This section introduces a statistical method for `recalibrating'
wind storm footprints, where recalibration describes deriving the
true distribution of wind gust speeds, given MetUM model output.
The proposed method is based on
polynomial regression between transformed gust speeds: the response
variable represents station observations and the explanatory
variable MetUM model output. All data within the footprint's
domain are used, ranging between storms from 154 to 1,224 stations,
depending on data availability. Gust speeds above 20ms^{-1}
are recalibrated. Where MetUM model gust speeds do not exceed
20ms^{-1}, the recalibrated footprint uses the original
MetUM model output. By assuming that the observations are
representative of the true gust speeds, the regression relationship
gives the distribution of true gust speeds given the MetUM
model's output.

A random effects model (Pinheiro and Bates, 2000) is used to allow multiple wind storm footprints to be recalibrated simultaneously, which is achieved by associating a separate random effect with each storm. This model is based on an underling polynomial relationship between observed and MetUM-model-simulated gust speeds, from which storm-specific relationships deviate according to some distributional assumptions and location-specific covariates. The random effects capture unmodelled differences between storms, one example being whether a storm is a sting jet or not. Not only does this allow a specific storm's footprint to be recalibrated, but storms without observational data can too, by integrating out the random effects, though this latter feature is not utilised here.

Note that as new storms are added to the catalogue, the additional observational data will affect the recalibration of the original 50 storms. This effect is minimal, however, so the original 50 recalibrated footprints will not be updated.

The notation adopted is that *Y _{j}(s)* is
the observed
maximum gust speed for storm

log j) ~
N(m
(log _{j}X(_{j}s),z(s)),
σ)
^{2} |
(1) |

where *z(s)* is a vector of known covariates for
location *s*, σ^{2} is a variance parameter, and the
mean has linear form.

*m _{j}*(log

where (*b _{j,0}, b_{j,1},
b_{j,2})^{T} ~
MVN((0, 0, 0)^{T}, Σ_{b})
(c_{j,0}, c_{j,1}, c_{j,2})^{T} ~
MVN((0 ...... 0)^{T}, Σ_{c}), β_{0},
β_{1},
β_{2}, γ_{0}, γ_{1}
and γ_{2}* are regression coefficients
and Σ

Let *z ^{T}*(

Recalibrated footprints for The Great Storm of
'87 (row 1) and Daria (row 2). Column 1 shows observed vs. MetUM
model maximum wind gust speeds for storm duration with recalibrated
mean (-----), 95% confidence ( - - - ) and 95% prediction
(⋅⋅⋅⋅)
intervals based on London and *y=x* (grey) superimposed.
Column 2 shows the mean recalibrated footprint, column 3 its ratio
to the original footprint and columns 4 and 5 2.5% and 97.5%
prediction bounds, respectively.

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