Aleixandre Verger
email: verger@creaf.uab.cat

Frédéric Baret

Marie Weiss

Iolanda Filella

Josep Peñuelas

GEOCLIM: a global climatology of LAI, FAPAR, and FCOVER from VEGETATION observations for 1999-2010

Keywords: Climatology, vegetation phenology, seasonal and interannual variability, biophysical variables, SPOT VEGETATION

Land-surface modelling would benefit significantly from improved characterisation of the seasonal variability of vegetation at a global scale. GEOCLIM, a global climatology of leaf area index (LAI), fraction of absorbed photosynthetically active radiation (FAPAR), both essential climate variables, and fraction of vegetation cover (FCOVER) is here derived from observations from the SPOT VEGETATION programme. Interannual average values from the GEOV1 Copernicus Global Land time series of biophysical products at 1-km resolution and 10-day frequency are computed for 1999 to 2010. GEOCLIM provides the baseline characteristics of the seasonal cycle of the annual vegetation phenology for each 1-km pixel on the globe. The associated standard deviation characterises the interannual variability. Temporal consistency and continuity is achieved by the accumulation of multi-year observations and the application of techniques for temporal smoothing and gap filling. Specific corrections are applied over cloudy tropical regions and high latitudes in the Northern Hemisphere where the low number of available observations compromises the reliability of estimates. Artefacts over evergreen broadleaf forests and areas of bare soil are corrected based on the expected limited seasonality. The GEOCLIM data set is demonstrated to be consistent, both spatially and temporally. GEOCLIM shows absolute differences lower than 0.5 compared with MODIS (GIMMS3g) climatology of LAI for more than 80% (90%) of land pixels, with higher discrepancies in tropical and boreal latitudes. ECOCLIMAP systematically produces higher LAI values. The phenological metric for the date of maximum foliar development derived from GEOCLIM is spatially consistent (correlation higher than 0.9) with those of MODIS, GIMMS3g, ECOCLIMAP and MCD12Q2 with average differences within 14 days at the global scale.

Introduction

The state and dynamics of vegetation play key roles in the carbon cycle and global climate. A set of essential climatic variables was identified as both accessible from remote sensing observations and involved in key processes [START_REF] Gcos | Implementation Plan for the Global Observing System for Climate in support to UNFCCC (2010 Update)[END_REF]. Among those relating to land surfaces, the leaf area index (LAI) and the fraction of absorbed photosynthetically active radiation (FAPAR) can be derived from observations in the reflective solar domain. These biophysical variables of vegetation are crucial in several processes, including photosynthesis, respiration, and transpiration. LAI is defined as one half the total area of green elements per unit area of horizontal ground [START_REF] Chen | Defining leaf area index for non-flat leaves[END_REF][START_REF] Gcos | Implementation Plan for the Global Observing System for Climate in support to UNFCCC (2010 Update)[END_REF]. It controls the exchanges of energy, water, and greenhouse gases between the land surface and the atmosphere. FAPAR is defined as the fraction of radiation absorbed by the canopy in the 400-700 nm spectral domain under specified conditions of illumination and is a main input in models of light-use efficiency [START_REF] Mccallum | Satellite-based terrestrial production efficiency modeling[END_REF]. The fraction of vegetation cover (FCOVER), defined as the fraction of the background covered by green vegetation as seen from the nadir, is also as a very pertinent variable that can be used in models of the surface-energy balance to separate the contribution of the soil from that of the canopy [START_REF] Gutman | The derivation of the green vegetation fraction from NOAA/AVHRR data for use in numerical weather prediction models[END_REF][START_REF] Su | Modeling Evapotranspiration during SMACEX: Comparing Two Approaches for Local-and Regional-Scale Prediction[END_REF]. LAI, FAPAR, and FCOVER are routinely estimated from sensors with medium resolution such as VEGETATION [START_REF] Baret | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part1: Principles of development and production[END_REF], Moderate Resolution Imaging Spectroradiometer (MODIS) [START_REF] Myneni | Global products of vegetation leaf area and absorbed PAR from year one of MODIS data[END_REF] and the Advanced Very High Resolution Radiometer (AVHRR) [START_REF] Zhu | Global Data Sets of Vegetation Leaf Area Index (LAI)3g and Fraction of Photosynthetically Active Radiation (FPAR)3g Derived from Global Inventory Modeling and Mapping Studies (GIMMS) Normalized Difference Vegetation Index (NDVI3g) for the Period 1981 to 2011[END_REF]. The European Copernicus Global Land Service delivers global LAI, FAPAR, and FCOVER products from SPOT VEGETATION data from 1999 to the present with a spatial sampling close to 1 km. The products, known as GEOV1 products, have benefitted from the development and validation of existing products [START_REF] Baret | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part1: Principles of development and production[END_REF]. [START_REF] Camacho | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part 2: Validation and intercomparison with reference products[END_REF] demonstrated that GEOV1 products were more accurate and precise than current products. Some land-surface models (LSMs) for simulating terrestrial water and carbon cycles use the spatiotemporal variation of LAI or FAPAR, described by different lookup tables depending on the type of vegetation (Viterbo and Beljaars 1995). The availability of satellite data in the last two decades describing the state and evolution of vegetation has allowed a better integration of biophysical variables into LSMs. Previous studies have demonstrated the improved performance of LSMs due to a better characterisation of the seasonal and interannual variability of vegetation functioning provided by the assimilation of satellite data. In particular, data assimilation yields a more realistic parameterisation in phenological models and reduces the models' prediction errors to 21 and 15% for FAPAR and LAI, respectively [START_REF] Stöckli | A global reanalysis of vegetation phenology[END_REF]. A number of studies have shown the potential of assimilating LAI observations to correct vegetation model states [START_REF] Demarty | Assimilation of global MODIS leaf area index retrievals within a terrestrial biosphere model[END_REF]Gu et al. 2006) and the implications of introducing the observed seasonal (van den Hurk et al. 2003) and interannual [START_REF] Guillevic | Influence of the Interannual Variability of Vegetation on the Surface Energy Balance-A Global Sensitivity Study[END_REF] variability of LAI in the annual cycle of hydrological fluxes. [START_REF] Boussetta | Impact of a satellite-derived leaf area index monthly climatology in a global numerical weather prediction model[END_REF] showed that the assimilation of a MODIS derived LAI monthly climatology, i.e. the interannual average of LAI time series (as opposed to a vegetationdependent constant LAI), in a model of global numerical weather prediction improved the forecast of near-surface (screen-level) air temperature and relative humidity through its effect on evapotranspiration. [START_REF] Barbu | Integrating ASCAT surface soil moisture and GEOV1 leaf area index into the SURFEX modelling platform: a land data assimilation application over France[END_REF] more recently demonstrated the potential of the assimilation of GEOV1 LAI into an ISBA-A-gs land-surface model to improve the monitoring of droughts. A LAI climatology was also useful for the identification of anomalies and trends in global vegetation [START_REF] Baret | Use of GEOV1 climatology based on 11 years of consistent observations: identification of trends at the global scale[END_REF][START_REF] Brandt | Local Vegetation Trends in the Sahel of Mali and Senegal Using Long Time Series FAPAR Satellite Products and Field Measurement (1982-2010[END_REF]Verger et al. 2014b;[START_REF] Kandasamy | A comparison of methods for smoothing and gap filling time series of remote sensing observations: application to MODIS LAI products[END_REF][START_REF] Zhu | Global Data Sets of Vegetation Leaf Area Index (LAI)3g and Fraction of Photosynthetically Active Radiation (FPAR)3g Derived from Global Inventory Modeling and Mapping Studies (GIMMS) Normalized Difference Vegetation Index (NDVI3g) for the Period 1981 to 2011[END_REF]. The climatology of the biophysical variables reveals the seasonality inherent to the land-cover type and improve land-cover classification (Verhegghen et al. 2014). A climatology gap filling can better cope with missing and noise-contaminated data than can standard temporal filters for most missing data or large gaps in a single annual time series of satellite data, which have a large impact on the accuracy of the phenological metrics extracted from the reconstructed time series [START_REF] Guyon | Monitoring elevation variations in leaf phenology of deciduous broadleaf forests from SPOT/VEGETATION time-series[END_REF][START_REF] Kandasamy | A comparison of methods for smoothing and gap filling time series of remote sensing observations: application to MODIS LAI products[END_REF][START_REF] Kandasamy | A comparison of methods for smoothing and gap filling time series of remote sensing observations: application to MODIS LAI products[END_REF]). Extraction of phenological information is also sensitive to the temporal [START_REF] Pouliot | Evaluation of compositing period and AVHRR and MERIS combination for improvement of spring phenology detection in deciduous forests[END_REF][START_REF] Zhang | Sensitivity of vegetation phenology detection to the temporal resolution of satellite data[END_REF]) and spatial [START_REF] Fisher | Cross-scalar satellite phenology from ground, Landsat, and MODIS data[END_REF][START_REF] Kovalskyy | The suitability of multi-temporal webenabled Landsat data NDVI for phenological monitoringa comparison with flux tower and MODIS NDVI[END_REF]) resolution of the satellite data. The climatology derived from time series of moderate spatial resolution sensors preserves the high temporal frequency mandatory for phenological studies [START_REF] Guyon | Monitoring elevation variations in leaf phenology of deciduous broadleaf forests from SPOT/VEGETATION time-series[END_REF]. Finally, the climatology information can make projections and improve the stability of near real time estimates [START_REF] Jiang | Modeling MODIS LAI time series using three statistical methods[END_REF]Verger et al. 2014a).

