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
Intensity-Duration-Frequency relationship, i.e.
= average rainfall intensity (mm/hr or inch/hr)
td = storm
= constants dependent on the units employed and the return
frequency of the storm.
The asymmetry of the hyetograph is
described by a parameter r
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 7-1,
i.e. with a maximum intensity
at the start of rainfall at
which then decreases monotonically with elapsed time
according to some function f(t)
which is, as yet, unknown. If the duration of such a storm is
then it is easy to see that the total volume of rainfall is
represented by the area under the curve from
The average rainfall intensity for such an event could be estimated as
= Volume/td as illustrated by
the shaded rectangle of Figure 7-1.
Figure 7-1 - Development
of the Chicago storm
Several storms with different durations
but with the same time distribution of intensity would produce values
which decrease as
increases, leading to the dotted curve of Figure 7-1. Thus:‑
If the average intensity
iave over an elapsed time
can be described by an empirical function such as equation [7-1], then
by combining [7-1] and [7-2], the functional form of
can be obtained by differentiation, i.e.
Now if the value of
is in the range
the time to peak intensity for a given duration is tp
The time distribution of rainfall intensity can then be defined in
terms of time after the peak
= (1-r).t and time before the
by the following two equations.
The solid curve of Figure 7-1 shows the
time distribution of rainfall using a value of
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 7-2 below. For convenience this curve is computed so that
is zero at
and is defined in terms of the elapsed time after and before
The expressions for volume after and before
are then given by equations [7-7] and [7-8] respectively.
Figure 7-2 –
Discretization of an integrated volume
values of rainfall intensity can now be obtained by defining a series
of 'slices' of equal timestep
The time step at the peak intensity is positioned relative to the
position so that it is disposed about the peak in the ratio
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
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.