ELEMENTS OF FREQUENCY ANALYSIS IN HYDROLOGY - Hydrology: An Introduction - page 521

500 1

400

S k + 1

2

300

3

200

4

100

5

6

7

0

0 100 200 300 400 500 600 700

S k

Fig. 13.6 Numbers of dry day sequences lasting at least k + 1 days, plotted against the numbers of sequences
lasting at least k days, as observed over the period 1951-1960 at Portland, Maine, on the basis of the
analysis by Hershfield (1970a). The dry days are defined for three thresholds of daily precipitation,
namely P < 0.254 mm (circles), P < 2.54 mm (squares), and P < 6.35 mm (triangles). The respective
values of the slopes of the best-fit straight lines through the origin are 0.708, 0.815 and 0.868; these are
nearly the same as the respective values of the conditional probabilities p obtained with the method of
moments, namely 0.708, 0.816 and 0.870. For clarification, values of k are indicated next to the points
of the seven shortest dry day sequences with P < 0.254 mm (circles).

or equal to k days; this is given by Equation (13.30), which can readily be shown to
yield 1 − p k . Conversely, the probability that a dry day sequence will last at least k + 1

days, that is, that it will be equal to or longer than k 1 days, is the complement of

(13.30), or p k . Similarly, the probability that a dry day sequence will last at least k days
is p k − 1 . Hence, the validity of the assumption, that p is independent of k , can be readily
examined empirically in any given situation by checking whether or not the ratio of the
number of sequences, that last at least k days, over the number of those that last at least
( k

+

− 1) days is a constant for all k ,or

S k
S k − 1 =

const . (13.31)

where the constant should in principle be equal to p . As an illustration, Figure 13.6 shows
a plot of the numbers of dry day sequences S k + 1 against S k observed at Portland, Maine,
over the period 1951-1960, on the basis of precipitation data analyzed by Hershfield
(1970a). A “dry” day or a day “without rain” was defined here as a day with precipitation
less than a certain finite threshold; three such thresholds were considered, namely days
with less than 0.254 mm, 2.54 mm and 6.35 mm. The slopes of the regression lines
through the origin are 0.708, 0.815 and 0.868, respectively, which should be reasonable
estimates of p for each of these three thresholds.
The more common way, however, to estimate the parameters is the method of
moments. Because the geometric distribution (13.29) has only one parameter, viz. p ,
a knowledge of the mean of k will suffice. The mean duration of a dry period is given by

N DD
N DS

k = (13.32)