Despite the significance of global phenology for earth system monitoring and modelling, there are few data sets that explicitly describe the annual vegetation cycle at global scale. [START_REF] Boussetta | Impact of a satellite-derived leaf area index monthly climatology in a global numerical weather prediction model[END_REF] derived a monthly LAI climatology from 2000-2008 MODIS observations to be used in a numerical weather prediction model as indicator of the leaf development stage. The ECOCLIMAP programme [START_REF] Faroux | ECOCLIMAP-II/Europe: a twofold database of ecosystems and surface parameters at 1 km resolution based on satellite information for use in land surface, meteorological and climate models[END_REF][START_REF] Masson | A global database of land surface parameters at 1km resolution in meteorological and climate models[END_REF]) is a dual database at 1 km resolution that includes an ecosystem classification and a coherent set of land surface parameters (including LAI, FAPAR and FCOVER) that are primarily mandatory in meteorological modelling for soil/vegetation-atmosphere transfer schemes. Other studies focus on the time variation of vegetation indices to propose a global normalized difference vegetation index (NDVI) and enhanced vegetation index (EVI) reference data set for land-surface phenology using 13 years of VEGETATION observations (Verhegghen et al. 2014) or to derive phenological metrics from VEGETATION [START_REF] Guyon | Monitoring elevation variations in leaf phenology of deciduous broadleaf forests from SPOT/VEGETATION time-series[END_REF], MODIS [START_REF] Ganguly | Land surface phenology from MODIS: Characterization of the Collection 5 global land cover dynamics product[END_REF][START_REF] Zhang | Monitoring vegetation phenology using MODIS[END_REF] or AVHRR [START_REF] Atzberger | Phenological Metrics Derived over the European Continent from NDVI3g Data and MODIS Time Series[END_REF]) time series.

The aim of this study is to provide a global climatology of LAI, FAPAR, and FCOVER for describing the seasonal and interannual variability of the vegetation cycle at the global scale. The derived climatology, GEOCLIM, will take advantage of the improvements in accuracy and temporal consistency provided by GEOV1 over existing products [START_REF] Camacho | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part 2: Validation and intercomparison with reference products[END_REF]. We propose to build a current climatology using a limited set of recent annual time-series since climate and land cover are changing with time. The time series, however, are expected to be sufficiently long for depicting the baseline annual cycle of the vegetation and for encompassing anomalies (Verhegghen et al. 2014). The climatology should simulate the data as closely as possible, i.e. it does not use existing land-cover maps or a model to describe the seasonal dynamics, thereby preventing the introduction of possible artefacts due to the lack of realism of the model used.

GEOV1 time series from 1999 to 2010, corresponding to 12 years of estimates of biophysical variables at a spatial resolution of 1 km and a frequency of 10 days are used. The climatology is computed for each pixel as the average for a given date in a year across all years of the time series.

The associated standard deviation characterises the interannual variability.

We will first describe the methodology and the data sets used to produce GEOCLIM. We will then evaluate GEOCLIM based on its main spatiotemporal features and its performance relative to other climatology data sets derived from AVHRR and MODIS data, with special emphasis on seasonality and the derived phenology.

GEOCLIM implementation

The generation of GEOCLIM was achieved from GEOV1 time series for the period 1999-2010 based on the interannual means of biophysical variables and the application of specific corrections.

The input data set and the steps required to produce GEOCLIM are described hereafter.

GEOV1 biophysical products

GEOV1 biophysical products provide global coverage of LAI, FAPAR, and FCOVER from 1998/12/24 to the present at a ground sampling distance of 1/112° (approximately 1 km at the equator) and 10-day steps. A neural-network machine-learning algorithm was used to estimate GEOV1 products [START_REF] Baret | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part1: Principles of development and production[END_REF]. Directionally normalised VEGETATION reflectances [START_REF] Roujean | A bidirectional reflectance model of the Earth's surface for the correction of remote sensing data[END_REF]) from the top of the canopy in the red, near-infrared, and short-wave infrared bands derived from the CYCLOPES processing line [START_REF] Baret | LAI, fAPAR and fCover CYCLOPES global products derived from VEGETATION. Part 1: Principles of the algorithm[END_REF] were used as inputs to the neural networks. Based on the validation results for the available biophysical products [START_REF] Garrigues | Validation and intercomparison of global Leaf Area Index products derived from remote sensing data[END_REF][START_REF] Baret | LAI, fAPAR and fCover CYCLOPES global products derived from VEGETATION. Part 1: Principles of the algorithm[END_REF]), the MODIS and CYCLOPES products were selected for the training process. The selected products were combined after re-projection onto the VEGETATION Plate-Carrée 1/112° grid, smoothed over time, interpolated at the 10-day frequency, combined, and eventually re-scaled to better fit the expected range of variation. Further details for the training of the neural networks and the generation of the product are provided in [START_REF] Baret | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part1: Principles of development and production[END_REF]. Recent validation studies indicated that GEOV1 outperformed existing products in both accuracy and precision [START_REF] Camacho | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part 2: Validation and intercomparison with reference products[END_REF]). GEOV1 products are freely available at www1.

Climatology generation

For the generation of GEOCLIM only the GEOV1 biophysical products with the best quality were selected according to the quality flags on snow, aerosol, reflectance input and biophysical output status [START_REF] Baret | Climatology of LAI, FAPAR and FCOVER products (V1)[END_REF]. The climatology is defined as the interannual mean of the best quality GEOV1 products accumulated for 1999-2010. It is generated at the pixel scale (1-km spatial resolution) and at a dekadal temporal step (a 10-day period, with 36 dekads per year) within a 30day compositing window (±15 days). The average values are then computed from three adjacent dekads instead of only one dekad along the 12-year period, which allows an increase in the number of points (12*3=36 compared to only 12), provides more robust and continuous estimates, and induces fewer artefacts because the dynamics of the products are approximately linear between the three dekads [START_REF] Baret | Climatology of LAI, FAPAR and FCOVER products (V1)[END_REF], as shown by [START_REF] Camacho | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part 2: Validation and intercomparison with reference products[END_REF] for the smoothness of the GEOV1 products. A temporal smoothing and gap-filling (TSGF) technique (Verger et al. 2011) was applied to correct the residual artefacts, especially when the GEOV1 products were systematically unavailable across the years due to cloud coverage, and to ensure continuity and consistency in GEOCLIM as phenological studies request. Gap filling was achieved by linear interpolation.

Temporal smoothing relied on an adaptive Savitzky-Golay second-degree polynomial fitting by processing three valid values on either side of the date (Verger et al. 2011). The compositing window may be asymmetric due to possible missing data. TSGF technique was demonstrated to improve other existing temporal filters for reconstruction satellite LAI time series in terms of the accuracy as compared to the original data by ensuring robustness to noise and missing data, while preventing over-smoothing [START_REF] Kandasamy | A comparison of methods for smoothing and gap filling time series of remote sensing observations: application to MODIS LAI products[END_REF]Verger et al. 2011;[START_REF] Kandasamy | A comparison of methods for smoothing and gap filling time series of remote sensing observations: application to MODIS LAI products[END_REF]). An example of a climatology computation is illustrated in Figure 1, and TSGF correction is illustrated in Figure 2a.

[Figure 1] [Figure 2]

Correction of specific artefacts

The generated climatology was then corrected for specific problematic behaviours based on available expert knowledge:

-Some artefacts were observed at northern high latitudes during the winter: anomalous seasonality and unexpected increases in LAI (FAPAR, FCOVER) (Figure 2b) with an artificial maximum peak in winter (Figure 2b) and high interannual variability resulting in high standard deviations (Figure 3a). These artefacts were mainly due to snow cover or very poor conditions of illumination that limited the number of valid observations and the reliability of the bidirectional reflectance model applied for the correction of VEGETATION data [START_REF] Roujean | A bidirectional reflectance model of the Earth's surface for the correction of remote sensing data[END_REF]) (Figures 2b,3b). The LAI (FAPAR, FCOVER) values are expected to be relatively stable and low due to the low temperatures, short days, and low illumination during winter at these high latitudes. To correct these artefacts at northern latitudes, the GEOCLIM inputs higher than the 20th percentile during winter (defined here as the period for which the sun zenith angle, SZA>70º at the time of VEGETATION overpass, i.e. around 10:30) were fixed at the minimum pixel values observed over the entire period. We used the minimum values by preferentially selecting the values computed from at least three valid observations because the quality of GEOCLIM inputs is highly correlated with the number of valid observations available for their composition (Figures 2b,3). We used the minimum value computed over all dekads if none of the dekads verified this condition. The areas where this specific correction was applied are shown in Figure 4a. Similar approaches based on representative winter values and thresholds to fill gaps and correct values affected by snow or poor illumination at high latitudes were also considered by [START_REF] Beck | Improved monitoring of vegetation dynamics at very high latitudes: A new method using MODIS NDVI[END_REF]; [START_REF] Delbart | Determination of phenological dates in boreal regions using normalized difference water index[END_REF]; [START_REF] Zhang | Climate controls on vegetation phenological patterns in northern mid-and high latitudes inferred from MODIS data[END_REF].

-Significant artefacts were also detected at equatorial and tropical latitudes due to aerosolcloud contamination that produced high instabilities, artificial seasonalities, and missing data in the GEOV1 products and, consequently, in the derived output (Figure 2c). The high standard deviations (Figure 3a) and the low number of available observations (Figures 2c,3b) appeared to be good indicators of the high uncertainty associated with the computed output over these tropical areas. Most of these cases corresponded to evergreen broadleaf forests (EBFs) (cf. Figures 3,4b), so a minimum seasonality and high LAI values should be observed. We thus identified a pixel as an EBF if the 90th percentile (P90) of the LAI output was >4.5 and the 20th percentile was >P90-1.5. This method for the detection of EBFs based only on GEOV1 products (Figure 4a) agreed well with the GLOBCOVER land-cover map [START_REF] Defourny | The first 300 m global land cover map for 2005 using ENVISAT MERIS time series: a product of the GlobCover system[END_REF]) (Figure 4b). For EBFs, the GEOCLIM values were fixed to the 90th percentile computed over the entire period (Figure 2c).

