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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-21-1313-2021</article-id><title-group><article-title>Extreme wind return periods from tropical cyclones <?xmltex \hack{\break}?> in Bangladesh: insights from a high-resolution convection-permitting numerical model</article-title><alt-title>Extreme wind return periods from tropical cyclones in
Bangladesh</alt-title>
      </title-group><?xmltex \runningtitle{Extreme wind return periods from tropical cyclones in
Bangladesh}?><?xmltex \runningauthor{H.~Steptoe and T.~Economou}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Steptoe</surname><given-names>Hamish</given-names></name>
          <email>hamish.steptoe@metoffice.gov.uk</email>
        <ext-link>https://orcid.org/0000-0003-3677-5951</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2 aff1">
          <name><surname>Economou</surname><given-names>Theodoros</given-names></name>
          
        <ext-link>https://orcid.org/0000-0001-8697-1518</ext-link></contrib>
        <aff id="aff1"><label>1</label><institution>Met Office, FitzRoy Road, Exeter, EX1 3PB, UK</institution>
        </aff>
        <aff id="aff2"><label>2</label><institution>College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">Hamish Steptoe (hamish.steptoe@metoffice.gov.uk)</corresp></author-notes><pub-date><day>29</day><month>April</month><year>2021</year></pub-date>
      
      <volume>21</volume>
      <issue>4</issue>
      <fpage>1313</fpage><lpage>1322</lpage>
      <history>
        <date date-type="received"><day>8</day><month>September</month><year>2020</year></date>
           <date date-type="accepted"><day>28</day><month>March</month><year>2021</year></date>
           <date date-type="rev-recd"><day>10</day><month>February</month><year>2021</year></date>
           <date date-type="rev-request"><day>12</day><month>October</month><year>2020</year></date>
      </history>
      <permissions>
        <copyright-statement>Copyright: © 2021 </copyright-statement>
        <copyright-year>2021</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="d1e98">We use high-resolution (4.4 <inline-formula><mml:math id="M1" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula>) numerical simulations of tropical
cyclones to produce exceedance probability estimates for extreme wind (gust) speeds over Bangladesh. For the first time, we estimate equivalent return periods up to and including a 1-in-200 year event, in a spatially coherent manner over all of Bangladesh, by using generalised additive models. We show that some northern provinces, up to 200 <inline-formula><mml:math id="M2" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> inland, may experience conditions equal to or exceeding a very severe cyclonic storm event (maximum wind speeds in <inline-formula><mml:math id="M3" display="inline"><mml:mrow><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">64</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M4" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) with a likelihood equal to coastal regions less than 50 <inline-formula><mml:math id="M5" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> inland. For the most severe super cyclonic storm events (<inline-formula><mml:math id="M6" display="inline"><mml:mrow><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">120</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M7" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>), event exceedance probabilities of 1-in-100 to 1-in-200 events remain limited to the coastlines of southern provinces only. We demonstrate how the Bayesian interpretation of the generalised additive model can facilitate a transparent decision-making framework for tropical cyclone warnings.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

