I Emmanuel

H Andrieu

E Leblois

N Janey

Influence of rainfall spatial variability on hydrological modelling: a simulation approach

Keywords: Rainfall spatial variability, Hydrological modelling, Rainfall simulator, Stream network model

The present work aims to quantify the effects of neglecting rainfall spatial variability for runoff modelling at the outlet of catchments of about ten to several hundred km². In order to overcome modelling and rainfall data errors, to control the rainfall variability as well as characteristics and hydrological behavior of catchments, we have proceeded by simulation. For this, a simulation chain has been used, including a stream network model, a rainfall simulator and a distributed hydrological model (with four production functions and a distributed transfer). The choice has been made not to be exhaustive but to study very contrasted situations. Results showed that it is difficult to obtain robust general conclusions by studying only a few rainfall events which might explain that different conclusions were drawn in previous case studies. However, some tendencies were observed. For the different studied rainfall configurations, the effect of neglecting rainfall variability is particularly important if: 1) the dimensionless ratio "velocity of rainfall field on celerity along flow path" is small, 2) the rainfall field direction is perpendicular to the catchment flow direction, 3) the catchment has an elongated shape and 4) an Horton or SCS production function.

INTRODUCTION

The link between rainfall space-time variability and hydrological modelling at a catchment outlet remains an open research issue in hydrology. It puts the following question: what is the effect of neglecting rainfall spatial variability in hydrological modelling? This question has been tackled by numerous studies and results are contrasted. For example, [START_REF] Adams | An analysis of the impact of the spatial variability in rainfall on runoff and sediment predictions from a distributed model[END_REF] notice that routing averaging effects remove the majority of the impact of rainfall spatial variability at the catchment scale (150 km²). The analysis of a storm event occurred in Germany on the Weisseritz catchment (384 km 2 ) led [START_REF] Tarolli | Rainfall space-time organization and orographic control on flash flood response: the Weisseritz event of August 13, 2002[END_REF] to the similar conclusion. However, [START_REF] Zoccatelli | Which rainfall spatial information for flash flood response modelling? A numerical investigation based on data from the Carphatian range, Romania[END_REF], who studied three extreme flash flood events on catchments of 36 km² to 167 km² in Romania, report that neglecting spatial rainfall variability results in a considerable loss of modelling efficiency in about 30% of the cases. The influence of the rainfall measurement error on runoff modelling can also be significant [START_REF] Quintero | A scenario-incorporating analysis of the propagation of uncertainty to flash flood simulations[END_REF], and contributes to hide the gain of accuracy allowed by a detailed knowledge of the rainfall variability. In this context, we suggest proceeding by simulation in order to evaluate for which catchments characteristics and for which rainfall types neglecting rainfall spatial variability has an effect on the hydrological modelling of catchments for which the surface runoff is the dominant process. Simulation enables us to explore a high number of situations in order to gain a general point of view. Moreover, it allows overcoming different sources of errors (rainfall and flows data, modelling) and controlling as much as possible rainfall variability and catchments characteristics (area, shape, hydrological behavior). In this way, we have developed a simulation chain composed of three independent modules: a stream network model, a rainfall simulator and a distributed hydrological model.

PRESENTATION OF THE SIMULATION CHAIN

Stream network model

The stream network model [START_REF] Janey | Modélisation et synthèse d'images d'arbres et de bassins fluviaux associant méthodes combinatoire et plongement automatique d'arbres et cartes planaires[END_REF][START_REF] Emmanuel | Evaluation de l'apport de la mesure de pluie par radar météorologique pour la modélisation pluie-débit de petits bassins versants[END_REF] is based on the diffusion limited aggregation (DLA) method which builds tree networks from the random walk of particles on a lattice. It allows the simulation of catchments of determined area with random irregular shape and different stream network organization. [START_REF] Emmanuel | Evaluation de l'apport de la mesure de pluie par radar météorologique pour la modélisation pluie-débit de petits bassins versants[END_REF] showed that the generated stream networks have morphometric properties similar to those of real networks. For a detailed presentation of the stream network model, the reader may refer to [START_REF] Emmanuel | Evaluation de l'apport de la mesure de pluie par radar météorologique pour la modélisation pluie-débit de petits bassins versants[END_REF].

Rainfall simulator

The rainfall simulator, called SAMPO, acronym for Simulation of Advected Mesoscale Precipitations and their Occurrence [START_REF] Leblois | Space-time simulation of intermittent rainfall with prescribed advection field: Adaptation of the turning band method[END_REF], is based on the Turning Band Method which has been adapted to simulate time series of rainfall fields at chosen spatial and temporal resolutions. The final simulated rainfall field is obtained by the product of two independent fields: 1) an indicator field of zero and nonzero rainfall pixels defining the outline of the rainy areas; 2) a field of nonzero rainfall defining the inner rainfall variability. The displacement of the rainfall field is defined by the direction and the velocity of the advection. One main characteristic of SAMPO is that it simulates realistic temporal evolution of rainfall fields of given types. For a detailed presentation of SAMPO, the reader may refer to [START_REF] Leblois | Space-time simulation of intermittent rainfall with prescribed advection field: Adaptation of the turning band method[END_REF].

Hydrological model

The catchment acts as a spatio-temporal filter which smooths the rainfall variability in time and space. The rainfall pattern is transformed on hillslopes into a net rainfall pattern which is aggregated and routed as flow in the stream network. In order to represent in a simple and robust way the most influential processes, the hydrological model is based on the following hypotheses: 1) the production function is applied at the catchment cell scale and reproduces the main types of runoff generation processes, 2) the surface runoff is the dominant process on hillslopes and the travel time on hillslopes is shorter than the travel time in the stream network, 3) the flow characteristics in the stream network are independent of net rainfall. According to these hypotheses, the production function gives a net rainfall pattern at the catchment cell (equivalent to a rainfall pixel) scale which is, in a second step, routed by the transfer function to the catchment outlet. Both functions are fully distributed at the catchment cell scale.

Production function

Rainfall variability influences particularly the fast response of catchments. Therefore, the hydrological model considers several production functions describing this fast response, supposed to correspond to surface runoff on hillslopes. The aim is to consider production functions which are widely used and for which rainfall variability acts in different ways allowing to obtain different patterns of net rainfall over the studied catchments. The following production functions have been considered to obtain different hydrological behavior: -constant runoff coefficient (denoted CRC): a constant runoff coefficient (RC in %) is applied to all the catchment cells; -a contributive area model (CA): the RC% cells of the catchment, displaying the highest topographic index, form the contributive areas. Those cells have a net rainfall of 100%; -US Soil Conservation Service (SCS): the potential maximum retention (S SCS in mm) is computed such that, at the event scale, RC% of the raw rainfall is transformed into net rainfall.

-a Horton model (H): the infiltration capacity (P H in mm/h) is computed such that, at the event scale, RC% of the raw rainfall is transformed into net rainfall. For a given cell and time step, with a raw rainfall P, the net rainfall is equal to 0 if P< P H , otherwise, its value is P -P H .

