × 30 cells of variable size, covering an area of 16 × 4.5 m 2 . The mesh used in the simulations and the digitized bathymetry are shown in Fig. 6. As initial conditions, free surface elevations at the reservoir and the river are set to 0.85 m, and 0.56 m, respectively. Zero velocities over the entire domain are considered at t=0. The open boundary condition is used at the downstream end of the river reach and close boundary conditions are applied to the sides of the computational domain. CFL condition is set to 0.9 in order to ensure the numerical stability of the simulations. The gate is instantly removed at t = 0 and the wave propagates downstream as seen in the experiments.

Results

The numerical model is able to simulate the dam-break event and the propagation of the bore wave over the river reach. Fig. 7 illustrates the propagation process showing 4 snapshots of the numerical computation, where the blue color represents the wet surface. As depicted in these figures, the bore propagates downstream of the river reach through the narrow and steep valley.

It takes approximately 10 seconds to the bore wave to reach the end of the river reach (≈14 m).

The entire flood wave propagates downstream along the complex geometry of the physical model, flooding and drying cells, and reaches a steady state after nearly 60 seconds. Fig. 8 shows the velocity vectors field 10 seconds after the gate opening; flow features observed in the laboratory experiments, such as recirculation and reflection due to topographical obstacles are observed in these numerical results. The main aspects of the rapidly varying flow measured in the experiments are thus reproduced by the numerical model.

Evolution of the bore wave is studied and compared to experimental data through time series of the computed free surface dynamics at the same locations measured in the experiments. Additionally, we have carried out numerical simulations using similar NSWE numerical models in order to assess and compare our model accuracy for similar discretization settings. The chosen numerical models are AnuGA [START_REF] Roberts | AnuGA User Manual[END_REF][START_REF] Mungkasi | Validation of anuga hydraulic model using exact solutions to shallow water wave problems[END_REF] and GeoClaw (Clawpack Development Team 2013;[START_REF] Berger | The geoclaw software for depthaveraged flows with adaptive refinement[END_REF]. Both models solve the NSWE using a well-balanced shock-capturing finite volume method, mainly differing in the discretization of the Guerra et al., February 11, 2014 domain; AnuGA model uses non-structured meshes, while GeoClaw uses an adaptive meshing approach. In order to define comparable grids for both simulations, we prepared meshes with a similar discretization sizes. For the AnuGA model, we defined a mesh of 4026 triangular elements.

GeoClaw model uses an adaptive mesh configuration that varies the distribution of element size with time, defining areas of refinement when needed. For this particular case, the refinement zone was controlled in order to mantain a number of grid nodes similar to our model and to AnuGA.

Figures showing the defined grids for AnuGA and GeoClaw along with the configuration for both simulations are presented in Appendix I.

Comparisons between measured and computed time series of the free surface evolution for the three models are presented in Fig. 9. Our analysis shows that the main features of the process, i.e. the arrival times, peak amplitudes and recession curves, are well reproduced by our numerical model and by GeoClaw. AnuGA results seem to be accurate in representing maximum amplitude, but for this particular case, the model overestimates the final free surface elevation.

Based on the results obtained in the simulations, we establish three quantities related to the inundation and propagation process that are studied, namely the overall agreement of the free surface elevation on time, the maximum amplitude of the bore at each measurement location and the arrival time of the wave front. For each of these variables, mean relative errors to the measurements were calculated at gauge locations; these relative errors are defined as follows:

Relative root mean square error (RRMSE)

A root mean square error is used to compare the experimental and numerical free surface elevation at each location and at the same time to highlight the locations were the larger differences were found. The relative root mean square error for a location k, and the average relative root mean square error considering all locations are defined respectively as:

RRM SE 2 k = 1 T T 0 η k m (t) -η k n (t) η k m (t) 2 dt ( 10
)
16 Guerra et al., February 11, 2014

RRM SE = 1 M n k=1 RRM SE 2 k 1/2 (11)
where T is the period of time considered in measurements and in simularions (60 s); M is the number of measurement points, and η k m and η k n are the measured and numerical free surface elevation at each k location respectively.

