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  <front>
    <journal-meta><journal-id journal-id-type="publisher">NHESS</journal-id><journal-title-group>
    <journal-title>Natural Hazards and Earth System Sciences</journal-title>
    <abbrev-journal-title abbrev-type="publisher">NHESS</abbrev-journal-title><abbrev-journal-title abbrev-type="nlm-ta">Nat. Hazards Earth Syst. Sci.</abbrev-journal-title>
  </journal-title-group><issn pub-type="epub">1684-9981</issn><publisher>
    <publisher-name>Copernicus Publications</publisher-name>
    <publisher-loc>Göttingen, Germany</publisher-loc>
  </publisher></journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.5194/nhess-19-1459-2019</article-id><title-group><article-title>Wildland fire potential outlooks for Portugal using<?xmltex \hack{\break}?> meteorological indices of fire danger</article-title><alt-title>Wildland fire potential outlooks for Portugal</alt-title>
      </title-group><?xmltex \runningtitle{Wildland fire potential outlooks for Portugal}?><?xmltex \runningauthor{S. A. Nunes et al.}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Nunes</surname><given-names>Sílvia A.</given-names></name>
          <email>sanunes@fc.ul.pt</email>
        <ext-link>https://orcid.org/0000-0001-8368-6447</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>DaCamara</surname><given-names>Carlos C.</given-names></name>
          
        <ext-link>https://orcid.org/0000-0003-1699-9886</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2">
          <name><surname>Turkman</surname><given-names>Kamil F.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Calado</surname><given-names>Teresa J.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1 aff3">
          <name><surname>Trigo</surname><given-names>Ricardo M.</given-names></name>
          
        <ext-link>https://orcid.org/0000-0002-4183-9852</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2">
          <name><surname>Turkman</surname><given-names>Maria A. A.</given-names></name>
          
        </contrib>
        <aff id="aff1"><label>1</label><institution>Instituto Dom Luiz (IDL), Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisbon, Portugal</institution>
        </aff>
        <aff id="aff2"><label>2</label><institution>DEIO-CEAUL, Universidade de Lisboa, 1749-016 Lisbon, Portugal</institution>
        </aff>
        <aff id="aff3"><label>3</label><institution>Departamento de Meteorologia, Universidade Federal do Rio de Janeiro, 21941-916, Rio de Janeiro, Brazil</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">Sílvia A. Nunes (sanunes@fc.ul.pt)</corresp></author-notes><pub-date><day>18</day><month>July</month><year>2019</year></pub-date>
      
      <volume>19</volume>
      <issue>7</issue>
      <fpage>1459</fpage><lpage>1470</lpage>
      <history>
        <date date-type="received"><day>15</day><month>March</month><year>2019</year></date>
           <date date-type="rev-request"><day>25</day><month>March</month><year>2019</year></date>
           <date date-type="accepted"><day>26</day><month>June</month><year>2019</year></date>
      </history>
      <permissions>
        <copyright-statement>Copyright: © 2019 </copyright-statement>
        <copyright-year>2019</copyright-year>
      <license license-type="open-access"><license-p>This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link></license-p></license></permissions><self-uri xlink:href="https://nhess.copernicus.org/articles/.html">This article is available from https://nhess.copernicus.org/articles/.html</self-uri><self-uri xlink:href="https://nhess.copernicus.org/articles/.pdf">The full text article is available as a PDF file from https://nhess.copernicus.org/articles/.pdf</self-uri>
      <abstract><title>Abstract</title>
    <p id="d1e138">Portugal is recurrently affected by large wildfire events that have serious
impacts at the socio-economic and environmental levels and dramatic
consequences associated with the loss of lives and the destruction of the
landscape. Accordingly, seasonal forecasts are required to assist fire
managers, thus contributing to alter the historically based purely reactive
response. In this context, we present and discuss a statistical model to
estimate the probability that the total burned area during summer will
exceed a given threshold. The statistical model uses meteorological
information that rates the accumulation of thermal and vegetation stress.
Outlooks for the 39-year study period (1980–2018) show that, when the
statistical model is applied from 26 May to 30 June, out of
the six severe years, only one year is not anticipated as potentially severe
and, out of the six weak years, only one is not anticipated as potentially
weak. The availability of outlooks of wildfire potential with an
anticipation of up to 1 month before the starting of the fire season, such
as the one proposed here, may serve to provide clear directions for the fire
community when planning prevention and combating fire events.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

