E Delavaud
email: elise.delavaud@sed.ethz.ch

F Cotton

C Beauval

S Drouet

S Akkar

M A Sandikkaya

M Segou

F Scherbaum

L Danciu

J Douglas

R Basili

E Faccioli

Keywords: Logic trees, Ground-motion prediction equations, Expert judgment, Model selection, Seismic hazard assessment

. Results of both approaches were taken into account to commonly select the smallest set of GMPEs to capture the uncertainty in ground-motion prediction in Europe. For stable continental regions, two models, both from Eastern North America, have been selected for shields and three GMPEs from active shallow crustal regions have been added for continental crust. For subduction zones, four models, all non-European have been chosen. Finally, for active shallow crustal regions, we selected four models, each of them from a different host region but only two of them were kept for long periods. In most cases, a common agreement has been also reached for the weights. In case of divergence, a sensitivity analysis of the weights on the seismic hazard has been conducted, showing that once the GMPEs have been selected, the associated set of weights has a smaller influence on the hazard.

 Basin, 1996Basin, -2000) )project, the Seismic HAzard haRmonization in Europe (SHARE) project (http://www.share-eu.org) is one of the large international research initiatives that have been recently launched to harmonize hazard estimates across political boundaries and to derive procedurally-consistent pan-national hazard models. As a regional program of the Global Earthquake Model (GEM) project (http://www.globalquakemodel.org), the SHARE project aims at defining methods for seismic hazard and loss assessment in the Euro-Mediterranean region which will become standards at global and regional scales.

The team responsible for ground-motion prediction in the SHARE project (WP4 group, see Table 1) has been working on the definition of a reference European model that captures the complete center, body and range of possible ground motions in Europe and tackles the unresolved question of regional variations in ground motions. The construction of logic trees that express this variability and the associated epistemic uncertainty is a multi-step procedure that required a common effort in characterizing ground shaking in Europe and identifying reliable equations for the prediction of ground motion parameters of interest together with measures of uncertainties.

Within the large geographical area covered by SHARE there is wide variation in terms of the magnitude and distance ranges that influence probabilistic seismic hazard assessment (PSHA). PSHA in Europe is not only controlled by large magnitudes. As a matter of fact, probabilistic disaggregation analyses indicate that in many seismically-active regions of Europe, the seismic hazard for 10% exceedence in 50 years level and short vibration periods (fundamental for Ultimate Limit State verifications of most structures) tends to be controlled by nearby earthquakes (distance < 20 km) in the 4.5to-5.5 magnitude range. [START_REF] Beauval | Can Strong Motion Observations be Used to Constrain Probabilistic Seismic Hazard Estimates?[END_REF] showed that in active regions of France, magnitudes 4 to 5 are also responsible for a non-negligible contribution to the hazard even for return periods as large as 10,000 years. The above observation is commonly found, for example, for Italian sites in active areas and, obviously, is particularly true for low seismicity regions, i.e. north of the Alps [START_REF] Faccioli | Seismic Hazard Mapping for Italy in Terms of Broadband Displacement Response Spectra[END_REF]. This is why ground-motion prediction equations (GMPEs) that include events with magnitudes down to Mw 4 in their datasets should also be considered. Concerning the influence of the larger magnitudes on hazard, evaluations tend to depend to some extent on the model of earthquake sources adopted, e. g. whether extended zones, or fault sources, or smoothed seismicity. In active regions of Southern Italy (notably the Calabrian Arc), where maximum historical magnitudes exceed 7 and which are certainly among the most active in the Mediterranean area, recent seismic hazard studies for sites lying within source zones indicate that magnitudes > 6.5 in the distance range within 20 km dominate hazard only at return periods of 5000 years and vibration periods of 1 s and larger [START_REF] Faccioli | Seismic Hazard Mapping for Italy in Terms of Broadband Displacement Response Spectra[END_REF]. In such regions, even at return periods as large as 1500 years and at vibration periods of 1 s hazard is typically controlled by magnitudes ≤ 6.0 in the short distance range. On the other hand, again considering Southern Italy, sites lying at few tens of km from the boundaries of model source zones are mostly affected by M > 6.5 starting from return periods of 1000 years or so, in the whole range of vibration periods. If smoothed seismicity representations are used instead, things change to some extent, in that -for a given return period -lower magnitudes tend to dominate.

In this paper, we describe the methodology that we adopted to build the logic tree for PSHA in Europe. This process is illustrated in Figure 1. The goal of this strategy is to identify the smallest set of GMPEs to capture the epistemic uncertainty in ground-motion prediction in Europe. The particularity of our approach is that we do not only take into account the judgment of experts to select and rank models but we also use data to guide our choice and weights. Thanks to an increasing amount of strong-motion data, data driven guidance is indeed now feasible and can give valuable information about the ability of GMPEs to predict ground motion in different regions (e.g., [START_REF] Drouet | Selection and ranking of ground motion models for seismic hazard analysis in the Pyrenees[END_REF][START_REF] Allen | Evaluation of Ground-Motion Modeling Techniques for Use in Global shakemap -A Critique of Instrumental Ground-Motion Prediction Equations, Peak Ground Motion to Macroseismic Intensity Conversions, and Macroseismic Intensity Predictions in Different Tectonic Settings[END_REF][START_REF] Delavaud | Testing the Global Applicability of Ground Motion Prediction Equations for Active Shallow Crustal Regions[END_REF].

The structure of this paper follows the adopted procedure. First, we show how a list of GMPEs for each tectonic regime was selected from the many existing models using exclusion criteria. In a second step, we describe the expert judgment, including the conditions imposed to the experts and the weights that they chose. We pay particular attention to the rationale behind the choices of the experts and expose what we learnt from them. The third step consists in the testing of the candidate GMPEs using data. GMPEs are ranked according to a criteria, the negative average sample log-likelihood, proposed by [START_REF] Scherbaum | Model Selection in Seismic Hazard Analysis: an Information-Theoretic Perspective[END_REF] and based on information theory. In particular, we estimated to what degree the data support or reject a model with respect to the state of non-informativeness defined by uniform weighting. The following section describes how GMPEs were selected taking expert judgment and data testing into account and different sets of weights proposed as a consensus within the SHARE WP4 group. Finally, we present the results of a sensitivity analysis of the weights on hazard maps to help us make a final choice. The goal of this paper is to make the SHARE GMPE logic tree methodology transparent and reproducible.

Pre-selection of Ground-Motion Prediction Equations

The pre-selection of GMPEs is first guided by the seismotectonic description of the area covered by the SHARE project (Figure 2). The SHARE source model also influences ground-motion prediction, especially in terms of distance calculation. This source model combines modern source types (area, fault and point sources) within a logic tree to account for the inherent uncertainty in the expert views on seismicity. The source logic tree considers the different source types within the principal methodologies used: the zone-based [START_REF] Cornell | Engineering seismic risk analysis[END_REF]) and the kernel-smoothed approach [START_REF] Grünthal | Earthquake model for the European-Mediterranean Region for the purpose of GEM1[END_REF][START_REF] Hiemer | Stochastic Earthquake Source Model: Satisfying Accepted Laws[END_REF]. Final details on the source models can be found within the reports of the SHARE project at http://www.share-eu.org or within a yet to be written manuscript on the new Euro-Mediterranean hazard model.

The pre-selection of GMPEs was realized from an already compiled list by [START_REF] Douglas | Further errata of and additions to "ground motion estimation equations 1964-2003[END_REF] that contains over 250 published ground-motion models, to retain a subset of the most robust equations for all the existing seismotectonic regimes in the wider European region. Six broad tectonic domains were identified for groundmotion prediction to represent the region covered by the SHARE project (Figure 2): stable continental regions (SCR) include the shield (Baltic) where Precambrian crystalline igneous or metamorphic rocks crop out and is characterized by low wave attenuation and a low deformation rate, and continental crust (most of Europe and Africa) with low deformation rate; oceanic crust include mainly the eastern Atlantic and small patches in the Mediterranean Sea; active shallow crustal regions (ASCR) mainly outline the plate boundaries but occur also in the continental interiors at places with significant rate of deformation; subduction zones (SZ) such as the Calabrian, Hellenic, and Cyprus arcs; areas of deep focus non-subduction earthquakes, such as Vrancea (Romania) or the Betics (Spain); and active volcanoes. For this pre-selection, it was decided to apply the seven exclusion criteria proposed by [START_REF] Cotton | Criteria for selecting and adjusting ground-motion models for specific target regions: Application to Central Europe and rock sites[END_REF], briefly: 1. the model is from a clearly irrelevant tectonic regime, 2. the model is not published in an international peer-reviewed journal, 3. the documentation of model and its underlying dataset is insufficient, 4. the model has been superseded by more recent publications, 5. the frequency range of the model is not appropriate for engineering application, 6. the model has an inappropriate functional form and 7. the regression method or regression coefficients are judged to be inappropriate. From the existing GMPEs, six models remained for SCR, eight for SZ, nineteen for ASCR including six regional or local models, one model for volcanic zones [START_REF] Mcverry | New Zealand acceleration response spectrum attenuation realtions for crustal and subduction zone earthquakes[END_REF] and one for areas of deep focus non-subduction earthquakes [START_REF] Sokolov | Ground-motion prediction equations for the intermediate depth Vrancea (Romania) earthquakes[END_REF]. No model for the prediction of ground motions from oceanic crustal earthquakes were available in the international literature, but models for ASCR and SCR have been suggested to account for such seismotectonic regimes. The engineering needs, evaluated at the beginning of the project, were taken into account by favoring models well calibrated in the period range between 0.02 s to 10 s. Most of existing models are however not applicable for periods greater than 3 s and, hence, a specific logic tree was built to ensure PSHA computations for periods between 3 s and 10 s.