Maximum amplitude of the wave

The maximum amplitude of the bore is an important flood variable since it is related with the destructive potential of the wave. This variable is defined as the difference between the maximun and the initial free surface elevation at the river reach. The mean relative error between experimental data and numerical results is estimated as follows:

∆H r = 1 M M k=1 H(η k m ) -H(η k n ) H(η m k ) (12)

Arrival time

The arrival time of the wave is an interesting parameter for defining evacuation plans as it indicates the available time to leave flood-prone areas and it is also a proxy for the celerity of the bore. The arrival time of at a location k is defined as the first instant where the signal surpasses the initial value by a certain threshold, which is defined here as 1 mm. The mean relative error of this variable over all the measuring sections is defined as:

∆T a = 1 M M k=1 T a (η k m ) -T a (η k n ) T a (η k m ) (13)
The calculated mean relative errors for each model are summarized in Fig. 10. The RRMSE for the three models shows a very good overall agreement between measurements and numerical predictions, with a 1.7% relative error for our model and a 2.1% relative error for AnuGA and GeoClaw, which is an acceptable error considering how demanding the experiment is. For our model, the highest relative errors are found in the gauges located upstream, near the gate. Guerra et al., February 11, 2014 It is found that in average, the maximum amplitude is underestimated by all models, being the variable with the higher relative error. However, in terms of arrival times of the wave front, predicted results are in excellent agreement with observations (less than 5% mean relative error for our model). Main differences were found in the first gauges closer to the reservoir with maximum local errors reaching 10%. These differences might be explained by the opening mechanism of the gate, which is frictionless and instantaneous in the simulation, but performed in a finite time in the experiments, where vertical velocities might also be generated and interfere with the dynamics of the wave downstream, thus increasing the maximum amplitude of the wave.

Summarizing, the results from our model show that it can correctly capture the time evolution of the free surface elevation, the arrival time of the bore and its maximum amplitude. For this singular case, our simulations show a decrease in the studied relative errors when comparing to the results of previously validated numerical models when a similar discretization setting of the domain is used.

CONCLUSIONS

In this investigation we have developed and validated a finite-volume numerical model to simulate extreme flows and rapid flooding over natural terrains and complex geometries. The numerical scheme successfully reproduces the flow hydrodynamics over rough and highly variable topographies, incorporating an accurate and robust treatment of bore dissipation and wet and dry process.

The method is based on algorithms proposed by [START_REF] Marche | Evaluation of well-balance borecapturing schemes for 2D wetting and drying processes[END_REF], and adapted here to solve the bed-slope source term and to incorporate friction by using the splitting semi-implicit scheme developed by [START_REF] Liang | Numerical resolution of well-balanced shallow water equations with complex source terms[END_REF]. An important advantage of this model is the simplicity and low cost of its implementation, yielding accurate results using coarse computational grids.

Two benchmark test cases are considered to illustrate the capabilities of our new model. Test cases involve the use of boundary-fitted grids, with frictional and varying bathymetry, and dambreak floods. The use of boundary fitted grids is shown for the case of the dam-break in a converging-diverging flume [START_REF] Bellos | Experimental investigation of two-dimensional dambreak induced flows[END_REF]. Numerical results are in excellent agreement with experimental data obtained by [START_REF] Bellos | Experimental investigation of two-dimensional dambreak induced flows[END_REF], showing the ability of the model to deal with 18 Guerra et al., February 11, 2014 a complex geometry and a rapidly varying flow. The process of run-up and run-down, and wetting and drying of the terrain as the wave propagates are reproduced in the numerical simulations of the benchmark cases analyzed in this investigation. This case also illustrates the stability and the well-balanced property of our model, since steady state is correctly reached due to frictional effects.

We report new experimental data for a dam-break wave propagating over the scaled physical model of a river reach with narrow and steep valleys. This experiment was specially designed to test the numerical model and to evaluate its ability for representing extreme and rapid flooding over natural conditions. For this case, we have also compared our results with those from two other NSWE numerical models, showing a better relative performance by our model in terms of the studied hydrodynamic variables when similar computational discretizations are employed.

The results reported in this research confirm that our numerical model is a robust and powerful tool that can be used to simulate high-volume flash-floods and significant inundation over dry terrain under realistic conditions, giving accurate results in terms of water depth evolution, discharge and inundated area. The model performance shows that it can become a useful tool for evaluating extreme and rapid flood events over complex bathymetries, and to assess their hazards in terms of inundation extent and depth, depth-averaged velocities, arrival time of peak discharge, etc. We expect that the model will be used as an instrument to develop new inundation hazard maps in coastal and riverine areas.

