Insights into the Drivers and Spatiotemporal Trends of Extreme Mediterranean Wildfires with Statistical Deep Learning

Jordan Richards aStatistics Program, Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

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Raphaël Huser aStatistics Program, Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

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Emanuele Bevacqua bDepartment of Computational Hydrosystems, Helmholtz Centre for Environmental Research–UFZ, Leipzig, Germany

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Jakob Zscheischler bDepartment of Computational Hydrosystems, Helmholtz Centre for Environmental Research–UFZ, Leipzig, Germany
cTechnische Universität Dresden, Dresden, Germany

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Abstract

Extreme wildfires continue to be a significant cause of human death and biodiversity destruction within countries that encompass the Mediterranean Basin. Recent worrying trends in wildfire activity (i.e., occurrence and spread) suggest that wildfires are likely to be highly impacted by climate change. To facilitate appropriate risk mitigation, it is imperative to identify the main drivers of extreme wildfires and assess their spatiotemporal trends, with a view to understanding the impacts of the changing climate on fire activity. To this end, we analyze the monthly burnt area due to wildfires over a region encompassing most of Europe and the Mediterranean Basin from 2001 to 2020 and identify high fire activity during this period in eastern Europe, Algeria, Italy, and Portugal. We build an extreme quantile regression model with a high-dimensional predictor set describing meteorological conditions, land-cover usage, and orography, for the domain. To model the complex relationships between the predictor variables and wildfires, we make use of a hybrid statistical deep learning framework that allows us to disentangle the effects of vapor pressure deficit (VPD), air temperature, and drought on wildfire activity. Our results highlight that while VPD, air temperature, and drought significantly affect wildfire occurrence, only VPD affects wildfire spread. Furthermore, to gain insights into the effect of climate trends on wildfires in the near future, we focus on the extreme wildfires in August 2001 and perturb VPD and temperature according to their observed trends. We find that, on average over Europe, trends in temperature (median over Europe: +0.04 K yr−1) lead to a relative increase of 17.1% and 1.6% in the expected frequency and severity, respectively, of wildfires in August 2001; similar analyses using VPD (median over Europe: +4.82 Pa yr−1) give respective increases of 1.2% and 3.6%. Our analysis finds evidence suggesting that global warming can lead to spatially nonuniform changes in wildfire activity.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jordan Richards, jordan.richards@kaust.edu.sa

Abstract

Extreme wildfires continue to be a significant cause of human death and biodiversity destruction within countries that encompass the Mediterranean Basin. Recent worrying trends in wildfire activity (i.e., occurrence and spread) suggest that wildfires are likely to be highly impacted by climate change. To facilitate appropriate risk mitigation, it is imperative to identify the main drivers of extreme wildfires and assess their spatiotemporal trends, with a view to understanding the impacts of the changing climate on fire activity. To this end, we analyze the monthly burnt area due to wildfires over a region encompassing most of Europe and the Mediterranean Basin from 2001 to 2020 and identify high fire activity during this period in eastern Europe, Algeria, Italy, and Portugal. We build an extreme quantile regression model with a high-dimensional predictor set describing meteorological conditions, land-cover usage, and orography, for the domain. To model the complex relationships between the predictor variables and wildfires, we make use of a hybrid statistical deep learning framework that allows us to disentangle the effects of vapor pressure deficit (VPD), air temperature, and drought on wildfire activity. Our results highlight that while VPD, air temperature, and drought significantly affect wildfire occurrence, only VPD affects wildfire spread. Furthermore, to gain insights into the effect of climate trends on wildfires in the near future, we focus on the extreme wildfires in August 2001 and perturb VPD and temperature according to their observed trends. We find that, on average over Europe, trends in temperature (median over Europe: +0.04 K yr−1) lead to a relative increase of 17.1% and 1.6% in the expected frequency and severity, respectively, of wildfires in August 2001; similar analyses using VPD (median over Europe: +4.82 Pa yr−1) give respective increases of 1.2% and 3.6%. Our analysis finds evidence suggesting that global warming can lead to spatially nonuniform changes in wildfire activity.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jordan Richards, jordan.richards@kaust.edu.sa

1. Introduction

With the frequency and severity of wildfires expected to be exasperated by climate change (Di Virgilio et al. 2019; Dupuy et al. 2020; Jones et al. 2020; Ruffault et al. 2020; Ribeiro et al. 2022), many countries with Mediterranean climates now experience regularly occurring and devastating wildfires. The combined effect of extensive heatwaves and droughts in 2022 led to Europe observing its second-largest annual burnt area on record by 4 August (Abnett 2022). Uncontrolled, wildfires pose a major risk to both environmental and ecological systems throughout the world; wildfires contribute significantly to global CO2 emissions (Liu et al. 2014; Copernicus 2021), with worrying trends predicted under a changing climate (De Sario et al. 2013; Knorr et al. 2016), and lead to destruction of biomass and biodiversity reduction among both plants and animals (Díaz-Delgado et al. 2002; Moreira and Russo 2007; Pausas et al. 2008; Bradshaw et al. 2011). Moreover, wildfires carry a number of anthropogenic health risks, with hundreds of human fatalities in the past few decades being directly attributed to European wildfires (San-Miguel-Ayanz et al. 2013; Kron et al. 2019; Molina-Terrén et al. 2019); the indirect consequences of wildfires on human health through increased air pollution and particulates are much more difficult to quantify (Jiménez-Guerrero et al. 2020; Weilnhammer et al. 2021). There exists a clear need for the development of robust statistical frameworks that can be used to facilitate the prevention and risk mitigation of European wildfires, particularly those that lead to extreme fuel consumption and burnt acreage, and thus, high economic cost and pollution. To that end, here we develop a model that identifies the drivers of Mediterranean Europe wildfire occurrence and extreme spread, while simultaneously producing risk maps that can be used to characterize high-risk areas.

The extremal characteristics of European wildfires have previously been studied using various statistical methodologies. de Zea Bermudez et al. (2009), Mendes et al. (2010), and Turkman et al. (2010) model sizes of individual wildfires in Portugal using the generalized Pareto distribution (GPD), with the latter two studies using a Bayesian hierarchical framework; a similar hierarchical model was applied to French wildfire sizes by Pimont et al. (2021). Point-process tools have been exploited for modeling occurrences of wildfires in Spain, using hurdle models (Serra et al. 2014), Portugal, via empirical clustering and kernel density estimation (Tonini et al. 2017), and France, using log-Gaussian Cox processes (Gabriel et al. 2017; Opitz et al. 2020; Koh et al. 2023). Ríos-Pena et al. (2018) adopt a zero-inflated semiadditive beta regression model for jointly modeling wildfire size and occurrence in Galicia. Wildfire size and impact are often characterized through measures of aggregated burnt area for spatiotemporal regions (Xi et al. 2019). While there exists some debate over appropriate probability distributions for wildfire sizes (Cumming 2001; Cui and Perera 2008; Hantson et al. 2016; Pereira and Turkman 2019), many studies have shown that burnt area is typically heavy tailed (Pereira and Turkman 2019; Koh et al. 2023; Richards and Huser 2022). As our focus is on modeling extreme wildfires, we employ the asymptotically justified GPD (Coles 2001).

Typical approaches designed to identify the drivers of wildfire risk often rely on regression-type statistical models; see, for example, Vilar del Hoyo et al. (2011), Vilar et al. (2016), Ríos-Pena et al. (2018), and Xi et al. (2019). While simple linear, and additive, regression models are computationally easy to fit and facilitate fast statistical inference, they cannot capture highly complex or nonlinear structure in data. Due to the complex and nonstationary nature of the climate in southern Europe and the Mediterranean Basin (Lionello et al. 2006), as well as the high diversity in land-cover types, that is, both fuel abundance and relative combustibility (San-Miguel-Ayanz et al. 2012; Malinowski et al. 2020), it is highly unlikely that simple regression models will be appropriate here. Recent advances in wildfire modeling have seen substantially better fitting models and predictive performance from machine learning and deep learning approaches [see, for example, Radke et al. (2019), Zhang et al. (2019), Bergado et al. (2021), Bjånes et al. (2021), Cisneros et al. (2023), and Koh (2023)] as these are significantly better than simple regression models at capturing complex structure in data and scale well to high-dimensional data. A more complex model is not necessarily guaranteed to provide a better fit to wildfire data, but we believe that this is a safe assumption to make as Richards and Huser (2022) showcase large gains in predictive power for a neural network–based model of extreme U.S. wildfire spread, relative to classical regression models. As their data share similar complexity with ours, we adapt aspects of their methodology to model European wildfires.

