STUDY OF FIXED-DURATION DESIGN RAINFALLS IN
AUSTRALIAN RAINFALL RUNOFF AND JOINT
PROBABILITY BASED DESIGN RAINFALLS
ABDUR RAUF and ATAUR RAHMAN
School of Engineering and Industrial Design, University of Western Sydney, Australia
Rainfall, in terms of its intensity, duration and temporal patterns, known as design
rainfall, is the basic input to many hydrological models. Estimation of design rainfall is
based on frequency analysis of historical rainfall data. The design rainfalls in Australia
are provided in its national guideline known as Australian Rainfall and Runoff (ARR) – a
guide to flood estimation, published by the Institution of Engineers Australia. An
alternative approach to ARR method is a joint probability based method, which considers
rainfall event duration as a random variable. This allows accounting for the joint
probabilistic nature of the rainfall duration and depth in a more rigorous manner. This
joint probability based design rainfalls can be used to estimate design floods using a Joint
Probability Approach, which considers the probabilistic behavior of the input variables
and model parameters in an explicit manner in the flood estimation. This paper
investigates the design rainfall estimates using the two above approaches using rainfall
data from Victoria state in Australia. This in particular examines the sampling properties
of the rainfall events in the two approaches to identify any systematic differences
between them. It has been found that ARR design rainfall estimates are generally higher
than the joint probability based estimates, however these differences vary with locations,
durations and average recurrence intervals. The results of this study offer insights into the
two methods of rainfall analysis, which will assist in developing a link between the two
types of design rainfalls.
INTRODUCTION
Design flood estimates are used for design of hydraulic structures, flood plain
management, and for various water resources, environmental and ecological studies. In
Australia, rainfall-based flood estimation methods are widely adopted. The currently
recommended method in Australia is known as the Design Event Approach [I. E. Aust.,
[1]], which considers the probabilistic nature of rainfall depth but ignores the
probabilistic behavior of other model inputs such as rainfall temporal pattern and losses.
A more holistic rainfall-based flood estimation method is derived distribution method
[Eagleson [2]], which is also known as the Joint Probability Approach. In recent years,
there have been significant researches on the Joint Probability Approach to rainfall
analysis and flood estimation in Australia [e.g. Rahman et al., [3, 4]; Kuczera et al., [5].
The method proposed by Rahman et al. [3] is based on a Monte Carlo Simulation, which
is a simplified form of the Joint Probability Approach.
1
2
Application of the Joint Probability Approach ideally requires periods of long
continuous rainfall data to derive marginal distributions of various rainfall characteristics
such as duration, average rainfall depth (intensity) and temporal pattern. Design rainfalls
are rainfall intensities, generally expressed in the form of intensity-frequency-duration
(IFD) curves and used as the basic input to many flood estimation models. Design rainfall
data in the form of IFD curves, based on the concept of Design Event Approach, are
available in Australian Rainfall and Runoff (ARR), which is the national guideline for
design flood estimation in Australia [1]. The design rainfall data in the Joint Probability
Approach are different to that of ARR because of different sampling properties of rainfall
events. The ARR method considers prefixed durations while the Joint Probability
Approach considers rainfall duration as a random variable.
The objective of this paper is to identify the relationship between ARR IFD and Joint
Probability IFD data. Any meaningful relationship between these two design rainfall
intensities will allow derivation of JP IFD data from ARR IFD data thus overcoming the
current difficulty in the direct derivation of Joint Probability IFD data, which generally
requires a longer period of pluviograph data.
OVERVIEW OF DESIGN RAINFALLS IN AUSTRALIA
The ARR IFD data were obtained in mid eighties based on a large number of pluviograph
stations distributed all over Australia. For an individual station, annual maximum series
of rainfall intensity for various durations (for example, 1, 6, 24, 48 and 72 hours) were
extracted and a Log Pearson Type 3 distribution was fitted to the annual series. The
design rainfall intensities of various average recurrence intervals (ARI) (for example, 1,
2, 5, 10, 20, 50 and 100 years) were computed from the fitted Log Pearson Type 3
distribution. Finally, the annual maximum series estimates were adjusted to obtain partial
duration series estimates.