<sec id="Ch1.S1" sec-type="intro">
  <label>1</label><title>Introduction</title>
      <p id="d1e171">Bangladesh is one of the most disaster-prone countries in the world, ranking
seventh in the 1999–2018 Long-Term Climate Risk Index (Eckstein et al.,
2019). Large portions of the population are exposed to the multiple natural
hazards, including those derived from tropical cyclones (TCs), such as
high winds, storm surge and flooding (e.g. Dilley et al., 2005).  In the last 30 years, TCs impacting Bangladesh, from the Bay of Bengal (BoB), have been responsible for damages of ca. USD 8.9 billion and affected 45 million people (EM-DAT, 2021), with average annual extreme-weather-event-related losses amounting to 1.8 <inline-formula><mml:math id="M8" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula> of GDP between 1990 and 2008 (International Monetary Fund, 2019b). The wider north Indian Ocean basin averages five cyclones per year (accounting for ca. 7 % of global tropical cyclone activity) (Sahoo and Bhaskaran, 2016); however, there is some indication of a decrease in TC frequency (Alam et al., 2003; Mohapatra et al., 2017; Rao, 2004; Singh et al., 2019) and an increase in cyclone intensity (Balaguru et al., 2014) that is projected to continue under a warming climate (Knutson et al., 2020).</p>
      <p id="d1e182">Recently, the International Monetary Fund (2019b) highlighted the early response Bangladesh is taking to the challenges posed by climate change; however, they also emphasise the importance of insurance mechanisms to enhance financial cover against impacts of natural disasters (International Monetary Fund, 2019a). Insurance facilitates disaster risk resilience and adaptation by transferring residual risk away from individuals and communities. Cost-effective and risk-informed sustainable development is based on the comprehensive understanding of hazards; the vulnerability of economies, societies and governments; and the exposure of society, people and belongings (UNDRR, 2019), but the lack of understanding of one or more of these components frequently limits the use of insurance mechanisms in many regions of the world most at risk from weather and climate hazards. This leaves significant populations around the world more vulnerable to the economic consequences of events that are otherwise manageable in countries with well-developed insurance markets (von Peter et al., 2012).</p>
      <?pagebreak page1314?><p id="d1e185">Detailed understanding of hazards is an essential part of understanding risk,
but a relatively sparse meteorological observational network and interrupted
non-continuous data records impose fundamental constraints on the description
of TC hazards. Simulations of tropical cyclones in the BoB remain challenging
for the current generation of seasonal forecasting systems (Camp et al.,
2015), global climate models (Shaevitz et al., 2014) and reanalyses (Hodges
et al., 2017), partly due the relatively coarse spatial and temporal
resolution of the numerical simulations. It is well understood that
large-scale thermodynamics and vertical wind shear have a significant impact on
TC intensity, but there are also numerous vortex, convective, turbulent and
frictional dissipative processes (e.g. Bryan and Rotunno, 2009; Nolan et al.,
2007; Tang et al., 2015 amongst others) that occur on much smaller scales and
also influence TC intensity, the impacts of which are not captured in low-resolution modelling. For example, extreme gusts associated with vigorous
(deep) convection will generally be underestimated without kilometre-scale
grid spacing that can explicitly resolve deep convection (e.g. Leutwyler
et al., 2017; Weisman et al., 1997). More generally, as summarised by
Leutwyler et al. (2017, and references therein), grid spacings of
<inline-formula><mml:math id="M9" display="inline"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> are comparable to the size of the particularly energetic
eddies in the planetary boundary layer.</p>
      <p id="d1e206">Previous insights into TC hazards affecting Bangladesh focus on compiling
catalogues of events (see Alam and Dominey-Howes, 2015, and references
therein), or apply statistical analysis to event catalogues (e.g.
Bandyopadhyay et al., 2018; Bhardwaj et al., 2020), and can only provide
limited insight into the spatial extent, variability and magnitude of events
based on first-hand eyewitness reports and limited observational
records. Other authors take a parametric wind-field approach, combing the
geostrophic (gradient) wind with a planetary boundary layer model to produce
hazard maps at kilometre-scale resolution (e.g. Done et al., 2020; Krien
et al., 2018; Tan and Fang, 2018); although this is a relatively
computationally inexpensive approach, the quality of the result appears highly
variable between global TC basins. Additionally, there are several holistic
risk assessment views that combine multiple sources of hazard data,
recognising that there are multiple hazards associated with TCs and that a
combined risk assessment is non-trivial. However, these techniques are often
limited to particular events (e.g. Hoque et al., 2016, 2019) or particular
areas (e.g. Alam et al., 2020). In both cases, the quality of hazard and/or
risk assessment is limited by available observational and track data.</p>
      <p id="d1e210">In this study we seek to improve our understanding of the historical extreme
gust speed hazard associated with recent TCs. To address the lack of
observation data in this region, we use the latest-generation Met Office
regional model over the BoB to simulate nine versions of 12 historical tropical
cyclone cases representing 1979–2019. This generates spatially and temporally
consistent counterfactual simulations (relative to observed TC cases), albeit
limited by the constraints of the model configuration and computational
resources. This ensemble configuration enhances our understanding of how each
cyclone may evolve if a similar event were to happen again. We combine the
ensemble information in a spatially coherent manner to produce hazard maps at
4.4 <inline-formula><mml:math id="M10" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> resolution over Bangladesh for extreme wind (gust)
hazards. Using Bayesian inference, we estimate gust speed exceedance intervals
(return periods) across all of Bangladesh and demonstrate how this
information can be directly integrated into a decision-making framework.</p>
</sec>
<sec id="Ch1.S2">
  <label>2</label><title>Numerical modelling and geospatial processing</title>
      <p id="d1e229">Tropical cyclone simulations are derived from a nine-member ensemble for 12
historical events, using the latest-generation Met Office Unified Model (Brown
et al., 2012) convection-permitting regional atmosphere configuration RAL2-T,
based on Bush et al. (2020) – hereafter referred to as RAL2. The RAL2
4.4 <inline-formula><mml:math id="M11" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> domain avoids placing model boundaries over the Himalayas and
covers Nepal, Bhutan, Myanmar, most of India, and parts of the Tibetan
Plateau. To ensure model stability over this mountainous terrain, the RAL2
model was run with a 30 <inline-formula><mml:math id="M12" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:math></inline-formula> time step. Each ensemble member requires a
24 <inline-formula><mml:math id="M13" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">h</mml:mi></mml:mrow></mml:math></inline-formula> spin-up period as the RAL2 model adjusts from weak initial
conditions taken from the ERA5 driving global model (of Hersbach et al.,
2020). This initial 24 <inline-formula><mml:math id="M14" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">h</mml:mi></mml:mrow></mml:math></inline-formula> period of model data is discarded in subsequent
analysis and data files. Thereafter, each ensemble member is free running for
a further 48 <inline-formula><mml:math id="M15" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">h</mml:mi></mml:mrow></mml:math></inline-formula>, with hourly boundary conditions provided by
ERA5. Collectively, the ensembles members sample a range of lead times before
landfall from 12–36 <inline-formula><mml:math id="M16" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">h</mml:mi></mml:mrow></mml:math></inline-formula>.</p>
      <p id="d1e281">The parameterised RAL2 gust diagnostic represents a prediction of the
3 <inline-formula><mml:math id="M17" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:math></inline-formula> average wind speed at every time step. The maximum of this
3 <inline-formula><mml:math id="M18" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:math></inline-formula> average speed over an hour is then taken to give the hourly
maximum 3 <inline-formula><mml:math id="M19" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">s</mml:mi></mml:mrow></mml:math></inline-formula> gust speed. While not truly resolving deep convection,
RAL2 is able to explicitly represent deep convective processes within the
resolved dynamics.  At these kilometre-scale resolutions the lower horizontal
size limit of convective cells is still set by the effective resolution of 5
to 10 times the grid length (Boutle et al., 2014; Skamarock, 2004), and
consequentially we expect that turbulent processes, as well as the dominant
turbulent length scale, will still be under resolved in this 4.4 <inline-formula><mml:math id="M20" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula>
dataset. The RAL2 model uses a gust parameterisation based on 10 <inline-formula><mml:math id="M21" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi></mml:mrow></mml:math></inline-formula> wind
speed with scaling proportional to the standard deviation of the horizontal
wind that also accounts for friction velocity, atmospheric stability and
roughness length (Lock et al., 2019).</p>
      <p id="d1e324">We use the ensemble output to first derive event “footprints” – a common
method within the catastrophe modelling community to define peak hazard
relating to a given event. In this case, footprints are based on the maximum
wind gust speed achieved within each model run of 48 <inline-formula><mml:math id="M22" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">h</mml:mi></mml:mrow></mml:math></inline-formula>, which
implicitly collapses the time dimension to leave a 2D gust field in<?pagebreak page1315?> a
longitude–latitude frame of reference. Although the original regional model
data in Steptoe et al. (2021) covers a significant portion of the BoB, we crop
the data to approximately 87.5 to 93.0<inline-formula><mml:math id="M23" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> E and 20.5 to
27.5<inline-formula><mml:math id="M24" display="inline"><mml:msup><mml:mi/><mml:mo>∘</mml:mo></mml:msup></mml:math></inline-formula> N.</p>
      <p id="d1e353">In general, median peak gust speeds from the RAL2 model ensemble are found to
be 22 to 43 <inline-formula><mml:math id="M25" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> faster compared to ERA5 reanalysis, but it is
known that extreme gusts associated with vigorous convection in ERA5 are
generally underestimated, sometimes by a factor of 2 (Owens and Hewson,
2018). For wind speed, the RAL2 median difference is 18 <inline-formula><mml:math id="M26" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>
faster compared to ERA5 and 5 and <inline-formula><mml:math id="M27" display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>3 <inline-formula><mml:math id="M28" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> compared to the
International Best Track Archive for Climate Stewardship data (IBTrACS, of
Knapp et al., 2010, 2018) for the India Meteorological Department and Central
Pacific Hurricane Center, Honolulu, regional forecast centres
respectively. Further details of the regional modelling process and validation
against IBTrACS and ERA5 reanalysis can be found in Steptoe et al. (2021).</p>
<sec id="Ch1.S2.SS1">
  <label>2.1</label><title>Generalised additive modelling (GAM)</title>
      <p id="d1e422">To summarise information from all nine regional model ensemble member footprints
into a coherent spatial summary of the tropical cyclone hazard, we use a
generalised additive model (GAM), after Hastie and Tibshirani (1986), based on
the R package <italic>mgcv</italic> of Wood (2017), as a flexible spatial regression
framework. GAMs are an extension of generalised linear modelling that use
smooth functions of covariates to build a linear predictor and have previously
been applied in similar geospatial natural hazard assessments, such as storm
count data over Europe (Youngman and Economou, 2017), spatial prediction of
maximum wind speed over Switzerland (Etienne et al., 2010) and return level
estimation for US wind gusts (Youngman, 2019).  In each case, these studies
incorporate spatial information into the GAMs formation, thereby implicitly
respecting the spatial interaction (autocorrelation) present in the source
data, and use the spatial dependence as a source of information.</p>
      <p id="d1e428">For our purposes, we use a Gaussian location-scale (GLS) model family (Wood
et al., 2016) to describe the natural logarithm (log) of the gust speed, where
both the mean and the log of the standard deviation are smooth functions of
predictors – in this case, longitude and latitude. Although other model
families were trialled (such as generalised extreme value and gamma
distributions), the GLS family was found to have the best trade-off between
computational efficiency and model fit. The general form of our GAM is

