S Akkar

J Douglas

C Di

K Campbell

P Somerville

F Cotton

W Silva

J Baker

Defining a consistent strategy to model groundmotion parameters for the GEM-PEER Global GMPEs Project

Keywords: Ground-motion prediction equations (GMPEs), seismic hazard assessment, predictor parameters of GMPEs

The project entitled Global Ground Motion Prediction Equations is funded by the Global Earthquake Model (GEM) Foundation and has the objective of recommending a harmonized suite of ground motion prediction equations (GMPEs) that can be used at the global and regional scales for seismic hazard analysis and loss estimation studies. As part of this project, T a s k 1 a e x p e r t s were commissioned to make recommendations on the critical aspects of seismological predictor parameters that are used by predictive model developers to estimate ground motions for different earthquake scenarios. It is hoped that these recommendations would lead to the optimum description of groundmotion models that can be used efficiently for reliable seismic hazard assessment studies.

INTRODUCTION

Ground-motion prediction equations (GMPEs) relate a ground-motion intensity measure to a set of explanatory variables describing the source, wave propagation path and site response. Although these estimator parameters mainly include the magnitude, style-of-faulting and source-to-site distance, there are significant efforts to include additional variables to model the ground-motion behavior in a more realistic way. These variables have different levels of influence on the estimated ground-motion intensity measure. Moreover, each of these variables is affected by different factors that in turn define the level of accuracy of the predictions. This paper is a summary of a report (Task 1a report: [START_REF] Akkar | Define a consistent strategy to model ground motion: consistency in model parameters (Task 1a)[END_REF] prepared under the framework of the "GEM-PEER Global Ground Motion Prediction Equations project. The overall aim of the project is to propose GMPEs for reliable regional and global seismic hazard assessment. Under this objective, Task 1a aimed to identify the significance of each predictor parameter used in the functional forms of GMPEs. To this end, members of Task 1a suggested a set of recommendations for the optimum performance of each major predictor parameter and highlighted the required features of these parameters as well as the predictive models that are believed lead to a consistent seismic hazard assessment. The companion article of [START_REF] Douglas | Compilation and critical review of GMPEs for the GEM-PEER Global GMPEs Project[END_REF] presents a pre-selection o f c a n d i d a t e G M P E s f o r t h e s a m e p r o j e c t b ased on the recommendations made in Task 1a.

CONSIDERATION OF GROUND-MOTION INTENSITY MEASURES

The ground-motion intensity measures (IMs) of interest in the predictive models should primarily respond to the needs of the engineering community. Both force-based and displacement-based design and seismic performance assessment codes/guidelines use the pseudo-spectral acceleration (PSA), spectral displacement (SD) as well as peak ground acceleration (PGA) and velocity (PGV) as IMs. The use of these IMs is expected to remain at least for the next decade. Therefore, GMPEs that consider these IMs are of prime importance for hazard assessment. In addition to the above mentioned IMs, GMPEs that consider other ground-motion parameters (e.g., equivalent number of cycles, strongmotion duration, spectral ratios at different damping values, horizontal-to-vertical spectral ratios, Arias intensity and cumulative absolute velocity (CAV)) are also important as they either provide supplementary information to conventional seismic design or more advanced design/risk assessment projects.

The reliability of IMs that are computed from processed records and are used in the derivation of GMPEs needs to be considered, because they affect the ground-motion estimates. The reliable estimation of long-period spectral ordinates is important for base-isolated structures as well as highrise buildings and long-span bridges. Consistent short-period vertical spectral ordinates become essential for the derivation of horizontal-to-vertical spectral ratios as well as for the design of shortperiod machinery components mounted on building systems. GMPEs that present a clear documentation on the specific features of the implemented processing procedure as well as describe the usable spectral period range of the processed data are recommended for hazard studies.

The major challenges in data processing schemes that rely on filtering are: (a) minimum interference of filter corner frequencies that result in stable peak ground-motion values and spectral ordinates computed from the processed records, (b) choice of filter cut-off frequencies that result in minimum manipulation in the actual frequency content of the ground motion, and (c) determination of reliable usable spectral period ranges where the influence of processing is minimal. Acausal (zero-phase) filters are the recommended filtering type (e.g., [START_REF] Boore | Effect of causal and acausal filters on elastic and inelastic response spectra[END_REF] in order to address the point raised in (a). If time-domain acausal filtering is used, it is necessary to add leading and trailing zero pads at the beginning and end of the recording to accommodate filter transients. Choice of filtering cut-off frequencies can be based on low-and high-frequency behavior of ground motions (e.g., [START_REF] Akkar | Influence of long-period filter cut-off on elastic spectral displacements[END_REF][START_REF] Douglas | High-frequency filtering of strong-motion records[END_REF]. This approach uses theoretical source spectrum behavior and aims at minimum manipulation of the ground-motion data. The usable period band should be as wide as possible to address a large range of engineering applications. There are alternative methods for the computation of usable spectral period range (e.g., [START_REF] Abrahamson | Empirical response spectral attenuation relations for shallow crustal earthquakes[END_REF][START_REF] Akkar | Influence of long-period filter cut-off on elastic spectral displacements[END_REF][START_REF] Douglas | High-frequency filtering of strong-motion records[END_REF]. The common conclusion of these procedures is that the long-period spectral ordinates are reliable up to a certain fraction of the highpass filter cut-off and that the low-pass filtering has minimal effects on short-period spectral ordinates. For example, [START_REF] Akkar | The high-frequency limit of usable response spectral ordinates from filtered analogue and digital strong-motion accelerograms[END_REF] suggest extending the short-period limit of pan-European strong-motion database to at least 0.02 s. The NGA ground-model models use 0.01 s as the shortest period for the calculation of spectral ordinates.

As noted above, recent research tends to extend the usable period range as much as possible to respond to engineering needs. On the other hand, most of the current GMPEs, except for those developed in the NGA project, lack such a wide period band. The NGA models are able to estimate spectral ordinates between 0.01 and 10 s. This limitation can require interpolation of the some of the candidate GMPEs for PSHA f o r s o m e i n t e r m e d i a t e p e r i o d s p r o v i d e d t h a t b o t h e n d s o f t h e interpolation interval are fixed. When interpolation is required, it can be done either directly on the predicted spectral ordinates or on the regression coefficients. The second alternative (e.g., Bommer et al., 2012) is more flexible, because it offers a wider range of applicability of the interpolated GMPEs (in particular for PSHA studies). Nevertheless caution should be exercised in this alternative if the functional form of the interpolated model is complex and contains a large number of coefficients. Although interpolation of GMPEs for missing spectral ordinates are recommended for hazard studies for reasons explained herein, the extrapolation of GMPEs to both short and long periods is strongly discouraged.

CONSIDERATION OF SEISMOTECTONIC REGIONS

Acceptable ground-motion models must consider the differences between the main tectonic regimes. The major tectonic regimes that are the focus of the vast majority of hazard studies are: (a) stable continental regions (SCRs), (b) subduction zones, (c) active shallow crustal regions (ASCRs), (d) volcanic regions, (e) earthquakes with mainly oceanic travel paths (e.g., Portugal) and (f) deep Vrancea-type earthquakes. The models should address the important features of fault mechanisms and source properties that characterize each tectonic regime. Furthermore, models should consider differences in the attenuation of ground-motion amplitudes and in important estimator parameters that can be of particular use for a seismotectonic region. During a hazard study, it is very important to consider the seismotectonic setting for which a model was developed.

There are a number of different regionalizations that describe SCRs. The most widely used classification, based on contrasts in age, tectonic origin and tectonic history, is the one developed by EPRI [START_REF] Johnston | The earthquakes of stable continental regions: Assessment of large earthquake potential[END_REF]. The EPRI classification was not based on observations of contrasting earthquake source or seismic wave propagation characteristics, although it implied that such contrasts might result from contrasts in tectonic origin and history. The developers of GMPE's for SCRs are now in the process of discovering the degree to which this conjecture is borne out in the actual earthquake source and seismic wave propagation characteristics of these regions. As part of a global effort, the classification needs to be consistent with the other regional hazard studies (e.g., in strain rates). For example, the tectonic regions can be globally separated into shield (cratonic -low deformation with low attenuation) and continental crust (non-cratonic -l o w d e f o r m a t i o n b u t significant attenuation). It is recommended to consider separate GMPEs for the cratonic regions and non-cratonic regions (e.g., [START_REF] Somerville | Source and ground motion models of Australian earthquakes[END_REF]. An attempt towards this suggestion is presented by [START_REF] Delavaud | Toward a ground-motion logic tree for probabilistic seismic hazard assessment in Europe[END_REF].