-Some artefacts were also detected in the raw output for areas of bare soil (BS) where the observed seasonality was of the same order of magnitude of the precision of the GEOV1 product (Figure 2d). A pixel was identified as BS if the 90th percentile of the LAI output was <0.05 (compare Figure 4a to the land-cover map in Figure 4b). For BS, the GEOCLIM values were fixed to the 50th percentile computed over the entire period (Figure 2d). The global phenology derived from the temporal seasonality of GEOCLIM is also investigated in this section. For simplicity, we focus on the date of maximum foliar development, i.e. the timing of the peak of the growing season in the LAI annual cycle [START_REF] Brown | Global phenological response to climate change in crop areas using satellite remote sensing of vegetation, humidity and temperature over 26 years[END_REF][START_REF] Jönsson | Seasonality extraction by function fitting to timeseries of satellite sensor data[END_REF]. In addition to the phenological metrics derived from the LAI climatologies, we use also the MCD12Q2 MODIS phenological product [START_REF] Zhang | Monitoring vegetation phenology using MODIS[END_REF].

Validation data sets

ECOCLIMAP product

ECOCLIMAP is a database at 1/112º resolution on a Plate-Carrée grid resolution that includes a classification of ecosystems and a consistent set of associated land-surface variables, including LAI, at 10-day temporal sampling [START_REF] Faroux | ECOCLIMAP-II/Europe: a twofold database of ecosystems and surface parameters at 1 km resolution based on satellite information for use in land surface, meteorological and climate models[END_REF]. We used the latest Open-ECOCLIMAP v1 The ECOCLIMAP LAI data set at the original 1/112º resolution was aggregated at 0.5º spatial sampling and averaged at monthly temporal sampling.

MODIS climatology

The MODIS Collection 5 Boston University (BU) LAI product at 0.25º latitude/longitude grid is the extracted best quality of standard MODIS LAI product based on MOD15A2 and MOD13A2 quality flags [START_REF] Samanta | Comment on "Drought-Induced Reduction in Global Terrestrial Net Primary Production from 2000 Through 2009[END_REF]. The standard MODIS LAI products relies on a biome dependent lookup table inversion of a radiative transfer model which ingests red and near infrared bidirectional reflectance factor values, their associated uncertainties, the view-illumination geometry, and biome type (within eight types based on MOD12Q1 land cover map). Further details on the retrieval algorithm are provided in [START_REF] Myneni | Global products of vegetation leaf area and absorbed PAR from year one of MODIS data[END_REF]; [START_REF] Yang | MODIS leaf area index products: from validation to algorithm improvement[END_REF]. Valid 1 km 8-day values are averaged to obtain monthly LAI [START_REF] Samanta | Comment on "Drought-Induced Reduction in Global Terrestrial Net Primary Production from 2000 Through 2009[END_REF]. The monthly LAI 1km sinusoidal product is aggregated and projected onto a 0.25º Plate-Carrée projection.

We derived a monthly MODIS climatology as the interannual average from 2000 to 2010 MODIS BU LAI product. The 0.25º product was aggregated to 0.5º spatial resolution for comparison purposes.

GIMMS3g climatology

The GIMMS3g LAI product derived from AVHRR data is available at 15-day temporal steps and 1/12º spatial resolution for the period July 1981 to December 2011. The principles used for the generation of this LAI data set are based on the use of neural networks which were trained first with GIMMS NDVI3g and MODIS LAI products for the overlapping period 2000-2009. The trained neural network algorithm is then applied using the land-cover class, the latitude and longitude coordinates, and the NDVI3g as the input data to generate the full temporal coverage of the GIMMS3g LAI data set. Further details on the algorithm for GIMMS3g retrieval can be found in [START_REF] Zhu | Global Data Sets of Vegetation Leaf Area Index (LAI)3g and Fraction of Photosynthetically Active Radiation (FPAR)3g Derived from Global Inventory Modeling and Mapping Studies (GIMMS) Normalized Difference Vegetation Index (NDVI3g) for the Period 1981 to 2011[END_REF].

We derived the GIMMS3g climatology as the interannual mean of GIMMS3g LAI time series at 15-day temporal step for the period 1999-2010. We aggregated the 1/12º products at 0.5º spatial resolution and averaged at monthly temporal sampling for comparison purposes.

MCD12Q2 product

MCD12Q2 (Collection 5) [START_REF] Ganguly | Land surface phenology from MODIS: Characterization of the Collection 5 global land cover dynamics product[END_REF][START_REF] Zhang | Monitoring vegetation phenology using MODIS[END_REF] provides global yearly vegetation phenologies at 500 m from 2001 to 2010 MODIS time series. The MCD12Q2 algorithm uses a series of piecewise logistic functions fitted over the annual cycle of EVI data [START_REF] Zhang | Monitoring vegetation phenology using MODIS[END_REF]). Among the transition dates provided by the MCD12Q2 product, the parameter "onset of greenness maximum" is used here for comparisons with the parameter "peak of growing season" derived from the LAI climatologies , i.e. the date for which the climatology reach its maximum value in the LAI annual cycle. The onset of maximal greenness conceptually corresponds to the transition date at which the annual cycle of the vegetation reaches maturity. This date is thus expected to be earlier than the date of maximum vegetation. [START_REF] Zhang | Global vegetation phenology from Moderate Resolution Imaging Spectroradiometer (MODIS): Evaluation of global patterns and comparison with in situ measurements[END_REF] compared the MCD12Q2 parameter to in-situ measurements and found that it corresponded to the time at which 85-90% of the individual leaves reached their final size.

The MCD12Q2 500 m sinusoidal product was projected onto a 0.5º Plate-Carrée projection using the MODIS re-projection tool (www3). Yearly MCD12Q2 values from 2001 to 2010 were then averaged to provide a typical phenology for comparison with the phenological metrics derived from the LAI climatology data sets.

Spatiotemporal consistency

The GEOCLIM biophysical variables had highly consistent spatial and temporal patterns (Figure 5a), in agreement with the global distributions of biomes (Figure 4b). The seasonal patterns of GEOCLIM LAI also reflected the expected regimes of vegetation at the global scale. Evergreen broadleaf forests exhibited null seasonality (Figure 5d) in the tropical belt where LAI was near 5 throughout the year (Figure 5b). Deserts also had no seasonality (Figure 5d) where LAI was near zero (Figure 5b). These results were expected given the forcing applied for evergreen broadleaf forests and bare soils (cf. Section 2.3). As expected, deciduous broadleaf forests and crops had the highest seasonalities (Figure 5d). The observed seasonality in needleleaf forests (Figure 5d) with a LAI ≤4 (Figure 5a) and means near 2 (Figure 5b) agreed with the observed seasonality of the understory layer, which can reach a LAI of ~2 or more in summer but which is often near zero in winter [START_REF] Chen | Leaf area index of boreal forests: theory, techniques and measurements[END_REF][START_REF] Jiao | Mapping global seasonal forest background reflectivity with Multi-angle Imaging Spectroradiometer data[END_REF][START_REF] Masson | A global database of land surface parameters at 1km resolution in meteorological and climate models[END_REF]).

The areas with the highest interannual variabilities in GEOCLIM (Figure 5c) corresponded to cropland in the USA and Eurasia, with intrinsic variabilities due to crop rotation or management, but also regions of severe drought and fire in South America, Africa, and Asia, regions of landcover change such as the deforestation in Amazonian and Indonesian tropical forests, and regions of extreme events such as drought and heat waves in Europe, eastern China, and Australia. High interannual variability, however, may also indicate a problem with the computed GEOCLIM value due to instabilities in the GEOV1 data or to insufficient available GEOV1 data, as observed in the Gulf of Guinea (compare Figures 3b and5c). In most regions, the interannual variability (Figure 5c) was significantly lower than the seasonal variability (Figure 5d), demonstrating that GEOCLIM provided a baseline vegetation annual cycle that was representative of the current phenology and that smoothed most of the anomalies.

[Figure 5]

Comparison with ECOCLIMAP, MODIS and GIMMS3g climatologies

The map of annual mean differences between GEOCLIM and the LAI climatologies derived from AVHRR and MODIS data (Figure 6) shows LAI differences of ±0.5 for 54%, 83% and 91% of the land pixels as compared with ECOCLIMAP, MODIS and GIMMS3g, respectively. GEOCLIM produced systematically lower values than ECOCLIMAP for the remaining 46% of pixels, with larger differences for dense forests (northern boreal and tropical forests) but with significant differences also for crops (e.g. USA and eastern Asia) (Figure 6a). These systematic negative bias of GEOCLIM as compared to ECOCLIMAP was evident across latitudes and along the year (Figure 7). GEOCLIM produced also systematically lower values than MODIS (Figure 6b) over tropical forests with differences ~0.5 along the year (Figure 7b) and over northern deciduous broadleaf forest during the maximum growing leaf development (lower frequencies for the maximum values in Figure 8c). On the contrary GEOCLIM produced slightly higher LAI values than GIMMS3g (Figure 6c) over Amazon and Indonesian evergreen broadleaf forests and over boreal needle leaf forests in Russia and USA during the winter time (Figure 7d).