As a second step, these pre-selected GMPEs have been analyzed and compared in order to identify their weaknesses and limitations in the light of the criteria set proposed by [START_REF] Bommer | On the Selection of Ground-Motion Prediction Equations for Seis-mic Hazard Analysis[END_REF]. Considering the rapid increase in published GMPEs for ASCR (in particular), [START_REF] Bommer | On the Selection of Ground-Motion Prediction Equations for Seis-mic Hazard Analysis[END_REF] updated the exclusion criteria of [START_REF] Cotton | Criteria for selecting and adjusting ground-motion models for specific target regions: Application to Central Europe and rock sites[END_REF] to reflect the state-ofthe-art in ground-motion prediction. The new exclusion criteria especially aim at identifying the robust and well-constrained models based on new quality standards in the formulation and derivation of models as well as considering their applicability range in terms of spectral ordinates, magnitude and distance. In particular, magnitude and distance ranges should be large enough so that the need for extrapolations when conducting PSHA is minimized. In addition, the number of earthquakes per magnitude and the number of records per different distance intervals should be maximized. A detailed comparative study between the models of each tectonic regime has been conducted. In particular, they compared the predicted ground-motion amplitudes by the GMPEs for different scenarios (in terms of magnitude, distance and period) and extracted the main characteristics of the models that are summarized in Table 2 for SCR, in Table 3 for SZ and in Table 4 for ASCR. In these tables, we report the type and range for the magnitude and the distance, the spectral period band as well as the inclusion of PGA and/or PGV estimations by the candidate GMPEs. We also indicate whether the site classification is based on a continuous function of v S30 or on generic site classes in terms of v S30 intervals. The tables also list the horizontal component definitions of the GMPEs that vary among the models. Moreover some of the models do not use styleof-faulting as a predictor variable. In order to combine the GMPEs within a logic tree framework, component and style-of-faulting adjustments have been performed. Horizontal components are converted using the conversion coefficients determined by [START_REF] Beyer | Relationships between Median Values and between Aleatory Variabilities for Different Definitions of the Horizontal Component of Motion[END_REF]. For models which do not consider the style-offaulting, adjustment factors depending on the proportions of normal and reverse events in the underlying database of each model are applied using the approach proposed in [START_REF] Bommer | Style-of-faulting in ground-motion prediction equations[END_REF]. These factors are given in the tables except for SZ models, which do not explicitly consider the style-of-faulting but differentiate inslab earthquakes from interface earthquakes, taking into account both the mechanism and the depth of the earthquake. The effects of the adjustment strategies are presented in [START_REF] Drouet | Deliverable 4.2: Regionally adjusted Ground Motion Prediction Equations (GMPE) for Europe[END_REF]. Finally, we did not take non-linear effects into account as the seismic hazard is computed for rock sites. Our model selection is thus focusing on the rock part of GMPEs.

As a result of the model analysis the members of the WP4 team discussed their concerns about the predictive ability and the weaknesses of the models. Reservations were expressed for the Next Generation Attenuation (NGA) models of [START_REF] Abrahamson | Summary of the Abrahamson & Silva NGA Ground Motion Relations[END_REF], [START_REF] Campbell | NGA Ground Motion Model for the Geometric Mean Horizontal Component of PGA, PGV, PGD and 5%-Damped Linear Elastic Response Spectra for Periods Ranging from 0.01 to 10 s[END_REF] and [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] because they require too many estimator parameters that are not available in all European earthquakeprone regions. This limitation can be partially overcome by using the a priori estimations of these unknown input parameters given by the methodology proposed in [START_REF] Kaklamanos | Estimating Unknown Input Parameters when Implementing the NGA Ground-Motion Prediction Equations in Engineering Practice[END_REF]. Models that suffer from restrictions in the definition of predictive variables were also questioned: the SZ model of [START_REF] Garcia | Inslab Earthquakes of Central Mexico: Peak Ground-Motion Parameters and Response Spectra[END_REF] only considers inslab earthquakes, [START_REF] Atkinson | Predicted Ground Motions for Great Interface Earthquakes in the Cascadia Subduction Zone[END_REF] only consider large interface earthquakes, the NGA model of [START_REF] Idriss | An NGA Empirical Model for Estimating the Horizontal Spectral Values Generated By Shallow Crustal Earthquakes[END_REF] does not cover sites with v S30 values less that 450 m/s and finally, [START_REF] Kanno | A New Attenuation Relation for Strong Ground Motion in Japan Based on Recorded Data[END_REF] do not provide proportions of normal and reverse faulting events thereby preventing style-of-faulting adjustments. The model of [START_REF] Cotton | On the Discrepancy of Recent European Ground-Motion Observations and Predictions from Empirical Models: Analysis of KiK-net Accelerometric Data and Point-Sources Stochastic Simulations[END_REF] was also questioned because it does not include style-of-faulting as a predictor variable and they might have included subduction interface events in their ASCR dataset since the selection of earthquakes was based on focal depth only. Finally, reservations were expressed for the models of [START_REF] Tavakoli | Empirical-Stochastic Ground-Motion Prediction for Eastern North America[END_REF] and of [START_REF] Ambraseys | Equation for the Estimation of Strong Ground Motions from Shallow Crustal Earthquakes Using Data from Europe and the Middle East: Horizontal Peak Ground Acceleration and Spectral Acceleration[END_REF] as they are similar to [START_REF] Campbell | Prediction of Strong Ground Motion Using the Hybrid Empirical Method and Its Use in the Development of Ground-Motion (Attenuation) Relations in Eastern North America[END_REF] and [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF] respectively, and for the model of Özbey et al. (2004) which was derived from data from the 1999 earthquakes in Turkey that were recorded within a limited area. Although these GMPEs suffer from the above restrictions, they were considered by the experts and in the testing.

There are obviously large differences in terms of number of GMPEs between the three tectonic regimes and also in terms of their host regions. For stable continental regions, all the GMPEs except [START_REF] Douglas | Ground-motion prediction equations for Southern Spain and Southern Norway obtained using the composite hybrid model perspective[END_REF] were derived from eastern North America (ENA) data whereas ASCR regime includes GMPEs from all over the world (although mainly California, Europe and Japan). The selection for subduction zones includes no European candidate ground-motion model.

Finally, we acknowledge that the pre-selected GM-PEs may not be calibrated homogeneously in all the parameter space relevant for PSHA in Europe, especially for low magnitudes (Mw < 5). The development of new GMPEs was beyond the scope of the present project and for this reason and the lack of homogeneous local datasets we have not applied the calibration of existing models using local data, e.g. following the method by [START_REF] Scasserra | A Comparison of NGA Ground-Motion Prediction Equations to Italian Data[END_REF]. We recognize that there is a need to start to apply these new methods, at least in Turkey and Italy, where local data are available.

The pre-selected GMPEs have been subsequently evaluated in two parallel steps. In one approach, a group of experts were asked to propose the logic tree weights for these GMPEs. In the other approach, ground-motion records, whenever available, were used to test the ability of the GMPEs to model the overall trends of the ground motions in each tectonic region. We present now these two complementary approaches.

Expert judgment

Since the 1970s, there has been an increasing interest in the use of expert judgment to help in decision making by formalizing and quantifying expert judgment on problems that involve uncertainties (e.g., [START_REF] Cooke | Experts in Uncertainty: Opinion and Subjective Probability in Science[END_REF][START_REF] Goossens | Expert judgment for a probabilistic accident consequence uncertainty analysis[END_REF]. Private agencies as well as academia have recourse to this approach especially when data are unavailable or inadequate, and this is why it has been, until recently, the only method used to select and weigh GMPEs for probabilistic seismic hazard assessment. Although guidance for expert judgment is given by the Senior Seismic Hazard Analysis Committee (SSHAC) in [START_REF] Budnitz | Recommendations for Probabilistic Seismic Hazard Analysis: Guidance on Uncertainty and Use of Experts[END_REF], there is no clear standard procedure for the selection and weighting of GMPEs by experts. In this section, we present the strategy that we adopted for the determination of logic-tree weights from a group of experts.

We composed a group of six experts working in five different countries in academia or public institutions in Europe (see Table 1). Six seemed to be a good number, to have enough different points of view without too much redundancy. They were chosen for their great experience with GMPEs (e.g. some of them developed GMPEs) and also for their experience of PSHA in specific countries such as Italy, France or Greece. Four people (the first four authors of the present paper) defined the guidelines and the processing of the expert judgment. We asked the experts to propose logic-tree weights expressing their degree of belief in the ability of candidate GMPEs to predict earthquake ground motions in different tectonic regimes in Europe. We did not guide the experts in how to assign their weights (e.g., whether all GMPEs should be weighted or not), since no clear methodology was available at that time. However, we asked them to explain the rationale behind their weighting strategy. Weights had to be assigned for each tectonic regime (with a differentiation between shield and continental crust for SCR and between interface and inslab earthquakes for SZ) for different ranges in spectral frequency (f), magnitude (Mw) and distance (d): f ≤ 1/3 Hz, 1/3 Hz <f ≤ 25 Hz or f > 25 Hz, Mw ≤ 5 or Mw > 5 and d ≤ 10 km, 10 km < d ≤ 100 km, 100 km < d ≤200 km or d > 200 km. By doing so, we wanted to see whether or not the experts would consider the frequency-magnitude-distance dependency for the logic-tree weights. Finally, the experts did not communicate with each other to have independent alternatives for logic-tree weights. They also did not know about the testing results conducted by using the empirical data (see next section). However, they were provided with the characteristics of the GMPEs are presented.

A lesson that we learned from the experts is that assigning weights to GMPEs is not straightforward, especially for such a large number of models and with distinctions in terms of frequency, magnitude and distance. We also realized that the time limit imposed for the entire decision process was quite limited for this task (five weeks). In particular, some experts raised the possibility of assigning zero weights to some of the GMPEs, indirectly raising the question of what logic tree weights represent. This led to a lively discussion between the WP4 members and the experts during a meeting where experts presented their weighting strategy. Experts had a common approach: they selected a set of models which enabled them to capture epistemic uncertainty as much as possible. For some of the experts, a small number of GMPEs (two to four) was sufficient (not all models are used although they could be appropriate). On the other hand, some experts selected many or all the candidate GMPEs assigning small weights (< 0.1) for the less favorable ones. Although logic trees are now widely used, we realized that it is not clear yet how weights should be assigned and what they should be assumed to represent. [START_REF] Scherbaum | Logic Tree Branch Weights and Probabilities: Summing Up to One Is Not Enough[END_REF] recently showed the importance of treating logic tree weights as probabilities instead of simply as generic quality measures of GMPEs, which are subsequently normalized. In particular, they show the danger of using a performance/grading matrix approach (independently assigning of grades to different quality criteria) where the normalization process can lead to an apparent insensitivity to the weights. In order to achieve consistency with a probabilistic framework, [START_REF] Scherbaum | Logic Tree Branch Weights and Probabilities: Summing Up to One Is Not Enough[END_REF] proposed to assign weights in a sequential fashion (e.g., if the first GMPE of three selected gets a 0.6 weight, then the sum of the weights for the two remaining models is 0.4).