There is also a need to set a consistent definition for differentiating between different types of earthquakes associated with a subduction zone (e.g., using depth and focal mechanism). Acceptable ground-motion models should take into consideration the differences existing among, for instance, interface and intraslab earthquakes, and provide separate sets of coefficients for each type of earthquake. This implies that the metadata should maintain accuracy in providing reliable information on earthquake depth, focal mechanism e t c . A S C R s are c h a r a c t e r i z e d b y a c t i v e t e c t o n i c s w i t h relatively high strain rates, generally close to plate boundaries, and earthquakes that occur in the upper 20 to 30 km of the crust generally on well-identified mature faults (e.g., California, Italy, Turkey, Greece and Japan). It is often considered that ground motions associated with this type of seismotectonic regime do not show strong regional dependencies, at least for moderate and large earthquakes. The vast majority of published GMPEs were derived for this type of area, partly because of the considerable amount of strong-motion data that are available. Volcanic earthquakes are unlikely to be larger than about M w 5.5 (and probably much smaller) and, therefore, the seismic hazard from this type of earthquake is unlikely to be important on a global scale. Although there are almost no ground-motion models specifically estimating the ground-motion amplitudes for earthquakes from volcanic activity, some GMPEs considered earthquake data from volcanic regions (e.g., [START_REF] Mcverry | New Zealand acceleration response spectrum attenuation relations for crustal and subduction zone earthquakes[END_REF][START_REF] Zhao | Geometric spreading functions and modeling of volcanic zones for strong-motion attenuation models derived from records in Japan[END_REF][START_REF] De Natale | Scaling of peak ground motions from digital recordings of small earthquakes at Campi Flegrei, southern Italy[END_REF] proposed a stochastic model for estimating the ground motions of volcanic events in Italy. In the framework of the GEM-PEER project, separate GMPEs may be adopted for regions of volcanic activity in which the historical seismicity is well-documented. Such GMPEs should be able to deal with events of shallow focal depth, from 2 to 5 km. While deep-focus non-subduction earthquakes have occurred or could occur in various seismic prone regions around the world (e.g., Vrancea (Romania) zone, New Hebrides Islands, Baja California, Bucaramanga (Colombia), eastern Anatolia, New Guinea and Pamir Hindu-Kush), there are practically no ground-motion predictive models for such regions. In fact, the only model that exists for the prediction of response spectral ordinates in such a region is the one developed by [START_REF] Sokolov | Ground-motion prediction equations for the intermediate depth Vrancea (Romania) earthquakes[END_REF], which is specific to the Vrancea region. In the absence of an adequate number of GMPEs for such regions, it may be suggested to use the GMPEs of intraslab earthquakes for hazard analysis. For offshore seismic hazard studies, a mixture of GMPEs from active and SCRs may be recommended as such regions also lack GMPEs that specifically feature the earthquake source and wave-propagation characteristics of these regions.

CONSIDERATION OF MAGNITUDE TYPE AND RANGE

Moment magnitude (M w ) has almost become the standard magnitude scale for developing GMPEs, since it can describe the energy release of an earthquake without suffering from saturation. Although the use of M w shows a significant increase during the last decade, GMPEs that use M L (local magnitude) as well as M s (surface-wave magnitude) still constitute a fairly large fraction in the overall number of available GMPEs. Models that use M L are generally local GMPEs, whereas many earlier global GMPEs as well as a few local ones have used M s . Therefore, conversions between magnitude scales are necessary when the magnitude definitions of the GMPEs differ. Although there are quite a few peer-reviewed empirical relations that convert various magnitude scales to a common magnitude definition (e.g., moment magnitude) [START_REF] Utsu | R e l a t i o n s h i p s b e t w e e n m a g n i t u d e s c a l e s[END_REF], they can differ significantly in estimating the target magnitude scale. These relations carry their own uncertainty due to different regression methods adopted in their calculation as well as the different approaches and databases used in the computation of magnitude scales by different seismic agencies.

State-of-the-art seismic-hazard analyses consider error propagation due to the aleatory and epistemic uncertainties inherent in magnitude scaling relationships when GMPEs that use different magnitude definitions have to be converted to a common magnitude format. The error propagation due to magnitude conversion has a larger influence at longer spectral periods and low annual exceedance probabilities in hazard because of their stronger magnitude dependence [START_REF] Bommer | On the use of logic trees for ground-motion prediction equations in seisrnic-hazard analysis[END_REF]. To avoid such complications, it is suggested to employ GMPEs that adopt M w as the magnitude scale. If there is a need to use GMPEs of different magnitude definitions, the analyst has to understand the modeling uncertainty stemming from magnitude conversions. Observing differences in spectral ordinate estimates after the application of alternative magnitude conversion relations can be one approach to understand the modeling uncertainty due to magnitude conversion.

Most of the GMPEs are based on data from moderate-to-large earthquakes with magnitudes equal to or greater than 5. Reasons behind this include: (a) existence of a significant number of analog and early digital accelerograms with magnitudes 5 and above, (b) general consensus among the engineering and seismological communities that most structures are not susceptible to damage for magnitude levels less than about 5, and (c) the increased uncertainty in the magnitude determination of small events. However, ground-motion models derived from moderate-to-large earthquakes tend to overestimate the ground motions from small magnitude events and at the lower magnitude limit of their original datasets, which is of particular importance when computing the hazard for regions of low-to-moderate seismicity (e.g., [START_REF] Chiou | Ground-motion attenuation model for small-to-moderate shallow crustal earthquakes in California and its implications on regionalization of groundmotion prediction models[END_REF]Campbell, 2011). Given the current improvements in strong-motion databases, the developers of new GMPEs may want to lower the magnitude limit to M w 4 or 4.5 so that ground motions near M w 5 are not overestimated. The GMPEs developed by [START_REF] Chiou | Ground-motion attenuation model for small-to-moderate shallow crustal earthquakes in California and its implications on regionalization of groundmotion prediction models[END_REF] and [START_REF] Atkinson | Modifications to existing ground-motion prediction equations in light of new data[END_REF] can be considered as the early attempts to cover smaller magnitudes in ground-motion modeling.

The upper magnitude limit of applicability of GMPEs generally ranges between 7 and 8 for the most recent models. This magnitude range reflects the maximum magnitudes that can be expected in most seismically-active regions. The recent devastating subduction-interface earthquakes (M w 8.8 Maule, Chile, and M w 9.0 Tohoku-oki, Japan) showed that maximum magnitudes could be well above M w 8 for such earthquakes. These new observations as, well as more refined geological and paleoseismic studies, will eventually push the upper magnitude level above 8 in future GMPEs pending the accumulation of such data in strong-motion databases. In view of the state of practice in the use of current GMPEs, the maximum magnitudes to be considered in SCRs and ASCRs should be in the range of M w 8. This upper bound should be at least M w 8.5 for most subduction-interface GMPEs.

CONSIDERATION OF SOURCE-TO-SITE DISTANCE MEASURES

The common source-to-site distance measures in GMPEs are R epi ( e p i c e n t r a l d i s t a n c e ) , R hyp (hypocentral distance), R rup (closest distance to rupture surface) and R JB (closest distance to horizontal projection of rupture to the surface, so called Joyner-Boore distance). The latter two distance metrics have become popular as they can address the variation of ground-motion amplitude more appropriately for large events at sites closer to the source. Both R rup and R JB require reliable finitefault (or rupture model) representations that are generally available for large-magnitude earthquakes or for some specific events that are studied individually. R epi and R hyp are still more accessible in the existing strong-motion databases, b e c a u s e seismic agencies routinely report the seismological parameters required for their calculation (although the depth parameter used for R hyp still contains significant uncertainty in current catalogs). Consequently, there is still a larger number of GMPEs that use either R epi or R hyp (point-source distance measures) Ground-motion models that employ extendedsource distance measures (R JB and R rup ) are often derived from global strong-motion databases, because such databases are generally developed under the framework of well-funded projects that can recruit experts on database compilation and processing. Many local GMPEs as well as some of the global GMPEs use R hyp or a combination of point-and extended-source distance pairs in their functional forms.

