M Vanesa Moreno

Pierre Laurent

Philippe Ciais

Florent Mouillot

Assessing satellite-derived fire patches with functional diversity trait methods

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

Introduction

Fire is the source of approximately one third of aerosol, greenhouse gas and other trace gas emissions. This biomass burning leads to major effects on the abundance and diversity of vegetation, carbon and water cycling, and as a result, the overall climate system [START_REF] Andreae | Emission of trace gases and aerosols from biomass burning -an updated assessment[END_REF][START_REF] Andreae | Emission of trace gases and aerosols from biomass burning[END_REF][START_REF] Bowman | Fire in the Earth System[END_REF][START_REF] Van Der Werf | Global fire emissions and the contribution of deforestation, savanna, forest, agricultural, and peat fires (1997-2009)[END_REF]. Spatial and temporal monitoring of fire disturbances is thus essential to benchmark the fire models used to assess fire-climate feedbacks and the impact of fires on climate and the carbon cycle. Indeed, fire disturbance is one of the Essential Climate Variables (ECVs) that is listed by the Global Climate Observation System (GCOS) as a critical component in the climate system [START_REF] Wmo | The Global Observing System for Climate: Implementation Needs 200[END_REF].

Out of the different satellite-derived fire disturbance products, burned areas are the most commonly used for the development and evaluation of fire modules embedded in Dynamic Global Vegetation Models (DGVMs) (Bistinas et al., 2014;[START_REF] Hantson | The status and challenge of global fire modelling[END_REF]Mouillot et al., 2014). However, burned-area products alone do not allow a process-oriented development of fire modules. For instance, a bias of ignition numbers can compensate a bias of fire size in a model, and still match observed burned area (e.g., [START_REF] Yue | Modelling the role of fires in the terrestrial carbon balance by incorporating SPITFIRE into the global vegetation model ORCHIDEE -Part 1: Simulating historical global burned area and fire regimes[END_REF].

One critical aspect of fire modules not yet evaluated on a systematic basis is how they simulate the spread of fires. Many models simulate fire spread based on Rothermel's equation [START_REF] Rothermel | A mathematical model for predicting fire spread in wildland fuels[END_REF]. This equation assumes that fires form an ellipse of burned area centred on each ignition point -it assumes homogeneous vegetation and topography, and a constant wind direction [START_REF] Rabin | The Fire Modeling Intercomparison Project (FireMIP), phase 1: Experimental and analytical protocols with detailed model descriptions[END_REF].

Fire-patch products have emerged from pixel-level burned-area products. These fire-patch products work on the principle that contiguous burned pixels occurring during the same period of time can be aggregated into a single patch. The critical parameter in those algorithms is called the cut-off (in days), defined as the maximum time period below which consecutive neighbouring burned pixels can be grouped into the same patch. For example, at the continental or local scale, [START_REF] Archibald | Identifying individual fires from satellite-derived burned area data[END_REF], and [START_REF] Nogueira | Can we go beyond burned area in the assessment of global remote sensing products with fire patch metrics[END_REF] used a cut-off of 8 days in Africa and central Brazil respectively, while [START_REF] Fusco | Detection rates and biases of fire observations from MODIS and agency reports in the conterminous United States[END_REF] used a cut-off of 9 days in the United States. At the global scale, [START_REF] Hantson | Global fire size distribution is driven by human impact and climate[END_REF] used a cutoff of 14 days, while Laurent et al. (2018a) tested cut-offs of 3, 5, 9, and 14 days to produce the global FRY v1.0 fire-patch functional traits database. A source of uncertainty in the determination of the cut-off parameter arises from the fact that burned areas derived from low Earth orbit satellite observations are not continuous in time. For example, the presence of clouds may mask burned-area pixels that occur between two consecutive observations with clear-sky conditions as implemented in new algorithms accounting for uncertainty in the date-of-burning [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF][START_REF] Campagnolo | A patch-based algorithm for global and daily burned area mapping[END_REF][START_REF] Oom | Highlighting biome-specific sensitivity of fire size distributions to time-gap parameter using a new algorithm for fire event individuation[END_REF].

Geostationary satellites might be better, but they are still affected by persistent clouds. The arbitrary choice of a cut-off value in fire-patch products and the current lack of assessment of their uncertainties [START_REF] Boschetti | Global validation of the collection 6 MODIS burned area product[END_REF][START_REF] Padilla | Validation of the 2008 MODIS-MCD45 global burned area product using stratified random sampling[END_REF] hinders the use of these products for quantitative evaluation of fire models.

Assessment of uncertainties in satellite-derived ECVs is essential if they are to be used effectively in climate modelling [START_REF] Wmo | The Global Observing System for Climate: Implementation Needs 200[END_REF]. Two assessments (validation and intercomparison) are required by GCOS: where validation is the process of assessing the accuracy of satellite-derived products using independent reference data [START_REF] Boschetti | A sampling method for the retrospective validation of global burned area products[END_REF]. Based on the GCOS scientific and technical requirements, the Land Product Validation (LPV) subgroup of the Committee on Earth Observation Satellites (CEOS) Working Group on Calibration and Validation (WGCV) is developing protocols for the assessment of uncertainties in products based on satellite observations at five different validation stages [START_REF] Boschetti | International Global Burned Area Satellite Product Validation Protocol. Part I -Production and standardization of validation reference data[END_REF]. Considerable effort has been put into the validation of moderate spatial resolution burned-area products at the global scale, reaching CEOS validation stage 3 [START_REF] Padilla | Validation of the 2008 MODIS-MCD45 global burned area product using stratified random sampling[END_REF]. The validation relies on the acknowledged cross-tabulation approach of quantifying omission and commission errors in addition to the overall accuracy using high-resolution imagery as reference data. [START_REF] Boschetti | A sampling method for the retrospective validation of global burned area products[END_REF].

However, the evaluation does not cover fire-patch morphology, which is needed to model fire spread processes. Fire spread can be subject greater error than is found in burned area, because pixels may be wrongly committed or omitted particularly at the fire-patch edge [START_REF] Humber | Assessing the Shape Accuracy of Coarse Resolution Burned Area Identifications[END_REF].

At the local scale, in central Brazil, [START_REF] Nogueira | Can we go beyond burned area in the assessment of global remote sensing products with fire patch metrics[END_REF] assessed overlapping fire patches using high-resolution Landsat imagery as reference data and estimated patch-to-patch correlations of their morphological features. In a similar approach, [START_REF] Chuvieco | Generation and analysis of a new global burned area product based on MODIS 250 m reflectance bands and thermal anomalies[END_REF][START_REF] Chuvieco | A new global burned area product for climate assessment of fire impacts[END_REF]) used forest service perimeters, while [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF] used high-resolution imagery and fire duration by combining perimeters of fire patches using VIIRS active fire detection. [START_REF] Fusco | Detection rates and biases of fire observations from MODIS and agency reports in the conterminous United States[END_REF] assessed the positive and negative rates of detection between fire patches and forest service reports in the United States. In this study, we propose an original statistical framework for fire ecology, to probe the spatial conservation of fire-patch morphology across sensors.