                <disp-formula specific-use="align" content-type="numbered"><mml:math id="M29" display="block"><mml:mtable displaystyle="true"><mml:mlabeledtr id="Ch1.E1"><mml:mtd><mml:mtext>1</mml:mtext></mml:mtd><mml:mtd><mml:mstyle class="stylechange" displaystyle="true"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>∼</mml:mo><mml:mtext>LogNormal</mml:mtext><mml:mo>(</mml:mo><mml:mi mathvariant="italic">μ</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:msup><mml:mo>)</mml:mo><mml:mn mathvariant="normal">2</mml:mn></mml:msup><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E2"><mml:mtd><mml:mtext>2</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>f</mml:mi><mml:mi mathvariant="italic">μ</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="italic">μ</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mtext>long</mml:mtext><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mtext>lat</mml:mtext><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr><mml:mlabeledtr id="Ch1.E3"><mml:mtd><mml:mtext>3</mml:mtext></mml:mtd><mml:mtd><mml:mstyle displaystyle="true" class="stylechange"/></mml:mtd><mml:mtd><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mtext>log</mml:mtext><mml:mo>(</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="italic">σ</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mtext>long</mml:mtext><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mtext>lat</mml:mtext><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mlabeledtr></mml:mtable></mml:math></disp-formula>