Whenever ground-motion models of different distance metrics are employed in probabilistic seismic hazard analysis (PSHA), the state-of-the-practice is to adjust the differences in distance measures to model consistent earthquake scenarios. Moreover, GMPEs that use extended source distance metrics should be modified properly in order to accurately estimate the hazard due to background seismicity for which earthquake scenarios are treated as point sources. For such distance conversions various proposed relationships may be used (e.g., [START_REF] Johnston | The earthquakes of stable continental regions: Assessment of large earthquake potential[END_REF][START_REF] Scherbaum | On the conversion of source-to-site distance measures for extended earthquake source models[END_REF]. The major challenge in employing distance conversion relations is the propagation of the error into the estimated ground motions. This uncertainty can lead to larger standard deviations (i.e., increased aleatory variability) in the GMPEs and may significantly increase the hazard [START_REF] Scherbaum | Composite ground-motion models and logic trees: methodology, sensitivities, and uncertainties[END_REF]. One approach to avoiding such large uncertainties in hazard calculations is to use hazard codes that are capable of modeling finite-fault sources in such a way that the commonly used distance metrics can be computed and that each GMPE can be used with its original distance definition [START_REF] Bommer | On the selection of ground-motion prediction equations for seismic hazard analysis[END_REF]. In a similar fashion, for hazard analysis concerning areal sources for which the earthquake scenarios are defined as points, GMPEs that use R JB or R rup can be employed consistently together with GMPEs that use point-source distance measures, if the hazard software is capable of simulating pseudo-ruptures for each earthquake scenario with random orientations and with dimensions determined from empirical relationships [START_REF] Bommer | On the selection of ground-motion prediction equations for seismic hazard analysis[END_REF]. Another reasonably simple method, as implemented by the U.S. Geological Survey, is to use look-up tables of pre-calculated average finite-source distance measures, assuming random azimuth and specific rupture geometries for each point source. As proposed by Bommer and Akkar (2012), a more pragmatic solution for using consistent distance metrics in hazard calculations is to develop GMPEs in pairs; one that uses an extended-source distance metric for PSHA applications to finite-fault sources and another that is based on either R epi or R hyp to properly consider the contribution of areal sources. The discussed methodologies in this paragraph are recommended for computing reliable hazard results while adjusting the ground-motion estimations of GMPEs that use different distance measures.

CONSIDERATION OF STYLE-OF-FAULTING

The consideration of style-of-faulting (SoF) as a predictive parameter in GMPEs is now common. As is often the case for empirical models, there can be strong trade-offs between the SoF factor and other predictive parameters such as source depth. Reliable source-depth information is important for identifying the rupture mechanisms as well as different tectonic regimes for subduction zones. Depth and fault mechanism, or centroid moment tensor (CMT) information, provides a good way to distinguish crustal, interface and intraslab subduction earthquakes. The classification schemes used for mapping the earthquakes into different fault mechanisms and the database distribution also affect the estimated ground-motion amplitudes for different types of faulting. Using the plunge of the P, T and B axes seems to be a reliable way of classifying different styles-of-faulting, although rake angle schemes yield consistent SoF classifications for many cases. Thus, use of the plunge of the P, T and B axes should be verified with the rake angle intervals as implemented by [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] in order to reduce the number of unclassified earthquakes in terms of SoF.

Currently most of the global GMPEs and a few local ones address SoF. The ground-motion models that include the SoF predictor parameters should be preferred in PSHA as long as they address correlations with other parameters that can also explain the observations and are proven to be efficient and accurate for the considered seismic region. [START_REF] Bommer | Style-of-faulting in ground-motion prediction equations[END_REF] proposed a scheme to incorporate SoF into GMPEs that either consider only one fault mechanism or are independent of fault mechanisms. Unless there are compelling reasons for using such adjusted GMPEs in the hazard analysis, the use of methodologies that externally adjust the estimated ground-motion amplitudes for SoF effects should be avoided, since such manipulations would increase the uncertainty in the estimated ground motions.

CONSIDERATION OF NEAR-FAULT EFFECTS (DIRECTIVITY / DIRECTIONALITY)

To accurately characterize near-fault ground motions, directivity and directionality effects should be taken into account. Directivity effects generally increase spectral accelerations for periods longer than approximately 0.5 s at locations where the rupture has propagated primarily towards the site of interest. At locations where the rupture has propagated primarily away from the site, spectral accelerations generally decrease. Forward-directivity effects are manifested in the presence of a shortduration velocity pulse, which is not modeled by most GMPEs. The directivity effect depends upon rupture direction and additional predictor parameters are needed to account for this effect. Most directivity models use a description of the amount of rupture that has ruptured towards the site of interest either as an absolute distance or a fraction of the total length of rupture. Directivity models require the specification of hypocenter location and the size and orientation of the rupture plane as well as the dependence of directivity effects on fault mechanism.

The directivity model developed by [START_REF] Somerville | Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity[END_REF] provides separate predictions of the strike-normal and strike-parallel components of motion, and was later modified by [START_REF] Abrahamson | s e i s z a r d a n a l y s i s[END_REF] for strike-slip earthquakes. These models are currently widely used for hazard calculations because they use simple parameters that are easy to implement. The [START_REF] Somerville | Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity[END_REF] model indicates a strong dependence of the ratio of strike-normal to average horizontal motions on magnitude and rupture length. The ratio of strike-normal to average horizontal motions is found to be period-dependent and becomes significant for periods greater than 0.6 s. The model by [START_REF] Somerville | Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity[END_REF] also includes systematic variations of duration in near-fault ground motions due to directivity.

Simple time-domain pulses can represent near-fault ground motions containing forward rupture directivity. Various studies developed equations relating the period of the pulse to the earthquake magnitude and the effective velocity of the pulse to the earthquake magnitude and distance (e.g., [START_REF] Somerville | Magnitude scaling of the near fault rupture directivity pulse[END_REF]. Although considerable differences exist between the predictions by different models, all of the correlations show a similar trend of pulse period increasing with earthquake magnitude. For large surface faulting earthquakes (e.g., the M w 7.6 Chi-Chi earthquake), the value of the pulse period may also be affected by the large permanent displacements accompanying the fault rupture.

There are also more complicated models for directivity, such as the one developed by [START_REF] Rowshandel | Directivity Correction for the Next Generation Attenuation (NGA) Relations[END_REF] within the NGA project, or the ones that are currently studied within the framework of the NGA-West2 project. The model proposed by [START_REF] Rowshandel | Directivity Correction for the Next Generation Attenuation (NGA) Relations[END_REF] is capable of handling every type of rupture direction, because they generalize the directivity predictors of [START_REF] Somerville | Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity[END_REF] by applying a surface integral over the fault for every receiver location. Based both on analysis of ground motion simulations and the database of recorded ground motions developed for the NGA project, [START_REF] Spudich | Directivity in NGA earthquake ground motions: Analysis using isochrone theory[END_REF] developed a new physically-based directivity model using isochrone theory. Their model provides an improved characterization of directivity over earlier models. However, the Spudich-Chiou model is most applicable to the NGA GMPEs and would likely be difficult to apply to other models. The Spudich-Chiou directivity model currently does not address differences between strike-normal and strike-parallel components. Under these discussions, the [START_REF] Somerville | Modification of Empirical Strong Ground Motion Attenuation Relations to Include the Amplitude and Duration Effects of Rupture Directivity[END_REF] and [START_REF] Abrahamson | s e i s z a r d a n a l y s i s[END_REF] studies seem to constitute the most suitable and practical models, while considering near-fault effects on ground-motion amplitudes, until the other mentioned models have been implemented widely to assess their level of practicality in a state-of-theart PSHA.

Most GMPEs estimate the average ("geometric mean") of the response spectra of two horizontal components. In some cases it may be of greater interest to know the maximum spectral value, over all possible orientations, at a given period. Various models are proposed for making such a "maximum direction" prediction (e.g., [START_REF] Boore | Orientation-Independent, Nongeometric-Mean Measures of Seismic Intensity from Two Horizontal Components of Motion[END_REF]. Loosely speaking, the maximum-direction spectral value will be about 1.2 times the geometric mean value at short periods and 1.3 times the geometric mean value at long periods.

CONSIDERATION OF OTHER IMPORTANT PARAMETERS: BASIN EFFECTS,

DEPTH TO TOP-OF-RUPTURE, Z 1.0 , Z 2.5 , NONLINEAR SITE EFFECTS AND V S30 -KAPPA CORRELATIONS

Many additional predictor parameters are important in the reliable estimation of ground motion and their consideration in GMPEs has proven to be useful (see discussions by NGA model developers).

For instance, parameters such as Z TOR (depth to top-of-rupture), Z 1.0 and Z 2.5 (depth from the ground surface at which shear-wave velocity attains values of 1.0 km/s and 2.5 km/s, respectively) are important for forward calculations, in particular for site-specific hazard analysis. The parameter R x , the shortest horizontal distance to the top edge of rupture measured perpendicular to the strike (site coordinate), has also an important role when considering hanging-wall effects. H o w e v e r , s u c h information is still limited in the metadata of strong-motion databases. Thus, consideration of such parameters in ground-motion models may still not be recommended unless the model developer assures their reliability. [START_REF] Kaklamanos | A framework for estimating unknown input parameters when implementing the NGA ground motion prediction equations in engineering practice[END_REF] propose default values for fault dips, Z TOR and other parameters used in the NGA models when they are unknown. However, care should be taken in evaluating how error propagates when considering these "default" values.