We evaluated the fire-patch functional traits products provided in the global FRY v1.0 database described by Laurent et al. (2018a). This database allows series of data to be calculated with different cut-offs of 3, 6, 9, and 14 days from two global burned-area products: MERIS Fire_cci version 4. 1 (FireCCI41;[START_REF] Chuvieco | A new global burned area product for climate assessment of fire impacts[END_REF] and MODIS MCD64A1 collection 6 (MCD64A1; [START_REF] Giglio | The Collection 6 MODIS burned area mapping algorithm and product[END_REF]. The MERIS and MODIS sensors have different temporal (~1 day with MODIS to 3 days with MERIS) and spatial (~300 m with MERIS to 500 m with MODIS) resolutions. These contrasting temporal and spatial resolutions allowed us to examine whether a higher temporal resolution or a higher spatial resolution leads to the best evaluation performance, and to specify future requests for new global burned-area developments. This issue is still open because Laurent et al. (2018a) performed only an intercomparison between the two sensors -i.e., they performed a CEOS validation stage 0, but no evaluation.

To assess whether the FRY products (Laurent et al., 2018a) provide realistic fire-patch traits, we applied a new functional diversity trait-based approach widely used in species assemblage comparative studies [START_REF] Mouillot | A functional approach reveals community responses to disturbances[END_REF][START_REF] Villéger | Decomposing functional β-diversity reveals that low functional β-diversity is driven by low functional turnover in European fish assemblages[END_REF][START_REF] Villéger | New multidimensional functional diversity indices for a multifaceted framework in functional ecology[END_REF]. The approach compares the functional space built on morphological traits of individual fire patches, of assemblages of fire patches ≥ 300 ha from the two FRY products with cut-offs of 3, 5, 9, and 14 days to reference data derived from forest service perimeters (Canadian Forest Service, 2014;U.S. Geological Survey, 2017) in North America during the period 2005-2011. This method is now considered as a statistical reference framework in community ecology; we tested its efficiency when applied to fire-patch assemblages defined by their morphological traits. We examined whether the FRY-FireCCI41 product based on the MERIS sensor with a lower temporal resolution and a higher spatial resolution is better or worse than the FRY-MCD64A1 based on the MODIS sensor regarding fire-patch diversity; and what are the most realistic cut-off parameter values to use in different subregions of North America.

Methods

Data

FRY v1.0 database

FRY is a global database of fire patches, defined by their morphological features, also known as their functional traits. These features are established from two satellite-derived pixel-level burned-area products and computed with four cut-off values (Laurent et al., 2018a[START_REF] Laurent | FRY: a global database of fire patch functional traits[END_REF]. The two FRY products correspond to the processing of global burned-area products from FireCCI41 and MCD64A1. Both burned-area products overlapped during the 2005-2011 time period. MCD64A1 [START_REF] Giglio | The Collection 6 MODIS burned area mapping algorithm and product[END_REF] is based on the MODIS sensor aboard the Terra and Aqua satellites, with a spatial resolution of 500 m and a temporal resolution of one day. It was created by hybrid algorithms that combine thermal information from the MODIS MCD14A1 collection 6 active fire product [START_REF] Giglio | The collection 6 MODIS active fire detection algorithm and fire products[END_REF] and post-fire reflectance. FireCCI41 [START_REF] Chuvieco | A new global burned area product for climate assessment of fire impacts[END_REF] uses the post-fire reflectance from the MERIS sensor, aboard the Envisat satellite with a spatial resolution of 300 m at nadir and a temporal resolution of three days.

The flood-fill algorithm used to determine the FRY patches aggregates neighbouring burned pixels with differences in the date-of-burning below a given time threshold, defined as cut-off values of 3, 5, 9, and 14 days. If the time elapsed between the date-of-burning of two neighbouring pixels exceeds the cutoff value, a new fire patch is created (Laurent et al., 2018a). FRY also aims to provide functional fire-patch traits as morphological descriptors relevant to the underlying spreading processes (Laurent et al., 2018a). For example, the traits include size and geometric features such as morphological complexity, as well as the elongation and direction of an ellipse fitted to each fire patch. We considered only fire patches ≥ 300 ha, because of the inaccurate morphologies of smaller patches [START_REF] Nogueira | Can we go beyond burned area in the assessment of global remote sensing products with fire patch metrics[END_REF], and higher omission and commission errors for small fires in burned-area products both globally [START_REF] Padilla | Comparing the accuracies of remote sensing global burned area products using stratified random sampling and estimation[END_REF][START_REF] Randerson | Global burned area and biomass burning emissions from small fires[END_REF][START_REF] Roteta | Development of a Sentinel-2 burned area algorithm: Generation of a small fire database for sub-Saharan Africa[END_REF], and in North America [START_REF] Sparks | An accuracy assessment of the MTBS burned area product for shrub-steppe fires in the northern Great Basin, United States[END_REF].

Reference data

As reference data, we used fire patches from the Canadian National Fire Database (CNFD; Canadian Forest Service, 2014) and the Wildland Fire Perimeters Database (WFPD; U.S. Geological Survey, 2017) recorded during the 2005-2011 period coincident with the FRY products. CNFD and WFPD provide homogeneous high spatial resolution daily fire-patch data collected by different agencies from ground measurements, aerial products or highresolution satellite sensors. They have been used previously as reference data for validation exercises based on their standard data assemblage protocol over North America [START_REF] Chuvieco | Generation and analysis of a new global burned area product based on MODIS 250 m reflectance bands and thermal anomalies[END_REF]. We checked both CNFD and WFPD fire patches to remove fire duplicates at interstate borders or the border between the United States and Canada, and those recorded by different agencies within the same country. Similarly, we also removed fire patches from WFPD empirically drawn as circles around active fires identified by MODIS sensors. We also grouped into a common patch some fires that had been split into spatially distinct patches in these databases; this process makes them compatible with the way FRY treats patches. To create the reference data, we rasterized CNFD and WFPD fire patches ≥ 300 ha to the spatial resolutions of each FRY product (i.e., 300 m and 500 m), and derived their functional traits following the same approach as in FRY (Laurent et al., 2018a).

Functional diversity trait-based approach

To assess the agreement between the FRY products and the reference data, we adopted the functional diversity trait-based statistical framework proposed by [START_REF] Mouillot | A functional approach reveals community responses to disturbances[END_REF] and [START_REF] Villéger | Decomposing functional β-diversity reveals that low functional β-diversity is driven by low functional turnover in European fish assemblages[END_REF][START_REF] Villéger | New multidimensional functional diversity indices for a multifaceted framework in functional ecology[END_REF]. This approach was initially developed in functional ecology to compare ecological communities (assemblages of individuals defined by their functional traits and abundances) based on functional traits (i.e., characteristics of organisms related by their functions, e.g., plant height for light interception; [START_REF] Garnier | Plant Functional Diversity, Plant Functional Diversity[END_REF]. We applied this framework to compare fire-patch products by considering each fire-patch product as a community and each fire patch as an individual defined by its morphological traits. This approach complements the usual omission and commission errors performed for accuracy assessment. Low commission and omission errors ensure most burned pixels are identified, and in turn, that fire patched overlap between products. Comparing fire-patch morphology reinforces this information by quantifying how the remaining small omission and commission errors affect burned pixels at the fire-patch boundary and in turn modify fire shapes. The approach includes four steps (Fig. 1). (2013) and [START_REF] Villéger | Decomposing functional β-diversity reveals that low functional β-diversity is driven by low functional turnover in European fish assemblages[END_REF][START_REF] Villéger | New multidimensional functional diversity indices for a multifaceted framework in functional ecology[END_REF] to assess satellite-derived fire patches in four steps.

