FLOOD02408

• County: Statewide
• Community: Nationwide
• Basin: Statewide
• Title: Design Hydrographs for Very Small Wathersheds from Rainfall
• Date: 7/1/1962
• Prepared By: USDA Hydrograph Laboratory, Civil Engineering Section Colorado State University
• Floodplain - Doc Type: Educational/Technical/Reference Information

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Preparing each trial mathematical hydrograph in-
volved substituting pairs of m and G values into
equation (1) to determine a series of q values at
various times for plotting.

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Statistical Parameters Describing Rainstorms

The manner in which rainfall intensity varied
with time was published (23) as a hyetograph along
with each hydrograph studied. Two such diagrams
are reproduced in Fig. 3. Also noted on the figure
are the three statistics, mean time, standard devia-
tion. and skewness I which summarize the major
features of the time distribution of the storm. They
comprise a new attempt at describing important
characteristics of a rainstorm with a few values.
One may visualize a skew bell-shaped curve with the
same statistical parameters of skewness, standard
deviation, mean, etc. as presenting a smoothed
version of the hyetograph. This elimination of very
short angular peaks and breaks in intensity is desir-
able since these steps occur at different times at
different points throughout the watershed. The over-
all input to the watershed is more likely to approach
a smooth curve than it would the steplike pattern
published for a single gage. Furthermore the natural
watershed immediately begins to destroy these angu-
lar peaks by storage.

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Thus the computation of moments, and thence
the standard deviation of time and skewness were
undertaken. It was considered that the statistical
parameters, 81 ' 82, and S3 ' adequately describe

for the watershed generally whether the storm was:
extremely peaked or relatively uniform in time, early-
peaking or late-peaking. To reflect the amount of
rain, the storm total and the average rainfall inten~
sity were added to the set. Three antecedent rainfall
amounts complete the statistical rainfall parameters
as listed under section IV of Table Z.

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Mean time, S1 - This may be regarded as

fixing the center of gravity of the hyetograph along
the time axis. Moments for each block of rainfall
about the starting time were summed and divided by
storm total to obtain this statistic 81 . It established

the axes MM in Fig. 3 about which the first moments
summed to zero.

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Standard deviation about mean time and skew-
ness - These 8tatistical Rainfall Parameters, 82 and

S 3 ,were computed from the hyetograph in a similar

way that they would be derived from a frequency hist-
ogram. If Vk 1 the k th moment about A (an
k
Ef.(x.-A)
1 1
F
= ~ - A 1 and v 2 = f(A) with a minimum

the k th moment about ;Z , is

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arbitrary reference value), is

; then

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v 0 = 1, v1
at A = x .
-k
Ef.(x.-x)
1 1
F

If I'k

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then:

,
the variance, u2 = v2 - v 1

u 3 = v 3 - 3 v 1 v 3 + 2 vt
the standard deviation,

s = ~= ~
2 2
"3 1 'fi-3
the momental skewness, S = - = - --=r
3 2 2 IT

Table 3 outlines a typical calculation. x is the mean
time described in the preceding section. The com-
putations were performed on a digital computer, as
rainfall data had been punched on cards for other
purposes. The program transformed the data so that
A became zero; whence v = x = S
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Multiple Linear Regression Analysis

In broad perspective the entire study was
composed of three facets.

1. It was necessary to describe each ob-
served hydrograph by ascribing numeri-
cal values to the three parameters W .
qo,andG.

2. Another set of variables had to be evalu-
ated which described the characteristics
of the watershed topography, of the soil
and current land use, of the rainstorm
causing each particular flood, and of the
antecedent conditions measured by pre-
ceding rainfall.

3. The final facet was to relate variables
obtained in 1 above to those obtained in 2.

The basic approach to be used in this third facet will
be discussed in the remainder of this chapter.

More specifically, regression equations had to
be developed from which each of W , q . and G
o
could be predicted (as a dependent variable) from a
few of the many so-called independent variables which
described the storm, the environment, and the ante-
cedent moisture conditions. The technique chosen to
establish the relationships between variables from
the two sets was the stepwise multiple regression
analysis. A description of this statistical technique.
I given before proceeding to the next chapter on data
analysis, should provide an understanding of why
many of the parameters wer e included as independent
variables.

Generalized procedure - The technique of
multiple regression analysis establishes a functional
relationship by which the dependent variable may be
approximately predicted from a number of independent
variables. An anticipated relationship is set up and
the least squares criteria is applied to empirical
observations of both dependent and independent vari-
abIes. This results in a system. of equations which
have to be solved simultaneously for the coefficients
of each term. Since there is one equation for each
variable, the computations become so cumbersome
as to require a digital computer. No attempt will be

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