<sec id="Ch1.S1" sec-type="intro">
  <label>1</label><title>Introduction</title>
      <p id="d1e150">Portugal is regularly affected by very large and destructive wildfires that
represent a serious threat to human lives and to the territory, and have
very strong and adverse impacts at the social, economic, ecologic and
environmental levels that include human casualties, the destruction of homes
and other structures, damages to forests, agricultural fields, shrublands,
and livestock, changes in the landscape and emission of greenhouse gases
(Costa et al., 2011; Pereira et al., 2011). According to the European
Commission technical report about forest fires in Europe (San-Miguel-Ayanz
et al., 2018), from 1980 to 2017, Portugal accounted for 26 % of the total
burned area in the five southern member states (Portugal, Spain, Italy,
Greece and France). However, this figure increases to 42 % when
restricting to the last decade. The extreme year of 2017 deserves a special
emphasis with Portugal accounting for 59 % of the burned area in the five
southern member states, despite representing 6 % of the total area of the
countries.</p>
      <p id="d1e153">As in Mediterranean Europe, fire activity in Portugal involves complex
interactions among climate, vegetation and humans (Lavorel et al., 2006;
Costa et al., 2010). Persistent warm and dry conditions followed by heat
spells in summer provide the optimal meteorological background to the onset
and spread of large fire events (Pereira et al., 2005; Trigo et al.,
2006; DaCamara et al., 2014) that take place in a landscape of cumulated
biomass and increased fuel connectivity as a result of agricultural
abandonment and forest expansion (Pausas and Fernández-Muñoz, 2012;
Fernandes et al., 2014; Viedma et al., 2015; Oliveira et al., 2017). In
addition, Portugal has the highest density of ignitions in southern Europe,
with human presence and activity being the key drivers of ignitions,
most of them associated with land management practices and inadequate use of
fire (Catry et al., 2009). Finally, the impacts of climate change cannot be
disregarded since a future warmer climate will instigate a larger number<?pagebreak page1460?> of
severe wildfire episodes (Flannigan et al.,
2013; Sousa et al., 2015). In this context, the catastrophic fires of June
2017 in Portugal (with 65 fatalities) associated with the strongest heatwave
ever observed over Iberia in June provide a stark reminder of the growing
likelihood of these events under the current warming climate
(Sánchez-Benítez et al., 2018).</p>
      <p id="d1e156">The magnitude of the problems related to fire activity in Portugal has
motivated the scientific and technical communities to cooperate with forest
managers, firefighters and the fire community with the aim of improving the
understanding of the fire regime in Portugal and develop better tools that
will help prevent and mitigate the impacts of severe fire events
(Collins et al., 2013; Oliveira et al., 2016).</p>
      <p id="d1e159">Outlooks of wildfire potential for different regions of the globe are
usually based on statistical approaches (e.g. Gudmundsson et al., 2014;
Turco et al., 2017) or, alternatively, using fire weather predictions a few
months in advance based on dynamical seasonal forecasts by atmospheric
circulation models (Anderson et al., 2007; Turco et al., 2018). For
instance, the European Forest Fire Information System (EFFIS), one of the
components of the Emergency Management Services in the EU Copernicus
programme, is currently disseminating maps (as experimental products) of
long-term seasonal forecasts of temperature and rainfall anomalies based on
the European Centre for Medium-Range Weather Forecasts (ECMWF) Seasonal
Forecasting System (System 4). However, as pointed out by Bedia et al. (2018), although there is significant skill in predicting 1 month ahead for
above-average summer fire weather in some parts of south-eastern Europe,
skill is in general quite poor elsewhere.</p>
      <p id="d1e163">As an alternative, the aim of this paper is to describe and discuss a model
that allows outlooks of wildfire potential in Portugal during the
fire season up to 1 month in advance based on a statistical approach that
integrates information about meteorological fire danger before and during
the fire season. The rationale is that soil moisture deficit and drought
have an impact on the increased frequency and can amplify the magnitude of
hot summer extreme events in the Mediterranean (Vautard et al., 2007;
Hirschi et al., 2011). Thus, persistent warm and dry conditions along the
pre-fire season induce thermal and water stress on vegetation, making the
landscape more prone to the occurrence of very severe fire episodes. At
the same time, these conditions increase the likelihood of heatwave spells that steer the
onset and propagation of large fires (Gudmundsson et al., 2014; Turco et
al., 2017). The procedure involves three steps. First we set up a null model
of burned area (BA) during the fire season (where no meteorological
information is incorporated); then we set up a diagnostic model of BA that
incorporates meteorological information during the pre-fire and the fire
seasons; finally we set up a prognostic model of BA that only incorporates
meteorological information before the fire season. We then show that both
the diagnostic and the prognostic models perform better than the null
model and that the loss in performance of the prognostic model (when
compared to the diagnostic one) is relatively small, namely in respect
to its capacity to anticipate fire seasons characterized by high amounts of
BA. Section 2 provides a description of data and methods, and results are
presented in Sect. 3. Finally, discussion of results and conclusions are
drawn in Sect. 4.</p>
</sec>
<sec id="Ch1.S2">
  <label>2</label><title>Data and methods</title>
      <p id="d1e174">The area of interest is defined as the territory of Portugal and the study
covers the 39-year period from 1980 to 2018. Data of BA consist of yearly
amounts of cumulated BA in July and August, hereby referred to as the fire
season. BA data are derived from the official Portuguese Rural Fire Database
provided by the national authority for forests (ICNF). The database contains
more than half a million records of fire events, with information of total
burned area, date and time of ignition and extinction, and spatial location
of the starting point. Details about the database are provided in Pereira et al. (2011). Cumulated BA in the fire season accounts for more than 70 % of the
total burned area in Portugal (Pereira et al., 2013), and more than 80 %
of extreme fire days (defined as the top 5 % in terms of radiative energy
released by wildfires) occur in July and August (DaCamara and Trigo, 2018).</p>
      <p id="d1e177">Information about meteorological fire danger consists of daily values of the
Daily Severity Index (DSR) covering the months of April to August. DSR is an
extension of the Canadian Forest Fire Weather Index System (CFFWIS) and
rates the difficulty of controlling fires (van Wagner, 1987). This index has
been successfully used to model BA variability in Portugal at daily and
monthly scales (Calado et al., 2008; Pereira et al., 2013). DSR results from
a direct transformation of the Fire Weather Index (FWI), the last of the six
components of CFFWIS, according to the relation <inline-formula><mml:math id="M1" display="inline"><mml:mrow><mml:mi mathvariant="normal">DSR</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.0272</mml:mn></mml:mrow></mml:math></inline-formula> (FWI)<inline-formula><mml:math id="M2" display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">1.77</mml:mn></mml:msup></mml:math></inline-formula>. This transformation weights
FWI sharply as it increases so that DSR becomes more suitable than FWI to be
cumulated or averaged. For each day, the six components are computed based
on consecutive daily observations of meteorological parameters of the
previous days, namely temperature, relative humidity, wind speed and
24 h cumulated precipitation (Wang et al., 2015). In this study,
meteorological parameters consist of gridded daily values at 12:00 UTC of 2 m temperature, relative humidity, wind speed and 24 h cumulated
precipitation that were obtained from the ERA-Interim reanalysis dataset (Dee et
al., 2011) issued by ECMWF. As described in Pinto et al. (2018), the
original ERA-Interim data were re-projected onto the normalized
geostationary projection (NGP) of Meteosat Second Generation (MSG)
(EUMETSAT, 1999), with an average pixel size of about 4 km <inline-formula><mml:math id="M3" display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 4 km
over Portugal. Daily values of DSR for Portugal were then obtained by
averaging over all grid points located within the study area.</p>
      <p id="d1e208">As discussed in Pereira et al. (2005), the interannual variability of BA in
Portugal is modulated by two kinds of<?pagebreak page1461?> meteorological factors, namely the
temperature and precipitation regimes during the pre-fire season and the
occurrence of hot and dry spells during the fire season. Following Nunes et
al. (2014), the period from April to August is accordingly divided into two
subperiods: (1) the pre-fire season that runs from 1 April (day 1) up
to 30 June (day 91) and (2) the fire season that runs from 1 July (day 1) up to 31 August (day 62).</p>
      <p id="d1e211">For each year of the period 1980–2018, meteorological fire danger at day <inline-formula><mml:math id="M4" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>
of the pre-fire season (pfs) of the considered year is rated by index
<inline-formula><mml:math id="M5" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, defined as the cumulative value of daily DSR since 1 April:
          <disp-formula id="Ch1.E1" content-type="numbered"><label>1</label><mml:math id="M6" display="block"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>d</mml:mi></mml:munderover><mml:msub><mml:mi mathvariant="normal">DSR</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width="0.25em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.25em"/><mml:mspace width="0.25em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.25em"/><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">…</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mo>,</mml:mo><mml:mn mathvariant="normal">91</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
        where <inline-formula><mml:math id="M7" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="normal">DSR</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the value of DSR at day <inline-formula><mml:math id="M8" display="inline"><mml:mi>i</mml:mi></mml:math></inline-formula>. As <inline-formula><mml:math id="M9" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> runs along the
pre-fire season of the year considered, progressively more information is
integrated in <inline-formula><mml:math id="M10" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> about past daily meteorological fire danger.</p>
      <p id="d1e343">In turn, meteorological fire danger of the fire season (fs) of each year of
the period 1980–2018 is rated by index <inline-formula><mml:math id="M11" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">fs</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, defined as the square root
of the mean squared anomalies performed over days characterized by a
positive anomaly of DSR:
          <disp-formula id="Ch1.E2" content-type="numbered"><label>2</label><mml:math id="M12" display="block"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">fs</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mn mathvariant="normal">62</mml:mn></mml:munderover><mml:mi>H</mml:mi><mml:mfenced close="]" open="["><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:msup><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mn mathvariant="normal">62</mml:mn></mml:munderover><mml:mi>H</mml:mi><mml:mfenced open="[" close="]"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mfrac></mml:mstyle></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
        where <inline-formula><mml:math id="M13" display="inline"><mml:mrow><mml:mi>H</mml:mi><mml:mo>[</mml:mo><mml:mi>x</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula> is the Heaviside step function (<inline-formula><mml:math id="M14" display="inline"><mml:mrow><mml:mi>H</mml:mi><mml:mfenced close="]" open="["><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>x</mml:mi><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>;</mml:mo><mml:mi>H</mml:mi><mml:mfenced open="[" close="]"><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi>x</mml:mi><mml:mo>≤</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and
          <disp-formula id="Ch1.E3" content-type="numbered"><label>3</label><mml:math id="M15" display="block"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="normal">DSR</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mrow><mml:msub><mml:mi mathvariant="normal">DSR</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula>
        is the anomaly of DSR at day <inline-formula><mml:math id="M16" display="inline"><mml:mi>j</mml:mi></mml:math></inline-formula> of the considered year, which is defined as
the departure of DSR from the climatological mean <inline-formula><mml:math id="M17" display="inline"><mml:mover accent="true"><mml:mrow><mml:msub><mml:mi mathvariant="normal">DSR</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:math></inline-formula> for day
<inline-formula><mml:math id="M18" display="inline"><mml:mi>j</mml:mi></mml:math></inline-formula> (as obtained by averaging DSR for that day over the 39-year period
1980–2018).</p>
      <p id="d1e548">The temperature and precipitation regimes during the pre-fire season are
quantified by <inline-formula><mml:math id="M19" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, where, for a given day <inline-formula><mml:math id="M20" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> of the pre-fire
season, large values indicate persistent warm and/or dry conditions up to
day <inline-formula><mml:math id="M21" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, inducing thermal and water stress on vegetation and a shortage of
water in the soil. The impact of hot and dry spells during the fire season
is in turn quantified by <inline-formula><mml:math id="M22" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">fs</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, which is very sensitive to the occurrence,
during the considered fire season, of very large positive anomalies of DSR
that are usually associated with heatwaves (Gudmundsson et al., 2014; Turco
et al., 2017).</p>
      <p id="d1e593">For each day <inline-formula><mml:math id="M23" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, the 39 values of <inline-formula><mml:math id="M24" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> for the period 1980–2018
were normalized by subtracting the respective sample mean and then dividing
by the sample standard deviation. Values of <inline-formula><mml:math id="M25" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">fs</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> were also normalized by
subtracting the sample mean and dividing by the sample standard deviation.
Obtained normalized indices of <inline-formula><mml:math id="M26" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M27" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">fs</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> will be hereby
denoted as <inline-formula><mml:math id="M28" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M29" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>, respectively.</p>
      <p id="d1e678">The following types of models are considered in this study:
<list list-type="order"><list-item>
      <p id="d1e683">a null model <inline-formula><mml:math id="M30" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="M31" display="inline"><mml:mi>X</mml:mi></mml:math></inline-formula> is the considered
variable and <inline-formula><mml:math id="M32" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M33" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> are the mean and the standard
deviation to be estimated, respectively;</p></list-item><list-item>
      <p id="d1e746">a nested model with two covariates <inline-formula><mml:math id="M34" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>;</mml:mo><mml:mi>a</mml:mi><mml:mo>×</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mo>×</mml:mo><mml:mi>B</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, where the mean of the normal
distribution linearly depends on covariates <inline-formula><mml:math id="M35" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula> and
<inline-formula><mml:math id="M36" display="inline"><mml:mi>B</mml:mi></mml:math></inline-formula>, with <inline-formula><mml:math id="M37" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M38" display="inline"><mml:mi>b</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M39" display="inline"><mml:mi>c</mml:mi></mml:math></inline-formula> being parameters to be estimated;</p></list-item><list-item>
      <p id="d1e833">the previous nested model then simplified to a nested model with just one
covariate <inline-formula><mml:math id="M40" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>;</mml:mo><mml:mi>p</mml:mi><mml:mo>×</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, where the mean
of the normal distribution linearly depends on covariate <inline-formula><mml:math id="M41" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula>,
and <inline-formula><mml:math id="M42" display="inline"><mml:mi>p</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M43" display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula> are parameters to be estimated.</p></list-item></list></p>
      <p id="d1e896">Using the maximum likelihood method (Wilks, 2011), estimates of parameters
<inline-formula><mml:math id="M44" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M45" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M46" display="inline"><mml:mi>p</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M47" display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M48" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M49" display="inline"><mml:mi>b</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M50" display="inline"><mml:mi>c</mml:mi></mml:math></inline-formula> are obtained as
follows from sample <inline-formula><mml:math id="M51" display="inline"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mo>(</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">…</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> where <inline-formula><mml:math id="M52" display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> is the size of
the sample.

