Global monthly burned area data were taken from the Global Fire Emissions Database (GFED) (Giglio et al., 2013) and the ESA Fire CCI datasets (Chuvieco et al., 2016). GFED version 4 provides monthly burned area time series at a 0.25∘ spatial resolution for the period 1995–2015 based on a combination of the MODIS burned area product (from 2000 onwards) with active fire observations from VIRS (Visible and Infrared Scanner) and ATSR (Along-Track Scanning Radiometer) (before 2000) (Giglio et al., 2013). Fire CCI version 4.1 provides burned area time series at 0.25∘ spatial resolution for the period 2005–2011 based on a combination of MERIS data and MODIS thermal anomalies (Alonso-Canas and Chuvieco, 2015; Chuvieco et al., 2016). Because of the longer temporal coverage, the GFED dataset was used as the response variable in model development and for model evaluation. The Fire CCI dataset was used as an independent burned area dataset in model evaluation. Differences between the two datasets reflect the uncertainty in satellite-derived burned area. For both datasets burned area is expressed as the fractional burned area of a 0.25∘ grid cell.

Land cover data were taken from the ESA land cover CCI product which provides three global land cover maps at 300 m spatial resolution covering the epochs 1998–2002, 2003–2007, and 2008–2012. We did not use the original land cover classification of the maps, but translated land cover classes into plant functional types (PFTs) to be comparable with the classification used in global vegetation models (Poulter et al., 2011). The translation followed largely the rules by Poulter et al. (2015a) with some modifications to avoid coverage of broad-leaved evergreen trees and shrubs in boreal and Arctic regions (Table A1). The following nine PFTs were derived: broadleaved evergreen tree and shrub (Tree.BE, Shrub.BE), broadleaved deciduous tree and shrub (Tree.BD, Shrub.BD), needle-leaved evergreen tree and shrub (Tree.NE, Shrub.NE), needle-leaved deciduous tree (Tree.ND), natural grass or herbaceous vegetation (Herb), and managed grasslands or crops (Crop). The land cover maps were spatially aggregated and expressed as the fractional coverage of PFTs within a 0.25∘ grid cell.

We further aggregated the coverage of PFTs within each 0.25∘ grid cell to the total coverages of trees (Tree = sum of all tree PFTs, Table 1), shrubs (Shrub), and herbaceous vegetation including croplands (HrbCrp = Herb + Crop). To potentially characterize fuel types based on the dominant leaf type, PFTs were further aggregated into needle-leaved (Needleleaf) and broadleaved vegetation (Broadleaf) vegetation.

As land cover distribution is affected by fires, the land cover maps may regionally contain effects of past fires. Consequently, it can happen that fire activity is explained by the impact of the actual fire activity already present in a land cover map. We tried to reduce this effect by shifting the land cover maps by 2 years. This means that the map for the epoch 1998–2002 is used for the years ≤ 2004, the map for the epoch 2003–2007 for the period 2005–2009, and the map for the period 2008–2012 for the years ≥ 2010. However, the three maps have only marginal temporal differences, so that the impact of assigning land cover maps to certain years is rather small.

We used monthly data of mean air temperature, diurnal temperature range (DTR), and monthly number of wet days from the Climate Research Unit (CRU) TS3.2 dataset (Harris et al., 2014). DTR has been long used as predictor for fire weather conditions because it is sensitive to stable weather conditions that are usually associated to low humidity and are supportive for fire activity (Bistinas et al., 2014; Venevsky et al., 2002). These datasets provide monthly climate time series at 0.5∘ resolution based on spatially interpolated weather station observations. Precipitation was taken from the Global Precipitation Climatology Center (GPCC) version 7 dataset (Schneider et al., 2015). All climate datasets were resampled to 0.25∘ using the nearest neighbour method in order to avoid smoothing of climate anomalies through alternative resampling methods such as bilinear interpolation.

We used the monthly values and long-term conditions of climate datasets as predictor variables (Table 1). As long-term conditions, we computed the mean temperature, mean diurnal temperature range, mean number of wet days, and total precipitation of the actual month and the 12 preceding months.

Surface soil moisture was taken from the ESA CCI soil moisture dataset (version 02.3 COMBINED) which is based on a merging of soil moisture products from various active and passive satellite sensors (Dorigo et al., 2015; Liu et al., 2011a, 2012). The dataset represents the upper soil layer (∼ 2 cm) and is available at a 0.25∘ spatial resolution and daily time step for the period 1979–2015. The long-term dynamic of the soil moisture dataset is consistent and environmentally plausible, as demonstrated in a comparison with precipitation, soil moisture, and Normalized Difference Vegetation Index trends from independent datasets or land surface models (Albergel et al., 2013; Dorigo et al., 2012).

As soil moisture cannot be accurately retrieved underneath dense (tropical) forests, estimates are not available in all regions, and thus the dataset has spatial gaps. We excluded such grid cells in the full analysis. Soil moisture time series were aggregated to monthly mean values. Temporal gaps in soil moisture time series were filled using a season-trend regression model as described in Forkel et al. (2013) and based on Verbesselt et al. (2010a, b), but without accounting for breakpoints. However, some years in some grid cells were excluded from the entire analysis if soil moisture estimates were only available for less than 3 months within this year.

We used the monthly soil moisture values and long-term soil moisture conditions as predictor variables (Table 1). Long-term soil moisture conditions were computed as the mean soil moisture of the actual month and the 12 preceding months.