            where <inline-formula><mml:math id="M30" display="inline"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is the response variable, namely gust speed for each ensemble member <inline-formula><mml:math id="M31" display="inline"><mml:mi>i</mml:mi></mml:math></inline-formula> in each grid cell <inline-formula><mml:math id="M32" display="inline"><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">…</mml:mi><mml:mi>N</mml:mi></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M33" display="inline"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">207</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mn mathvariant="normal">081</mml:mn></mml:mrow></mml:math></inline-formula>. <inline-formula><mml:math id="M34" display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="italic">μ</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (a function of the mean) and <inline-formula><mml:math id="M35" display="inline"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi mathvariant="italic">σ</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (a function of the variance) are each defined as thin-plate regression splines (Wood, 2003) – isotropic smooth functions of covariates long<inline-formula><mml:math id="M36" display="inline"><mml:msub><mml:mi/><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> and lat<inline-formula><mml:math id="M37" display="inline"><mml:msub><mml:mi/><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> (longitude and latitude respectively). Each smooth function requires a user-defined maximum amount of desired flexibility (wiggliness), traditionally quantified by the number of knots. This flexibility is objectively penalised within <italic>mgcv</italic> to avoid overfitting, while optimally explaining the trends in the data (Wood, 2003). Trial and error shows that <inline-formula><mml:math id="M38" display="inline"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mn mathvariant="normal">600</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> knots are required to construct thin-plate spline basis functions that avoid over-smoothing given the resolution of the regional model data. Under this model formulation, the mean <inline-formula><mml:math id="M39" display="inline"><mml:mrow><mml:mi mathvariant="italic">μ</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> can be interpreted as an aggregated prediction across the ensemble members.</p>
      <p id="d1e725">The smooth model parameters are estimated using restricted maximum likelihood
(REML). However, once the model is fitted, it can be shown that it has a
Bayesian interpretation. In particular, the coefficients of the smooth
functions are assumed to have a multivariate normal prior distribution, whose
covariance matrix determines the wiggliness penalisation (see Wood, 2017, for
further details). A Gaussian approximation of the posterior distribution for
the coefficients then provides a multivariate normal distribution as the
posterior (Gelman et al., 2013). In practice, once a GAM model is fitted to
each named storm, under the Bayesian interpretation, we obtain 1000
simulations from the posterior distribution of the smooth function
coefficients via random draws from a multivariate normal distribution
(MVN). The MVN mean vectors are the REML coefficient estimates, and the MVN
covariance is derived as a function of the covariance matrix of the sampling
distribution of the model coefficients. In Bayesian inference, sampling from
the posterior distribution implies we can then derive samples from the
posterior predictive distribution of gust speed for each grid cell,
<inline-formula><mml:math id="M40" display="inline"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>. The predictive distribution, a unique feature of
Bayesian inference, fully quantifies estimation uncertainty and variability in
gust speed across ensemble members. We take 1000 samples from the posterior
predictive distribution and construct prediction intervals based on the
empirical quantiles of these samples. To aggregate gust information from all
ensembles of all named storms, we pool the 1000 posterior predictive
simulations from each event into a total of 12 000 samples from the
predictive distribution of gust speed across all 12
events. Figure 1 summarises the key parts of this process.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1"><?xmltex \currentcnt{1}?><?xmltex \def\figurename{Figure}?><label>Figure 1</label><caption><p id="d1e748">Summary of generalised additive modelling and the derivation of the
posterior predictive gust speed distribution. The posterior predictive
distribution is derived for each grid cell of the regional model domain.
Gust speed prediction intervals are found from the percentiles of the
posterior predictive distribution.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/21/1313/2021/nhess-21-1313-2021-f01.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2" specific-use="star"><?xmltex \currentcnt{2}?><?xmltex \def\figurename{Figure}?><label>Figure 2</label><caption><p id="d1e759">Detrended quantile–quantile (worm) plots for each GAM model per
storm. We discretise the quantiles into 50 bins (open circles). The red dashed
line represents zero deviance between data and theoretical quantiles defined
in the GAM. Where model quantiles deviate below (above) the zero deviance
line, this implies that the model predictions are overestimated
(underestimated) relative to the data: for any given theoretical model
quantile, the data quantile is lower (higher). Deviance residuals respect the
model family used when fitting the GAM and are calculated via the simulation
method of Augustin et al. (2012).</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/21/1313/2021/nhess-21-1313-2021-f02.png"/>

        </fig>

      <p id="d1e768">Assessing the GAM specification for <inline-formula><mml:math id="M41" display="inline"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> with detrended
quantile–quantile (worm) plots (based on the method of Augustin et al., 2012),
Fig. 2 shows that generally storms are well represented. For some storms (such
as Aila, BOB01, BOB07, Bulbul, Rashmi and TC01B) there is a tendency for the
GAM to overestimate the tails of the distribution (positive kurtosis) relative
to the 4.4 <inline-formula><mml:math id="M42" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> data, as indicated by quantile–quantile plot points
falling below the zero residual line. In these cases, the GAM will
overestimate extremes. Akash is the only storm where maximum gust speeds are
likely to be underestimated in the GAM relative to the 4.4 <inline-formula><mml:math id="M43" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> data,
but only for extreme upper-tail gust speeds.  Checking for the consistency of
variance over the range of predictor values,<?pagebreak page1316?> shows that the distribution of
the residuals is stationary for both longitude and latitude (not shown).</p>
</sec>
</sec>
<sec id="Ch1.S3">
  <label>3</label><title>Tropical cyclones in Bangladesh</title>
      <p id="d1e813">Aggregating the 12 historical tropical cyclones ensembles, Fig. 3 shows the
50th, 95th and 99th percentiles of the posterior predictive maximum gust speed
distribution across Bangladesh. Based on historical cases, the provinces of
Chittagong, Barisal and Khulna are most exposed to high wind speed associated
with tropical cyclone gusts, whilst Sylhet and Rajshahi are least exposed. The
cities of Chittagong and Cox's Bazar are particularly at risk of maximum
tropical cyclone gust speeds exceeding 45 <inline-formula><mml:math id="M44" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (87 <inline-formula><mml:math id="M45" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>)
and 60 <inline-formula><mml:math id="M46" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (116 <inline-formula><mml:math id="M47" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) respectively, in 5 <inline-formula><mml:math id="M48" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula> of
events making landfall. Maximum gust speeds in Dhaka are likely to reach
35 <inline-formula><mml:math id="M49" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (68 <inline-formula><mml:math id="M50" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) in 1 <inline-formula><mml:math id="M51" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula> of events,
25 <inline-formula><mml:math id="M52" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (48 <inline-formula><mml:math id="M53" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) in 5 <inline-formula><mml:math id="M54" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula> to 50 <inline-formula><mml:math id="M55" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula> of
events. We note that despite the northern provinces of Rajshahi, Rangpur and
Mymensingh being over 200 <inline-formula><mml:math id="M56" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> inland, they experience 95th and 99th
percentile gust speeds greater than those observed in the populated provincial
capitals of Dhaka, Barisal and Khulna. These extreme percentiles reflect the
influence of cyclones Fani (May 2019) and Aila (May 2009) which had strong
persistent in-land tracks.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3" specific-use="star"><?xmltex \currentcnt{3}?><?xmltex \def\figurename{Figure}?><label>Figure 3</label><caption><p id="d1e960">Gust speed exceedance thresholds for the 50th <bold>(a)</bold>,
95th <bold>(b)</bold> and 99th <bold>(c)</bold> percentile credible intervals. The
50th, 95th and 99th percentiles represent the maximum gust speeds expected from a 1-in-2, 1-in-20 and 1-in-100 event respectively (conditional on a tropical cyclone making landfall over Bangladesh). These credible intervals are based on the posterior model distribution derived from all 12 named tropical cyclones, conditional on a tropical cyclone making landfall in Bangladesh. The 20–60 <inline-formula><mml:math id="M57" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> gust speed range roughly corresponds to a range of 39–117 <inline-formula><mml:math id="M58" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>, equivalent to the cyclonic to super cyclonic storm classification used in Bangladesh. Province boundaries are outlined in white, with the 18 most populated towns and cities marked by circles.</p></caption>
        <?xmltex \igopts{width=483.69685pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/21/1313/2021/nhess-21-1313-2021-f03.png"/>

