Geoscience Reference
In-Depth Information
T
r
1.01
1.1
1.5 2
3
4
5
10
20
50
100
200
400
x
3
−
2
−
1
0
1
2
4
5
6
y
Fig. 13.3 Probability paper with the abscissa based on the first asymptote for largest values. Both the
y
-scale
and the
T
r
-scale are shown. (See Example 13.1.)
to fit a mathematical probability distribution function to the available data; indeed such
functions provide a smooth and succinct description of the data and they allow the
formulation of objective confidence criteria. Moreover, although the procedure may be
hazardous and should only be applied with great caution, a mathematical function may
also allow some degree of extrapolation to estimate probabilities outside the range of the
available data.
The application of most theoretical distribution functions can be justified on strict
probabilistic considerations. Unfortunately, it is rare that such considerations are rigor-
ously valid for data sets of hydrologic concern, and in most cases the actual mathematical
form of the distribution function, that represents the population, is unknown. Thus the
best that can be hoped for is that the selected distribution is simple enough and also
physically plausible to be useful in practice.
Several procedures are available to determine the parameters in these distributions.
Among them the method of maximum likelihood is commonly considered to be the
best, in principle; however, several studies (see, for example, Stedinger, 1980; Martins
and Stedinger, 2000) have revealed that this is not always the case, particularly in small
samples. The present treatment concentrates mainly on the method of moments, which is
usually simpler to apply. An additional feature of the method of moments is that it weights
the larger observations more heavily, so that it may be more suitable in the analysis of
large values. In the following two sections a few of the more common functions are
considered which have been useful in the analysis of hydrologic events.