[Figure 6]

Despite the large discrepancies in the magnitude of LAI between the different datasets, due in part to the differences in sensors and retrieval algorithms, seasonality and its phasing generally agreed well (Figure 7). Seasonality was inverted in the Southern Hemisphere relative to the Northern Hemisphere (compare Figures 7a and7c). In the Northern Hemisphere, LAI seasonality decreased in the length of season (active growth period) with latitude (compare Figures 7c and7d). In the tropical latitudes (-20-10º) characterized by very limited seasonality GEOCLIM and GIMMS3g systematically showed lower values than MODIS and ECOCLIMAP (Figure 7b). The largest differences in terms of seasonality were in the Northern Hemisphere at high latitudes (40-70º) where ECOCLIMAP produced longer growing seasons as compared to other LAI datasets and higher values in the period of active growth (Figure 7d). Nevertheless, all the data sets agreed well for the base level of LAI during the dormancy period for the 40-70º latitudes validating a posteriori the reliability of the specific correction applied in winter (SZA>70º) to the GEOCLIM values for high northern latitudes (Section 2.3).

[Figure 7]

Histograms of the LAIs (Figure 8) indicated very similar distributions between the different LAI datasets for shrubs/savannah/bare soil. Some similarities in the position of the maximum frequency were also observed for crops and grassland. Some discrepancies, however, were observed: ECOCLIMAP produced low frequencies for LAIs of zero at the expense of higher intermediate values, while GEOCLIM, MODIS and GIMMS3g produced a smoother transition. Forests had the largest discrepancies between the different data sets. GEOCLIM and GIMMS3g produced a bimodal distribution for deciduous broadleaf forests, with a peak for low values (LAI near 1) corresponding to the dormant period of the vegetation in winter and a second mode for the period of active growth with values higher than 6 in few occasions. MODIS produced also a peak for LAI near 1 and a smooth transition up to maximum values around 6.5. ECOCLIMAP produced an even distribution for deciduous forests, but with unrealistic peaks. Evergreen broadleaf forests had relatively consistent narrow distributions between GEOCLIM, ECOCLIMAP and MODIS but with significant differences in the magnitudes (i.e. similar shapes but shifted distributions). The LAI modes were 5 for GEOCLIM, and 6 for ECOCLIMAP and MODIS. GIMMS3g produced broader distributions with the LAI mode ~4. Needleleaf forests had similar distributions for GEOCLIM, MODIS and GIMMS3g but with higher frequencies for low values compared to those in ECOCLIMAP. The LAI mode around 1 for GEOCLIM (MODIS and GIMMS3g) for deciduous broadleaf forests (Figure 8c) and needleleaf forests (Figure 8e) corresponded to the winter LAI value and reproduced the expected seasonality in northern high latitudes (Figure 7d) while ECOCLIMAP produced unrealistic LAI distributions and peaks' locations (Figures 8c and8e).

[Figure 8]

Assessment of global phenology

The spatial pattern of the phenology derived by GEOCLIM (Figure 9) reflected the distributions of climate and biome type (Figure 4b). Seasonality was strongly dependent on temperature at northern latitudes >30º, and the timing of maximum greenness had a clear latitudinal gradient indicating a delay in the date of peak development with latitude (Figure 10). In other regions, seasonality had more complex spatial patterns that were driven mostly by biome type, land use, and the seasonal variation in rainfall (Figure 9).

The phenological metrics (Figure 10) were spatially consistent in the timing of the maximum of the growing season as derived from the different data sets and particularly between GEOCLIM and MODIS (GIMMS3g) with uncertainties of around 14 days in terms of RMSE, bias of less than 1 day, a correlation higher than 0.95 and a slope of the linear regression close to the unity (Table 1).

The phenology derived from ECOCLIMAP was also highly spatially consistent with GEOCLIM (correlation about 0.9, slope close to the unity and bias about 6 days, Table1) but it diverged to some degree (uncertainties of about one month in terms of RMSE), mostly in the Southern Hemisphere (Figure 10) in regions with a limited seasonality (Figure 5d). As expected (Section 3.1.4), the phenological phase for the "onset of greenness maximum" retrieved in MCD12Q2 occurred earlier than the peak date in GEOCLIM (bias of 14 days, Table 1) due to differences in the definitions of the phenological metrics. The phenological metrics derived from GEOCLIM constitutes an intermediate solution across latitudes between MODIS, GIMMS3g, ECOCLIMAP and MCD12Q2 for the date of maximum foliar development (Figure 10)

[Figure 9] [Figure 10] [Table 1]

Discussion

Twelve years (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010) of data from GEOV1 LAI, FAPAR, and FCOVER products were used to compute GEOCLIM outputs for the interannual average seasonal cycle at a pixel scale. The main assumptions were that (i) no land-cover change or abrupt disturbance leading to a change in the phenological annual cycle occurred for the period considered and (ii) the time series were sufficiently long to reduce the sensitivities of the averages to anomalies. Specific correction were applied at northern high latitudes, bare soils and evergreen broadleaf forests to overcome problems associated, respectively, with strong bidirectional effects and snow cover, precision and signal to noise ratio, and aerosol-cloud contamination (Section 2.3). The identification of bare soil and evergreen broadleaf forests was completely driven by the data avoiding possible miss-classification errors introduced by external land cover map information though a good spatial consistency with GLOBCOVER map was observed. In these problematic cases, GEOCLIM was forced to fixed values derived from the input data at the pixel level under the following hypothesis: (i) minimum vegetation activity in winter time at northern latitudes, and no seasonality in (ii) desert areas and (iii) evergreen broadleaf forests where the vegetation is respectively low (LAI~0) and high (LAI~5) throughout the year. The last hypothesis constitutes an oversimplification of the reality because of the possible seasonality of evergreen broadleaf forests. The high uncertainty associated with the data due to poor atmospheric correction and very high cloud occurrence in equatorial and tropical latitudes prevented the extraction of meaningful phenology at the resolution of the individual pixels of 1 km. The high spatial and temporal resolution of forthcoming Sentinel2 sensors should improve the monitoring of vegetation in these problematic areas.

GEOCLIM was indirectly validated based on the comparison with AVHRR and MODIS derived climatologies of LAI. Multitemporal ground data would be preferable for validating GEOCLIM but unfortunately were rarely available. GEOCLIM showed a high agreement with MODIS (GIMMS3g) climatology of LAI and absolute differences were higher than the Global Climate Observing System (GCOS 2010) requirements for accuracy, i.e. 0.5 LAI, only in northern boreal and tropical forests representing less than 20% (10%) of land pixels. GEOCLIM systematically produced lower values than MODIS over evergreen broadleaf forest as also observed in the comparison between GEOV1 and MODIS [START_REF] Camacho | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part 2: Validation and intercomparison with reference products[END_REF][START_REF] Fang | Characterization and intercomparison of global moderate resolution leaf area index (LAI) products: Analysis of climatologies and theoretical uncertainties[END_REF]. The difficult observational conditions in tropical latitudes with persistent clouds can cause irregularities in the solution and thus variable but systematic underestimations of LAI (Verger et al. 2011). The specific correction applied to GEOCLIM removed the instabilities in the solution but cannot correct possible biases in the magnitude of original GEOV1 products used as input data for GEOCLIM. Previous studies have also shown that GEOV1 products produce slightly higher values than MODIS for needleleaf forest in winter [START_REF] Fang | Characterization and intercomparison of global moderate resolution leaf area index (LAI) products: Analysis of climatologies and theoretical uncertainties[END_REF]. The specific correction applied in GEOCLIM at northern high latitudes reduced these differences but may result in some underestimation of the seasonal amplitude in winter time. Accurate estimation of LAI in needleleaf forests in winter is challenging because contamination by clouds and snow limits the reliability of the reflectances used as inputs in the algorithms [START_REF] Camacho | GEOV1: LAI, FAPAR Essential Climate Variables and FCOVER global time series capitalizing over existing products. Part 2: Validation and intercomparison with reference products[END_REF]. Further, the strong bidirectional effects of surface-reflectance at very high latitudes are not well simulated by the radiative transfer models currently used for product generation (Yang et al. 2006a). In addition, the understory and foliage clumping are not well accounted for [START_REF] Jiao | Mapping global seasonal forest background reflectivity with Multi-angle Imaging Spectroradiometer data[END_REF][START_REF] Pisek | Impacts of including forest understory brightness and foliage clumping information from multiangular measurements on leaf area index mapping over North America[END_REF]). LAI values were systematically higher for ECOCLIMAP than for GEOCLIM, MODIS and GIMMS3g. [START_REF] Boussetta | Impact of a satellite-derived leaf area index monthly climatology in a global numerical weather prediction model[END_REF] reported similar higher LAI values for ECOCLIMAP than for MODIS. [START_REF] Garrigues | Validation and intercomparison of global Leaf Area Index products derived from remote sensing data[END_REF] also reported large positive biases for ECOCLIMAP compared with CYCLOPES, MODIS and GLOBCARBON and with ground measurements. The differences in the temporal period, input data and sensors (VEGETATION for GEOCLIM and AVHRR and MODIS for ECOCLIMAP, Section 3.1) can partially account for the significant discrepancies between GEOCLIM and ECOCLIMAP although the relatively good agreement of GEOCLIM with MODIS and GIMMS3g AVHRR derived LAI climatologies indicates that the major source of discrepancies are related to the retrieval algorithms. The linear relationship between NDVI and LAI used to retrieve the ECOCLIMAP product for pixels out of Europe [START_REF] Masson | A global database of land surface parameters at 1km resolution in meteorological and climate models[END_REF]) may have introduced some overestimation because the LAI-NDVI relationship is exponential and saturates at medium to high values (e.g. [START_REF] Myneni | Global products of vegetation leaf area and absorbed PAR from year one of MODIS data[END_REF]). Since ECOCLIMAP assumes low spatial variability within each class of land cover, it is limited to capture the LAI spatial variability as compared to other LAI datasets [START_REF] Garrigues | Validation and intercomparison of global Leaf Area Index products derived from remote sensing data[END_REF]). Nevertheless, identifying the source of the differences between ECOCLIMAP and the other LAI datasets being analyzed would require further attention and it is out of the scope of this paper.