Modeling frameworks for fitting GPDs with deep learning exist (Rietsch et al. 2013; Carreau and Bengio 2007; Carreau and Vrac 2011; Ceresetti et al. 2012; Pasche and Engelke 2022; Wilson et al. 2022), but typically these models cannot be used to identify the drivers of risk; standard neural networks often lack interpretability due to their large number of trainable parameters. Recently, Richards and Huser (2022) proposed the partially interpretable neural network (PINN) framework for semiparametric regression, with the influence of a subset of predictors modeled using “interpretable” parametric, or semiparametric, functions and the influence of the rest of the predictors modeled using nonparametric neural networks (see section 3c); they used this framework to fit an extreme-value point-process model to U.S. wildfire data, and we extend their approach to create a bespoke GPD model for extreme European and Mediterranean wildfires. While Richards and Huser (2022) focused on identifying drivers and estimating extreme burnt area quantiles, we here also study spatiotemporal trends and climate change impacts.

The paper is outlined as follows: In section 2, we introduce the data used in our study of extreme European wildfires. Section 3 details the extreme-value deep learning model used to perform our analyses, with details of the GPD and PINN frameworks provided in sections 3b and 3c, respectively. Our analyses are presented in section 4 with separate consideration given for the interpretable results, wildfire risk assessment, and climate change impacts, in sections 4b–4d. We conclude the paper in section 5.

2. Data

As the impact of wildfires is not directly observable, we quantify it through a measure of burnt area, which is a useful proxy for both fuel consumption and emissions (Koh et al. 2023). Let {Y(s,t):sS,tT} be the aggregated burnt area (BA), measured in square kilometers, for a spatiotemporal grid box (s, t) indexed by a spatial domain SR2 and temporal domain TR+, and let {X(s,t):sS,tT} denote a d-dimensional space–time process of chosen predictor variables where X(s, t) = [X1(s, t), …, Xd(s, t)]T for all (s,t)S×T and for dN. We denote observations of X(s, t) and Y (s, t) by x(s, t) and y(s, t), respectively, and note that typically observations are recorded at a finite collection of space–time locations; we denote these space–time observation locations by ωS×T.

Our burnt area data are derived from version 5.1 of the Fire Climate Change Initiative (FireCCI) dataset (Lizundia-Loiola et al. 2020), which is generated by Moderate Resolution Imaging Spectroradiometer (MODIS) 250-m reflectance data and guided by active fire detection on a grid of pixels with a spatial resolution of 1 km2 (Otón et al. 2021). Our data have a spatial resolution of 0.25° × 0.25° with a temporal resolution of 1 month, and the observation period covers 2001–20. Figure 1 gives the monthly count and median area of nonzero BA values across the study region S, taken to be a regular grid of pixels over large areas of southern Europe and countries encompassing the Mediterranean Basin (see Fig. 2). We choose to model only the months of June–November inclusive, which is the period when most wildfires occur. Figure 1 highlights differences in the temporal structure in wildfire occurrences and spread and motivates us to model them separately (see section 3d).

Fig. 1.
Fig. 1.

Monthly counts (brown) and median (blue) of nonzero BA across the entire spatial domain (km2). The time series spans all months in 2001–20, inclusive.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. 2.
Fig. 2.

Maps of (a) observed log{1 + Y(s, t)} [BA; log(km2)], (b) 3-month SPI (unitless), (c) proportion of grassland coverage (unitless), (d) 2-m air temperature (K), (e) log{1 + λ(s, t)} [burnable area; log(km2)], and (f) dominant land-cover class for August 2001. Note that spurious values of SPI subceeding −3 have been truncated.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

We build a regression model with d = 38 predictors of three classes: meteorological, orographical, and land coverage. Thirteen meteorological variables are provided by the monthly ERA5 reanalysis on single levels (Hersbach et al. 2019), available through the Copernicus Climate Data Service, which is given as monthly averages on a 0.25° × 0.25° grid. Eleven variables are provided directly from ERA5: both eastern and northern components of wind velocity at 10 m above ground level (m s−1), temperature at 2 m above ground level (K; see Fig. 2d), potential evaporation (m), evaporation (m of water equivalent), surface pressure (Pa), surface net solar, and thermal, radiation (J m−2), and snowmelt, snowfall, and snow evaporation (m of water equivalent for all three). Monthly total precipitation (m) is used to derive a 3-month standardized precipitation index (SPI; unitless), which is illustrated in Fig. 2b; SPI was derived using the standardized precipitation evapotranspiration index (SPEI) package in R under the assumption that the data follow a gamma distribution.1 Hourly temperature and dewpoint temperature at 2 m above ground level (K) are used to derive monthly vapor pressure deficit (VPD, measured in Pa), which refers to the difference (deficit) between the amount of moisture in the air and how much moisture the air can hold when it is saturated. Note that 2-m dewpoint temperature and total precipitation are not included in the model to reduce colinearity among the predictors.

The four orographical predictors are latitude and longitude coordinates, and the mean and standard deviation of the elevation (m), for each grid cell. Elevation estimates are derived using a densely sampled gridded output from the R package “elevatr” (Hollister et al. 2017), which accesses Amazon Web Services Terrain Tiles (https://registry.opendata.aws/terrain-tiles/); the standard deviation of the elevation is here used as a proxy for terrain roughness.

We also used land-cover variables that describe the proportion of a grid cell composed of one of 21 different types, including water, tree species, urban areas, and grassland (see Fig. 2c); for a full list of labels, see Bontemps et al. (2015). We derive these predictors using a gridded land-cover map, of spatial resolution 300 m, that is also produced by Copernicus and is available through their Climate Data Service. For all 0.25° × 0.25° grid cells, the proportion of land-cover types is derived from the high-resolution land-cover product by counting the number of 300 m × 300 m cells of each type that fall within the boundaries of the larger grid cell. These predictors are dynamic and updated at the start of every year.

Alongside values of BA, FireCCI provides the “fraction of burnable area” for (s, t), that is, the fraction of a spatiotemporal grid box composed of burnable land-cover types;2; we use this to derive a measure of burnable area, denoted {λ(s, t)}, by counting the number of “burnable” cells from the high-resolution, 300 m × 300 m, product. Note that λ(s, t) is not constant over time t and the number of spatial locations with λ(s, t) > 0 decreases monotonically from 10 083 to 10 075 with t. We assume that wildfires cannot form at any space–time locations with λ(s, t) = 0, as no fuel is present, and so we treat observations y(s, t) at these locations as missing3 for all analyses (see Fig. 2); this leaves 1 209 066 observations. We further note that Y(s, t) must satisfy Y(s, t) ≤ λ(s, t) for all (s,t)S×T.

In Fig. 2, we illustrate observations of BA, λ(s, t), and selected predictors for August 2001. We focus on this month as (i) it exhibits one of the largest total burnt area values across the entire observation period (see Fig. 1) and (ii) it serves as a reference period when we investigate the impacts of climate change on the wildfire distribution in section 4d. Figure 2f provides the dominant land-cover class for each grid cell. Land-cover types are allocated to one of five classes: water, bare, and urban areas, with the remaining vegetation types classified as either tree or nontree; for each grid cell, we then plot the land-cover class which provides the largest proportion of its land cover. We observe that Europe is mostly dominated by nontree land-cover types, with forests typically being dominant in areas with lower average temperatures (see Fig. 2d).