To account for effects of vegetation phenology, biomass, or vegetation water content on fire activity, we used the GIMMS3g FAPAR (Zhu et al., 2013) and a VOD dataset (Liu et al., 2011b). GIMMS3g FAPAR is a long-term multi-sensor merged dataset of FAPAR and is based on the GIMMS3g NDVI (Normalized Difference Vegetation Index) dataset with a spatial resolution of 1/12∘ and a temporal resolution of 16 days for the period 1981 to 2012 (Pinzon and Tucker, 2014). GIMMS3g FAPAR was aggregated to 0.25∘ spatial resolution and averaged to monthly time steps. VOD by Liu et al. (2011b) is a long-term harmonized dataset from several passive microwave sensors. The VOD dataset has a spatial resolution of 0.25∘ and a monthly temporal resolution for the period 1988–2012.

Permanent gaps in FAPAR or VOD time series (mostly gaps occurring in winter at northern latitudes) were filled with the minimum value of each time series (Forkel et al., 2015) and remaining gaps were filled using the season-trend regression model (Forkel et al., 2013).

We used the monthly FAPAR or VOD values of the precedent month as predictor variables because the vegetation of the actual month is likely affected by the fire event which we aim to explain. Additionally, we computed the mean FAPAR and VOD of the 12 precedent months as long-term vegetation state predictor variables.

We used satellite-based datasets on population density and socioeconomic development as predictor variables for burned area.

Population density (PD) was taken from the Global Rural-Urban Mapping Project (GRUMP) V1 dataset (Balk et al., 2006). This dataset is based on (sub-)national population statistics, satellite observations of night-time lights, and the spatial distribution of cities to provide estimates of population density on a 1 km grid for the years 1990, 1995, and 2000. The dataset was aggregated to 0.25∘. The dataset was temporally interpolated between 1990 and 2000 and extrapolated between 2000 and 2011 for each grid cell to achieve a full coverage for the period 1997–2011. The interpolated time series is the median estimate from three interpolation methods (repeating the last value as a constant, linear interpolation, spline interpolation). This allowed us to make use of the temporal information of the population density dataset.

As an indicator of socioeconomic development, we used the Night Light Development Index (NLDI) (Elvidge et al., 2012). The NLDI is derived from satellite observations of light emissions during night and an independent estimate of population density. The NLDI ranges between 0 (light emissions equally distributed among people, highest development) and 1 (light emissions concentrated on one person, lowest development). The NLDI is highly correlated with electrification rates and the human development index (Elvidge et al., 2012). The dataset is available at a 0.25∘ spatial resolution for the year 2006. The NLDI is not available for grid cells without a population or without detected night lights, which introduces gaps into the global NLDI map. We filled these gaps with a value of 1.01 (indicating very low development or natural ecosystems) in order to not introduce spatial gaps of the NLDI dataset into the empirical modelling of burned area.

SOFIA is a data-driven fire model approach that allows us to test several alternative functional relationships and associated variables to predict fractional burned area. The basic structure of SOFIA fire models is inspired by SIMFIRE (simple fire model) which uses empirical relationships to estimate fire frequency from vegetation (i.e. FAPAR), fire weather conditions, and socioeconomic variables (Knorr et al., 2014). In SOFIA we generalize the SIMFIRE approach by using and testing several alternative predictor variables as controls for fire activity. Each SOFIA model structure is based on the assumption that potentially the entire vegetated area can burn, but burning is actually restricted by several functional relationships to controlling factors: BAt=∑g=1g=NAg⋅fg,t, where BA is the fractional burned area of a grid cell at time step t, A is the fractional coverage of land cover group g, and fg is a factor that controls fire spread (0 = fully restricted burning and 1 = unconstrained burning) for a specific land cover group. Land cover groups g can for example be classified according to growth forms (trees, shrubs, grasses, crops), plant functional types (PFTs), or any other potentially meaningful separation of land cover. The factor fg is a product of individual functions that represent climatic, environmental, and socioeconomic controls on fire: fg=∏i=1i=Nfxi,g,f(xi,g)=min1,max⁡g,i1+e-sli,g×x-x0i,g, where x is the value of an environmental or socioeconomic variable i; and max, sl and x0 are parameters of a logistic function. We used the minimum value from 1 and the logistic function, and included max as a free parameter to allow the representation of exponential relationships within the basic structure of logistic functions. Parameters of the logistic functions can be either defined per vegetation cover group or as global parameters. Variables x can be for example vegetation state variables such as FAPAR or VOD to represent fuel loads, climate variables such as the number of wet days or diurnal temperature range to represent fire weather conditions, and socioeconomic variables such as population density or NLDI to represent human effects on fire activity. Consequently, the development of an actual SOFIA model requires two steps, namely the definition of a model structure (i.e. selection of candidate predictor variables, Sect. 3.2) and the estimation of the model parameters (Sect. 3.3).

SOFIA models allow us to reproduce the typical right-tailed distribution of burned area (i.e. many grid cells and months with no burned area in comparison to relatively few grid cells and months with fire activity). The underlying functional relationships can take step-wise, linear, sigmoidal, or exponential shapes depending on the parameters of the logistic functions (Fig. 1). Similar model structures like SOFIA where a response variable is controlled by a product of several functions have been previously applied in environmental modelling, for example, in light-use efficiency models to simulate NPP (Cai et al., 2014; Nemani et al., 2003) or in phenology models to simulate leaf development (Forkel et al., 2014; Jolly et al., 2005; Stöckli et al., 2011). The response value of the functional relationship can also be used to map sensitivities of burned area to environmental or socioeconomic variables. Such a mapping of controls was previously done for plant productivity (Nemani et al., 2003) and phenology (Forkel et al., 2014; Jolly et al., 2005) based on red–green–blue (RGB) composite maps. Here we will demonstrate how this approach can be used to investigate spatial patterns of sensitivities between burned area and climatic, environmental, and socioeconomic controls on fire activity.