Nonlinear site effects are currently estimated by using the average shear-wave velocity in the upper 30 m of the soil profile (V S30 ). Measured V S30 values are still rare in many strong-motion databases and many ground-motion equations still consider generic soil classes to model site effects. The average V S30 and kappa (high-frequency filter commonly considered to be related to near-surface attenuation) on rock can significantly affect short-period response spectral ordinates. GMPEs developed for different parts of the world have used different rock definitions and consequently require calibration to a specific rock classification. This is reasonably straightforward for models that treat V S30 as a variable. V S30 and kappa are generally considered to be correlated (e.g., [START_REF] Van Houtte | Analysis of the origins of (Kappa) to compute hard rock to rock adjustment factors for GMPEs[END_REF]. Therefore, a coupled V S30 -kappa correlation, despite large uncertainties, can be used to calibrate GMPEs for a specific rock definition [START_REF] Van Houtte | Analysis of the origins of (Kappa) to compute hard rock to rock adjustment factors for GMPEs[END_REF]. In the context of this project such relationships are encouraged to provide guidance in hazard analysis to describe rock conditions in different parts of the world.

CONSIDERATION OF DISPERSION ABOUT MEDIAN

The idealizations imposed in the ground-motion models result in differences between the observed and estimated ground-motion IMs. This dispersion is described by the standard deviation (sigma) associated with the GMPE. Sigma is usually interpreted as describing the aleatory variability of ground motion, but it can also convey information on the epistemic (modeling) uncertainty due to differences in functional forms. The value of sigma has a significant impact on the seismic hazard analysis, in particular, at low annual exceedance rates (Bommer and Abrahamson, 2006).

The quality, selection and record processing of the strong-motion data can affect the accuracy of predicted ground-motion parameter that in turn can result in high values of sigma [START_REF] Strasser | Sigma: issues, insights, and challenges[END_REF]. The uneven distribution of the database can also increase the value of sigma. The reliability of metadata information of the strong-motion database will contribute to sigma as well. For example [START_REF] Abrahamson | Summary of the Abrahamson and Silva NGA ground motion relations[END_REF] indicated that consideration of measurement errors in magnitude and depth-to-top of rupture can reduce the inter-event variability at long periods. In fact, recent GMPEs consider the magnitude effect on the dispersion about median estimations, because a decrease in magnitude seems to inflate sigma due to the larger errors involved in the magnitude, location and depth of smaller events (e.g., [START_REF] Abrahamson | Summary of the Abrahamson and Silva NGA ground motion relations[END_REF][START_REF] Chiou | An NGA model for the average horizontal component of peak ground motion and response spectra[END_REF]. There are counter observations finding no significant evidence for the dependence of sigma on magnitude, which may be related to the specific features of the database used in developing the GMPE. Other than magnitude, some studies find the dependence of sigma on distance and soil behavior. Increased nonlinear behavior of soil deposits also leads to a reduction in intra-event variability. While some estimator parameters contribute to the level of sigma as summarized above, [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] indicated that inclusion of SoF does not have a significant impact on sigma. However, disregarding SoF effect on sigma can be disadvantageous for site-specific hazard studies that are dominated by a specific rupture type. Inclusion of additional predictor variables in the GMPEs can reduce the sigma, but this effort requires well-constrained metadata information. Sigma can also be controlled by studying the behavior of its individual components of variability associated with a single event or a single station (e.g., [START_REF] Rodriguez-Marek | Analysis of single-station standard deviation using the Kik-net data[END_REF]. Such studies can be of use for site-specific hazard studies.

A proper consideration of the influence of the previously mentioned predictor parameters (and similar ones) in functional forms as well as the use of high-quality strong-motion databases and the use of efficient regression techniques (e.g., maximum-likelihood methods) in the development of GMPEs that address the variation of different components of sigma are recommended.

CONSIDERATION OF SIMULATION-BASED GMPEs

The earthquake source involves a complex rupture process on a fault, seismic waves arrive at the site by propagation through a complex waveguide, and complex local geology can have an important influence on the recorded ground motions. Strong-motion simulation methods have the advantage over empirical methods by allowing the incorporation of information about earthquake source, seismic wave propagation, and local site characteristics that are specific to the region and to a specific site location. Modern seismic hazard analysis for regions of sparse data (e.g., SCRs or ASCRs with limited number of strong-motion recordings) generally requires GMPEs that are based on groundmotion simulations, because the available observational data in such regions are insufficient to properly constrain the parameters in a GMPE. Even in tectonically active regions, the ground motions that dominate seismic hazards at return periods used in seismic design are often for large magnitudes and close distances; there are few records in the strong-motion databases at such magnitudes and distances, although this limitation is being reduced. Synthetic accelerograms can also supplement the strong-motion database where the data distribution is poorly constrained in terms of independent predictor parameters such as magnitude, distance, SoF etc. Stochastic (either point-or finite-source) (e.g., [START_REF] Boore | Prediction of ground motion using the stochastic method[END_REF][START_REF] Motazedian | Stochastic finite-fault model based on dynamic corner frequency[END_REF] and broadband simulation methods (e.g., [START_REF] Graves | Broadband Ground-Motion Simulation Using a Hybrid Approach[END_REF][START_REF] Somerville | Source and ground motion models of Australian earthquakes[END_REF] can be used to develop GMPEs for determining the deterministic and probabilistic estimates of design ground motion. In addition to recent simulationbased GMPEs, the hybrid empirical method has become popular (e.g., Campbell, 2 0 0 3 ). This approach estimates ground motions in regions of limited data (target regions) by calibrating reliable GMPEs from regions of abundant data (host regions).

All of these methods depend on a detailed parameterization where the assessment of simulated ground motions and related uncertainties may be difficult for regions that are poorly covered by seismological networks and studies. The ideal target regions are, therefore, those with a rate of seismicity such that the path and site properties can be determined f r o m s m a l l m a g n i t u d e e a r t h q u a k e s . I n P S H A implementation for regions that lack strong-motion data, GMPEs derived from point-source and extended-source stochastic simulations (that properly address epistemic uncertainty) are appealing as these methods are easy to apply. The stochastic methods can also be used efficiently in hybrid empirical and composite techniques, which can also be considered for PSHA studies of regions lacking GMPEs due to sparse strong-motion data. The consistent use of stochastic methods in PSHA depends on their suitability for the region under consideration. Their validity is suggested to be checked with few available regional strong-motion records, assuming that such data exists. Provided that adequate knowledge of the seismological parameters is available, the hybrid techniques to simulate broadband ground motions can be used in site-specific PSHA studies. Such methods can also account for the epistemic uncertainty and are capable of addressing the propagation of uncertainties on the simulated ground motion parameters.

CONCLUSION

The recommendations made throughout the paper are believed to be important f o r c o n s i s t e n t modeling of ground motions as well as for increased performance of each predictor parameter. These recommendations constitute one of the building blocks for stable and reliable seismic hazard assessment at regional and global scales. A comprehensive list of references that cannot be cited due to space limitations in this paper can be found in [START_REF] Akkar | Define a consistent strategy to model ground motion: consistency in model parameters (Task 1a)[END_REF]. The list of recommended GMPEs that generally meet the criteria presented in this paper are available in [START_REF] Douglas | Compilation and critical review of GMPEs for the GEM-PEER Global GMPEs Project[END_REF].

AKCNOWLEDGEMENT

This study was funded by the GEM Foundation as part of the Pacific Earthquake Engineering Research Center's (PEER's) Global GMPEs project. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the sponsors. The first author acknowledges the valuable recommendations of Dr. David M. Boore and Dr. Norman N. Abrahamson.


the three-dimensional structure of the scene by using the measurements provided by one or more sensors over time (e.g. vision sensors, ego-motion sensors, range sensors). Solving this problem consists in simultaneously performing self motion perception (sMP) and depth perception (DP). In the case of visual measurements only, the SfM has been solved up to a scale [START_REF] Chiuso | Structure from Motion Causally Integrated Over Time[END_REF][START_REF] Davison | MonoSLAM: Real-Time Single Camera SLAM[END_REF][START_REF] Hartley | In Defense of the Eight-Point Algorithm[END_REF][START_REF] Longuet-Higgins | A computer algorithm for reconstructing a scene from two projections[END_REF][START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF] and a closed form solution has also been derived [START_REF] Hartley | In Defense of the Eight-Point Algorithm[END_REF][START_REF] Longuet-Higgins | A computer algorithm for reconstructing a scene from two projections[END_REF][START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF], allowing the determination of the three-dimensional structure of the scene, without the need for any prior knowledge. The case of inertial and visual measurements has particular interest and has been investigated by many disciplines, both in the framework of computer science [START_REF] Bryson | Observability Analysis and Active Control for Airbone SLAM[END_REF][START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF][START_REF] Strelow | Motion estimation from image and inertial measurements[END_REF] and in the framework of neuroscience (the visual-vestibular integration for sMP [START_REF] Berthoz | Perception of Linear Horizontal Self-Motion Induced by Peripheral Vision (Linearvection) Basic Characteristics and Visual-Vestibular Interactions[END_REF][START_REF] Fetsch | Visual-vestibular cue integration for heading perception: Applications of optimal cue integration theory[END_REF][START_REF] Macneilage | Computational Approaches to Spatial Orientation: From Transfer Functions to Dynamic Bayesian Inference[END_REF][START_REF] Zupan | Using sensory weighting to model the influence of canal, otolith and visual cues on spatial orientation and eye movements[END_REF] and for DP [START_REF] Dokka | Estimating distance during self-motion: a role for visual-vestibular interactions[END_REF]). Prior work has answered the question of which are the observable modes, i.e. the states that can be determined by fusing visual and inertial measurements [START_REF] Bryson | Observability Analysis and Active Control for Airbone SLAM[END_REF][START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF][START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. The questions of how to compute these states in the absence of a prior, and of how many solutions are possible, have only been answered very recently [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF].