3. Model

a. Overview

As our aim is to gain insight into the drivers of both wildfire occurrence and extreme spread, we propose the following two-stage model; we first estimate the probability of wildfire occurrence p0(s,t):=Pr{Y(s,t)>0|X(s,t)=x(s,t)} using a logistic partially interpretable neural network (see section 3c). We then independently model the extremal behavior of wildfire spread, that is, the distribution of BA given that a fire has occurred; this is given by Y(s, t)|{Y(s, t) > 0, X(s, t) = x(s, t)}, which we model using the so-called peaks-over-threshold approach, detailed in section 3b. Our choice to model these two components separately is motivated by observable differences in their spatial and temporal patterns (see Figs. 1 and 4). Estimates from the two models can then be combined to do inference on the full distribution of burnt area, Y(s, t)|X(s, t), by noting that Pr{Y(s, t) ≤ y|X(s, t) = x(s, t)} equals
1p0(s,t)+p0(s,t)Pr{Y(s,t)y|Y(s,t)>0,X(s,t)=x(s,t)}.
Note that, in this way, our full model for BA combines separate models for the occurrence probability and for the spread.

b. POT model

The peaks-over-threshold (POT) approach is a widely applied framework for modeling the upper tails of a random variable; see, for example, Pickands (1975), Davison and Smith (1990), and Coles (2001). For a random variable Y, we first assume that there exists some high threshold u such that the distribution of (Yu)|(Y > u) is characterized by the generalized Pareto distribution, denoted by GPD(σu, ξ), σu>0,ξR, with distribution function H(y) = (1 + ξy/σu)−1/ξ and support y ≥ 0 for ξ ≥ 0 and 0 ≤ y ≤ −σu/ξ for ξ < 0. Note that the scale parameter σu is dependent on u, that is, it changes as we condition upon Y exceeding a higher threshold; this makes it difficult to interpret estimates of σu if u varies over space–time locations (s, t). To that end, we exploit the threshold-stability law (Coles 2001) of the GPD and reparameterize the distribution in terms of a scale parameter σ = σuξu > 0, which is independent of u.

The shape parameter ξ controls the limit of the upper tail of Y: for ξ < 0 and ξ ≥ 0, we have that Y has a bounded, and infinite, upper tail, respectively. If ξ ≥ 1, then Y is very heavy tailed with infinite expected value. This property is considered to be inappropriate for environmental applications, and so we constrain ξ < 1 throughout. A number of studies have shown that wildfire burnt areas are heavy tailed (with ξ > 0; see Pereira and Turkman 2019; Koh et al. 2023; Richards and Huser 2022), but our data satisfy the natural physical constraint that burnt areas are bounded above by the available burnable area [i.e., y(s, t) < λ(s, t) for all (s,t)S×T; see Fig. B1 in appendix B]. This suggests that a model with bounded upper tail (i.e., ξ < 0) would, in principle, be more appropriate. However, it can be computationally problematic to fit such a model due to issues regarding numerical instability (Smith 1985) and parameter-dependent support of the distribution, which makes inference with neural networks particularly difficult (Richards and Huser 2022); we thus further constrain ξ > 0 throughout. We note that, in theory, our model is inconsistent with the physical characteristics of the data generating process, but, in practice, the upper-tail behavior of the data is approximated sufficiently well provided that one does not extrapolate too far into the upper tail. At observable levels, we find that a model with ξ > 0 fits our data better than a model with ξ < 0 and excellent model fits are obtained with ξ > 0; see section 4c; the threshold u would need to be excessively high to see the benefit of imposing ξ < 0.

We model the extremal behavior of nonzero spread Y(s, t)|{Y(s, t) > 0, X(s, t) = x(s, t)} using the POT approach, with a spatiotemporally varying exceedance threshold u(s, t) > 0 and a scale parameter σ(s, t) > 0 such that they are both functions of predictor set X(s, t) described in section 2; similarly to Koh et al. (2023), we fix ξ over all space and time for computational efficiency. Hence, we have {Y(s, t) − u(s, t)}|{Y(s, t) > u(s, t), X(s, t) = x(s, t)} ∼ GPD{σ(s, t), ξ}; models for u(s, t) and σ(s, t) are detailed in section 3c. While u(s, t) is difficult to interpret, the scale σ(s, t) can be interpreted as a measure of the conditional severity of extreme wildfire spread, should a fire occur at site s in month t. The distribution function of Y(s, t)|{Y(s, t) > 0, X(s, t) = x(s, t)}, denoted by F(s,t),0(y) for all (s,t)S×T, is given by
F(s,t),0(y)=1pu(s,t)[1+ξ{yu(s,t)}σ(s,t)+ξu(s,t)]1/ξ,ify>u(s,t),
where pu(s, t) := Pr{Y(s, t) > u(s, t)|Y(s, t) > 0, X(s, t) = x(s, t)} ∈ [0, 1]. As our interest lies only in the extremes of Y(s, t), for yu(s, t), we estimate F(s,t),0(y) empirically; here, we use the estimator
F^(s,t),0(y)={1p^u(s,t)}[(s,t)ω1{y(s,t)y}]×[(s,t)ω1{y(s,t)u(s,t)}]1,
where 1 denotes the indicator function and p^u(s,t) denotes an estimate of pu(s, t), with observations pooled across all space–time observation locations ω. An alternative approach to an empirical estimator could be to use a parametric distribution for the bulk and lower tail of Y(s, t)|Y(s, t) > 0; see, for example, Carreau and Bengio (2007) and Opitz et al. (2018). Similar zero-inflated models with empirical bulk and GPD upper tails have been employed by Richards et al. (2022) and D’Arcy et al. (2023) for modeling extreme precipitation and U.S. wildfire burnt areas, respectively.

Typically, we would estimate the threshold u(s, t) as some high τ quantile of nonzero spread Y(s, t)|{Y(s, t) > 0}, for τ ∈ (0, 1). If u(s, t) is assumed known, then it follows that pu(s, t) = 1 − τ for all (s, t); however, given the highly nonstationary model that we wish to fit and the complexity of the data, it may be inappropriate to assume that u(s, t) is exactly the required τ quantile. Instead, we use a neural network to model pu(s, t) (see section 3c), which leads to improved model fits through increased flexibility of (2). [To illustrate the components of (2), Fig. B3 in appendix B gives maps of observed Y(s, t), for August 2001, and the corresponding estimates of u(s, t) (with τ = 0.4), pu(s, t) and exceedances Y(s, t) − u(s, t).]

c. Partially interpretable neural networks

As in Richards and Huser (2022), we represent model parameters using PINNs. Consider a general parameter θ(s, t), which is a function of space–time locations (s,t)S×T; for example, we may take one of θ := σ, θ := u, θ := p0, or θ := pu, for σ, u, and pu, defined in (2), and p0 defined in section 3a. For all (s, t), we divide the predictor process {X(s, t)} into two complementary components: “interpreted” and “noninterpreted” predictors, denoted by {XI(s,t)} and {XN(s,t)}, respectively, where for I ∈ {0, 1, …, d − 1} we have that XI(s,t)RI and XN(s,t)RdI are distinct subvectors of X(s, t), with observations of the components denoted by xI(s,t) and xN(s,t), respectively; we also note that the indices mapping x(s, t) to xI(s,t) are consistent across all (s, t) and similarly for xN(s,t). We then represent θ as
θ(s,t)=C(s,t)h[mI{xI(s,t)}+mN{xN(s,t)}],
for functions mI:RIR,mN:RdIR, a link function h:RR controlling the range of θ, and a scaling function C(s,t):(S,T)R. In cases where θ ∈ (0, 1) (e.g., for θ := p0 or θ := pu), we set h to the logistic function, that is, h(x) = 1/2 + tanh(x/2)/2. When θ > 0, for example, for θ := u or θ := σ, we take h(x) = exp(x).
Richards and Huser (2022) assert that the function mI must be interpretable, that is, it must be simple and either parametric, for example, linear, or semiparametric (e.g., additive). We choose to adopt thin-plate splines (Wood 2003) to independently model the effect of each component of xI(s,t)=[xI,1(s,t),,xI,I(s,t)]T on mI; that is, we let
mI{xI(s,t)}=j=1ImI,j{xI,j(s,t)},
where for all j = 1, …, I, each mI,j is a thin-plate spline of xI,j. The function mN is assumed to be of unknown structure and highly nonlinear and is approximated via a neural network (NN) with a suitable architecture. As neural networks are not readily interpretable, we should choose xI(s,t) to be the set of predictors for which we wish to infer their influence on the parameter θ; the remaining dI predictors form xN. In the case where mI(s,t)=0 for all (s, t) or I = 0, we describe the model as “fully NN.”

