Geoscience Reference
In-Depth Information
different assumed probability distributions should be made and the mean value
of the resulting ensemble of estimates used, with the range of estimates given
using different assumptions then providing an approximate lowest estimate of the
error implicit in the calculation. An estimated probability given by the statistics
of extremes approach is compromised if the sample of precipitation in the
observational record used is not representative of the precipitation climate at the
location of interest. It also involves the implicit assumption that the precipitation
climate is not changing.
Conditional probabilities
Up to this point the analysis perspective we have adopted has been that precipita-
tion events are independent of each other. In reality, however, precipitation cli-
mates often can be viewed as having 'wet spells' and 'dry spells. This is evident in
the fact that precipitation data commonly reveal seasonal dependency and some-
times evidence of periodicity. A major cause of shorter term persistence is the fact
that the weather conditions and weather systems involved in the production of
precipitation themselves have some persistence. Indeed, much of the skill in
weather forecasting depends on this fact.
The presence of persistence in the precipitation measured at a particular loca-
tion can be sought, and the extent to which it occurs quantified, using simple
stochastic techniques. A long time series of observed precipitation is analyzed to
count the number of times a 'wet day' or 'rain day' follows a preceding period
comprising a specified number of days with rain or a specified number of days
without rain. In this way a set of conditional probabilities are derived that charac-
terize the level of persistence in the data record. Figure 13.14 shows a simple exam-
ple. In this case the probability that a wet period will be further extended by a day
with rain, or a dry period further extended by a day without rain is plotted for
Selangor, Malaysia.
The probabilities shown in Fig. 13.13 are examples of
conditional probabilities
,
in this case for daily rainfall. The four relevant basic conditional probabilities are
the probability that day
n
is wet if day (
n
−
1) was wet; the probability that day
n
is
wet if day (
n
−
1) was dry; the probability that day
n
is dry if day (
n
−
1) was dry; and
the probability that day
n
is dry if day (
n
1) was wet. Such conditional probabili-
ties (or those of greater complexity) are the basis of a widely used mathematical
model of persistence, the Markov chain. This model, whose detailed description is
beyond the scope of this text, begins by calculating a number of equations derived
to calculate the probability of a wet day or days after a wet or a dry day, of wet or
dry spells of different lengths, etc. One important application of models of persis-
tence such as the Markov chain model is to generate synthetic time series of pre-
cipitation which have the same level of persistence as the data series originally
used to calibrate the model. Such long time series might be used in design studies
for hydraulic structures.
−