Functional trait (Ftrait) is a morphological feature of the fire patch. Functional diversity indices (FDIs) include functional fire-patch number (Ffp), functional richness (Fric) and functional dispersion (Fdis). Continental scale tile-to-tile Standardized Major Axis (SMA) regression of FDIs, and tile-level functional beta diversity (Fβ), functional turnover (Fturn) and functional nestedness (Fnest) were used in the assessment. For a simple graphical display, the first step shows tiles from only one product and the reference data. Similarly, the second, third and fourth steps show a functional space. This space has only two functional traits in one tile; in addition, the third step-Fric and the fourth steps show convex hull volumes as a circle. A Venn diagram illustrates these cases.

First step: sample the fire-patch products

To sample the fire-patch products by 2° tiles across the continent, we grouped the list of fire patches belonging to a set of 10 products according to their central geographic coordinates into 2° x 2° spatial tiles (Fig. 1; First step). This tile resolution allows us to get tiles with more than three fire patches in each product and optimizes computing time. This method has previously been used for global analysis [START_REF] Hantson | Global fire size distribution is driven by human impact and climate[END_REF][START_REF] Laurent | Varying relationships between fire radiative power and fire size at a global scale[END_REF]. We assigned the geographic coordinates of each tile (bottom left corner) over the entire sample area as the tile ID name. Similarly, we used the cut-off values from each FRY product and the spatial resolution from each reference data set as the product ID name.

Second step: build the functional space

To build the functional space of the fire patches for each satellite-derived product and cut-off value within each tile, we selected the functional traits of size, elongation, and direction, from the FRY database as the three essential non-correlated fire spread traits, which are independent of pixel resolution (Fig. 1; Second step). Before building the multidimensional Euclidean space, we normalized the functional traits of size and elongation applying the Box-Cox method using the 'boxcoxfit' R function from the 'geoR' R library [START_REF] Ribeiro | geoR: Analysis of Geostatistical Data[END_REF]. For size, the function became the reciprocal square root transformation, and for elongation, the function became the exponential transformation.

Likewise, functional traits were overall standardized to generate a trait distribution with a mean that equalled 0 and a variance that equalled 1 using the 'decostand' R function from the 'vegan' R library [START_REF] Oksanen | vegan: Community Ecology Package, R package version 2.4-0[END_REF]. Standardizing implies that all traits are considered equally important for the analysis and makes functional spaces comparable within and across products. The three continuous transformed and scaled functional traits were suitable for building the functional space without using ordination methods to reduce or synthesize axes, as is required when using more than three dimensions [START_REF] Maire | How many dimensions are needed to accurately assess functional diversity? A pragmatic approach for assessing the quality of functional spaces[END_REF].

Then, we located each fire patch according to its functional trait values into the functional space built on the three selected functional trait axes. The method then compares the FRY products with the reference data based on the dissimilarities between their functional volumes.

Third step: assessment of the functional structure

To assess the functional structure of the fire-patch products, defined as the distribution of individual fires in the functional space built within each tile and for each product, we first computed three functional diversity indices:

• Functional fire-patch index is the number of patches per tile (Fig. 1; Third step-Ffp).

Accurately conserving functional fire patches from satellite-derived products means that the pixel aggregation method accurately represents the fractionation of total burned area into patch number. It complements the fire size distribution index widely used in global (e.g., [START_REF] Hantson | Global fire size distribution is driven by human impact and climate[END_REF]Laurent et al., 2018a) and regional studies (e.g., [START_REF] Malamud | Forest Fires: An Example of Self-Organized Critical Behavior[END_REF][START_REF] Moreno | Wildfire frequency-area statistics in Spain[END_REF].

• The functional richness is the convex hull volume in the functional space driven by the range of extreme values of patch traits in a product (Fig. 1; Third step-Fric).

Differences in functional richness between the products correspond to differences in the convex hull volume upon gathering all the fire patches belonging to the product.

Conserving functional richness from satellite-derived products means that convex hull volumes driven by extreme values on three traits (size, elongation and direction) are accurately represented -and independently of the patch morphological features inside this convex hull volume.

• Functional dispersion is the mean distance of individual fire patches to the average position of all the fire patches in the multidimensional functional space (Fig. 1; Third step-Fdis). The differences in functional dispersion represent differences in the deviation of fire-patch trait values from the centre of the functional space filled by the product. Conserving functional dispersion from satellite-derived products means that the distribution of individual fire-patch traits within the functional space is accurately represented, so we can expect same fire-patch morphologies beside the extreme values as tested with functional richness.

The values of functional richness range between 0 and 1 because it is the proportion of the functional space filled by the fire patches present in a product compared to the functional space filled by all the fire patches present in all the products to be evaluated. Similarly, functional dispersion ranges between 0 and 1 because it is scaled by the maximum value possible considering all the fire patches present in a product. We computed both indices using the 'dbFD' R function from the 'FD' R library [START_REF] Laliberté | FD: measuring functional diversity from multiple traits, and other tools for functional ecology[END_REF][START_REF] Laliberte | A distance-based framework for measuring functional diversity from multiple traits[END_REF].

We then regressed across tiles the functional diversity indices computed from each FRY product against the ones computed from the reference data using the standardized major axis (SMA) method [START_REF] Warton | Bivariate line-fitting methods for allometry[END_REF] to assess their spatial extent. The aim here is to test whether the continental pattern of fire diversity within tiles is conserved across the products; and thus if satellite-derived pyrodiversity indices are robust indicators of fire regimes. SMA error is measured perpendicular to the SMA line -not along the y-axis as in ordinary least squares regression. It is the appropriate method for summarising a bivariate relationship between products, given uncertainties in both global products [START_REF] Boschetti | Global validation of the collection 6 MODIS burned area product[END_REF][START_REF] Padilla | Validation of the 2008 MODIS-MCD45 global burned area product using stratified random sampling[END_REF] and forest service perimeters (e.g., [START_REF] Short | Sources and implications of bias and uncertainty in a century of US wildfire activity data[END_REF][START_REF] Short | A spatial database of wildfires in the United States, 1992-2011[END_REF].

Bivariate SMA lines with a squared correlation coefficient (r 2 ) that were significant at the 95% level were inferred, assuming x and y observations are independent. The inference consisted of testing if the slope equals 1 and the intercept equals 0 at the 95% significance level. SMA lines were estimated and inferred using the 'sma' R function from the 'smatr' R library [START_REF] Warton | smatr 3-an R package for estimation and inference about allometric lines[END_REF] for each functional diversity index, of which functional fire-patch number and dispersion were log10-transformed for an approximately normal distribution.

Two additional SMA lines were estimated and inferred in the boreal and temperate regions defined in the Global Fire Emission Database (GFED; [START_REF] Giglio | Analysis of daily, monthly, and annual burned area using the fourth-generation global fire emissions database (GFED4)[END_REF] across North America to further assess the relationships by region with potentially contrasting results because of different longitudes and fire types.

