Hydrological Sciences Journal
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The magnitude of the hydrological frequency
factor in maximum rainfall estimation /
Valeur du facteur de fréquence hydrologique
dans l'estimation de la hauteur maximale de
précipitations
DAVID M. HERSHFIELD
To cite this article: DAVID M. HERSHFIELD (1981) The magnitude of the hydrological frequency
factor in maximum rainfall estimation / Valeur du facteur de fréquence hydrologique dans
l'estimation de la hauteur maximale de précipitations, Hydrological Sciences Journal, 26:2,
171-177, DOI: 10.1080/02626668109490874
To link to this article: https://doi.org/10.1080/02626668109490874
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Hydrological Sciences-Bulletin-des Sciences Hydrologiques, 26, 2, 6/1981
The magnitude of the hydrological frequency
factor in maximum rainfall estimation
DAVID M. HERSHFIELD
Hydrology Laboratory,
Maryland 20705, USA
USDA-SEA-AR,
Beltsville,
ABSTRACT
Chow's hydrological frequency factor, K, is
used to compare and relate results and attach probabilities to several sets of maximum rainfall data. K is
primarily a function of the recurrence interval for a
particular probability distribution. K is displayed as a
function of the mean of the annual maxima for both
official and unofficial rainfall observations and the
probable maximum precipitation (PMP) and PMP/2 for the 6
and 24 h durations. The magnitude of the unofficial
observations appears to have a strong influence on the
level of PMP. The geographical distribution of K for the
latter is displayed on maps for the eastern US and ranges
from about 20 in the south to more than 35 in the north
for PMP and from about 7.5 to more than 17.5 for PMP/2.
Probability seems to have very little meaning for PMP or
for the largest unofficial observations because these
values are so rare that their return periods are several
orders of magnitude greater than the length of record
upon which they are based.
Valeur du facteur de fréquence hydrologique dans
l'estimation de la hauteur maximale de précipitations
RESUME
Le facteur de fréquence hydrologique de Chow, K,
est utilisé pour comparer et rattacher les résultats aux
probabilités qui leur sont liées pour plusieurs séries de
données de précipitation maximales. K est essentiellement
une fonction de l'intervalle de récurrence pour une
distribution particulière de probabilité. K est présenté
comme une fonction de la moyenne des précipitations
annuelles maximales pour l'ensemble des observations
régulières et occasionnelles des précipitations et de la
précipitation maximale probable (PMP) et PMP/2 pour des
durées de 6 et de 24 h; La valeur des observations
occasionnelles semble manifestement avoir une forte
influence sur le niveau de la PMP. La distribution
géographique de K pour cette dernière est présentée sur
des cartes pour l'est des Etats Unis. K varie de 20
environs dans le sud jusqu'à plus de 35 pour la PMP et de
7.5 environs à plus de 17.5 pour PMP/2. Les probabilités
semble avoir très peu de sens pour la PMP ou pour les plus
fortes observations occasionnelles car les valeurs sont si
rares que leurs périodes de retour sont de plusieurs ordres de grandeur plus élevés que la longueur de la période
d'observations sur laquelle sont effectués leurs calculs.
171
172
David M. Hershfield
INTRODUCTION
As a result of the failure of several dams in recent years, there
has been renewed interest in examining the design standards and
requirements for dam safety. This includes the magnitude of the
rainfall component and its meaning in terms of a frequency index or
recurrence interval. Engineers use various approaches for estimating
the rainfall factor. The frequency approach has been fairly well
established, and although different procedures (combination of curvefitting and probability distribution) might provide different values,
the analysis is generally straightforward and amenable to objective
review. Other procedures that do not consider frequency but extend
and increase maximum storm rainfall experience for an area are much
more difficult to evaluate in terms of risk and uncertainty. One
example is the concept of probable maximum precipitation (PMP)
which was developed more than 40 years ago. The conventional
procedure is primarily concerned with a combination of physical and
subjective reasoning to estimate the "upper limit" of precipitation
or a search for certainty. Formal statistical methods for estimating probabilities of various parameters for a PMP model have little
or no role in the conventional PMP procedure. However, since there
is no known physical upper bound to precipitation, there is a finite
probability of an extreme condition being exceeded.