Transfer function

In most flood routing applications, the Saint-Venant equations can be simplified and lead to the diffusive wave model. This model depends on two parameters: the celerity (C) and the diffusivity (D). Generally, these two parameters are function of the discharge. In the particular case where C and D are assumed to be constant during the rainfall event and in the case of a semi-infinite channel, the diffusive wave model admits an analytical solution: the Hayami model which is a linear model [START_REF] Moussa | Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow[END_REF]. For sake of simplicity, it is assumed that the flow velocities on hillslope and in the stream network are equal. As the hillslope travel time is assumed to be much less than the stream network travel time, this simplification weakly affects the total travel time. Therefore, net rainfall of each catchment cell u is routed to the outlet of the catchment using the Hayami kernel function [START_REF] Moussa | Analytical Hayami solution for the diffusive wave flood routing problem with lateral inflow[END_REF] defined as:

( ) 2 / 3 2 / 1 2 4 exp 2 ) ( t L Ct Ct L D CL D L = t K u u u u u ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - π (1)
with t the time, L the stream path length (in m) from the considered cell u to the catchment outlet, C the celerity (in m/s) and D the diffusivity (in m 2 /s).

The flow at the catchment outlet is given by the expression:

∑ = = - n u 1 u u n 3 ) t ( K * ) t , u ( P 3600 S 10 = ) t ( Q (2)
with Q(t) the flow (in m 3 /s), S the cell area (in m²), t the time, n the number of cells belonging to the catchment, P n (u,t) the net rainfall of the cell u (in mm/h), K u (t) the Kernel function of cell u (in s -1 ) defined in equation 1. * represents the convolution product.

PRESENTATION OF THE CASE STUDY

In this study, the presented simulation chain is used to evaluate for which catchments characteristics and for which rainfall types neglecting rainfall spatial variability has an effect on flow modelling at the catchment outlet. The proposed approach is original. We have made the choice to work on contrasting situations. For this purpose, we have simulated time series of rainfall fields with a temporal and a spatial variability more or less important. Those time series of rainfall fields are used to force catchments of different area, shape and hydrological behavior which are also simulated. This section presents the characteristics of the simulated catchments and of their hydrological behavior as well as the characteristics of the rainfall events. The criteria used to evaluate the effect of ignoring rainfall spatial variability on flow hydrographs are discussed in the last part of this section.

Characteristics of the simulated catchments and of their hydrological behavior

Four areas of 10 km², 30 km², 90 km² and 270 km² have been considered in order to cover the range of areas of urban or peri-urban catchments of fast hydrological response which depends highly on the rainfall input.

The stream network model can simulate catchments of random and irregular shapes [START_REF] Emmanuel | Evaluation de l'apport de la mesure de pluie par radar météorologique pour la modélisation pluie-débit de petits bassins versants[END_REF]. However, in this study we have decided to work with given, regular and schematic shapes (Figure 1). The influence of working with regular and schematic shapes rather than with irregular and random shapes is discussed in the results section. Our objective being to consider contrasting shapes of different elongations, three shapes have been considered (Figure 1): an elongated shape, an oblate shape and a nearly circular shape (named hereafter the intermediate shape). Those three shapes have been defined so that their elongation coefficient is, respectively, less than 1 (equal to 0.6), higher than 1 (equal to 1.2), close to 1 (equal to 0.9). The elongation coefficient is equal to the ratio between the diameter of the circle having the same area as the catchment and the catchment maximum length [START_REF] Schumm | Evolution of drainage systems and slopes in badlands at Perth Amboy, New Jersey[END_REF]. The four production functions have been constrained to restore the same event runoff coefficient (RC), in order to obtain, for each simulated rainfall event, the same runoff volume and thus to make the results comparable. In this case study, RC has been taken equal to 0.3. The transfer being independent of the net rainfall value, the value of RC does not influence the transfer.

The studied catchments are divided in 250 m x 250 m cells (same areas as rainfall pixels, cf. following part) and represent hillslopes. Three values of celerity have been considered: C = 0.5 m/s, C = 1 m/s and C = 2 m/s. As [START_REF] Moussa | Criteria for the choice of flood-routing methods in natural channels[END_REF] indicate that the diffusive wave model is more sensitive to the parameter C than to D, a value of D = 500 m²/s has been adopted. The values of C and D have been chosen in order to cover the range of response time of small catchments up to about 300 km 2 (cf. Table 1).

The following "reference" case has been defined: a catchment of 90 km² area, of intermediate shape with a SCS production function, with C = 1 m/s. The values of area and C have been chosen to correspond to the intermediate value of each parameter range. The SCS production function has been chosen as it is widely used. In order to evaluate the sensitivity to a given parameter, only the values of this specific parameter (area, shape, production function, celerity) are to vary, while the three other parameters remain equal to the reference values. A total of 11 catchments scenarios are obtained (called hereafter by the names contained in Table 1) which characteristics are summarized in Table 1.

Characteristics of the simulated rainfall events

SAMPO has been used to simulate rainfall events with a spatial resolution of 250 m x 250 m over a window of 30 km x 30 km (higher than the spatial extent of all considered catchments). In this case study, a rainfall event corresponds to a time series of rainfall fields of 5 min temporal resolution and of 12h00 duration (corresponding to the highest concentration time of the different catchments, cf. Table 1). During the rainfall-runoff simulations, the catchments centers of gravity are located in the center of the simulation window.

Our goal being to consider contrasted rainfall types more or less variable, six rainfall types (RT1 to RT6) have been considered with RT1 less variable than RT2, which is less variable than RT3, etc. A simulated image of each rainfall field type is shown on Figure 2. The second objective is to consider time series of rainfall fields which last more or less over the catchment. For this, for each rainfall type, 3 advection velocities have been defined: 2.5 (denoted V2.5), 5 (V5) and 10 m/s (V10). Finally, different rainfall fields' directions have been studied by considering three directions of advection: 1) from upstream to downstream, 2) from downstream to upstream and 3) perpendicular to the main flows direction of the catchment and downward, which is equivalent to the direction upward as the considered catchments are symmetrical.

Therefore, in this case study, 54 configurations of rainfall have been defined as well as 11 catchments scenarios that are used to evaluate the effect of neglecting rainfall spatial variability on flow modelling at the catchment outlet.

Criteria used to evaluate the effect of neglecting rainfall spatial variability

The objective is to quantify the effect of neglecting rainfall spatial variability on rainfall-runoff hydrographs. Thus, two spatial rainfall resolutions have been considered: 1) the simulated rainfall spatial resolution (spatial resolution of 250 m x 250 m, called "distributed rainfall"), and 2) spatially uniform rainfall (called "average rainfall") and their associated obtained hydrographs have been compared. Average rainfall is computed, at each time step, by averaging the distributed rainfall of each catchment cell. Thus, at each time step, each catchment cell receives the same amount of raw rainfall. Moreover, the total event raw rainfall over the catchment is the same both for the distributed and the average rainfall.