Here we derive a very simple and intuitive derivation of the solution introduced in [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. We show that the SfM problem can have a unique solution or two distinct solutions or infinite solutions depending on the trajectory, on the number of point-features and on the number of monocular images where the same point-features are seen. Our results are relevant in all the applications which need to solve the SfM problem with low-cost sensors and which do not demand any infrastructure. Additionally, our results could play an important role in neuroscience by providing a new insight on the process of vestibular and visual integration. Key-words: Sensor Fusion, Inertial Sensors, Vision, Non linear Observability, Structure from Motion Résumé : Le problème structure from motion (SfM) consiste en déterminer la structure à trois dimensions de la scène en utilisant les mesures fournies par un ou plusieurs capteurs progressivement (par exemple capteurs optiques, capteurs d'ego-mouvement, capteurs de distance). Résoudre ce problème consiste en résoudre simultanément la perception du mouvement (sMP) et la perception de profondeur (DP). Dans le cas de mesures visuelles seulement, le SfM a été résolu [START_REF] Chiuso | Structure from Motion Causally Integrated Over Time[END_REF][START_REF] Davison | MonoSLAM: Real-Time Single Camera SLAM[END_REF][START_REF] Hartley | In Defense of the Eight-Point Algorithm[END_REF][START_REF] Longuet-Higgins | A computer algorithm for reconstructing a scene from two projections[END_REF][START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF] et une solution analytique a été aussi dérivée [START_REF] Hartley | In Defense of the Eight-Point Algorithm[END_REF][START_REF] Longuet-Higgins | A computer algorithm for reconstructing a scene from two projections[END_REF][START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF], permettant la détermination de la structure de la scène à moins de l'echelle, sans le besoin d'une connaissance a priori. Le cas de capteurs inertielles et visuelles a un intérêt particulier et a été examiné par beaucoup de disciplines, soit dans le cadre des sciences informatiques [START_REF] Bryson | Observability Analysis and Active Control for Airbone SLAM[END_REF][START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF][START_REF] Strelow | Motion estimation from image and inertial measurements[END_REF] soit dans le cadre des neurosciences (l'intégration visuel-vestibulaire pour le sMP [START_REF] Berthoz | Perception of Linear Horizontal Self-Motion Induced by Peripheral Vision (Linearvection) Basic Characteristics and Visual-Vestibular Interactions[END_REF][START_REF] Fetsch | Visual-vestibular cue integration for heading perception: Applications of optimal cue integration theory[END_REF][START_REF] Macneilage | Computational Approaches to Spatial Orientation: From Transfer Functions to Dynamic Bayesian Inference[END_REF][START_REF] Zupan | Using sensory weighting to model the influence of canal, otolith and visual cues on spatial orientation and eye movements[END_REF] et pour le DP [START_REF] Dokka | Estimating distance during self-motion: a role for visual-vestibular interactions[END_REF]). Beaucoup d'effort a été fait pour détecter les modes observables, c.-à-d. les états qui peuvent être déterminés en fusionnant données visuelles et inertielles [START_REF] Bryson | Observability Analysis and Active Control for Airbone SLAM[END_REF][START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF][START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. La question de comment estimer la valeur de ces états en absence d'une connaissance a priori, et de combien de solutions sont possibles, a été répondu seulement très récemment [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. Ici nous proposons une dérivation très simple et intuitive de la solution introduite dans [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. Nous montrons que le problème de SfM peut avoir une solution unique, deux solutions ou encore infinies solutions. Nos résultats sont fondamentales dans toutes les applications qui demandent de résoudre le problème SfM avec des détecteurs à bas prix et qui n'exige pas de l'infrastructure. En plus, nos résultats pourraient jouer un rôle important dans la neuroscience en fournissant une nouvelle prospective sur le processus d'intégration vestibulaire et visuelle.

Mots-clés : Fusion Sensoriel, Capteurs inertiels, Vision, Observabilité, Structure from Motion

Introduction

Recent works on visual-inertial navigation have analytically derived the observable states and the set of parameters which are identifiable when fusing monocular vision with inertial sensing [START_REF] Bryson | Observability Analysis and Active Control for Airbone SLAM[END_REF][START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF][START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. Specifically, it has been shown that the velocity, the absolute scale, the gravity vector in the local frame and the bias-vectors which affect the inertial measurements, are observable modes. On the other hand, the vast majority of existing techniques for navigation using visual and IMU measurements employ either a recursive Bayesian estimation approach [START_REF] Bryson | Observability Analysis and Active Control for Airbone SLAM[END_REF][START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF], or a smoothing formulation [START_REF] Strelow | Motion estimation from image and inertial measurements[END_REF]. Both types of methods require a prior estimate for the initial state: since both rely on linearization of the nonlinear measurement models, lack of a good initial estimate can lead to divergence. This important limitation would be eliminated by introducing a deterministic solution, i.e., by analytically expressing all the observable modes in terms of the measurements provided by the sensors during a short timeinterval. A first derivation of this solution has been introduced in [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF]. Here we propose an alternative and more intuitive derivation.

Derivation

We consider a given time interval [t 0 , T ]. Our goal is to express in closed-form all the observable modes at time t 0 only in terms of the visual and inertial measurements obtained during [t 0 , T ]. In order to characterize the motion of a body moving in a 3D environment we introduce a global frame. Its z-axis points vertically upwards. As we will see, for the next derivation we do not need to better define this global frame. We will adopt lower-case letters to denote vectors in this frame (e.g. the gravity is g = [0, 0, -g] T , where g 9.8 ms -2 ). We also introduce a local frame on the body and we will adopt upper-case letters to denote vectors in this frame. Since this local frame is time dependent, we adopt the following notation: W t (τ ) will be the vector with global coordinates w(τ ) in the local frame at time t. Additionally, we will denote with C t2 t1 the matrix which characterizes the rotation occurred during the time interval (t 1 , t 2 ). We have:

W t1 (τ ) = C t2 t1 W t2 (τ ) and (C t2 t1 ) -1 = C t1 t2 .
Finally, C t will denote the rotation matrix between the global frame and the local frame at time t, i.e., w(τ

) = C t W t (τ ).
The position of the body r at any time t ∈ [t 0 , T ] satisfies the following equation:

r(t) = r(t 0 ) + v(t 0 )∆t + g ∆t 2 2 + C t0 S t0 (t) (1)
where v ≡ dr dt , ∆t ≡ t -t 0 , S t0 (t) ≡ t t0 τ t0 C ξ t0 A ξ (ξ)dξdτ . Note that the matrix C ξ t0 can be obtained from the angular speed during the interval [t 0 , ξ] provided by the gyroscopes [START_REF] Farrell | Aided Navigation: GPS and High Rate Sensors[END_REF] and A ξ (ξ) is the acceleration provided by the accelerometers [START_REF] Farrell | Aided Navigation: GPS and High Rate Sensors[END_REF]. In other words, C ξ t0 A ξ (ξ) is the acceleration of the vehicle at the time ξ ∈ (t 0 , T ) expressed in the vehicle frame at time t 0 . The vector S t0 (t) can be obtained at any time by integrating the data provided by the gyroscopes and the accelerometers delivered during the interval [t 0 , ξ].

