Intensity-Frequency-Duration Analysis
If rainfall data from a recording raingauge is available for long periods such as 25 years or more, the frequency of occurrence of a given intensity can also be determined. Then we obtain the intensity-frequency-duration relationships. Such relationships may be established for different parts of the year, e.g. a month, a season or the full year. The procedure to obtain such relationships for the year is described in this section. The method for parts of the year is similar.
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The entire rainfall record in a year is analysed to find the maximum intensities for various durations. Thus each storm gives one value of maximum intensity for a given duration. The largest of all such values is taken to be the maximum intensity in that year for that duration. Likewise the annual maximum intensity is obtained for different duration. Similar analysis yields the annual maximum intensities for various durations in different years. It will then be observed that the annual maximum intensity for any given duration is not the same every year but it varies from year to year. In other words it behaves as a random variable. So, if 25 years of record is available then there will be 25 values of the maximum intensity of any given duration, which constitute a sample of the random variable. These 25 values of any one duration can be subjected to a frequency analysis. Often the observed frequency distribution is well fitted by a Gumbel distribution. A fit to a theoretical distribution function like the Gumbel distribution is required if maximum intensities at return periods larger than can be obtained from the observed distribution are at stake. Similar frequency analysis is carried out for other durations. Then from the results of this analysis graphs of maximum rainfall intensity against the return period for various durations such as those shown in the next figure can be developed.
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Figure: Intensity-frequency-duration curves
By reading for each duration at distinct return periods the intensities intensity-duration curves can be made. For this the rainfall intensities for various duration at concurrent return periods are connected as shown in the Figure.
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Figure: Intensity-frequency-duration curves for various return periods
From the curves of figure above the maximum intensity of rainfall for any duration and for any return period can be read out.
Alternatively, for any given return period an equation of the form.
05 - Time Series Analysis^image108.gif! (1)
can be fitted between the maximum intensity and duration
where
I = intensity of rainfall (mm/hr)
D = duration (hrs)
c, a, b = coefficients to be determined through regression analysis.
One can write for return periods T~1~ , T~2~ , etc.:
05 - Time Series Analysis^image110.gif! (2)
where c~1~ , a~1~ and b~1~ refer to return period T~1~ and c~2~ , a~2~ and b~2~ are applicable for return period T~2~ , etc. Generally, it will be observed that the coefficients a and b are approximately the same for all the return periods and only c is different for different return periods. In such a case one general equation may be developed for all the return periods as given by:
05 - Time Series Analysis^image112.gif! (3)
where T is the return period in years and K and d are the regression coefficients for a given location. If a and b are not same for all the return periods, then an individual equation for each return period may be used.
When the intensity-frequency-duration analysis is carried out for a number of locations in a region, the relationships may be given in the form of equation (3) with a different set of regression coefficients for each location. Alternatively, they may be presented in the form of maps (with each map depicting maximum rainfall depths for different combinations of one return period and one duration) which can be more conveniently used especially when one is dealing with large areas. Such maps are called isopluvial maps.
Annual maximum and annual exceedance series
In the procedure presented above annual maximum series of rainfall intensities were considered. For frequency analysis a distinction is to be made between annual maximum and annual exceedance series. The latter is derived from a partial duration series, which is defined as a series of data above a threshold. The maximum values between each upcrossing and the next downcrossing are considered in a partial duration series. The threshold should be taken high enough to make successive maximums serially independent or a time horizon is to be considered around the local maximum to eliminate lower maximums exceeding the threshold but which are within the time horizon. If the threshold is taken such that the number of values in the partial duration series becomes equal to the number of years selected then the partial duration series is called annual exceedance series.
Since annual maximum series consider only the maximum value each year, it may happen that the annual maximum in a year is less than the second or even third largest independent maximum in another year. Hence, the values at the lower end of the annual exceedance series will be higher than those of the annual maximum series. Consequently, the return period derived for a particular I(D) based on annual maximum series will be larger than one would have obtained from annual exceedances. The following relation exists between the return period based on annual maximum and annual exceedance series (Chow, 1964):
05 - Time Series Analysis^image114.gif! (4)
where:
T~E~ ~~ = return period for annual exceedance series
T = return period for annual maximum series
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Figure Definition of partial duration series
The ratio T~E~ /T is shown in the next figure. It is observed that the ratio approaches 1 for large T. Generally, when T < 20 years T has to be adjusted to T~E~ for design purposes. Particularly for urban drainage design, where low return periods are used, this correction is of importance.
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Figure: Relation between return periods annual maximum (T) and annual exceedance series (T~E~ ).
In HYMOS annual maximum series are used in the development of intensity-duration-frequency curves, which are fitted by a Gumbel distribution. Equation (4) is used to transform T into T~E~ for T < 20 years. Results can either be presented for distinct values of T or of T~E~ .
Data Requirements
Series codes
Series can be selected by clicking the series in the series codes list-box. Only one series may be selected at a time. Preferably the series should not contain missing values, missing values in the analysis period will be treated as zero values and can influence greatly the results.
Select intensity interval
The analysis timesteps must always be equal or greater than the selected series timestep. The maximum intensities are computed for the durations entered in this spreadsheet. The values entered in the first column are the replicators of the input timestep. The durations in the second column are computed by HYMOS. If the timestep of the input series is for example hour then a value 1 means that the first computed duration is one hour, a value 2 means a duration of 2 hours, etc..
Sub period
In the sub-period textboxes the start and end date of the sub-period can be entered. Computation of input data for the month of July requires a start date of "01-07" and an end date of "01-08". For the analysis of complete years the start date of the sub-period must be "01-01" and the end date of the sub-period must also be "01-01".
Return Periods
For each duration at distinct return periods the intensities intensity-duration curves will be made. The return periods can be entered in this spreadsheet.
Constant in plotting formula
The plotting of ordered data is done according to a general plotting position function:
P = (m-a) / (N+1-2a).
Constant 'a' is an input variable and is default set to 0.44, according to Gringorton. This is the best plotting position function for the Gumbel distribution.
Output Options
A selection can be made to make a computation of:
- Intensity Frequency curves
- Intensity Duration Frequency curves with return periods for annual maximum series
- Intensity Duration Frequency curves with return periods for annual exceendance series
When the functions is executed, the function results can be analysed by pressing the <Report> button, or <Graph> button on the HYMOS main form.