In order to obtain robust results, 50 rainfall events have been simulated for each of the 54 configurations of rainfall. Thus, for each of the 11 catchments scenarios and for each of the 54 configurations of rainfall, 50 pairs of hydrographs are obtained, each pair regrouping a "distributed hydrograph" obtained from the distributed rainfall and an "average hydrograph" obtained from the average rainfall. The distributed hydrographs are taken as reference. The differences between each pair of hydrographs (distributed and average) give an indication about the errors in runoff response

Q Q Q = RAE - (3)
with Q Dmax the maximum value of the distributed hydrograph and Q Amax the maximum value on the average hydrograph. For each case, 50 values of RAE are obtained. The value that is presented is the quantile 80% (denoted RAE 80 in %). The quantile RAE 80 means that 20% of the studied values are higher than RAE 80 . The highest are the values of RAE 80 , the biggest are the differences between distributed and average hydrographs.

RESULTS

In this paper, results of catchment shape and production functions are not presented but only given in the Conclusion.

Influence of a given, schematic and regular catchment shape and of stream networks organization

The proposed methodology simulates the influence of rainfall variability on three schematic catchments of regular shapes with one stream network organization. This simplification allows us to limit the number of simulations. It is based on the assumption that all the catchments of a given area with the same elongation behave similarly regarding rainfall variability. The validity of the proposed methodology has been confirmed in two steps: 1) 10 catchments of 90 km² and of elongation coefficient equals to the one of the reference catchment (i.e. equals to 0.9) have been simulated (an example is given on Figure 3). Those 10 catchments have been submitted to the 50 rainfall events of each of the six rainfall types (with V5 and an upstream-downstream direction). The RAE 80 have been computed on the obtained pairs of hydrographs. It appears that RAE 80 computed for the irregular shape catchments are very close to those of the reference catchment (Figure 3). Therefore, the schematic and regular chosen shapes can be considered, in the framework of this study, as representative of irregular shapes of same elongation coefficient. 2) 100 reference catchments of different stream networks organization have been submitted to 50 events of RT3 type (with V5 and an upstream-downstream direction). The obtained RAE 80 are very similar: they range between 11.4% and 12.8%. As the precise organization of the stream network has only a small influence in our study, we have considered only one stream network organization for each catchment scenario.

General remarks on the RAE distributions

In addition to the RAE 80 , the mean and the standard deviation associated to the RAE distribution have been computed for the 11 catchments scenarios with a rainfall type RT5, a velocity V2.5 and a perpendicular direction (Figure 4). This rainfall configuration engenders, for all catchments scenarios, the highest differences between distributed and average hydrographs. Figure 4 gives, for each catchment scenario, an indication on the highest observed differences between distributed and average hydrographs. Those differences are the highest for the scenario El (cf. Table 1), for which the mean of RAE equals 34% and RAE 80 equals to 45%. This means that for the catchment with an elongated shape forced by a rainfall type RT5, moving at V2.5 in the perpendicular direction of the catchment, the maximum value of the 50 detailed hydrographs and the corresponding value on the average hydrographs differ with a mean of 34%. In addition, 20% of the studied pairs present an RAE higher than 45%. Moreover, the computed standard deviations are high. The variation coefficients (ratio between the standard deviation and the average) range, in this example, between 100% for the scenario A10 and 43% for the scenario El. These results show that there is a high dispersion of the RAE around their mean. The dispersion inside a scenario is higher than the dispersion between the different scenarios. This observation is verified for all the rainfall types' configurations. It clearly confirms that it is difficult to obtain reliable general conclusions, on the influence of neglecting rainfall spatial variability, from only the study of a few rainfall events. It also contributes to explain why the study of a limited number of events can reach to different conclusions.

Influence of the rainfall type (spatial extent of the rainfall fields) evaluated on the reference catchment

The influence of the rainfall type is illustrated here, for the reference scenario, with V5 and an upstream-downstream direction. As the advection velocity is the same for all rainfall types, the evaluation of the influence of rainfall type is equivalent to the evaluation of the influence of the spatial extent of rainfall fields. RAE 80 is close for RT1, RT2 and RT3 (equals to 10 -12%) and increases for RT4 (21%). It then remains constant for RT5 (21%) and decreases for RT6 (11%) (Figure 5). Those evolutions can be explained in the following way. For the less variable rainfall fields (RT1 and RT2), rainfall is spatially uniform over the catchment (Figure 5). In this case, the differences between distributed and average hydrographs are small. When the spatial variability of rainfall increases (RT4 and RT5), rainfall fields generate contrast over the catchment which results in an increase of RAE 80 (an example is given on Figure 5, where it rains only on a part of the catchment). Finally, for the most variable rainfall fields (RT6), the high variability (Figure 5) appears significantly filtered by the catchment and thus an average knowledge of the rainfall spatial variability is not very different from a distributed knowledge.

Influence of the rainfall direction evaluated on the reference catchment

The influence of rainfall fields' direction is illustrated for the reference catchment scenario with V5. RAE 80 are the highest for the perpendicular direction. RAE 80 reaches a value of 28% for RT4 (Figure 6). Moreover, RAE 80 is smaller than 12% for a downstream-upstream direction.

For the downstream-upstream or upstream-downstream directions, from a time step to another, the rainfall field will pass over different isochrones lines and the spatial variations of rainfall will be filtered by the catchment. For the perpendicular direction, the same rainfall field will stay on the same isochrones line during several time steps. This persistence seems to engender more contrast between a detailed and an average knowledge of rainfall.

We can also notice that the observations made in the previous section concerning the influence of the rainfall type remain verified for the three considered directions. Whatever the rainfall type, the lowest is the ratio V/C; the highest are the differences between the distributed and the average hydrographs. For example, for RT4, RAE 80 equals to 33% for a ratio of 1.25 against 5% for a ratio of 20. Moreover, when the ratio is high (equals to 10 and 20); RAE 80 remains small whatever the rainfall type (equals at a maximum to 12% for a ratio of 10 and to 5% for a ratio of 20). For a ratio of 20, the rainfall type does not have any more influence. This could be explained by the fact that for high values of rainfall field velocity or for small values of catchment celerity, the catchment filters the spatial variations of the successive rainfall fields. Therefore, an average knowledge of rainfall becomes close to a distributed one.

Those observations are made for rainfall fields with an upstream-downstream direction. However, they remain valid whatever the rainfall field direction. The only difference is that, as observed in the previous part, RAE 80 have higher values for a perpendicular direction and smaller values for a downstream-upstream direction.

Influence of the catchment area

Figure 8 presents the evolution of RAE 80 for the four considered scenario A270, A90 (Ref), A30 and A10 with a ratio V/C equals to 5 (V equals to 5 m/s and C to 1 m/s) and with an upstreamdownstream rainfall field direction.

Values of RAE 80 increase with the catchment area. For example, for RT4, RAE 80 equals to 26% for the 270 km² catchment against 10% for the 10 km² catchment. Thus, for the considered rainfall fields, the biggest is the catchment; the highest are the differences between average and distributed hydrographs. Figure 8 illustrates this aspect through an example. The spatial structure of the rainfall field makes it weakly variable over the catchments of 10 km² and 30 km². Some significant spatial contrasts appear over the 90 km² catchment and become important for the 270 km² catchment, for which the differences between distributed and average hydrographs are the highest. We can note that those tendencies are verified for all combinations of rainfall fields' directions and values of V/C, with however different values of RAE 80 .