We finally qualified the SMA lines to find the line of best fit following a statistical criterion proposed previously to assess simulations [START_REF] Mesplé | Evaluation of simple statistical criteria to qualify a simulation[END_REF] with five possible outcomes, ordered by increasing bias: an unbiased relationship (slope ≈ 1, intercept ≈ 0) when two products entirely agree (Fig. 1; Third step-a); a constantly biased relationship (slope ≈ 1, intercept ≠ 0) when one product is systematically overestimating or underestimating the other (Fig. 1; Third step-b); a proportionally biased relationship (slope ≠ 1, intercept ≈ 0) when one product gives values that are higher or lower than those from the other by an amount that is proportional to the value of the observations (Fig. 1; Third step-c); and an additional outcome, the combination of both a constantly and proportionally biased relationship (slope ≠ 1, intercept ≠ 0; Fig. 1; Third step-d). We graded the lines of best fit according to r 2 when several lines fall into the same outcome (Fig. 1; Third step-e). Finally, no significant relationship occurs when the correlation is not significant (p > 0.05).

Fourth step: assessment of the functional dissimilarity

As a final step, we assessed the functional dissimilarity between fire-patch products within tiles to capture whether or not two convex hull volumes entirely overlap (best agreement with similar trait value range). We estimated and decomposed the functional beta diversity into its two components of turnover and nestedness [START_REF] Villéger | Decomposing functional β-diversity reveals that low functional β-diversity is driven by low functional turnover in European fish assemblages[END_REF]; Figure 1; Fourth step):

• Functional beta diversity is the dissimilarity in convex hull volumes in the fire-patch functional space between two products. It is the sum of functional turnover and functional nestedness. Functional beta diversity equals 0 for perfect overlap (Fig. 1;

Fourth step-a) and equals 1 for a total mismatch. Between these two extremes, different values of functional beta diversity result from different values of functional turnover and functional nestedness (Fig. 1; Fourth step-e and f).

• Functional turnover is the proportion of convex hull volumes in the functional space not shared between two products, driven by differences in trait values. When functional beta diversity equals 0, functional turnover also equals 0 because the two volumes perfectly overlap. When functional beta diversity equals 1, functional turnover also equals 1, if the two similar convex hull volumes are in total mismatch (Fig. 1; Fourth step-b and c), or equals 0 if one volume is included inside the other one.

• Functional nestedness is the proportion of convex hull volumes in the functional space shared between two products, driven by differences in functional richness.

Functional nestedness is the difference between functional beta diversity and functional turnover. When functional beta diversity equals 1, functional nestedness also equals 1 if one volume fills only a small portion of the other (Fig. 1; Fourth stepd).

We computed functional beta diversity and its two components using the all.intersect R function from the 'rcdd' R library [START_REF] Geyer | rcdd: Computational Geometry[END_REF]. Differences in functional dissimilarities across cut-off values and within cut-off values were estimated using ANOVA analysis and means were compared using Fisher's least significant difference test, both of them at the 95% significance level -these are available using the 'aov' R function from 'stats' R library (R Core Team, 2017) and the 'LSD.test' R function from 'agricolae' R library [START_REF] De Mendiburu | Agricolae: Statistical procedures for agricultural research. R package version 1[END_REF].

Results

We show below how indices of fire-patch morphological trait diversity are conserved between the satellite-derived products and the reference data across 141 tiles in North America (mainland United State and Canada), including 68 tiles (48%) in Boreal North America (BONA) and 73 tiles (52%) in Temperate North America (TENA).

Functional structure

When we compared the functional diversity indices across tiles in North America between the FRY products and the reference data, we found that the functional structure in the FRY-MCD64A1 product was more correlated with the reference data and had smaller bias than the FRY-FireCCI41 product (Figs. 2 and3).

Functional fire-patch number relationships revealed significant correlations between the FRY products and reference data across North America, BONA and TENA, with all the cut-off values. Correlations with the reference data were stronger in the FRY-MCD64A1 product than in the FRY-FireCCI41 product, particularly in BONA. In the FRY-MCD64A1 product r 2 varied from 0.21 to 0.45 (p < 0.001) whereas in the FRY-FireCCI41 product, r 2 varied from 0.20 to 0.33 (p < 0.001). Such relationships were all proportionally biased across North America in both FRY products, but unbiased in both BONA and TENA in the FRY-MCD64A1 product, and unbiased only in TENA in the FRY-FireCCI41 product. In both FRY products, correlations increased with increasing cut-off values across North America and BONA, but decreased slightly with increasing cut-off values in TENA. Based on the higher r 2 (p < 0.001), the line of best fit between both FRY products and the reference data was with a cut-off of 14 days across North America (r 2 = 0.28 and r 2 = 0.31 for the FRY-FireCCI41 and FRY-MCD64A1 products respectively) and BONA (r 2 = 0.33 and r 2 = 0.45 for the FRY-FireCCI41 and FRY-MCD64A1 products respectively), whereas best fit was with a cut-off of 3 days in TENA (r 2 = 0.23 and r 2 = 0.29 for the FRY-FireCCI41 and FRY-MCD64A1 products respectively) as well as with cut-offs of 5 and 9 days in FRY-FireCCI41. In turn, fire-patch number is well conserved in both FRY products with high cut-off values across North America and BONA, yet small cut-off values in TENA. Overall, better performance was found with the FRY-MCD64A1 product.

Continental functional richness relationships highlighted the differences between the FRY products and reference data in the chosen 3D trait space. In the FRY-FireCCI41 product, r 2 varied from 0.04 to 0.03 (p < 0.05), and such weak relationships with the reference data were significant only with cut-offs of 9 and 14 days across North America. In the FRY-MCD64A1 product, r 2 varied from 0.08 to 0.19 (p < 0.01) and was significant with all cut-offs across North America and TENA, and with cut-offs of above 5 days in BONA. In both FRY products, all the relationships with the reference data were constantly biased, and the correlations were stronger in the FRY-MCD64A1 product than in the FRY-FireCCI4 product.

Additionally, the correlations increased with increasing cut-off values across North America and BONA but decreased slightly with increasing cut-off values in TENA. Based on the higher r 2 (p < 0.001), the line of best fit between the FRY-MCD64A1 product and the reference data was with a cut-off of 14 days across North America (r 2 = 0.19) and with cutoffs of both 9 and 14 days in BONA (r 2 = 0.15), and with a cut-off of 3 days in TENA (r 2 = 0.15). Whereas the line of best fit between the FRY-FireCCI41 product and reference data was with a cut-off of 14 days across North America (r 2 = 0.04). We conclude here that the 3D volumes, driven by extreme values of fire size, elongation and direction, are less spatially conserved than fire-patch number but still significant with better performances by the FRY-MCD64A1 product with high cut-off values across North America and BONA as well as small cut-off values in TENA.

Finally, functional dispersion relationships between the FRY products and the reference data highlighted an additional better performance by the FRY-MCD64A1 product over FRY-FireCCI41. In the FRY-FireCCI41 product, r 2 varied from 0.06 to 0.08 (p < 0.05), and such relationships with the reference data were significantly correlated only with high cut-off values in TENA. With the FRY-MCD64A1 product, values of r 2 were slightly higher, and varied from 0.05 to 0.12 (p < 0.05). They were significant for all the cut-off values across North America and BONA. In both FRY products, all the relationships with the reference data were likewise unbiased. Furthermore, correlations increased with increasing cut-off values across North America and BONA, but decreased slightly with increasing cut-off values in TENA. Based on the higher r 2 (p < 0.001), the line of best fit between the FRY-MCD64A1 product and the reference data was with a cut-off of 14 days across North America (r 2 = 0.07), and with cut-offs of 9 and 14 days in BONA (r 2 = 0.12). However, the line of best fit between the FRY-FireCCI41 product and reference data was with a cut-off of 9 days in TENA (r 2 = 0.08). We therefore conclude that the distribution of fire patches within the 3D volumes conserves less information in the FRY products, but they are still significant and the overall performance is better in the FRY-MCD64A1 product with high cut-off values. to the number of fire patches (Ffp), middle column to functional richness (Fric) and the right column to functional dispersion (Fdis). The upper panel shows the Standardized Major Axis (SMA) line statistics. The squared correlation coefficient is represented by a rectangle and a filled rectangle when it is significant (p < 0.05). In addition, for every significant squared correlation coefficient, the SMA slope represented by a triangle and a filled triangle when it is significant (p > 0.05), SMA intercept represented by a point and a filled point when it is significant (p > 0.05), and confidence intervals at the 95% level are provided. The lower panel shows the scatter plots with SMA lines for every significant squared correlation coefficient (p < 0.05). The horizontal axis presents the functional diversity indices in the reference data and the vertical axis in the FRY-MCD64A1 cut-off values. Both axes are in log10-scale for Ffp and Fdis. Each point is a 2° tile containing multiple patches.