Further research should focus on the development of improved LAI datasets with due attention to areas (boreal and tropical latitudes) and periods (winter time) where higher uncertainties exist [START_REF] Fang | Characterization and intercomparison of global moderate resolution leaf area index (LAI) products: Analysis of climatologies and theoretical uncertainties[END_REF]. In these cases characterized by high level of noise and missing data, the use of the climatology and temporal smoothing and gap filling techniques applied at daily estimates of biophysical variables may increase the robustness of the solution as compared to the classical composition techniques (Verger et al. 2014a).

The phenological metrics derived from GEOCLIM was highly spatially consistent (correlation higher than 0.9) with MODIS and AVHRR derived phenologies, including ECOCLIMAP ones, for the date of maximum foliar development with differences lower than six days in all cases except when compared with MCD12Q2 product (systematic biases of 14 days) due to the differences in the definition. A standardization in the definitions of the phenological metrics appears necessary (White et al. 2009). Disentangling the mechanisms that govern the seasonal and interannual variability in phenology and vegetation-climate dynamics at the global scale would require further analysis.

Conclusions

This article has presented and provided a first quality assessment of GEOCLIM-a global climatology of LAI, FAPAR and FCOVER-from the multiannual time-series of 10-days and 1-km GEOV1 products built from 1999 to 2010. Results showed GEOCLIM was temporally consistent for the seasonality across biomes and latitudes. GEOCLIM showed a high agreement with MODIS and AVHRR climatologies of LAI: differences within the Global Climate Observing System requirements, i.e. ±0.5 LAI, in more than 80% (90%) of GEOCLIM land pixels compared with MODIS (GIMMS3g). ECOCLIMAP systematically produced higher LAI values. Further research should focus on tropical and boreal latitudes where higher uncertainties exist in the LAI datasets.

The phenology of the timing of maximum foliar development derived from GEOCLIM constituted an intermediate solution between those of GIMMS3g, MODIS, ECOCLIMAP and MCD12Q2: differences within 14 days and spatial correlation >0.9 at the global scale.

The GEOCLIM data set is continuous, both spatially and temporally, and it can be used for a wide range of land-biosphere applications. It may contribute to a better characterisation of the seasonal variability of vegetation in global land-surface models. It provides the baseline characteristics of the seasonal cycle of LAI, FAPAR, and FCOVER for the identification of anomalies and trends in global vegetation. for maximum foliar development derived from GEOCLIM and mean differences as compared to the phenological metrics derived from ECOCLIMAP, MODIS, GIMMS3g and MCD12Q2.

List of Figure Captions

Table 1. The root mean square error (RMSE), bias, standard deviation (σ), correlation coefficient (R), and slope of the regression line through the origin for comparisons between the dates of maximum foliar development derived from GEOCLIM, ECOCLIMAP, and MCD12Q2 at a global scale at 0.5º. The areas of bare soil and evergreen broadleaf forests (Figure 4a) with insufficient seasonality for computing the phenological metrics and 10% outliers were not included.

RMSE

  This section assesses the performance of GEOCLIM. The validation data sets are first described.The spatial and temporal consistencies of GEOCLIM are discussed and evaluated by comparison with climatologies derived from both AVHRR and MODIS data. For the sake of brevity, the results focus on LAI, because of the three variables LAI, FAPAR, and FCOVER, LAI is used most by the scientific community. The comparison was performed at 0.5º spatial sampling on a Plate-Carrée grid. The 0.5º spatial resolution corresponds to the typical resolution of global models and reduces computation time. For comparison purposes, the different LAI data sets were averaged at monthly time step because one-month corresponds to the lowest temporal sampling among the validation datasets.

  version, available since June 2014 at www2. It combines the global database of the first version, ECOCLIMAP-I, and an upgraded version, ECOCLIMAP-II, for Europe. ECOCLIMAP-I contains 215 ecosystems obtained by combining existing land covers, climatic maps, and NDVI seasonal profiles from AVHRR data acquired between April 1992 and March 1993 (Masson et al. 2003). For each class of vegetation, the maximum and minimum LAI values are fixed based on in-situ knowledge, and the annual cycle of LAI is constrained by the NDVI AVHRR temporal profiles using a linear relationship between NDVI and LAI. The second version, ECOCLIMAP-II, contains 573 ecosystems across Europe based on more recent land-cover maps, and the annual LAI profiles are derived from MODIS Collection 5 for the years 2002-2006[START_REF] Faroux | ECOCLIMAP-II/Europe: a twofold database of ecosystems and surface parameters at 1 km resolution based on satellite information for use in land surface, meteorological and climate models[END_REF]).

Figure 1 .

 1 Figure 1. Illustration of climatology computation over a grassland site (17.98ºS, 16.82ºE). (a) Time

Figure 2 .

 2 Figure 2. Illustration of TSGF correction for LAI for four GLOBCOVER biome classes (Defourny

Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .

 34567 Figure 3. Global maps of (a) the maximum standard deviations (Max. SD) of interannual LAI

Figure 8 .

 8 Figure 8. Distributions of GEOCLIM, ECOCLIMAP, MODIS and GIMMS3g LAIs per biomes

Figure 9 .Figure 10 .

 910 Figure 9. Global map of the day of the year (DOY) for maximum foliar development (date of peak

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .

 12345 Figure 1. Illustration of climatology computation over a grassland site (17.98ºS, 16.82ºE). (a) Time

Figure 7 .

 7 Figure 7. Temporal profiles of GEOCLIM LAI averaged for 30º latitudinal bands and mean LAI

Figure 8 .

 8 Figure 8. Distributions of GEOCLIM, ECOCLIMAP, MODIS and GIMMS3g LAIs per biomes

Figure 9 .Figure 10 .

 910 Figure 9. Global map of the day of the year (DOY) for maximum foliar development (date of peak

  This is the author's version of a work that was accepted for publication in Remote sensing of environment (Elsevier). Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Remote sensing and environment, vol. 166 (set. 2015),126-137. DOI 10.1016126-137. DOI 10. /j.rse.2015.05.027 .05.027

Acknowledgements

This research was partially supported by the European Earth observation programme Copernicus (388533), Pôle Thématique Surfaces Continentales THEIA (CNES 140570) and the FP7 geoland2 (218795), GIOBIO (32-566) and LONGLOVE (32-594) projects. This research was also supported by the Spanish Government grant CGL2013-48074-P, the Catalan Government grant SGR 2014-274, and the European Research Council Synergy grant ERC-2013-SyG-610028 IMBALANCE-P. Aleixandre Verger is the recipient of a Juan de la Cierva postdoctoral fellowship from the Spanish Ministry of Science and Innovation. The authors are indebted to Ranga Myneni, Taejin Park, the Boston University, MODIS, CNES, VITO, and CNRM-GAME (Météo France, CNRS) teams for providing access to the data.

WWW sites www1: GEOV1 Biophysical Products. http://land.copernicus.eu/global (last accessed 22 June2015) www2: ECOCLIMAP code and data https://opensource.cnrm-game-meteo.fr/projects/ecoclimap (last accessed 22 June2015) www3: MODIS Reprojection Tool https://lpdaac.usgs.gov/tools/modis_reprojection_tool (last accessed 22 June2015)

van den Hurk, B.J.J.M., Viterbo, P., & Los, S.O. (2003). Impact of leaf area index seasonality on the annual land surface evaporation in a global circulation model. Journal of Geophysical Research: Atmospheres, 108, 4191 Verger, A., Baret, F., & Weiss, M. (2011). A multisensor fusion approach to improve LAI time series. Remote Sensing of Enviroment, 115, 2460Enviroment, 115, -2470 Verger, A., Baret , F., & Weiss, M. (2014a). Near real time vegetation monitoring at global scale. Observations and Remote Sensing,7,[3473][3474][3475][3476][3477][3478][3479][3480][3481] Verger, A., Baret, F., Weiss, M., Filella, I., & Peñuelas, J. (2014b). Quantification of leaf area index anomalies from global VEGETATION observations. In, Global vegetation monitoring and modeling (available at https://colloque.inra.fr/gv2m/Oral-Sessions). Avignon (France) Verger, A., Baret , F., Weiss, M., Kandasamy, S., & Vermote, E. (2013). The CACAO method for smoothing, gap filling and characterizing seasonal anomalies in satellite time series. IEEE transactions on Geoscience and Remote Sensing, 51, 1963Sensing, 51, -1972 Verhegghen

IEEE Journal of Selected Topics in Applied Earth


Lucas Roquet

Fernando Fernandes

Santos

Paolo Rech

Marcello Traiola

Olivier Sentieys

Angeliki Kritikakou

Cross-Layer Reliability Evaluation and Efficient Hardening of Large Vision Transformers Models

Vision Transformers (ViTs) are highly accurate Machine Learning (ML) models. However, their large size and complexity increase the expected error rate due to hardware faults. Measuring the error rate of large ViT models is challenging, as conventional microarchitectural fault simulations can take years to produce statistically significant data. This paper proposes a two-level evaluation based on data collected through more than 70 hours of neutron beam experiments and more than 600 hours of software fault simulation. We consider 12 ViT models executed in 2 NVIDIA GPU architectures. We first characterize the fault model in ViT's kernels to identify the faults more likely to propagate to the output. We then design dedicated procedures efficiently integrated into the ViT to locate and correct these faults. We propose Maximum corrupted Malicious values (MaxiMals), an experimentally tuned low-cost mitigation solution to reduce the impact of transient faults on ViTs. We demonstrate that MaxiMals can correct 90.7% of critical failures, with execution time overheads as low as 5.61%.