              <disp-formula specific-use="gather" content-type="numbered"><mml:math id="M53" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E4"><mml:mtd><mml:mtext>4</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:msub><mml:mover accent="true"><mml:mi mathvariant="italic">μ</mml:mi><mml:mo mathvariant="normal" stretchy="false">^</mml:mo></mml:mover><mml:mi>X</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mstyle><mml:mi mathvariant="normal">Σ</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E5"><mml:mtd><mml:mtext>5</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:msubsup><mml:mover accent="true"><mml:mi mathvariant="italic">σ</mml:mi><mml:mo mathvariant="normal" stretchy="false">^</mml:mo></mml:mover><mml:mi>X</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mn mathvariant="normal">1</mml:mn><mml:mi>n</mml:mi></mml:mfrac></mml:mstyle><mml:mi mathvariant="normal">Σ</mml:mi><mml:msup><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mover accent="true"><mml:mi mathvariant="italic">μ</mml:mi><mml:mo stretchy="false" mathvariant="normal">^</mml:mo></mml:mover><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E6"><mml:mtd><mml:mtext>6</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">var</mml:mi><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi>A</mml:mi><mml:mo>)</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>B</mml:mi><mml:mo>)</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">var</mml:mi><mml:mfenced open="(" close=")"><mml:mi>A</mml:mi></mml:mfenced><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">var</mml:mi><mml:mfenced close=")" open="("><mml:mi>B</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mo>[</mml:mo><mml:mi mathvariant="normal">cov</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi>B</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mo>]</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E7"><mml:mtd><mml:mtext>7</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">var</mml:mi><mml:mfenced open="(" close=")"><mml:mi>A</mml:mi></mml:mfenced><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi mathvariant="normal">cov</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>B</mml:mi></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mi mathvariant="normal">cov</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi>B</mml:mi></mml:mrow></mml:mfenced><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant="normal">var</mml:mi><mml:mfenced open="(" close=")"><mml:mi>A</mml:mi></mml:mfenced><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi mathvariant="normal">var</mml:mi><mml:mfenced open="(" close=")"><mml:mi>B</mml:mi></mml:mfenced><mml:mo>-</mml:mo><mml:mo>[</mml:mo><mml:mi mathvariant="normal">cov</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mo>]</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E8"><mml:mtd><mml:mtext>8</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mi>b</mml:mi><mml:mover accent="true"><mml:mi>B</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E9"><mml:mtd><mml:mtext>9</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">cov</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>A</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi mathvariant="normal">var</mml:mi><mml:mfenced close=")" open="("><mml:mi>A</mml:mi></mml:mfenced></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E10"><mml:mtd><mml:mtext>10</mml:mtext></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

          Here <inline-formula><mml:math id="M54" display="inline"><mml:mover accent="true"><mml:mrow><mml:mo>(</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mo>)</mml:mo></mml:mrow><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:math></inline-formula>, <inline-formula><mml:math id="M55" display="inline"><mml:mrow><mml:mi mathvariant="normal">var</mml:mi><mml:mo>(</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M56" display="inline"><mml:mrow><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> respectively denote the mean,
the variance and the covariance operators.</p>
      <p id="d1e1406">In the case of the null model, the Lilliefors test (Conover, 1980) is used
to test the null hypothesis that the data come from a normal distribution.
In the case of the nested models with covariates, the likelihood ratio test
(Wilks, 1938) is used to decide on the null hypothesis that the null model
is to be retained (against the alternate model under consideration).</p>
</sec>
<sec id="Ch1.S3">
  <label>3</label><title>Results</title>
<sec id="Ch1.S3.SS1">
  <label>3.1</label><title>Null model</title>
      <p id="d1e1424">The time series of cumulated BA in the fire season for the period 1980–2018
(Fig. 1) presents very large interannual variability, the extremely high
amounts of 2003 and 2005 contrasting with the extremely low values that were
observed in 1983, 1988, 1997 and 2008. It is worth noting that 2017, the
year with the largest record in total BA (circa 450 000 ha),<?pagebreak page1462?> only ranks
fourth when restricting to July and August because the largest fire events
took place outside of the fire season, in June and October
(Sánchez-Benítez et al., 2018).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1" specific-use="star"><?xmltex \currentcnt{1}?><label>Figure 1</label><caption><p id="d1e1429">Time series of yearly cumulated BA (ha) in the fire season (July
and August).</p></caption>
          <?xmltex \igopts{width=398.338583pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f01.png"/>