Functional dissimilarity

We compared the overlap of functional spaces within tiles between FRY products and the reference data, as a complementary indicator of functional similarities. We found that the functional dissimilarities were lower between the FRY-MCD64A1 product and reference data than between the FRY-FireCCI41 product and reference data (Figs. 4, 5, 6 and 7 and Table S4). This result confirms the better performance of the FRY-MCD64A1 product in conserving fire-patch morphology with more overlapping functional spaces.

Functional beta diversity in the FRY-MCD64A1 product (beta diversity from 0.22 to 0.96) was lower than in the FRY-FireCCI41 product (beta diversity from 0.24 to 0.97). That means a higher overlap between functional spaces between the reference data and the FRY-MCD64A1 product than between the reference data and the FRY-FireCCI41 product. In both FRY products, the functional beta diversity was on average higher in BONA than in TENA and significantly different (p < 0.05), indicating lower agreement with reference data in BONA. For example, the overlap between functional spaces in California (Figs. 5 and7 in columns labelled California) on the 3 traits was higher than in Quebec (e.g., Figs. 5 and 7 in columns labelled Quebec), particularly on the size axis when related to the fire-patch elongation. We also found the functional beta diversity decreased with increasing cut-off values in both FRY products across North America and BONA and TENA (Figs. 4 and6).

That means an increase in the overlap between functional spaces. However, only in the FRY-FireCCI41 product, was functional beta diversity derived from fire patches with a cut-off of 3 days significantly lower (p < 0.05) than that with a cut-off of 14 days across North America and in BONA.

In both FRY products, functional beta diversity was mostly driven by functional turnover, indicating a low overlap of functional spaces between the FRY products and reference data; this was due to a translation of functional spaces rather than the inclusion of a smaller one into the other. We observed that functional turnover decreased with increasing cut-off values, indicating higher overlap between functional spaces, and in turn, a better agreement between functional spaces of fire patches generated with high cut-off values and functional spaces generated with reference data. Furthermore, functional turnover was on average also significantly (p < 0.05) higher in BONA than in TENA, thereby indicating lower agreement with reference data in BONA. For example, on the size axis when related to the elongation in Quebec (e.g., Figs. 5 and 7 in columns labelled Quebec) there is a tendency for translation in functional spaces, especially in the FRY-FireCCI41 product. This tile in Quebec in the FRY-MCD64A1 product is also a clear example of how functional dissimilarities complement functional diversity indices. Here the two assemblages have similar functional volumes and individual distribution in this volume but translated, thus functional spaces do not overlap.

Functional nestedness varied slightly with cut-off values across North America but decreased with increasing cut-off values in BONA and increased with increasing cut-off values in TENA. This occured in both FRY products, indicating the inclusion of one functional space into the other and in agreement with decreasing turnover. While in the FRY-FireCCI41 product functional nestedness on average was higher in TENA because of differences in functional richness between the product and reference data; in the FRY-MCD64A1 product, it varied slightly between regions. However, none of those differences were significant. For example, in Florida (e.g., Figs. 5 and 7 in columns labelled Florida), there are considerable differences in functional spaces due to their different volumes leading to the inclusion of the smaller volume of the reference data and thus a high nestedness.

Discussion

A new statistical framework

We evaluated the conservation of fire-patch morphology delivered in the satellite-derived FRY database against reference data using a new functional trait-based approach. The approach is a generic and synthetic surrogate to patch-patch regressions on morphological traits [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF][START_REF] Chuvieco | Generation and analysis of a new global burned area product based on MODIS 250 m reflectance bands and thermal anomalies[END_REF][START_REF] Chuvieco | A new global burned area product for climate assessment of fire impacts[END_REF][START_REF] Nogueira | Can we go beyond burned area in the assessment of global remote sensing products with fire patch metrics[END_REF], and goes beyond classical omission and commission errors [START_REF] Fusco | Detection rates and biases of fire observations from MODIS and agency reports in the conterminous United States[END_REF] used previously in comparative pyrogeography.

Our approach has the advantage of assessing fire-patch accuracy using a synthetic view of traits. Although here we built a functional space from three traits, more traits could be included using ordination methods to reduce the data [START_REF] Maire | How many dimensions are needed to accurately assess functional diversity? A pragmatic approach for assessing the quality of functional spaces[END_REF]. However, because the approach is sensitive to the addition of traits as well as to their correlations [START_REF] Zhu | Trait choice profoundly affected the ecological conclusions drawn from functional diversity measures[END_REF], it is, critical to make conscious decisions when selecting the traits [START_REF] Gitay | What are functional types and how should we seek them?[END_REF]). Here we selected three non-correlated essential traits related to fire spread.

We did not select functional traits of morphological complexity (e.g., perimeter/area ratio, fractal dimension) as they are highly dependent on resolution.

The functional trait-based approach enables an assessment of satellite-derived fire patches both across and within tiles, complementing each other. Across 2° tiles, it allows the assessment of the spatial conservation of functional fire-patch diversity measured by indices, such as functional fire patches, richness and dispersion between the two different global satellite sensor products and the reference data. Within tiles, it allows the assessment of the dissimilarities of functional diversity of fire-patch assemblages and detect where mismatching occurs. Within tiles, we could also detect whether beta diversity was driven by functional trait value differences or by richness differences. Although it is not a direct outcome, it also allows us to explore which functional traits drive dissimilarities within tiles by visual checking (e.g., Figs. 5 and7).

In addition to fire patches, the approach proposed can be applied to assess emerging satellite-derived trait products such as Essential Biodiversity Variables (EBVs; [START_REF] Kissling | Towards global data products of Essential Biodiversity Variables on species traits[END_REF][START_REF] Pereira | Essential biodiversity variables[END_REF]. The approach is also suitable for large-and small-scale [START_REF] Carmona | Traits Without Borders: Integrating Functional Diversity Across Scales[END_REF] pyrogeography studies such as the recent study of Africa by [START_REF] Hempson | Continent-level drivers of African pyrodiversity[END_REF]. The availability of fire-patch traits (Laurent et al., 2018a), is opening up a new field of research on functional pyrogeography, by linking biogeography and landscape ecology indices within a functional ecology statistical framework.

