FLOOD06389

• County: Statewide
• Community: State of Colorado
• Basin: Statewide
• Title: Urban Storm Runoff Inlet Hydrograph Study - Volume 4
• Date: 5/1/1975
• Prepared By: Cheng-lung Chen
• Floodplain - Doc Type: Flood Mitigation/Flood Warning/Watershed Restoration

SYNTHETIC HYETOGRAPH EQUATIONS

A synthetic hyetograph to be developed for
design of drainage facilities is not the one seen or
measured in an actual storm. It is rather a temporal
pattern with average variation of intensities within the
design burst and the most likely sequence of these
varying intensities. We all recognize that such a syn-
thetic hyetograph derived from the rainfall intensity-
duration-frequency data does not generally repre-
sent the rainfall in complete storms, but rather
represents intense bursts within storms. It is possible
that a number of bursts may come at any time within
the actual storms with much, or little, additional rain
falling in between the bursts. Therefore, without the
recorded intense bursts of a given duration, it is
unlikely that a hyetograph can be constructed which
represents the average rainfall in complete storms,

Since an elemental urban highway cross-section,
as described before, is a small watershed with drainage
area less than 0,1 square mile, it may be justified to
develop a synthetic hyetograph which represents an
intense burst rather than a complete stann of a long
duration which consists of a series of single intense
bursts. This in a sense is equivalent to assuming that a
synthetic hyetograph for urban highway watersheds is
a single-burst storm pattern having the duration of
ralnfall equal to the time of concentration for an
entire storm sewer system which drains storm water
collected at all dralnage inlets. The basic equations to
be used are the rainfall intensity-duration-frequency
relationships, Eq. 6 or 7, which can be formulated by
using rainfall data in Weather Bureau Technical Paper
No, 25 or 40.

Formulation of Hyetograph Equations

As mentioned in the previous section, the
rainfall intensity-duratlon.frequency relationship, Eq.
6 or 7, can be used to derive a synthetic hyetograph
equation which produces the same total amount (or
depth) of rainfall as described by the corresponding
intensity-duration-frequency curve up to any time, t,
for given frequency (or return period), F.
Mathematically it is expressed as

J t r dr = r t ....................... (8)
o av

in which r is the ralnfall intensity in inches per hour
at any time in the synthetic storm; r is the

integration variable for time; and fav is the average
rainfall intensity in inches per hour and is assumed to
be expressible in the form of Eq, 6 or 7, Because of
the reasons as stated previously, only Eq. 6 is used in
the present analysis.

The rainfall intensity-duration-frequency
formula, Eq. 6, has a parameter, b, the value of which
may be either positive or negative. In the case of a
negative b, Eq. 6 is not valid for td S Ibl. A
preliminary analysis of rainfall data obtained from
Weather Bureau Technical Paper Nos. 25 and 40 has
indicated that a positive b mainly applies to a large
section of the country-perhaps to the portion east of
the Rocky Mountains-while a negative b generally
applies to the west of the Rocky Mountains.
However. in some special meteorological areas such as
Hawaii (Weather Bureau Technical Paper No, 43,
1962; Cheng and UlU, 1973), the value of b was
found to be, or almost, zero. In light of the variety of
the b vaIue to be found in nature, it is suggested that
Eq. 6 be modified to include the cases of negatlve and
zero b. Therefore, Eq, 6 is expressed in the following
form in order that the b value is always kept positive,

a
r = . . . . . . . . . . . . . .. . . . . . . . . (9)
av (td z b)c

Substituting the expression of r av (from Eq. 9 after
changing its td to t) into Eq. 8 and then
differentiating the result with respect to t yields a set
of hyetograph equations for three types of storm
patterns, which are defined according to the skewness
of the pattern,"'1 , (l.e" the ratio of the time before
the peak to the total time duration). Because of the
nature of Eq. 9 which also breaks down for tdS bin
the case of -b, the hyetograph equations need to be
separately formulated for both cases of +b and -b.

Hyetograph equations for positive b

Three types of storm patterns are specified by
using the different values of "'1, A completely
advanced (initial burst) type storm pattern has "'1 = 0
and a completely delayed (fmal burst) type storm
pattern, "'1 = 1. Both types which seldom occur in
nature may be regarded as extreme cases of the third
type, namely an intermediate type storm pattern,
which has 0 < "'1 < 1, Hyetograph equations for the
three types are derived and listed as follows:

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