Cédric Le Bastard

Student Member, IEEE Jingjing Pan
email: jingjing.pan1@etu.univ-nantes.fr

Member, IEEE Yide Wang

Shreedhar Meng Sun

Savant Todkar

Vincent Baltazart

Nicolas Pinel

Amine Ihamouten

Xavier Derobert

Christophe Bourlier

Vincent Baltazar

Xavier Dérobert

A Linear Prediction and Support Vector Regression-Based Debonding Detection Method Using Step-Frequency Ground Penetrating Radar

Keywords: Debondings, detection, linear prediction (LP), pavements, step-frequency ground penetrating radar (SF-GPR), support vector regression

published or not. The documents may come

rapid nondestructive probing of roadways (the roadways are assumed to be horizontally stratified). The vertical pavement structure can be deduced by backscattered echoes, and more precisely by the estimated time delays and amplitudes of the measured GPR signals. Amplitude estimation is used to retrieve the wave speed within each layer. This letter focuses on the detection of thin interlayer debondings within pavement structures. These debondings have a substantial influence on the residual life of the pavement, and thus, early detection is important for pavement maintenance. Nowadays, a GPR allows to detect significant interlayer debondings [START_REF] Saarenketo | Road evaluation with ground penetrating radar[END_REF]- [START_REF] Marecos | Assessing the pavement subgrade by combining different non-destructive methods[END_REF]. However, the detection of thin debondings still remains unresolved due to the very small thicknesses of the debonding layers. This challenge remains important for end users (clients or awarding authorities) to optimize the road structures' maintenance once such defects are detected at an early age. This letter presents a new method to detect the thin debondings within pavement structures with the stepfrequency ground penetrating radar (SF-GPR).

In the context of thin interlayer debondings, i.e., overlapped echoes, electromagnetic inversion [START_REF] Benedetto | Civil Engineering Applications of Ground Penetrating Radar[END_REF], subspace methods [START_REF] Qu | Time-delay estimation for ground penetrating radar using ESPRIT with improved spatial smoothingtechnique[END_REF], [START_REF] Meschino | Detection of subsurface metallic utilities by means of a SAP technique: Comparing MUSIC-and SVM-based approaches[END_REF], compressive sensing methods [START_REF] Li | Enhanced GPR signal for layered media time-delay estimation in low-SNR scenario[END_REF], deconvolution methods [START_REF] Zhao | Development of regularization methods on simulated ground-penetrating radar signals to predict thin asphalt overlay thickness[END_REF], and supervised machine learning-based regression methods [START_REF] Meschino | Detection of subsurface metallic utilities by means of a SAP technique: Comparing MUSIC-and SVM-based approaches[END_REF], [START_REF] Bastard | Time delay and permittivity estimation by ground-penetrating radar with support vector regression[END_REF] are some of the methods that are able to estimate the amplitudes and time delays of backscattered echoes. However, estimation methods, such as subspace methods or supervised machine learning-based regression methods, assume, either explicitly or implicitly, that the number of echoes (or the number of interfaces) is known beforehand. Some criteria (such as Akaike's information criterion or minimum description length [START_REF] Badawy | Estimating the number of sources in white Gaussian noise: Simple eigenvalues based approaches[END_REF]) can be used to detect the number of echoes. Nevertheless, in practice, these criteria are not efficient enough due to the high coherence of the echoes. Compressive sensing methods are also a family of promising methods, but their behavior and performance depend on the dictionaries used and the regularization parameters. The behavior of deconvolution methods also depends on the regularization parameters.

Furthermore, buried object detection methods using linear prediction (LP) are proposed in [START_REF] Van Genderen | Imaging of stepped frequency continuous wave GPR data using the Yule-Walker parametric method[END_REF]- [START_REF] Yoldemir | A least squares approach to buried object detection using ground penetrating radar[END_REF]. LP can be used A c c e p t e d m a n u s c r i t p t to predict the next GPR signal using previous observations. An object is detected when the measured signal is different from its prediction. Yoldemir and Sezgin [START_REF] Yoldemir | A least squares approach to buried object detection using ground penetrating radar[END_REF] proposed a leastsquares-based LP approach, which assumes that the number of time samples is larger than the number of spatial samples (number of A-scans). This assumption can be perfectly valid in the case of pulse GPR in the time domain. Unfortunately, this assumption may not hold true in the case of SF-GPR. Recently, Pan et al. [START_REF] Pan | Time-delay estimation using ground-penetrating radar with a support vector regression-based linear prediction method[END_REF] proposed a new forward-backward linear prediction (FBLP) method to estimate the time delays of backscattered GPR echoes with a limited number of frequency samples and A-scans. To achieve this, they proposed to combine the FBLP and support vector regression (SVR) theories. Based on the structural risk minimization principle, the SVR serves as a good machine learning algorithm for sparse modeling [START_REF] Vapnik | Statistical Learning Theory[END_REF]. Due to the potential advantages, it is of great interest to extend the detection method of [START_REF] Yoldemir | A least squares approach to buried object detection using ground penetrating radar[END_REF] by using the techniques in [START_REF] Pan | Time-delay estimation using ground-penetrating radar with a support vector regression-based linear prediction method[END_REF]. Thus, the proposed detection method requires only a small number of frequency samples and A-scans.

The rest of this letter is organized as follows. Section II presents the adopted signal model. Section III briefly presents the conventional LP and the proposed LP-SVR detection methods. Section IV shows the performance of the proposed method with experimental data. Section V gives the concluding remarks.

II. SIGNAL MODEL

The signal model from [START_REF] Qu | Time-delay estimation for ground penetrating radar using ESPRIT with improved spatial smoothingtechnique[END_REF] and [START_REF] Pan | Time-delay estimation using ground-penetrating radar with a support vector regression-based linear prediction method[END_REF] is adopted in this letter. It represents the backscattered radar echoes (time-shifted and attenuated replicas of the transmitted signal) from lossless stratified media. In order to use the principle of LP methods, the whitening process that divides each received frequency sample by the corresponding radar pulse is applied. Thus, the signal model used within frequency bandwidth B with N discrete frequencies can be written as follows:

r ( f n ) = g( f n ) e( f n ) = K k=1 s k exp(-j 2π f n t k ) + u( f n ) (1) with u( f n ) = (m( f n )/e( f n ))
; K represents the number of echoes; e( f n ) and m( f n ) are the GPR pulse and an additive Gaussian white noise with zero mean and variance σ 2 at frequency f n , respectively. Frequency f n is defined as

f n = f 1 +(n -1) f , with n = 1, 2 .
. . , N, f 1 the lowest frequency, and f the frequency shift. s k and t k are the amplitude and time delay of the kth backscattered echo, respectively. For N discrete frequencies within the bandwidth B, the whitened received signal r can be written in the following vector form:

r = As + u (2)
where r = [r ( f 1 ), r ( f 2 ), . . . , r ( f N )] T is the data vector representing the measurements from a step frequency radar to which the whitening procedure is applied; the superscript T denotes the transpose operation; A = [a 1 , a 2 , . . . , a K ] is the mode matrix and a k , the mode vector, is defined as:

a k = [e -j 2π f 1 t k , e -j 2π f 2 t k , . . . , e -j 2π f N t k ] T ; s = [s 1 , s 2 , . . . , s K ] T
is the vector composed of the reflected echoes' amplitudes; u = [u( f 1 ), u( f 2 ), . . . , u( f N )] T is the complex noise vector.

For a B-scan composed of L A-scans, we can define matrix R, in the frequency domain, as follows:

R = [r 1 , r 2 , . . . , r L ] (3)
with r i the i th A-scan in the frequency domain and i = 1, 2 . . . , L.

III. LINEAR PREDICTION DETECTION METHOD

A. Conventional LP

The LP-based detection method of [START_REF] Yoldemir | A least squares approach to buried object detection using ground penetrating radar[END_REF] developed in the time domain is briefly presented here in the frequency domain. It is used to predict the next whitened GPR signal y = [y( f 1 ), y( f 2 ), . . . , y( f N )] T from the previous observations R. Debonding is detected when the whitened measured signal,

r i = z = [z( f 1 ), z( f 2 ), . . . , z( f N )] T , with i > L,
is different from the prediction y. LP estimates the unknown samples with a linear combination of the known observations by minimizing the mean square prediction error, e, as follows:

Rω = y (4) e = z -y (5)
where y ∈ C N×1 , R ∈ C N×L , and ω ∈ C L×1 . The weighting coefficient vector ω can be estimated by the least-squares approach

ω = (R H R) -1 R H z (6)
where the superscript H denotes the conjugate transpose operation. The Euclidean norm of the error, e 2 , indicates the possible occurrence of a defect (debonding). The larger the error is, the higher the chance of detection will be. However, the inversion of R H R requires the inequality constraint: N > L. As discussed in Section I, the assumption made in [START_REF] Yoldemir | A least squares approach to buried object detection using ground penetrating radar[END_REF] (i.e., N > L) is not necessarily valid when using SF-GPR.