CONCLUSION

The objective of this paper has been to quantify the effects of neglecting rainfall spatial variability on the hydrological modelling of catchments of about ten to several hundred km². A simulation approach has been undertaken. The choice has been made not to be exhaustive but to study contrasted situations. Therefore, different configurations of rainfall fields have been considered by simulating time series of rainfall fields with different spatial extents, different velocities and different directions. Those time series of rainfall fields have been associated with different scenarios of catchments, each scenario having a catchment with determined area, shape and hydrological behavior. It appeared that the effects of neglecting rainfall variability were particularly important if: 1) the dimensionless ratio "velocity of rainfall field on celerity along flow path" is small, 2) the rainfall field direction is perpendicular to the catchment flow direction and also if 3) the catchment presents an elongated shape and 4) a Horton or SCS production function. Both last results are not presented in this paper. Results also showed that it is difficult to obtain general conclusions by studying only a few events which contribute to explain the different conclusions reached in the literature. At last, in order to verify that simulation results are close to reality, all results will have to be confronted to those obtained on real catchments with some real rainfall measurements.

Figure 1 :

 1 Figure 1: 90 km² simulated catchments with an elongated (left), oblate (middle) and intermediate shape (right). For the figure, the stream networks are represented with a drainage area threshold of 0.5 km². The main flow direction is indicated by the arrow.

Figure 2 :

 2 Figure 2: Example, for each rainfall type, of a simulated rainfall field over the simulation window (30 km x 30 km) at a temporal resolution of 5 min and at a spatial resolution of 250 m x 250 m.

  neglecting rainfall spatial variability. The differences between each pair of hydrographs are assessed by the relative absolute error (RAE):

Figure 3 :

 3 Figure 3: Example of a catchment of irregular shape of 90 km² and of elongation coefficient equals to 0.9. Evolution of RAE 80 (in %), computed for the reference scenario, in function of the rainfall type for the catchment of schematic and regular shape (circle in bold) and for the 10 catchments of irregular shapes (points).

Figure 4 :

 4 Figure 4: Evolution of the mean (circle) more or less the standard deviation (lines) (up). Evolution of RAE 80 (points) (down). The statistical scores are computed for each catchment scenarios for RT5 with V2.5 and a perpendicular rainfall field direction.

Figure 5 :

 5 Figure 5: Evolution of RAE 80 (in %), computed for the reference scenario, in function of the rainfall type (RT1 to RT6). Rainfall fields have a velocity V5 and an upstream-downstream direction. Example of a simulated image of rainfall field RT1, RT4 and RT6 over the simulation window (the color bar is the same as the one of Figure 2). The 90 km² catchment of intermediate shape is represented.

Figure 6 :

 6 Figure 6: Evolution of RAE 80 (in %), for the reference scenario, in function of the rainfall type for the three considered rainfall fields' directions: downstream-upstream (plus), upstream-downstream (circle) and perpendicular (points).

  catchment response time -rainfall field velocityThe analysis has been performed for the 90 km² catchment, of intermediate shape, with a SCS production function and with different values of ratio of the rainfall field velocity (V) by the catchment celerity (C) of 2.5, 5 and 10. Those three values of ratio can be obtained for different combinations of values of V and C. We have noticed that for the same value of the ratio V/C, the results are very similar. Thus, those results give rise to a temporal invariant: V/C. Figure 7 presents the RAE 80 evolution for different values of V/C. The different simulations enable to obtain five values of ratio equal to 1.25, 2.5, 5, 10 and 20. The values of 1.25 and 20 are obtained only with one combination of V and C.

Figure 7 :

 7 Figure 7: Evolution of RAE 80 (in %), for the 90 km² catchment with an intermediate shape and a SCS production function, for a ratio V/C equals to 1.25 (plus), 2.5 (points), 5 (circles), 10 (stars) and 20 (crosses). The rainfall fields' direction is upstream-downstream.

Figure 8 :

 8 Figure 8: Evolution of RAE 80 (in %) in function of the rainfall type for the scenarios: A270 (points), Ref (circles), A30 (plus) and A10 (stars) with V5 and an upstream-downstream direction. Catchments of 270 km², 90 km², 30 km² and 10 km² are represented over a RT2 rainfall field image.

Table 1 :

 1 Characteristics and name of the different considered catchments. Lmax corresponds to the length of the longest hydrological path. Values relative to the reference catchment are in bold.

			Fixed parameters			Obtained characteristics
	Name	Area (km²)	Shape	Production function	C (m/s)	Lmax (km)	Response time (h)	Concentration time (h)
	Reference	90	Intermediate	SCS	1	12.25	2h00	5h45
	A10	10	Intermediate	SCS	1	3.75	0.5h	2h40
	A30	30	Intermediate	SCS	1	7	1h00	3h50
	A270	270	Intermediate	SCS	1	21.25	4h00	8h50
	El	90	Elongated	SCS	1	18.25	3h25	7h45
	Ob	90	Oblate	SCS	1	9	1h45	4h50
	CRC	90	Intermediate	CRC	1	12.25	2h00	5h45
	CA	90	Intermediate	CA	1	12.25	2h00	5h45
	Horton	90	Intermediate	Horton	1	12.25	2h00	5h45
	C05	90	Intermediate	SCS	0.5	12.25	3h25	12h00
	C2	90	Intermediate	SCS	2	12.25	1h10	2h30


Mathieu Mangeot-Lerebours

Gilles Sérasset
email: gilles.serasset@imag.fr

Papillon Lexical Database Project: Monolingual Dictionaries & Interlingual Links

This paper presents a new research and development project called Papillon. It started as a French-Japanese cooperation between laboratories GETA/CLIPS (Grenoble, France) and NII (Tokyo, Japan). Its goal is to build a multilingual lexical database and to extract from it digital bilingual dictionaries. The database is built with monolingual dictionaries, one for each language of the database, linked to an interlingual dictionary. The pivot architecture of the database is based on Gilles Sérasset's Ph.D. thesis. The structure of the monolingual dictionaries is based on the lexical work done by Igor Mel'cuk and Alain Polguère. From the lexical database, it is planned to derive user customized bilingual dictionaries in multiple target formats. It will be possible to generate human usage dictionaries as well as specialized dictionaries for machine translation software. These dictionaries will be available under the terms of an open source license. This project, initiated by some computational linguists, aims at being useful and open to all those who are interested in Japanese and French. It is also opened to any other language. Moreover, the pivot architecture of the database will facilitate the addition of new languages and save translation efforts.