To test appropriate controlling factors and related predictor variables in SOFIA models, we defined several alternative model structures. Each SOFIA model uses a specific land cover grouping scheme and several functional relationships for fire activity.

We tested different land cover grouping schemes to assess the required complexity of SOFIA models to regionalize model parameters. As grouping schemes we either used growth forms (“GrowthForm” including the variables Tree, Shrub, and HrbCrp; Table 1), growth forms with crops separated from herbaceous vegetation (“GrowthFormCrop” including Tree, Shrub, Herb, and Crop), leaf types (“LeafType” including Needleleaf, Broadleaf, Herb, and Crop), or PFTs (“PFT” using the nine PFTs). Differences between GrowthForm and GrowthFormCrop will allow assessment of whether a separation of croplands from herbaceous vegetation is necessary to explain fire activity. The LeafType grouping scheme may potentially be useful because needles usually decompose more slowly than broadleaves and thus form larger pools of litter fuel. Differences between GrowthFormCrop and LeafType allow assessment of whether model parameters should be separated rather by growth form or by leaf type. The PFT land cover grouping scheme is finally used to assess whether the interaction of growth forms and leaf types is required to regionalize model parameters.

We defined five controlling factors on fire activity and assigned several corresponding predictor variables to each controlling factor to evaluate the following required components of SOFIA models.

Human influences represent potential relations between socioeconomic indicators and burned area. As predictor variables we used either population density with a global parameter set (PD), NLDI with a global parameter set (NLDI), or NLDI with parameters that vary per land cover group (NLDI.g).

Temperature effects represent potential relations between diurnal temperature range (CRU.DTR.orig) or long-term air temperature (CRU.T.annual) and burned area.

Direct wetness effects represent the obvious restriction of fire activity by wet conditions. We included either the current month's number of wet days (CRU.WET.orig), precipitation (GPCC.P.orig), or surface soil moisture (CCI.SM.orig).

Direct vegetation effects represent potential relations between the precedent vegetation state and burned area. Therefore we either used previous month's FAPAR (GIMMS.FAPAR.pre) or VOD (Liu.VOD.pre) as predictor variables.

Long-term wetness or vegetation effects represent potential relations between long-term averaged precedent conditions of wetness or vegetation variables and burned area. Several reasons exist to test long-term averaged predictor variables as structural components of SOFIA models. Firstly, long-term conditions of precipitation and soil moisture are strongly linked to plant productivity especially in semi-arid ecosystems and thus might represent variations in vegetation and fuel loads. Secondly, long-term conditions of FAPAR and VOD are more closely related to vegetation coverage or biomass and thus might better represent fuel loads than the actual monthly values. As predictor variables for long-term conditions, we used aggregated values from the 12 precedent months for the number of wet days (CRU.WET.annual), precipitation (GPCC.P.annual), soil moisture (CCI.SM.annual), FAPAR (GIMMS.FAPAR.annual), or VOD (Liu.VOD.annual).

We also allowed that a certain controlling factor is not included in a model to test whether this controlling factor is generally needed in the SOFIA model. This set-up of controlling factors and associated predictor variables allows the definition of several candidate model structures (Table A2). For example, SOFIA model SF.124421 (the coding is described in Table A2) used growth forms as a land cover grouping scheme, the NLDI for human influences, diurnal temperature range as a temperature effect, the number of wet days as a direct wetness effect, the previous month's FAPAR as a direct vegetation effect, and long-term precedent VOD as a long-term vegetation effect (Fig. 1). The model structure determines the complexity which we assess here based on the number of controlling factors within a SOFIA model and on the number of parameters N in a model (N= number of controlling factors × number of land cover groups × 3 parameters). We required that SOFIA models include at least three controlling factors and have fewer than 100 parameters. This results in 2712 candidate SOFIA models. We optimized and evaluated 95 randomly selected models from the set of candidate models (Table A2). Although this selection does not allow a full factorial assessment of controlling factors and predictor variables in SOFIA models, it is a trade-off between computational feasibility and an assessment of the tendency of a factor regarding model performance.

We used the random forest machine learning approach to evaluate if the basic structure of SOFIA models is flexible enough to predict burned area or if a more flexible modelling approach can reach higher performances. Random forest is a regression approach that can consider non-linear, non-monotonic and abrupt, and non-additive relations between multiple predictor variables and a response variable (Breiman, 2001). Random forest is an ensemble of multiple regression trees that are trained based on the response variable. Each tree uses a randomly selected set of predictor variables and data points (Breiman, 2001). Random forest was already previously applied to identify controls on vegetation dynamics and on fire activity (Aldersley et al., 2011; Archibald et al., 2009). We used 500 trees per random forest. For the training of the random forest, we used the same data subset that was also used to optimize SOFIA models (Sect. 3.3.3). The analysis was performed using the randomForest package in R (Liaw and Wiener, 2002).

We performed three different random forest model experiments. Model experiment RF1 used all predictor variables from Table 1 to explore the potential performance of the used datasets to predict burned area. Model experiment RF2 used all predictor variables except for the variables from the soil moisture dataset in order to apply random forest globally and to compare the results with SOFIA independently of the spatial gaps of the soil moisture dataset. Model experiment RF.124421 uses the same predictor variables as SOFIA model SF.124421 (i.e. CCI.LC.Tree/Shrub/HrbCrp, NLDI, CRU.WET.orig, Liu.VOD.annual, GIMMS.FAPAR.pre, CRU.DTR.orig) in order to compare the performance of the two model approaches based on the same predictor variables.