The first conclusion of the experts was that the number of selected GMPEs should be kept as small as possible (between two and five) to prevent the logic tree for ground-motion prediction being too complex, which is especially important for such a wide area considered by the SHARE project. In addition, most of the experts gave weights that are independent of the magnitude, distance and frequency, except for long periods (3 s < T ≤ 10s) for ASCR. The main motivation behind this choice was to prevent having a discontinuity due to the transition from one logic tree to another one in the uniform hazard spectra produced by PSHA. The experts selected GMPEs which are sufficiently robust to cover a wide range of magnitudes, distances and spectral periods. Such GMPEs are indeed better able to capture the magnitude scaling of ground motion that decreases when magnitude increases [START_REF] Cotton | On the Discrepancy of Recent European Ground-Motion Observations and Predictions from Empirical Models: Analysis of KiK-net Accelerometric Data and Point-Sources Stochastic Simulations[END_REF][START_REF] Atkinson | Observations on Regional Variability in Ground-Motion Amplitudes for Small-to-Moderate Earthquakes in North America[END_REF]. Moreover, [START_REF] Bommer | The Influence of Magnitude Range on Empirical Ground-Motion Prediction[END_REF] strongly recommended not to apply GMPEs outside and even close to their magnitude limits. Finally, GMPEs developed from limited datasets are more likely to incorporate random earthquake effects (biases) into their models. Therefore, global predictive models were preferred as compared to regional ones. Finally, experts assigned equal weights for the models that they are not familiar with or for which they lack sufficient information.

For stable continental regions, selecting the models was a particular challenge as all but one are derived for ENA and inadequate information about their applicability for Europe is known. Experts made no distinctions between shield and continental crust. Of the six GMPEs proposed for SCR, the expert selection ranged between retaining three or all six. The SCR models by [START_REF] Campbell | Prediction of Strong Ground Motion Using the Hybrid Empirical Method and Its Use in the Development of Ground-Motion (Attenuation) Relations in Eastern North America[END_REF] and [START_REF] Toro | Model of strong ground motions from earthquakes in central and eastern North America: best estimates and uncertainties[END_REF] were selected by all experts. The choice of GMPEs for subduction zones was also a challenging task since none of the pre-selected GMPEs were derived using empir-ical data from Europe. Due to this particular reason, the experts preferred choosing the global SZ models that cover a wide range of magnitude and distance intervals. Only one expert considered interface and inslab models separately in ranking whereas three experts considered the spectral period ranges while assigning weights. The experts selected two to five GM-PEs among the eight models proposed for SZ. Only one model was selected by all the experts, the model of [START_REF] Atkinson | Empirical Ground-Motion Relations for Subduction Zone Earthquakes and Their Application to Cascadia and Other Regions[END_REF]. Contrary to the SCR and SZ regions, the excessive number of GMPEs for the shallow crustal active regions challenged the expert judgment. Regional ASCR models (e.g., [START_REF] Massa | Empirical Ground-Motion Prediction Equations for Northern Italy Using Weak-and Strong-Motion Amplitudes, Frequency Content, and Duration Parameters[END_REF][START_REF] Kalkan | Empirical Attenuation Equations for Vertical Ground Motion in Turkey[END_REF][START_REF] Danciu | Engineering Ground-Motion Parameters Attenuation Relationships for Greece[END_REF] were either excluded or given small weights (less than 0.1). The expert choices lean towards global and pan-European models in the ranking of ASCR GMPEs. Of the entire expert group only two of them considered magnitude-distance-frequency ranges while assigning weights. The experts selected between three to ten GMPEs among the 18 candidate models. The GMPEs by [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF] and of [START_REF] Boore | Ground Motion Prediction Equations for the Average Horizontal Component of PGA, PGV, and 5%-Damped PSA at Spectral Periods between 0.01 s and 10.0 s[END_REF] were the commonly selected models by all experts. Tables 8, 9 and10 summarize the choice of each expert for the three tectonic regimes under the section Category based on expert judgment.

Data Testing

To complement the expert opinions described above, testing of the candidate GMPEs against empirical data was undertaken. The goal of this phase was to judge the applicability of candidate models by evaluating their probability to have generated the available data. We used the data-driven method developed by [START_REF] Scherbaum | Model Selection in Seismic Hazard Analysis: an Information-Theoretic Perspective[END_REF] that implemented an information-theoretic approach for the selection and the ranking of GMPEs. The method derived a ranking criterion from the Kullback-Leibler (KL) divergence, which denotes the information loss when a model g defined as a distribution is used to approximate a reference model f [START_REF] Burnham | Model Selection and Multimodel Inference: a Practical Information-Theoretic Approach[END_REF]. The KL divergence between two models represented by their probability density functions f and g is defined as:

D(f, g) = E f [log 2 (f )] -E f [log 2 (g)] (1)
where E f is the statistical expectation taken with respect to f . In the case of GMPE selection, f represents the data-generating process (nature) and is only known through observations. Consequently, the term E f [log 2 (f )] called the self-information of f cannot be calculated. However, the second term, -E f [log 2 (g)], can still be approximated via the observations. This approximation is the negative average sample log-likelihood, noted LLH and defined by:

LLH(g, x) := - 1 N N i=1 log 2 (g(x i )) (2)
where x = {x i },i = 1, ..., N are the empirical data and g(x i ) is the likelihood that model g has produced the observation x i . In the case of GMPE selection, g is the probability density function given by a GMPE to predict the observation produced by an earthquake defined by a magnitude M (and by other characteristics such as the style of faulting) at a site i that is located at a distance R from the source. We used the LLH divergence as a criterion to rank the candidate GMPEs. Due to its negative sign, the negative average sample log-likelihood is not a measure of closeness but a measure of the distance between a model and the data-generating process. A small LLH indicates that the candidate model is close to the process that has generated the data while a large LLH corresponds to a model that is less likely of having generated the data.

In order to interpret the rankings, weights obtained from the LLH values were compared to the uniform weight w unif = 1 M , where M is the number of GM-PEs. This comparison tells us to what degree the data support or reject a model with respect to the state of non-informativeness. It is expressed by the data support index (DSI) which gives the percentage by which the weight of a model is increased (positive DSI) or decreased (negative DSI) by data. The DSI of model g i with LLH-value based weight w i is:

DSI i = 100 w i -w unif w unif , (3)
where

w i = 2 -LLH(gi,x) K k=1 2 -LLH(g k ,x) (4)
This ranking method has been recently used by [START_REF] Delavaud | Testing the Global Applicability of Ground Motion Prediction Equations for Active Shallow Crustal Regions[END_REF] to test the global applicability of GMPEs for active shallow crustal regions. The LLH divergence was computed for eleven GMPEs for different regions and magnitude and distance ranges to assess their validity domain.

The LLH-based weights defined by eq. ( 4) cannot be automatically regarded as probabilities as the LLH values are independently determined for each model (Kolmogorovs axioms of mutual exclusiveness and collective exhaustiveness are not respected) and only subsequently made to sum up to one (see [START_REF] Scherbaum | Logic Tree Branch Weights and Probabilities: Summing Up to One Is Not Enough[END_REF] for more details about this subject). Therefore, we advise not to directly use them as logic tree weights but to use them in combination with expert judgment. The purpose of using empirical data was not to replace expert judgment but rather to help the judgment process by providing additional information about the applicability of GMPEs, especially in regions where no indigenous model exists.

Although the amount of empirical ground-motion data is rapidly increasing worldwide, we faced some limitations while testing GMPEs with the compiled data. We considered the distribution of empirical groundmotion data in terms of magnitude and distance. We also accounted for the reduction in available data size in terms of their usable period range in order to reduce the filter cut-off influence on the spectral ordinates. The country-based distribution of the data was also considered to examine the similarities between the original datasets of the tested GMPEs and our database used for testing these GMPEs because we wanted to prevent biased evaluations due to similarities between our dataset and those of the GMPEs.

A homogenous dataset was not available for SCR and, therefore no tests for this tectonic regime were made.

For subduction zones, we had available a restricted dataset that only consisted of inslab strike-slip earthquakes along the Hellenic arc with a total number of 65 recordings. Moment magnitudes of SZ data range from 5.2 to 6.7, their depth mainly varies from 40 km to 90 km and the hypocentral distances are mostly from 70 km to 300 km. All the GMPEs from Table 3 have been tested against this Greek dataset except for the models of [START_REF] Atkinson | Predicted Ground Motions for Great Interface Earthquakes in the Cascadia Subduction Zone[END_REF] and [START_REF] Garcia | Inslab Earthquakes of Central Mexico: Peak Ground-Motion Parameters and Response Spectra[END_REF]. The former model only considers interface events with magnitudes greater than 7.5 whereas the latter model is only derived for inslab earthquakes and hence its applicability for PSHA is limited. None of the tested GMPEs used Greek data for their derivations. Rankings have been performed for pseudo-spectral accelerations (PSAs) at spectral periods between 0.05 s and 2 s. Table 5 shows the ranking using the chosen spectral periods. The first 2 models in the ranking are the models of [START_REF] Lin | Ground-Motion Attenuation Relationships for Subduction Zone Earthquakes in Northeastern Taiwan[END_REF] and [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF]. These modes were derived from the data pertaining to Northern Taiwan and Japan, respectively. They are equally supported by the Greek data with a DSI of about 29% each. On contrary, the global model of Atkinson and Boore (2003) appears particularly inconsistent with the present Greek dataset with a DSI of about -50%. The [START_REF] Atkinson | Empirical Ground-Motion Relations for Subduction Zone Earthquakes and Their Application to Cascadia and Other Regions[END_REF] model was sensitive to the changes in the period range considered as it predicted the Greek ground motions fairly well for periods below 0.16s. The overall results of the testing for subduction zones are summarized in Table 9 under the section Category based on data-testing results. They are presented in more detail in [START_REF] Beauval | GMPEs for PSHA calculations: the case of the Greek subduction zone (and other zones around the world)[END_REF].