We use representation (4) for each of the parameters p0, pu, u, and σ, albeit with different predictor components xI and xN and functions h, mI, and mN; see section 4a for full details. The scaling function C(s, t) also differs between parameters; for p0 and pu, we simply set C(s, t) = 1 for all (s, t). However, for u and σ, we found improvements in model fit when setting C(s, t) = λ(s, t), that is, by using burnable area as an offset term. With this representation, the predictor set x(s, t) directly influences properties of the distribution of the proportion of burnable area that is burnt by wildfires, rather than the absolute value of burnt area; this accounts for the high variability in λ(s, t) observed over the observed domain ωS×T. For models with C(s, t) = 1, we include λ(s, t) as an extra predictor in xN(s,t), raising the number of predictors in these models to d = 39.

Different types of neural network exist for estimating mN in (4), which capture different structures in xN. For estimating GPD parameters, Rietsch et al. (2013), Carreau and Bengio (2007), Carreau and Vrac (2011), and Wilson et al. (2022) employ feed-forward neural networks, while Pasche and Engelke (2022) use recurrent neural networks built with long short-term memory (LSTM) layers (Hochreiter and Schmidhuber 1997). In their application, Richards and Huser (2022) favor the use of feed forward over recurrent layers for estimating mN within the PINN framework, due to the high computational demand and a large number of trainable parameters required for training the latter; we thus adopt their approach and construct a feed-forward PINN. We use only feed-forward NNs in our model but employ different types for each parameter [i.e., p0 in (1) and σ, u, and pu, in (2); see section 4a for details]. We use either a convolutional neural network (CNN) or densely connected NN; for details of the two types, see appendix A.

d. Inference

We use a two-stage procedure to construct a model for the full distribution of burnt area Y(s, t)|{X(s, t) = x(s, t)}. We first simultaneously estimate u(s, t), the τ quantile of nonzero spread Y(s, t)|{Y(s, t) > 0, X(s, t) = x(s, t)}, and occurrence probability p0(s, t); we then use estimates of u(s, t) to estimate pu(s, t) and the GPD parameters σ(s, t) and ξ. Combining the aforementioned functional parameter estimates with the empirical estimator F(s,t),0*, defined in (3), allows us to then estimate the distribution function of burnt area Y(s, t)|{X(s, t) = x(s, t)}; a schematic illustrating this modeling procedure is presented in Fig. 3.

Fig. 3.
Fig. 3.

Schematic of methodology to estimate the distribution of BA Y|(X = x). Dashed lines denote a connection through the exceedance threshold u only. Blue, red, and green boxes denote data, parameters, and models, respectively. The space–time index (s, t) has been dropped from notation for brevity.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

The neural networks in each parameter model are trained using the pinnEV package (Richards 2022), which utilizes the R interface to Keras (Allaire and Chollet 2021). We use the adaptive moment estimation (Adam) algorithm (Kingma and Ba 2014),4 to minimize the negative log-likelihood associated with each model; for p0 and pu, this is equivalent to the binary cross-entropy loss or the negative log-likelihood for a Bernoulli random variable; for u(s, t), we use the tilted quantile loss (Koenker 2005)
l{u(s,t);y(s,t)}=(s,t)ωmax[τ{y(s,t)u(s,t)}, (τ1){y(s,t)u(s,t)}].
For σ(s, t) and ξ, we minimize the negative log-likelihood for the GPD (Coles 2001). In cases where y(s, t) is treated as missing, we remove its influence from the loss function. For all model fits, we use a validation and testing scheme to reduce overfitting and improve out-of-sample prediction, with details provided in appendix A; for a review of cross-validation techniques for interpreting machine learning-based spatiotemporal models, see Sweet et al. (2023). We do not use any form of dropout during training, as the NNs we use are relatively simple (see section 4a). To quantify all parameter uncertainty, we utilize a stationary bootstrap scheme (Politis and Romano 1994) with an expected block size of 2 months (see appendix A) and with the entire model reestimated for each bootstrap sample.

4. Results

a. Overview

We choose to interpret the effect of I = 3 variables on the occurrence probability of wildfires and the σ parameter determining extreme spread; these are VPD, temperature, and 3-month SPI, which were chosen as these are important drivers of fire occurrence and are strongly impacted by climate change in the Mediterranean (Giorgi and Lionello 2008; Bevacqua et al. 2022). As covariate effects on the parameters pu and u are difficult to interpret, we estimate both using a fully NN model, that is, we do not interpret the effect of any predictors and set xN(s,t):=x(s,t) for all (s, t); this improves model fits through added model flexibility. To assess the impact associated with the interpreted predictors, we model p0 and σ using the PINN framework described in section 3c. Before training, predictors are standardized by subtracting and dividing by their marginal means and standard deviations to improve numerical stability.

To perform model selection and determine the optimal hyperparameters and network architecture, we evaluate scores and diagnostics on test data using the estimated parameters from the fitted models for each bootstrap sample; recall that the test data are not used in model fitting. For p0(s, t) and pu(s, t), we compare model fits using the area under the receiver operating characteristic curve (AUC). For details on comparing fits of the GPD PINN model (2), see appendix A.

All neural networks are trained with minibatch size equal to the number of observations; a model checkpoint is saved at each of 10 000 epochs and only the estimate that minimizes the validation loss is returned. Under our model selection scheme, we determine that the optimal architecture for estimating mN in (4) differs between the four parameters. As inference on p0 uses all observations of burnt area Y(s, t), it requires the most complex architecture: a five-layered CNN (see appendix A) with 16 filters/nodes per layer. The threshold u(s, t) is estimated with nonexceedance probability τ = 0.4 and uses a two-layered CNN with four filters/nodes per layer. We use densely connected NNs to estimate pu and σ; precisely, we use a five-layered network with 16 nodes per layer and a four-layered network with 10 nodes per layer for the former and latter, respectively.

We use 250 bootstrap samples to assess model and parameter uncertainty and present results for the quantities of interest (e.g., quantiles and model parameters) as the empirical median and pointwise quantile estimates across all bootstrap samples. The splines used to model mI are estimated with knots taken to be marginal quantiles of the predictors {x(s, t) : (s, t) ∈ ω} with equally spaced probabilities; splines for p0 and σ use 10 and 6 knots, respectively. Inference on mI is conducted by considering the individual contribution of each interpreted predictor to mI, denoted by m^I,j(xj) for j = 1, 2, 3. Estimates of each spline m^I,j(xj) for each bootstrap sample are centered by subtracting the value of the spline evaluated at the median of the predictor values; it is important to center the curves as we are interested in relative changes in mI, not its absolute value, and it circumvents issues related to identifiability between mI and mN (Richards and Huser 2022). Uncertainty in the centered spline estimates is then presented using functional boxplots (Sun and Genton 2011); these can be considered to be analogs of the classical box-and-whisker plots for functional data. The central region gives an indication of the “central” 50% of curves, with the median taken to be the most central. Maximum envelopes (blue curves in Fig. 5) encompass all functions not identified as outliers. For full details on the definition of “centrality” and identification of outliers for functional data, see details in Sun and Genton (2011).

As an exploratory analysis, we present a climatology of observed, and modeled, wildfire occurrence and spread, in Fig. 4. Here, we present sitewise medians of model estimates of wildfire occurrence probability [i.e., p0(s, t)], and conditional spread intensity [i.e., σ(s, t)], as well as sitewise empirical estimates of p0(s, t) and median nonzero spread Y(s, t)|Y(s, t) > 0. Excellent fit is obtained for the occurrence probability, and extreme spread, models, with details of goodness of fit discussed in section 4c. All metrics are averaged over all months and years in the observation period; model outputs are averaged over all bootstrap samples and empirical estimates of p0(s, t) are derived by counting the mean number of wildfire occurrences within the observation period. Figure 4 highlights areas where fires are likely to occur in any given month and the average sitewise intensity of those wildfires. During 2001–20, areas of concern include the Nile Delta and Turkey, where the probability of wildfire occurrence is very low (see Figs. 4a,c), but extreme wildfires were particularly intense (Figs. 4b,d), and Ukraine, which exhibits both high probability of occurrence and intensity; the Alps and Carpathians experience very few, if any, wildfires, and the estimated intensity in these regions is relatively low. We note that, given the short observation period of the data, there is some uncertainty around these estimates and the inferences drawn may not apply to other 20-yr periods.

Fig. 4.
Fig. 4.