Introduction

Transformers are state-of-the-art ML models that excel in various autonomous system tasks such as language processing, image classification, radar processing, and instance segmentation. When used in vision applications, ViTs selectively focus on essential features in a given frame using attention mechanisms instead of treating all features equally. Thanks to their ability to learn a wide range of concepts from data, ViTs are especially useful for complex applications like autonomous driving [START_REF] Fang | Eva-02: A visual representation for neon genesis[END_REF] and industrial automation [START_REF] Chen | Compound fault diagnosis for industrial robots based on dual-transformer networks[END_REF]. However, given the complexity of the models and the high number of parameters (which can exceed 1 trillion [START_REF] Fedus | Switch transformers: Scaling to trillion parameter models with simple and efficient sparsity[END_REF]), ViTs need to be executed on large hardware accelerators, such as Graphic Processing Units (GPUs). GPUs are the most suitable hardware architecture to train and use large ViT models due to their flexibility and high-performance computing capabilities. GPU vendors have significantly improved their products' computing power, frameworks, and hardware reliability. Modern GPUs feature a tailored Single Error Correction Double Error Detection (SECDED) Error Correction Codes (ECC) in the main GPU memories [START_REF] Sullivan | Characterizing And Mitigating Soft Errors in GPU DRAM[END_REF]. Despite ECC in main memories, as we show in this paper, GPUs executing ViTs can still present high neutron-induced fault rates due to their extensive computing resources. Additionally, the probability of multiple parallel units being simultaneously affected compromises the reliability of ViT-based systems, posing a threat to ViT-based autonomous safety-critical applications.

Assessing the reliability of ViTs is exceptionally challenging due to the complexity of both the hardware and software framework. While radiation experiments are used as a realistic error rate estimation source [START_REF] Hari | Making convolutions resilient via algorithmbased error detection techniques[END_REF], they do not allow for tracking fault propagation. Contrarily, with fault simulation, we can pinpoint specific fault sites [START_REF] Papadimitriou | Demystifying the system vulnerability stack: Transient fault effects across the layers[END_REF]- [START_REF] Gavarini | Evaluation and mitigation of faults affecting swin transformers[END_REF]. However, fault models are synthetic and provided by the user. It is thus essential to ensure that simulation fault models are realistic to avoid drawing misleading conclusions or implementing ineffective hardening solutions. Recent works show that the results obtained from different fault simulation levels, such as microarchitectural and software, can vary up to one order of magnitude [START_REF] Papadimitriou | Demystifying the system vulnerability stack: Transient fault effects across the layers[END_REF], if the fault model does not realistically represent real-world errors. In this paper, we adopt a cross-layer analysis (novel for ViTs), combining radiation and software fault simulation approaches, to obtain realistic and traceable insights into ViT reliability.

Unfortunately, traditional mitigation strategies, such as modular redundancy [START_REF] Baek | FT-DeepNets: Fault-Tolerant Convolutional Neural Networks With Kernel-Based Duplication[END_REF] and Algorithm-Based Fault Tolerance (ABFT) [START_REF] Hari | Making convolutions resilient via algorithmbased error detection techniques[END_REF], become nearly impractical when adapted to large Transformers, since there are billions (even trillions) of parameters and gigabytes of memory to manage and protect. Even a simple float value restriction, applied across all layers of a ViT model, imposes a significant 68.61% overhead [START_REF] Gavarini | Evaluation and mitigation of faults affecting swin transformers[END_REF]. Thus, novel and effective solutions are required to enhance ViT's reliability, such as the one we propose in this paper.

We propose protecting against faults most likely to affect ViT's accuracy. To identify the faults to correct, we expose two NVIDIA GPU architectures to neutron beams for over 70 hours (equivalent to 700,000 years of terrestrial operation), measuring the error rate of 6 large ViT models (with and without ECC) and characterizing the fault model of ViT kernels. Then, we conduct over 600 hours of software fault simulation using the NVIDIA Fault Injector [START_REF] Tsai | NVBitFI: Dynamic Fault Injection for GPUs[END_REF] tuned with the experimentally observed fault models. We track fault propagation and identify the faults that are more likely to corrupt the ViT inference. Based on our cross-layer analysis, we propose a low-cost and effective fault-tolerant mechanism, named Maximum corrupted Malicious values (MaxiMals), that corrects only the critical corrupted floating-point values in the inference. Notably, MaxiMals incurs a low average overhead of 5.61% in execution time and requires minimal model modifications while reducing up to 90.7% of misclassifications (61.41% on average). Specifically, our contributions include: • A reliability evaluation of 6 ViTs on 2 GPU architectures using a neutron beam, discussing the dependence of ViT error rates on complexity and trade-off of enabling ECC.

Background and Contributions

The ViT model, introduced by Dosovitskiy et al. [START_REF] Dosovitskiy | An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale[END_REF], improves image classification accuracy by treating input images as sequences of patches. These patches are transformed using linear transformations before being fed into the model. Additionally, the ViTs have similar structures across their variants like EVA2 [START_REF] Fang | Eva-02: A visual representation for neon genesis[END_REF], SwinV2 [START_REF] Liu | Swin transformer v2: Scaling up capacity and resolution[END_REF], and MaxViT [START_REF] Tu | MaxViT: Multi-axis vision transformer[END_REF]. Hardware accelerators for ViT employed on autonomous systems are susceptible to soft errors caused by faults induced by ionizing particles, such as high-energy neutrons [START_REF] Mahatme | Comparison of Combinational and Sequential Error Rates for a Deep Submicron Process[END_REF]. These errors may not damage the device physically but can significantly impact the output of ViT models, potentially changing their final classifications. When not masked, soft errors can propagate to the software level and cause Detected Unrecoverable Errors (DUEs) or Silent Data Corruptions (SDCs). DUEs hang the program or crash the entire system, while SDCs allow the application to complete its execution but with an incorrect output and, without a fault-tolerance method, the failure remains undetected. Particularly concerning ViT models, SDCs can be further categorized into Tolerable SDCs, which modify the model output but not the classification outcome, or Critical SDCs, which causes the model to change the top 1 classification probability, resulting in misclassification. We focus on correcting Critical SDCs only, to provide efficient mitigation solutions.

In light of the complexity of ViT models and their high memory and computational requirements, we have opted for reliability evaluation methods that generate accurate data in a feasible amount of time: physical injection with a neutron beam and instructionlevel fault simulation. Although lower levels of fault simulation, such as RTL and microarchitectural, can produce more precise outcomes [START_REF] Papadimitriou | Demystifying the system vulnerability stack: Transient fault effects across the layers[END_REF], recent studies indicate that characterizing all the requisite fault simulations for large ML models could take months to years [START_REF] Lew | Analyzing machine learning workloads using a detailed GPU simulator[END_REF]. For example, simulating a tiny 5-layer neural network that is around 5.10 3 times smaller than the smallest model we assessed (ViT BS32-224) takes approximately 1.2 hours in a GPU microarchitectural simulator [START_REF] Lew | Analyzing machine learning workloads using a detailed GPU simulator[END_REF]. It would require ≈7 centuries to simulate 1,000 faults on a GPU microarchitectural simulator for ViT BS32-224.

In order to provide a comprehensive evaluation, we adopt a crosslayer strategy that consists of observing, with beam experiments, how (and how often) the hardware fault propagates to a software visible state. Then, we propagate this effect in software to track how it modifies the ViT output. Combining data from neutron beam experiments with instruction-level simulations therefore provides valuable insights into the reliability of large ViT models, and how to enhance their fault tolerance at the application level.

ABFT [START_REF] Hari | Making convolutions resilient via algorithmbased error detection techniques[END_REF] and value restriction [START_REF] Chen | A Low-cost Fault Corrector for Deep Neural Networks through Range Restriction[END_REF], [START_REF] Hoang | FT-ClipAct: Resilience Analysis of Deep Neural Networks and Improving Their Fault Tolerance Using Clipped Activation[END_REF] are established approaches to prevent Critical SDC on ML models. Interestingly, conventional strategies for large Transformers lead to high overheads due to their resource-demanding nature. Researchers have adapted ABFT [START_REF] Ma | Error Resilient Transformers: A Novel Soft Error Vulnerability Guided Approach to Error Checking and Suppression[END_REF] and range restriction [START_REF] Gavarini | Evaluation and mitigation of faults affecting swin transformers[END_REF] for ViTs. However, a simple range restriction approach for all the layers of a ViT model can add up to 68.61% overhead on execution time [START_REF] Gavarini | Evaluation and mitigation of faults affecting swin transformers[END_REF]. To address this issue, our proposed MaxiMals approach is an experimentally tuned method at the application level that increases fault tolerance for large ViT models with low overhead. This is achieved by targeting only critical faults. We refrain from suggesting hardware design changes, resulting in costly hardware modifications that could affect performance and design time. Instead, we efficiently manage hardware faults at the application level.

Evaluation Methodology

This section describes our experimental methodology, error rate metrics, and reveals how ViT features affect reliability.