For active shallow crustal regions, we considered two databases that are composed of recordings from Europe (DB1) and from other ASCR around the world (DB2). The majority of recordings in DB1 are in the magnitude and distance range: 4 < Mw < 7 and 1 km < RJB <200 km, respectively. Due to the lack of large magnitude events in DB1, GMPEs were also tested against DB2 which is mainly composed of magnitudes between 6 and 8. The main assumption while running these analyses was the weak regional dependence of GMPEs. Both databases were extracted from the SHARE strong-motion databank (Akkar et al., 2010) that is compiled from various original databases [START_REF] Ambraseys | Internet-Site for European Strong-Motion Data[END_REF][START_REF] Luzi | ITACA (ITalian ACcelerometric Archive): A Web Portal for the Dissemination of Italian Strong-motion Data[END_REF]Chiou et al., 2008;[START_REF] Cotton | On the Discrepancy of Recent European Ground-Motion Observations and Predictions from Empirical Models: Analysis of KiK-net Accelerometric Data and Point-Sources Stochastic Simulations[END_REF][START_REF] Sandikkaya | Site classification of Turkish national strong-motion stations[END_REF]). The European dataset is presented in Figure 3. It contains 1533 recordings, mainly from Turkey. The major reason for the larger number of Turkish recordings is the fact that the other European databases do not contain site classification in terms of v S30 and lack complete information on some of the distance measures used by the candidate GMPEs. Other observations mostly come from Italy and Greece. The non-European dataset is presented in Figure 4. It mostly contains recordings from the USA and Taiwan that were extracted from the NGA database. Rankings based on PSAs at five spectral periods (0.1 s, 0.2 s, 0.5 s, 1 s and 2s) are presented in Tables 6 and7 for DB1 and DB2 respectively. A total of 14 GMPEs have been tested: all the models in Table 4 except for the GM-PEs of [START_REF] Idriss | An NGA Empirical Model for Estimating the Horizontal Spectral Values Generated By Shallow Crustal Earthquakes[END_REF], [START_REF] Kanno | A New Attenuation Relation for Strong Ground Motion in Japan Based on Recorded Data[END_REF], Özbey et al. (2004), [START_REF] Mcverry | New Zealand acceleration response spectrum attenuation realtions for crustal and subduction zone earthquakes[END_REF] and [START_REF] Pankow | The SEA99 Ground-Motion Predictive Relations for Extensional Tectonic Regimes: Revisions and a New Peak Ground Velocity Relation[END_REF]. The Italian model of [START_REF] Bindi | Horizontal and vertical ground motion prediction equations derived from the italian accelerometric archive (ITACA)[END_REF] is ranked first, closely followed by two non-European models, [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] and [START_REF] Cotton | On the Discrepancy of Recent European Ground-Motion Observations and Predictions from Empirical Models: Analysis of KiK-net Accelerometric Data and Point-Sources Stochastic Simulations[END_REF], with DSIs larger than 60%. The testing results of the [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF] model appear to be quite robust as it is well ranked for both datasets DB1 and DB2. The model of [START_REF] Kalkan | Empirical Attenuation Equations for Vertical Ground Motion in Turkey[END_REF] and the NGA models are better able to predict ground motions from the large earthquakes that compose DB2 than ground motions from DB1. The results of the data testing for active shallow crustal regions are summarized in Table 10 under the section Category based on data-testing results.

Ground-Motion Logic Tree

Based on the results of both the expert judgment and the testing, a consensus set of GMPEs was determined for each tectonic regime. Tables 8, 9 and 10 were used to guide the selection, especially when expert judgment and empirical data testing results were available. Models supported by the empirical data testing and the experts' choices (first category) were selected while the models that were not supported by the data testing and not chosen by the experts (fourth category) have been rejected. For rest of the models (second and third categories), discussions were held between the experts and ground-motion modeling group to decide on their rejection or selection. Weights were also determined but different propositions were retained for sensitivity analyses.

For stable continental regions, for which no testing could be performed due to a lack of data, a distinction between shield and continental crust was taken into account. For shields, the two models supported by all the experts were selected with equal weights: [START_REF] Toro | Modification of the Toro et al[END_REF] model which is an updated version of [START_REF] Toro | Model of strong ground motions from earthquakes in central and eastern North America: best estimates and uncertainties[END_REF] and the model by [START_REF] Campbell | Prediction of Strong Ground Motion Using the Hybrid Empirical Method and Its Use in the Development of Ground-Motion (Attenuation) Relations in Eastern North America[END_REF]. For continental crust, three GMPEs for ASCR were adopted [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF][START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF][START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] in addition to the models of [START_REF] Toro | Modification of the Toro et al[END_REF] and [START_REF] Campbell | Prediction of Strong Ground Motion Using the Hybrid Empirical Method and Its Use in the Development of Ground-Motion (Attenuation) Relations in Eastern North America[END_REF]. This accounts for uncertainty in knowing if ground motions in continental crust are more like those in ASCR or in SCR. Equal weights were assigned to the five GMPEs selected for continental crust. The [START_REF] Toro | Modification of the Toro et al[END_REF] and [START_REF] Campbell | Prediction of Strong Ground Motion Using the Hybrid Empirical Method and Its Use in the Development of Ground-Motion (Attenuation) Relations in Eastern North America[END_REF] models were decided to be adjusted for the generic rock site condition in Europe, established within the framework of SHARE project, that is described with a shearwave velocity of v rock = 800 m/s and κ = 0.03 following [START_REF] Van Houtte | Analysis of the Origins of κ (Kappa) to Compute Hard Rock to Rock Adjustment Factors for GM-PEs[END_REF]. The proposed weighting scheme for SCR is presented in Table 8.

For subduction zones (Table 9), the results of the empirical data testing and the choices of the experts were quite divergent. Only the model of [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF] was supported by the majority of experts and the data-testing results. This model has been selected. The [START_REF] Mcverry | New Zealand acceleration response spectrum attenuation realtions for crustal and subduction zone earthquakes[END_REF] model was neither supported by the experts nor the empirical data testing and consequentially it was rejected. The models of [START_REF] Atkinson | Empirical Ground-Motion Relations for Subduction Zone Earthquakes and Their Application to Cascadia and Other Regions[END_REF] and [START_REF] Youngs | Strong ground motion attenuation relationships for subduction zone earthquakes[END_REF] that were the consensus selection of the experts as well as the [START_REF] Lin | Ground-Motion Attenuation Relationships for Subduction Zone Earthquakes in Northeastern Taiwan[END_REF] and [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF] models that were ranked the best in the testing were considered in the weighting scheme. A weight of 0.4 was assigned to the [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF] model whereas the other 3 models were weighted equally with 0.2. An other weighting scheme where the four selected GMPEs have equal weights has been proposed. We decided to make a sensitivity study to investigate the differences between the two weighting schemes. No difference in weights be-tween interface and inslab earthquakes was made, due to a lack of data. However, this may be a critical issue in PSHA and further research is needed in the future. The proposed weighting schemes are presented in Table 9.

For active shallow crustal regions (Table 10), two models were supported by both the experts and the empirical data testing: the models of [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF] and [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF]. These models were directly considered in the logic-tree weighting scheme. In addition, we selected two other models: the [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF] model, which was favored by the majority of the experts, and the NGA model of [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF], which was supported by the testing using non-European recordings. This selection is in agreement with the recent study conducted by [START_REF] Delavaud | Testing the Global Applicability of Ground Motion Prediction Equations for Active Shallow Crustal Regions[END_REF] to test the global applicability of GMPEs using the global dataset of [START_REF] Allen | Evaluation of Ground-Motion Modeling Techniques for Use in Global shakemap -A Critique of Instrumental Ground-Motion Prediction Equations, Peak Ground Motion to Macroseismic Intensity Conversions, and Macroseismic Intensity Predictions in Different Tectonic Settings[END_REF]. For PGA and PSA at 1 Hz in Europe and the Middle East, the four best-ranked models according to [START_REF] Delavaud | Testing the Global Applicability of Ground Motion Prediction Equations for Active Shallow Crustal Regions[END_REF] are the models of [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF], [START_REF] Chiou | Ground-Motion Attenuation Model for Small-To-Moderate Shallow Crustal Earthquakes in California and Its Implications on Regionalization of Ground-Motion Prediction Models[END_REF], [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] and [START_REF] Berge-Thierry | New empirical response spectral attenuation laws for moderate european earthquakes[END_REF] that are followed by the models of [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] and [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF]. Note that the model of [START_REF] Chiou | Ground-Motion Attenuation Model for Small-To-Moderate Shallow Crustal Earthquakes in California and Its Implications on Regionalization of Ground-Motion Prediction Models[END_REF] is an extension of the [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] (2003). The model of [START_REF] Chiou | Ground-Motion Attenuation Model for Small-To-Moderate Shallow Crustal Earthquakes in California and Its Implications on Regionalization of Ground-Motion Prediction Models[END_REF] was not published when we started the selection of GMPEs. It is also defined only for a limited number of periods. We considered moreover that the model of [START_REF] Berge-Thierry | New empirical response spectral attenuation laws for moderate european earthquakes[END_REF] is obsolete. For longer periods, between 3 s and 10 s, only the models of [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] and [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] were selected and given equal weights (0.5) because the Akkar and Bommer (2010) and [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF] models are not defined for periods up to 10 s. For periods lower than 3s, different weighting schemes have been proposed, as shown in Table 10. The weighting scheme WS6 was however preferred, that assigns higher weights to the two models that are supported by both the experts and the empirical data testing. It was also decided to make a sensitivity analysis to better understand the influence of the weights on the computed hazard.

For active regions in oceanic crust, GMPEs from ASCR were chosen (Table 11). For areas of deep focus non-subduction earthquakes, such as Vrancea (Roma-nia) or the Betics (Spain), we decided to use the GM-PEs selected for subduction instead of the empirical model of [START_REF] Sokolov | Ground-motion prediction equations for the intermediate depth Vrancea (Romania) earthquakes[END_REF], which is directly derived from the recordings of Vrancea. The model by [START_REF] Sokolov | Ground-motion prediction equations for the intermediate depth Vrancea (Romania) earthquakes[END_REF] is too complex for regional PSHA because it models azimuthal variations of ground motion (Table 12). Finally, for volcanic zones, it was decided to adopt an approach similar to that implemented in Italy when creating the currently applied set of seismic hazard maps (see [START_REF] Montaldo | Treatment of ground-motion predictive relationships for the reference seismic hazard map of Italy[END_REF]) rather than using the model of [START_REF] Mcverry | New Zealand acceleration response spectrum attenuation realtions for crustal and subduction zone earthquakes[END_REF], which only seeks to model higher attenuation in volcanic zones rather than ground motions from volcano-related events. This approach consists of introducing separate GMPEs within a seismic source zone of limited extension surrounding a volcano with well-documented historical evidence of damaging earthquakes, such as the Mount Etna volcano, in Sicily. Such GMPEs should be able to deal with events of shallow focal depth, from 2 to 5 km. An analysis expressly carried out on acceleration records from recent moderate earthquakes (3.2 < M w <4.5) of recent years in Italy, notably in the Mount Amiata (southern Tuscany) and Mount Etna areas, has shown that the use of the recent attenuation model of [START_REF] Faccioli | New ground motion prediction equations for t > 1s and their influence on seismic hazard assessment[END_REF] was appropriate, and its adoption was therefore agreed to by WP4 of SHARE (Table 13).