Introduction

There are few French-Japanese usage dictionaries, which are really usable and useful for French speakers. The main problem is that the original Japanese script and the rômaji phonetic transcription are present together only in very small dictionaries. Also, dictionaries never contain numeric specifiers, which are as important in Japanese as gender and number in French. On the other hand, the information available in paper dictionaries does not exist in machine-readable forms, or is not accessible on line. The lack of bilingual resources is also an obstacle to develop linguistic software applications, for which adapted dictionaries are a need. As an example, Nippon Telegraph and Telephone in Japan or Lexiquest in France have to develop their own dictionaries in a separate and time-consuming effort. In the academic world, this implies that applications that have been created for French and Japanese offer only a reduced scope, while good English-Japanese pieces of software are available. Nevertheless, it is a true fact that Japan is very interested in the French language. Conversely, a growing number of French individuals invest much energy to learn Japanese. There is a vacuum to be filled. The leveraging of communication that Internet offers allows one to think that a convenient digital dictionary could be produced by a general cooperation between linguists, translators, computer scientists, etc., working together through Internet. A similar project between English and Japanese has been active for about a decade. This project has allowed the effective building of a free Japanese-English dictionary, available through an Internet server. This Edict project has been created and supported by Pr. Jim Breen from Monash University, Australia (see bookmark 2). The current JMDict dictionary comprises now 70,000 entries of common vocabulary, a specific kanji dictionary, and around twenty specialized dictionaries (biology, law, etc). A different project, fed by volunteers, is supported by NEC Corporation. Its aim is to increase the dictionaries used by the NEC translation tool (see bookmark 3), and to bring in new entries on a constant way. We should also mention the SAIKAM project [1], (see bookmark 5) cooperation between NII (Tokyo, Japan) and NECTEC (Bangkok, Thailand) active since about 5 years, where Thai students working or having worked in Japan have built a sizable Japanese-Thai online dictionary through Internet. In such a context, the GETA/CLIPS laboratory (Grenoble, France) and the National Institute of Informatics (Tokyo, Japan) started a research and development project in order to plan and implement a French-English-Japanese lexical database. Here are described the architecture of the database, the structure of the entries and the methodology adopted for the project.

General View of the Database

The lexical database is built on the one hand by integrating existing resources and on the other hand by writing and correcting new entries (see Figure 1).

Once the database is homogeneous, users will be able to extract their own customized dictionaries dynamically from the database and to interact with them.

Internal Architecture of the Database

The database will be built using a pivot architecture based on Dr. G. Sérasset's Ph.D.

thesis [START_REF] Gilles | Interlingual Lexical Organisation for Multilingual Lexical Databases in NADIA[END_REF] and experimented by Dr. E. Blanc in PARAX (Blanc 1999). The monolingual dictionaries will be linked only through a pivot dictionary of interlingual links called acceptions. These acceptions will also be linked together by refinement links. They may also be translated into the UNL language (UNL 1996), (see bookmark 6).

Each sense or meaning of each entry of a monolingual dictionary is linked to one or more acceptions of the pivot dictionary. For example, in French "! carte! " has two meanings: "! carte à jouer (card)! " and "! carte géographique (map)! ". The entry "! carte! " will consequently be linked to two "lexies" (corresponding to 2 word senses) in the French monolingual dictionary, which in turn will be linked to 2 acceptions in the pivot dictionary: in the example, the first has number 343, with the corresponding UNL "UW" (universal word) "card(icl>play)", and the second one has number 345, with UW "map(fld>geography)" (see Figure 2).

Structure of the monolingual dictionaries

The structure of the entries or microstructure of the monolingual dictionaries is based on the structure used for the formal lexical database DiCo [START_REF] Polguère | La théorie Sens-Texte[END_REF] The dictionaries will be encoded in XML to facilitate readability and conversion into various target formats.

Building methodology

The building methodology of the lexical database builds on one hand on the reuse of existing data, the French-English-Malay dictionary [START_REF] Yvan | Kamus Perancis-Melayu Dewan, dictionnaire français-malais[END_REF], (see bookmark 1) and the Japanese-English dictionary of Jim Breen (see bookmark 2), and on the other hand on the contribution of volunteers working through the Internet. Different steps are planned: The first step is the integration of existing resources. It consists in preparing a "lexical soup" by merging the two dictionaries thanks to the presence of English. This merging operation will produce correct as well as incorrect acceptions (interlingual links). These wrong acceptions will be corrected or deleted by lexicologists. Then the voluntary contributors will index new entries and the lexicologists will correct and • specialists of one language will write the monolingual entries; • people with good knowledge of French and Japanese like translators will work on the links between the monolingual entries and the acceptions; • people with good knowledge of UNL will translate the acceptions into UNL (UNL 1996).

Encourage voluntary contributions

One of the main idea in Papillon project is to encourage voluntary contribution. These contributions will be accepted via a "community web site" where any user should be able to! :

• Communicate and discuss about the available material • Consult dictionaries • Correct dictionaries • Contribute

-By giving personal dictionaries -By providing entries to other dictionaries A mockup of this web site is currently developed in Java that integrates web services with XML processing. About 300 detailed French and some Japanese entries are currently integrated in the database (in XML form). The web side gives access to these entries by providing the users with a dynamically generated form obtained via XSL transformations. Different XSL transformations are provided that give access to different views. Up to now, we only develop a complete view inspired by the Explanatory and Combinatory Dictionary (ECD) developed by Igor Mel'cuk (see Figure 3).

Dictionaries produced

Several monolingual or bilingual dictionaries can then be extracted from the database. Different types are needed: for human use, via database and plug in functionalities or via usual dictionary formats, and for machine use.

For human use, via database and plug in functionalities

Persons that interact in foreigner languages often can access computers. One of the aims of this dictionary is then to provide them with a direct help, within their editor, browser, or their daily used personal digital assistant.

For human use, via usual dictionary formats

We plan to automatically derive from the database digital presentations for web consultation and paper edition. The FeM (see bookmark 1) and JMDict (see bookmark 2) formats are the first targeted formats.

For machine use

The terminology resources available for building lingware (linguistic software) are almost null between Japanese and French. The rare available ones have to be radically restructured and augmented. The orientation of the Papillon lexical database towards possible use by machines will encourage the realization of lingware including both languages, by providing a first support for such projects.

Conclusion

The pivot architecture allows an easy integration of new languages because the reuse of existing links will save a lot of time consuming efforts. The Thai language is already about to integrate the project through a cooperation with Kasetsart University (KU/Thailand), and National Electronics and Computer Technology Center (NECTEC/Thailand).

The open source license makes all the data available to anyone. Furthermore, we will be able to generate multiple formats from the lexical database. Finally, it should be stressed that such an endeavor will not only need the dedication of as many volunteer contributors as possible, but some stable support, in the form of a server and, more difficult, of a central team of experts charged of "refining the raw ore" of individual contributions.

That team does not have to be in a single place, but convenient groupware tools should be developed for it.

Figure 2 .

 2 Figure 2. Lexical architecture of the papillon database

Figure 3 .

 3 Figure 3. Web page dynamically generated for French entry "meurtre" (murder)


R Falconi

A Durán

M Núñez-Regueiro

R Escudero

Pressure effects in PrT 2 B 2 C (T = Co, Ni, Pt): Applied and chemical pressure

Keywords:

High-pressure electrical resistivity ρ ab measurements on intermetallic Pr(Co, Ni, Pt)2B2C compounds were performed down to 2K. At room pressure the ρ ab (T ) curves for the non superconducting Pr(Co, Ni)2B2C compounds exhibit magnetic correlations at about 10 and 4 K, respectively. At low temperatures, PrCo2B2C shows a large spin-dependent electron scattering in comparison to PrNi2B2C. Under applied pressure the magnetic scattering tends to be suppressed more effectively in PrCo2B2C than in PrNi2B2C. The low temperature behavior of ρ ab (T, P ) for PrNi2B2C and PrCo2B2C suggests a spin fluctuations mechanism. In the other hand PrPt2B2C compound shows superconductivity at about 6 K and under pressure its superconducting transition temperature tends to be degraded at a rate dTC /dP = -0.34 K/GPa, as expected in compounds with transition metals. The experimental results in Co, Ni and Pt based compounds are analyzed from the point of view of the external and chemical internal pressure effects.