B. Proposed Method: LP-SVR

The objective of LP is to estimate the weighting vector ω. When the weighting vector ω is estimated, we can predict the whitened A-scan y using (4). In the context where the number of A-scans (L) is larger than the number of frequency samples, the weighting vector cannot be well-estimated and the original LP method cannot work. Notice that (4) represents a typical form of SVR. Thus, we propose to use the SVR theory to estimate the weighting vector ω without the inequality constraint (N > L). The SVR method aims to find a hyperplane that fits the data within a deviation less than a given value . The -insensitive loss function is used here [START_REF] Martínez-Ramón | Support Vector Machines for Antenna Array Processing and Electromagnetics[END_REF]. The optimization problem can be written as follows [START_REF] Pan | Time-delay estimation using ground-penetrating radar with a support vector regression-based linear prediction method[END_REF]:

min ω,ξ r , ξ r ,ξ i , ξ i 1 2 ω 2 + C N n=1 ξ r n + ξr n + ξ i n + ξ i n s.t. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Re y n -x H n ω + ξ r n , n = 1, . . . , N Re -y n + x H n ω + ξr n , n = 1, . . . , N Im y n -x H n ω + ξ i n , n = 1, . . . , N Im -y n + x H n ω + ξ i n , n = 1, . . . , N ξ r n , ξ x n , ξ i n , ξ i n 0, n = 1, . . . , N (7)
where x n is the nth column of R H and y n = y( f n ); ξ r n and ξr n stand, respectively, for positive and negative errors in the real part of the output y n , while ξ i n and ξ i n stand for the corresponding imaginary part; Re(•) and Im(•) denote the real and imaginary parts, respectively; the value C controls the tradeoff between the structural and empirical errors. To solve [START_REF] Meschino | Detection of subsurface metallic utilities by means of a SAP technique: Comparing MUSIC-and SVM-based approaches[END_REF], we employ the Lagrangian multipliers and the Karush-Kuhn-Tucker theorem with Wirtinger's calculus on the complex variable ω [START_REF] Pan | Time-delay estimation using ground-penetrating radar with a support vector regression-based linear prediction method[END_REF]. Thus, the weighting vector ω can be written as follows:

ω = N n=1 ((a n -ân ) + i (b n -bn ))x n ( 8
Im(RR H ) Re(RR H ) a - â b -b -1 T (a + â + b + b) + Re(y T )(a -â) + Im(y T )(b -b) s.t. 0 ≤ a n , ân , b n , bn ≤ C, n = 1, . . . , N ( 9
)
where 1 is an all-one column vector with N elements. In the calculation, a small identity term γ I is added in the cost function in the case of ill-conditional inaccuracies [START_REF] Martínez-Ramón | Support Vector Machines for Antenna Array Processing and Electromagnetics[END_REF]. Thus, the weighting vector ω can be obtained.

Compared with the conventional LP, the proposed LP-SVR needs additional computation. However, the number of frequencies and A-scans is limited. Thus, the increase of computational burden is very low. A comparison of the average execution time between LP and LP-SVR will be given in Section IV.

IV. EXPERIMENTAL RESULTS

In this section, the proposed method is tested with two experimental data sets from IFSTTAR carousel [START_REF] Sun | Time delay and interface roughness estimation using modified ESPRIT with interpolated spatial smoothing technique[END_REF], [START_REF] Simonin | Case study of detection of artificial defects in an experimental pavement structure using 3D GPR systems[END_REF]. Two media are probed with SF-GPR. The first probed area is composed of two and three layers, as shown in Fig. 1. Layers 1 and 3 are asphalt surface pavements and layer 2 is made of sand. Layer 2 represents artificial debondings with a thickness about 0.5-1 cm. The measurements have been carried out at 10-, 50-, and 200-Kcycles loading stages [START_REF] Sun | Time delay and interface roughness estimation using modified ESPRIT with interpolated spatial smoothing technique[END_REF]. A quasi-monostatic step frequency radar with transmitter (Tx) and receiver (Rx) antennas close to each other is used. The distance between Tx and Rx is 20 cm and it is constant during the measurements. For each experiment, the height of the antennas is set to 70 cm above the ground level. A frequency bandwidth of 10 GHz, B = [0.8-10.8] GHz, with N = 11 samples, is used. Before using the proposed method, we apply two preprocessing techniques. First, a time filtering is used to delete the air wave. Second, data whitening by the pulse is applied [START_REF] Sun | Time delay and interface roughness estimation using modified ESPRIT with interpolated spatial smoothing technique[END_REF]. The radar pulse is measured from a metal plane. The first time delay (first interface) is considered as constant in the B-scan. The number of A-scans (L) used to estimate the weighting vector ω is set to 20. The SVR-related parameters are set like in [START_REF] Pan | Time-delay estimation using ground-penetrating radar with a support vector regression-based linear prediction method[END_REF], i.e., = 0, C = 1, and γ = 10 -6 .

Figs. Euclidean norm (NEEN), defined as e 2 /max(e 2 ), with LP and LP-SVR. The method detects the debonding when the NEEN is greater than a certain threshold. Following various measurements, we have observed that the detection threshold should be set between 0.4 and 0.5. Thus, we have chosen to set the detection threshold to 0.5, which is represented in Figs. Fig. 5 shows the NEEN with LP-SVR for 10-Kcycle loading stage with respect to the number of previous A-scans, L. We can see that the results are almost the same with different values of L. Debonding is detected from the 35th or 36th A-scan.

Fig. 6 shows the NEEN from LP and LP-SVR for 10-Kcycle loading Here, we consider the case where N > L. The behavior of both the methods is almost the same in this context. Debonding is detected from the 35th A-scan. Furthermore, the average execution time for one A-scan is 2.4 × 10 -4 and 0.08 s, for LP and LP-SVR, respectively.

To analyze the behavior of LP-SVR for thin debondings, the second experiment is carried out on a medium composed of two and three layers, as shown in Fig. 1. In this case, layer 2 is a thin air gap (occurring due to the uncoat interface shows that only LP-SVR is able to detect the debonding presence from the 30th A-scan.

The proposed LP-SVR method has also been tested on simulated data for a medium composed of three interfaces of which the first one is horizontal and the last two ones are modeled by a wedge model. The results indicate the effectiveness of the proposed method in detecting debondings with different layer thicknesses.

V. CONCLUSION

In this letter, we propose a new method based on LP and SVR theories to detect thin debondings within pavement structures using SF-GPR. Unlike the conventional LP method, the proposed method is not constrained by the number of frequency samples and A-scans. In the particular context where the number of frequency samples is smaller than the number of A-scans (N < L), the proposed method outperforms the conventional LP method. The experimental results have shown the efficiency of the proposed method. In perspective, future work will be focused on the theoretical evaluation of the detection threshold.

A c c e p t e d m a n u s c r i t p t

T

  ) with a n , ân , b n , and bn , the Lagrangian multipliers. Define a = [a 1 , . . . , a N ] T , â = [â 1 , . . . , âN ] T , b = [b 1 , . . . , b N ] T , and b = [ b1 , . . . , bN ] T . The Lagrange multiplier vectors a, â, b, and b can be computed from the maximization problem as follows: Re(RR H ) -Im(RR H )

Fig. 1 .

 1 Fig. 1. Horizontal stratified medium.

Fig. 2 .

 2 Fig. 2. (a) B-scan from SF-GPR in the time domain, 10 Kcycles; B = [0.8 -10.8] GHz. (b) Detection results from LP and LP-SVR; L = 20 and N = 11.

Fig. 3 .

 3 Fig.3. Experimental results for the same parameters as in Fig.2, excepted for the number of Kcycles; here, 50 Kcycles.

  2(a), 3(a), and 4(a) show the B-scans in the time domain obtained from SF-GPR at various Kcycle loading stages. Figs. 2(b), 3(b), and 4(b) show the normalized error

A c c e

  p t e d m a n u s c r i t p t

Fig. 4 .

 4 Fig. 4. Experimental results for the same parameters as in Fig. 2, excepted for the number of Kcycles; here, 200 Kcycles.

Fig. 5 .