      </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4" specific-use="star"><?xmltex \currentcnt{4}?><?xmltex \def\figurename{Figure}?><label>Figure 4</label><caption><p id="d1e1005">Event exceedance probabilities for a severe cyclonic storm <bold>(a)</bold>,
very severe cyclonic storm <bold>(b)</bold> and super cyclonic storm <bold>(c)</bold> WMO
tropical cyclone classifications used in the Bay of Bengal (WMO,
2018). Event exceedance probabilities show the likelihood of a maximum
tropical cyclone gust speed being greater than or equal to the corresponding
classification wind threshold, conditional on a tropical cyclone making
landfall over Bangladesh. An exceedance threshold of 50 <inline-formula><mml:math id="M59" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula> (0.5 <inline-formula><mml:math id="M60" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula>)
represent a 1-in-2 (1-in-200) chance of a tropical cyclone exceeding a given
threshold. Areas where the exceedance probability is <inline-formula><mml:math id="M61" display="inline"><mml:mrow><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">50</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M62" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula>
(<inline-formula><mml:math id="M63" display="inline"><mml:mrow><mml:mo>&lt;</mml:mo><mml:mn mathvariant="normal">0.5</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M64" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">%</mml:mi></mml:mrow></mml:math></inline-formula>) are shaded black (grey). Province boundaries are
outlined in white, with the 18 most populated towns and cities marked by
circles.</p></caption>
        <?xmltex \igopts{width=483.69685pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/21/1313/2021/nhess-21-1313-2021-f04.png"/>

      </fig>

      <p id="d1e1077">The gust speed hazard can also be considered in terms of the probability of
exceeding a threshold. Using WMO thresholds for tropical cyclone wind speeds
(WMO, 2018), Fig. 4 shows that significant areas of southern provinces
(Khulna, Barisal and Chittagong) will experience maximum wind speed in excess of “severe” cyclonic storm condition <inline-formula><mml:math id="M65" display="inline"><mml:mrow><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">25</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M66" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (48 <inline-formula><mml:math id="M67" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) with a probability of 20 %–50 % per tropical cyclone event. At higher wind speeds, only areas within 30 <inline-formula><mml:math id="M68" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> of the coastline are predicted to experience gust speeds in excess of “very severe” cyclonic storm conditions <inline-formula><mml:math id="M69" display="inline"><mml:mrow><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">33</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M70" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (64 <inline-formula><mml:math id="M71" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) with the same likelihood (20 %–50 % per event). Wind speeds in excess of “super cyclonic” conditions
<inline-formula><mml:math id="M72" display="inline"><mml:mrow><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">62</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M73" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> (120 <inline-formula><mml:math id="M74" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) are predicted to be exceeded with a likelihood of 0.5 %–5 % per event in limited areas south of Chittagong, with a small area in the vicinity of Cox's Bazar seeing exceedances of 5 %–10 % per event.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5"><?xmltex \currentcnt{5}?><?xmltex \def\figurename{Figure}?><label>Figure 5</label><caption><p id="d1e1196">Exceedance probability curves for 18 of the most populated towns
and cities in Bangladesh (grey lines), with four key cities highlighted: Dhaka
(orange), Comilla (blue), Chittagong (green) and Cox's Bazar (red). For
reference, the minimum and maximum range of exceedance probabilities (across
all of Bangladesh) are represented by the dashed lines. Note that storm
exceedance probability is shown on a log scale.</p></caption>
        <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/21/1313/2021/nhess-21-1313-2021-f05.png"/>