        </fig>

      <p id="d1e1438">The decimal logarithm of cumulated BA in the fire season follows a normal
distribution model (Fig. 2); the null hypothesis that the sample of <inline-formula><mml:math id="M57" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula> comes from a normal distribution is not rejected at the 5 %
significance level by the Lilliefors test (<inline-formula><mml:math id="M58" display="inline"><mml:mi>p</mml:mi></mml:math></inline-formula> value of 0.21). We therefore have
the following null model:
            <disp-formula id="Ch1.E11" content-type="numbered"><label>11</label><mml:math id="M59" display="block"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
          with <inline-formula><mml:math id="M60" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">4.69</mml:mn></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M61" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.46</mml:mn></mml:mrow></mml:math></inline-formula> as using the maximum
likelihood method.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2"><?xmltex \currentcnt{2}?><label>Figure 2</label><caption><p id="d1e1540">Normal probability plot comparing the sample of <inline-formula><mml:math id="M62" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula> to the normal distribution. The groups of severe and
weak years are marked in red and green respectively and the years are
identified by the two digits next to the symbols.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f02.png"/>

        </fig>

      <p id="d1e1562">Values of <inline-formula><mml:math id="M63" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula> above percentile 80 of the model (highlighted in
red in Fig. 2) and below percentile 20 (highlighted in green) present larger
departures from the fitted normal distribution than the remaining years;
these two groups are classified respectively as severe years (1991, 1995,
1998, 2003, 2005 and 2017) and as weak years (1983, 1988, 1997, 2007, 2008
and 2014). The remaining years (marked in black in Fig. 2) are classified as
moderate.</p>
</sec>
<sec id="Ch1.S3.SS2">
  <label>3.2</label><title>Model with two covariates</title>
      <p id="d1e1586">Departures of severe and weak years from the fitted normal distribution
suggest that other factors, namely meteorological ones, are playing a role
in the interannual variability of BA (Pereira et al., 2005). Following the
approach proposed by Nunes et al. (2014), for a given fixed day <inline-formula><mml:math id="M64" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, where <inline-formula><mml:math id="M65" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>
is chosen between 26 May  (<inline-formula><mml:math id="M66" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">56</mml:mn></mml:mrow></mml:math></inline-formula>) and 30 June  (<inline-formula><mml:math id="M67" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">91</mml:mn></mml:mrow></mml:math></inline-formula>), we
tested an alternate model that incorporates information about meteorological
fire danger in the pre-fire and the fire seasons. The starting date of 26 May  was set a posteriori, as the day when the alternate model becomes
statistically significant and therefore the information provided is relevant
for the users. Accordingly, using Eqs. (1)–(3), we fitted to the sample
of <inline-formula><mml:math id="M68" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula> a normal distribution with the mean linearly depending
on covariate <inline-formula><mml:math id="M69" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> for the considered fixed day <inline-formula><mml:math id="M70" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, and on covariate
<inline-formula><mml:math id="M71" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>; i.e. we tested the following model:
            <disp-formula id="Ch1.E12" content-type="numbered"><label>12</label><mml:math id="M72" display="block"><mml:mrow><mml:mtable columnspacing="1em" rowspacing="0.2ex" class="split" displaystyle="true" columnalign="right left"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mi>a</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced><mml:mo>×</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced><mml:mo>×</mml:mo><mml:mi mathvariant="italic">χ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          It is worth noting that index <inline-formula><mml:math id="M73" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> is fixed in Eq. (12) and is only used
to identify the fixed day of the pre-fire season when covariate <inline-formula><mml:math id="M74" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> is computed. Maximum likelihood estimates of coefficients <inline-formula><mml:math id="M75" display="inline"><mml:mrow><mml:mi>a</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>
and <inline-formula><mml:math id="M76" display="inline"><mml:mrow><mml:mi>b</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> for the considered fixed day <inline-formula><mml:math id="M77" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> are then obtained according to
Eqs. (6) and (7). However, parameter <inline-formula><mml:math id="M78" display="inline"><mml:mi>c</mml:mi></mml:math></inline-formula> in Eq. (12) does not
depend on chosen day <inline-formula><mml:math id="M79" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> since, according to Eq. (8), <inline-formula><mml:math id="M80" display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> for
all days because <inline-formula><mml:math id="M81" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M82" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> have zero mean (since
they are normalized).</p>
      <p id="d1e1849">An alternate model with covariates <inline-formula><mml:math id="M83" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M84" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> is
accordingly fitted for each fixed day <inline-formula><mml:math id="M85" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, and the null hypothesis that the
null model is to be retained (against the alternate nested model) is
assessed using the likelihood ratio test. For all days between 26 May
<inline-formula><mml:math id="M86" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">56</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and 30 June (<inline-formula><mml:math id="M87" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">91</mml:mn></mml:mrow></mml:math></inline-formula>), the null hypothesis is rejected at the
5 % significance level, with the <inline-formula><mml:math id="M88" display="inline"><mml:mi>p</mml:mi></mml:math></inline-formula> values steeply decreasing from <inline-formula><mml:math id="M89" display="inline"><mml:mrow><mml:mn mathvariant="normal">9</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> to <inline-formula><mml:math id="M90" display="inline"><mml:mrow><mml:mn mathvariant="normal">6</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn mathvariant="normal">10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>. In turn, when <inline-formula><mml:math id="M91" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> progresses along the
pre-fire season, obtained values of <inline-formula><mml:math id="M92" display="inline"><mml:mrow><mml:mi>a</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> increase whereas corresponding
values of <inline-formula><mml:math id="M93" display="inline"><mml:mrow><mml:mi>b</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> slightly decrease (Fig. 3).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3"><?xmltex \currentcnt{3}?><label>Figure 3</label><caption><p id="d1e1986">Temporal evolution of coefficients <inline-formula><mml:math id="M94" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M95" display="inline"><mml:mi>b</mml:mi></mml:math></inline-formula>
of the normal model with two covariates.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f03.png"/>

        </fig>

      <p id="d1e2010">For each day <inline-formula><mml:math id="M96" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, performance of the fitted alternate model may be assessed
by representing each year in space <inline-formula><mml:math id="M97" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi><mml:mi mathvariant="italic">χ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> framed by covariates <inline-formula><mml:math id="M98" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M99" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> over a background of probability of exceedance of a given
fixed threshold, e.g. of the mean value <inline-formula><mml:math id="M100" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> of the null model, i.e.
<inline-formula><mml:math id="M101" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="italic">χ</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mo>[</mml:mo><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>;</mml:mo><mml:mi>a</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced><mml:mo>×</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced><mml:mo>×</mml:mo><mml:mi mathvariant="italic">χ</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>]</mml:mo></mml:mrow></mml:math></inline-formula>. Results of models fitted on 26 May (<inline-formula><mml:math id="M102" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">56</mml:mn></mml:mrow></mml:math></inline-formula>), 15 June
(<inline-formula><mml:math id="M103" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">76</mml:mn></mml:mrow></mml:math></inline-formula>) and 30 June (<inline-formula><mml:math id="M104" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">91</mml:mn></mml:mrow></mml:math></inline-formula>) are shown in Fig. 4. Severe (weak)
years tend to spread over the upper right (lower left) quadrants of the
space indicating that they are associated with high (low) values of both <inline-formula><mml:math id="M105" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M106" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>. Severe (weak) years are also associated with high (low)
values of <inline-formula><mml:math id="M107" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and, except for 26 May  and 15 June 2007,
the groups of severe and weak years are fully separated by contour
<inline-formula><mml:math id="M108" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.5</mml:mn></mml:mrow></mml:math></inline-formula> (which, in fact, corresponds to the probability of exceedance
of <inline-formula><mml:math id="M109" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in the null model, where meteorological factors are not taken
into account). Values of <inline-formula><mml:math id="M110" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> associated with severe years tend to
gradually increase along the pre-fire season and, on 30 June, five
out the six severe years present values above 0.7 (a threshold that is
barely surpassed by just two of the 27 moderate years). Finally, it is worth
noting that the contour lines of <inline-formula><mml:math id="M111" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="italic">χ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula>
become steeper along the pre-fire season, a result in agreement with the
steep increase in parameter <inline-formula><mml:math id="M112" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula> (Fig. 3).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4" specific-use="star"><?xmltex \currentcnt{4}?><label>Figure 4</label><caption><p id="d1e2279">Scatter plots of the 39-year sample in the space of covariates
<inline-formula><mml:math id="M113" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> vs. <inline-formula><mml:math id="M114" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> for <bold>(a)</bold> 26 May (<inline-formula><mml:math id="M115" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">56</mml:mn></mml:mrow></mml:math></inline-formula>), <bold>(b)</bold> 15 June (<inline-formula><mml:math id="M116" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">76</mml:mn></mml:mrow></mml:math></inline-formula>) and <bold>(c)</bold> 30 June (<inline-formula><mml:math id="M117" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">91</mml:mn></mml:mrow></mml:math></inline-formula>). Groups of severe (weak) years are identified by
the red (green) circles. The straight lines are contours of
<inline-formula><mml:math id="M118" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="italic">χ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula>.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f04.png"/>