 5 Fig. 5. LP-SVR results from different number of previous A-scans (L); 10 Kcycles, B = [0.8 -10.8] GHz, and N = 11.

  2(b), 3(b), 4(b), 5, 6, and 7(b) by the dotted horizontal gray line. In Figs. 2-4, L is superior to N and as a consequence LP cannot work. Only LP-SVR is able to detect the debonding presence. We can see that LP-SVR allows to detect the presence of the debonding from the 35th A-scan, the 34th A-scan, and the 31th A-scan for 10-, 50-, and 200-Kcycle loading stages, respectively. These results have been obtained by a computer with a processor unit (CPU) at 2.3 GHz and 8GB RAM. The average execution time for one A-scan is about 0.022 s for Figs. 2(b), 3(b), and 4(b).

Fig. 6 .

 6 Fig. 6. Detection results from LP-SVR and LP; 10 Kcycles, B = [0.8 -10.8] GHz, N = 30, and L = 20.

Fig. 7 .

 7 Fig. 7. (a) B-scan from SF-GPR, B = [0.8 -10.8] GHz, tack-free interface. (b) Detection results from LP and the proposed method; L = 20, B = [1 -6] GHz, and N = 11.

A c c e

  p t e d m a n u s c r i t p t


Y Fargier

S Palma Lopes

B29 2D-Electrical Resistivity Imaging for Sike Survey: Impact of the a Priori Information Management

In France, EDF (Électricité De France) is in charge of managing the large hydraulic structure stock dedicated to producing hydroelectricity. This responsability is associated with high economical and safety issues. Electrical Resistivity Imaging (ERI) is being widely used on earth embankment dikes, either for extensive survey of large sections (a few km) or for higher resolution surveys on shorter sections (a few 100m).

However, the processing of extensive longitudinal survey with conventional 2.5D software is likely to yield image artefacts. This study aims to show that a bad management of the dike complexity in the inversion process is the origin of these artifacts.

To overcome this limitation, a new inversion algorithm specially created to take into account the complex behaviour of the dike is presented. This development aims at filling the gap between more classical 2D and purely 3D approaches, by optimising a compromise between acquisition cost and model reliability. Applied to real data sets this tool supplies a new understanding of the interpretation capability of the inversion results.

Introduction

In France, EDF (Électricité De France) is in charge of managing the large hydraulic structure stock dedicated to producing hydroelectricity. This responsability is associated with high economical and safety issues. Overtopping and internal erosion are the main mechanisms of dikes failure and represent more than 90% of the accidents [START_REF] Foster | The statistics of embankment dam failures and accidents[END_REF][START_REF] Fell | Internal erosion of dams and their foundations[END_REF].

Among other geophysical methods, DC-electrical resistivity imaging (ERI) is being widely used on earth embankment dikes, either for extensive survey of large sections (a few km) or for higher resolution surveys on shorter sections (a few 100m) previously identified as higher hydraulic risk zones [START_REF] Fauchard | Geophysical and geotechnical methods for diagnosing flood protection dikes. Guide for implementation and interpretation[END_REF]. This method is also selected owing to its high sensitivity to internal erosion consequences (modifications of clay content, water content, temperature and porosity) [START_REF] Johansson | Seepage monitoring in embankment dams[END_REF][START_REF] Sjödahl | [END_REF].

However, using ERI in a conventional manner is likely to yield image artefacts [START_REF] Sjödahl | [END_REF] when the processing is realized with 2.5D inversion software. This study aims to show that a bad management of the dike complexity during the inversion process is the origin of these artefacts. To overcome this limitation, a new inversion algorithm specially created to take into account the complex behaviour of the dike is presented. This development aims at filling the gap between more classical 2D and purely 3D approaches, by optimising a compromise between acquisition cost and model reliability.

In this paper, we first briefly present the consequences of the complex behaviour of the dike on a 2D longitudinal ERI campaign. Then, principle of the new inversion codes is presented. Then a special importance is given to the a priori information that can be inferred from the preliminary study and the way to manage this information during the inversion process. This methodology is applied to 2D ERI campaign acquired on a real hydraulic structure. Results show that the complex behaviour of the dike can be better take into account (presence of a water reservoir, topography of the dike) with this method and supply a relevant way to process 2D-ERI "high outputs" campaign.

Pitfalls of a 2D-ERI longitudinal survey on a dike

For cost effectiveness reasons, ERI is usually applied in a "classical" way: a set of equidistant electrodes is aligned along the longitudinal direction on the dike crest, slope, or toe. However, the geo-electrical behaviour of dikes is fully 3D due to the internal composition and the topography. Moreover, 2D-ERI measurement campaigns are mostly interpreted using 2.5D inversion softwares (e.g. Res2dinv) which can lead to 3D artefacts on the results.

In the framework of dike survey, it is important to differentiate 3D effects due to :

1. The presence of external media (e.g. the water reservoir, a concrete facing) or the topography of the dike. 2. The internal resistivity variation of the medium (e.g. the presence of a core, a foundation).

The reason why it is important to differentiate these two effects is that the purpose of imaging is the dike and not the water reservoir. In addition, the topography of the structure and the resistivity of the water reservoir are elements that can be measured directly without having to carry out destructive measurements.

Figure 1 presents the result of a numerical test which aims to quantify the effect of the topography of a dike and of the water reservoir on a measure. The effects of inter electrode spacing and the distance (d) between the quadrupole and the water reservoir are also studied. The result indicates that the effect of the water reservoir is higher when the distance between the quadrupole and the reservoir decreases and can scale up to 35 % when inter-electrode spacing is 20 m. Measurements quadrupole correspond to a "wenner alpha" protocol. To compute the apparent resistivity a conventional geometrical factor is Near Surface Geoscience 2012 -18 th European Meeting of Environmental and Engineering Geophysics Paris, France, 3-5 September 2012 employed (2πa). Consequently, the strong effect of the topography and the water reservoir must be taken into account to avoid image artefacts and allow a robust result of inversion.

A 2D+ inversion procedure

The previous paragraph has presented some 3D effects intrinsic to a 2D-ERT procedure on a dike. It has also showed that these effects can be partly quantified. The objective of this paragraph is to present a cost effectiveness method taking most of the 3D effects into account.

The diffusion of an electrical field is a highly non-linear problem and a normalisation technique cannot completely take into account this nonlinearity [START_REF] Marescot | A general approach for DC apparent resistivity evaluation on arbitrarily shaped 3D structures[END_REF]. Therefore, it is necessary to develop inversion methods capable of taking this non-linearity into account. To limit the financial cost of the acquisition and the computational cost of inversion, new inversion codes specifically adapted to the dikes problem are developed. These codes are named '2D + ' and permit a 2D inversion that accounts for a part of the full 3D geo-electrical behaviour of a dike (The topography effect and water reservoir effects are taken into account).

To take into account this kind of information (e.g. topography, resistivity of the reservoir), a new methodology for meshing the direct problem and discretizing the inverse problem is used. The method, inspired by the triple grid method [START_REF] Günther | Three-dimensional modelling and inversion of dcresistivity data incorporating topography -II. Inversion[END_REF], consist to first create a discretization based on our a priori on the medium (Figure 2.a) and then to mesh this medium to solve the direct problem (Figure 2.b). However, this principle, although allowing an explicit integration of information on the environment, leads a higher inversion cost.

To solve the direct problem a finite element method is selected and meshes are deformed to simulate the topography. For the inversion, following the assumption that internal variations of the dike must be smooth, an OCCAM inversion type algorithm is implemented (Constable et al., 1992).

In conclusion, this new code provides a cost effectiveness manner to invert data obtain in high output conditions (2D-ERI longitudinal, figure 3.a) and permit a take into account of all external 3D effects (water reservoir, topography). A previous parametric study showed that a concrete facing does not have a significant effect on the measurement compared with the topography and the reservoir.

Near Surface Geoscience 2012 -18 th European Meeting of Environmental and Engineering Geophysics Paris, France, 3-5 September 2012

2D-ERI longitudinal survey on a real hydraulic structure: results and discussion

A survey in high output conditions is realized on real hydraulic structure of EDF (Figure 3.a). 96 electrodes were positioned every 5 m at the centre of the crest of the hydraulic structure. A Wenner-Schlumberger acquisition procedure was used to acquire data. Picture 3(a) shows the location of the electrode cable and the position of the water reservoir.