We simulated burned area with the SPITFIRE (spread and intensity of fire) fire module within the JSBACH (Jena Scheme for Biosphere-Atmosphere Coupling in Hamburg) land surface model in order to compare the performance of SOFIA models to a state-of-the art global vegetation-fire model. This comparison potentially allows us to provide suggestions for the further development of global vegetation-fire models.

JSBACH is the land component of the MPI (Max Planck Institute for Meteorology) Earth system model (Raddatz et al., 2007). SPITFIRE is a physically based fire module that simulates fire ignitions (based on lightning and population density), fire spread, and fire effects depending on weather conditions, vegetation type and structure, fuel moisture, and fuel size (Thonicke et al., 2010). SPITFIRE was originally developed for the LPJ (Lund-Potsdam-Jena) dynamic global vegetation model (Thonicke et al., 2010). For the implementation of SPITFIRE in JSBACH, two parameters in SPITFIRE were adjusted, one related to human ignitions and the other related to the drying of fuels (Lasslop et al., 2014). Additionally, the relation between wind speed and the rate of fire spread was modified (Lasslop et al., 2015) and a decrease in fire duration with increasing population density was implemented (Hantson et al., 2015a).

JSBACH was applied at a spatial resolution of 1.875∘ × 1.875∘. JSBACH runs on a half-hourly time step, while the SPITFIRE module is called at daily time steps. A detailed description of the simulation set-up is given in the FireMIP (fire model inter-comparison project) protocol, from which we use JSBACH baseline simulation SF1 (Rabin et al., 2017). Following a spin-up period to equilibrate carbon pools (continued until the slow carbon pool varied less than 1 % between consecutive 50-year periods), a transient simulation was started in 1700. Data on land use (Hurtt et al., 2011) and population density (Goldewijk et al., 2010) were used starting in 1700 and interpolated to annual resolution. The CO2 concentration of the atmosphere was provided starting from 1750 at annual resolution (Le Quéré et al., 2014). CO2 concentration before 1750 was set to the value of 1750. Climate forcing is based on the CRUNCEPv5 dataset (1901–2013) (Wei et al., 2014). Climate data were recycled over the years 1901–1920 before 1901.

The optimized candidate SOFIA models covered wide ranges of complexities and performances (Fig. 2, Table 2). The best-performing SOFIA models reasonably explained the monthly spatial–temporal patterns of fractional burned area (i.e. up to IoA = 0.45 for SF.230512, Table 2) but underestimated the observed variance (i.e. negative FV, best FV = -1.44 for SF.204422). Although the comparison of the GFED and Fire CCI burned area datasets showed only a moderate agreement (IoA = 0.85 and FV = 0.06), the performance of SOFIA models was similar for both datasets (Table 2). The performance of SOFIA models was very similar for the optimization and evaluation data subsets, which shows that SOFIA models can be robustly applied to different spatial and temporal domains. The SOFIA model with the lowest AIC considered only three controlling factors and had 21 parameters (SF.124002, Table A2). However, this model reached only a poor performance (IoA = 0.29 and FV = -1.68 in the optimization subset). Consequently, this model is not suited to simulating global fire activity. Therefore we selected the best SOFIA models according to both performance (IoA ≥ 0.4) and AIC (AIC ≤ 200) (Fig. 2a). The four best SOFIA models had different combinations of predictor variables, which demonstrates the equifinality in predicting global burned area. However, the results show that SOFIA models were robust enough to predict global monthly fractional burned area for different spatial and temporal domains and using different datasets.

We also tested if alternative cost functions in the optimization of SOFIA models would reduce the underestimation of the observed variance of burned area. The tested alternative cost functions explicitly accounted for variance, burned area anomalies, or were based on transformed burned area values (Table A3). Although a cost function based on IoA and FV reached better performances in terms of IoA (best IoA = 0.45 against CCI.BA in the evaluation subset) and reproduced the observed variance of burned area (FV = 0 against GFED in the training subset), the resulting model overestimated mean fractional burned area which is reflected by a high SSE (Table A3). Other alternative cost functions resulted in weaker performances than the default SSE cost function. Consequently, we used the SSE-based cost function for the optimization of all SOFIA models.

The performance of SOFIA models varied with model complexity. SOFIA models that used a higher number of controlling factors (n = 4 or 5) had on average a better performance than models with only three factors (Fig. 2b). However, very complex SOFIA models with a high number of parameters (n = 70–90) did not necessarily result in higher performances than models with an average number of parameters (n = 30–70, Fig. 2c). Models with a low number of parameters (n < 30) had on average low performances, but we also found some SOFIA models with few parameters that reached good performances (e.g. SF.124021 with only 30 parameters, Table A2). The four best SOFIA models had between 30 and 50 parameters. The number of parameters in SOFIA models was mostly affected by the choice of a certain land cover grouping scheme to regionalize model parameters. Models that used the GrowthForm (three groups), GrowthFormCrop or LeafType (both four groups) grouping schemes reached much lower AIC values than models that used the PFT grouping scheme (with nine PFTs) (Fig. 2a). These results demonstrate that SOFIA models with a higher number of predictor variables but a medium number of model parameters reached the best performances in predicting global monthly spatial–temporal patterns of burned area.