Future research using the model developed in this investigation will focus on the study of complex engineering and geophysical flows. Large-scale coastal flows, such as tsunami propagations [START_REF] Yamazaki | Shelf resonance and impact of near field tsunami generated by the 2010 Chile earthquake[END_REF], will also be studied by incorporating the Coriolis effect into the model. Also, advanced sediment transport and morphodynamic models [START_REF] Cao | Computational dam-break hydraulics over erodible sediment bed[END_REF][START_REF] Vasquez | Modeling bed changes in meandering rivers using triangular finite elements[END_REF]) will be added to the basic equations of the flow to study erosion and sedimentation processes in fluvial and coastal environments. Finally, this model will be employed to investigate density-coupled flows, incorporating the transport of active and passive contaminants in rivers and estuaries [START_REF] Loose | Finite volume modeling of variable density shallow-water flow equations for a well-mixed estuary: application to the rio Maipo estuary in central Chile[END_REF].

AnuGA

  ) at the boundaries of the computational domain, we have implemented and tested in the model three types of boundary conditions: i.) Transmissive or open boundary, allowing the information to freely leave the domain without propagating spurious information back to the domain; ii.) Solid wall or close boundary, that imposes no discharge through the boundary of the domain; and iii.) Absorbing/Generating boundary condition, which relies on the work of[START_REF] Sanders | Non-reflecting boundary flux function for finite volume shallow water models[END_REF] and[START_REF] Cienfuegos | A fourth order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations -Part II : Boundary Conditions and validation[END_REF], which allow to prescribe inflow discharge or free surface information at the boundary, such as incoming waves or stage-discharge relationships, and freely evacuate back-traveling waves. Finally, the stability of the numerical model is controlled by the Courant-Friedrich-Lewy criterion (CFL)[START_REF] Toro | Shock-Capturing Methods for Free-Surface Shallow Flows[END_REF]).

  dam is located at the end of the contraction, at a distance of 8.5 m from the upstream boundary of the flume. Initial conditions consist of a water depth upstream of the dam of 0.30 m, null flow velocities, and dry terrain downstream of the dam. A no-flow boundary condition was imposed at the sidewalls and at the upstream boundary of the flume. At the downstream end of the channel an open boundary condition is applied to allow all the information to exit the domain without

FIG. 3 .

 3 FIG. 1. Discretization cell and numerical fluxes. L and R denote the left and right boundaries of the cell,and + signs are the left and right sides of the cell interface.

  FIG. 5. Dam break Experience: (a) Physical model and measurement instruments, upstream view, (b) Longitudinal profile of the river reach bottom elevation, initial conditions and measurement points

  FIG. 11. Physical model discretization grids: (a) Triangular mesh of 4060 elements for AnuGA model, (b) Initial adaptive mesh with three levels of refinement for Geo-Claw model

Guerra et al., February 11, 2014

ACKNOWLEDGEMENTS

This investigation was supported by ECOS-Conicyt project C07U01. Additional support has been provided by Conicyt Fondef project D11i1119 and Conicyt/Fondap program 15110017. The equipment for the laboratory experiments was supplied by the Instituto Nacional de Hidráulica. The authors also thank the technical support provided by Enrique Rosa and the work of Domenico Sciolla and Eduardo González in the development of the experiments.

20 Guerra et al., February 11, 2014

APPENDIX I.

In this section we present additional information about the configuration used for the numerical simulations of the Dam-break over a physical model using AnuGA and GeoClaw numerical models. Fig. 11 present the mesh configuration for each model; Fig. 11(a) presents the triangular mesh of 4026 required by the AnuGA model and Fig. 11(b) presents the initial adaptive mesh used for the simulation with GeoClaw. The element size distribution of the AnuGa and of our model considered approximately a 60% of the elements under 100 cm 2 in size, being both distributions comparable. For the simulation using GeoClaw, the time evolution of the distribution of elements was controled in order to maintain the number of the finer elements aroud 4000, which are 69.4 cm 2 in size. For both models, CFL number was set equal to 0.95 in order to achieve numerical stability.