        </fig>

      <p id="d1e2375">Results obtained suggest that the likelihood of a given year to belong to
the severe or to the weak groups may be estimated based on the value of
<inline-formula><mml:math id="M119" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> of each year as estimated by the fitted model on chosen day <inline-formula><mml:math id="M120" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>
(Fig. 5). For a given year, values of <inline-formula><mml:math id="M121" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> change very slowly from
day to day. This is to be expected since (1) the models are all fitted to the
same sample of <inline-formula><mml:math id="M122" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula>, (2) covariate <inline-formula><mml:math id="M123" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> is the same in all
models and (3) covariate <inline-formula><mml:math id="M124" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> has high serial correlation since,
according to Eq. (1), index <inline-formula><mml:math id="M125" display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi mathvariant="normal">pfs</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> accumulates values of DSR since
1 April up to the considered day. Therefore, any decision taken for a given
year, based on the respective estimate of <inline-formula><mml:math id="M126" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> from the model fitted
on day <inline-formula><mml:math id="M127" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, is not expected to drastically change in the next few days
unless there is an incoming sequence of very high (or very low) daily values
of DSR that will considerably change <inline-formula><mml:math id="M128" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and therefore <inline-formula><mml:math id="M129" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5" specific-use="star"><?xmltex \currentcnt{5}?><label>Figure 5</label><caption><p id="d1e2528">Temporal evolution of <inline-formula><mml:math id="M130" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi mathvariant="italic">χ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> during the pre-fire season (26 May to 30 June). Severe (weak) years are identified by the red (green)
curves. The black horizontal lines represent thresholds used in Type A
(<inline-formula><mml:math id="M131" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.5</mml:mn></mml:mrow></mml:math></inline-formula>) and Type B
(<inline-formula><mml:math id="M132" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.7</mml:mn></mml:mrow></mml:math></inline-formula>) decisions. Vertical dashed lines
delimit the phase where Type A and Type B decisions were checked.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f05.png"/>

        </fig>

      <p id="d1e2587">The following two types of decisions are tested.
<list list-type="bullet"><list-item>
      <p id="d1e2592"><italic>Type A</italic>. If <inline-formula><mml:math id="M133" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0.5</mml:mn></mml:mrow></mml:math></inline-formula> at day <inline-formula><mml:math id="M134" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> then the year is not classified as weak
(i.e. it is either moderate or severe); otherwise, if <inline-formula><mml:math id="M135" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>≤</mml:mo><mml:mn mathvariant="normal">0.5</mml:mn></mml:mrow></mml:math></inline-formula> then
the year is not classified as severe (i.e. it is either moderate or weak).</p></list-item><list-item>
      <p id="d1e2635"><italic>Type B</italic>. If <inline-formula><mml:math id="M136" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0.7</mml:mn></mml:mrow></mml:math></inline-formula> at day <inline-formula><mml:math id="M137" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> then the year is classified as severe.</p></list-item></list>
Results obtained indicate that decision types A and B should be applied
to different periods of the pre-fire season from 26 May
to 30 June (Table 1) and from 18 to 30 June (Table 2).
Furthermore, for each type of decision, two phases are identified in the
respective periods, the days of each phase being characterized by the same
decisions taken for each year. In the case of Type A decisions, there is just
one wrong decision (2007 is incorrectly classified as not being a weak year)
during phase A1 (26 May to 22 June) and all decisions are
correct during phase A2 (23 to 30 June). In the case of Type
B decisions, and for the entire period (18 to 30 June), all
severe years but one (1998) are correctly classified as severe; conversely, three moderate years are incorrectly classified as severe during phase
B1 (18 to 22 June) and this decreases to two during
phase B2 (23 to 30 June).</p>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T1"><?xmltex \currentcnt{1}?><label>Table 1</label><caption><p id="d1e2667">Performance assessment of Type A decisions during the pre-fire
season (26 May to 30 June) based on the model with two
covariates.</p></caption><oasis:table frame="topbot"><?xmltex \begin{scaleboxenv}{0.94}[0.94]?><oasis:tgroup cols="3">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left"/>
     <oasis:colspec colnum="3" colname="col3" align="left"/>
     <oasis:thead>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">Severe years</oasis:entry>
         <oasis:entry colname="col3">Weak years</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">incorrectly classified</oasis:entry>
         <oasis:entry colname="col3">incorrectly classified</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">as non-severe</oasis:entry>
         <oasis:entry colname="col3">as non-weak</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>
         <oasis:entry colname="col1">Phase A1</oasis:entry>
         <oasis:entry colname="col2">None</oasis:entry>
         <oasis:entry colname="col3">2007</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1">(26 May–22 June)</oasis:entry>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">Phase A2</oasis:entry>
         <oasis:entry colname="col2">None</oasis:entry>
         <oasis:entry colname="col3">None</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">(23–30 June)</oasis:entry>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup><?xmltex \end{scaleboxenv}?></oasis:table></table-wrap>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T2" specific-use="star"><?xmltex \currentcnt{2}?><label>Table 2</label><caption><p id="d1e2767">As in Table 1 but for performance assessment of Type B decisions for
18 to 30 June.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="4">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="right"/>
     <oasis:colspec colnum="3" colname="col3" align="right"/>
     <oasis:colspec colnum="4" colname="col4" align="right"/>
     <oasis:thead>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">Severe years correctly</oasis:entry>
         <oasis:entry colname="col3">Severe years not</oasis:entry>
         <oasis:entry colname="col4">Moderate years incorrectly</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">classified as severe</oasis:entry>
         <oasis:entry colname="col3">classified as severe</oasis:entry>
         <oasis:entry colname="col4">classified as severe</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>
         <oasis:entry colname="col1">Phase B1</oasis:entry>
         <oasis:entry colname="col2">1991, 1995, 2003,</oasis:entry>
         <oasis:entry colname="col3">1998</oasis:entry>
         <oasis:entry colname="col4">1987, 1992, 2010</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1">(18–22 June)</oasis:entry>
         <oasis:entry colname="col2">2005, 2017</oasis:entry>
         <oasis:entry colname="col3"/>
         <oasis:entry colname="col4"/>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">Phase B2</oasis:entry>
         <oasis:entry colname="col2">1991, 1995, 2003,</oasis:entry>
         <oasis:entry colname="col3">1998</oasis:entry>
         <oasis:entry colname="col4">1992, 2010</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">(23–30 June)</oasis:entry>
         <oasis:entry colname="col2">2005, 2017</oasis:entry>
         <oasis:entry colname="col3"/>
         <oasis:entry colname="col4"/>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

