chapter - Atmospheric and Oceanic Science

Statistical analysis of extreme events in a non-stationary context
The models for non-stationarity mentioned above for the parameters of probability
distributions have all had parameters which describe linear variations; thus
in µ(t) = α + βt, the parameters α and β describe a linear variation, and the same
is true of the harmonic model of the preceding paragraph. Even the model for nonstationarity
in dispersion, σ(t) = exp (α + βt), can be converted to a linear form by
taking logarithms. Models with this in-built linearity will not be appropriate for
modelling every kind of statistical non-stationarity in hydrological extremes, however.
One instance requiring an alternative approach would be where consideration
of the physical processes that give rise to the non-stationarity suggests that the
change in (say) mean value will eventually achieve a stable value. For example,
non-stationarity in annual maximum discharges recorded at a flow-gauging site
may be a consequence of deforestation in upstream areas; deforestation might be
expected to cause annual maximum discharges to fluctuate about a mean value that
is higher than that the mean which existed before deforestation began, giving rise
to a change in the form of a trend towards a “plateau”. If it were reasonable to
assume that deforestation is the only influence driving the non-stationarity, a more
complex model - perhaps a GEV in which the parameter µ(t) = α + β exp( - k t) -
could be appropriate. The three parameters α, β and k, (together with the GEV dispersion
and shape parameters σ and ξ) can be estimated by Maximum Likelihood,
following the general procedure briefly outlined in Section 16.3 above. Excellent
software packages are available (e.g., Matlab®) for such calculations.
15.5. Records with missing values
In records of rainfall intensity, it is quite common for years to be incomplete.
This is not a problem if the POT approach is used, but causes difficulties for the
Block method. To avoid rejection of years for which part of the record is missing,
one approach is the following, assuming that there are N years of record, some of
which are incomplete.
(I) Taking each month in turn, select the monthly maxima for all months that
have no data missing. Thus for January, abstract the data from all years for which
the January record is complete. (If r of the Januaries are incomplete, the number of
monthly maxima for January will be N-r). Repeat for each of the 12 months.
(II) Fit GEV distributions to each month in turn; denote the fitted cumulative
probability distributions by F1(x, µ1, σ1, ξ1)… F12(x, µ12, σ12, ξ12).
(III) The cumulative probability distribution of the annual maximum intensity
is then the product given by
FAnual(x)= F1(x, µ1, σ1, ξ1). F2(x, µ2, σ2, ξ2)… F12(x, µ12, σ12, ξ12)
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