List of


Chantal Enguehard
email: enguehard@irin.univ-nantes.fr

Béatrice Daille
email: daille@irin.univ-nantes.fr

Emmanuel Morin
email: morin@irin.univ-nantes.fr

Tools for Terminology Processing

Keywords: Terminology processing, term acquisition, term recognition, term structuring

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

INTRODUCTION

The terminology of a domain is an important resource for researchers from different areas: lexicographers, translators, and also domain specialists (engineers, technicians, redactors of technical documentation, etc.). In addition, specialists in Natural Language Processing have realized that this knowledge could be useful for a numbes of other applications, such as indexing, automatic or machine-aided translation, text generation, semi-automatic abstraction, edition of hypertext systems, ontology building, etc. However, collections of terms are rare because they are difficult to constitute: terminologists have to study large amounts of documentation about the targeted domain to extract the terms, give their definition, and structure them following a hierarchical classification. In the 90's, the compilation of large electronic corpora made it possible to build systems to carry out different tasks in terminology processing automatically, especially terminology acquisition, terminology recognition, and terminology structuring. We present three tools which can assist in these different terminology processing tasks, and we observe their capability of multilingual terminology processing.

TERM ACQUISITION

Many different professional groups are interested in the automatic acquisition of terminology through corpus exploitation to achieve a number of goals: building of glossaries, vocabularies and terminological dictionaries, text indexing, automatic translation, building of knowledge databases, edition of hypertext systems, etc. The appearance of large electronic corpora in the 90s rendered the development of terminology extractors which can assist terminological work. Different approaches arose: linguistic (CLARIT (ref. [START_REF] Evans | Noun-phrase analysis in unrestricted text for information retrieval[END_REF], LEXTER (ref. [START_REF] Bourigault | LEXTER, a Natural Language Processing tool for terminology extraction[END_REF], TERMS (ref. [START_REF] Justeson | Technical terminology: some linguistic properties and an algorithm for identification in text[END_REF]), heuristic or statistical (ANA (ref. [START_REF] Enguehard | Automatic Natural Acquisition of a Terminology[END_REF])), or hybrid linguistic-statistical (ACABIT (ref. [START_REF] Daille | Study and Implementation of Combined Techniques for Automatic Extraction of Terminology[END_REF], NEURAL (ref. 6). The systems we present analyse electronic texts issued from a specialised domain and produce a list of candidate-terms. These candidate-terms have to be validated by a terminologist or a specialist of the domain). These systems are especially designed for two tasks:

Terminology mining

Terminology mining refers to several functionalities:  To propose a list of candidate-terms ranked from the most representative of the corpus to the least.

This list could be used to build specialised dictionaries relative to a new domain;  To propose a list of candidate-terms for which a morphological variation has been encountered and thus reflect a more advance lexicalisation. This list could be added to update an existing reference list.

Automatic Indexing

Automatic Indexing is a subtask of information retrieval which assigns an indexing language to texts (ref. 7). This language consists of a set of descriptors which must reflect the textual document in a discriminatory and a non-ambiguous way. There are two modes of indexing:  Controlled indexing where indexes are occurrences of terms from a controlled vocabulary;  Free Indexing where indexes are chosen without consideration for an authority list. With free indexing, the descriptors, which are simple words, are problematic because of their ambiguity (ref. [START_REF] Salton | Automatic Text Processing[END_REF], contrary, key-phrase indexes are not ambiguous, but pose problems because of their variations (ref. 9).

ACABIT

ACABIT is a term extractor which takes as input a tagged corpus with part of speech and lemma which has been structured with XML tags to identify the main parts of the text: title, abstract and sentences. ACABIT proposes as output a rank list of multi-word terms for each text. The program scans the corpus, counts and extracts collocations whose syntax characterizes base-terms or one of their variants. This is done with shallow parsing using local grammars based on regular expressions (ref. [START_REF] Basili | Acquisition of Selectional Patterns in Sublanguage[END_REF]). These grammars use the morphosyntactic information associated with the words of the corpus after tagging. The different occurrences are grouped as pairs formed by lemmas or morphologically derived lemmas obtained though a morphological analysis of the candidate-term.

These pairs are sorted following an association measure which takes into account the frequence of the co-occurrences.