Optimal cut-off values for fire-patch morphology

The primary outcome of our analysis is that satellite-derived fire-patch morphology conservation varied with the cut-off value. Out of the four cut-off values assessed, high cutoff values generated fire patches with morphologies closest to the reference data in North America and particularly in BONA (Figs. 2, 3, 4 and 6, and Table S4). However, in TENA, the correlations slightly decreased with increasing cut-off values, but differences were not significant. Still, the unexpected decrease of correlation with increasing cut-off values in TENA in the FRY products may be because small fires, such as sugarcane fires in Florida [START_REF] Mccarty | The spatial and temporal distribution of crop residue burning in the contiguous United States[END_REF] mistakenly reached the threshold of 300 ha with increasing cut-off values (Figs. S1, S3, S5, and S7, and Tables S1 andS2), but were unreported in the reference data (Figs. S9, S11, S13 and S14, and Table S3). Additionally, the omission and commission errors of satellite-derived burned-area products [START_REF] Boschetti | Global validation of the collection 6 MODIS burned area product[END_REF][START_REF] Padilla | Validation of the 2008 MODIS-MCD45 global burned area product using stratified random sampling[END_REF][START_REF] Randerson | Global burned area and biomass burning emissions from small fires[END_REF][START_REF] Roteta | Development of a Sentinel-2 burned area algorithm: Generation of a small fire database for sub-Saharan Africa[END_REF], together with the uncertainties in the date-of-burning, can cause fragmentation of large fires leading to more fire patches (Figs. S13, S14, S15 and S16), and thus weak structural relationships between the FRY products and reference data. Furthermore, the functional dissimilarities were lower on average with a cut-off of 14 days across North America and in BONA and TENA (Figs. 4 and6, and Table S4). Previous observations led to an estimated cut-off of 10 days in the same regions, which was the highest of the assessed values (4, 6, 8, and 10 days; [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF], and is in agreement with our results.

Fire-patch conservation also varied with region. For a given cut-off value, functional dissimilarities of FRY products with reference data were higher in BONA than TENA.

Interestingly, the spatial relationships on functional structure (Figs. 2 and3) were stronger in BONA than in TENA. That means that despite presenting a similar regional pattern of functional fire-patch diversity values between the FRY products and the reference data, functional fire-patch traits were regionally dissimilar.

For example, in the FRY-MCD64A1 product, functional richness was constantly biased across North America and in BONA and TENA (Fig. 3), and functional dissimilarities were higher because of trait value differences (Fig. 6 and Table S4). Those trait value differences, particularity for extremes, were more notable in BONA (e.g., Figs. 5 and 7 in Quebec and Figs. S13 andS14). In contrast, functional dissimilarities driven by functional richness differences and thus convex hull volume differences were more notable in TENA (e.g., Figs.

5 and 7 in columns labelled Florida and Figs. S13 andS14), showing contrasting performances on fire-patch morphology conservation between BONA and TENA. The fact that the functional turnover contributed more on average than functional nestedness also supports that result (e.g., Figs. 5 and 7 in columns labelled Quebec). This result based on our new statistical framework agrees well with previous studies using products of burned area (e.g., [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF][START_REF] Hantson | Global fire size distribution is driven by human impact and climate[END_REF] and active fire [START_REF] Oom | Highlighting biome-specific sensitivity of fire size distributions to time-gap parameter using a new algorithm for fire event individuation[END_REF] for fire-patch identification, arguing that the suitability of cut-off values can change regionally.

High temporal resolution for fire-patch morphology

We found that the MODIS sensors with the highest temporal resolution better capture firepatch morphology than the MERIS sensor that has a higher spatial resolution. Fire patches in the FRY-MCD64A1 product based on MODIS sensors with a temporal resolution of 1 day were closer to the reference data than in the FRY-FireCCI41 product (Figs. 4 and 5,and Table S4) based on the MERIS sensor that had a temporal resolution of 3 days. Furthermore, the FRY-MCD64A1 product showed a stronger spatial correlation of the functional structure (Fig. 3) and lower functional dissimilarities (Fig. 6, and Table S4) with the reference data than the FRY-FireCCI41 product (Figs. 4 and5, andTable S4). For example, in BONA, where fires tend to be extensive and spread for weeks and even months [START_REF] Wang | The potential and realized spread of wildfires across Canada[END_REF], fire-patch morphology was better conserved in the FRY-MCD64A1 product than in the FRY-FireCCI41 product (Figs. 2 and 3,and Figs. 5 and 7 in columns labelled Quebec), in which uncertainties in the date-of-burning at the pixel-level in the raw data can make large fires appear fragmented (Figs. S13 andS14).

Our results suggest that high temporal resolution reduces uncertainties in the date-ofburning in fast-spreading fires (e.g., grassland fires) or short duration fires (e.g., cropland fires or fires near human settlements where firefighters react quickly). For example, fire patch number in the FRY-FireCCI41 product related to cropland fires in Idaho and, in particular, sugarcane fires in Florida [START_REF] Mccarty | The spatial and temporal distribution of crop residue burning in the contiguous United States[END_REF] (Figs. S1,S3, and S13 and Table S1) were approximately one-third of those in the FRY-MCD64A1 product (Figs. S5, S7 and S14, and Table S2). These results suggest that the number of burned pixels in these cropland systems are higher in FRY-FireCCI41 and affects patch assemblage.

No other studies have assessed the fire patches derived from the same two burned-area products used here. However, they have been assessed for the MCD64A1 product alone [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF][START_REF] Fusco | Detection rates and biases of fire observations from MODIS and agency reports in the conterminous United States[END_REF] and for previous versions of both products [START_REF] Nogueira | Can we go beyond burned area in the assessment of global remote sensing products with fire patch metrics[END_REF] in which the results from the two satellite sensors depended on traits.

Our assessment agrees well with the results of the validation of those burned-area products [START_REF] Padilla | Validation of the 2008 MODIS-MCD45 global burned area product using stratified random sampling[END_REF]. Thus, satellite sensors with high temporal resolutions are more likely to characterize fire patches accurately and thus would be a more cost-effective observation system to consider in future ECV requirement updates [START_REF] Mouillot | Ten years of global burned area products from spaceborne remote sensing-A review: Analysis of user needs and recommendations for future developments[END_REF][START_REF] Wmo | The Global Observing System for Climate: Implementation Needs 200[END_REF]. This result implies that the low temporal resolution MERIS sensor should be abandoned in the newly delivered FireCCI51 burned-area product. A merger of the high temporal resolution data from the MODIS sensors at its highest spatial resolution of 250 m should now act as the reference ESA Fire CCI product [START_REF] Chuvieco | Generation and analysis of a new global burned area product based on MODIS 250 m reflectance bands and thermal anomalies[END_REF].