Maps of sitewise (a) empirical fire occurrence probability p0(s, t) (unitless) and medians of (b) observed log{1 + Y(s, t)}|Y(s, t) > 0 [conditional spread; log(km2)], (c) estimated p0(s, t) (unitless), and (d) estimated log{1 + σ(s, t)} [conditional spread severity; log(km2)]. Metrics are averages over all months and years in the observation period.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

b. Interpretable results

Figure 5 gives estimates m^I,j,j=1,2,3, for both the occurrence probability p0 and conditional spread severity σ. As the link function h in (4) is taken to be the logistic and exponential functions for p0 and σ, respectively, the scales on the y axes in the top and bottom rows of Fig. 5 indicate the contribution of each predictor to the log-odds of p0 and logσ, respectively. We can roughly determine if there is a “significant” effect of a predictor on a parameter by observing whether the magenta regions in Fig. 5 encompass the horizontal line intersecting the origin. We observe a positive relationship between temperature and p0, and the converse holds for the SPI drought index; this is in line with the fact that hotter and drier climates produce readily ignitable fuel. The results for vapor pressure deficit are less easy to interpret. We first note that an initial increase in VPD above zero leads to an increase in p0. As VPD increases, the air gets relatively drier; hence, we find agreement with the results for the SPI. However, as VPD increases above approximately 1500 Pa, we begin to observe a reduction in p0 with VPD. This may be due to a lack of availability of burnable fuel, as regions with high values of monthly VPD typically have arid climates with little vegetation. Overall, note that temperature has the strongest impact on p0 relative to VPD and SPI (comparing the scales of the y axes in Fig. 5).

Fig. 5.
Fig. 5.

Functional boxplots of estimated additive function contributions mI,j(x),j=1,2,3 to the (top) log-odds of fire occurrence probability p0 and (bottom) log conditional spread severity log(σ); an increase of one on the y axis corresponds to an increase in one of these parameter functions. Blue curves denote maximum envelopes, and the black curve gives the median function over all bootstrap samples. The red dashed curves represent outlier candidates, and the 50% central region is given in magenta. Red triangles denote the locations of knots, and the red horizontal lines intersect mI,j(x)=0.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

We observe major differences in the spline results for p0 and σ, suggesting that the interpreted predictors do not impact these two parameters in a similar fashion. The scale parameter σ can be intuited as a measure of conditional wildfire spread severity; relatively large values of σ suggest that, given the occurrence of a wildfire, the magnitude of extreme wildfire spread above the threshold u is likely to be larger. Similar results are found for the effect of VPD on σ as for VPD on occurrence probability p0; an initial positive relationship between VPD and conditional spread severity is observed as VPD increases above zero, but the reverse holds as VPD increases above approximately 2500 Pa. While there appears to be a significant effect of VPD on σ, the same does not seem to hold for 2-m air temperature and 3-month SPI. For temperature, we observe that temperatures larger than the median do not lead to a significant increase in σ. At the lower end, we observe a small negative relationship between temperatures below the median and σ, with lower temperatures leading to an increase in conditional wildfire spread severity. This is an interesting result that may be caused by the differences in the distribution of land-cover types in regions with lower monthly temperatures (see Fig. 2f).

Figure 5 suggests that an increase in SPI values greater than zero leads to a small but significant decrease in σ. Oddly, we find no significant effect for changes in SPI less than zero, which corresponds to space–time locations experiencing less 3-monthly rainfall relative to the average conditions, that is, drought conditions; this seems counterintuitive as we would expect drought conditions to facilitate extreme wildfire spread. The GPD models extreme wildfire spread conditioned on the prerequisite of a wildfire occurrence; this decomposition reveals that drought conditions may facilitate only wildfire occurrence. Once the drought conditions are such that a wildfire ignites, further changes do not impact extremes of the distribution of the subsequent spread. Moreover, the lack of a significant effect of air temperature and SPI on σ may be evidence to suggest that much of the nonstationarity in the upper tail of the spread distribution can be accommodated by the nonstationary threshold model u(s, t); with the variability captured in u(s, t), the parameter function σ, and hence the distribution of extreme spread, is almost stationary with respect to the predictors. Note that these inferences are made for the upper tails of the spread distribution only; consideration for nonextreme spread is outside of the scope of our analyses.

c. Risk assessment

Figure 6 provides maps of observations of log{1 + Y(s, t)} (burnt area) and the modeled probability of fire occurrence, that is, the median estimated p0(s, t), for fixed t corresponding to chosen months. Here, we consider 4 months: August 2001, August 2008, October 2016, and November 2020. The first 2 months are considered as across the observation period these were the most devastating in terms of the total burnt area across the entirety of the spatial domain; we also consider October 2016 as this month observed the largest recorded value of BA5 and November 2020, which had the lowest total wildfire spread across all months.

Fig. 6.
Fig. 6.

(left) Observed log{1 + Y(s, t)} [BA; log(km2)] and (right) bootstrap median estimated fire occurrence probability p0(s, t) (unitless) for (a),(e) August 2001, (b),(f) August 2008, (c),(g) October 2017, and (d),(h) November 2020.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

For each bootstrap model fit, we evaluate the AUC for the fitted occurrence probability models on both the entire original data and the test data for the bootstrap sample, that is, an out-of-sample estimate. The median values (2.5% and 97.5% quantiles) across all bootstrap samples are 0.947 (0.944, 0.949) and 0.948 (0.940, 0.955), for the original and test data, respectively, suggesting that the chosen architecture provides an excellent predictor for the occurrence of wildfires. These values, alongside the maps of estimated p0 given in Fig. 6, suggest that the model predicts well the probability of wildfire occurrence; we find agreement in the predicted p0 values and the observations of occurrence, for the period 2001–20. For the chosen months, notable areas of concern include large portions of eastern Europe, including Ukraine, Romania, Bulgaria, and Serbia, as well as the north of Portugal and the Galicia region of Spain, as these regularly experience large estimates of p0; we also observe high probabilities of wildfire occurrence in northern Algeria, Turkey, and Italy.

To assess the fit of conditional model (2) for extremes of wildfire spread, we provide a pooled quantile–quantile (Q–Q) plot (Heffernan and Tawn 2001) (see Fig. B2 of appendix B); the estimated models are used to transform all original observations of nonzero spread Y(s, t)|Y(s, t) > 0 onto standard exponential margins, and we then compare theoretical quantiles against empirical ones derived using the fitted model; this procedure is repeated for each bootstrap sample and we observe good fits, particularly in the upper tails, as the estimated 95% tolerance bands include the diagonal. Despite the physical constraint that the burnt area must subcede the total burnable area for a grid box, that is, Y(s, t) ≤ λ(s, t), the distribution of Mediterranean wildfire spread is well approximated by a heavy-tailed distribution, with the median of the shape parameter estimates (2.5% and 97.5% quantiles) being estimated as 0.322 (0.280, 0.353). Richards and Huser (2022) find similar values for the shape parameter estimates of burnt area due to U.S. wildfires; however, they model the square root of the response, rather than its unadulterated counterpart, which suggests that wildfire spread due to Mediterranean wildfires is considerably lighter tailed than those occurring in the United States.

Figure 7 provides maps of the bootstrap median estimated log{1 + σ(s, t)} (conditional spread severity) and the bootstrap median of estimated 95% quantiles of log{1 + Y(s, t)}|X(s, t) (burnt area), for the four considered months; we note that the latter metric concerns quantiles of all values of burnt area, not just strictly positive values (i.e., spread), and so can be considered as a measure of compound risk that combines both the probability of wildfire occurrence and the spread distribution. Through Figs. 6 and 7, we observe that there is not necessarily a one-to-one correspondence between wildfire occurrence probability, conditional spread severity, and compound risk, that is, locations with high p0 also have high σ and high 95% quantile estimates. Notable regions include the northern parts of Africa, particularly the Nile Delta in Egypt, as well as parts of Spain; here, we see that the climate conditions and fuel type suggest a particularly high wildfire spread severity across the months, but as p0 remains low in these locations, they exhibit relatively low compound risk. We quantify uncertainty for the maps in Figs. 6 and 7 via the 2.5% and 97.5% bootstrap quantiles of occurrence probability p0(s, t), conditional spread severity σ(s, t), and the estimated 95% quantile of log{1 + Y(s, t)}|X(s, t) (burnt area); these are provided in Figs. B4B6, respectively, of appendix B.

Fig. 7.
Fig. 7.