System Under Test: We performed fault simulations and beam experiments on two NVIDIA GPU architectures, Pascal (Quadro P2000) and Ampere (RTX A2000). The Quadro P2000 is built with TSMC 16𝑛𝑚 FinFET, featuring an L1 cache of 48KB per Streaming Multiprocessor (SM), an L2 cache of 1280 KB, and 1024 CUDA cores. The RTX A2000 is built with TSMC 7𝑛𝑚 FinFET, featuring an L1 cache of 128 KB per SM, an L2 cache of 3MB, and 3328 CUDA cores. Both GPUs have 256 KB registers per SM and a power consumption of up to 75W. Our beam experiments only focus on GPU core errors (beam spot set to 2cm diameter to avoid affecting onboard DRAM). The RTX A2000 has SECDED ECC to protect the register file and cache memories. Tests were conducted with ECC ON for some ViTs to assess ECC efficacy.

We evaluated 12 ViT models from the HuggingFace library (v0.8.19) [START_REF] Wightman | Huggingface[END_REF]. The models belong to 4 families: Original ViT [START_REF] Dosovitskiy | An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale[END_REF], EVA2 [START_REF] Fang | Eva-02: A visual representation for neon genesis[END_REF], SwinV2 [START_REF] Liu | Swin transformer v2: Scaling up capacity and resolution[END_REF], and MaxViT [START_REF] Tu | MaxViT: Multi-axis vision transformer[END_REF]. The models differ in size and input patches. For the experiments, we used a Python program with PyTorch v2.0.0 to load the ViT and perform inferences on a batch of random images from the ImageNet dataset [START_REF] Deng | ImageNet: A large-scale hierarchical image database[END_REF]. Table 1 shows essential features of the evaluated models, including their GPU memory usage, accuracy, execution time, and the min and max Identity layers' output values used for the MaxiMals implementation.

Physical Fault Injection: We measured the neutron-induced error rate on a subset of ViTs from Table 1 by exposing the GPUs to a neutron beam. As we could not assess all 12 configurations due to beam time limitations, the original ViT configurations were prioritized. The beam experiments provide the Failure In Time (FIT), which is calculated by dividing the number of errors by the neutron fluence and then multiplying by the terrestrial neutron flux (13 × 10 9 ). The experiments were done at the ChipIR facility of the Rutherford Appleton Laboratory, UK. Figure 1 shows the installed setup, consisting of GPUs aligned with the neutron beam and connected to the motherboard. The beam setup utilizes Python scripts to monitor and execute ViT models on a server outside the beam room, while the software is designed to recover from device hangs and restart the program if it fails to respond within a set timeframe. The same ViT model is run on the GPU for several iterations, and any differences between the outputs and a previously saved output are recorded as Tolerable SDC or Critical SDC. The codes used on the beam experiments are disclosed 1 . Software Fault Simulation: We used the NVIDIA Bit Fault Injector (NVBitFI) [START_REF] Tsai | NVBitFI: Dynamic Fault Injection for GPUs[END_REF] for the instruction-level fault simulation. NVBitFI allows simulating faults at the Shader Assembly level (SASS), i.e., GPU kernel at the assembly level. NVBitFI allows choosing different fault models and sites to evaluate the ViT models. While injecting random bit flips in software fault injection is not accurate [START_REF] Papadimitriou | Demystifying the system vulnerability stack: Transient fault effects across the layers[END_REF], injecting in software an experimentally-tuned fault model (i.e., the observed manifestation of the hardware fault in a software visible state) has been proved accurate for GPUs [START_REF] Bolchini | Fast and accurate error simulation for cnns against soft errors[END_REF]. We inject faults in general-purpose registers, memory load instructions, and arithmetic floating-point operations. We used different bit fault models derived from beam experiments, detailed in Section 4. We simulate 1,750 faults per ViT model, equivalent to more than 600 hours of simulations. The failures (SDCs and DUEs) are counted similarly to beam experiments. With fault simulations, we measure the Program Vulnerability Factor (PVF) for each ViT model. The PVF is the probability of an injected fault propagating from the assembly instruction to the application output [START_REF] Sridharan | Eliminating microarchitectural dependency from Architectural Vulnerability[END_REF].

Fault Effects on ViT Models

In this section, we analyze the ViT's FIT rate and identify the causes of Critical SDCs. We show that single-bit flip fault simulations are insufficient for ViT reliability assessment.

ViT' FIT rate: Figure 2 shows the experimentally measured SDC (Tolerable and Critical) and DUE FIT rates for the tested ViTs configurations. Values are reported with 95% confidence intervals considering a Poisson distribution.

All ViTs show high DUE FIT rates, on average, 20.50 for Pascal and 32.57 for Ampere ECC OFF. Investigating the cause for DUEs, we found that, when ECC is OFF, on average, 59.39% of the DUEs for Pascal and 78.14% of the DUEs for Ampere are caused by memory faults, such as incorrect memory address accesses and unaligned memory operations. The differences in the DUE rate for Pascal and Ampere are attributed to the lower resources available on Pascal, forcing more global memory accesses and warp scheduling stress. On Ampere GPU, when ECC is ON, the DUE FIT rate increases by As shown in Figure 2, whereas the DUE FIT rate does not significantly depend on the model, the SDC FIT rates directly depend on the ViT model complexity and the related GPU resource utilization. The average SDC FIT rate for the Pascal GPU is 6.66, with the highest SDC FIT rate being 9.55 (EVA2 B14-448). For Ampere GPU, when ECC is OFF, the average SDC FIT is 30.74, and the highest is 50.62 (H14-224). The trend is less evident for Pascal, for which we observe only a slight increase in the SDC rate. This is justified by resource saturation in the Pascal GPU (even the smaller ViT uses all GPU resources), while in the Ampere GPU, bigger models use more parallel resources, increasing the SDC FIT rate.

The likelihood of Critical SDCs occurring depends on various ViT characteristics like weight values, accuracy, and activation layers. Despite the high accuracy and significant data redundancy of ViTs, Critical SDCs can still occur, as shown in Figure 2. This is particularly evident in the case of large models, such as EVA2 B14-448 and ViT L14-224, which exhibit a percentage of Critical SDC of 33.33% and 37.33%, respectively. Models with many residual connections and linear operations, like EVA2, can have higher criticality than other models due to the ease with which errors can propagate between the layers. Interestingly, H14-224 has a low Critical SDC rate when executed with ECC OFF, 7.14% for Pascal and 6.17% for Ampere. This can be attributed to the model having a massive amount of parameters (2.48GB), making it less likely for the corruption of a single or a small group of parameters to result in Critical SDCs.

When ECC is ON, the SDC FIT rates are reduced by 71.51%. Nonetheless, even with ECC ON, the ViT L14-224 still experiences Critical SDCs in 10.0% of the cases. ECC does not provide full protection from Critical SDCs and increases the DUE rate. Thus, alternative hardening solutions are required.

ViT' Fault Models: We analyze the reliability of the most common ViT kernels (MLP, Attention, and the Encoder Block) to unveil the leading causes of Critical SDCs. Table 2 shows data from beam experiments for the kernels extracted from the L14-224 model on a single inference on Pascal GPU. We compute SDC and DUE FIT rates, the percentage of Not a Number (NaN ) and ±infinity (inf ) values observed in all the experiments, and the maximum difference between fault-free and corrupted outputs, for each kernel. The SDC FIT rates for kernels (on average, 20.49) are higher than the FIT rate of most ViT modules. This is not surprising, since an SDC in a kernel of the ViT still needs to propagate through the ViT model, and it can still be masked in the downstream Encoder Blocks. That is, we are not yet considering the propagation probability of these faults in the ViT model. Conversely, the DUE FIT rate is comparable to the ViT model's (30.08 on average). This is because a DUE in a kernel will hang the kernel execution and, consequently, the application in which the kernel is being used. In other words, when a crash/hang occurs, it cannot be masked.

The MLP kernel produces no inf /NaN values. The MLP algorithm is a sequence of multiplying and accumulating instructions and being a simpler algorithm, MLP has less chance of generating inf /NaN values. Contrarily, the Attention kernel is composed of softmax and division operations. Those operations demand many cycles to compute and are more prone to yield inf /NaN values, leading to the highest percentage of corrupted values. Lastly, the ViT Block reveals much lower inf /NaN percentages in the output than the Attention kernel. Attention produces many inf /NaN values, but, due to masking, these values may not propagate through to the final output of the ViT Block. If corrupted values reach the output of the Block, they can potentially affect the classification of the entire ViT model.

Software Fault Simulation: To realistically evaluate which faults -among those observed in beam experiments on the kernels -propagate and generate Critical SDCs, we conducted fault simulation campaigns employing multiple fault models. Moreover, the goal is to identify the models that better represent the neutroninduced faulty effect. For instance, for FP32 instructions, we used a customized version of NVBitFI to simulate a more complex fault model that writes a random value on all the threads on a GPU warp. Multiple threads fault models accurately emulate GPU faults that threaten ML models' reliability, as demonstrated in [START_REF] Bolchini | Fast and accurate error simulation for cnns against soft errors[END_REF].

Figure 3 shows the average probability for the fault models injected in the 4 ViT families to induce a Critical SDC. We also plot the results for the MaxiMals hardened version (described in Section 6) to introduce the efficacy of our method. We include in Figure 3 the average Critical SDC rate obtained with neutron beam experiments (for Original ViT and EVA2). The similarity of the rates obtained with beam and fault injection further strengthens the accuracy of our fault simulation. Figure 3 shows that single-bit flips injected at the instruction level do not provide a realistic evaluation for ViT models since most are masked and produce a Critical SDC rate close to zero, well off the rate obtained with beam experiments. Note that only 3% of single-bit flip fault injections resulted in values higher than 10 6 after the fault mask was applied to the target register. In contrast, when random values were used, 45% of the fault mask immediately produced values higher than 10 6 . Random and warp random values closely match the beam experiment results. Single particle corruption in shared memory or warp scheduler can cause faults leading to inf, NaN, and large values. If these faults spread to ViT structures, they may lead to Critical SDCs. Our hardening strategy is designed based on these observations.