Figures 3(b) presents an inversion result obtained with the 2D + code. This result is presented at the third iteration with an RMS error of 0.62 %. The discretization of the inverse problem consists of approximately 3500 cells for the dike and one cell for the water reservoir following the principle presented figure 2. The result highlights a resistive layer between 1 and 8 m depth (R1) and two conductive anomalies (A1 and A2) can be interpreted inside the R2 layer. In conclusion, usual artefacts due to the presence of the effect of the water reservoir (that appear with conventional 2.5D software) are inexistent suggesting the superiority of this result.

Impact of the prior information on the inversion result

The way the a priori information (shape and resistivity of the water reservoir) is incorporated into the inversion process can have a large impact on the inversion result.

Near Surface Geoscience 2012 -18 th European Meeting of Environmental and Engineering Geophysics Paris, France, 3-5 September 2012

During the study, we focused a part of our attention on how the a priori information must be implemented in function of the quantity of information and its quality.

Indeed, when the medium is not well known, the inversion must let more freedom (with a low damping factor) and/or with a high number of inversion cells. Conversely, in our case, several measurements have indicated that the resistivity of the reservoir was strongly homogeneous (because of a current in the reservoir). This knowledge has led us to not discretize the reservoir into several cells and impose strongly the resistivity of the reservoir with a high damping factor. Finally, a study concerning the robustness of the result (resolution matrix method) [START_REF] Menke | Geophysical data analysis: Discrete inverse theory[END_REF] indicates that imposing the resistivity of the reservoir increases the model resolution of the cell composing the dike and thus the robustness of the imaging result.

Conclusions & prospects

2D-ERI is a widely used method for surveying and monitoring hydraulic structures. However, processed in a conventional manner, this method can lead to misinterpretation of the imaging result.

To prevent these pitfalls, a new method, called 2D + , is presented. The objective of this method is to provide a treatment that can reflect the complexity of the structure without increasing the cost of measurement on the field.

This method is applied to real data acquired on a hydraulic earth structure. Results show that this method improves the robustness of the inversion process and thus the reliability of the interpretation. This procedure is now commonly applied to high-output 2D surveys, and we hope that it will be soon available for 3D high-resolution survey and monitoring of short stretches of dikes.

Figure 1

 1 This figure presents (a) a scheme of the model used for the numerical test and (b) the result of the apparent resistivities simulated on the model at different distances of the reservoir and for various electrode separations. Three distinct regions are shown (I,II,III).

Figure 2

 2 figure (a) presents the discretization of the inverse problem. The water reservoir (represented in blue) is simulated with just one inversion cell. The pink cell illustrates one of the deformed inversion cells composing the dike. The illustration (b) presents the finite element mesh used to resolve the direct problem.

Figure 3

 3 Figure presents (a) a picture showing the dike and the electrode cable location (in orange) and (b) the 2D + inversion result from a Wenner-Schlumberger dataset obtained on medium (a). Two layers are differentiated (R1 and R2) and two anomalies must be identified (A1 and A2).

Acknowledgements

We would like to thanks EDF -CIH and more particularly Jean Robert Courivaud and Julien Cintract for the funding of part of the work and for the GPS assistance on-site.


Théo Zimmermann Πr
email: theo@irif.fr

Challenges in the collaborative development of a complex mathematical software and its ecosystem

Keywords: engineering → Open source model, Maintaining software, Documentation, Software version control, Open source software, release management, mathematical software, Coq, proof assistant

This is a contribution to the OpenSym 2018 Doctoral Symposium. This paper describes my PhD objectives. As an insider in the Coq development team, I've worked at making the release process of the Coq proof assistant smoother and more automated, at opening the development to external contributions, and at shaping the ecosystem around Coq. I'm intending to evaluate how well-known software engineering techniques and results about open source software communities apply in the specific case of the proof assistant I'm studying.

It serves as the basis of large research projects involving dozens of researchers and receiving millions in funding. 1 It is also a tool for industrial verification projects.

This has some consequences for Coq. On the one hand, it is not just a research prototype anymore but a software on which the work of many people depends. It must therefore meet their expectations (in terms of compatibility between versions, improved performance, improved support for some usages) while continuing at the same time to be an object of study for researchers. On the other hand, it now has a large community of users which can be seen as a crowd of potential contributors.

The problem I am thus interested in is how the development of Coq can adapt to meet the new expectations while taking the most of the new opportunities arising from a more open development.

As an insider in the Coq development team since 2015, I have been a direct witness and important actor in the evolution towards a more open development, which encourages external contributions.

GOALS

The evolution in the development process to guarantee more stability had started before I arrived with the nomination of a "release manager" and the objective to have time-based release cycles. I was, nevertheless, a witness of the first time-based release and a main actor of the second and third ones. My first contribution is my participation to this evolution. In my thesis, I will analyze the evolution and evaluate its consequences on the development of new features (was it hindered? did it impact the research around Coq?) and on the satisfaction of big users' needs (has the system gained in stability and has this been useful to them?).

The real opening of the development of Coq (the proof language and the software itself by opposition to extensions such as libraries or plugins) to external contributions started with the first Coq Coding Sprint (later renamed Coq Implementors Workshop) to which I was a participant. In the following years, the development became more and more transparent and open to external contributions. My second contribution is my action in favor of such external contributions and a more open development team. In my thesis, I will analyze this evolution and evaluate how it helped get valuable contributions from new active contributors (how did it help get new contributors, who were they, and how useful were their contributions given the inherent complexity of the Coq software?).

The ecosystem around the Coq proof assistant (in which I include external packages such as libraries, plugins and editor support packages, external documentation in the form of tutorials, books, Q&A. . . ) evolved as well, as a consequence of the growth of the user community and changes in the distribution of Coq packages that had started before I arrived. We passed from a centralized model to a more distributed model. In the centralized model, library and plugin developers were solicited to send their packages to the Coq development team who would then take care of maintaining them and distributing them further (in an archive of so-called "user-contribs"). In the new model, they are self-maintaining their packages and encouraged to directly submit them to a package repository to reach users who are then able to install them using a specific package manager. My third contribution will be to the organization and shaping of this ecosystem by involving the community more and more, especially in the long-term maintenance of Coq packages and their diffusion. I have launched a project for collaborative long-term maintenance of Coq packages called coq-community 2 inspired by the similar and already successful elm-community. 3 A cross-cutting goal is to evaluate how well-known software engineering techniques and results about open source software communities apply in the specific case of the proof assistant I am studying.

METHODS

To propose amendments to our development processes, I observe what issues arise with the previous way of doing. For instance, a lot of breaking changes were implemented and merged without being documented in the changelog, just because developers and reviewers forgot, and we got complaints from users. Then, most of the time, I propose and I test alternative ways based on standard practices in open source development. In the above example, there were several possible solutions (used in other projects): enforcing that pull requests always come with a changelog entry by adding an automatic check; automatically generating a changelog using information from GitHub pull requests and commit messages; or simply reminding this requirement to pull request authors using a pull request template with a few checkboxes. I implemented the latter because it was the lighter solution and it turned out to be pretty efficient (although exact measurements are still to be done). However, there is still a problem with some experimented developers stripping the template without reading it because they think it's not for them, and we will see in the future if there is a need to move to one of the other two solutions.

Furthermore, the open source way is in constant evolution itself so there is no definite best solution, and all open source development teams are experimenting a bit at the same time. It is quite useful to keep informed of what others are doing, and to be ready to reuse their tools (and to share ours). The GitHub platform itself, which we are using, is evolving quite fast these days, and we regularly experiment with new features to see how they fit in our practices and if they are of any help.

Sometimes, just reporting a problem in our development processes and having it discussed is enough for good ideas to emerge. This kind of meta-issues really took off after the move from our old bug tracker to GitHub issues (described in the results section). Documenting processes provides clear and immediate benefits such 2 https://github.com/coq-community 3 http://elm-community.github.io/ as helping new contributors gain in confidence or making it easier to swap some special roles like "release manager". But it can also be a good opportunity to discuss these processes. Indeed, more senior developers are often questioning the adequacy of the new processes and lack of proper communication around some changes sometimes created frustration or confusion.

I also intend to have repetitive tasks automated more systematically. I was particularly inspired in this aspect by the talk "Cyborg Teams: Training machines to be Open Source contributors" by Stef Walter at FOSDEM'18. Lots of such automations are natively proposed by the GitHub platform while some others can be easily added because they have been implemented and distributed by other developers. However, we must be ready to adapt them or develop our own when what is proposed cannot fit in easily in our processes, would impose too radical changes or wouldn't help with respect to our objectives. For instance, I created a methodology to track pull request backporting using a GitHub project (a Kaban-like board with columns and cards). When GitHub added automation features to these projects, I started using them to manage this more efficiently, but because these features were quite limited, I ended up developing a bot that moves cards around for me and saves me a lot of time. This will also make it easier to pass on this role to someone else.