With logic trees containing between two and five GMPEs for each tectonic regimes (except for volcanic zones), the SHARE WP4 group, aimed at better taking epistemic uncertainties into account. In comparison, for the SESAME project which provided the first homogeneous assessment of seismic hazard for the whole Mediterranean region, only three GMPEs were considered: the model of [START_REF] Ambraseys | Prediction of Horizontal Response Spec-tra in Eurpope[END_REF] for all crustal sources, the model of [START_REF] Musson | Probabilistic seismic hazard maps for the North Balkan region[END_REF] for Vrancea and the model of [START_REF] Papaioannou | Time-Independent and Time-Dependent Seismic Hazard in Greece Based on Seismogenic Sources[END_REF] for intermediate-depth seismic activity sources in the Hellenic Arc [START_REF] Jiménez | Unified seismic hazard modelling throughout the Mediterranean region[END_REF].

Sensitivity Analysis

Logic tree and sensitivity analyses are generally run simultaneously. The results of a preliminary sensitivity analysis usually offer guidance for an iterative revision of the final logic tree structure [START_REF] Scherbaum | Composite Ground-Motion Models and Logic-Trees: Methodology[END_REF][START_REF] Bommer | The use and misuse of logic-trees in probabilistic seismic hazard analysis[END_REF]. As different weighting schemes were proposed for SZ and ASCR, a sensitivity analysis was necessary to explore the impact of the assigned weights on the final hazard results. We briefly report the main results of this analysis and refer to [START_REF] Danciu | Sensitivity Analysis of Logic Tree Weights within the SHARE Project[END_REF] for more details.

The sensitivity analysis was performed by considering the selected GMPEs for each tectonic regime (as presented in Tables 9 and10) and a set of fictitious seismic sources. Two types of seismic sources were considered: an area source zone (ASZ) and a fault source (FS). A fictitious ASZ was used to simulate the seismicity in an ASCR and also on subduction intraslab sources. A virtual FS was used to model an active shallow crustal fault as well as a subduction interface source.

Four ground motion intensity measures were considered: peak ground acceleration, peak ground velocity (PGV) and pseudo-acceleration spectra at the spectral periods of 0.2s and 1.0s. Seismic hazard maps for each weighting scheme were obtained for two reference return periods: 475 and 2475 years; the former corresponding to 10% probability of exceedance in 50 years, and the latter indicates a 2% probability of exceedance in 50 years. Additionally, individual hazard maps for each GMPE were produced. All the resulting hazard maps were estimated for a reference rock site, defined by v S30 = 800m/s. The seismic hazard software [START_REF] Pagani | GEM1 Hazard: Description of input models, calculation engine and main results[END_REF] developed within the prototype of GEM, namely the GEM1 project, was used for the calculations. Details of the source characterization and hazard calculation settings are presented in more detail in [START_REF] Danciu | Sensitivity Analysis of Logic Tree Weights within the SHARE Project[END_REF].

The difference between the hazard maps of the preferred weighting schemes and those computed from alternative weights was quantified by the mean of the percentage differences. Percentage difference for each grid point was computed by:

PerDiff(%) = WS (proposed) -WS (preferred) WS (preferred) × 100 (5)
where WS (preferred) are the estimated expected ground motion at each grid point from the preferred weighting scheme and WS (proposed) contains the values using the alternative weighting schemes. For example, for active crustal regions the preferred weighting scheme is WS6 and the alternative weighting schemes are WS1, WS2, WS3, WS4, WS5 and WS7 as reported in Table 10. Maps showing the percentage difference were produced for all ground-motion intensity measures and for the specified return periods. Due to the limited space, the percentage difference maps are not reproduced herein but a summary of the percentage differences for PGA, PSA (0.2s) and PSA (1s) for different weighting schemes are summarized in Table 14 for ASCR and in Table 15 for SZ.

For ASCR, the sensitivity analysis shows that the absolute difference between the preferred and the proposed weighting schemes varies within the range of 5% to 10% for the ASZ and 5 to 15% for the FS. Most of the differences occur when the proposed weighting scheme WS7 is compared with the preferred one (WS6). The examination of individual hazard maps for each selected GMPE suggest that the equations proposed by [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF] and [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] dominate the hazard in the areal zones of active shallow crustal tectonic regimes because they yield higher values when compared to the other remaining two GM-PEs (roughly 30% larger). [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] and [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] predict larger ground motions in shallow crustal fault sources (roughly 60% larger than the other two GMPEs), thus dominating the hazard for such cases.

In the case of subduction zone the percentage difference varies between 4 and 10%. The differences decrease with increasing period values for subduction inslab sources. Contrary to this observation the hazard differences become larger as the vibration period increases for subduction interface cases. By contrast, for subduction interface sources the difference increases as the return periods increase. [START_REF] Zhao | Attenuation Relations of Strong Ground Motion in Japan Using Site Classification Based on Predominant Period[END_REF] and [START_REF] Youngs | Strong ground motion attenuation relationships for subduction zone earthquakes[END_REF] were found to be responsible for the larger values in the seismic hazard for the subduction sources (predicting roughly six times larger ground motions than the other two GMPEs).

The overall results of the sensitivity analysis on different weighting schemes suggest that for the considered seismic sources there is a moderate impact on the hazard results. [START_REF] Sabetta | Sensitivity of PSHA results to ground motion prediction relations and logic-tree weights[END_REF] also concluded that if four or more GMPEs are used in the logic tree, the assigned weights do not significantly affect the hazard results. [START_REF] Scherbaum | Composite Ground-Motion Models and Logic-Trees: Methodology[END_REF] also indicated that the selection of the GMPEs seems to be more important than the choice of weighting strategy.

In essence, the practical conclusion of this preliminary sensitivity analysis led us to keep the present structure of the ground motion logic tree together with the preferred weights.

Conclusions

Although it is now common practice to treat uncertainty in ground motion prediction with a logic tree approach, there is no standard procedure that describes how the tree should be constructed. In this paper, we shared our experience on this subject by presenting the strategy that was adopted to build a logic tree for Europe within the SHARE project. This task involved roughly a dozen institutions with the goal, in a limited amount of time (18 months), to commonly define a logic tree that would capture the center, body and range of ground motion in six different tectonic regimes in the Euro-Mediterranean region.

The principal idea that guided our strategy was to gather as much knowledge as possible from independent sources and different methods. Based on the characteristics of the available GMPEs determined by [START_REF] Douglas | Further errata of and additions to "ground motion estimation equations 1964-2003[END_REF], recently updated by [START_REF] Douglas | Ground-motion prediction equations 1964-2010[END_REF], we first identified the best candidates using the rejection criteria of [START_REF] Cotton | Criteria for selecting and adjusting ground-motion models for specific target regions: Application to Central Europe and rock sites[END_REF] and [START_REF] Bommer | On the Selection of Ground-Motion Prediction Equations for Seis-mic Hazard Analysis[END_REF]. Afterwards, expert judgment highlighted sets of GMPEs that were, according to the experts, capable of capturing epistemic uncertainties, while testing using observational data showed GMPEs capable of closely predicting past ground motions. The integration of these different approaches was undertaken to propose logic trees that were then subjected to a sensitivity analysis to see the impact on the seismic hazard.

From this experience, we have learnt lessons and identified weaknesses in our methodology. First a great effort should be dedicated to the collection of data and meta-data in order to get as much information as possible from the GMPE testing. In our case, data were not sufficient to cover the center, body and range of ground motions in Europe. Secondly, the procedure for the selection and weighting of GMPEs by experts should be clearly defined. Within the SHARE project, most experts required more guidance and information (e.g., what do weights represent?, how will they be used afterwards?). Selecting GMPEs and assigning weights is still not an obvious task, although [START_REF] Scherbaum | Logic Tree Branch Weights and Probabilities: Summing Up to One Is Not Enough[END_REF] have recently proposed a method. To build a logic tree is not to give a quality measure to each candidate GMPE independently from the others but rather to identify the set of models that together, with a certain weighting, can capture the perceived epistemic uncertainty. In the context of the SHARE project that covers a large area, the set of GMPEs had to be the smallest one.

The robustness of the proposed logic tree is a crucial property. For ASCR, the SHARE GMPE selection is in agreement with the data testing of [START_REF] Delavaud | Testing the Global Applicability of Ground Motion Prediction Equations for Active Shallow Crustal Regions[END_REF] who used an independent dataset to rank candidate GMPEs for Europe and Middle East. Their study also showed that for this particular dataset, the recent model of [START_REF] Chiou | Ground-Motion Attenuation Model for Small-To-Moderate Shallow Crustal Earthquakes in California and Its Implications on Regionalization of Ground-Motion Prediction Models[END_REF] but also the model of [START_REF] Berge-Thierry | New empirical response spectral attenuation laws for moderate european earthquakes[END_REF] were both able to predict ground motion in Europe and Middle East reasonably well. Regular updates of the logic tree should be planned to take new data and new GMPEs into account.

Each project is unique and it is important to be aware of its particularities. However, we think that the procedure described above is reproducible and that at least it contributed to the reflections on the way a logic tree for ground-motion prediction should be built.

Acknowledgements The main part of this work has been funded by the EC-Research Framework programme FP7, Seismic Hazard Harmonization in Europe (SHARE), contract number 226967. The authors would like to warmly thank Julian Bommer, Hilmar Bungum and Fabian Bonilla for their expert role and their key contribution to the ground-motion prediction equation evaluation and weighting. We thank an anonymous reviewer and Jochen Woessner for their comments on the first draft of this article. This paper also benefited from the feedbacks and interaction with Donat Fäh, Ben Edwards, Marco Pagani, Kyriazis Pitilakis, Pierre-Yves Bard and Carola Di Alessandro. This work strongly benefited from the constant support of Jochen Woessner and Domenico Giardini, manager and coordinator of the SHARE FP7 project. Description of the ground-motion prediction equations for active crustal regions.