Bootstrap median estimated (left) log{1 + σ(s, t)} [conditional spread severity; log(km2)] and (right) 95% quantile of log{1 + Y(s, t)}|X(s, t) [BA; log(km2)] for (a),(e) August 2001, (b),(f) August 2008, (c),(g) October 2017, and (d),(h) November 2020.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

d. Impacts of long-term climate trends

To gain insights into the impact of climate trends on extreme wildfire events, we focus on the month of August 2001, which was chosen as it exhibits the highest total burnt area throughout the observation period. We estimate how the distribution of wildfires in August 2001 may have looked like under observed changes in the interpreted predictors during 2001–20 (Maraun et al. 2022). Given that we ultimately aim at gaining insights into how climate change–driven trends in interpretable predictors will affect wildfires, here we considered observed trends in temperature and VPD only. This is because observed temperature and VPD trends are already generally in line with trends expected from anthropogenic climate change (Hawkins and Sutton 2012), while observed SPI trends are yet largely dominated by internal climate variability. In other words, climate change trends in SPI are expected to emerge from the noise of internal climate variability in the future (Maraun 2013; Zappa et al. 2021). Overall, considering trends in VPD and temperature in August, which are critical drivers of wildfire changes, allows for gaining insights into the ongoing impact of climate change–driven trends on fire activity. However, we note that changes in wintertime and spring conditions, as well as snowpack changes, will further shape wildfire activity changes.

In practice, we investigate the impact of changing temperature and VPD on the wildfire distribution estimates (Bevacqua et al. 2020). For each bootstrap sample, we estimate the model using input predictors x determined by one of two scenarios: scenario (i), we use the observed conditions in August 2001 (as in Figs. 6 and 7), and scenario (ii), the value of VPD at each site is perturbed by adding, to the observed values in August 2001, the long-term trends in August values of VPD calculated over the period 2001–20. We then derive the sitewise differences in estimates of two distribution-related metrics that arise under these two scenarios; we investigate changes in p0(s, t), which describes the occurrence probability of wildfires, and q0.9+(s,t), the 90% quantile of nonzero spread Y(s, t)|{X(s, t), Y(s, t) > 0}, which is a measure of intensity, conditional on fire occurrence, derived using parameters6 u and σ. We consider q0.9+, rather than quantiles of unconditional burnt area (i.e., only Y), as changes in q0.9+ are driven by changes in u and σ only. Changes in p0 have no impact on q0.9+, and so we are able to disentangle the effects of long-term climate trends on the frequency, and severity, of Mediterranean wildfires. Note that in scenario (ii) above, only values of VPD are changed, while the other d − 1 predictor variables remain the same; in this way, estimates from scenario (ii) give an indication as to how the distribution of wildfires in August 2001 may have looked like with VPD conditions having experienced the 20-yr observed trend. This procedure is repeated with 2-m air temperature replacing VPD as the chosen perturbed predictor.

We predict values of 2-m air temperature and VPD for August 2020 at each site sS using expectation regression, under the assumption that the variable follows a linear temporal trend (i.e., a simple linear regression model with only year as a covariate). Trends are estimated using the ERA5 data for August only, across 2001–20. Although the trends are assumed to be heterogeneous over S and are estimated at each site independently, we pool data from adjacent grid cells to improve estimation. The median values (2.5% and 97.5% quantiles) of the estimated yearly sitewise changes, across the entire domain, are +4.82 Pa (−4.29, 26.4) for VPD and 0.042 K (−0.012, 0.103) for air temperature. Maps of the estimated trends (i.e., the regression coefficients from the fitted linear models) are provided in Fig. 8 for both air temperature and VPD. As discussed above, in line with climate change trends (Zappa et al. 2021), for both variables, we observe generally positive trends across the entirety of Mediterranean Europe. Negative trend estimates are rare; Fig. 8 does illustrate a slight predicted reduction in both VPD and 2-m air temperature in northern Libya and in VPD only in northern Italy and west Turkey.

Fig. 8.
Fig. 8.

(a) Sitewise estimated trends (change per year) in August VPD (Pa) observed over the period 2001–20. Under these trends, maps of the sitewise (bootstrap median) changes in estimates of the (b) occurrence probability p0(s, t) (unitless) and (c) log{1+q0.9+(s,t)} [log(km2)] [withq0.9+(s,t), the 90% quantile of the spread Y(s, t)|{Y(s, t) > 0, X(s, t)}, a measure of wildfire intensity] for the month of August 2001. (d)–(f) As in (a)–(c), but using trends in August 2-m air temperature (K).

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Figure 8 presents maps of the median sitewise differences (for separate perturbations in VPD and air temperature) in p0 and q0.9+ across all bootstrap model fits. Positive increases in p0 and q0.9+ correspond to an increased expected frequency and intensity, respectively, of wildfires under observed trends for VPD and air temperature. As noted in section 4c, there is no clear one-to-one correspondence between wildfire occurrence and spread; areas that exhibit positive trends in wildfire frequency do not necessarily exhibit positive trends in intensity, and vice versa. Under perturbations in VPD, we observe a small increase in the intensity of extreme wildfires across some of Europe (Fig. 8c) but observe little increase in the expected frequency of wildfires anywhere in the domain (Fig. 8b); p0 increases only in Ukraine and Romania. Our model predicts reductions in p0 in regions such as northern Algeria and southern Spain, with relatively large reductions observed in eastern Turkey and Syria (Fig. 8b); for the latter two areas, a concomitant reduction in wildfire intensity is also predicted (see decreasing q0.9+ in Fig. 8c). Figure 8c illustrates surprisingly large reductions in wildfire intensity in eastern Ukraine and western Russia, as well as small decreases in parts of Algeria, Morocco, and northern Saudi Arabia. Concerning relative, rather than absolute, differences, the median (2.5% and 97.5% quantiles) of the sitewise estimates in frequency and severity were 1.2% (−29.4%, 16.4%) and 3.6% (−9.5%, 12.3%), respectively.

Figures 8e and 8f show a much stronger signal in both occurrence probability p0 and conditional spread intensity q0.9+ when air temperature is perturbed. Under changes in 2-m air temperature, the frequency of August wildfires appears to be increasing at a much faster rate across Europe and North Africa, with no areas exhibiting an obvious reduction in their respective p0 estimates. Unlike the estimated intensity trends observed under changes in VPD, Fig. 8f illustrates that changes in temperature appear to adversely affect Ukraine and western Russia; we observe strong positive changes in q0.9+ in these regions, compared to Fig. 8c. Figure 8f also illustrates positive trends in q0.9+ across the vast majority of Europe, with no regions in the domain exhibiting a reduction in wildfire intensity. The median values (2.5% and 97.5% quantiles) of relative sitewise changes in frequency and severity were 17.1% (−3.1%, 60.6%) and 1.6% (−5.0%, 6.9%), respectively. This suggests that, under our observed warming trends, fires are becoming more frequent and serve, but not at a proportional rate. Uncertainty in Figs. 8b, 8c, 8e, and 8f is quantified via the 2.5% and 97.5% bootstrap quantiles of changes in p0 and q0.9+ for the two scenarios; these are presented in Fig. B7 of appendix B. Figures B7a, B7c, B7i, and B7k cast some doubt on the significance of the results observed in Figs. 8b and 8c, as most of the sitewise 95% confidence intervals include zero; the converse holds for Figs. 8e and 8f, suggesting that the worrying positive trends observed for p0 and q0.9+, under perturbations in air temperature, are significant. A similar analysis is performed for August 2008, August 2015, October 2017, and November 2020 and presented in Figs. B8B11 of appendix B. For the August months, we observe consistency with the results for August 2001 (see Fig. 8). In October 2017 and November 2020, we observe stronger inhomogeneity, compared to August, in changes of p0 and q0.9+ across the domain. Under perturbations in air temperature, we observe that many countries in the south of the domain experience a reduction in q0.9+, while p0 values do not generally change significantly. On the contrary, perturbations in VPD lead to increases in both p0 and q0.9+ across much of the spatial domain.

