This study covers an area of 80 500 km2 located in the southeast of France (Fig. ). Figure a presents the main geographical regions, which can be classified as Mediterranean lowlands (Provence, Languedoc-Roussillon, Maritime Alps), hinterlands, and foothills (Southern Alps, Cévennes), and mountainous areas (the Alps, Corsica, Massif Central, and eastern Pyrénées), which corresponds to high elevations (>1000 m a.s.l.). Mediterranean areas have a fairly dry and warm climate (mean rainfall <700 mm yr-1 and temperature >13 ∘C) and, as a consequence, they are conducive to fire activity. The frequency of wildfires is smaller in hinterland and mountain areas due to a high mean rainfall (>700 mm yr-1) and medium temperatures (<13 ∘C). Strong and dry winds also increase the summer fire activity along the Mediterranean coastline fuel dryness, larger fire rate of spread;.

This study area encompasses different bioclimates, different vegetation fuels, and different levels of human activity (which generate about 95 % of ignitions), leading to different fire activities. For this reason we define three main homogeneous subregions based on fire activity and bioclimatic variables, so-called pyroclimatic regions PCr, adapted from; see Fig. c. PCr-1 contains most fire hot spots of maximal fire activity with many fires (notably the largest ones) and large amounts of BA. It includes Corsica, Provence, and the Maritime Alps and concerns 64 % of all fires and 67 % of the total BA in southern France. In addition, it contains 63 % of the largest fires (>100 ha) and 74 % of the total BA of these fires. This is due to the combination of high fire weather danger, medium to high fuel amount, and connectivity in the landscape as well as a very high density of human-caused ignitions by negligence, by accident, or by intent . This region is undoubtedly prone to large fires, which occur mostly in summer. PCr-2 has medium to high fire activity and covers the Mediterranean hinterland and mid-elevation mountains. PCr-3 has low fire activity and few large fires. It corresponds to high mountains with a wet and cool climate.

We use the national georeferenced fire database called Prométhée , which contains information on more than 106 000 wildfires. It corresponds to wildfire characteristics recorded by fire crews and foresters after each intervention from 1973 to 2016. Prométhée gives standardized information on each fire, including the day and hour of ignition, its location on a 2×2 km grid (with a total of 20 142 pixels), and the final BA in hectares.

Figure  presents the temporal evolution of the number of large fires (>100 ha) for each pyroclimatic region. Clearly, the fire activity is the highest in PCr-1, and PCr-3 has a limited number of large fires in comparison. From these time series, it is also evident that the fire policy set up in 1994 reduced the number of large fires in all regions.

Extreme value theory indicates that, under some mild conditions, the GEV distribution is the limiting distribution of maxima applied to very large blocks. In our case, the blocks correspond to the wildfire samples for each year. We assume that the wildfire samples are sufficiently large, so that annual maxima of BA can be considered to follow a GEV distribution. This has already been shown to be adequate in a very large number of applications, for example in hydrology , finance and insurance , physics , mountain hazards , or coastal engineering to cite a few. The cumulative distribution function (cdf) of the GEV is F(x|θ)=exp⁡{-(1+ξx-μσ)-1/ξ}ifξ≠0and,1+ξ(x-μ)/σ>0exp⁡{-exp⁡(-(x-μ)/σ)}ifξ=0,x∈IR, where θ={μ,σ,ξ} is the set of parameters of the GEV distribution, μ∈IR the location parameter, σ>0 the scale parameter, and ξ∈IR the shape parameter. When ξ<0, the distribution has an upper tail and x<μ-σ/ξ. When ξ=0, the distribution is unbounded. When ξ>0, the distribution has a lower bound (x>μ-σ/ξ) and when ξ increases, the distribution tail becomes heavier and very large values of x (here the BA) are more likely. The corresponding density function is f(x|θ)=1σ(1+ξx-μσ)-1/ξ-1ifξ≠0,exp⁡{-(1+ξx-μσ)-1/ξ}1σexp⁡-x-μσ×exp⁡{-exp⁡-x-μσ}ifξ=0.

For the GEV distribution, the quantile Q(p|θ) corresponding to a probability p and a set of parameters θ is easily obtained:

Q(p|θ)=μ+σ{(log⁡(1/p)-ξ)-1}/ξifξ≠0,μ+σlog⁡(1/log⁡(1/p))ifξ=0.

For a return period T (e.g., T=20 years), the return level (e.g., 100 ha, 1000 ha) is thus expressed by Q(p=1-1/T|θ).

In this study, statistical inference is performed using Bayesian methods see, e.g., monographs by. A Bayesian framework has several advantages compared to standard frequentist approaches. First, explicit a priori assumptions can be made about the parameters to estimate. Second, Bayesian methods provide a direct assessment of the uncertainty related to the parameter estimation. Generally, frequentist methods only provide confidence intervals based on theoretical results that hold true for very large samples (i.e., asymptotically). Third, in Bayesian methods, the uncertainty of a quantity of interest can be quantified by means of the predictive distribution. Here as well, frequentist methods provide similar theoretical results, which hold true under some restrictive conditions.