Uncertainties and the way forward

We obtained low correlations between the functional diversity indices derived from the satellite-derived FRY products and reference data. This result causes us to question both the quality of the burned-area products at the pixel-level and the fire patches that resulted from the pixel aggregation flood-fill algorithm. Despite being widely used as the only homogeneous source of global information on burned area, spatial intercomparisons of satellite-derived burned-area products [START_REF] Humber | Spatial and temporal intercomparison of four global burned area products[END_REF] with satellite-derived reference data based on fine-scale [START_REF] Boschetti | Global validation of the collection 6 MODIS burned area product[END_REF][START_REF] Boschetti | A stratified random sampling design in space and time for regional to global scale burned area product validation[END_REF][START_REF] Rodrigues | How well do global burned area products represent fire patterns in the Brazilian Savannas biome? An accuracy assessment of the MCD64 collections[END_REF][START_REF] Sparks | An accuracy assessment of the MTBS burned area product for shrub-steppe fires in the northern Great Basin, United States[END_REF] or local inventories [START_REF] Fusco | Detection rates and biases of fire observations from MODIS and agency reports in the conterminous United States[END_REF][START_REF] Turco | A comparison of remotely-sensed and inventory datasets for burned area in Mediterranean Europe[END_REF], remain spatially poorly correlated regarding burned area only, with intrinsic theoretical uncertainty varying between 12.78% and 13.90% in TENA and between 53.53% and 65.87% in BONA [START_REF] Brennan | Theoretical uncertainties for global satellite-derived burned area estimates[END_REF]. Thus, we expected a low correlation between observations and fire-patch morphology; this expectation was based on the already low spatial correlation between the data and the burned-area products, and significant commission and omission errors in both the MCD64A1 [START_REF] Boschetti | Global validation of the collection 6 MODIS burned area product[END_REF] and FireCCI41 [START_REF] Padilla | Comparing the accuracies of remote sensing global burned area products using stratified random sampling and estimation[END_REF] burned-area products. Yet we still find that our correlation coefficients capture a realistic estimation of the agreement on fire-patch morphologies. Fire-patch morphologies are more sensitive to omission and commission errors at the pixel level, but produce different fire shapes as a consequence of boundary misrepresentation [START_REF] Humber | Assessing the Shape Accuracy of Coarse Resolution Burned Area Identifications[END_REF]. Thus, we suggest that, besides increasing the effort in detecting small missing fires (e.g., [START_REF] Roteta | Development of a Sentinel-2 burned area algorithm: Generation of a small fire database for sub-Saharan Africa[END_REF], more effort should be devoted to better boundary delimitation.

We attribute this low agreement to remaining uncertainties on the date-of-burning and fire location at the pixel-level in the raw satellite data, but we also acknowledge issues in the reference data (e.g., [START_REF] Short | Sources and implications of bias and uncertainty in a century of US wildfire activity data[END_REF][START_REF] Short | A spatial database of wildfires in the United States, 1992-2011[END_REF] and discrepancies in pixel-aggregation methods [START_REF] Andela | The Global Fire Atlas of individual fire size, duration, speed and direction[END_REF][START_REF] Artés | A global wildfire dataset for the analysis of fire regimes and fire behaviour[END_REF][START_REF] Campagnolo | A patch-based algorithm for global and daily burned area mapping[END_REF][START_REF] Oom | Highlighting biome-specific sensitivity of fire size distributions to time-gap parameter using a new algorithm for fire event individuation[END_REF]. The flood-fill algorithm used in FRY (Laurent et al., 2018a) can over-aggregate pixels, and fail to split concomitant fire-starts merging into one final single fire patch. This phenomenon is most likely to occur in a savanna ecosystem, so we would expect it to occur less frequently in North America [START_REF] Oom | Highlighting biome-specific sensitivity of fire size distributions to time-gap parameter using a new algorithm for fire event individuation[END_REF]. However, we anticipate improvements in fire patch delineation to result from a more efficient method of pixel aggregation -new satellitederived fire patches are still emerging.

The trait-based approach used here has been widely used in community ecology over the past few decades [START_REF] Chao | An attribute-diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures[END_REF]. [START_REF] Gitay | What are functional types and how should we seek them?[END_REF] have discussed the critical issues regarding the choice of traits to consider and how to seek them; [START_REF] Maire | How many dimensions are needed to accurately assess functional diversity? A pragmatic approach for assessing the quality of functional spaces[END_REF] the sensitivity of indices to the selection and reduction of traits; and [START_REF] Zhu | Trait choice profoundly affected the ecological conclusions drawn from functional diversity measures[END_REF] heir correlations. We acknowledge that combining the assessments of the functional structures and dissimilarities is crucial because fire-patch assemblages can lead to similar functional diversity indices [START_REF] Chao | An attribute-diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures[END_REF]. Furthermore, by using complementary indices as we did, it is possible to efficiently differentiate assembly processes [START_REF] Münkemüller | From diversity indices to community assembly processes: a test with simulated data[END_REF].

For future trait-based assessments, we suggest both the spatial location and date-of-burning of fire patches should be considered as essential fire regime traits. Although we omitted the spatial location of the fire patches from our analysis, that was because we expected low commission and omission errors at the boundaries of fire patches, i.e., we assumed that most overlapping patches would be captured, an assumption based on the work of [START_REF] Nogueira | Can we go beyond burned area in the assessment of global remote sensing products with fire patch metrics[END_REF], who found more than 70% of fire patches to be ≥ 300 ha identified. For example, [START_REF] Hempson | Continent-level drivers of African pyrodiversity[END_REF] considered the fire return interval as a trait to examine the spatial and temporal overlap between patches and [START_REF] Wiegand | Spatially Explicit Metrics of Species Diversity, Functional Diversity, and Phylogenetic Diversity: Insights into Plant Community Assembly Processes[END_REF] implemented the spatial location of individuals in plant diversity. Fire return interval can thus be used for more comprehensive regional pyrodiversity analysis.

Although our approach benefits from its versatility across scales [START_REF] Carmona | Traits Without Borders: Integrating Functional Diversity Across Scales[END_REF], the sensitivity analysis of tile size by biome remains for future studies. For example, [START_REF] Matthews | sars: an R package for fitting, evaluating and comparing species-area relationship models[END_REF] explored the species-area relationships to determine the spatial resolution capturing all species types. Similarity, [START_REF] Turco | A comparison of remotely-sensed and inventory datasets for burned area in Mediterranean Europe[END_REF] observed that increasing tile resolution improved correlations between satellite-derived burned area and reference data in Europe.

Despite those limitations, the functional trait-based approach proposed here presents a significant advance in fire patch trait assessment -it allows a comprehensive assessment of fire spread diversity to be derived from the FRY products across North America.

Conclusions

We implemented and tested the efficiency of a functional trait-based approach for the new topic of satellite-derived fire-patch morphology assessment. We showed that this method can be used for assessing satellite-derived fire patches beyond assessments based on commission and omission errors in satellite-derived burned-area products. The approach relies on fire-patch traits and with just a few indices can synthesize an intercomparison of burned-area products or pyroregions. The approach opens up a new and promising field of research.

These results demonstrate that the more reliable fire-patch maps can be created by using a fixed cut-off value -in North America that cut-off value is 14 days. Furthermore, satellite sensor monitoring at a higher temporal resolution will lead to improvements in burned-area mapping and the identification of fire patches; thus, high frequency monitoring from satellites is the most cost-effective climate observation system.

(FDIs) include functional fire-patch number (Ffp), functional richness (Fric) and functional dispersion (Fdis). Continental scale tile-to-tile Standardized Major Axis (SMA) regression of FDIs, and tile-level functional beta diversity (Fβ), functional turnover (Fturn) and functional nestedness (Fnest) were used in the assessment. For a simple graphical display, the first step shows tiles from only one product and the reference data. Similarly, the second step shows the fire patches present in a functional space of only two traits in one tile, and the fourth step shows functional spaces as a circle. A Venn diagram illustrates these cases. Table S2. Statistical summary of functional diversity indices (FDIs) in the FRY v1.0 MCD64A1 product in North America (NA), Boreal North America (BONA) and Temperate North America (TENA). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Fig. 1 :

 1 Fig. 1: Theoretical functional diversity trait-based approach adapted from Mouillot et al.

Fig. 2 .