Random forest models reached slightly better performances than the best-performing SOFIA models. The random forest model based on all variables reached very good performance in training (IoA = 0.95 for RF1) and moderate performances in the evaluation subset (IoA = 0.58 for RF1, Table 2). The random forest models with (RF1) and without (RF2) soil moisture variables reached similar performances. Similar to the SOFIA models, the employed random forests underestimated the observed variance. However, when using random forest with the same set of predictor variables as SOFIA (RF.124421 vs. SF.124421), random forest reached even weaker performances (IoA = 0.4, FV = -1.7 in evaluation against CCI burned area) than the corresponding SOFIA model. Thus the highly flexible structure of the random forest machine learning approach did not necessarily result in a much better performance than the best-performing SOFIA models. Consequently, the SOFIA approach offers enough flexibility to assess different controlling factors and its functional relationships to predict burned area.

The performance of SOFIA models depended on the controlling factor and associated predictor variables that were used in model structures (Fig. 3). The choice of a certain land cover grouping scheme in SOFIA models to regionalize model parameters had only weak effects on model performance (Fig. 3a). Although models based on the GrowthForm scheme had on average weaker performances than models based on land cover grouping schemes with croplands, the best SOFIA models were not related to a certain land cover grouping scheme.

Including human influences as controlling factors in SOFIA models did not improve model performance (Fig. 3b). The best models either did not consider human influences or considered human influences through NLDI as global controlling function. However, NLDI did in average not contribute to higher performances. SOFIA models that used population density had on average weaker performance than SOFIA models that used NLDI or that did not consider human influences. The weaker performance of population density as component in SOFIA models could be caused by the general model structure in which potential burned area equals the total vegetated area: As highly populated areas are usually associated with low vegetation cover, potential burned area is low as well, and thus population density does not provide further information. However, the SOFIA models (SF.314511) revealed a global decline of burned area with increasing population density (Fig. A4), a finding which is in agreement with previous studies (Andela et al., 2017; Bistinas et al., 2014; Knorr et al., 2014). Although two of the best SOFIA models did not contain any variable for human influences (SF.204422, SF.203512), they however considered the fractional coverage of croplands in the used land cover grouping scheme. Consequently, these two models considered human influence on fire indirectly through the coverage of croplands. These results suggest that human influences on fire activity can be relatively interchangeably described in SOFIA models by the coverage of croplands, NLDI, or population density.

Considering temperature variables in SOFIA models caused on average better model performances than model structures without temperature variables (Fig. 3c). However, we also found one model without a temperature control that reached good performance (SF.233210, Table A2). All of the best-performing models included a diurnal temperature range or pre-fire annual mean temperature as controlling factors. These results show that temperature-related variables are important predictors in SOFIA.

The consideration of direct wetness effects in SOFIA models had the largest positive impact on model performance (Fig. 3d). Models that did not consider direct wetness effects had lower performances than models that used soil moisture, precipitation, or the number of wet days. Especially models based on the number of wet days reached significant higher IoA than models without direct wetness effects (Wilcoxon rank sum test, p≤0.05). Consequently, direct wetness effects on fire activity were a required component of SOFIA models to predict burned area.

Including or not including direct vegetation controls did not lead to a significant change in the performance of the SOFIA models (Fig. 3e). The best models either did not consider direct vegetation effects (SF.324202) or used pre-fire FAPAR (SF.204422, SF.124421) or pre-fire VOD (SF.203512). This suggests that precedent FAPAR and VOD conditions did not provide additional information to predict burned area in SOFIA models.

On the contrary, considering long-term wetness or vegetation effects in SOFIA models caused significantly higher model performances than not considering these effects (Fig. 3f). Especially SOFIA models that used pre-fire annual precipitation or VOD reached significantly higher IoA. Models with long-term effects based on soil moisture, the number of wet days, or FAPAR had on average similar performances to models without long-term effects. However, we also found some good models that used long-term conditions of FAPAR (e.g. SF.203512). These results demonstrate that long-term conditions in vegetation productivity (reflected by annual precipitation) or vegetation structure (reflected by VOD or FAPAR) were required components of SOFIA models to predict burned area.

Based on the performances of the different controlling factors and associated predictor variables, the ideal SOFIA model should include the NLDI as a human influence, one variable to account for temperature effects, the number of wet days as a direct wetness effect, and pre-fire annual conditions of precipitation or VOD as long-term wetness/vegetation effects. This ideal model structure is realized in two of the best-performing SOFIA models (SF.124421 and SF.324202, Fig. 3). The choices of a certain land cover grouping scheme or of a direct vegetation effect are secondary components of SOFIA model structures. The distribution of model parameters in SF.124421 after optimization reflects the fact that parameters for the functional relationships with the NLDI, the number of wet days, and VOD were well constrained and thus were the most sensitive parameters within this model to estimate global monthly burned area dynamics. These parameter estimates and distributions could potentially be used as prior parameter estimates to further constrain SOFIA models.

The underlying functional relationships in SOFIA models allow us to map the sensitivities of burned area to human, vegetation, and climate variables. To demonstrate such a potential application of a SOFIA model, we mapped mean responses from each functional relationship for the period 1997–2011 from SOFIA model SF.124421 (Fig. 6). Based on this model, human influences (i.e. the NLDI) restricted burned area in most parts of Europe and southern Russia, eastern and south-eastern Asia, India, central and eastern North America, south-eastern South America, southern Australia, and New Zealand (Fig. 6a). These regions correspond to the most populated and developed regions of the world. This pattern was caused by the underlying functional relationship of SF.124421 where NLDI < 1 (i.e. developed regions) restricted and NLDI > 1 (i.e. unpopulated regions or natural ecosystems) allowed fire activity (Fig. 1b). These results indicate a predominant restricting effect of humans on fire activity.