The principle purpose of this paper is to compare and relate
results from several sets of data, in terms of a frequency factor.
This involves standardizing the residual (maximum rainfall minus
mean of annual maxima) by dividing it by the standard deviation to
obtain the frequency factor. The frequency factor is analogous to
the normal deviate or reduced variate when normally distributed or
extreme-value data are analysed, respectively.
ANALYSIS
Chow (1954) proposed equations (1) or (2) as the "general equations
for hydrologie frequency analysis" because they are applicable to
many probability distributions in hydrology where interest is
primarily in the right-hand side of the distribution.
x = x + sxK
(1)
x/x = 1 + C V K
(2)
x is the magnitude of a rainfall at a particular probability level,
x is the mean of the time series being used in the analysis, s x
equals the standard deviation of the time series, C v is the coefficient of variation and equals sx/x, and K is the frequency factor,
which varies with the probability distributions.
The frequency factor, K, is principally a function of the
recurrence interval for a particular probability distribution. For
example, when analysing a time series of annual maximum 24-h
rainfalls by the Gumbel procedure (Gumbel, 1958), where the coefficient of skew is constant, K is about 3.2 for the lOO-year value and
averages about 3.1 for the lognormal distribution. Thus, K can be
used both as an index to compare the relative frequency of rainfalls
Magnitude of the hydrological frequency factor
173
at different stations or for computing the rainfall probability if a
distribution is assumed. The statistic, C v , is useful when analysing extreme rainfalls for a particular duration because it displays
a distinct geographical pattern, that to a large extent, transcends
the normal sampling fluctuations and can be related to meteorological
factors on a broad scale.
In an earlier paper, Hershfield (1961a) used equation (1) in
combination with daily data from about 1000 official stations in
many countries and found that 15K enveloped all the observed maxima
in each series. The maximum in the individual series was not used
to compute the mean and standard deviation. Subsequent research
revealed that K varied inversely with the mean of the series of
annual maxima (Hershfield, 1965). A third modification was
presented in analytical form for three durations with a slightly
increased value of K (Hershfield, 1977).
Figure 1 displays two curves which envelop K as a function of the
mean of the annual maximum rainfalls. The lower curve envelops the
maximum K values computed from series of annual maximum daily rainfalls observed at official stations. The 2-year and 100-year 24-h
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Unofficial
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Official
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MEAN ANN MAX
6-hr RAINFALL(mm)
Fig. 1 Relationship between frequency factor K and the mean of the annual
maximum 24-h rainfall series, official and unofficial observations and selected grid
values of PMP.
174
David M. Hershfield
values estimated from generalized point-rainfall frequency maps of
TP40 (Hershfield, 1961b) were used as input for relationships to
compute the mean and standard deviation. The upper curve envelops
the maximum 10 square-mile unofficial observations in the US from
Hydrometeorological
Report
no.
51
(Schreiner & Riedel, 1978) for
the 1878-1972 period. Most of the latter rainfalls were not
observed in raingauges but were estimated ex post facto by observers
who inspected all available containers in the area of the storm for
detailed rainfall information. This post-storm observational practice is generally called a "bucket survey".
The x's represent PMP estimates extracted from maps for selected
points on a uniform grid system (Schreiner & Riedel, 1978).
Proximity of the PMP values to the upper curve strongly suggests the
influence of the unofficial observations on the level of PMP.
Figures 2 and 3 portray the geographical distribution of K for
Fig. 2
Geographical distribution of K for the 6-h duration.
Fig. 3
Geographical distribution of K for the 24-h duration.
the eastern US for two durations, as computed using a combination of
25.9 km 2 PMP data in Hydrometeorological
Report
no.
51,
point
Magnitude of the hydrological frequency factor
175
frequency data in TP40, and equation (1).
Hydrometeorological
Report
no. 51 contains a set of generalized maps of PMP for several
durations and areas for the eastern US. This is the third report in
the Hydrometeorological
Report
series for the same region and
subject matter published in the past 30 years.