In addition, ACABIT proposes semantic classes of candidate-terms based on the study of relational adjectives.

ANA

ANA is a term extractor which takes as input a raw data corpus and proposes as output a rank list of terms for each text or for the entire corpus. This system has been specially design to treat bad quality corpora (for instance written in telegraphic style) which can not be tagged. It does not make any hypothesis on the form of terms.

This incremental system uses a bootstrap (a few frequent terms), a list of functional words, and a list of lexical scheme words.

The system 1. recognises the terms in the texts (and the candidate-terms) included in the bootstrap, 2. collects the context of these recognized terms (through a window of n words), 3. studies these contexts to infer some candidate-terms which will be included in the bootstrap, 4. returns to step 1 unless a stop condition is filled (for instance, there is no more discovery of candidate-terms).

The inference of new candidate-terms follows three patterns. We illustrate their presentation by examples where terms are written in capital letters, functional words belong to the set {"a" "any" "for" "in" "is" "may" "of" "or" "the" "this" "to"}, there is one lexical scheme: "of", and six terms constitute the bootstrap: "WOOD" "COLOUR" "BEECH" "TIMBER", "DIESEL", "ENGINE". 1. Two terms (or candidate-terms) of the bootstrap appear in the same window A candidate-term is qualified when two existing terms (or candidate-terms) of the bootstrap appear frequently with almost the same arrangement. The most frequent arrangement becomes a candidate-term Example : ... "the" "DIESEL", "ENGINE" "is" ... … "this" "DIESEL", "ENGINE" "has" ... ... "a" "DIESEL", "ENGINE" "never" ... The study of these three contexts of "DIESEL" and "ENGINE" will qualify "DIESEL ENGINE" as a candidate-term. 2. A term (or candidate-term) of the bootstrap and a lexical scheme word appear in the same window A candidate-term is qualified when a term (or candidate-term) of the bootstrap appears frequently with a lexical scheme word and with a word. This word then becomes a new candidate-term. Example : ... "any" "shade" "of" "WOOD" "could" ... ... "this" "shade" "of" "WOOD" "is" ... ... "the" "shade" "of" "BEECH" "may" ... ... "new" "shade" "of" "TIMBER"... ... "same" "shade" "of" "WOOD" "in" ... The study of these contexts will qualify "SHADE" as candidate-term.

A term (or candidate-terms) of the bootstrap without any lexical scheme nor second term

A candidate-term is qualified when a term (or candidate-terms) of the bootstrap appears frequently with a word without any lexical scheme word nor second term.. This word then becomes a new candidate-term. Example : ... "use" "any" "soft" "WOODS" "to" "make" "this" ... ... "buy" "this" "soft" "WOODS" "or" "plastic" "for" ... ... "cheapest" "soft" "WOODS" "comes" "from" ... The study of these contexts will qualify "SOFT WOODS" as a candidate-term.

Thus, the ANA system is capable of extracting single-word terms and multi-word terms. It can also enrich an existing terminology.

The adaptiveness of the system to the corpus (and to the language of the corpus) is improved by the possible inference of the lexical scheme words as has been suggested in previous work (ref . 11).

TERM RECOGNITION

In a corpus, terms do not always appear with the reference form. For instance, we can meet some slight variations due to diacritic signs (accent, hyphenation) or in the plural, but also some deep variations are not so rare as the transformation of a compound word into a verbal form.

It is now admitted that variations represent 25% of the occurrences of terms. It is especially important to identify them for indexing, or for text mining.

Surprisingly, research especially dedicated to this area had been rare over the last 20 years. We can cite Jacquemin who worked specifically on the recognition of terms and their variations and built the FASTR system based on a fine linguistic study (ref. 12). In English, in addition to common linguistic variations (singular/plural for instance), four types of term variations can be recognized: coordinations (recognition of simple and complex terms is the coordination of recognition of simple terms and recognition of complex terms), modifications/substitutions (book on heart malformations and book on malformations), permutations (blood flow or flow of blood), and morphosyntactic variations (colourless eyes, eyes without colour). . The FASTR system uses the formalism of unification grammar in which lexical rules are composed of lists of constraints on features and of a structure on categories which is out of context. The rules impose some constraints that must be respected, for example, genre and number. In addition to this basic formalism, meta-rules allow the manipulation of the lexical rules.