0.2

Table 9 Expert choices, data-based testing results and logic trees for subduction zones. Names in bold are the selected models. The preferred weighting scheme (WS) is shown in bold.

Category based on expert judgment Models

Models supported by all the experts

0.40

Selected models for periods 3s < T ≤ 10s WS [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] 0.5 [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] 0.5

Table 12 Weighting scheme for Vrancea (same as for SZ).

The preferred weighting scheme (WS) is shown in bold.

  rupture, RJB: Joyner-Boore distance[START_REF] Abrahamson | Overview[END_REF] 2 Style-of-faulting: N: normal, R: reverse, T: thrust, U: undefined 3 pN: proportion of normal faulting, pR: proportion of reverse faulting needed for the style of faulting adjustment method of[START_REF] Bommer | Style-of-faulting in ground-motion prediction equations[END_REF] 4 Horizontal component type. GMEAN: geometric mean, GMRotI50: new geometric mean used in the PEER NGA Project

  , RJB: Joyner-Boore, HYP: hypocentral, EPI: epicentral distance (Abrahamson and Shedlock, 1997) 2 cont. function: continuous function of v S30 3 Style-of-faulting: N: normal, R: reverse, T: thrust, O: oblic, U: undefined 4 pN: proportion of normal faulting, pR: proportion of reverse faulting needed for the style of faulting adjustment method of Bommer et al. (2003) 5 Horizontal component type. GMEAN: geometric mean, LRGENV: larger envelop, VectComp: vectorial composition, GMRotI50: new geometric mean used in the PEER NGA Project

Atkinson and Boore ( 2003 )

 2003 Models supported by a majority of experts[START_REF] Youngs | Strong ground motion attenuation relationships for subduction zone earthquakes[END_REF]) Zhao et al. (2006) Category based on data-testing resultsModelsModels supported by the testing[START_REF] Lin | Ground-Motion Attenuation Relationships for Subduction Zone Earthquakes in Northeastern Taiwan[END_REF] for long periods (T > 0.16s) Zhao et al. (2006) Models supported by the testing Atkinson and Boore (2003) for short periods (T ≤ 0.16s) Zhao et al. (2006) Category based on expert judgment Models and data-testing resultsModels supported by the data testing and the experts choices Models chosen by a majorityZhao et al. (2006) of experts and supported by the data-testing results Models chosen by a minority McVerry et al. (2006) of experts or with a low Atkinson and Macias (2009) data-testing result Garcia et al. (2005) Models not supported by the data-testing and not chosen by the experts

⊲⊲Fig. 1

 1 Fig.1Process adopted for the construction of the groundmotion logic tree for Europe

Fig. 2 Fig. 3 Fig. 4

 234 Fig. 2 Seismotectonic map of the Euro-Mediterranean area developed for the SHARE project by WP3.2. 1: SCR, shield (a) and continental crust (b); 2: oceanic crust; 3: ASCR, compression-dominated areas (a) including thrust or reverse faulting, associated transcurrent faulting (e.g. tear faults), and contractional structures in the upper plate of subduction zones (e.g. accretionary wedges), extension-dominated areas (b) including associated transcurrent faulting, major strikeslip faults and transforms (c), and mid oceanic ridges (d); 4: subduction zones shown by contours at 50 km depth interval of the dipping slab; 5: areas of deep-focus non-subduction earthquakes; 6: active volcanoes and other thermal/magmatic features.

  model for small-to-moderate magnitudes. It is consistent with the Chiou and Youngs (2008) model for large magnitudes (Mw ≥ 6) but has new coefficients for smaller magnitudes. This result suggests that Delavaud et al. (2012) would have ranked first the four models that we selected for ASCR if they had not considered the model of Chiou et al. (2010) and the model of Berge-Thierry et al.

Table 1

 1 SHARE WP4 group involved in the building of the logic tree for ground-motion prediction.

Table 2

 2 Description of the ground-motion prediction equations for stable continental regions.

Table 3

 3 Description of the ground-motion prediction equations for subduction areas.

Table 5

 5 Ranking of the candidate GMPEs for subduction zones based on LLH values for PSA at 0.05s, 0.3s, 0.5s, 0.8s, 1s, 1.5s and 2s

	Subduction zones -PSA 0.05s to 2s
	Rank LLH	DSI	Model
	1	1.979 29.57	Lin and Lee (2008)
	2	1.988 28.76	Zhao et al. (2006)
	3	2.206 10.71	Youngs et al. (1997)
	4	2.499 -9.641 Kanno et al. (2006)
	5	2.500 -9.704 McVerry et al. (2006)
	6	3.344 -49.70 Atkinson and Boore (2003)

Table 6

 6 Ranking of the candidate GMPEs for ASCR based on LLH values for PSA at 0.1s, 0.2s, 0.5s, 1s and 2s from DB1 (European database)

	Active shallow crustal regions -DB1 -PSA 0.1s to 2s
	Rank LLH	DSI	Model
	1	2.378 68.29	Bindi et al. (2010)
	2	2.396 66.20	Cauzzi and Faccioli (2008)
	3	2.427 62.67	Cotton et al. (2008)
	4	2.588 45.49	Akkar and Bommer (2010)
	5	2.680 36.50	Douglas et al. (2006)
	6	2.800 25.61	Zhao et al. (2006)
	7	2.938 14.15	Chiou and Youngs (2008)
	8	3.158 1.99	Ambraseys et al. (2005)
	9	3.271 -9.38	Danciu and Tselentis (2008)
	10	3.869 -40.13 Abrahamson and Silva (2008)
	11	4.121 -49.72 Boore and Atkinson (2008)
	12	4.785 -68.27 Campbell and Bozorgnia (2008)
	13	4.921 -71.12 Kalkan and Gülkan (2004)
	14	5.332 -78.28 Massa et al. (2008)

Table 7

 7 Ranking of the candidate GMPEs for ASCR based

	on LLH values for PSA at 0.1s, 0.2s, 0.5s, 1s and 2s from DB2
	(non-European database)
	Active shallow crustal regions -DB2 -PSA 0.1s to 2s
	Rank LLH	DSI	Model
	1	1.558 29.49	Akkar and Bommer (2010)
	2	1.592 26.43	Chiou and Youngs (2008)
	3	1.620 24.00	Boore and Atkinson (2008)
	4	1.672 19.65	Abrahamson and Silva (2008)
	5	1.678 19.15	Kalkan and Gülkan (2004)
	6	1.710 16.45	Campbell and Bozorgnia (2008)
	7	1.761 12.50	Bindi et al. (2010)
	8	1.813 8.477	Danciu and Tselentis (2008)
	9	1.835 6.81	Cauzzi and Faccioli (2008)
	10	1.850 5.75	Zhao et al. (2006)
	11	2.331 -24.25 Ambraseys et al. (2005)
	12	2.545 -34.67 Douglas et al. (2006)
	13	2.897 -48.82 Cotton et al. (2008)
	14	3.288 -60.97 Massa et al. (2008)

Table 8

 8 Expert choices and the final logic trees for stable continental regions. Names in bold are the selected models. WS stands for weighting scheme.

	Category based on expert judgment	Models
	Models supported by all the experts	Campbell (2003)
			Toro et al. (1997)
	Models chosen by a majority of experts Atkinson and Boore (2006)
			Douglas et al. (2006)
			Atkinson (2008)
	Models chosen by a minority of experts
	Models not chosen by the experts	Tavakoli and Pezeshk (2005)
	Selected models for shield	WS
	Campbell (2003)	0.5
	Toro (2002)	0.5
	Selected models for continental crust	WS
	Campbell (2003) adjusted to 800m/s	0.2
	Toro (2002) adjusted to 800m/s	0.2
	Akkar and Bommer (2010)	0.2
	Cauzzi and Faccioli (2008)		0.2
	Chiou and Youngs (2008)

Table 10

 10 Expert choices, data-based testing results and logic trees for active shallow crustal regions. Names in bold are the selected models. The preferred weighting scheme (WS) is shown in bold.

	Category based on expert judgment	Models
	Models supported by all the experts	Boore and Atkinson (2008)

Table 11

 11 Weighting scheme for active regions in oceanic crust (same as for ASCR). The preferred weighting scheme (WS) is shown in bold.

	Selected models for periods T ≤ 3s	WS1	WS2	WS3	WS4	WS5	WS6	WS7
	Akkar and Bommer (2010)	0.30	0.30	0.40	0.25	0.30	0.35	0.10
	Cauzzi and Faccioli (2008)	0.30	0.20	0.20	0.25	0.25	0.35	0.40
	Zhao et al. (2006)	0.20	0.20	0.20	0.25	0.20	0.10	0.10
	Chiou and Youngs (2008)	0.20	0.30		0.25	0.25	0.20

Table 13

 13 Weighting scheme for volcanic zones

	Selected Models	WS
	Faccioli et al. (2010)	1

Table 14

 14 Percentage differences for ASCR for three groundmotion intensity measure types, two seismic source types and two return periods. The maximum for each ground-motion intensity measure type, source type and return period is highlighted in bold.

	Weighting Schemes	Area Source	Fault Source
		475 years	2475 years	475 years	2475 years
		PGA
	Preferred WS vs. WS1	3.27%	3.36%	4.22%	3.97%
	Preferred WS vs. WS2 (+)7.68%	5.95%	5.38%	5.88%
	Preferred WS vs. WS3	6.96%	5.20%	7.82%	6.66%
	Preferred WS vs. WS4	6.39%	5.43%	5.79%	5.10%
	Preferred WS vs. WS5	5.45%	4.33%	4.57%	4.25%
	Preferred WS vs. WS7	4.44%	(-)6.49%	(-)8.59%	(-)9.51%
		PSA (0.2s)
	Preferred WS vs. WS1	3.28%	3.55%	4.21%	4.60%
	Preferred WS vs. WS2	7.80%	6.84%	6.42%	5.79%
	Preferred WS vs. WS3	8.93%	7.86%	10.71%	9.78%
	Preferred WS vs. WS4	5.91%	6.00%	5.63%	5.87%
	Preferred WS vs. WS5	5.49%	5.12%	5.07%	5.02%
	Preferred WS vs. WS7 (-)11.04%	(-)8.83%	(-)14.57%	(-)14.26%
		PSA (1s)
	Preferred WS vs. WS1	3.53%	4.49%	3.31%	4.34%
	Preferred WS vs. WS2	3.59%	7.31%	2.65%	6.38%
	Preferred WS vs. WS3	3.73%	7.35%	2.48%	4.56%
	Preferred WS vs. WS4	4.68%	(-)7.50%	4.72%	7.46%
	Preferred WS vs. WS5	3.02%	5.89%	2.88%	5.32%
	Preferred WS vs. WS7 (+)4.95%	4.60%	(-)5.88%	(-)7.48%

Table 15

 15 Percentage difference values for SZ for three ground-motion intensity measure types, two seismic source types, and two return periods. The maximum for each source type and return period is highlighted in bold.