The salient feature of the maps of Figs 2 and 3 is the large
values of K or the large number of standard deviations that must be
added to the mean of the annual maximum 6 and 24 h series to obtain
their respective probable maximum precipitation. Another feature is
that the distribution of K generally increases nonuniformly with
increasing latitude. It should also be noted that the relationships
between K, mean of the annual maximum rainfalls and unofficial
observations shown in Fig. 1 can provide patterns similar in both
magnitude and configuration to those illustrated in Figs 2 and 3.
There is no unique relationship between areal and point rainfall,
but there is a consistent bias between the two quantities with the
latter always larger than the areal values. Thus, if an adjustment
from areal to point values had been made, the values of K shown for
PMP would be increased further.
If it is assumed that a particular extreme-value distribution is
applicable to all stations in the eastern US, and the distribution
meets other requirements, such as equal coefficients of skew and
lengths of record, then the PMP in the southern US has a greater
probability of occurring than in the north. The extremely rare
probabilities, in terms of return periods, are displayed in Table 1
for the Gumbel procedure.
Table 1
Relationship between K and return period
K
Return period (years)
10
20
40
6.7 X 10s
2.5 X 101
3.4 X 102:
K values for one-half of the magnitude of the PMP are displayed
on the maps of Figs 4 and 5. The magnitude of K is reduced to
about one-half of those associated with PMP, yet for all practical
purposes, they have geographical patterns similar to those for PMP.
The extremely high value of K in the Ohio region is a result of two
factors - the relatively low C v and the high PMP. The magnitude of
the PMP value in this region was strongly influenced by the
Smethport, Pennsylvania, storm of 17-18 July 1942, which was
estimated to have averaged 742 mm over 25.9 km in 24 h.
DISCUSSION
This paper has been concerned with making comparisons on both an
index and probability basis of PMP, unofficial observations, and
rainfall frequency data and not the details of a particular
procedure for estimating return periods or PMP. Both sets of data
176
David M. Hershfield
Fig. 4
Geographical distribution of PMP/2 for the 6-h duration.
Fig. 5
Geographical distribution of PMP/2 for the 24-h duration.
cover a period of about 100 years. The enveloping relationships
which show K as a function of the magnitude of the unofficial rainfall observations and PMP for areas in the eastern US suggest that
the magnitude of PMP is largely influenced by the magnitude of the
former. The PMP and unofficial maximum rainfalls are so much larger
than the rainfalls that make up the extreme value point rainfall
series used in the more common frequency analyses that attempts to
assign a probability to these maxima have very little meaning because
they are so rare. In fact, their return periods are several orders
of magnitude greater than the length of record upon which they are
based. Thus, the return periods of 10 2 2 and 10 al years of Table 1
are relatively meaningless. No extreme-value distribution should be
extrapolated at these levels in the context given.
REFERENCES
Chow, V. T. (1954) The log-probability law and its engineering
applications. Proc. ASCE 80, Paper no. 536, 1-25.
Magnitude of the hydrological frequency factor
177
Gumbel, E. J. (1958) Statistics
of Extremes.
Columbia University
Press.
Hershfield, D. M. (1961a) Estimating the probable maximum precipitation. J. Hydraul. Div. ASCE 87 (HY5), 99-116.
Hershfield, D. M. (1961b) Rainfall frequency atlas of the United
States for durations from 30 minutes to 24 hours and return
periods, 1 to 100 years. US Weather Bureau Tech. Pap. no. 40,
Washington DC.
Hershfield, D. M. (1965) Method for estimating probable maximum
rainfall. J. Am. Wat. Wks Ass. 57, 965-972.
Hershfield, D. M. (1977) Some tools for hydrometeorologists. Preprint volume. Second Conference on Hydrometeorology,
October 1977,
Toronto, Ontario, Canada, 79-82.
Schreiner, L. C. & Riedel, J. T. (1978) Probable maximum precipitation estimates. United States East of 105th Meridian.
Hydrometeorological
Report no. 51.
Received
10 October
1980