MaxiMals

The previous sections established that inf, NaN, and large values pose a risk to ViT's reliability. We modified the Identity layers within the ViT Block to prevent the propagation of corrupted values that can generate Critical SDCs. Fig. 4a shows a standard ViT Encoder Block, while Fig. 4b shows the modified model structures needed to implement MaxiMals.

Using simple object-oriented programming techniques, the Max-iMals approach can be easily implemented for any ViT structure. We create a child class (HardenedIdentity) that extends the default Identity layer class. Replacing the default Identity object with the extended HardenedIdentity allows us to effortlessly harden 12 different models described in Table 1 without any compatibility problem. Then, we execute all the ViTs on the entire ImageNet validation dataset, store the minimum and maximum output values on the Identity layers, and use them as bounds to filter corrupted values. To avoid changing values that are lower/higher false positives, we multiply the profile values by 1.3. If the corrupted value is detected, it is replaced by the lowest or highest value in the case of ±inf, and 0 in the case of NaN. MaxiMals can be applied to any of the 120,000 models available on the HuggingFace library that uses the default PyTorch modules.

Identity layers neither perform any arithmetic operations nor have learnable parameters. Thus, the performance impact of the MaxiMals is proportional to the number of ViT Identity layers.

Experimental Validation

In this section, we present the validation of MaxiMals. Figure 5 depicts the SDC (Tolerable and Critical) and DUE PVFs for the 12 tested ViTs models, unhardened and hardened versions. The fault simulations are performed both on Pascal (Fig. 5a) and Ampere (Fig. 5b) GPUs. While the unhardened original ViT models exhibit the highest SDC PVF (on average, 38.97% on Pascal GPU and 34.20% on Ampere GPU), more complex ViTs like unhardened MaxViT effectively mask more faults. MaxViT has the lowest SDC PVF, on average 24.22% on Pascal GPU and 30.51% on Ampere GPU. However, despite the masking ability of more complex models, the Critical SDCs still propagate. For the unhardened models, the MaxViT and EVA2 show the highest Critical SDC PVF, on average, 16.05% for MaxViT and 14.35% for EVA2. Due to their distinct Transformers architecture, these models are especially susceptible to faults such as random and warp random values. MaxViT and EVA2 introduce many improvements, such as different normalization layers (e.g., sub-LN), additional connections for positional information injection, and convolution blocks before Attention layers. These additional modules facilitate the propagation of corrupted values, leading to an increase in Critical SDCs. In contrast, the SwinV2 model manifests the lowest Critical SDC PVF, on average, 2.40%. SwinV2's patches are organized using a "shifted window" that slides through the input, creating overlapped patches, which add more redundancies to the represented data, leading to a more reliable model.

MaxiMals reduces the Critical SDC, on average, from 8.50% to 3.28%. For the most complex models, EVA2 and MaxViT, our approach lowers the Critical SDC PVF, on average, to 1.65% and 8.54%, respectively. For EVA2 B14-448, the Critical SDCs are reduced by 90.70% (from 12.91% to 1.20%). The notable success of the MaxiMals technique in models like EVA2 and MaxViT can be attributed to the Ampere Pascal Ampere ViT 1.7%-3.5% 5.1%-10.0% 1.2%-2.0% 1.0%-1.7% EVA2 1.3%-5.3% 2.3%-9.4% 0.4%-1.5% 0.4%-1.8% SwinV2 0.1%-0.5% 0.3%-0.8% 0.1%-0.1% 0.1%-0.1% MaxViT 9.5%-10.7% 15.3%-16.0% 2.7%-2.7% 3.1%-3.1% number of Identity layers (necessary for their complex architecture), allowing the filtering of corrupted values at a higher frequency.

MaxiMals exhibits lower effectiveness in reducing the Critical SDCs for SwinV2. The average Critical SDC for SwinV2 models is reduced by 21.82%. SwinV2 possesses an intriguing characteristic of having few Identity layers compared to other models (only 5), lowering the effectiveness of our approach. This observation aligns with earlier studies that applied value restrictions to SwinV2 [START_REF] Gavarini | Evaluation and mitigation of faults affecting swin transformers[END_REF]. To implement value restriction on SwinV2 models, comprehensive modifications across all layers of the ViT would be necessary, incurring an impractical overhead of 68.61% [START_REF] Gavarini | Evaluation and mitigation of faults affecting swin transformers[END_REF].

Figure 3 illustrates the efficacy of MaxiMals approach in dealing with various fault models that generate Critical SDCs. Our proposed hardening approach is effective in these cases, as it reduces the Critical SDC, on average, of 65.87% for random value and 57.31% for warp random value. Note that our method can be further enhanced for models like SwinV2, where the Identity layers are less frequent. In the SwinV2 scenario, the Blocks can be selectively strengthened by increasing the frequency of value restriction operations based on their criticality without affecting performance.

Ultimately, the proposed MaxiMals has kept the DUE PVF on fault simulation the same for the unhardened and hardened models, on average, 2.55% and 2.48%, respectively.

Conclusions

We conducted an extensive analysis to understand how transient errors induced by neutrons can affect ViT models, potentially leading to Critical SDCs. Despite ViT's high accuracy, these models are notably resource-intensive and exhibit high SDC and DUE rates. Our comprehensive fault propagation analysis shows the impact of different types of faults on ViT models' classification accuracy. Our findings suggest that single-bit flip faults have minimal impact, while severe faults can significantly degrade accuracy. To address this, we developed a fault tolerance approach called MaxiMals, tailored explicitly to ViT models. Our approach reduces Critical SDCs and improves fault tolerance for complex ViT models.

1Figure 1 .

 1 Figure 1. P2000 (P) and A2000 (A) GPUs on ChipIR beamline.

Figure 2 .

 2 Figure 2. Pascal and Ampere GPUs experimentally measured Tolerable and Critical SDC, and DUE FIT rates for the ViTs.

Figure 3 .

 3 Figure 3. Average Critical SDC per fault model and ViT type.

Figure 4 .

 4 Figure 4. Unhardened and Hardened ViT Blocks.

•

  The ViT neutron-induced fault model characterization and how transient faults affect ViT kernels. • A detailed analysis of the fault propagation in ViT models based on fault simulation to unveil which faults affect ViT's ability to classify images correctly. • An efficient hardening technique with minimal model modifications and low execution time overhead.

Table 1 .

 1 ViTs' memory size, accuracy, execution times for Pascal (P.) and Ampere (A.) GPUs and profiled values.

		Size Acc. Time (ms) Value Range
		(MB) (%) P.	A.	min	max
		BS32-224 339 73.6 31	7	-32.8	37.6
	ViT [11]	B16-224 333 84.5 135 24 B16-384 340 85.4 461 72 L14-224 1164 87.9 488 106 -231.3 124.6 -35.1 63.5 -55.6 67.0
		H14-224 2479 88.2 1644 216 -83.4	90.9
	EVA2	B14-448 350 88.6 812 161 -904.2 483.7
	[1]	L14-448 1176 89.9 2686 509 -342.6 327.6
	SwinV2 [12]	B-256 372 86.2 182 53 B-384 431 87.1 579 172 -18.6 -18.6 L-256 787 86.9 404 92 -22.5	19.0 18.5 22.7
	MaxViT	L-384 845 87.9 938 282 -66806.8 35259.4
	[13]	L-512 856 88.0 1762 518 -83250.6 40806.9

Table 2 .

 2 Experimental data of ViT kernels in the neutron beam experiments for Pascal GPU.

					FIT Rate	Inf/NaN	Max val.
				SDC	DUE	(%)	diff
		MLP	22.2 ± 6.7 24.8 ± 7.1	0.0%	1.3 × 10 34
		Attention 13.9 ± 3.2 26.2 ± 4.4	10.5% 6.0 × 10 34
		Block	25.2 ± 3.6 39.14 ± 4.5	2.9%	1.0 × 10 37
						Unhardened	Hardened
		50%	Neutron Beam	Single Bit-Flip	Random Value	Warp Random Value
	SDC	40%
	Critical	30%
	Average	10% 20%
		0%		ViT		EVA2	SwinV2	Max ViT

Table 3

 3 shows the execution time and additional instructions overheads added by MaxiMals. Our method has a very low overhead in terms of execution time, on average, 5.61%. The worst case is the MaxViT models, which have 384 Identity layers, with a time over-

head of 16.09%. We also use NVIDIA profiling tools (Nvprof/Nsight Systems) to measure the GPU-executed instructions for each ViT model for more precise measurements. MaxiMals increases by up to 3.19% the number of executed instructions.

Table 3 .

 3 Added overheads for ViT model families (min-max)

	Execution Time	Instructions
	Pascal

Acknowledgment

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 899546 with the support of the Brittany Region and under the RADNEXT grant agreement No 101008126 [22]. This project is partially funded by ANR FASY (ANR-21-CE25-0008-01) and ANR RE-TRUSTING (ANR-21-CE24-0015-02). ChipIR and RADNEXT provided and supported neutron beam time experiments (DOI https://doi.org/10.5286/ISIS.E.RB2300036). We acknowledge the researchers Dr. Christopher Frost, Dr. Carlo Cazzaniga, and Dr. Maria Kastriotou, who helped with neutron experiments.