	Weighting Schemes	Subduction Inslab	Subduction Interface
		475 years 2475 years 475 years 2475 years
		PGA
	Preferred WS vs. WS2	(-)8.80%	6.81%	6.87%	(-)6.97%
		PSA (0.2s)
	Preferred WS vs. WS2	(-)8.70%	8.51%	6.83%	(-)8.00%
		PSA (1s)
	Preferred WS vs. WS2	(-)4.00%	1.67%	5.45%

RUP: rupture, HYP: hypocentral distance[START_REF] Abrahamson | Overview[END_REF]

B: inslab earthquakes, F: interface earthquakes

cont. function: continuous function of v S30

Horizontal component type. GMEAN: geometric mean, VectComp: vectorial composition

Coordinators and Elise Delavaud ISTerre, Grenoble, France / ETH, Zürich, Switzerland Methodology definition Fabrice Cotton ISTerre, Grenoble, France Sinan Akkar METU, Ankara, Turkey Frank Scherbaum Inst. of Earth and Env. Science, Potsdam, Germany Experts Julian Bommer Imperial College, London, UK Fabian Bonilla IFSSTAR, Paris, France Hilmar Bungum NORSAR/ICG, Kjeller, Norway John Douglas BRGM, Orléans, France Ezio Faccioli Politecnico di Milano, Milano, Italy Nikos Theodoulidis ITSAK, Thessaloniki, Greece Participants Laurentiu Danciu (Sensitivity Analysis) ETH, Zürich, Switzerland Céline Beauval (Data testing) ISTerre, Grenoble, France Stéphane Drouet (Models pre-selection) ISTerre, Grenoble, France John Douglas (Models pre-selection) BRGM, Orléans, France Roberto Basili (Zonation map) INGV, Roma, Italy Abdullah Sandikkaya (Data testing) METU, Ankara, Turkey Margaret Segou (Data testing) METU, Ankara, Turkey Ezio Faccioli (Volcanic zones) Politecnico di Milano, Milano, Italy Nikos Theodoulidis (Data testing) ITSAK, Thessaloniki, Greece Donat Fäh ETH, Zürich, Switzerland Benjamin Edwards ETH, Zürich, Switzerland Pierre-Yves Bard ISTerre, Grenoble, France Kyriazis Pitilakis University of Thessaloniki, Thessaloniki, Greece Marco Pagani GEM Foundation, Pavia, Italy

 [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF]Models supported by a majority of experts [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF]Faccioli (2008) Zhao et al. (2006) Category based on data-testing results

Models

Models supported by the testing (European dataset) Models supported by the data testing and by the experts choices Models chosen by a majority of experts and [START_REF] Akkar | Empirical Equations for the Prediction of PGA, PGV, and Spectral Accelerations in Europe, the Mediterranean Region, and the Middle East[END_REF] supported by the data-testing results [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] Models chosen by a minority of experts or with a low [START_REF] Abrahamson | Summary of the Abrahamson & Silva NGA Ground Motion Relations[END_REF]

0.40

Selected models for periods 3s < T ≤ 10s WS [START_REF] Cauzzi | Broadband (0.05 to 20 s) prediction of displacement response spectra based on worldwide digital records[END_REF] 0.5 [START_REF] Chiou | An NGA Model for the Average Horizontal Component of Peak Ground Motion and Response Spectra[END_REF] 0.5


Cyril Roussillon

Aurélien Gonzalez

Joan Solà

Jean-Marie Codol

Nicolas Mansard

Simon Lacroix

Michel Devy

RT-SLAM: a generic and real-time visual SLAM implementation

This article presents a new open-source C++ implementation to solve the SLAM problem, which is focused on genericity, versatility and high execution speed. It is based on an original object oriented architecture, that allows the combination of numerous sensors and landmark types, and the integration of various approaches proposed in the literature. The system capacities are illustrated by the presentation of an inertial/vision SLAM approach, for which several improvements over existing methods have been introduced, and that copes with very high dynamic motions. Results with a hand-held camera are presented.

Motivation

Progresses in image processing, the growth of available computing power and the proposition of approaches to deal with bearings-only observations have made visual SLAM very popular, particularly since the demonstration of a real-time implementation by Davison in 2003 [START_REF] Davison | Real-time simultaneous localisation and mapping with a single camera[END_REF]. The Extended Kalman Filter (EKF) is widely used to solve the SLAM estimation problem, but it has recently been challenged by global optimization methods that have shown superior precision for large scale SLAM. Yet EKF still has the advantage of simplicity and faster processing for problems of limited size [START_REF] Strasdat | Real-time monocular slam: Why filter?[END_REF], and can be combined with global optimization methods [START_REF] Estrada | Hierarchical slam: Real-time accurate mapping of large environments[END_REF] to take the best of both worlds.

Monocular EKF-SLAM reached maturity in 2006 with solutions for initializing landmarks [START_REF] Kwok | An efficient multiple hypothesis filter for bearingonly slam[END_REF] [12] [START_REF] Montiel | Unified inverse depth parametrization for monocular slam[END_REF]. Various methods for landmark parametrization have been analyzed [START_REF] Sola | Consistency of the monocular ekf-slam algorithm for three different landmark parametrizations[END_REF], and the literature now abounds with contributions to the problem.

This article presents RT-SLAM 1 , a software framework aimed at fulfilling two essential requirements. The first one is the need for a generic, efficient and flexible development tool, that allows to easily develop, evaluate and test various approaches or improvements. The second one is the need for practical solutions for live experiments on robots, for which localization and mapping require real-time execution and robustness. RT-SLAM is a C++ implementation based on Extended Kalman Filter (EKF) that allows to easily change robot, sensor and landmark models. It runs up to 60 Hz with 640 × 480 images, withstanding highly dynamic motions, which is required for instance on humanoid or high speed all terrain robots. RT-SLAM is available as open source software at http://rtslam.openrobots.org.

Section 2 details the architecture of RT-SLAM and section 3 presents some of the techniques currently integrated within, to define an efficient inertial/visual SLAM approach. Section 4 analyzes results obtained with a hand-held system assessed thanks to ground truth, and section 5 concludes the article by a presentation of prospects.

Overall Architecture

Fig. 1 presents the main objects defined in RT-SLAM. They encompass the basic concepts of a SLAM solution: the world or environment contains maps; maps contain robots and landmarks; robots have sensors; sensors make observations of landmarks. Each of these objects is abstract and can have different implementations. They can also contain other objects that may themselves be generic. Map. The maps contain an optimization or estimation engine: for now RT-SLAM uses a standard formulation of EKF-SLAM. Since this solution is very well documented in the literature [START_REF] Davison | Monoslam: Real-time single camera slam[END_REF], it is not detailed in depth here. Indirect indexing within Boost's ublas C++ structures is intensively used to exploit the sparsity of the problem and the symmetry of the covariances matrices.

Robot. Robots can be of different type according to the way their state is represented and their prediction model. The latter can be either a simple kinematic model (constant velocity, constant acceleration, . . . ) or a proprioceptive sensor (odometric, inertial, . . . ), as illustrated section 3.3. The proprioceptive sensor is an example of generic object contained in robot objects, as different hardware can provide the same function.

Sensor. Similarly to robots, sensors can also have different models (perspective camera, panoramic catadioptric camera, laser, . . . ), and contain a generic exteroceptive sensor hardware object (firewire camera, USB camera, . . . ). In addition, as sensors belong to the map, their state can be estimated: this opens the possibility for estimating other parameters such as extrinsic calibration, time delays, biases and gain errors, and the like.

Landmark. Landmarks can be of different type (points, lines, planes, . . . ), and each type can have different state parametrization (Euclidean point, inverse depth point, . . . ). Moreover the parametrization of a landmark can change over time, as explained section 3.2. A landmark also contains a descriptor used for data association, which is a dual description to the state representation.

As shown Fig. 1(a), it is worth noticing that landmark objects are common to the different sensors, all of them being able to observe the same landmark (provided they have compatible descriptors for this landmark of course). This allows to greatly improve the observability of landmarks compared to a system where the sensors are strictly independent. In the particular case of two cameras for instance, landmarks can be used even if they are only visible from one camera or if they are too far away for a stereovision process to observe their depth (this process was introduced in [START_REF] Sola | Bicamslam: Two times mono is more than stereo[END_REF] as BiCam SLAM).

Observation. In RT-SLAM, the notion of observation plays a predominant role. An observation is a real object containing both methods and data. One observation is instantiated for every sensor-landmark pair, regardless of the sensor having actually observed the landmark or not, and has the same lifetime as the associated landmark. The methods it contains are the conventional projection and back-projection models (that depend on the associated sensor and landmark models), while the stored data consist of results and intermediary variables such as Jacobian matrices, predicted and observed appearances, innovations, event counters and others, that allow to greatly simplify and optimize computations.

Managers. In order to achieve full genericity wrt landmark types, in particular to allow the concurrent use of different landmark types for one sensor, two different manager objects are added: data manager and map manager. Their implementations define a given management strategy, while their instantiations are dedicated to a certain landmark type. The data manager processes the sensors raw data, providing observations of the external landmarks. For this purpose, it exploits some raw data processors (for feature detection and tracking), and decides which observations are to be corrected and in which order, according to the quantity of information they bring and their quality. For example it can apply an active search strategy and try to eliminate outliers as described in section 3.1. The map manager keeps the map clean, with relevant information, and at a manageable size, by removing landmarks according to their quality and the given policy (e.g. visual odometry where landmarks are discarded once they are not observed, or multimap slam where maps are "closed" according to given criteria). These managers communicate together: for example, the data manager may ask the map manager if there is enough space in the map to start a new initialization process, and to allocate the appropriate space for the new landmark.

3 Inertial/visual SLAM within RT-SLAM

Active search and one-point RANSAC

The strategy currently implemented in RT-SLAM's data manager to deal with observations is an astute combination of active search [START_REF] Davison | Monoslam: Real-time single camera slam[END_REF] and outliers rejection using one-point RANSAC [START_REF] Civera | 1-point ransac for ekfbased structure from motion[END_REF].