To involve the user community more, the best way is sometimes to directly call for contributions: on Coq-Club, the Coq users' mailing list, like I did once to get a volunteer to migrate the Coq FAQ, which was bitrotting, to the Coq wiki; or at the Coq workshop, like I intend to do to find participants to the coq-community collaborative maintenance project.

This collaborative maintenance project is also giving me a different perspective and I intend to conduct some improvement upstream to help package maintainers. Two specific questions are how to setup CI for testing these packages with various versions of Coq; and how to distribute Coq packages in the most efficient way. I'm currently exploring a new approach to these two questions based on the Nix package manager. 4 I have been teaching Coq for two years and this too is giving me a different perspective. Next year, I'd like to push my students to contribute to Coq at their own level: that is by highlighting what are the common problems that beginners encounter when they use Coq for the first few months. This kind of feedback is very valuable and usually quite difficult to get because users who open most of the issues are users who feel already quite confident about their knowledge of the system.

To evaluate the impact of changes in our processes, I would like to get objective measures as often as possible. Most of the time, I will collect these indicators from gross statistics obtained through GitHub APIs. Ideally, we want to compare them both to the same statistics before a change was implemented and to equivalent statistics over other open source projects when we can extract such statistics from previous research papers. For instance, there is an interesting analysis to be done on the predicting factors of the time it will take for a pull request to get merged, and a comparison to Challenges in the collaborative development of a complex mathematical software OpenSym '18, August 22-24, 2018, Paris, France do with the general results obtained in "Wait for It: Determinants of pull request evaluation latency on GitHub" by Yu et al. 5 When this is necessary, this should be augmented with qualitative evaluation by interviewing interested parties or by detailed exploration of specific examples.

Literature search and exploration of other important open source projects will also get a central place in my thesis to be able to present a comparative evaluation of our methods and to import well functioning methods from other projects.

RESULTS

As mentioned in the background section, I have contributed to the new release process: I encouraged the use of milestones and labels (which took much more importance after the migration to GitHub issues, see below); and, more importantly, I was put in charge of the stable v8.7 branch at a time when we switched from a model of preparing bug fixes for stable branches and merging them into the development branch to a model where almost all pull requests target the development branch and bug fixes and documentation updates are backported to the active stable branch. In this position, I participated in the beta and final releases of Coq version 8.7, and I managed the patch-level releases for this version. This gave me a good overview of the whole release process, which I formalized and documented.

Another evolution towards more stability was the switch to a pull-based development model (including for core contributors). This allowed for better code reviews and systematic automatic testing (a.k.a continuous integration). In a first phase of this switch, which spanned two releases and more than a year, a single person was in charge of merging all pull requests after ensuring they had been reviewed. This was too inefficient and too time-consuming for the person in charge, so we recently switched to a distributed merging process based on maintainership of components. I helped put in place this new process. While it had clear immediate benefits, we can start to observe patterns on the time it takes for a pull request to get merged depending on the component it affects (and thus whose maintainer is in charge). It will be interesting to do a detailed data analysis of this switch and its consequences.

After observing the many drawbacks of our previous bug tracker, I managed to convince the rest of the development team to move to GitHub issues. One important aspect of the migration was to keep the existing bugs and their bug numbers as often as possible. I conducted the migration by adapting a script that had been designed for much smaller scale migrations and I used it successfully to migrate the 4900 existing bug reports to GitHub issues. Only 500 bug reports, whose numbers were taken by existing pull requests, had to be renumbered. I documented this experience in a blog post available at https://theoz.im/bugzilla and shared the migration script. 6 I think this migration had a positive impact both on the opening to new contributors (as most of them already had a GitHub account, especially the students and the contributors to other open source projects) and on existing developers. One of my arguments in favor of the migration was that communication was hindered by 5 https://doi.org/10.1109/MSR.2015.42 6 My improvements are now being integrated upstream, thanks in particular to the involvement of a reader of my blog post who used my improved version of the script to perform a similar migration.

the heavy user interface of the previous bug tracker and it seems indeed that online communication among developers was more fluid afterwards. This would however require a real evaluation which I plan to conduct.

MY QUESTIONS

My advisor has no particular experience in research on open collaboration (nor software engineering): he is rather a specialist of the Coq proof assistant. Therefore, it is very important for me to get feedback and advice on my methods.

I am also interested to learn how useful my case study would be judged by specialists of the open collaboration domain and to get advice on where and how to publish relevant parts.

The DeepSpec project https://deepspec.org involves four top US universities, lists 78 collaborators and received $10 million in funding from the NSF.

https://nixos.org/nix/


Aïna Chalabaev
email: aina.chalabaev@univ-grenoble-alpes.fr

Imed Rémi Radel

Ben Mahmoud

Bernard Massiera

Thomas Deroche

Fabienne D'arripe-Longueville

Is motivation for marathon a protective factor or a risk factor of injury?

Keywords: Self-determined motivation, perceived susceptibility to injury, competitive sport

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

Is motivation for marathon a protective factor or a risk factor of injury?

Background

It is well established that physical activity, defined as any bodily movement produced by skeletal muscles that requires energy expenditure, has beneficial effects on health, as it can reduce the risk of cardiovascular diseases, diabetes, or cancer (World Health Organization, 2010). However, while research highlights the benefits of moderate physical activity (i.e., activity that requires a moderate amount of effort and noticeably accelerates the heart rate), intense physical activity (i.e., activity that necessitates a large amount of effort and causes rapid breathing and a substantial increase in heart rate) can be damaging for health [START_REF] Landolfi | Exercise addiction[END_REF]. For example, a large-scale study indicated a dose effect with moderate runners having the lowest mortality rate, while strenuous runners had a similar mortality rate to the sedentary group [START_REF] Schnohr | Dose of jogging and long-term mortality: the Copenhagen City Heart Study[END_REF]. There is also evidence that sport (i.e., the competitive form of physical activity) enhances risks of musculoskeletal injuries (e.g., Van Mechelen et al., 1992).

It is therefore crucial to identify the risk and preventive factors of injuries in this context. The present research investigated this question in marathon, an intense competitive sport.

Models of injury prevention in sports have identified internal (e.g., physical fitness, previous injuries, psychological variables) and external (e.g., equipment, weather conditions) risk factors (e.g., [START_REF] Meeuwisse | A dynamic model of etiology in sport injury: The recursive nature of risk and causation[END_REF], among which perceived susceptibility to injury (i.e., the perceived likelihood that an injury will occur, [START_REF] Deroche | Social cognitive determinants of the intention to wear safety gear among adult in-line skaters[END_REF]. In running, people who evaluate their susceptibility to injury as high are more able to detect early warning signals, leading them to take preventive measures (e.g., reducing training intensity), and to thereby avoid injuries [START_REF] Ekenman | Stress fractures of the tibia: Can personality traits help us detect the injury-prone athlete?[END_REF]. Studies have identified past experiences of injury [START_REF] Williams-Avery | Injuries and use of protective equipment among college in-line skaters[END_REF], stress [START_REF] Williams | Psychosocial antecedents of sport injury: Review and critique of the stress and injury model[END_REF], and personality variables (e.g., [START_REF] Deroche | Predictors of perceived susceptibility to sport-related injury[END_REF] as predictors of perceived susceptibility; however, few studies have investigated its motivational factors [START_REF] Stephan | Predictors of perceived susceptibility to sport-related injury among competitive runners: The role of previous experience, neuroticism, and passion for running[END_REF]. The goal of this research was to examine this question based on self-determination theory (e.g., [START_REF] Ryan | On assimilating identities to the self: a self-determination perspective on integration and integrity with cultures[END_REF]. This model distinguishes different motives that vary depending on how they satisfy the need for autonomy. When motivation is autonomous, individuals experience a sense of personal choice and autonomy when behaving and feel their actions represent their true self (e.g., a person running for the satisfaction felt when mastering his/her abilities). Intrinsic motivation (i.e., enjoyment in the task itself) is the highest form of such motivation, that also encompasses identified regulation, which is adopted when the individual values the behaviour and its consequences (e.g., running for one's health). In contrast, when motivation is controlled, people feel pressured into behaving by external forces (e.g., a person running for the prestige associated with the activity). External regulation (i.e., when the behaviour is driven by external demands) represents the most controlled form of motivation, that also includes introjected regulation, which is more internalized but still involves feelings of pressure to engage in the activity. Finally, amotivation represents lacking the intention to act.