Temperature effects in SF.124421, expressed as diurnal temperature range, allowed fire activity mostly in the semi-deserts of western North America, in the Sahel, and in Australia, and had a moderate restriction effect in tropical forests and the tundra (Fig. 6b). These spatial patterns were caused by the controlling function that had a strong sigmoidal increase in fire activity with a diurnal temperature range in shrublands and allowed moderate fire activity in herbaceous vegetation and croplands (Fig. 1c).

Direct wetness effects, expressed as the number of wet days, generally allowed fire activity in all forest regions and moderately restricted fire activity in the rest of the world (Fig. 6c). The underlying controlling function in SF.124421 showed no sensitivity for forests, a weak positive relation in herbaceous vegetation and croplands, and a strong exponential decrease in fire activity with an increasing number of wet days in shrublands (Fig. 1d).

As a direct vegetation effect, pre-fire FAPAR restricted fire activity in herbaceous vegetation and croplands of central North America, central Asia, the northern Sahel, the Kalahari, central Australia, and parts of South America (Fig. 6d). On the other hand, pre-fire FAPAR supported fire activity mostly in the southern Sahel and northern and eastern Australia. These patterns were caused by a general strong restriction of fire activity with pre-fire FAPAR in herbaceous vegetation and croplands and an exponential increase in fire activity with increasing pre-fire FAPAR in shrublands in SF.124421 (Fig. 1e).

As a long-term vegetation effect, 12-month precedent mean vegetation optical depth strongly supported fire activity in central North America, central Asia, the Tibetan Plateau, the Sahel, parts of India, the Kalahari, Australia (except the interior), and northern Patagonia (Fig. 6e). In all other regions, annual VOD had a moderate effect on fire activity in SF.124421. The underlying controlling function in SF.124421 showed an exponential increase in fire activity with annual VOD in shrublands, an exponential decrease with annual VOD in herbaceous vegetation and croplands, and a strong restriction across all VOD ranges for trees (Fig. 1f). The diverging responses with annual VOD in shrublands and herbaceous vegetation indicate that fire activity increases with higher vegetation density or biomass in shrublands but decreases with increasing vegetation water content in herbaceous vegetation, respectively. Additionally, the general restriction of fire activity with VOD for trees indicates that fire activity is restricted by vegetation density or high vegetation water content in forests.

We further combined the controlling functions of SF.124421 to investigate combined controls on fire activity. Therefore we created a red–green–blue composite map in which the red channel contains the NLDI functional relationship, the green channel contains the mean of the direct (precedent month FAPAR) and long-term vegetation (12-month precedent VOD) effect, and the blue channel contains the climate effects (mean response of functional relationships to the number of wet days and diurnal temperature range) from SF.124421 (Fig. 6f). Generally, bright colours on this map indicate a strong restriction of fire activity (small burned area) and dark colours indicate that fire activity is allowed (large burned area). Regionally, different combinations of socioeconomic, vegetation, and climate factors controlled fire activity. Socioeconomic development dominantly restricted fire activity in western North America and in populated regions of boreal forests (red colours). Vegetation predominantly suppressed fire activity in southern boreal and tropical forests (green colours). Primarily climate conditions and secondly socioeconomic development restricted fire activity in semi-deserts of the northern Sahel, central Asia, the Kalahari, and south-western Australia (purple colours). Socioeconomic development and climate equally suppressed fire activity in the Mediterranean, India, eastern Asia, and eastern South America (pink colour). Both socioeconomic development and vegetation conditions suppressed fire activity in most parts of Europe, central and eastern North America, and eastern China (yellow/orange colours). Both climate and vegetation conditions suppressed fire activity in the tundra and in central Australia (cyan colours). All factors moderately supported fire activity in boreal forests and strongly support fire activity in large parts of the Sahel, southern Africa, northern Australia, and western North America (dark colours). We want to point out that these sensitivities might look different if SOFIA models with alternative but adequate model structures are applied for such an analysis. However, the results highlight that fire activity is controlled by regionally diverse and complex interactions of human, vegetation, and climate factors.

We developed the SOFIA modelling approach as a framework to explore the importance of and the functional relationships between different predictor variables and burned area while relying on relatively simple model structures. The best SOFIA models reached globally average performances but outperformed the JSBACH-SPITFIRE state-of-the-art process-oriented vegetation-fire model. We interpret the globally medium and regionally varying performances as current upper limits that can be reached with the used predictor datasets and variables because the more flexible and highly adaptive machine learning algorithm random forest did not achieve much higher performance in the evaluation data subset. These upper limits in model performance might be for several reasons.

Uncertainties in the observations for the predictor and response variables inhibit the development of models with high performance. For example, we found regionally partly large differences between the two burned area datasets, especially in northern regions. These uncertainties originate from differences in sensor characteristics and in the ability of the used algorithms to detect small fires.

Other processes and variables are important for the spread of fires but cannot be resolved at the used spatial and temporal resolution. For example, on local to regional scales the spread of fire is controlled by landscape structure and topography whereas climatic controls are usually more important on larger scales (Archibald et al., 2009; Z. Liu et al., 2013; Parisien et al., 2010). Most of the regional controls can likely not be resolved at the used spatial resolution (0.25∘) although this resolution is already higher than the resolution of most global vegetation-fire models. Also wind speed and direction is an important control on the spread of fires on short temporal scales but this effect cannot accurately be represented based on monthly data (Bistinas et al., 2014).