5. Conclusions

We develop a hybrid statistical deep learning framework that combines asymptotically justified extreme-value models, generalized additive regression, and neural networks, to investigate the drivers of both wildfire occurrence and extreme spread in Mediterranean Europe. The impact of vapor pressure deficit (VPD), 2-m air temperature, and a 3-month standardized precipitation index (SPI), on model parameters determining the occurrence probability, and the scale of extreme spread, of wildfires is accommodated through the use of thin-plate splines, while the effect of a large number of other covariates determining meteorological and land surface conditions is modeled using neural networks. We investigate spatiotemporal trends in wildfire frequency and intensity relative to perturbations in VPD and air temperature by estimating their spatially localized linear trends; these are then fed into the model to determine how these trends would have affected wildfire occurrence probabilities and the spread distribution parameter for the month of August 2001, which is the month with the highest observed total burnt area.

Our analysis reveals that different drivers impact Mediterranean Europe wildfire occurrence and extreme spread. While VPD, air temperature, and SPI all affect the former, only VPD appears to have a significant effect on extreme wildfire spread; although similar conclusions about wildfire occurrence have been drawn by many studies, for example, by Turco et al. (2014, 2017), Ruffault et al. (2018), Parente et al. (2019), and de Dios et al. (2021, 2022), the conclusions regarding extreme spread are less consistent with the literature, which further supports our claim, in section 4b, that nonstationarity in the upper tails of the spread distribution are captured in the model for u(s, t). As the threshold u(s, t) is not easily interpretable, we chose not to represent this aspect of the model using the PINN framework described in section 3c; in order to better understand the impact of the interpretable predictors on the extreme values of wildfire spread, we may be better suited replacing the GPD tail model with an entire bulk-tail model that foregoes prespecification of an exceedance threshold u(s, t), such as those proposed by Papastathopoulos and Tawn (2013), Naveau et al. (2016), and Stein (2021). We further find that, when comparing month-specific maps of estimated wildfire occurrence probability and extreme quantiles of wildfire spread across Mediterranean Europe, the meteorological and land surface conditions that lead to high estimates of occurrence probabilities do not necessarily lead to high estimates of spread quantiles. Maps of estimated extreme quantiles of wildfire spread can facilitate risk assessment as they provide a means of identifying high-risk areas of Europe; we highlight areas of major concern in eastern Europe and the Iberian Peninsula. By focusing on the extreme wildfires in August 2001, we find that the impact on wildfire activity of ongoing trends in VPD and temperature, which are critical in view of climate change, may not be homogeneous across Mediterranean Europe; while many regions exhibit worrying positive trends in wildfire frequency and intensity, some locations do exhibit a reduction in extreme wildfire risk. We further find that ongoing trends in VPD and temperature may lead to substantially different changes in the expected frequency and severity of wildfires. For wildfires in August 2001 and on average over Europe, observed trends in air temperature lead to a relative increase of 17.1% and 1.6% in the expected frequency and severity, respectively, while changes in VPD suggest respective increases of 1.2% and 3.6%.

1

The data include a small percentage (<0.24%) of unrealistically small values of SPI (<−3) at certain locations in arid climates with very little rainfall where the gamma assumption used to derive SPI is not appropriate. We found that these spurious values did not have a significant impact on the model fits.

2

Nonburnable land-cover types include permanent bodies of water, snow, and ice as well as urban and bare areas. All other land-cover types are burnable.

3

Handling of missing observations is described in section 3d.

4

With default hyperparameters; see https://keras.io/api/optimizers/adam/ (accessed 23 May 2023).

5

The largest observed value of BA was 416.9 km2 occurring within the Lousã and Oliveira do Hospital municipalities of Portugal (Ribeiro et al. 2020).

6

To derive BA quantiles, we also use values of the parameter pu; we keep this fixed with respect to changes in VPD and 2-m air temperature.

A1

A 3-month block was chosen to reduce computational expense.

Acknowledgments.

The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Awards OSR-CRG2020-4394 and ORA-2022-5336. Support from the KAUST Supercomputing Laboratory is gratefully acknowledged. This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement 101003469. J. Z. acknowledges funding from the Helmholtz Initiative and Networking Fund (Young Investigator Group COMPOUNDX, Grant Agreement VH-NG-1537). The authors thank the three reviewers of this paper for their helpful feedback.

Data availability statement.

The data that support our findings and the code for fitting the PINN models are both available in the R package pinnEV (Richards 2022).

APPENDIX A

Model Details

a. Neural network architecture

We now describe a feed-forward neural network with J hidden layers, each with nj neurons for j = 1, …, J; first, note that xN(s,t)=[x1(s,t),,xId(s,t)]T for all (s,t)S×T. We describe the output of layer j, neuron i, through the function mj,i(s,t). Define the vector of outputs from layer j by mj(s,t)=[mj,1(s,t),,mj,nj(s,t)]T, with the convention that n0 = Id and m0,i(s, t) := xi(s, t) for i = 1, …, Id. The output from the final layer is
mN{x(s,t)}=(w(J+1))TmJ(s,t)+b(J+1),
for a vector of weights w(J+1)RnJ and a bias b(J+1)R. The output of the jth layer at the ith node, for j = 1, …, J and i = 1, …, nj, can be written as
mj,i(s,t)=aj{gj,i[{mj1(s,t):sS}]+b(j,i)},
for a bias b(j,i)R, a fixed activation function aj, and a function gj,i that is dependent on some real weights. As advocated by Glorot et al. (2011), we use only the rectified linear unit (ReLu) activation function and set aj(x) = max{0, x} for all j = 1, …, J. If (A1) is a densely connected layer, then gj,i[{mj1(s,t):sS}]=(w(j,i))Tmj1(s,t) for weights w(j,i)Rnj1 and for all i = 1, …, nj.
Through specification of gj,i, we can define (A1) as a convolutional layer; however, these can only be applied in the case where S can be represented as a D1 × D2 grid of locations. In such a case, a convolution filter with prespecified dimension d1,j × d2,j for d1,j, d2,j odd integers, d1,jD1, d2,jD2, can be created for the jth layer; we keep the filter dimension consistent over all layers and subsequently drop the subscript j from the notation. Define a neighborhood L(s) for sS as the set of locations that form a d1 × d2 grid with site s at the center and let M(j−1,i)(s, t) be a d1 × d2 matrix created by evaluating mj−1,i(s, t) on the grid L(s). For a d1 × d2 weight matrix W(j,i) with real entries, the ith convolutional filter for layer j is of the form Cj,i{mj1,i(s,t)}=k=1d1l=1d2W(j,i)(k,l){M(j1,i)(s,t)}(k,l) with operations taken entrywise; here, A(k, l) denotes the (k, l)th entry of matrix A. The layer (A1) is then considered convolutional if
gj,i[{mj1(s,t):sS}]=k=1nj1Cj,k{mj1,k(s,t)},
for all i = 1, …, nj. To apply a convolutional filter at the boundaries of S, we pad each input map, that is, {xi(s,t):sS} for i = 1, …, d, by mapping S to a new domain S*, taken to be a {D1 + d1 − 1} × {D2 + d2 − 1} grid of sites with S at the center. Then, for all i = 1, …, d, we set xi(s, t) = 0 for all s within distance (d1 − 1)/2 and (d2 − 1)/2 of the first and second dimension, respectively, of the boundaries of S*; a similar padding is used for all {mj,k(s,t):sS} for k = 1, …, nj (i.e., the output from the jth layer). Throughout the main text, we follow Richards and Huser (2022) and use only 3 × 3 filters, with d1 = d2 = 3. We denote neural networks that use only convolutional or densely connected layers as convolutional neural networks (CNNs) and densely connected NNs, respectively.

b. Uncertainty assessment

To quantify parameter uncertainty, we utilize a stationary bootstrap scheme (Politis and Romano 1994) with expected block size k. To create a single bootstrap sample, we repeat the following until obtaining a sample of length greater than or equal to |T|; draw a starting time t*T uniformly at random and a block size K from a geometric distribution with expectation k, and then add the block of observations {y(s,t):sS,t{t*,,t*+K1}} to the bootstrap sample. In cases where t* is generated with t*+K1>|T|, we instead add {y(s,t):sS,t{1,,t*+K|T|1}{t*,,|T|}}; the sample is then truncated to have length |T|. The full model is estimated for each bootstrap sample.