I. INTRODUCTION

The phenomenon of superconductivity appearing in compounds having magnetic elements, has received noticeable attention during the last three decades due to the great variety of exotic electronic and magnetic correlations it involves. Particularly interesting for this topic, has been the discovery of the quaternary intermetallic compounds; RNi 2 B 2 C, (R = rare earths, Y, Sc, Th) [1,2] where coexistence of antiferromagnetism and superconductivity has been observed as for example in R = (Tm, Er, Ho, Dy, Lu) [2][3][4][5][6][7]. As far as we know several RT 2 B 2 C intermetallics compounds, with different rare earths (R) and transition metal (T) combinations, have been synthesized [5,[7][8][9][10][11][12][13][14] and most of them, such as HoNi 2 B 2 C [15], show superconductivity in spite of the presence of the rare earth magnetic element.

The influence of the transition metal magnetism in the magnetic properties of these compounds, seem to be of minor importance compared to that of the rare earth ions, whose magnetic moments apparently impose the magnetic ordering at all. Thus in some borocarbides with 3d transition elements such as Ni and Co, neutrondiffraction measurements [16], and electronic transport measurements [11,17], have revealed that no significant magnetic moment develops in the T sites. Local structure studies at the Ni site, using Mossbauer spectroscopy on 57 Fe doped (1 at %) samples, also support this fact [18]. Interestingly, the T elements play an indirect role in the magnetism of magnetic RT 2 B 2 C systems through the spatial dependent indirect RKKY exchange interac-tions [6,[START_REF] Gratz | Handbook on the Physics and Chemistry of Rare Earths[END_REF], that govern the magnetic ordering in these compounds. On the other hand, the electronic influence of the T elements in the superconductivity of RT 2 B 2 C is more relevant than that for the R , B and C elements. This is particularly true for the Ni based borocarbide superconductors, where the density of state at the Fermi level is mainly due to the Ni 3d bands [START_REF] Loureiro | [END_REF]. On the contrary, in a comparative study of the structure and superconducting properties of RNi 2 B 2 C, Loureiro et al. [START_REF] Loureiro | [END_REF], showed that the superconducting state is more strongly affected by the magnetism of the R ion than for the Rion size, at least for R between Dy and Tm. However, the role of the magnetism and ion-size of T elements in the superconductivity of RT 2 B 2 C when magnetic R ions are presents, is not clear yet. In this work we aboard the study of a particularly interesting case: the PrT 2 B 2 C compounds, with T = Ni, Co, and Pt which have revealed many peculiarities. PrNi 2 B 2 C, and PrCo 2 B 2 C do not superconduct as measured down to 0.3 K [21], however, PrPt 2 B 2 C does superconduct at 6 K, even in the magnetic Pr +3 ion presence [22,23]. Noticeably, PrPt 2 B 2 C does not show any magnetic ordering at low temperatures [23] but in contrast, Pr-Ni and Pr-Co based borocarbides, develop antiferromagnetic ordering at about 4 and 8.5 K, respectively [24,25]. Recently, magnetoresistance and specific heat studies in Pr(Co, Pt) 2 B 2 C [25,26] have pointed out that spin fluctuation mechanism is involved in the electronic behavior of these two compounds. However, although evidence for spin fluctuations can be deduced from certain features in the electronic transport measurements, the interpretation of those proper-ties could be not so clear. High pressure experiments in spin fluctuators such as RCo 2 [27], CeNi 5 [28], and UPt 3 [29] have proven to be a useful tool in order to make clear if a spin fluctuation mechanism is occurring in such systems. The aim of this paper is to enlighten the influence of the chemical and external applied pressure on the superconducting state and magnetic scattering at low temperature for the three Pr-based borocarbides: Pr(Ni, Co, Pt) 2 B 2 C. We analyzed the changes of the resistivity as a function of pressure and temperature. We assume that interactions between itinerant electrons plays an important role in the low temperature resistivity characteristic, and those can be modified by applied external or internal chemical pressure.

II. EXPERIMENTAL DETAILS

Three compounds were prepared: samples of PrCo 2 B 2 C, PrNi 2 B 2 C, and PrPt 2 B 2 C. The single crystals were grown by cold copper crucible method as described by Durán et al [25]. All samples were characterized by X-ray diffraction using a Bruker P4 diffractometer, with monochromatized Mo-Kα radiation. The cell parameters were: a = 3.6156 Å, and c = 10.3507 Å for PrCo 2 B 2 C, a = 3.6996 Å, and c = 9.9885 Å for PrNi 2 B 2 C, and a = 3.8373 Å, c = 10.761 Å for PrPt 2 B 2 C samples. Resistivity measurements in the a-b plane were performed by the four-probe technique using gold wires of 10 µm diameter as electrical contacts. Pressure experiments were performed by using a micro-cryogenic diamond anvil cell MCDAC (cell piston-cylinder type Be-Cu cell) consisting of two diamonds, each of 0.5 mm culet size. A Cu-Be gasket was preindented and a 150 µm diameter hole was drilled at the center. The samples used have dimensions of about 80 × 15 × 40 µm 3 and were placed in the gasket hole. The transmitting pressure medium was MgO powder. The metallic gasket was electrically insulated, pressing over it Al 2 O 3 powder of 1 µm grain size. As the MCDAC pressure increases, the wires used to measure the electrical resistance may be cut off at the edge of the diamonds because of the diamond indentation. To reduce this problem, we used a thin aluminum foil placed under the four gold wires, with this set up frequently we reached high quasi-hydrostatic pressures in the range about 6 GPa. Also, in order to prevent motion of the sample and of the electrical leads, at the initial compressing, a thin mylar film was placed over them. Additional pressure experiments in polycrystalline Pr(Ni, Pt) 2 B 2 C compounds up to about 21 GPa were made using a sintered diamond Bridgman anvil apparatus, with a pyrophyllite gasket and two steatite disks as the pressure medium. For determination of the pressures a Pb manometer were used.

III. RESULTS AND DISCUSSION

It is known that the RNi 2 B 2 C compounds crystallize in the tetragonal body-centered structure (space group I4 /mmm) and when the rare earth atom radii is increased (R goes from La to Lu), the c-parameter becomes larger whereas the a-parameter decreases [16,30]. This structural behavior can be accounted by for the rigidity of the B-C and Ni-B bonds and the variable tetrahedral angle in the NiB 4 unit. Distortions of this tetrahedral unit are claimed to be a decisive parameter for Tc in non magnetic or antiferromagnetic RNi 2 B 2 C and RNiBC compounds [31]. In the case when the size of the transition element is increased, maintaining the same rare earth element, the structural behavior of the unit cell seems to be slightly different. Fig. 1 displays the T-T shortest length between 3d ions, particularly in the Pr(Ni, Co, Pt) 2 B 2 C compounds and that for the metal T. Also it shows the behavior of the a and c parameters for each compound. We note that increasing the ionic radii T-size causes an increasing of the c-parameter and an anomaly behavior of the a-parameter for the PrCo 2 B 2 C compound. At first glance and according to the figure, this anomaly is correlated with the variation of the T-T shortest bond in the framework of the PrT 2 B 2 C structure, and not with the ionic radii size of the T element.