      </fig>

      <p id="d1e1205">In addition to specific thresholds, exceedance probability curves (Fig. 5)
summarise information for gust speeds up to 80 <inline-formula><mml:math id="M75" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>
(155 <inline-formula><mml:math id="M76" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">kn</mml:mi></mml:mrow></mml:math></inline-formula>) for 18 of the most populated towns and cities in Bangladesh
(grey lines) with four key cities highlighted. The coastal cities of Cox's Bazar and Chittagong are unsurprisingly the population centres most exposed to high gust speeds. Chittagong and Cox's Bazar are roughly 2.5 and 4.8 times more likely to experience tropical cyclones exceeding very severe cyclonic storm conditions than Dhaka for a landfalling cyclone.</p>
<sec id="Ch1.S3.SS1">
  <label>3.1</label><title>Decision-making under uncertainty</title>
      <p id="d1e1240">By defining a loss function, it is possible to exploit the information in the
Bayesian posterior predictive distributions to create a warning model based on
decision theory (Lindley, 1991). Following Economou et al. (2016), defining a
loss function <inline-formula><mml:math id="M77" display="inline"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> to quantify the consequences of the various actions <inline-formula><mml:math id="M78" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula>
(e.g. issuing warnings) that could be taken in the event of a landfalling TC
of varying intensities <inline-formula><mml:math id="M79" display="inline"><mml:mi>x</mml:mi></mml:math></inline-formula> (see Table 1 for an example of four discrete gust
categories), provides a method of mapping predictive information onto an
action. The optimum action <inline-formula><mml:math id="M80" display="inline"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>, given some predictive information <inline-formula><mml:math id="M81" display="inline"><mml:mi>y</mml:mi></mml:math></inline-formula> (i.e. predictions of gust speed <inline-formula><mml:math id="M82" display="inline"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> from the GAM), is one
that minimises the loss <inline-formula><mml:math id="M83" display="inline"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> taking into account the uncertainty in the
predictive information, expressed as the probability of TC intensity <inline-formula><mml:math id="M84" display="inline"><mml:mi>x</mml:mi></mml:math></inline-formula> given
predictive information <inline-formula><mml:math id="M85" display="inline"><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>:

                <disp-formula id="Ch1.E4" content-type="numbered"><label>4</label><mml:math id="M86" display="block"><mml:mstyle displaystyle="true" class="stylechange"/><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:msup><mml:mi>a</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>arg⁡</mml:mi><mml:mo movablelimits="false">min⁡</mml:mo><mml:munder><mml:mo movablelimits="false">∫</mml:mo><mml:mi>x</mml:mi></mml:munder><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>a</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>x</mml:mi><mml:mi mathvariant="normal">|</mml:mi><mml:mi>y</mml:mi><mml:mo>)</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>x</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

          In practice, <inline-formula><mml:math id="M87" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mi mathvariant="normal">|</mml:mi><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> can be easily computed from the predictive samples
from the GAM, while the loss function <inline-formula><mml:math id="M88" display="inline"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is defined subjectively.
Defining <inline-formula><mml:math id="M89" display="inline"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> is a non-trivial process, as it should encapsulate the
relative cost of false-positive (i.e. where action against a TC was taken but
the<?pagebreak page1317?> TC did not occur) and false-negative (i.e. where no action was taken but
the TC did occur) events. For the purposes of demonstrating the principle of
this approach, we define a dummy loss function in Table 1, based on the four
TC warning levels used in Bangladesh (WMO, 2018). Here relative loss is
defined on a 100-point scale, where 0 equates to no loss associated with a
given landfalling event, and 100 equates to maximum loss. Evacuation typically
takes places at the “great danger” level.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6"><?xmltex \currentcnt{6}?><?xmltex \def\figurename{Figure}?><label>Figure 6</label><caption><p id="d1e1470">Example warning status given an impending landfalling tropical
cyclone over Bangladesh. These warnings represent the most effective action
minimising the loss as defined in Table 1.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://nhess.copernicus.org/articles/21/1313/2021/nhess-21-1313-2021-f06.png"/>

        </fig>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T1"><?xmltex \currentcnt{1}?><label>Table 1</label><caption><p id="d1e1482">Dummy loss function for actions associated with four Bangladesh TC
warning levels and their associated wind speed intensity. In this case loss
is defined on a 100-point scale, where 0 means no loss and 100 means maximum
loss, associated with a given landfall TC event.</p></caption><oasis:table frame="topbot"><?xmltex \begin{scaleboxenv}{.95}[.95]?><oasis:tgroup cols="6">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left" colsep="1"/>
     <oasis:colspec colnum="3" colname="col3" align="right"/>
     <oasis:colspec colnum="4" colname="col4" align="right"/>
     <oasis:colspec colnum="5" colname="col5" align="center"/>
     <oasis:colspec colnum="6" colname="col6" align="center"/>
     <oasis:thead>
       <oasis:row>

         <oasis:entry namest="col1" nameend="col2" colsep="1">Loss function </oasis:entry>

         <oasis:entry namest="col3" nameend="col6" align="center">Warning level <inline-formula><mml:math id="M90" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula></oasis:entry>

       </oasis:row>
       <oasis:row>

         <oasis:entry colname="col1"/>

         <oasis:entry colname="col2"/>

         <oasis:entry colname="col3">OK</oasis:entry>

         <oasis:entry colname="col4">Warning</oasis:entry>

         <oasis:entry colname="col5">Disaster</oasis:entry>

         <oasis:entry colname="col6">Great</oasis:entry>

       </oasis:row>
       <oasis:row rowsep="1">

         <oasis:entry colname="col1"/>

         <oasis:entry colname="col2"/>

         <oasis:entry colname="col3"/>

         <oasis:entry colname="col4"/>

         <oasis:entry colname="col5"/>

         <oasis:entry colname="col6">danger</oasis:entry>

       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>

         <?xmltex \rotentry?><oasis:entry colname="col1" morerows="3">Event <inline-formula><mml:math id="M91" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula></oasis:entry>