A Bayesian analysis combines the information in the data represented by the likelihood function with prior knowledge about the parameter. Parameter estimation is made through the posterior distribution, which is computed using Bayes' theorem Π(θ|D)∝Π(D|θ)×Π(θ), where Π(θ|D) is the posterior distribution of the GEV parameters θ, Π(D|θ) is the likelihood function, and Π(θ) is the prior distribution of θ. In this expression, it is assumed that the normalizing constant of the posterior distribution is unknown.

The likelihood function represents the conditional density of the data D, here the logarithms of the maximum BA, given the parameters θ, and describes the plausibility of observing the data D given θ. Here, we assume that the logarithm of these maxima are independently distributed (i.e., the maximum BA in a year y does not influence the maximum of the year y+1) and that they follow a GEV distribution. The likelihood is thus given by Π(D|θ)=∏i=1Nf(xi|μ,σ,ξ), where xi,i,…,N are the the logarithms of the maximum BA observed during N years. Here, following , we choose to work with the logarithm of the maximum BA in order to solve scale issues. Indeed, raw BAs are measures of surface, i.e., two-dimensional data, and very large BA values (e.g., 1000 ha) are a lot larger than large BA values (e.g., 100 ha). This scale issue becomes obvious when the distribution of the raw BA is represented, this distribution being extremely skewed. As a consequence of these very skewed distributions, extreme value distributions with an infinite variance (i.e., when xi>0.5) are often obtained see when they are applied to raw BA. Applying a log transformation to maximum BA is a convenient way to avoid this problem.

The prior distribution Π(θ) represents our a priori knowledge about the parameters θ. Here, we choose the following.

Location parameter μ∼N(0,100000): vague prior.

The posterior distribution Π(θ|D) can be sampled using a Markov chain Monte Carlo (MCMC) algorithm . Here, we choose to apply the Metropolis–Hastings algorithm using the function MCMCmetrop1R from the MCMCpack package in R software . In this algorithm, candidates are randomly generated from a proposal distribution and accepted as a new sample according to an acceptance rate, which is computed using the posterior distribution Π(θ|D). The main difficulty with the Metropolis–Hastings algorithm is to find an adequate proposal distribution, leading to a reasonable acceptance rate. In this work, the multivariate normal proposal distribution is scaled using the covariance matrix of the parameter estimates when the maximum-likelihood method is applied, using the function fgev from the evd package in R. For each estimation, we produce a burn-in sample size of 100 000, and the retained sample used to represent the posterior distribution is M=10000, for which a thinning interval of 10 is applied in order to reduce the autocorrelation inherent in MCMC chains produced with the Metropolis–Hastings algorithm.

Draws from the posterior distribution Π(θ|D) can be used to estimate other unknown quantities. Specifically, the predictive distribution of a quantity y=g(⋅|θ), function of some arguments and depending on the GEV parameters θ, is expressed as seeEq. 1.4 Π(y|D)=∫θg(⋅|θ)Π(θ|D)dθ.

Hence, it fairly propagates the uncertainty related to parameter estimation on the quantities of interest. However, a closed expression of the predictive distribution can rarely be obtained, and it is often estimated using the draws from the posterior distribution: Π^(y|D)=1M∑ig(⋅|θ(i)), where θ(i) is the ith draw from the posterior distribution. For example, the predictive distribution of a quantile q=Q(p|θ) (see Eq. ) can be estimated by applying Eq. () with g(⋅|θ(i))=Q(p|θ(i)), i.e., π^(q)=1M∑iQ(p|θ(i)).

Likewise, the predictive density of a maximum log-BA x̃ can be estimated with the expression π^(x̃)=1M∑if(x̃|θ(i)), which can be used in turn to estimate specific probabilities, such as the exceedance probability of some critical log-BA xc. For example, the probability of having a BA x exceeding 10000 ha in a year is obtained as Pr⁡(x>10000)=∫log(10000)∞π^(x̃)dx̃, where x̃ is integrated over all values exceeding 10000 ha, on the logarithm scale.

Numerous statistical tests aim at testing an assumption of equality between two distributions (for example the test of Kolmogorov–Smirnov). These tests are adequate when two samples are compared and when we want to test if the sampling distributions can be considered equivalent. However, it must be noticed that these statistical tests are very powerful with large samples, which means that in this case, a small difference will be considered highly significant. As a result, comparing posterior distributions with these statistical tests generally results in a rejection of the assumption of equality, even when the distributions are very similar.