Let us suppose that N f point-features are observed, simultaneously. Let us denote their position in the physical world with p i , i = 1, ..., N f . According to our notation, P i t (t) will denote their position at time t in the local frame at time t. We have: p i = r(t) + C t0 C t t0 P i t (t) and, by using (1), we obtain the following fundamental equation:

P i t (t) = C t0 t P i t0 (t 0 ) -V t0 (t 0 )∆t -G t0 (t 0 ) ∆t 2 2 -S t0 (t) (2)
Our goal is to determine the observable modes, i.e., P i t0 (t 0 ) (i = 1, ..., N f ), V t0 (t 0 ) and G t0 (t 0 ). We remark that the determination of G t0 (t 0 ) is equivalent to the knowledge of the roll and pitch angles at the time t 0 together with the magnitude of the gravity (g). Let us denote byX the (3N f + 6)-column vector whose entries are the unknowns (X ≡ [P 1 t0 (t 0 ), P 2 t0 (t 0 ), ..., P Nf t0 (t 0 ), V t0 (t 0 ), G t0 (t 0 )] T ). Equation [START_REF] Berthoz | Perception of Linear Horizontal Self-Motion Induced by Peripheral Vision (Linearvection) Basic Characteristics and Visual-Vestibular Interactions[END_REF] states that the position of any point-feature in the local frame at time t is affine in X. Let us suppose that the N f point-features are observed in n im consecutive images. A single image provides the bearing angles of all the point-features in the local frame. In other words, an image taken at the time t provides all the vectors P i t (t) up to a scale. Since the data provided by the gyroscopes during the interval (t 0 , T ) allow us to build the matrix C t t0 , having the vectors P i t (t) up to a scale, allows us to also know the vectors P i t0 (t) = C t t0 P i t (t) up to a scale. From equation (2) we easily obtain:

P i t0 (t) = P i t0 (t 0 ) -V t0 (t 0 )∆t -G t0 (t 0 ) ∆t 2 2 -S t0 (t) (3)
namely, also the vector P i t0 (t), which can be obtained up to a scale by using the camera and gyroscope data in the interval (t 0 , T ), is affine in X.

Knowing a vector up to a scale means to know the ratios among its components. Therefore, the j th image provides, for any point-feature (i = 1, ..., N f ), the two ratios y ij , where t 1 < t 2 < .... < t nim are the times when the n im camera images are taken. Hence we have:

[P i t0 (t j )] x = [P i t0 (t j )] z y ij 1 [P i t0 (t j )] y = [P i t0 (t j )] z y ij 2 (4)
By using (3) it is easy to realize that these are 2n im N f linear equations in the unknowns. In other words, we obtain the following linear system:

αX = β ( 5
)
where α is a 2n im N f × (3N f + 6) matrix and β is a (2n im N f )-column vector. The entries of α and β are directly obtained from the bearing measurements obtained from the images and the gyroscopes and the accelerometers measurements during the interval (t 0 , T ).

Number of solutions

In order to understand whether the problem can be solved or not it is necessary to compute the rank of α. In particular, we are interested in understanding how this rank depends on the motion, on the number of observed point-features and on the number of camera images. We have performed an observability analysis [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF][START_REF] Martinelli | Vision and IMU Data Fusion: Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination[END_REF] based on the rank criterion [START_REF] Hermann | Nonlinear Controllability and Observability[END_REF][START_REF] Isidori | Nonlinear Control Systems[END_REF][START_REF] Martinelli | State Estimation Based on the Concept of Continuous Symmetry and Observability Analysis: the Case of Calibration[END_REF]. The computation was carried INRIA out by using the symbolic computational tool of matlab in order to obtain the expression of the Lie derivatives up to the 4 th order together with their gradients.

We have extended the results obtained in literature [START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF][START_REF] Kelly | Visual-inertial simultaneous localization, mapping and sensor-to-sensor self-calibration[END_REF] by proving that the observability of the modes holds for any number of point-features (in particular, it holds even in the case of one point). More importantly for our purposes here, we proved that:

1. the observability of all the modes requires the body to move with a varying acceleration and to observe the same point-features in at least four images;

2. when the body moves with constant acceleration all the modes are observable with the exception of the magnitude of the gravity;

3. when the magnitude of the gravity is a priori known and the vehicle moves with constant speed all the remaining modes are observable up to a scale;

4. in general, by only having three images, all the modes are observable with the exception of the magnitude of the gravity.

From these results, we obtain that α cannot be full rank when n im ≤ 3 and/or when the body moves with constant acceleration. On the other hand, the cases n im = 3, N f ≥ 2 and the case when the vehicle moves with a constant (non null) acceleration, can still be solved in closed form by exploiting an a priori knowledge of the magnitude of the gravity (which is the unique non-observable mode for these cases). In these cases, since only one mode (g) is unobservable, the matrix α has in general a null space of dimension one (we found special cases where the null space has dimension larger than one, e.g., when the pointfeatures and the body positions where the images are taken are coplanar [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF] ).

Let us denote by n the unit vector spanning this null space. The linear system αX = β has infinite solutions. Each solution satisfies the following equation: X = α * β + γn, being α * the pseudoinverse of α and γ a scalar number [START_REF] Meyer | Matrix Analysis and Applied Linear Algebra[END_REF]. The determination of γ is obtained by enforcing the constraint that the norm of the vector formed by the last three elements of X is equal to g.

| ¶ (α * β + γn)| = g, ¶ ≡ [0 3×(3N f +3) , I 3 ] (6)
where 0 n×m is the n × m matrix whose entries are all zero and I 3 is the identity 3 × 3 matrix. The equation in ( 6) is a quadratic polynomial in γ and has two real roots. Hence, we obtain in general two distinct solutions for X.

When n im ≥ 4 and for a motion with a varying acceleration, the matrix α is full rank apart the case n im = 4, N f = 1 (actually, there are special cases where α is not full rank, e.g., when the point-features and the four body positions are coplanar [START_REF] Martinelli | Closed-Form Solutions for Attitude, Speed, Absolute Scale and Bias Determination by Fusing Vision and Inertial Measurements[END_REF]). In this latter case the number of equations is 8 and the number of unknowns 9. As in the cases n im = 3, N f ≥ 2 the problem has in general two distinct solutions which can be determined by solving [START_REF] Cornilleau-Peres | The visual perception of threedimensional shape from self-motion and object-motion[END_REF] where ¶ ≡ [0 3×6 , I 3 ].

Table 1 summarizes these results by providing the number of solutions case by case. Figures 1 and2

Extension to the biased case

The closed-form solution here derived can be easily extended to the case when the accelerometers measurements are biased. In this case, equation (2) becomes:

P i t0 (t) = P i t0 (t 0 )-V t0 (t 0 )∆t-G t0 (t 0 ) ∆t 2 2 - t t0 τ t0
C ξ t0 dξdτ A bias -S t0 (t) [START_REF] Davison | MonoSLAM: Real-Time Single Camera SLAM[END_REF] and the same procedure previously described can be adopted by considering the extended state X E ≡ [X, A bias ] T , which includes the three accelerometer bias-components.

Discussion and conclusion

The most useful applications of the closed-form solution here derived will be in all the applicative domains which need to solve the SfM problem with low-cost sensors and which do not demand any infrastructure (e.g., in GPS denied environment). In these contexts, there is often the need to perform the estimation without any prior knowledge. Typical examples of applicative domains are the emergent fields of space robotics [START_REF] Mourikis | Vision-Aided Inertial Navigation for Spacecraft Entry, Descent, and Landing[END_REF], humanoid robotics and unmanned aerial navigation in urban-like environments [START_REF] Weiss | Monocular-SLAM-Based Navigation for Autonomous Micro Helicopters in GPS-Denied Environments[END_REF], where the use of the GPS is often forbidden. The closed-form solution could also play an important role in the framework of neuroscience by providing a new insight on the process of vestibular and visual integration for DP and sMP. The influence of extra retinal cues in depth perception has extensively been investigated in the last decades [START_REF] Longuet-Higgins | The interpretation of a moving retinal image[END_REF][START_REF] Macneilage | Vestibular heading discrimination and sensitivity to linear acceleration in head and world coordinates[END_REF][START_REF] Cornilleau-Peres | The visual perception of threedimensional shape from self-motion and object-motion[END_REF][START_REF] Wexler | Self-motion and perception of stationary objects[END_REF]. In particular, a very recent study investigates this problem by performing trials with passive head movements [START_REF] Dokka | Estimating distance during self-motion: a role for visual-vestibular interactions[END_REF]. The conclusion of this study is that the combination of retinal image with vestibular signals can provide rudimentary ability to depth perception. Our findings could provide a new insight to this integration mechanism for depth perception since, according to the closed-solution here derived, by combining retinal image with vestibular signals it is possible to determine the scale factor even without any knowledge about the initial speed. Our findings also show that it is possible to easily distinguish linear acceleration from gravity. Specifically, the closed form solution performs this determination by a very simple matrix inversion. This problem has also been investigated in neuroscience [START_REF] Angelaki | Neurons compute internal models of the physical laws of motion[END_REF][START_REF] Merfeld | Humans use internal models to estimate gravity and linear acceleration[END_REF]. Our results could provide a new insight to this mechanism since they clearly characterize the conditions (type of motion, features layout) under which this determination can be performed.