         <oasis:entry colname="col2"><inline-formula><mml:math id="M92" display="inline"><mml:mrow><mml:mo>&lt;</mml:mo><mml:mn mathvariant="normal">14</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M93" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></oasis:entry>

         <oasis:entry colname="col3">0</oasis:entry>

         <oasis:entry colname="col4">5</oasis:entry>

         <oasis:entry colname="col5">15</oasis:entry>

         <oasis:entry colname="col6">20</oasis:entry>

       </oasis:row>
       <oasis:row>

         <oasis:entry colname="col2"><inline-formula><mml:math id="M94" display="inline"><mml:mrow><mml:mn mathvariant="normal">14</mml:mn><mml:mo>≥</mml:mo><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>&lt;</mml:mo><mml:mn mathvariant="normal">17</mml:mn></mml:mrow></mml:math></inline-formula></oasis:entry>

         <oasis:entry colname="col3">50</oasis:entry>

         <oasis:entry colname="col4">10</oasis:entry>

         <oasis:entry colname="col5">20</oasis:entry>

         <oasis:entry colname="col6">25</oasis:entry>

       </oasis:row>
       <oasis:row>

         <oasis:entry colname="col2"><inline-formula><mml:math id="M95" display="inline"><mml:mrow><mml:mn mathvariant="normal">17</mml:mn><mml:mo>≥</mml:mo><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>&lt;</mml:mo><mml:mn mathvariant="normal">25</mml:mn></mml:mrow></mml:math></inline-formula></oasis:entry>

         <oasis:entry colname="col3">80</oasis:entry>

         <oasis:entry colname="col4">50</oasis:entry>

         <oasis:entry colname="col5">25</oasis:entry>

         <oasis:entry colname="col6">30</oasis:entry>

       </oasis:row>
       <oasis:row>

         <oasis:entry colname="col2"><inline-formula><mml:math id="M96" display="inline"><mml:mrow><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">25</mml:mn></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="M97" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">m</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula></oasis:entry>

         <oasis:entry colname="col3">100</oasis:entry>

         <oasis:entry colname="col4">100</oasis:entry>

         <oasis:entry colname="col5">80</oasis:entry>

         <oasis:entry colname="col6">40</oasis:entry>

       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup><?xmltex \end{scaleboxenv}?></oasis:table></table-wrap>

      <?pagebreak page1319?><p id="d1e1753">Figure 6 illustrates the optimal warning that should be issued based on
Table 1 and the range of gust speed information summarised by our GAM. This
can be interpreted as the default optimal action to take for planning and
preparation purposes, and in this case, the northern extent of TC risk, as
highlighted in Figs. 2 and 3, is again reflected in the warning level, but in
practice separate loss functions could be defined for each province or for
different economic sectors of society. By understanding the exposure,
vulnerability and decision-making process of each user, bespoke warnings could
be issued. For operational forecasting purposes, the optimal action (<inline-formula><mml:math id="M98" display="inline"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>)
would be updated once forecast information of a TC becomes available specific
to an impending event. Actions are strongly conditioned by the loss function
and the accuracy of the gust speed information, but our aim here is
to demonstrate a proof-of-concept transparent workflow that clearly translates
hazards into actions and which is equally applicable to short-term numerical
weather prediction information as it is to hazard maps derived from historical
events.</p>
</sec>
<sec id="Ch1.S3.SS2">
  <label>3.2</label><title>Limitations</title>
      <p id="d1e1775">Despite the ensemble simulation framework, our analysis is still restricted to
only 12 historical cases, which represent the recent 40-year period. The
number of events was determined by the availability of source data (ERA5) for
driving the regional model (RAL2), for TC events that made landfall over
Bangladesh – in this case limited to the period of ERA5 data availability,
which at the time of analysis extended back to 1979. Given the relatively low
ERA5 resolution (31 <inline-formula><mml:math id="M99" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula>), we selected TCs defined as at least a
Category 1 event in the IBTrACS database to be sure they would be
identifiable within the low-resolution ERA5 data and could be downscaled by
the RAL2 model.</p>
      <p id="d1e1786">The initial conditions posed in the regional model play a significant role in
determining the outcome of each event. In forecasting situations this is
desirable behaviour: well-chosen initial conditions ensure the model retains a
realistic representation of reality. Even though the modelling domain that
produced these 4.4 <inline-formula><mml:math id="M100" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> data had the freedom to deviate in a physically
plausible way (see Steptoe et al., 2021), it does not have the ability to
sample the full spectrum of possible BoB tropical cyclone events. Simulations
driven by a wider range of initial conditions, derived from a wider range of
historical cases, would improve the sample size of cyclonic conditions on
which this analysis is based. Note that this would not necessarily reduce
uncertainty in exceedance thresholds (in a frequentist paradigm), but it would
update our view (i.e. our posterior estimate) of what is credible within the
continuum of possible tropical cyclone events. In Bayesian parlance, our
posterior view of Bangladesh tropical cyclones would become our new prior
belief if subsequent simulation data became available.</p>
      <p id="d1e1797">A different limitation is posed by the initial aggregation of the
4.4 <inline-formula><mml:math id="M101" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> model over time. This removes our ability to draw inferences on
annual occurrence of (or longer-term variability in) TC events. This means
that our estimates of exceedance probabilities are conditional on a tropical
cyclone event actually impacting Bangladesh. For the purposes of risk
assessment, we do not feel this limitation is significant – current
generation weather forecast models are capable of accurately predicting the
landfall location and track of tropical cyclones in the BoB many days in
advance (e.g. Mohanty et al., 2021; Singh and Bhaskaran, 2020). It should
also be noted that, due to the computation expense of the 4.4 <inline-formula><mml:math id="M102" display="inline"><mml:mrow class="unit"><mml:mi mathvariant="normal">km</mml:mi></mml:mrow></mml:math></inline-formula> data
simulation, we only chose events that specifically impacted Bangladesh, so
conclusions cannot be drawn on the frequency of other TCs within the wider BoB
region.</p>
</sec>
</sec>
<sec id="Ch1.S4" sec-type="conclusions">
  <label>4</label><title>Summary and conclusions</title>
      <p id="d1e1825">Generalised additive models (GAMs) provide a useful framework for condensing
spatial hazard information in an interpretable way, from multiple numerical
model simulations, into a single spatially coherent hazard map. Using a
restricted maximum likelihood approach to fit the GAM allows us to interpret
model predictions in a Bayesian fashion that logically provides credible
exceedance estimates. High-resolution convection-permitting numerical
predictions of 12 historical cyclone events, in an ensemble model set-up,
give an improved sense of the plausibility and likelihood of possible extreme
events without being constrained by the lack of observational history in this
region. Combining ensemble simulations with a GAM then allows us to robustly
quantify the likelihood of maximum gust speed exceedances in a spatially
coherent manner.</p>
      <p id="d1e1828">Our new maps of exceedance intervals show that north-western provinces of
Bangladesh are relatively exposed to high-wind-speed hazards – in some areas
the exceedance probabilities are equal to those experienced along the
coast. Our hazard-to-decision-making framework suggests that these areas may
need to be considered in an equivalent manner to coastal regions from a
disaster risk reduction perspective. In coastal areas of Cox's Bazar and
Chittagong we show super cyclonic conditions may occur as frequently as
1-in-20 to 1-in-100 years. We hope that these kilometre-scale hazard maps
facilitate one part of the risk assessment chain to improve local ability to
make effective risk management and risk transfer decisions. Future work to
co-produce a proper loss function, given wind speed thresholds, would
facilitate a method of transparent operational decision-making that could be
used as the basis of an operational warning system.</p>
</sec>