There is a lack of global observations that directly represent fuel loads, fuel moisture, or modes of human fire usage. For example, all of the used predictor variables are only proxies for fuel loads (FAPAR or VOD) or fuel moisture (surface soil moisture), but do not directly represent such fuel conditions. Similarly, data on population density or socioeconomic development are used as proxies for human effects on fire, but cannot represent the complex social, economic, and cultural practices and policies of human fire use and management.

The four best SOFIA models reached similar performances in savannas and tropical croplands, and in tropical forests, which demonstrates the equifinality in fire modelling. Equifinality, i.e. the presence of multiple adequate models and parameter sets that result in very similar responses, is a general problem in environmental modelling (Beven, 2006). General approaches to avoid equifinal models are the use of multiple datasets of the same variable to account for errors or uncertainties in model forcing or reference data, the testing of different cost functions to constrain certain parameters, the inclusion of prior parameter uncertainties in the cost function, or the application of models to new observational data or under different conditions (Beven, 2006; Beven and Binley, 2014; Williams et al., 2009). In our analysis, we were able to rule out three of four initially equifinal SOFIA models based on the application of these models to the global data and by regional comparisons against two burned area datasets. The results from the optimized SOFIA models allow extraction of parameter values and ranges for each functional relationship. To give an example, parameters that control the functional relationship with (1) socioeconomic development (NLDI), with (2) diurnal temperature range and the number of wet days in shrublands, and with (3) VOD were well constrained in SOFIA model SF.124421 (Fig. A3). These parameters could potentially be used as prior parameter values in a more constrained analysis in the future. The presence of equifinality in SOFIA model structures suggests the inclusion of such prior parameter uncertainties for each functional relationship to better constrain individual SOFIA models. This technique can be applied in future generations of individual SOFIA models by using the current versions as prior parameter estimates and uncertainties.

The derived SOFIA models and the spatial patterns of sensitivities show a sharp decline in burned area with increasing socioeconomic development or population density and thus agree with previous studies that show a primarily negative effect of human activities, population density, or croplands on burned area (Andela et al., 2017; Archibald et al., 2013; Bistinas et al., 2014; Chuvieco and Justice, 2010; Knorr et al., 2014). Strikingly, our results suggest that human effects on global burned area can be expressed by either cropland area, NLDI, or population density, but the combination of these factors did not improve the performances of SOFIA models. These variables all serve as proxies for the negative relationship between humans and burned area, but do not directly describe human activities of fire use or suppression. For example, regional studies have shown that various information on infrastructure, land use, and other relevant socioeconomic indicators are important to predict fire activity (Archibald et al., 2009; Arndt et al., 2013; Parisien et al., 2016). However, such spatially and temporally resolved datasets and assessments are missing for the global scale. Certainly, our results do not imply that croplands are unimportant for the global variability of burned area. Agricultural fires account for around 10 % of all global fires (Korontzi et al., 2006) and for around 5 % of global burned area (Giglio et al., 2013) and are used to remove harvest residues or to fertilize soils. However, croplands show more small fires than large fires (Hantson et al., 2015b). As we here used the GFED burned area datasets that were not corrected for small fires (Giglio et al., 2013), small agricultural fires are likely misrepresented in this dataset and thus cannot be accurately analysed within the SOFIA approach. The representation of agricultural fires in a global fire model needs to account for various land use patterns and practices that go far beyond natural climate–vegetation relationships (Le Page et al., 2015; Magi et al., 2012; Rabin et al., 2015). By taking into account this complexity, agricultural fires are often not represented in global vegetation-fire models because they do not directly affect natural vegetation and carbon cycle dynamics (Hantson et al., 2016), unless agricultural fires escape to nearby forests (Cano-Crespo et al., 2015). In summary, an improved representation of human effects on fire in global vegetation-fire models is currently lacking since globally consistent, temporally and spatial resolved, relevant information on infrastructure and socioeconomics is not available.

Direct wetness effects, especially based on the number of wet days, were the component of SOFIA models that contributed most to model performance (Fig. 3). These results are in agreement with previous results that identified the number of dry days (the inverse of the number of wet days) as an important variable to predict fire activity (Bistinas et al., 2014). Especially for shrublands, we identified strong exponential relationships with the number of wet days and the diurnal temperature range. Currently, shrubs are not considered in all ecosystem models (e.g. not in models of the LPJ family, Sitch et al., 2003), which suggests the need to implement and parameterize shrub PFTs to improve simulations of fire activity. The number of wet days and the diurnal temperature range are also used in process-oriented fire models like SPITFIRE to compute the Nesterov index (a fire weather index) and fuel moisture content (Thonicke et al., 2010). Here we confirm that the use of the diurnal temperature range and the number of wet days are appropriate predictor variables to simulate fuel moisture conditions and thus fire activity. However, while the Nesterov index is used as a fire weather index in many fire modules of global vegetation models (Lasslop et al., 2014; Prentice et al., 2011; Thonicke et al., 2010; Venevsky et al., 2002; Yue et al., 2014), studies on forest fire management rely more often on alternative fire weather indices such as from the Canadian Forest Fire Weather Index (FWI) (Bedia et al., 2012; Stocks et al., 1989). We also show that direct wetness effects can be represented by satellite-derived surface soil moisture. Additionally, several other indices have been derived from satellite data to estimate fuel moisture conditions (Yebra et al., 2013). Consequently, it is necessary to systematically compare the predictive power of fire weather indices, satellite-derived and reanalysis-based surface soil moisture data, and soil moisture schemes of ecosystem models to potentially improve the direct effect of wet conditions on fire activity in global vegetation-fire models.