Fig. 2 shows the normalized electrical resistivity in and it has been associated with a decrease of the magnetic scattering of the conduction electrons by rare earth ions ( [32] and reference therein). However, according to the results present below, it is possible that other mechanism involving conduction electrons could also develop at low temperature. The case for PrPt 2 B 2 C is quite different, after following an upward curvature it becomes superconducting at about 6 K. Magnetic and heat capacity measurements in this compound [23], have revealed a nonmagnetic ground state for Pr ions due to CEF effects, which is claimed to be the reason for superconductivity. The inset of the Fig. 2 shows the resistivity behavior from 60 to 2 K for the three compounds at room pressure. At first glance, increasing the transition metal radius corresponds to a major resistivity droop at low temperature. The ρ ab (T ) curve for PrCo 2 B 2 C from about 2 to 8 K shows a clear T 2 -law dependence with a cuadratic coefficient A, equal to 0.08 µΩcm/K 2 , see Fig. 3. This low-temperature resistivity behavior is similar to the observed in heavy fermion systems, as for example; YbNi 2 B 2 C and UPt 3 compounds [33,34], and could be attributed to spin fluctuations [26,[START_REF] Moriya | Spin Fluctuations in Itinerant Electron Magnetism[END_REF]. Thus, in a similar compound but simpler, RCo 2 , a T 2 dependence has been found at low temperatures, which is due to spin fluctuating characteristics [27]. The fact that the magnitude of the cuadratic coefficient A of ρ(T ) for PrCo 2 B 2 C is of the order of that for RCo 2 ([36], found this coefficient as big as three orders of magnitude but in polycrystalline PrCo 2 B 2 C) suggests that spin fluctuating could be the responsible mechanism for the low temperature ρ(T ) behavior in this compound. Using the universal relation for heavy fermion compounds; A/γ 2 = 1.0x10 -5 µΩ cm(moleK) 2 /mJ 2 [37] the resulted Sommerfeld coefficient is γ = 89.4 mJ/mol-K 2 which is a low value compared with that for PrNi for the Pr-Ni based compound does not change as the pressure increases up to 5.3 GPa. The linear behavior of ρ ab (T ) (extending from about 100 K to room temperature) is attributed to electron-phonon scattering and under the applied pressures, it shows a slope decreasing from 0.089 µΩcm/K to 0.069 µΩcm/K. According to the inset of Fig. 4, the smooth drop of ρ ab (T ) at low temperature, which has been related to the decrease of magnetic scattering [32], is reduced with applied pressures increasing up to 5.3 GPa. In the case of PrCo 2 B 2 C, as can be seen in panel b of Fig. 5, the pressure effects are more stronger than in PrNi 2 B 2 C, mainly in the low temperature regime. From room pressure to about 1.7 GPa the high temperature behavior of ρ ab (T ), from 300 K to about 75 K, remains without appreciable changes and with almost a constant slope of 0.30 µΩcm/K. In an opposite way to PrNi 2 B 2 C, the low temperature curvature of ρ ab (T ), which is also associated with magnetic correlations, tends strongly to be suppressed by pressure. This tendency has also been observed by Massalami et. al. [START_REF] Massalami | [END_REF] applying pressures up to about 1.2 GPa, the maximum pressure value they applied. Similar to their results, we also observed that the cuadratic behavior of ρ ab (T ) at low temperatures is maintained under 1.2 GPa. However, we found, applying pressures higher than 1.7 GPa on this compound, a distinctive characteristic, namely the change of the low temperature curvature from concave to convex and the complete disappearance of the resistivity drop above 2.9 GPa (see panel b in Fig. 5). Interestingly, at this pressure the T 2 behavior disappeared and instead there is an appearing of a type plateau zone in ρ ab (T ) which start about 15 K and extend down to 1.8 K, the lowest temperature available in our experiments. Increasing the pressure up to 4.4 GPa, this zone of constant resistivity is extended from 1.8 K up to 20 K. At this pressure, the overall high temperature behavior of ρ ab (T ) remains almost with the same slope of about 0.30 µΩcm/K.

The above experimental facts reveal that the electronic properties of PrCo 2 B 2 C are more pressure sensitive than that for PrNi 2 B 2 C, mainly at the low temperature regime. Interestingly, we note that the shape of ρ ab (T ) for PrCo 2 B 2 C under pressure tends to be qualitatively similar to that for nonmagnetic YCo 2 B 2 C at room pressure [11]. Thus, it seems the effect of pressure in this compound is to suppress the magnetic correlations which originate the low temperature scattering behavior.

According to Fig 5, the resistivity can be fitted to a T 2 behavior only for a region of relatively low temperatures, interestingly under pressure the fitting range is extended. As already we pointed out, at atmospheric pressure the fitting goes from 2 up to about 8 K with a cuadratic coefficient A, equal to 0.08 µΩcm/K 2 . At 1.7 GPa the interval extends from 2 to about 19.5 K, with a value of A decreased to 0.018 µΩcm/K 2 . Once the applied pressure reaches the value of 2.9 GPa, it was not more possible to fit a cuadratic function to the curvature of ρ ab (T ) because the like plateau zone, also presented for the curves at 3.7 and 4.4 GPa. This ρ ab (T ) behavior at low temperature and pressures is accounted by the spin fluctuation scenario [START_REF] Rossiter | The Electrical Resistivity of Metals and Alloys[END_REF], which also take in account the decreasing of the A parameter with the applied pressure. Additionally, relatively low magnetic fields decreases the A parameter in a linear form in PrCo 2 B 2 C [26], which has been claimed to be due to quenching of the spin fluctuation. Similar results have been found in other systems as for example, Ce 0.8 (Pr, Nd)Ni 5 , indicating that spin fluctuations tends to be suppressed both by pressure and magnetic fields [START_REF] Reiffers | [END_REF]41]. The microscopic character of the state resulting from applying a magnetic field is completely different from that obtained with applying pressure; the first remains magnetic, whereas the last tends to be a real non-magnetic state; one where no microscopic magnetic moments exists. This is important because in the spin fluctuations model the state above T sf (where the spin fluctuation appears and which coincide with the temperature below which a T 2 law in resistivity is valid) is non the non magnetic state (like in the stoner model) but a magnetic state, where local moments still exist, but long range order tend to be destroyed by the fluctuations.

The collective modes described by the spin fluctuations can readily be excited at relatively low temperature, where the stoner excitations are still very small but we assume they can be suppressed by two factors: intense magnetic field and/or external applied pressure. Pressure increases the correlation there exist between f ions and promotes the itinerance of f electrons. As a result the fdensity of state near the Fermi level is lowered modifying the electron structure and influencing thus the prevailing long range order between the band electrons.