Research has shown that an autonomous motivation may be beneficial following serious injury (for a review see [START_REF] Podlog | The psychosocial aspects of a return to sport following serious injury: A review of the literature from a self-determination perspective[END_REF], and may positively predict sport injury prevention intentions [START_REF] Chan | Self-determined forms of motivation predict sport injury prevention and rehabilitation intentions[END_REF]. However, [START_REF] Chan | Self-determined forms of motivation predict sport injury prevention and rehabilitation intentions[END_REF] study examined the motivation toward health behaviours, and not toward sport. In the present research, we hypothesized that self-determined motivation toward marathon predicts perceived susceptibility to injury. To our knowledge, no study has examined this relationship, but related research on harmonious passion (strong inclination toward an activity) may provide support to this hypothesis. Harmoniously passionate individuals are less likely to evaluate their probability to be injured as high [START_REF] Stephan | Predictors of perceived susceptibility to sport-related injury among competitive runners: The role of previous experience, neuroticism, and passion for running[END_REF], suggesting that they are in control of injury occurrence through their ability to adopt preventive behaviours when detecting early warning signals [START_REF] Deroche | Psychological mediators of the sport injury-perceived risk relationship[END_REF]. Both harmonious passion and self-determined motivation are characterized by an autonomous regulation of an activity representing the person's identity [START_REF] Vallerand | Passion in sport: A look at determinants and affective experiences[END_REF], suggesting that self-determined individuals may also be more able to adopt preventive behaviours than non-self-determined individuals. Even if they feel early signs suggesting the occurrence of injury, controlled individuals may persevere in order to prove something to others or to gain prestige, being in turn more likely to be injured [START_REF] Masters | The relations of cognitive strategies with injury, motivation, and performance among marathon runners: Results from two studies[END_REF].

Based on Stephan et al.'s (2009) results, we predicted a negative relationship between self-determined motivation for marathon and perceived susceptibility to injury. However, although this relationship may reflect the ability of self-determined individuals to adopt preventive behaviours, it could in contrast reflect a lower perception of the risks associated with sport participation, leading them to take less preventive measures to avoid injuries (e.g., [START_REF] Ekenman | Stress fractures of the tibia: Can personality traits help us detect the injury-prone athlete?[END_REF]. This alternative hypothesis rests on evidence that autonomous motivation is associated with positive emotions (e.g., [START_REF] Puente | Exercisers' perceptions of their fitness instructor's interacting style, perceived competence, and autonomy as a function of self-determined regulation to exercise, enjoyment, affect, and exercise frequency[END_REF], low stress levels and better scores of global indicators of well-being [START_REF] Maltby | The relationship between exercise motives and psychological well-being[END_REF], even in the acute phase of exercise (i.e., a subset of physical activity that is planned, structured, and repetitive and aims at improving or maintaining physical fitness) (e.g., [START_REF] Guérin | The moderating influence of situational motivation on the relationship between preferred exercise and positive affect[END_REF]. Given that emotions play a prominent role in risky decision making [START_REF] Loewenstein | Risk as feelings[END_REF], autonomous individuals could be less likely to perceive the negative aspects of their sport, such as the risk it presents for their health. Related work on exercise addiction provides support for this hypothesis. According to [START_REF] Sachs | Running addiction[END_REF], people who exercise for extrinsic reasons do not consider their activity as a central part of their lives, and may therefore not suffer withdrawal symptoms when they cannot exercise. In contrast, people who exercise for intrinsic reasons may view exercise as a central part of their lives, and experience disturbing deprivation sensations when they are unable to exercise. They may therefore be more likely to continue exercising regardless of warning signals of physical injury or injury itself.

The goal of this research was to disentangle these competing hypotheses. In Study 1, we predicted that the more marathon runners are self-determined for marathon, the less they perceive themselves as susceptible to be injured when running a marathon. Study 2 investigated two competing explanations of this prediction, by examining the mediational role of situational susceptibility to adopt risky behaviours (the tendency to act in a certain way under hypothetical situations), which has been shown to reliably predict health behaviours (e.g., [START_REF] Lazuras | Predictors of doping intentions in elite-level athletes: A social cognition approach[END_REF]. If self-determined motivation negatively predicts the perceived susceptibility to adopt risky behaviours (e.g., keep running through pain) during the marathon, and if this susceptibility mediates the relationship between self-determination and perceived susceptibility to injury, this would suggest that self-determined motivation for marathon is health-protective. In contrast, if self-determined motivation positively predicts the perceived susceptibility to keep running through pain, and if this relationship is mediated by perceived susceptibility to injury, this would suggest that self-determined motivation for marathon is health-threatening.

In addition to examine the degree of self-determination, we investigated the role of each form of motivation separately, as they may differentially affect injury-related variables.

For example, while identified regulation and intrinsic motivation are both autonomous forms of motivation, the former could be health-protective when the person runs for health reasons, while the latter could be health-threatening when the person runs to surpass him/herself. Moreover, investigating each form of motivation separately allowed us to examine whether the relationship between self-determination and injury-related variables was driven by autonomous and/or controlled forms of motivation.

Study 1

Materials and methods

Participants and Procedure. The day before the 'Marathon des Alpes-Maritimes 2009" (Nice, France), we invited marathon runners when they went out of the registration office to take part to our study. The only exclusion criterion was to not speak French as native language. After being informed of the study, 378 marathon runners (106 women; 142 competitive runners, 207 regular leisure-time runners, 26 occasional leisure-time runners;

Mage=43 years, SD=9.76) agreed to take part in this study on a voluntarily basis. They filled out a 20-minute long computerized anonymous questionnaire before being debriefed and thanked. The protocol followed the institutional rules in terms of ethics and informed consent was obtained.

Measures

Self-determined motivation for marathon was measured by the Sports Motivation

Scale [START_REF] Brière | Développement et validation d'une mesure de motivation intrinsèque, extrinsèque et d'amotivation en contexte sportif: l'Echelle de motivation dans les sports (EMS) [Development and validation of a measure of intrinsic and extrinsic motivation and amotivation in sports context: the Sports Motivation Scale[END_REF], composed of six subscales measuring the forms of motivation postulated by the self-determination theory [START_REF] Ryan | On assimilating identities to the self: a self-determination perspective on integration and integrity with cultures[END_REF]. After reading the following statement "I participate in marathon:…", participants rated the extent to which each item corresponded to their own reasons to participate in marathon running on a 7-point scale from

(1) never to (7) always. Four items measured intrinsic motivation to gain knowledge and intrinsic motivation to accomplishment (e.g., "for the satisfaction I experience while I am mastering my abilities"), four items measured intrinsic motivation to experience stimulation (e.g., "for the excitement I feel when I am really involved in the activity"), four items measured identified extrinsic motivation (e.g., "because what I learn in this activity will be useful later"), four items measured introjected extrinsic motivation (e.g., "because I would feel guilty if I could not succeed in this activity"), five items measured external regulation motivation (e.g., "to be seen in a good light by others"), and five items measured amotivation (e.g., "I do not know why I run this marathon, if I could, I would get exempted"). All items were presented in a random order. The average of items of the subscales was calculated, as each one presented an acceptable internal consistency (.70<as>.81).

Each motivation was then separately submitted to the analyses, but we also merged them to assess the degree of self-determination, by giving each subscale a specific weight according to its respective place on the self-determination continuum, multiplying this weight by the score of the subscale, and adding the scores of all subscales to derive a single score (e.g., [START_REF] Li | Testing the simplex assumption underlying the Sport Motivation Scale: A structural equation modeling analysis[END_REF]. Specifically, the scores for the three types of intrinsic motivation were averaged and assigned the highest positive weight (+2) because intrinsic motivation is the highest self-determined form of motivation. Identified extrinsic motivation, a selfdetermined type of extrinsic motivation, was assigned a lower weight (+1). These positive weights represent the degree of autonomous motivation. The scores for external regulation and introjection were averaged and assigned a negative weight (-1), and amotivation, which represents the absence of self-determination, was weighted highly negatively (-2). These negative weights represent the degree of controlled motivation. Support for the validity and reliability of this type of composite index has been obtained in several studies (e.g., [START_REF] Vallerand | An integrative analysis of intrinsic and extrinsic motivation in sport[END_REF]). However, one limit of such index is that it does not distinguish between people with high autonomous and high controlled motivations, from people with low autonomous and low controlled motivations. In order to address this limit, we also considered the autonomous and controlled motivation indices separately.

Perceived susceptibility to marathon-related injury.