INRIA

INRIA

1 Table 1 :

 11 illustrate two cases when the SfM problem has two distinct solutions. The body configurations and the position of the point-features in the global frame are shown. The two solutions are in blue and red. The body configuration at the initial time is the same for both the solutions and is in black. Fig 1 regards the case of one feature seen in four images. Fig 2 displays the case of constant acceleration: the case of three point-features in Cases Number of Solutions Varying Acceleration Point-features and motion non-coplanar Unique Solution n im = 4, N f ≥ 2 ; n im ≥ 5, ∀N f Varying Acceleration Point-features and motion non-coplanar Two Solutions n im = 3, N f ≥ 2; n im = 4, N f = 1 Constant and non null Acceleration Point-features and motion non-coplanar Two Solutions n im = 3, N f ≥ 2; n im ≥ 4, ∀N f Null Acceleration Any Point-features layout Infinite Solutions ∀n im , ∀N f Any Motion Any Point-features layout Infinite Solutions n im ≤ 2, ∀N f ; n im = 3, N f = Number of distinct solutions for the SfM problem with visual and inertial sensing seven images has been considered and the seven poses of the body at the time when the images are taken is shown in the figure together with the position of the features.

Figure 1 :

 1 Figure 1: Illustration of two distinct solutions for the case n im = 4, N f = 1 (star symbols indicate the position of the point-feature respectively for the two solutions).

Figure 2 :

 2 Figure 2: Illustration of two distinct solutions for the case of constant acceleration (star symbols indicate the position of the point-features respectively for the two solutions).

RR n°0123456789


J Douglas

F Cotton

C Di

D M Boore

N A Abrahamson

S Akkar

Compilation and critical review of GMPEs for the GEM-PEER Global GMPEs Project

Keywords: ground motion prediction equations, attenuation relations, Global Earthquake Model, seismic hazard assessment, tectonic regionalization

Ground-motion prediction equations (GMPEs) relate a ground-motion parameter (e.g. peak ground acceleration, PGA) to a set of explanatory variables describing the source, wave propagation path and site conditions. In the past five decades many hundreds of GMPEs for the prediction of PGA and linear elastic response spectral ordinates have been published. We discuss the pre-selection of GMPEs undertaken within the framework of the GEM-PEER Global GMPEs Project. The pre-selection criteria adopted were consistent with the current state-ofthe-art in ground-motion characterization and sought to retain only the most robust GMPEs. Consideration of broad tectonic regionalization (e.g. shallow crustal seismicity in tectonically-active areas, stable continental regions and subduction zones) was made but it was assumed (based on previous studies) that strong regional differences were not present within these tectonic classes. In total about thirty GMPEs were pre-selected for closer inspection and testing to obtain a final set of ground-motion models.

INTRODUCTION

As discussed in the other articles in this proceedings on the GEM Global GMPEs project (http://peer.berkeley.edu/globalgmpe/) [START_REF] Akkar | Defining a consistent strategy to model ground-motion parameters for the GEM-PEER Global GMPEs Project[END_REF]Stewart et al., 2012a), the goal of this twoyear initiative is to provide to the Global Earthquake Model (GEM) a set of ground-motion models with worldwide applicability and based on a consensus view of dozens of international experts. These models should enable the prediction of peak ground acceleration (PGA) and linear elastic (pseudo-)spectral acceleration (PSA) for: the structural period range of main engineering interest; magnitudes from the lower limit generally considered in seismic hazard assessments (typical M w 5) up to the largest earthquakes possible (roughly M w 9.5 for subduction events); the source-to-site distances from the closest possible distance (i.e. 0km, right next to the rupture) to the farthest distance considered important in hazard assessments (possibly 1 000km in stable continental regions). The focus of this project is on the selection (and possible adjustment) of pre-existing alreadyparameterized ground-motion prediction equations (GMPEs) and not on the development of new models. This focus means that Task 2 of the project (the subject of this paper) concerning the preselection from the hundreds of available models to obtain a more manageable number for closer examination and testing within Task 3 (the subject of the companion article by [START_REF] Stewart | Selection or derivation of a global set of GMPEs for the GEM-PEER Global GMPEs Project[END_REF] is a key step.

Since the publication of the first ground-motion model in the form of an equation with magnitude and source-to-site distance by [START_REF] Esteva | Espectros de temblores a distancias moderadas y grandes[END_REF], the number of GMPEs has increased dramatically and over a dozen new studies are published every year (Figure 1). This high publication rate has been driven by, for example: increased recording (through lower-cost digital instruments and denser networks) and availability of strong-motion data [through online open-access databases, such as the Internet Site for European Strong-motion Data [START_REF] Ambraseys | Internet site for European strong-motion data[END_REF]], more journals and conferences publishing engineering seismology research, and large-scale initiatives, such as the Next Generation Attenuation (NGA) project (Powers et al., 2008). The latest compendium of published GMPEs by [START_REF] Douglas | Ground-motion prediction equations 1964-2010[END_REF] lists the characteristics of 289 empirical GMPEs for the prediction of PGA and 188 empirical models for the prediction of elastic response spectral ordinates. In addition, this report lists many dozens of simulation-based models to estimate these parameters. This abundance of models, however, creates a difficulty. On one hand, it is feasible from a practical point of view to carefully consider only a small fraction (less than 10%) of all available GMPEs in any project but, on the other hand, predictions of the median ground motions from the available GMPEs show a large (and not noticeably narrowing) dispersion (Figure 2), which needs to be considered since it demonstrates high epistemic uncertainty in ground-motion prediction. Consequently, a set of objective selection criteria need to be applied to the list of available models to pre-select GMPEs that are the most appropriate for the aims of a given project. For ease of application, these criteria should only require examination of the original references and should not involve numerical evaluation or testing of models against data, which can only be performed for a dozen models at most. These criteria were discussed by the experts comprising the Task 2 working group (the authors of this paper) and subsequently applied by the working group to the lists of models given in [START_REF] Douglas | Ground-motion prediction equations 1964-2010[END_REF]. The discussion process was conducted through a series of conference calls and email exchange in order to obtain a consensus view, which was also objective so it can be supported by the wider community (within the GEM Global GMPEs project and beyond). The working group benefitted from the experience gained in GMPEs pre-selection for the projects: PEGASOS [START_REF] Cotton | Criteria for selecting and adjusting ground-motion models for specific target regions: Application to central Europe and rock sites[END_REF], SHARE [START_REF] Douglas | Pre-selection of ground-motion prediction equations for WP4 of SHARE[END_REF], GEM1 [START_REF] Douglas | Selection of ground-motion prediction equations for GEM1[END_REF] and PEGASOS Refinement [START_REF] Bommer | On the selection of groundmotion prediction equations for seismic hazard analysis[END_REF].

Because a GMPE excluded during this stage could not be subsequently re-instated, care was taken to avoid applying criteria that are too strict at this step. About thirty GMPEs were finally pre-selected within Task 2, from the models summarized in [START_REF] Douglas | Ground-motion prediction equations 1964-2010[END_REF], for closer inspection and testing during Task 3 to obtain a final set of ground-motion models. This subsequent winnowing process is described in the companion paper by [START_REF] Stewart | Selection or derivation of a global set of GMPEs for the GEM-PEER Global GMPEs Project[END_REF].

PRE-SELECTION CRITERIA

The purpose of this section is to present the final pre-selection criteria for all the seismotectonic regimes present on Earth. Consideration of broad tectonic regionalization (e.g. shallow crustal seismicity in active areas, stable continental regions and subduction zones) was made but it was assumed (based on previous studies) that strong regional differences were not present within these tectonic classes, although we retained a suite of models within each class to account for epistemic uncertainty. We considered these broad tectonic regimes when pre-selecting GMPEs:

• Stable continental regions (SCRs), which can possibly be divided further into shield and continental/foreland;

• Subduction zones, which includes intraslab and interface earthquakes (and potentially fore-arc and back-arc locations);

• Active regions with shallow crustal seismicity;

• Volcanic zones;

• Areas of deep focus non-subduction earthquakes, such as Vrancea (Romania);

• Areas where the travel paths are mainly through oceanic crust, such as coastal Portugal.

The pre-selection criteria adopted in this study are consistent with the companion paper [START_REF] Akkar | Defining a consistent strategy to model ground-motion parameters for the GEM-PEER Global GMPEs Project[END_REF], which describes and suggests predictive parameters to use in ground-motion modeling and their appropriate values.

As discussed above, due to the vast number of available GMPEs within the literature it is necessary to define criteria to winnow down the models to a more manageable number although recognizing the necessity to retain sufficient models to account for epistemic uncertainty in the prediction of shaking.

For this pre-selection it was decided to apply the seven 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]. It should be noted that these criteria have been updated by [START_REF] Bommer | On the selection of groundmotion prediction equations for seismic hazard analysis[END_REF] to make them more objective and stronger. It was decided, however, that for pre-selection the [START_REF] Bommer | On the selection of groundmotion prediction equations for seismic hazard analysis[END_REF] proposals would be too strict. The seven exclusion criteria applied to the compilation (both empirical and simulation-based GMPEs) of Douglas ( 2011) are:

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.

