Hydrological Theory
Derivation Of The Chicago Storm


From the MIDUSS Version 2
Reference Manual  Chapter 7
(c) Copyright Alan A. Smith Inc. 
The synthetic hyetograph computed by the
Chicago method is based on the parameters of an assumed
IntensityDurationFrequency relationship, i.e.
where
i
= average rainfall intensity (mm/hr or inch/hr)
t_{d} = storm
duration (minutes)
a,b,c
= constants dependent on the units employed and the return
frequency of the storm.
The asymmetry of the hyetograph is
described by a parameter r
(where
0 < r < 1)
which defines that point within the storm duration td at which the
rainfall intensity is a maximum.
Imagine a rainfall distribution (with
respect to time) such as that shown by the dashed curve of Figure 71,
i.e. with a maximum intensity
imax
at the start of rainfall at
t=0,
which then decreases monotonically with elapsed time
t,
according to some function f(t)
which is, as yet, unknown. If the duration of such a storm is
td
then it is easy to see that the total volume of rainfall is
represented by the area under the curve from
t=0
to
t=td.
The average rainfall intensity for such an event could be estimated as
iave
= Volume/td as illustrated by
the shaded rectangle of Figure 71.
Figure 71  Development
of the Chicago storm
Several storms with different durations
td
but with the same time distribution of intensity would produce values
of
iave
which decrease as
td
increases, leading to the dotted curve of Figure 71. Thus:‑
If the average intensity
iave over an elapsed time
t
can be described by an empirical function such as equation [71], then
by combining [71] and [72], the functional form of
f(t)
can be obtained by differentiation, i.e.
or
Now if the value of
r
is in the range
0<r<1
the time to peak intensity for a given duration is tp
= r.t
The time distribution of rainfall intensity can then be defined in
terms of time after the peak
ta
= (1r).t and time before the
peak
tb
= r.t
by the following two equations.
The solid curve of Figure 71 shows the
time distribution of rainfall using a value of
r
greater than zero (r
= 0.4 approximately).
Calculation of the
discretized rainfall hyetograph is carried out by integrating
these equations to obtain a curve of accumulated volume as illustrated
in Figure 72 below. For convenience this curve is computed so that
volume V
is zero at
t=tp
and is defined in terms of the elapsed time after and before
tp.
The expressions for volume after and before
tp
are then given by equations [77] and [78] respectively.
Figure 72 –
Discretization of an integrated volume
curve.
Discretized
values of rainfall intensity can now be obtained by defining a series
of 'slices' of equal timestep
Dt.
The time step at the peak intensity is positioned relative to the
tp
position so that it is disposed about the peak in the ratio
r
to (1r).
In general, this means that the
commencement of the storm may not be precisely defined by t = ‑ r td
and the storm duration is therefore not disposed about the peak
exactly in the ratio
r
to (1‑ r).
However because the rainfall
intensities at the extremities of the storm are generally very small
this approximation is unlikely to lead to significant error.