As an alternative, different measures have been proposed by statisticians in order to quantify the similarity between two distributions, or, in other words, how much two distributions overlap (e.g., Mahalanobis distance, Kullback–Leibler divergence). In this paper, we apply the Bhattacharyya coefficient . This coefficient is easily normalized between 0 and 1, which facilitates its interpretation. Its expression is given by BC(p,q)=∫p(x)q(x)dx, where p(x) and q(x) are the probability density functions to be compared. We have 0≤BC≤1, with BC=0 indicating that the distributions do not overlap and BC=1 indicating a complete similarity.

For the sake of comparison with statistical tests, when this coefficient is applied to two normal distributions N(0,1) and N(2,1) (shift of 2 standard deviations), it has a value of 0.61. Thus, two distributions can be considered similar when the Bhattacharyya coefficient exceeds 0.61. This reference value, while subjective, gives a more precise idea of the interpretation of the Bhattacharyya coefficient.

For each of the three regions PCr-1, PCr-2, and PCr-3, we collect the annual maxima of BA. Following the methodology described in Sect. , we fit a GEV distribution to the logarithm of annual maxima of BA for the periods 1973–1994 and 1995–2016 in order to assess the impact of the fire policy established in 1994. Table  presents the acceptance rates of the Metropolis–Hastings algorithm, which lie between 0.32 and 0.42. These acceptance rates are reasonable and correspond to common requirements. Indeed, acceptance rates between 0.23 and 0.5 are usually advised in order to obtain a fast convergence of the algorithm . A visual assessment of the traces of the sampled parameters did not reveal any convergence issue (results not shown).

Taking the mean of the posterior distributions as point estimates, Fig. shows the QQ plots of the fitted GEV distributions for each region and each period. This figure compares observed maxima and quantiles from the fitted GEV distribution corresponding to empirical probabilities of observed maxima. Points are close to the 1:1 line, which indicates that these adjustments are reasonable.

Posterior distribution estimates for the extreme value parameters μ, σ, and ξ, before and after 1994 and for each region, are shown in Fig.  and provide a direct assessment of the parameter uncertainty. Large differences are obtained for some parameters and some regions. For example, for region PCr-1, the location parameter μ has clearly decreased between the two periods, indicating that the fitted distribution has globally shifted toward smaller BAs after 1994. The scale parameter has decreased in regions PCr-2 and PCr-3 and slightly increased in region PCr-1. The shape parameter exhibits a slight increase in all regions. However, parameter uncertainty is large and the posterior distributions before and after 1994 overlap, which limits the interpretation of this increase.

Table  quantifies these differences using the Bhattacharyya coefficient. As indicated above, this coefficient measures the similarity between two distributions. It is equal to 0 when the distributions do not overlap and 1 if they are identical. Table  indicates that, for region PCr-1, posterior distributions for μ before and after 1994 are clearly different, whereas they are roughly similar for the parameters σ and ξ, confirming the global shift of the distribution mentioned above. For region PCr-2, only σ has a significant change before and after 1994, while no difference is revealed for region PCr-3.

For a return period T (e.g., T=100 years), the corresponding BA estimate is directly obtained, for a set of parameters θ, using Eq. . With the Bayesian approach, we can obtain the predictive distribution estimate of these return levels by applying Eq. () to all MCMC samples of parameters θ(i),i=1,…,M drawn from the posterior distribution (see Eq. ).

Figure compares the predictive distributions of BA corresponding to return periods ranging from 2 years to 50 years, for periods before (in red) and after 1994 (in blue), for the three regions. For all regions, and particularly for region PCr-1, BAs corresponding to the smallest return periods (e.g., 2 years) have decreased, which demonstrates the efficiency of the fire policy established in 1994. However, for region PCr-1, the 90 % credible intervals overlap for higher return periods (e.g., 50 years). For this critical region, considering the large uncertainties associated with high return periods, no clear conclusion can be drawn in terms of decrease or increase in extreme return levels. The corresponding Bhattacharyya coefficients are given in Table  and indeed show that return levels have only changed significantly for return periods smaller than 10 years. For region PCr-2, the decrease in the return levels is more important, even for higher return periods (see Table ). Finally, there is no noticeable change for region PCr-3.

The probability of having very large fires (>10000 ha) in a year is estimated using Eq. () and is given in Table . These estimates indicate a clear decrease in this probability before and after 1994 in regions PCr-1 (from 0.076 to 0.037) and PCr-2 (from 0.040 to 0.008). In PCr-2, the probability of having such large fires remains small (from 0.009 to 0.006).

Table  reports the BA corresponding to high return periods (20 and 50 years), for the recent period 1995–2016. As an indicator of the central tendency of the predictive distributions of return levels, Table  reports the medians, i.e., the quantile 0.5. Median return levels correspond to BAs smaller than 5000 ha in regions PCr-2 and PCr-3. This is not the case in region PCr-1, for which BAs corresponding to 20- and 50-year return periods are very large (respectively 7050 and 12 510 ha). It can be seen that the 90 % credible interval corresponding to the 50-year return level of 12 510 ha in PCr-1 is very wide (from 5400 to 68 000 ha), which limits the interpretation of this estimate.