Criterion 1 was applied to retain only models relevant for the broad classes listed above (e.g. only subduction zone models were considered for these regions). Criterion 2 was applied to reject GMPEs that had not been published in a journal that is listed by ISI Web of Knowledge, which is a standard reference for bibliographic information, except for models for SCRs, which are often only published in the grey literature, and a few other special cases where it was felt that a model that failed this criterion was potentially interesting for further study within Task 3. Criterion 3 was applied to reject those studies that do not provide detailed information on the dataset used to derive the GMPEs presented. Criterion 4 has been applied to reject GMPEs for areas for which more recent models have been published using larger datasets, even if the more recent models have not been derived by the same author teams. For example, the model of [START_REF] Field | A modified ground-motion attenuation relationship for southern California that accounts for detailed site classification and a basin-depth effect[END_REF] for southern California has been rejected since the data he used are a subset of the NGA database used by the NGA teams in developing their models. Criterion 5 leads to all peak ground acceleration (PGA)-only models being rejected as well as those that do not provide coefficients for periods less than 0.04s (25Hz) (that can be assumed to approximate PGA) and up to at least 2s (0.5Hz). This criterion removes models such as that by [START_REF] Ghasemi | An empirical spectral ground-motion model for Iran[END_REF], who do not provide coefficients for periods less than 0.05s, and the GMPEs by [START_REF] Bommer | The influence of magnitude range on empirical ground-motion prediction[END_REF], who do not provide coefficients for periods greater than 0.5s. Criterion 6 has been applied to exclude models that do not use moment magnitude (M w ) (since there are difficulties and uncertainties in converting between other magnitude scales, especially M L , and M w , is the standard magnitude scale for seismic hazard assessments) and to exclude models that do not allow the prediction of ground motions at rock sites (e.g. [START_REF] Crouse | Ground-motion attenuation equations for earthquakes on the Cascadia subduction zones[END_REF]. Criterion 7 has been applied, in particular, to exclude those models based on simulations whose standard deviations were computed without taking into account modeling variability (e.g. [START_REF] Hwang | Attenuation relations of ground motion for rock and soil sites in eastern United States[END_REF]. Also models that are not thought to be provide reliable predictions over a wide range of magnitudes (roughly M w 5 to 7.5) and near-source distances to at least 100km are rejected since this is the focus of seismic hazard assessments in GEM. The choices here reflect the decision from Task 1a to reject models that would require more than a 0.5 unit extrapolation of the maximum M w for which the GMPEs were derived. This means that many local and regional models are rejected since they often only provide reliable predictions up to about M w 6.5. Because of the global nature of the project, we aimed to select models for different geographical regions even if they were of varying quality rather than to select many models for the same region (this is particularly true for SCRs where most robust models are for eastern North America).

Note that stochastic models without fitted functional forms were also considered even though these would require functions to be fitted to predictions for easy use within GEM. Models derived using the hybrid empirical-stochastic approach of [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] have also been considered. Non-parametric models (e.g. those represented by neural networks) were not considered because of the difficulty in using them in practice and because of doubts over their extrapolation outside the range of data.

Overall the aim was to retain between five and ten models per tectonic type for further examination.

PRE-SELECTED MODELS

Table 3.1 lists the models pre-selected in Task 2 for the three main seismotectonic regimes. Details on the pre-selection procedure and the pre-selected models are given in the report by [START_REF] Douglas | Pre-selection of ground-motion prediction equations (Task 2), PEER GEM -Global GMPEs Task 2 WG[END_REF]. It should be noted that this pre-selection was performed in April 2011. During the duration of the GEM Global GMPEs project many new GMPEs will be published (by considering the publication rate shown in Figure 1) but it is not planned within the project to repeat this pre-selection to account for these new models. This exclusion of the most recent GMPEs is an inherent limitation of all seismic hazard assessments.

As an example of the dispersion in the predicted response spectra from the pre-selected GMPEs, Figure 3 shows spectra estimated using the three sets of GMPEs for three of the scenarios considered by [START_REF] Douglas | Consistency of ground-motion predictions from the past four decades[END_REF], although for rock rather than stiff soil conditions. These scenarios are roughly in the barycentre of available strong-motion data and, consequently, the differences amongst the predicted spectra should be lowest here. The differences amongst the spectra will probably be greater closer to the source and for larger earthquakes because of a lack of data to constrain the models and a lack of knowledge on how earthquake shaking scales with magnitude and distance. These comparisons show that even after careful pre-selection of models based on data from the bestobserved regions (e.g. California and Japan) there is still much epistemic uncertainty in the prediction of earthquake response spectra. It will be the challenge of Task 3 of the GEM Global GMPEs project to propose a reduced set of ground-motion models that capture this epistemic uncertainty but without overestimating it. The interested reader is referred to the companion paper by [START_REF] Stewart | Site Effects in Parametric Ground Motion Models for the GEM-PEER Global GMPEs Project[END_REF] on how this challenge is being faced for subduction zones.

Very few GMPEs have been published for the remaining seismotectonic regimes [volcanic zones; areas of deep focus non-subduction earthquakes, such as Vrancea (Romania); and areas where the travel paths are mainly through oceanic crust, such as coastal Portugal] and the majority of these do not pass the pre-selection criteria. Therefore, within Task 3 it is planned to propose that models finally selected for the other three regimes are adjusted to make them applicable for the prediction of ground motions in these special areas. It is admitted that this work-around is not an optimal solution but we prefer proposing a suite of robust models to account for high epistemic uncertainty rather than choosing a single local (and potentially poorly-constrained) model for each regime that gives the false impression of low uncertainty.

Figure 1 .

 1 Figure 1. Number of published GMPEs per year (histogram) and cumulatively since 1964 (blue line).

Figure 2 .

 2 Figure 2. Predicted PGA and SA(1 s) (unfilled black circles) for a M w 6 strike-slip earthquake at 20 km on a NEHRP C site against publication date for over 250 models published in the literature. Filled red circles indicate models published in peer-reviewed journals and for which basic information on the used dataset is available. Also shown are the median PGA and SA(1 s) within five-year intervals (black line) and the median ±1 standard deviation (dashed black lines) based on averaging predictions. From Douglas (2010).

Table 3.1 List of pre-selected models in Task 2 of the GEM -PEER Global GMPEs project Stable Continental Atkinson (2008) as modified by Atkinson & Boore (2011): Referenced empirical model for eastern North America Atkinson & Boore (2006) as modified by Atkinson & Boore (2011): Extended stochastic model for eastern North America Campbell (2003): Hybrid model for eastern North America Douglas et al. (2006): Hybrid model for southern Norway Frankel et al. (1996) as parameterized by EPRI (2004): Stochastic model for eastern North America Pezeshk et al. (2011): Hybrid model for eastern North America Raghu Kanth & Iyengar (2006, 2007): Peninsular India Silva et al. (2002): Stochastic model for eastern North America Somerville et al. (2009): Simulation-based models for Australia Toro et al. (1997), originally published in EPRI (1993), modified by Toro (2002): Stochastic model for eastern North America Subduction Abrahamson et al. (2012): Worldwide Arroyo et al. (2010): Interface model for Mexico (complementary to Garcia et al., 2005)) Atkinson & Boore (2003): Worldwide Garcia et al. (2005): Intraslab model for Mexico (complementary to Arroyo et al., 2010) Kanno et al. (2006): Japan Lin & Lee (2008): Taiwan McVerry et al. (2006): New Zealand Youngs et al. (1997): Worldwide Zhao et al. (2006) with modifications by Zhao (2010): Japan Shallow crustal in tectonically active regions Abrahamson & Silva (2008): NGA model using worldwide data Akkar & Bommer (2010): Model using Mediterranean and Middle Eastern data Boore & Atkinson (2008) as modified by Atkinson & Boore (2011): NGA model using worldwide data Campbell & Bozorgnia (2008) : NGA model using worldwide data Cauzzi & Faccioli (2008) as updated by Faccioli et al. (2010): Model using worldwide data (mainly Japanese) Chiou & Youngs (2008): NGA model using worldwide data Kanno et al. (2006): Model using mainly Japanese data McVerry et al. (2006): Model using mainly New Zealand data Zhao et al. (2006): Model using mainly Japanese data

Shallow crustal earthquakes in active tectonic regions (M w 6 at r JB =20km)

Stable continental earthquakes (M w 6 at r JB =20km) Subduction interface earthquakes (Mw 6.5 at r rup =50km) Figure 3. Predicted elastic pseudo-acceleration response spectra from for a rock site for (top): a strike-slip shallow crustal earthquake of Mw 6 at a distance to the surface projection of the rupture of 20km, (middle): an Mw 6 earthquake in a stable continential region at a distance to the surface projection of the rupture of 20km, and (bottom): an Mw 6.5 interface earthquake in a subduction zone at a rupture distance of 50km.

AKCNOWLEDGEMENT

This study was funded by the GEM Foundation as part of the Pacific Earthquake Engineering Research Center's (PEER's) Global GMPEs project. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the sponsors