The goal of active search is to minimize the quantity of raw data processing by constraining the search in the area where the landmarks are likely to be found. Observations outside of this 3σ observation uncertainty ellipse would be anyway considered incompatible with the filter and ignored by the gating process. In addition active search gives the possibility to decide anytime to stop matching and updating landmarks with the current available data, thus enabling hard real-time constraints. We extended the active search strategy to landmark initialization: each sensor strives to maintain features in its whole field of view using a randomly moving grid of fixed size, and feature detection is limited to empty cells of the grid. Furthermore the good repartition of features in the field of view ensures a better observability of the motions.

Outliers can come from matching errors in raw data or mobile objects in the scene. Gating is not always discriminative enough to eliminate them, particularly right after the prediction step when the observation uncertainty ellipses can be quite large -unfortunately at this time the filter is very sensitive to faulty corrections because it can mistakenly make all the following observations incompatible. To prevent faulty observations, outliers are rejected using a one-point RANSAC process. It is a modification of RANSAC, that uses the Kalman filter to obtain a whole model with less points than otherwise needed, and provides a set of strongly compatible observations that are then readily corrected. Contrary to [START_REF] Civera | 1-point ransac for ekfbased structure from motion[END_REF] where data association is assumed given when applying the algorithm, we do the data association along with the one-point RANSAC process: this allows to look for features in the very small strongly compatible area rather than the whole observation uncertainty ellipse, and to save additional time for raw data processing.

Landmark parametrizations and reparametrization

In order to solve the problem of adding to the EKF a point with unknown distance and whose uncertainty cannot be represented by a Gaussian distribution, point landmarks parametrizations and initialization strategies for monocular EKF-SLAM have been well studied [START_REF] Davison | Real-time simultaneous localisation and mapping with a single camera[END_REF] [8] [START_REF] Montiel | Unified inverse depth parametrization for monocular slam[END_REF]. The solutions now widely accepted are undelayed initialization techniques with inverse depth parametrization. Anchored Homogeneous Point [START_REF] Sola | Consistency of the monocular ekf-slam algorithm for three different landmark parametrizations[END_REF] parametrization is currently used in RT-SLAM.

The drawback of inverse depth parametrizations is that they describe a landmark by at least 6 variables in the stochastic map, compared to only 3 for an Euclidean point (x y z)

T . Memory and temporal complexity being quadratic with the map size for EKF, there is a factor of 4 to save in time and memory by reparametrizing landmarks that have converged enough [START_REF] Civera | Inverse Depth to Depth Conversion for Monocular SLAM[END_REF]. The map manager uses the linearity criterion proposed in [START_REF] Montiel | Unified inverse depth parametrization for monocular slam[END_REF] to control this process.

Motion prediction

The easiest solution for EKF-SLAM is to use a robot kinematic model to do the filter prediction, such as a constant velocity model: R = (p q v w) T where p and q are respectively the position and quaternion orientation of the robot, and v and w are its linear and angular velocities.

The advantage of such a model is that it does not require complicated hardware setup (only a simple camera), but its strong limitation is that the scale factor is not observable. A second camera with a known baseline can provide a proper scale factor, but one can also use a proprioceptive sensor for the prediction step. Furthermore it usually provides a far better prediction with smaller uncertainties than a simple kinematic model, which brings several benefits: 1. search areas for matching are smaller, so processing is faster, 2. linearization points are more accurate, so SLAM precision is better, or one can reduce the framerate to decrease CPU load for equivalent quality, 3. it allows to withstand high motion dynamics.

In the case of an Inertial Measurement Unit (IMU), the robot state is then:

R = (p q v a b w b g) T
where a b and w b are the accelerometers and gyrometers biases, and g the 3D gravity vector. Indeed it is better for linearity reasons to estimate the direction of g rather than constraining the robot orientation to enforce g to be vertical.

A special care has to be taken for the conversion of the noise from continous time (provided by the manufacturer in the sensor's datasheet) to discrete time. As the perturbation is continuous white noise, the variance of the discrete noise grows linearly with the integration period.

Image processing

Point extraction. Point extraction is based on Harris detector with several optimizations. Some of them are approximations: a minimal derivative mask [-1, 0, 1] is used, as well as a square and constant convolution mask, in order to minimize operations. This allows the use of integral images [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF] to efficiently compute the convolutions. Additional optimizations are related to active search (section 3.1): only one new feature is searched in a small region of interest, which eliminates the costly steps of thresholding and sub-maxima suppression.

Point matching. Point matching is based on Zero-mean Normalized Cross Correlation (ZNCC), also with several optimizations. Integral images are used to compute means and variances, and a hierarchical search is made (two searches at half and full resolution are sufficient). We also implemented bounded partial correlation [START_REF] Di Stefano | Zncc-based template matching using bounded partial correlation[END_REF] in order to interrupt the correlation score computation when there is no more hope to obtain a better score than the threshold or the best one up to now. To be robust to viewpoint changes and to track landmarks longer, tracking is made by comparing the initial appearance of the landmark with its current predicted appearance [START_REF] Davison | Monoslam: Real-time single camera slam[END_REF].

Results

Fig. 2 shows the hardware setup that has been used for the experiments. It is composed of a firewire camera rigidly tied to an IMU, on which four motion capture markers used to measure the ground truth are fixed. Two different sequences are used, referred to as low dynamic and high dynamic sequences. Both were acquired indoor with artificial lights only, with an image framerate of 50 Hz synchronized to the inertial data rate of 100 Hz. The motion capture markers are localized with a precision of approximately 1 mm, so the ground truth has a precision of σ xyz = 1 mm in position and σ wpr = 0.57 • in angle (baseline of 20 cm).

Movies illustrating a run of every experiment are provided at: http://rtslam.openrobots.org/Material/ICVS2011.

Constant velocity model

The inertial data is here not used, and the prediction is made according to a constant velocity model. Fig. 3 presents the estimated trajectory and the ground truth for the low dynamic sequence, and Fig. 4 shows the absolute errors for the same run. Besides the global scale factor which is manually corrected, the camera trajectory is properly estimated. The movie shows that loops are correctly closed, even after a full revolution.

The position error raises up to 10 cm after quick rotations of the camera: this is due to a slight drift of the scale factor caused by the integration of newer landmarks (when closing loops, the scale factor is restimated closer to its initial value). Also, uncertainty ellipses remain relatively large throughout the sequence: these issues are solved by the use of an IMU to predict the motions.

4.2 Inertial/visual SLAM Fig. 5 shows the trajectory estimated with the high dynamic sequence, and Fig. 6 and7 show the behavior of inertial SLAM. Here, all of the 6 degrees of freedom are successively excited, then y and yaw are excited with extreme dynamics: the yaw angular velocity goes up to 400 deg/s, the rate of change of angular velocity exceeds 3,000 deg/s 2 , and accelerations reach 3 g. It is interesting to note that the time when SLAM diverges (around t = 65 s) corresponds to motions for which the angular velocity exceeds the limit of the IMU (300 deg/s) and where its output is saturated.

The IMU now allows to observe the scale factor, and at the same time reduces the observation uncertainty ellipses and thus eases the active search procedure. Conversely, the divergence of the SLAM process at the end of the sequence illustrates what happens when vision stops stabilizing the IMU: it quickly diverges because the biases and the gravity cannot be estimated anymore.

Outlook

We have presented a complete SLAM architecture whose genericity and performance make it both a useful experimentation tool and efficient solution for robot localization. The following extensions are currently being developed: using a second camera to improve landmarks observability, a multimap approach to cope with large scale trajectories and multirobot collaborative localization, and the use of line landmarks complementarily to points to provide more meaningful maps and to ease map matching for loop closure. Fig. 6. Errors of inertial/visual SLAM estimated over 100 runs on the dynamic sequence, as a function of time (in seconds) -the difference between each run is due to the randomized landmark selection, see section 3.1. All the runs remain in the neighborhood of the 99% confidence area. The angular ground truth uncertainty is not precisely known: its theoretical value is both overestimated and predominant over SLAM precision in such a small area. Fig. 7. Inertial/visual SLAM NEES [START_REF] Bailey | Consistency of the ekf-slam algorithm[END_REF] over 100 runs on the dynamic sequence. The average NEES is quickly out of the corresponding 99% confidence area, but as the 100 runs were made on the same sequence they are not independant and the average NEES should rather be compared to the simple NEES confidence area. The filter appears to be very conservative with angles but this is due to the overestimated ground truth uncertainty as explained in Fig. 6.

The architecture of RT-SLAM allows to easily integrate such developments, and also to consider additional motion sensors: to increase the robustness of the system, it is indeed essential to consider the various sensors usually found on board a robot (odometry, gyrometers, GPS). Eventually, it would be interesting to make RT-SLAM completely generic wrt the estimation engine as well, in order to be able to use global optimization techniques in place of filtering.

  (a) Main objects in a SLAM context. Different robots Rob have several sensors Sen, providing observations Obs of landmarks Lmk. States of robots, sensors and landmarks are stored in the stochastic map Map. There is one observation per sensor-landmark pair. (b) Objects hierarchy in RT-SLAM. Each individual map M in the world W contains robots R and landmarks L. A robot has sensors S, and an observation O is created for every pair of sensor and landmark. In order to allow full genericity, map managers MM and data managers DM are introduced.

Fig. 1 .

 1 Fig. 1. The main objects in RT-SLAM

Fig. 2 .

 2 Fig. 2. The experimental setup composed of a Flea2 camera and an XSens MTi IMU, and screen captures of the 3D and 2D display of RT-SLAM.

Fig. 3 .

 3 Fig. 3. Illustration of low dynamic trajectory, constant velocity SLAM (with scale factor of 2.05 manually corrected). Estimated trajectory parameters and ground truth, as a function of time (in seconds).

Fig. 4 .

 4 Fig.4. Errors of the estimated parameters of the trajectory shown Fig.3. The 99% confidence area corresponds to 2.57σ bounds. The SLAM 99% confidence area, that does not include ground truth uncertainty, is also shown.

Fig. 5 .

 5 Fig. 5. Illustration of high dynamic trajectory, inertial/visual SLAM. Estimated trajectory parameters and ground truth, as a function of time (in seconds).

RT-SLAM stands for "Real-Time SLAM"