 2 Fig. 2. Relationships of functional diversity indices between the FRY v1.0 FireCCI41 product and the reference data. Different colours identify data across tiles (n) in North America (NA),

Fig. 3 .

 3 Fig. 3. Relationships of functional diversity indices between the FRY v1.0 MCD64A1 product and the reference data. Different colours identify data across tiles (n) in North America (NA),

Fig. 4 .

 4 Fig. 4. Functional dissimilarities within tiles between the FRY v1.0 FireCCI41 product and the reference data in North America (NA), Boreal North America (BONA) and Temperate

Fig. 5 .Fig. 6 .

 56 Fig. 5. Examples of functional dissimilarities within tiles between FRY v1.0 FireCCI41 (blue volume) and the reference data (green volume) in California (Tile 34N, -120W), Florida (Tile 26N, -82W) and Quebec (Tile 50N, -72W) for specific traits considered: size, direction,

Fig. 7 .

 7 Fig. 7. Examples of functional dissimilarities within tiles between FRY v1.0 MCD64A1 (red volume) and the reference data (green volume) in California (Tile 34N, -120W), Florida (Tile 26N, -82W) and Quebec (Tile 50N, -72W) for specific traits considered: size, direction,

Fig. 2 .

 2 Fig. 2. Relationships of functional diversity indices between the FRY v1.0 FireCCI41 product and the reference data. Each point is a 2° tile containing multiple patches. Left column refers to the number of fire patches (Ffp), middle column to functional richness (Fric) and the right column to functional dispersion (Fdis). The squared correlation coefficient (r 2 ) is provided. In addition, for every significant r 2 (p < 0.05), the Standardized major axis (SMA) lines, SMA slope (b), SMA intercept (a), and confidence intervals at the 95% level (95% CI) are provided. Different colours identify data across tiles (n) in North America (NA) Boreal North America (BONA) and Temperate North America (TENA). The horizontal-axis presents the functional diversity index in the reference data and the vertical-axis in the FRY-FireCCI41 cut-off values. Both axes are in log10-scale for Ffp and Fdis.

Fig. 3 .

 3 Fig. 3. Relationships of functional diversity indices between the FRY v1.0 MCD64A1 product and the reference data. Each point is a 2° tile containing multiple patches. Left column refersto the number of fire patches (Ffp), middle column to functional richness (Fric) and the right column to functional dispersion (Fdis). The squared correlation coefficient (r 2 ) is provided. In addition, for every significant r 2 (p < 0.05), the Standardized major axis (SMA) lines, SMA slope (b), SMA intercept (a), and confidence intervals at the 95% level (95% CI) are provided. Different colours identify data across tiles (n) in North America (NA), Boreal North America (BONA) and Temperate North America (TENA). The horizontal axis presents the

Fig. 4 .

 4 Fig. 4. Functional dissimilarities within tiles between the FRY v1.0 FireCCI41 product and the reference data in North America (NA), Boreal North America (BONA) and Temperate

Fig. 5 .

 5 Fig. 5. Examples of functional dissimilarities within tiles between FRY v1.0 FireCCI41 (blue volume) and the reference data (green volume) in California (Tile 34N, -120W), Florida (Tile 26N, -82W) and Quebec (Tile 50N, -72W) for specific traits considered: size, direction, elongation of patches.

Fig. 6 .

 6 Fig.6. Functional dissimilarities within tiles between the FRY v1.0 MCD64A1 product and the reference data in North America (NA), Boreal North America (BONA) and Temperate

Fig. 7 .

 7 Fig. 7. Examples of functional dissimilarities within tiles between FRY v1.0 MCD64A1 (red volume) and the reference data (green volume) in California (Tile 34N, -120W), Florida (Tile 26N, -82W) and Quebec (Tile 50N, -72W) for specific traits considered: size, direction, elongation of patches.

Fig. S7 .

 S7 Fig. S7. Functional diversity indices in the FRY v1.0 MCD64A1 product in Florida (Tile 26N, -82W). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Fig. S8 .

 S8 Fig. S8. Functional diversity indices in the FRY v1.0 MCD64A1 product in Quebec (Tile 50N, -72W). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Fig. S9 .

 S9 Fig. S9. Functional diversity indices in the reference data at two spatial resolutions (i.e., 300 m and 500 m) in North America (mainland United State and Canada). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Fig. S10 .

 S10 Fig. S10. Functional diversity indices in the reference data at two spatial resolutions (i.e., 300 m and 500 m) in California (Tile 34N, -120W). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Fig. S11 .

 S11 Fig. S11. Functional diversity indices in the reference data at two spatial resolutions (i.e., 300 m and 500 m) in Florida (Tile 26N, -82W). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Fig. S12 .

 S12 Fig. S12. Functional diversity indices in the reference data at two spatial resolutions (i.e., 300 m and 500 m) in Quebec (Tile 50N, -72W). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Acknowledgements

The ESA Fire_cci project [Contract Nº 4000115006/15/I-NB] supported this research and P.

Laurent's work. M.V. Moreno's work was supported by UM Post-Doc programme cofinanced by the ESA Fire_cci project, and an IRD post-doctoral fellowship. We thank D.

Mouillot for his feedback on the methodology and results. We also thank the forest services for providing information about the fire perimeters used in this study.

Data 6, 296. https://doi.org/10.1038/s41597-019-0312-2 Bistinas, I., Harrison, S.P., Prentice, I.C., Pereira, J.M.C., 2014. Causal relationships versus emergent patterns in the global controls of fire frequency. Biogeosciences 11, 5087-

List of Figure Captions

Fig. 1: Theoretical functional diversity trait-based approach adapted from [START_REF] Mouillot | A functional approach reveals community responses to disturbances[END_REF] and [START_REF] Villéger | Decomposing functional β-diversity reveals that low functional β-diversity is driven by low functional turnover in European fish assemblages[END_REF][START_REF] Villéger | New multidimensional functional diversity indices for a multifaceted framework in functional ecology[END_REF] to assess satellite-derived fire patches in four steps. Table S1. Statistical summary of functional diversity indices (FDIs) in the FRY v1.0 FireCCI41 product across North America (NA), Boreal North America (BONA) and Temperate North America (TENA). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis). Table S3. Statistical summary of functional diversity indices (FDIs) in the reference data at two spatial resolutions (i.e., 300 m and 500 m) in North America (NA), Boreal North America (BONA) and Temperate North America (TENA). Functional fire patches (Ffp), functional richness (Fric) and functional dispersion (Fdis).

Table S4. Statistical summary of functional dissimilarities between the FRY v1.0 products and the reference data at two spatial resolutions (i.e., 300 m and 500 m) in North America (NA), Boreal North America (BONA) and Temperate North America (TENA). Functional beta diversity (Fbeta) and its two components of turnover (Fturn) and nestedness (Fnest).

Fig. S13.

Frequency density of fire-patch size in the FRY v1.0 FireCCI41 product (blue points) and reference data at 300 m resolution (green points) across North America (NA), Boreal North America (BONA) and Temperate North America (TENA). The frequency density of fire-patch size was calculated as in [START_REF] Moreno | Wildfire frequency-area statistics in Spain[END_REF]. Boreal North America (BONA) and Temperate North America (TENA). The frequency density of fire-patch size was calculated as in [START_REF] Moreno | Wildfire frequency-area statistics in Spain[END_REF].