      
      </body>
    <back><notes notes-type="codeavailability"><title>Code availability</title>

      <?pagebreak page1320?><p id="d1e1836">Python, R and data analysis code, including the fitted GAM model, is available at <ext-link xlink:href="https://doi.org/10.5281/zenodo.3953772" ext-link-type="DOI">10.5281/zenodo.3953772</ext-link> (Steptoe, 2020).</p>
  </notes><notes notes-type="dataavailability"><title>Data availability</title>

      <p id="d1e1845">The data used in this study are available at <ext-link xlink:href="https://doi.org/10.5281/zenodo.3600201" ext-link-type="DOI">10.5281/zenodo.3600201</ext-link> (Steptoe et al., 2020) and released under CC-BY 4.0.</p>
  </notes><notes notes-type="authorcontribution"><title>Author contributions</title>

      <p id="d1e1854">HS prepared the manuscript, with input from TE, and undertook the data analysis. TE and HS jointly developed and coded the GAM model. TE developed and coded the decision-making framework used in Sect. 3.1.</p>
  </notes><notes notes-type="competinginterests"><title>Competing interests</title>

      <p id="d1e1860">The authors declare that they have no conflict of interest.</p>
  </notes><ack><title>Acknowledgements</title><p id="d1e1866">This study is part of the Oasis Platform for Climate and Catastrophe Risk
Assessment – Asia (<uri>https://www.international-climate-initiative.com/en/details/project/oasis-platform-for-climate-and-catastrophe-risk-assessment-asia-18_II_165-3018</uri>,
last access: 28 April 2021).</p><p id="d1e1871">The authors thank Saiful Islam, Erasmo Buonomo, Richard Jones, Jane Strachan, Tamara Janes and two anonymous reviewers for comments that improved early versions of this paper.</p></ack><notes notes-type="financialsupport"><title>Financial support</title>

      <p id="d1e1876">This research has been supported by the International Climate Initiative (IKI) supported by the Federal Ministry for the Environment, Nature Conservation and Nuclear Safety, based on a decision of the German Bundestag.</p>
  </notes><notes notes-type="reviewstatement"><title>Review statement</title>

      <p id="d1e1882">This paper was edited by Animesh Gain and reviewed by two anonymous referees.</p>
  </notes><ref-list>
    <title>References</title>

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    <!--<article-title-html>Extreme wind return periods from tropical cyclones  in Bangladesh: insights from a high-resolution convection-permitting numerical model</article-title-html>
<abstract-html><p>We use high-resolution (4.4&thinsp;km) numerical simulations of tropical
cyclones to produce exceedance probability estimates for extreme wind (gust) speeds over Bangladesh. For the first time, we estimate equivalent return periods up to and including a 1-in-200 year event, in a spatially coherent manner over all of Bangladesh, by using generalised additive models. We show that some northern provinces, up to 200&thinsp;km inland, may experience conditions equal to or exceeding a very severe cyclonic storm event (maximum wind speeds in  ≥ 64&thinsp;kn) with a likelihood equal to coastal regions less than 50&thinsp;km inland. For the most severe super cyclonic storm events ( ≥ 120&thinsp;kn), event exceedance probabilities of 1-in-100 to 1-in-200 events remain limited to the coastlines of southern provinces only. We demonstrate how the Bayesian interpretation of the generalised additive model can facilitate a transparent decision-making framework for tropical cyclone warnings.</p></abstract-html>
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