</sec>
<?pagebreak page1463?><sec id="Ch1.S3.SS3">
  <label>3.3</label><title>Model with one covariate</title>
      <?pagebreak page1464?><p id="d1e2882">Despite its usefulness in characterizing the role played by meteorological
factors during the pre-fire and the fire seasons, the model discussed in the
last subsection cannot be used to anticipate the likelihood of a given fire
season given that one of the two covariates (<inline-formula><mml:math id="M138" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>) is derived from daily
information during that same fire season. However, results obtained in the
previous subsection indicate that the role played by covariate <inline-formula><mml:math id="M139" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> in
each fitted model becomes more and more relevant when day <inline-formula><mml:math id="M140" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> progresses
along the pre-fire season as suggested by the steady increase in coefficient
<inline-formula><mml:math id="M141" display="inline"><mml:mrow><mml:mi>a</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> that even becomes larger than <inline-formula><mml:math id="M142" display="inline"><mml:mrow><mml:mi>b</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> after 25 June (Fig. 3),
as well as by the increase in slope of contour lines of <inline-formula><mml:math id="M143" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>(d) (Fig. 4).</p>
      <p id="d1e2953">Therefore, for each considered day <inline-formula><mml:math id="M144" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, we fitted to the sample of <inline-formula><mml:math id="M145" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula> the following normal model, now with the mean linearly depending
just on covariate <inline-formula><mml:math id="M146" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>:
            <disp-formula id="Ch1.E13" content-type="numbered"><label>13</label><mml:math id="M147" display="block"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi><mml:mo>;</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi>p</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced><mml:mo>×</mml:mo><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>
          Maximum likelihood estimates of model parameters <inline-formula><mml:math id="M148" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M149" display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula>
are obtained using Eqs. (9) and (10) and it may be noted that again
parameter <inline-formula><mml:math id="M150" display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula> is the same for all fitted models, with <inline-formula><mml:math id="M151" display="inline"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">μ</mml:mi><mml:mi mathvariant="normal">BA</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> because
<inline-formula><mml:math id="M152" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> has a zero mean. As in the case of coefficient <inline-formula><mml:math id="M153" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula> in the model
with two covariates, coefficient <inline-formula><mml:math id="M154" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> increases as the considered day <inline-formula><mml:math id="M155" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>
progresses along the pre-fire season (Fig. 6). For each day <inline-formula><mml:math id="M156" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> the
relative role played by covariates <inline-formula><mml:math id="M157" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M158" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> may be assessed by
comparing the log-likelihood ratio statistics <inline-formula><mml:math id="M159" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M160" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="M161" display="inline"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula>,
<inline-formula><mml:math id="M162" display="inline"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M163" display="inline"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> are the likelihood functions of the null model, the
model with one covariate and the model with two covariates (Fig. 7). These
two ratios represent the increases in likelihood of the sample of BA when
replacing the null model (with no covariates) by the model with covariate
<inline-formula><mml:math id="M164" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula> and then the latter model by the model with covariates <inline-formula><mml:math id="M165" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula> and
<inline-formula><mml:math id="M166" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6"><?xmltex \currentcnt{6}?><label>Figure 6</label><caption><p id="d1e3270">Temporal evolution of coefficient <inline-formula><mml:math id="M167" display="inline"><mml:mi>p</mml:mi></mml:math></inline-formula> of the normal model
with one covariate.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f06.png"/>

        </fig>

      <p id="d1e3287">Values of <inline-formula><mml:math id="M168" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> increase as the considered
day <inline-formula><mml:math id="M169" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> progresses along the pre-fire season, contrasting with the behaviour
of <inline-formula><mml:math id="M170" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> where a decrease (albeit more
moderate) is observed. The former ratio even becomes larger than the latter
from 23 to 30 June; however, it may be noted<?pagebreak page1465?> that, for models
fitted on days between 26 May and 13 June, values of <inline-formula><mml:math id="M171" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> are smaller than the critical
value at the 5 % significance level, indicating that the null model is to
be retained (against the alternate model with one covariate). Nevertheless,
the model with one covariate is still tested along the entire pre-fire
season (26 May to 30 June) because, as shown in Fig. 4a, b, the severe (weak) groups tend to present for the most
part high (low) values of covariate <inline-formula><mml:math id="M172" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>. Reinforcing the relevance of
this covariate when estimating the likelihood of a given year to belong to
the severe class, we find that, when, for each considered fixed day <inline-formula><mml:math id="M173" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>, we
restrict to years with <inline-formula><mml:math id="M174" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> larger than the respective
daily median, there is a positive correlation between <inline-formula><mml:math id="M175" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M176" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>
that, except from 5 to 12 June, is significant at the 5 %
level (Fig. 8).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F7"><?xmltex \currentcnt{7}?><label>Figure 7</label><caption><p id="d1e3424">Temporal evolution of the log-likelihood ratio statistics
<inline-formula><mml:math id="M177" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> between the null
model and the model with one covariate and <inline-formula><mml:math id="M178" display="inline"><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>ln⁡</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:math></inline-formula> between the model with one covariate and
the model with two covariates. The horizontal dashed line represents the
critical value for the statistic at the 5 % significance level.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f07.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F8"><?xmltex \currentcnt{8}?><label>Figure 8</label><caption><p id="d1e3487">Temporal evolution of the correlation between <inline-formula><mml:math id="M179" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced close=")" open="("><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M180" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> from 26 May  to 30 June for all years (dotted curve) and restricting to years
<inline-formula><mml:math id="M181" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mfenced open="(" close=")"><mml:mi>d</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> larger than the respective median
(solid curve). Black circles in the solid curve identify values of
correlation that are significant at the 5 % level.</p></caption>
          <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f08.png"/>

        </fig>

      <p id="d1e3525">As with the model with two covariates, a given year is anticipated as
belonging to the severe or the weak groups by making Type A and Type B
decisions based on the value of <inline-formula><mml:math id="M182" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> of each year as estimated by
the fitted model (with one covariate) on that fixed day <inline-formula><mml:math id="M183" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>. It is worth
noting the use of the wording “is anticipated as” (instead of “is
classified as” employed in the previous section), which is meant to enhance
the prognostic character of the model with covariate <inline-formula><mml:math id="M184" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>. Since the
model with one covariate has lower variability than the model with two
covariates, the threshold of 0.7 (previously used in Type B decisions) is
now lowered to 0.66 (Fig. 9).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F9" specific-use="star"><?xmltex \currentcnt{9}?><label>Figure 9</label><caption><p id="d1e3562">Temporal evolution of <inline-formula><mml:math id="M185" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mi mathvariant="italic">ψ</mml:mi></mml:mfenced></mml:mrow></mml:math></inline-formula> during the pre-fire season (26 May to 30 June). Severe (weak) years are identified by the red (green) curves.
The black horizontal lines represent thresholds used in Type A
(<inline-formula><mml:math id="M186" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.5</mml:mn></mml:mrow></mml:math></inline-formula>) and Type B
(<inline-formula><mml:math id="M187" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.66</mml:mn></mml:mrow></mml:math></inline-formula>) decisions. Vertical dashed lines
delimit the phase where Type A and Type B decisions were checked.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f09.png"/>