Flexible-equality of terms

The flexible-equality of terms (ref. [START_REF] Enguehard | Flexible-equality of terms: definition and evaluation[END_REF]) is a mathematical operator which determines if two terms should be considered as equivalent. This operator uses a list of functional words (empty words) and depends on a parameter : k.

Flexible-equality of strings

The flexible-equality of two strings w and w' is based on the calculus of the minimum editing distance (ref. [START_REF] Levenshtein | Binary codes capable of correcting deletions, insertions and reversals[END_REF]. dist(w 1,i , w' 1,i ) = min ( dist(w 1,i-1 ,w' 1,i ) + q, dist(w 1,i , w' 1,j-1 ) + q, dist(w 1,i-1 , w' 1,j-1 ) + p . dist(w i,i , w' j,j ) )

with  w n,m being the sub-string beginning at the nth letter and finishing after the mth letter of the word w.  dist(x, y) = 1 if x = y = 0 otherwise  q : cost of the insertion / deletion of one letter,  p : cost of the substitution of one letter by another. Generally, a substitution is considered as a deletion followed by an insertion, thus p=2q. This algorithm has a high complexity: (n 3 ). It can be easily transformed into another algorithm with a (n 2 ) complexity by using a vector to memorize intermediary results (ref. [START_REF] Stephen | String searching algorithms[END_REF].

We define the weighted distance (WD) of two words w and w' as the minimum editing distance weighted by the sum of the length of the two words.

WD(w, w') = dist(w, w') /(|w| + |w'|) with  |w| the length of the word w (in number of letters)

 dist(w, w') = dist(w 1,|w| , w' 1,|w'| )
This distance varies from 0, when words are strictly equal to, 1, when words are completely different.

We define the flexible-equality of strings (noted |~) as follows: w |~ w' <=> WD(w, w')  1 / k with 1/k being a threshold to be determined 1/k represents the proportion of variation acceptable in a word.

Flexible-equality of complex terms

The flexible-equality of strings cannot be used directly on compound words because of the high complexity of the algorithm. Thus we define an operator for this compound words. This operator requires the succession of 3 steps :

Segmentation of the strings into words

The segmentation of a string into words is processed through the definition of a list of symbols which can be encountered in a string. This list includes all the letters of the considered alphabet (minuscules, majuscules and accentuated ones), the digits, and sometimes the minus charactere '-'. A term X is defined as an ordered list of words x i : X = x i,n .

example: "colour of a hammer" is considered as the ordered list ("colour", "of", "a", "hammer")

Deletion of the functional words

Functional words (empty words) are defined in the list. In English we would find in this list articles ("a", "the", "this", etc.), prepositions ("of", "through", etc.), pronouns ("they", "who', etc.), etc. We call restriction of a string X, and note R(X), the ordered list of words of X which are not functional words. example: X = "colour of a hammer" R(X) = ("colour", "hammer") "of" and "a" being functional words.

Respective comparisons of the words

Two strings are flexible-equal (noted '||~') if the words of their restriction are flexible-equal. example: Here there is an m forgotten in hamer in the string Y X = "colour of a hammer" R(X) = ("colour", "hammer") Y = "colour of any hamer" R(Y) = ("colour", "hamer") ("colour" |~ "colour" and "hammer" |~ "hamer")  (X ||~ Y)

We define the distance (D) between two terms X and Y as the average of the distances between their restriction, words being compared respectively to their rank. This definition is useful when several candidates are flexible-equal. In such a case the one with the smallest distance will be chosen. In the field of terminology structuring, the main direction of research is the semantic classification from distributional analysis. Generally speaking, these methods are robust to extract classes between words, but have some disadvantages:

TERM ORGANISATION

1. Clusters obtained with such techniques are not a priori significant (these clusters must be explained). 2. Clusters contain heterogeneous linguistic entities (for instance, a word and its antonym can be included in the same clusters). 3. Conceptual similarity, as well as semantic proximity, are "neutral" links which does not yield links labelled by semantic predicates.

Conversely, links between words that result from automatic acquisition of relevant predicative or discursive patterns are fine and accurate, but the acquisition of these patterns is a tedious task that requires substantial manual work. For our work, we use a low-cost system (called: Promethee) which extracts and uses lexico-syntactic patterns to acquire semantic relations between terms.