Long-term vegetation effects contributed strongly to the performance of SOFIA models and thus indicate an important role of vegetation dynamics in the spatial–temporal variability of fire activity. Consequently, global vegetation models require a good representation of vegetation distribution and dynamics to realistically simulate fire activity. Vegetation distribution can be improved either through the prescription of high-quality land cover maps in land surface models or by improving model structures and by constraining model parameters that affect vegetation dynamics in DGVMs. For both approaches, time-variant, e.g. annually resolved, land cover maps would be very valuable for realistically reflecting vegetation dynamics. However, it is currently unclear how realistic land cover dynamics are represented for example by the three epochs of the ESA CCI land cover maps or by annual or seasonal maps of the MODIS land cover product (Broxton et al., 2014). Hence intensified efforts are required to check the plausibility of land cover changes in current and upcoming time-variant land cover maps.

SOFIA models with a long-term effect of VOD had better performances than models without this effect. The good performance of SOFIA models with VOD as predictor variable likely reflects variability in fuel loads because VOD is sensitive to vegetation density and biomass (Andela et al., 2013; Liu et al., 2015). The importance of VOD suggests that processes such as carbon allocation, turnover and vegetation mortality which all control biomass dynamics need to be carefully assessed in global vegetation models in order to accurately simulate fuel loads and hence fire activity. The finding of a strong restriction of fire activity with VOD in forests corresponds to previous findings that show that woody vegetation tends to restrict burned area either because moist wood is more difficult to ignite than dry grass or litter, or because forests provide generally more moist conditions (Kelley and Harrison, 2014). Fire activity increases with biomass at low vegetation densities and strongly decreases with increasing biomass and very high vegetation densities but the actual fire activity is enhanced or restricted by moisture conditions (Krawchuk and Moritz, 2011; Murphy et al., 2011). Consequently, the SOFIA approach and the identified sensitivities of fire activity with direct wetness effects and with VOD confirm and implement previous conceptual models where fire activity follows a biomass gradient and is modulated by moisture conditions (Krawchuk and Moritz, 2011; Murphy et al., 2011).

The better performance of SOFIA models compared to JSBACH-SPITFIRE and the generally good performance especially in temperate and tropical regions demonstrate the potential of the SOFIA approach to improve global vegetation-fire models. The SOFIA approach can be potentially adapted to more complex global vegetation-fire models such as SPITFIRE. Thereby the functional relationships in SOFIA models should rely on forcing datasets (e.g. temperature, precipitation) and simulated state variables (e.g. litter and soil moisture, biomass compartments, litter stocks, vegetation structure) of the vegetation models. This also allows the representation of feedbacks of changing vegetation conditions on fire activity. By applying the SOFIA approach to forcing and state variables of a process-oriented vegetation model, more adequate predictor variables could be potentially identified and finally model performance could be improved.

In order to represent realistic vegetation-fire interactions, vegetation models need to satisfactorily reproduce observed patterns and dynamics of fuel moisture and vegetation state variables. Consequently, it is necessary to test and improve global vegetation-fire models against multiple observational datasets that cover various aspects of vegetation-fire interactions: for example, satellite datasets on land cover, FAPAR, VOD, biomass (Avitabile et al., 2016; Saatchi et al., 2011; Thurner et al., 2014), and estimates of litter fuels (Pettinari and Chuvieco, 2016) may be useful to constrain vegetation dynamics, biomass allocation, and fuel loads; datasets on surface soil moisture, VOD, and evapotranspiration (Tramontana et al., 2016) may be useful to test hydrological schemes and to constrain fuel moisture; and datasets on burned area, fire size (Hantson et al., 2015b), fire radiative power, fuel consumption (Andela et al., 2016; van Leeuwen et al., 2014), or separations between natural and agricultural fires (Korontzi et al., 2006; Le Page et al., 2010; Magi et al., 2012) may be useful for constraining fire behaviour. Such datasets are currently under-exploited in the development of global vegetation-fire models because (1) they were still missing at the time of model development (Thonicke et al., 2001), (2) there is only little experience in applying formal model–data integration approaches within global fire modelling, or (3) no appropriate model components or observation operators exist that link for example modelled fuel moisture with satellite-derived surface soil moisture or modelled biomass compartments with VOD. For example, it is currently unclear which physiological processes, morphological plant components, and ecosystem structures contribute to a certain VOD signal (Vreugdenhil et al., 2016a). Consequently, it is necessary to better understand the plant and ecosystem controls on VOD to improve global vegetation-fire models.

Previously developed global fire models commonly used observed data for model evaluation, but did not undertake a formal model–data integration cycle from the definition of model structures, model parameter estimation, to model evaluation, and potentially back to a re-formulation of model structures by using observational data. In our study we firstly applied the full model–data integration cycle to derive an optimal structure for an empirical global fire model to predict global burned area. However, in order to apply model–data integration for global process-oriented vegetation-fire models, multiple datasets on vegetation, hydrological, and fire-related variables should be used to realistically constrain vegetation-fire interactions. Hence there is a need to develop appropriate observation operators and to extend currently existing model–data integration frameworks of global vegetation models (Forkel et al., 2014; Kaminski et al., 2013; MacBean et al., 2016; Schürmann et al., 2016) to the corresponding fire modules in order to formally assess model structures and to constrain model parameters. In summary, model–data integration frameworks need to be developed that make use of multiple satellite datasets on vegetation and moisture proxies in order to improve the representation of fire in global vegetation models and thus to better understand interactions of fire with ecosystems and the atmosphere within the Earth system.