Abstract:
Extreme Value Theory nds application in problems concerning low probability but high
consequence events. In hydrology the study of heavy rainfall is important in regional
ood
risk assessment. In particular, the N-year return level is a key output of an extreme value
analysis, hence care needs to be taken to ensure that the model is accurate and that the
level of imprecision in the parameter estimates is made explicit.
Rainfall is a process that evolves over time and space. Therefore, it is anticipated that
at extreme levels the process would continue to show temporal and spatial correlation. In
this study interest is in whether any trends in heavy rainfall can be detected for the Western
Cape. The focus is on obtaining the 50-year daily winter rainfall return level and investigating
whether this quantity is homogenous over the study area. The study is carried out in
two stages.
In the rst stage, the point process approach to extreme value theory is applied to arrive
at the return level estimates at each of the fteen sites. Stationarity is assumed for the
series at each station, thus an issue to deal with is that of short-range temporal correlation of
threshold exceedances. The proportion of exceedances is found to be smaller (approximately
0.01) for stations towards the east such as Jonkersberg, Plettenbergbay and Tygerhoek.
This can be attributed to rainfall values being mostly low, with few instances where large
amounts of rainfall were observed. Looking at the parameters of the point process extreme
value model, the location parameter estimate appears stable over the region in contrast to
the scale parameter estimate which shows an increase towards in a south easterly direction.
While the model is shown to t exceedances at each station adequately, the degree of uncertainty
is large for stations such as Tygerhoek, where the maximum observed rainfall value is
approximately twice as large as the high rainfall values. This situation was also observed at
other stations and in such cases removal of these high rainfall values was avoided to minimize
the risk of obtaining inaccurate return level estimates. The key result is an N-year rainfall
return level estimate at each site. Interest is in mapping an estimate of the 50-year daily
winter rainfall return level, however to evaluate the adequacy of the model at each site the
25-year return level is considered since a 25 year return period is well within the range of the
observed data. The 25-year daily winter rainfall return level estimate for Ladismith is the
smallest at 22:42 mm. This can be attributed to the station's generally low observed winter
rainfall values. In contrast, the return level estimate for Tygerhoek is high, almost six times
larger than that of Ladismith at 119:16 mm. Visually design values show di erences between
sites, therefore it is of interest to investigate whether these di erences can be modelled.
The second stage is the geostatistical analysis of the 50-year 24-hour rainfall return level The aim here is to quantify the degree of spatial variation in the 50-year 24-hour rainfall
return level estimates and to use that association to predict values at unobserved sites within
the study region. A tool for quantifying spatial variation is the variogram model. Estimation
of the parameters of this model require a su ciently large sample, which is a challenge in
this study since there is only fteen stations and therefore only fteen observations for the
geostatistical analysis. To address this challenge, observations are expanded in space and
time and then standardized and to create a larger pool of data from which the variogram is
estimated. The obtained estimates are used in ordinary and universal kriging to derive the
50-year 24-hour winter rainfall return level maps. It is shown that 50-year daily winter design
rainfall over most of the Western Cape lies between 40 mm and 80 mm, but rises sharply as
one moves towards the east coast of the region. This is largely due to the in
uence of large
design values obtained for Tygerhoek. In ordinary kriging prediction uncertainty is lowest
around observed values and is large if the distance from these points increases. Overall, prediction
uncertainty maps show that ordinary kriging performs better than universal kriging
where a linear regional trend in design values is included.