The Promethee system

The Promethee system, for corpus-based information extraction, has two functionalities: 1. A corpus-based acquisition of lexico-syntactic patterns with respect to a specific conceptual relation. 2. The extraction of pairs of conceptually related terms through a database of lexico-syntactic patterns.

These functionalities are implemented in three main modules:

Lexical Preprocessor

The lexical pre-processor receives raw texts as input. First, the text is tokenized (recognition of word and sentence boundaries), tagged and lemmatized. Noun phrases, acronyms and sequences of noun phrases are detected through regular expressions. The output of the lexical pre-processor is an enriched text with SGML tags.

Shallow Parser and Classifier

This module extracts lexico-syntactic patterns relative to semantic relationships. This phase has been inspired by the works of P. Hearst (ref. 18) and implemented by a shallow parser associated with a classifier.

Information Extractor

The information extractor acquires pairs of conceptually related terms by using a database of lexicosyntactic patterns. This database can be the output of the shallow parser and classifier, or may be manually specified.

The shallow parser is complemented with a classifier for the purpose of discovering new patterns through corpus exploration. This procedure is inspired by P. Hearst (ref. 18) and consists of the following seven steps: 1. Manually select a representative conceptual relation, e.g. the hypernym relation. 2. Collect a list of pairs of terms linked by the previous relation. This list of pairs of terms can be extracted from a thesaurus, a knowledge base or may be manually specified. For instance, the hypernym relation « neocortex is-a-kind-of vulnerable area » is used. 3. Find sentences in which conceptually related lemmatized terms occur. These sentences are lemmatized, and noun phrases are identified. They are represented as lexico-syntactic expressions.

For instance, the previous relation HYPERNYM (vulnerable area,neocortex ) is used to extract the sentence: Neuronal damage was found in the selectively vulnerable areas such as neocortex, striatum, hippocampus and thalamus from a medical corpus.The sentence is then transformed into the following lexico-syntactic expression 1 : NP find in NP such as LIST 4. Find a common environment that generalizes the lexico-syntactic expressions extracted at the third step. This environment is calculated with the help of a function of similarity and a procedure of generalization that produce candidate lexico-syntactic patterns. For instance, from the previous At this level, two significant points make our technique different from Hearst's methodology: 1. A common environment relative to a set of sentences is extracted automatically by the former method and manually by Hearst. 2. The expert evaluation of candidate lexico-syntactic patterns or pairs of terms is absolutely necessary since all candidate patterns or pairs of terms do not denote the target relationship. This evaluation is not mentioned by Hearst.

The Promethee system has been used to extract discursive patterns relative to (1) a generic relation (hyponymy relation from a 1.3-million word French agricultural corpus and a 1.56-million word English medical corpus) and ( 2) two specific relations (merge and produce relations from a 0.9-million word English news stories corpus). Links extracted by the promethee system have a high precision, but cover only a part of targeted links occurring in the corpus.

FUTURE DEVELOPMENTS

Multilinguality

ACABIT, ANA and Promethee were designed for French, their adaptation to other languages mainly depends on the linguistic features of the targeted language. Previous experiments showed that this adaptation should be easy for non-agglutinative indo-european languages. For instance the adaptation to English language requires very slight adaptations, such as the new definition of base terms for ACABIT, new lists of functional words and lexical scheme words for ANA, Promethee would need more adaptations, for instance a tagger, a lemmatizer anda noun phrase extractor. The treatment of languages not belonging to Roman languages is also possible: we have customised ACABIT and ANA systems for Malagazy texts in previous work (ref. [START_REF] Daille | Traitement automatique de la terminologie en langue malgache[END_REF]. The Malagasy language is very close to the Indonesian language. It has been written since the 19 th century with the latin alphabet. Term identification from Malagazy texts presents some difficulties due to some of the linguistic characteristics of this language, for example the combinatorial variation, the agglutination of possessive and personal pronouns, the non-formal categories of the adverbs, and the homonymy between functional words and semantic words. However we succeeded in defining pre-treatments which allowed the utilization of both systems. The adaptation of the flexible-equality operator will depend on the morphological feature of the considered language.