c. Validation and testing

For all model fits, we use a validation and testing scheme to reduce overfitting and improve out-of-sample prediction. Before estimating any parameters, each bootstrap sample is partitioned into an 80–10–10 split for training, validation, and testing data. Partitioning the data is performed at random; we assign data for testing and validation by removing space–time clusters of observations. For each distinct 3-month blockA1 of observed space–time locations (i.e., ω), we simulate a standard space–time Gaussian process {Z(s, t)} with separable correlation function ρ{(si,ti),(sj,tj)}=exp(sisj*/λs)exp(|titj|*/λt), where * denotes the (geodesic) great-Earth distance in kilometers. The range parameters λs, λt > 0 are tunable hyperparameters that control the average size of a space–time cluster. In the application described in section 4, we set λs = 240 and λt = 5, as this gave reasonably large clusters. After simulating a single realization of Z(s, t) for each 3-month block, we assign observations y(s, t) to validation if the realization z(s, t) falls below the 10% quantile of {z(s, t) : (s, t) ∈ ω)}: similarly, we assign to testing observations y(s, t) for which z(s, t) falls above the 90% quantile of {z(s, t) : (s, t) ∈ ω)}. Models are then fitted using only the training and validation data; each neural network is trained for a finite number of epochs, by minimizing the associated loss evaluated on the training dataset. However, the model state after each epoch is recorded, and then, the final model fit is taken to be that which minimizes the validation loss, that is, the loss evaluated on the validation data, across all epochs. The testing data are not used in model fitting whatsoever; instead, we evaluate the associated model loss on these data to perform model selection (i.e., choosing the optimal architecture, see section 4a). Note that while we use different training/validation/testing partitions for each bootstrap sample, the same partition is used to fit the four models p0, u, pu, and σ with a single bootstrap sample.

To compare fits of model (2) for conditional spread Y(s, t)|Y(s, t) > 0, we utilize the threshold-weighted continuous ranked probability score (twCRPS) (Gneiting and Ranjan 2011), which is a proper scoring rule. Define PS×T as the set of out-of-sample space–time locations used for testing and let F^(s,t),0 denote estimates of F(s,t),0. Then, for a sequence of increasing thresholds {υ1,,υnυ} and a weight function r(x)=r˜(x)/r˜(υnυ),r˜(x)=1[1+(x+1)2/10000]1/4, the empirical estimator of the twCRPS, provided by Opitz (2023), is
twCRPS=(s,t)Pi=1nυr(υi)[1{y(s,t)υi}F^(s,t),0(υi)]2,
with r(x) putting more weight on higher-valued predictions. We use 17 irregularly spaced thresholds ranging from υ1 = 1 to υ17 = 750 > max(s,t)∈ω{y(s, t)}

APPENDIX B

Additional Figures

Appendix B provides additional figures. Figure B1 provides histograms of the observed response, BA. Figure B2 illustrates the goodness-of fit of the GPD PINN model with a standardized Q–Q plot. Figure B3 gives maps of the observed response for August 2001, as well as the corresponding estimates of the GPD PINN parameters. Figures B4B6 provide uncertainty estimates for p0, σ, and the 95% quantile of log{1 + Y(s, t)}|X(s, t) for August 2001. Figure B7 provides uncertainty estimates for the climate trend results showcased in Fig. 8. Figures B8B11 repeat the climate trends analysis (Fig. 8) for August 2008, August 2015, October 2017, and November 2020, respectively.

Fig. B1.
Fig. B1.

Histogram of all observations of (left) BA (km2) and (right) BA normalized by the burnable area (unitless). Note that the x axis is on the log scale.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B2.
Fig. B2.

Q–Q plot for the pooled marginal fit of fire spread on standard exponential margins, averaged across all bootstrap samples. The 95% tolerance bounds are given by the dashed lines. Black points give the median quantiles across all samples, with the quantile levels ranging from 0.5 to a value corresponding to the maximum observed value.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B3.
Fig. B3.

Maps of (a) observed log{1 + Y(s, t)} [BA; log(km2)] and corresponding estimates of (b) log{1 + u(s, t)} [log(km2)], (c) log{1 + Y(s, t) − u(s, t)}|Y(s, t) > u(s, t) [exceedances; log(km2)], and (d) pu(s, t) (unitless) for August 2001. Note that u(s, t) is the 40% quantile of nonzero spread Y(s, t)|Y(s, t) > 0. Estimates are from the first bootstrap sample; see section 4a of the main text.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B4.
Fig. B4.

The (left) 2.5%, (center) 50%, and (right) 97.5% bootstrap quantiles of estimated fire occurrence probability p0(s, t) (unitless) for (a),(e),(i) August 2001, (b),(f),(j) August 2008, (c),(g),(k) October 2017, and (d),(h),(l) November 2020.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B5.
Fig. B5.

The (left) 2.5%, (center) 50%, and (right) 97.5% bootstrap quantiles of estimated log{1 + σ(s, t)} [conditional spread severity; log(km2)] for (a),(e),(i) August 2001, (b),(f),(j) August 2008, (c),(g),(k) October 2017, and (d),(h),(l) November 2020.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B6.
Fig. B6.

The (left) 2.5%, (center) 50%, and (right) 97.5% bootstrap quantiles of estimated 95% quantile for log{1 + Y(s, t)}|X(s, t) [BA; log(km2)] for (a),(e),(i) August 2001, (b),(f),(j) August 2008, (c),(g),(k) October 2017, and (d),(h),(l) November 2020.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B7.
Fig. B7.

The (left) 2.5%, (center) 50%, and (right) 97.5% bootstrap quantiles of sitewise estimated changes, under predicted trends in VPD (Pa) for the period 2001–20, in the fire occurrence probability p0(s, t) (unitless) for (a),(e),(i) August 2001 (see also Fig. 4). (b),(f),(j) As in (a), (e), and (i), but for predicted trends in 2-m air temperature (K). (c),(g),(k) and (d),(h),(l) As in (a),(e),(i) and (b),(f),(j), but instead illustrating changes in log{1+q0.9+(s,t)} [intensity; log(km2)], with q0.9+(s,t) the 90% quantile of spread Y(s, t)|{Y(s, t) > 0, X(s, t)}.

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B8.
Fig. B8.

(a) Sitewise estimated trends (change per year) in August VPD (Pa) observed over the period 2001–20. Under these trends, maps of the sitewise (bootstrap median) changes in estimates of the (b) occurrence probability p0(s, t) (unitless) and (c) log{1+q0.9+(s,t)} [log(km2)] [withq0.9+(s,t), the 90% quantile of the spread Y(s, t)|{Y(s, t) > 0, X(s, t)}, a measure of wildfire intensity] for the month of August 2008. (d)–(f) As in (a)–(c), but using trends in August 2-m air temperature (K).

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B9.
Fig. B9.

(a) Sitewise estimated trends (change per year) in August VPD (Pa) observed over the period 2001–20. Under these trends, maps of the sitewise (bootstrap median) changes in estimates of the (b) occurrence probability p0(s, t) (unitless) and (c) log{1+q0.9+(s,t)} [log(km2)] [with q0.9+(s,t), the 90% quantile of the spread Y(s, t)|{Y(s, t) > 0, X(s, t)}, a measure of wildfire intensity] for the month of August 2015. (d)–(f) As in (a)–(c), but using trends in August 2-m air temperature (K).

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B10.
Fig. B10.

(a) Sitewise estimated trends (change per year) in October VPD (Pa) observed over the period 2001–20. Under these trends, maps of the sitewise (bootstrap median) changes in estimates of the (b) occurrence probability p0(s, t) (unitless) and (c) log{1+q0.9+(s,t)} [log(km2)] [with q0.9+(s,t), the 90% quantile of the spread Y(s, t)|{Y(s, t) > 0, X(s, t)}, a measure of wildfire intensity] for the month of October 2017. (d)–(f) As in (a)–(c), but using trends in October 2-m air temperature (K).

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

Fig. B11.
Fig. B11.

(a) Sitewise estimated trends (change per year) in November VPD (Pa) observed over the period 2001–20. Under these trends, maps of the sitewise (bootstrap median) changes in estimates of the (b) occurrence probability p0(s, t) (unitless) and (c) log{1+q0.9+(s,t)} [log(km2)] [with q0.9+(s,t), the 90% quantile of the spread Y(s, t)|{Y(s, t) > 0, X(s, t)}, a measure of wildfire intensity] for the month of November 2020. (d)–(f) As in (a)–(c), but using trends in November 2-m air temperature (K).

Citation: Artificial Intelligence for the Earth Systems 2, 4; 10.1175/AIES-D-22-0095.1

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