        </fig>

      <p id="d1e3615">Three phases (A1, A2 and A3) are used to assess Type A decisions (Table 3).
In all phases, only 1998 (out of the six severe years) is incorrectly
anticipated as a non-severe year. In turn, the number of weak years
incorrectly anticipated as non-weak decreases from three in phase A1 (26 to 30 May) to two in phase A2 (31 May to 5 June)
and then to just one in phase A3 (6 to 30 June). Assessment of
Type B decisions (Table 4) is performed in two phases (B1 and B2). Regarding
the severe years correctly classified as severe, it is worth pointing out
that, apart from phase B1 beginning 3 d later (on 21 June
instead of 18 June), there is no decrease in performance from the model
with two covariates, 1998 being again the only missed severe year in phases
B1 and B2. Finally, no virtual decrease is found in performance in the
number of moderate years incorrectly anticipated as severe, that, as with
the model with two covariates, decrease from three in phase B1 (21 to 26 June) to two in phase B2 (27 to 30 June).</p>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T3"><?xmltex \currentcnt{3}?><label>Table 3</label><caption><p id="d1e3621">As in Table 1 but for the model with one covariate.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="3">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="right"/>
     <oasis:colspec colnum="3" colname="col3" align="right"/>
     <oasis:thead>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">Severe years</oasis:entry>
         <oasis:entry colname="col3">Weak years</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">incorrectly  antici-</oasis:entry>
         <oasis:entry colname="col3">incorrectly antici-</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">pated as non-severe</oasis:entry>
         <oasis:entry colname="col3">pated as non-weak</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>
         <oasis:entry colname="col1">Phase A1</oasis:entry>
         <oasis:entry colname="col2">1998</oasis:entry>
         <oasis:entry colname="col3">1983, 1997, 2014</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1">(26–30 May)</oasis:entry>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">Phase A2</oasis:entry>
         <oasis:entry colname="col2">1998</oasis:entry>
         <oasis:entry colname="col3">1997, 2014</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1">(31 May–5 June)</oasis:entry>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">Phase A3</oasis:entry>
         <oasis:entry colname="col2">1998</oasis:entry>
         <oasis:entry colname="col3">2014</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">(6–30 June)</oasis:entry>
         <oasis:entry colname="col2"/>
         <oasis:entry colname="col3"/>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T4" specific-use="star"><?xmltex \currentcnt{4}?><label>Table 4</label><caption><p id="d1e3744">As in Table 2 but for the model with one covariate.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="4">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="right"/>
     <oasis:colspec colnum="3" colname="col3" align="right"/>
     <oasis:colspec colnum="4" colname="col4" align="right"/>
     <oasis:thead>
       <oasis:row>
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">Severe years correctly</oasis:entry>
         <oasis:entry colname="col3">Severe years not</oasis:entry>
         <oasis:entry colname="col4">Moderate years incorrectly</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1"/>
         <oasis:entry colname="col2">anticipated as severe</oasis:entry>
         <oasis:entry colname="col3">anticipated as severe</oasis:entry>
         <oasis:entry colname="col4">anticipated as severe</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>
         <oasis:entry colname="col1">Phase B1</oasis:entry>
         <oasis:entry colname="col2">1991, 1995, 2003,</oasis:entry>
         <oasis:entry colname="col3">1998</oasis:entry>
         <oasis:entry colname="col4">1992, 2006, 2015</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">
         <oasis:entry colname="col1">(21–26 June)</oasis:entry>
         <oasis:entry colname="col2">2005, 2017</oasis:entry>
         <oasis:entry colname="col3"/>
         <oasis:entry colname="col4"/>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">Phase B2</oasis:entry>
         <oasis:entry colname="col2">1991, 1995, 2003,</oasis:entry>
         <oasis:entry colname="col3">1998</oasis:entry>
         <oasis:entry colname="col4">1992, 2015</oasis:entry>
       </oasis:row>
       <oasis:row>
         <oasis:entry colname="col1">(27–30 June)</oasis:entry>
         <oasis:entry colname="col2">2005, 2017</oasis:entry>
         <oasis:entry colname="col3"/>
         <oasis:entry colname="col4"/>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

</sec>
</sec>
<sec id="Ch1.S4" sec-type="conclusions">
  <label>4</label><title>Discussion and conclusions</title>
      <p id="d1e3860">The increasing dimension of the impacts of extremely large wildfires that
have been affecting Portugal in the last decades calls for the improvement
of diagnostic and prognostic tools designed to assist forest and fire
managers in making better<?pagebreak page1466?> decisions on both prevention and combat. For this
purpose, we set up a model that allows outlooks, up to 1 month
ahead, of wildfire potential in Portugal during the fire season (defined as
July and August).</p>
      <p id="d1e3863">The model consists of a normal distribution of the decimal logarithm of the
yearly cumulated values of BA during the fire season (<inline-formula><mml:math id="M188" display="inline"><mml:mrow><mml:msub><mml:mi>log⁡</mml:mi><mml:mn mathvariant="normal">10</mml:mn></mml:msub><mml:mi mathvariant="normal">BA</mml:mi></mml:mrow></mml:math></inline-formula>)
where the mean of the distribution linearly depends on a covariate <inline-formula><mml:math id="M189" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, defined at a given fixed day <inline-formula><mml:math id="M190" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula> of the pre-fire season, which
responds to the accumulation of thermal and water stress up to the chosen
day <inline-formula><mml:math id="M191" display="inline"><mml:mi>d</mml:mi></mml:math></inline-formula>. This model with one covariate results from the simplification of a
model with two covariates <inline-formula><mml:math id="M192" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M193" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>, the latter
covariate being sensitive to the occurrence of hot and dry spells associated
with strong winds that are the key triggering mechanism for the onset and
spread of very large fire events (Amraoui et al., 2013). The rationale
behind setting up the model with covariate <inline-formula><mml:math id="M194" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is twofold: (1) long
periods of warm and dry weather during the pre-fire season induce high
levels of vegetation stress that increase the probability of occurrence of
large fire events as shown for Portugal (Trigo et al., 2006) and even the
Mediterranean basin (Gudmundsson et al., 2014; Turco et al., 2017); (2) in
fact, it is now well-known that soil moisture deficit and drought
(associated with large values of <inline-formula><mml:math id="M195" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>) have an impact on hot summer
extremes in Mediterranean regions (Vautard et al., 2007; Hirschi et al.,
2011), a feature that fostered the development of both statistical and
dynamical seasonal fire forecasting approaches in the Mediterranean region
(Gudmundsson et al., 2014; Turco et al., 2017, 2018).</p>
      <p id="d1e3950">Performance of the model was evaluated for the 39-year period 1980–2018 by
testing two types of decisions based on the temporal evolution of <inline-formula><mml:math id="M196" display="inline"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi mathvariant="normal">exc</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>:
for Type A, the decision is on whether a given year is classified as not
being severe or as not being weak, whereas for Type B it is on whether the
year is classified as severe. Type A decisions are made during the entire
pre-fire season (26 May to 30 June) and Type B decisions are
restricted to the end of the pre-fire season (21 to 30 June)
when the role played by the cumulation of vegetation stress becomes as
relevant as the one by the occurrence of extreme weather events during the
fire season. In the case of Type A decisions, only 1998 (out of the six severe
years) was incorrectly anticipated as non-severe. For Type B decisions, all
severe years but 1998 were correctly anticipated as severe, reflecting the
prominent role of cumulated vegetation stress at the end of the pre-fire
season in favouring the occurrence of severe years. However, a test similar
to Type B, but applied to weak years, has poor performance and this may be
viewed as an indication that low values of <inline-formula><mml:math id="M197" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula> along the pre-fire season
have a moderate impact on the likelihood of a following fire season with a
low value of <inline-formula><mml:math id="M198" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> and therefore on the likelihood of having a weak year.</p>
      <p id="d1e3978">Since both Type A and Type B decisions have shown to be incorrect for the
fire season of 1998, the temporal evolution of daily values of DSR is worth
being analysed along the pre-fire and the fire seasons (Fig. 10).
Except for two peaks in the last 10 d of June, the pre-fire season of
1998 is<?pagebreak page1467?> dominated by moderate daily values of DSR which are very close to
zero between 10 May and early June, a period characterized by
significant amounts of rainfall. Therefore, by the end of May, vegetation
was not subject to either thermal or water stress and this translates into
low values of <inline-formula><mml:math id="M199" display="inline"><mml:mrow><mml:mi mathvariant="italic">ψ</mml:mi><mml:mo>(</mml:mo><mml:mi>d</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> along the pre-fire season (Fig. 4). This
unstressed condition of vegetation explains why the two large peaks of daily
DSR in the second half of June did not trigger any large fire events as well
as why the large peak of DSR at the end of the first half of July was
followed by days of moderate burned area. However, the meteorological
conditions drastically changed in August, with the first half of the month
being dominated by values of daily DSR well above the median that culminated
with a sequence of days with DSR much higher than percentile 90, and the
second half presenting a peak of DSR very close to percentile 90. The impact
of this high number of extreme days translates into a large value of <inline-formula><mml:math id="M200" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula>
for 1998, the seventh largest in the 39 years analysed and the third of the
six severe years. Striking a vegetation stressed by the warm and dry
conditions that prevailed since the beginning of August, the two peaks of
DSR triggered two sequences of days affected by large fire events that made
1998 rank fifth in cumulated BA in 1980–2018.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F10"><?xmltex \currentcnt{10}?><label>Figure 10</label><caption><p id="d1e4005">Temporal evolution of daily values of DSR for 1998 (red curve) and of the daily values of the median (solid black curve) and of percentile
90 (dotted black curve) for the period 1980–2018. The dark grey bars
indicate the cumulated daily values of BA (in hectares) for Portugal. The
vertical black line subdivides the period into pre-fire and fire seasons.</p></caption>
        <?xmltex \igopts{width=241.848425pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/19/1459/2019/nhess-19-1459-2019-f10.png"/>