Participants responded to the four items used in past research [START_REF] Deroche | Predictors of perceived susceptibility to sport-related injury[END_REF] to measure their perceived susceptibility to experience injury during the marathon. This scale includes three absolute risk items: "What do you believe is the chance that you will get an injury during the marathon", "How susceptible do you feel you will get an injury during the marathon?", and "What do you think the likelihood is that you will get an injury during the marathon?", and one comparative risk item: "What do you believe your chances are of getting an injury during the marathon compared to other runners?". For the absolute risk items, participants responded on a 1-7 scale (item 1: from 1 "no chance of being injured" to 7 "certain to be injured"; item 2: from 1 "not at all susceptible" to 7 "very susceptible"; item 3: from 1 "less than a 10% chance" to 7 "100% chance"). For the comparative risk item, participants answered on a scale ranging from 1 "a lot lower" to 7 "a lot higher". The scale showed good reliability in the present study (a=.83). A principal-axis factor analysis showed that all items loaded on a single factor that explained 66.85% of the variance, with factor loadings ranging from .80 to .84.

Results and Discussion

Descriptive statistics and correlations among the measures are presented in Table 1. In line with the hypotheses, results showed that the self-determination index was negatively related to perceived susceptibility to injury, r=-.11, p=.035. In other words, the more participants were self-determined for marathon, the less they perceived themselves as susceptible to be injured during the race. More specifically, results indicated that while the autonomous motivation index was not significantly correlated to perceived susceptibility to injury, r=-.01, p=.871, the controlled motivation index was significantly related to this variable, r=.13, p=.011. In other words, the more runners adopted a controlled motivation toward marathon, the more they perceived themselves as susceptible to be injured during the race. Furthermore, when each form of motivation was considered separately, results showed that only external regulation (r=.11, p=.036) and amotivation (r=.12, p=.02) were significantly correlated with perceived susceptibility to injury.

Given that perceived susceptibility to injury is an important factor of health preventive behaviours (e.g., [START_REF] Ekenman | Stress fractures of the tibia: Can personality traits help us detect the injury-prone athlete?[END_REF], these results suggest that self-determined motivation for sports may be associated with health-related outcomes, and that this association may be driven mostly by controlled forms of motivation. However, they do not indicate whether self-determined motivation is beneficial or harmful. On the one hand, a low perceived susceptibility to injury may reflect high control of injury occurrence due to the appropriate adoption of precautionary behaviours. On the other hand, it may reflect a lower perception of the risks associated with running a marathon, which may lead to more risky behaviours. Study 2 aimed at identifying which of these hypotheses may explain the relationships between selfdetermined motivation and perceived susceptibility to injury observed in Study 1, by examining the mediating role of perceived susceptibility to adopt a risky behaviour during the race. In addition, Study 2 included past experiences of injury as a control variable, as it is an important predictor of perceived susceptibility to injury [START_REF] Williams-Avery | Injuries and use of protective equipment among college in-line skaters[END_REF]. Finally, Study 2 used a different measure of self-determined motivation to further ascertain its relationship with perceived susceptibility to injury. A situational scale was chosen to better reflect the motivational state of the participants for the competition of interest. Participants had run 6.4 marathons on average (SD=10.18) before this one, with a mean best performance of 218.38 min (SD=42.71). The protocol followed the institutional rules in terms of ethics and informed consent was obtained.

Study 2

Materials and methods

Participants and

Measures.

Self-determined motivation for marathon was measured by the Situational

Motivation Scale (SIMS, [START_REF] Guay | On the assessment of situational intrinsic and extrinsic motivation: The Situational Motivation Scale (SIMS)[END_REF], recommended for the assessment of selfdetermined motivation in a specific event as it is the case in our study. The SIMS is composed of four subscales and begins with the following sentence: "Why do you participate in this marathon?" Four items measured intrinsic motivation (e.g., "because I think that this activity is interesting"), four items measured identified extrinsic motivation (e.g., "because I am doing it for my own good"), four items measured external regulation motivation (e.g., "because I am supposed to do it"), and four items measured amotivation (e.g., "I don't know; I don't see what this activity brings me"). Participants answered on a 7-point scale from (1) totally disagree to (7) totally agree. In this study, each subscale presented an acceptable internal consistency (.72< as >.88). The average of items of each subscale was then used to calculate the self-determination index (e.g., [START_REF] Li | Testing the simplex assumption underlying the Sport Motivation Scale: A structural equation modeling analysis[END_REF]. The intrinsic motivation score was assigned the highest positive weight (+2). Identified regulation was assigned a lower weight (+1). The score for external regulation was assigned a negative weight (-1), and amotivation was weighted highly negatively (-2).

Perceived susceptibility to marathon-related injury.

Participants responded to the same four items used in Study 1. The scale showed good reliability (a=.86). A principal-axis factor analysis showed that items loaded on a single factor that explained 74.98% of the variance, with factor loadings ranging from .81 to .91.

Situational susceptibility to adopt a risky behaviour. In line with measures developed

in research on injury prevention [START_REF] Deroche | Athletes' inclination to play through pain: A coping perspective[END_REF] and doping [START_REF] Lazuras | Predictors of doping intentions in elite-level athletes: A social cognition approach[END_REF], participants responded to the following question: "During the marathon, if you feel a pain that you think might lead to injury, please indicate until what pain intensity you would continue running at the same pace." Answers were provided on a 7-point scale ranging from (1) very low to (7) very high. Assessing susceptibility with a one-item measure is a common practice which has proved to be valid in the exercise domain [START_REF] Courneya | Predicting repeated behavior from intention: The issue of scale correspondence[END_REF].

Past experiences of injury. Participants responded to the following question (e.g., [START_REF] Deroche | Predictors of perceived susceptibility to sport-related injury[END_REF]: "During the past 12 months, how many times have you been injured while running (injury necessitating to stop running for at least one session)?" Data analysis. Multiple regression analyses were carried out to test our hypotheses.

We first examined whether motivational variables predicted perceived susceptibility to injury and to keep running through pain, by testing the role of the self-determination index, then the role of autonomous and controlled motivation separately, and finally the role of each form of motivation. Next, the mediating role of perceived susceptibility to keep running through pain was tested with the a and b joint significance test [START_REF] Cohen | Applied multiple regression/correlation analyses for the behavioral sciences[END_REF]. The joint significance test consists in testing mediation by estimating a (effect of the predictor on the process) and b (effect of the process on the outcome, controlling for the predictor), and testing them individually against zero. There is a significant indirect effect when both a and b are significant, and a significant mediation when a, b, and the total effect of the predictor on the criterion are significant. This test was chosen because it is the only test that does not suffer from the Type I error issue, and it is statistically as powerful as other commonly used tests, such as the percentile bootstrap and numerical integration tests [START_REF] Judd | Mediation and moderation[END_REF].

Results and Discussion

Descriptive statistics and correlations among the measures are presented in Table 2.

First, as in Study 1, results showed that the self-determination index negatively predicted perceived susceptibility to injury, β=-.10, p=.049, but Study 2 goes beyond Study 1 by observing this relationship after controlling for past experiences of injury, β=.38, p<.001.

Importantly, the self-determination index also negatively predicted perceived susceptibility to keep running through pain during the marathon, β=-.11, p=.045, after controlling for past experiences of injury, β=.16, p=.003. In other words, the more runners were self-determined toward marathon, the less they perceived themselves as susceptible to adopt a risky behaviour during the race. Moreover, when the self-determination index, perceived susceptibility to keep running through pain, and past experiences of injury were simultaneously entered as predictors of perceived susceptibility to injury, the perceived susceptibility to keep running through pain was a significant and positive predictor, β=.22, p<.001, along with past experiences of injury, β=.34, p<.001, whereas self-determination was not significant, β=-.07, p=.153. In sum, given that the predictor (self-determination) significantly affected the criterion (perceived susceptibility to injury) and the mediator (perceived susceptibility to keep running through pain), and that the mediator significantly affected the criterion after controlling for the predictor, this suggests that the perceived susceptibility to keep running through pain mediated the relationship between self-determination and perceived susceptibility to injury. In other words, the more runners were self-determined for marathon, the less they perceived themselves as susceptible to take the risk to be injured during the race; in turn, the less they were susceptible to adopt a risky behaviour, the less they perceived a risk to be injured during the race.

Next, we examined the role of autonomous and controlled motivation separately.