Acceptation of new formats

XML format

ACABIT accepts texts designed with an XML format and also produces terms designated by an XML format. This capability could be added to ANA and to the supple_equality operator.

Unicode

Unicode offers the possibility to code any symbol used by any language in the world. The adaptation of our systems to this standard is crucial for their multilingual future.

Conclusion

Constituting terminologies of different technical or scientific domains and in different languages is crucial because it allows people to conceptualize and understand techniques and sciences in their own languages. These linguistic resources represent a major contribution to the increase of the education level of a population. Nevertheless Terminology processing is gaining a real place in Natural Language Processing technologies. Nervertheless, this linguistic effort is too often not processed, essentially because of its costs. The last decade saw the appearance of new tools, destined to help in the collecting, retrieving and structuring terms. Recent advances in the normalisation of linguistic symbols (with Unicode), and linguistic study of the targeted languages features should allow the adaptation of these existing tools to different languages.

  -terms which are extracted from the corpus belong to a special type of co-occurrence:  the co-occurrence is oriented and follows the linear order of the text;  it is composed of two lexical units which do not belong to the class of functional words such as prepositions, articles, etc.;  it matches one of the morphosyntactic patterns of what we will call ''base terms'', or one of their possible variations.

	The patterns for base terms are:
	 [Noun1 Adj]
	emballage biodégradable	(biodegradable package)
	 [Noun1 Noun2]
	ions calcium
	 [Noun1 (Prep (Det)) Noun2]
	protéine de poissons	(fish protein)
	chimioprophylaxie au rifampine	(rifampicin chemoprophylaxis)
	 [Noun1 à Vinf]
	viandes à griller	(grill meat)
	These base structures are not frozen structures and accept several variations. Those which are taken
	into account are:
	 Inflexional and Internal morphosyntactic variants:
	-graphic and orthographic variants which gather together predictable inflexional variants:
	conservation de produit	(product preservation)
	conservations de produit	(product preservation)
	or not:
	conservation de produits (products preservation)
	and case differences.
	-variations of the preposition:
	chromatographie en colonne	(column chromatography)
	chromatographie sur colonne	(chromatography on column)
	-optional character of the preposition and of the article:
	fixation azote	(nitrogen fixation)
	fixation d'azote	(fixation of nitrogen)
	fixation de l'azote	(fixation of the nitrogen)
	 Internal modification variants: insertion inside the base-term structure of a modifier such as the
	adjective inside the Noun1 (Prep (Det)) Noun2 structure:
	lait de brebis	(goat's milk)
	lait cru de brebis	(milk straight from the goat)
	 Coordinational variants: coordination of base term structures:
	alimentation humaine	(human diet)
	alimentation animale et humaine	(human and animal diet)
	 Predicative variants: the predicative role of the adjective:
	pectine méthylée	(methylate pectin)
	ces pectines sont méthylées	(these pectins are methylated)
	 Morphosyntactic variants: derivational variation that keeps the synonymy of the base term:
	acidité du sang	(acidity of the blood)}
	acidité sanguine	(blood acidity)

C.

Enguehard

, B. Daille, E. Morin, "Tools for Terminology Processing", The Indo-European Conference on Multilingual Communications Technologies (IEMCT), R. K. Arora, M. Kulkarni, H. Darbari, Tata McGraw-Hill, pp.218-229, Pune, India, June 2002. Candidate

  NP stands for a noun phrase, and LIST for a succession of noun phrases C. Enguehard, B. Daille, E. Morin, "Tools for Terminology Processing", The Indo-European Conference on Multilingual Communications Technologies (IEMCT), R. K. Arora, M. Kulkarni, H. Darbari, Tata McGraw-Hill, pp.218-229, Pune, India, June 2002. expression, and at least one other similar one, the following candidate lexico-syntactic pattern is deduced: NP such as LIST 5. Validate candidate lexico-syntactic patterns by an expert. 6. Use these validated patterns to extract additional candidate pairs of terms. 7. Validate candidate pairs of terms by an expert, and go to step 3.

1

  C.Enguehard, B. Daille, E. Morin, "Tools for Terminology Processing", The Indo-European Conference on Multilingual Communications Technologies (IEMCT), R. K. Arora, M. Kulkarni, H. Darbari, Tata McGraw-Hill, pp.218-229, Pune, India, June 2002.