      </fig>

      <p id="d1e4014">As shown in Fig. 4, 1998 is the only severe year when a value of <inline-formula><mml:math id="M201" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>
at the last day of the pre-fire season (30 June) below the mean minus
1 standard deviation is followed by a value of <inline-formula><mml:math id="M202" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> above the mean plus
1 standard deviation. The exceptional character of 1998 is reinforced by
the fact that, even when considering all 39 years, this condition is
fulfilled in only 2 years, i.e. 1998 and 1985 (a moderate year). However,
despite the extreme meteorological conditions observed in August, 1998 just
ranks fifth among the six severe years, suggesting that the mild
conditions during the pre-fire season still had a mitigating role. Conversely, the exceptionality of 1998 puts an emphasis on the importance of
short-term extreme events as triggers of large-fire events, a feature that
is not considered when restricting to the information provided by maps of
long-term seasonal forecasts of temperature and rainfall anomalies,
undermining the usefulness of current seasonal forecasting outputs for
wildland fire potential outlooks.</p>
      <p id="d1e4031">Several improvements to the proposed prognostic model (with covariate <inline-formula><mml:math id="M203" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>) are currently being considered. First, no advantage was taken of the fact
that covariates <inline-formula><mml:math id="M204" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M205" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> are positively correlated (when <inline-formula><mml:math id="M206" display="inline"><mml:mi mathvariant="italic">ψ</mml:mi></mml:math></inline-formula>
is larger than the median). This information may be added by setting up a
model of the distribution of <inline-formula><mml:math id="M207" display="inline"><mml:mi mathvariant="italic">χ</mml:mi></mml:math></inline-formula> using information about meteorological
conditions along the pre-fire season and then incorporating this information
in the prognostic model. Second, the model was developed for the entire
territory of Portugal, not taking into account regional characteristics of
climate, land cover and fire regime. It is therefore worth setting up
regional models over areas with distinct pyrogeographical characteristics.
Finally, the outlooks with respect to the months of July and August and given the
observed trend to have a longer fire season it would be very useful to
extend them until October. For instance, as pointed out by Beighley and Hyde (2018), 36 % of the total area burned in 2009–2017 was outside the period
from July to September, a fraction that is 3 times larger than the value
of 12 % observed when considering the period 2001–2008.</p>
      <p id="d1e4069">A beta version of the proposed model has been experimentally running since
May 2017 and the correctness of the outlooks (made in real time) about 2017
being a severe year (based on a Type B decision) and about 2018 not being
severe (based on a Type A decision) give confidence about the potential of
the model to be operationally used to anticipate the occurrence of severe
years.</p>
      <?pagebreak page1468?><p id="d1e4072">Daily outlooks are currently available at CeaseFire (<uri>http://idlcc.fc.ul.pt/CeaseFire/index.php</uri>, last access: 15 July 2019), a website designed to
integrate and disseminate relevant meteorological information to the user
fire community by means of a simple, fast and user-friendly interface
(Evans, 2018). Developed by Instituto Dom Luiz (IDL) at the Faculty of
Sciences of the University of Lisbon (Portugal), the platform relies on data
provided in near-real time by LSA SAF, the EUMETSAT Satellite Application
Facility for Land Surface Analysis (Trigo et al., 2011). Currently there are
about 900 registered users in the CeaseFire platform, most of them from
national authorities and services, as well as from municipalities and
private companies, namely from the paper and pulp industry.</p>
</sec>

      
      </body>
    <back><notes notes-type="dataavailability"><title>Data availability</title>

      <p id="d1e4083">The meteorological data used in this study were obtained from products of the LSA-SAF Fire Risk Map (LSA-504), produced and disseminated by EUMETSAT Satellite Application Facility on Land Surface Analysis (LSA-SAF, 2019); in accordance with EUMETSAT data policy, the LSA-SAF data are granted to every interested user free of charge. The burned area data are publicly available from the national fire record database (ICNF, 2019).</p>
  </notes><notes notes-type="authorcontribution"><title>Author contributions</title>

      <p id="d1e4089">SAN performed the study and wrote the paper; CDC, RMT and TJC contributed to
the model development and revised the paper; KFT and MAAT contributed to the
model validation and revised the paper.</p>
  </notes><notes notes-type="competinginterests"><title>Competing interests</title>

      <p id="d1e4095">The authors declare that they have no conflict of interest.</p>
  </notes><notes notes-type="sistatement"><title>Special issue statement</title>

      <p id="d1e4101">This article is part of the special issue “Spatial and temporal patterns of wildfires: models, theory, and reality”. It is a result of the conference EGU 2017, Vienna, Austria, 23–28 April 2017.</p>
  </notes><ack><title>Acknowledgements</title><p id="d1e4107">Research was developed within the framework of the EUMETSAT Satellite Application Facility for Land Surface Analysis (LSA SAF) and the FCT project “Forecasting fire probability and characteristics for a habitable pyroenvironment (FireCast)” under grant no. PCIF/GRF/0204/2017.</p></ack><notes notes-type="financialsupport"><title>Financial support</title>

      <p id="d1e4112">This research has been partially supported by The Navigator Company under contract CEASEFIRE.</p>
  </notes><notes notes-type="reviewstatement"><title>Review statement</title>

      <p id="d1e4118">This paper was edited by Marj Tonini and reviewed by two anonymous referees.</p>
  </notes><ref-list>
    <title>References</title>

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    <!--<article-title-html>Wildland fire potential outlooks for Portugal using meteorological indices of fire danger</article-title-html>
<abstract-html><p>Portugal is recurrently affected by large wildfire events that have serious
impacts at the socio-economic and environmental levels and dramatic
consequences associated with the loss of lives and the destruction of the
landscape. Accordingly, seasonal forecasts are required to assist fire
managers, thus contributing to alter the historically based purely reactive
response. In this context, we present and discuss a statistical model to
estimate the probability that the total burned area during summer will
exceed a given threshold. The statistical model uses meteorological
information that rates the accumulation of thermal and vegetation stress.
Outlooks for the 39-year study period (1980–2018) show that, when the
statistical model is applied from 26 May to 30 June, out of
the six severe years, only one year is not anticipated as potentially severe
and, out of the six weak years, only one is not anticipated as potentially
weak. The availability of outlooks of wildfire potential with an
anticipation of up to 1 month before the starting of the fire season, such
as the one proposed here, may serve to provide clear directions for the fire
community when planning prevention and combating fire events.</p></abstract-html>
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