Multiple regression analyses showed that neither autonomous motivation, β=-.07, p=.172, nor controlled motivation, β=.07, p=.176, significantly predicted perceived susceptibility to injury, after controlling for past experiences of injury, β=.38, p<.001. However, while autonomous motivation did not significantly predict perceived susceptibility to keep running through pain, β=-.05, p=.329, controlled motivation marginally predicted it, β=.10, p=.074, after controlling for past experiences of injury, β=.16, p=.004. In other words, the more runners endorsed a controlled motivation toward marathon, the more they tended to be susceptible to keep running through pain. Given that the a path was significant, we examined whether the b path was also significant. Results confirmed that the perceived susceptibility to keep running through pain significantly predicted perceived susceptibility to injury, β=.22, p<.001, after controlling for controlled motivation, β=.05, p=.342, and past experiences of injury, β=.34, p<.001. Overall, these results suggest an indirect effect of controlled motivation on perceived susceptibility to injury, through its impact on perceived susceptibility to keep running through pain.

Finally, we examined the role of each form of motivation separately. Concerning perceived susceptibility to injury, results indicated that intrinsic motivation (β=-.06, p=.215), identified regulation (β=-.08, p=.106), and external regulation (β=-.00, p=.985) were not significant predictors (after controlling for past experiences of injury, β=.38, p<.001), while amotivation was a marginal predictor (β=.10, p=.054). In other words, the more runners were amotivated, the more they tended to perceive they were at risk of injury. Results concerning perceived susceptibility to keep running through pain showed that it was not significantly predicted by neither intrinsic motivation (β=-.05, p=.367), identified regulation (β=-.07, p=.222), nor amotivation (β=.07, p=.174), after controlling for past experiences of injury, β=.16, p=.004. In contrast, external regulation significantly predicted perceived susceptibility to keep running through pain (β=.11, p=.049, after controlling for past experiences of injury, β=.15, p=.006). In other words, the more runners adopted an external regulation, the more they were susceptible to adopt a risky behaviour during the race. Given that the a path was significant, we examined the b path. Results confirmed that the perceived susceptibility to keep running through pain significantly predicted perceived susceptibility to injury, β=.23, p<.001, after controlling for external regulation, β=-.03, p=.563, and past experiences of injury, β=.34, p<.001. Overall, these results suggest an indirect effect of external regulation on perceived susceptibility to injury, through its impact on perceived susceptibility to keep running through pain.

Discussion

The present study examined the role of self-determined motivation in predicting injury prevention-related variables during an intense competitive sport (i.e., marathon). Results of two studies showed that the more runners were self-determined for marathon, the less they perceived a risk to be injured during the race. Two competing explanations of this relationship were examined. On the one hand, self-determined individuals engage in the activity willingly and have thus a flexible behavioural engagement [START_REF] Ryan | On assimilating identities to the self: a self-determination perspective on integration and integrity with cultures[END_REF]. Therefore, they may be able to initiate preventive actions as soon as they feel that the activity is hazardous for them, resulting in a low perceived susceptibility to injury (e.g., [START_REF] Masters | The relations of cognitive strategies with injury, motivation, and performance among marathon runners: Results from two studies[END_REF]. In this case, adopting a self-determined motivation would be health protective. A concurrent explanation is that self-determined individuals may be less likely to perceive negative aspects of their sport such as the risk it may present for their health (e.g., [START_REF] Sachs | Running addiction[END_REF]. One could posit that self-determined motivation negatively predicted perceived susceptibility because self-determined individuals were less likely to perceive the risk running a marathon may present for their health. In this case, self-determined motivation could be health damaging.

The results provided support for an adaptive interpretation of the negative relationship between self-determined motivation and perceived susceptibility to injury. Study 2 showed that the more runners were self-determined, the less they were susceptible to engage in a risky behaviour during the race (i.e., keeping running through pain), which was a proxy of perceived susceptibility to injury. In addition, the predictive role of self-determination was mostly driven by controlled forms of motivation, and more particularly external regulation. In other words, the less runners were externally regulated, the less they felt susceptible to keep running through pain, and the less they perceived themselves as susceptible to be injured.

These findings suggest that self-determined individuals, who have more control over their behaviours, could pay more attention to their bodily signs of pain and fatigue, while non self-determined individuals, who feel coerced by external pressures would be more exposed to possible injuries. For example, if a runner feels an alerting pain during the race, this person might not want to abandon if he/she is motivated for external reasons such as the pride to finish a marathon while it is more likely that this person stops if he/she runs for the pleasure, as there is no pleasure to run with a continuous feeling of pain. Taken together, these findings indicate that self-determined motivation for sport may be a factor of engagement in preventive behaviours with regard to injury. However, this research presents some limitations.

We did not measure the occurrence of actual injuries nor actual preventive behaviours, but instead situational susceptibility to adopt a risky behaviour. When questioning participants about their feeling of pain that might lead to injury, they may have understood the question differently, depending on their self-definition of injury. Finally, we used a cross-sectional design, which does not allow us to draw conclusions about causality. Future research should be conducted to confirm these results based on a prospective design and behavioural measures (i.e., objective injuries and objective preventive behaviours).

Perspective

While the role of self-determination for engaging in healthy behaviours has been examined (e.g., [START_REF] Chan | Self-determined forms of motivation predict sport injury prevention and rehabilitation intentions[END_REF], the present study went a step further by showing that self-determination for sport may also predict adoption of preventive behaviours during competition. Concerning practical implications, our findings can provide a valuable contribution to injury prevention in marathon. While aerobic exercise has many health benefits, marathon running is also quite constraining for the body with a yearly incidence rates for injury reported to be as high as 90% in those training for marathons [START_REF] Fredericson | Epidemiology and aetiology of marathon running injuries[END_REF]. By adopting self-determined motivations, runners can reduce their amount of risky behaviors and in turn the likelihood to be injured. A strength of self-determination theory is to consider motivation as depending on contextual factors, which may improve one's feelings of self-determination in sport. For example, by acting in an autonomy supportive way, the coach and the relatives can lead the runner to focus more on the personal benefits of running (e.g., in terms of health) and the satisfaction and pleasure experienced during the race (see [START_REF] Mageau | The coach-athlete relationship: A motivational model[END_REF]. These strategies may be used in a promising manner in the domain of injury prevention in competitive sports.

  Procedure. The procedure of Study 2 was mostly the same as in Study 1. It was carried out during the following edition of the "Marathon des Alpes-Maritimes" in 2010. Three-hundred-and-thirty-nine participants (79 women; Mage=43.02 years, SD=10.18) took part in Study 2 by filling out a computerized anonymous questionnaire. Among them, 118 participants ran everyday and 222 ran 2-3 times a week, during 73 minutes on average (SD=23.92), and since 12 years on average (SD=9.53).

Table 1

 1 Descriptive Statistics and Correlations Among the Variables (Study 1) .21 .04 .53 *** .48 *** .48 *** -6. Introjected Regulation 3.70 1.34 .06 .30 *** .33 *** .41 *** .49 *** -7. External Regulation 2.79 1.36 .11 * .28 *** .23 *** .25 *** .52 *** .57 *** -.57 *** .67 *** .58 *** .41 *** -.05 -.15 ** -.66 *** .67 *** -.54 *** -Note. Sex was coded as follows: male = 1, female = 2. * p < .05; ** p < .01; *** p < .001. Descriptive Statistics and Correlations Among the Variables (Study 2) Note. Sex was coded as follows: male = 1, female = 2. † † p < .10; † p < .06; * p < .05; ** p < .01; *** p < .001.

	Variable	Mean SD	1	2	3	4	5	6	7	8	9	10	11	12
	1. Perceived susceptibility to injury	2.43 1.09 -
	2. Intrinsic Motivation Knowledge	4.03 1.36 -.05	-
	3. Intrinsic Motivation Achievement	5.17 1.09 -.02 .64 ***	-
	4. Intrinsic Motivation Stimulation	5.23 1.16 -.01 .45 *** .64 ***	-
	5. Identified Regulation 3.63 18. Amotivation 1.69 0.95 .12 * .04	-.13 *	-.08	.19 ***	.28 *** .36 ***	-
	9. Autonomous Motivation Index	13.25 2.92 -.01 .81 *** .82 *** .76 *** .83 ***	.49 *** .43 *** .04	-
	10. Controlled Motivation Index	6.63 2.58 .13 * .18 *** .05	.11 *	.40 ***	.61 *** .67 *** .90 *** .27 ***	-
	11. Self-determination Index 6.62 3.34 -.11 12. Sex ---.01 .07	.11 *	.10	.09	.04	.02	.11 *	.03	.08	-	-

*

Acknowledgement

The authors wish to express their gratitude to "Azur Sport Organisation" and to the undergraduate students who helped in collecting the data.