In order to know if the ρ(T ) of PrNi 2 B 2 C could follow a similar behavior to that for its isomorphs PrCo 2 B 2 C at higher pressures, we carried out measurements for polycrystalline sample at several pressures up to 21.5 GPa. Interestingly, we found there is a marked tendency of ρ(T ) at low temperature to behave similar to that for PrCo 2 B 2 C (see Fig. 6). At about 13.5 GPa there is a change from negative to positive curvature of ρ(T ) at temperatures lower than about 15 K. This curvature change was also found in Pr-Co system but at about 1.7 GPa. It is important to mention that from the ρ ab (T = 260K, P ) curves for PrNi 2 B 2 C (see inset of Fig. 6), and PrCo 2 B 2 C (not shown), we discarded some structural phase changes that could be related to these effects. The fact that Pr-Ni system requires more pressure to behaves almost in the same way to Pr-Co system at low temperature, could be related to changes in the c-parameter. The main difference in the unit cell of these two compounds stands from this parameter, which is biggest in Pr-Co system and related to the modifiable tetrahedral B-T-B angle. On the other hand, it is known that for the spin fluctuator YMn 2 , the existence of a magnetic moment on Mn sites depends largely on the interatomic Mn-Mn distance [42]. Above a critical distance there exists a magnetic moment. Such distance plays a key-role in determining the magnetic properties and is sensitive to external or internal perturbations. The case for Pt in PrPt 2 B 2 C could be similar. As we already point out this compound shows an upward curvature in ρ(T ) at high temperature which has been related to crystalline electric field (CEF) effects [25]. We make resistivity measurements for polycrystalline PrPt 2 B 2 C under several pressures up to 21.5GPa (see fig. 7). As it can be observed, the negative curvature of ρ(T ) at high temperature is not appreciably modified under pressure and the main changes are at low temperatures. The superconducting transition temperature, Tc, decreases at the rate dTc/dP = -0.34 K/ GPa (see Fig. 8). It seems there is not correlation with the decreasing of Tc and the unmodified curvature related to CEF effects. A positive magnetoresistance at low temperature in this compound has been associated with spin fluctuation [25], however although this picture follows the same trends of Pr(Ni,Co) 2 B 2 C, further investigations are inquired in order to clarify this matter.

IV. CONCLUSIONS

High

FIG. 1 :

 1 FIG. 1: (Color online) Variation of the T-T shortest length between 3d ions for the transition metal T and PrT2B2C compounds, with T = (Co, Ni, Pt). It also shows the behavior of the cell parameters as a function of the T-size.

FIG. 3 :

 3 FIG.3:(Color online) Linear fit to ρ(T 2 ) from 2 to 8 K for PrCo2B2C. The low temperature behavior of ρ(T) follows a T 2 low with a cuadratic coefficient A = 0.08 µΩcm/K 2 . Inset shows the ρ(T ) behavior at low temperatures.

  Figs.4 and 5shown the ρ ab (T ) curves as a function of pressure for PrNi 2 B 2 C and PrCo 2 B 2 C single crystals, respectively. As we can see, these compounds reveal different pressure behaviors. The overall trend of ρ ab (T )

FIG. 5 :

 5 FIG. 5: (Color online) In a) it is presented ρ ab (T ) for PrCo2B2C single crystal under several pressures up to 4.4 GPa. Panel b) shows a view of ρ ab (T ) at low temperatures and for different pressure values.

FIG. 6 :

 6 FIG. 6: (Color online) Normalized R(T)/R(40 K) curves at several pressures up to 2.5 GPa for polycrystalline PrNi2B2C. Insert shows the behavior of ρ(P ) at 260 K.

FIG. 7 :

 7 FIG. 7: (Color online) R(T ) curves for polycrystalline PrPt2B2C, measured up to 21.5 GPa. High pressure tends to destruct superconductivity. Insert shows the low temperature behavior of R(T ).

FIG. 8 :

 8 FIG. 8: (Color online) Decreasing of the superconducting transition temperature for PrPt2B2C, as function of pressure. The rate of decreasing of the transition looks normal for a d electronic compound.

  These notable differences may signal a clearly distinctive influence of the crystalline field at high temperatures. A gradual but pronounced drop in resistivity comes to disturb the linear variation to about 8 and 20 K for PrNi 2 B 2 C, and PrCo 2 B 2 C, respectively. Such resistivity behavior at relatively low temperature is typical for magnetic elements of the RNi 2 B 2 C series,

FIG. 2: (Color online) Normalized resistivity at 295 K ρ ab (T )/ρ ab (295 K)) of PrNi 2B2C, PrCo2B2C, and PrPt2B2C compounds at room pressure. The three systems present metallic conductivity from room temperature to down 25 K. Inset shows the low temperature variation of the normalized resistivity. the a-b plane as a function of temperature for PrNi 2 B 2 C, PrCo 2 B 2 C, and PrPt 2 B 2 C single crystals at room pressure. The three compounds present metallic characteristic. At low temperature Ni and Co based compounds show notable similarities, but are not superconductors, whereas PrPt 2 B 2 C has a sharp superconducting temperature at about 6 K. The residual resistivity ratios RRR for the first two compounds are 9 and 33 respectively, whereas for PrPt 2 B 2 C is 5.5. The residual resistivity ρ 0 of all three is sample dependent, varying between 5 and 25 µΩcm. The main panel of Fig. 2 shows some interesting characteristics: Ni and Co based compounds present a notable positive curvature from about 150 to 50 K, whereas Pt compound presents a wide bump from about 250 to 20 K.

  GPa are able to change drastically the low temperature resistivity behavior of PrCo 2 B 2 C, but it requires ∼ 13.0 GPa in order to attain similar changes for PrNi 2 B 2 C. This means that the low temperature electronic transport properties of PrCo 2 B 2 C are more pressure sensitive than that for the isomorphous PrNi 2 B 2 C. Evidence for spin fluctuation in PrCo 2 B 2 C is reported from the cuadratic behavior of its resistivity at low temperature, and from the decreasing of the cuadratic coefficient as a function of pressure. For PrCo 2 B 2 C, the magnetic scattering related to spin disorder is suppressed at 2.9 GPa, but it remained observable at less under 5.3 GPa for PrNi 2 B 2 C. For the case of the superconductor PrPt 2 B 2 C, pressure does not modify the ρ(T ) curvature related to CEF effects, but decreases Tc at the rate dTc/dP = -0.34 K/GPa. Finally, although these conclusions are no decisive, we believe they would stimulate further experimental and theoretical studies for a better understanding of the pressure effects in the RT 2 B 2 C compounds, which is far for complete.

					foremost fact we found is that applied pressures of
					about 4.0
	pressure	resistivity	measurements	in
	Pr(Co,Ni,Pt) 2 B 2 C has been made.	The first and

The authors acknowledge the MCBT, Institut Néel, (CNRS) & UJ for the time granted to perform some high pressure experiments. R. F. thanks to SEP-PROMEP, UJATAB-CA175 for support. R. E. thanks UNAM-DGAPA, project No IN-101107. We thank to S. Bernès for the crystallographic studies and F. Silvar for liquid He supply.