In the Monte Carlo Simulation Technique, to provide the basis for a rigorous
assessment of flood probabilities, a new storm event definition is required that produces
rainfall events of random durations. Two different storm event definitions can be used: a
‘complete storm’ and a ‘storm-core’ within each complete storm (the most intense part of
the storm). A complete storm is defined as a period of significant rain preceded and
followed by an arbitrarily defined period of dry hours (e.g. 6 hours). The corresponding
storm-core is selected as the period within a complete storm that has the highest rainfall
intensity ratio compared to the 2-year ARI ARR design rainfall intensity. The method of
Rahman et al. [3] allows selection of 4 to 7 most significant partial duration series rainfall
storm-core events on average per year. The selected storm-core events are then analysed
to develop Joint Probability IFD curves using the following method.

The range of storm-core durations (Dc) are divided into a number of class intervals
(with a representative duration being selected for each class except the 1h class), e.g.
2-3h (representative duration 2h), 4-12h (6h), 13-36h (24h).
3



For the data in each class interval (except the 1h), a linear regression line is fitted
between log(Dc) and log(Ic). Here Ic is the average rainfall intensity of storm-core.
The slope of the fitted regression line is used to adjust the intensities for all durations
within the interval to the representative duration.
An exponential distribution is fitted to the partial series of the adjusted intensities
within the class interval, and design intensity values Ic(ARI) are computed for ARIs
of 2, 5, 10, 20, 50 and 100 years.
For a selected ARI, the computed Ic(ARI) values for each duration range are used to
fit a second-degree polynomial between log(Dc) and log(Ic). For the stations analyzed
in this study and for all the selected ARIs, the observed R2 values were found to be
greater than 98%. These polynomials can be used to obtain for each selected ARI a
value of rainfall intensity Ic for a duration Dc (1h  Dc  100h).
STUDY AREA AND DATA
The selected study area is the Victorian
state of Australia. Victoria occupies the
southeast corner of the continent
between latitudes 34 and 39 south and
longitudes 141 and 150 east. It covers
an area of 227,600 km2. From the
database of pluviograph stations in
Victoria, a total of 91stations having
longest periods of good quality data
were selected for the study. The
locations of the selected stations are
shown in Figure 1.
Figure 1. The Study Area – Victoria State in
Australia and selected pluviograph stations
locations
RESULTS
Selection of storm events
The ARR method uses prefixed durations and allows selection of same rainfall spells for
various durations in the formation of annual maximum series. For example, a rainfall
spell of 3-hour duration can be included as a part in the 12-hour duration. This makes
various data points across various durations less independent. To examine how frequently
the same rainfall spell can be repeated in the data series across various durations, a term
‘commonality’ is used here. The ‘commonality’ of storm event of a standard duration is
used to measure the frequency of repetition of the storm event of that duration in the
storm events of subsequent longer durations. Standard durations used here are 1, 2, 3, 6,
12, 24 and 48 hours. The ‘commonality’ can be better explained using Table 1, which
uses an asterisk (*) to indicate common rainfall spell in two durations. For Station 82042,
4
there is a common spell of rainfall for the storm durations of 1- and 2-hour, similarly for
3- and 6-hour storms.
Table 1. Storm duration commonality matrix for Stations 82042 and 84005
Storm duration (hours)
82042 Strathbogie PO (Year 1996)
1
1
2
3
6
12
24
48
84005 Buchan PO (Year 1964)
72
*
1
2
3
6
12
24
*
*
*
*
*
2
*
*
*
3
*
*
*
*
*
1
2
3
*
6
6
12
*
24
*
*
12
*
*
24
48
48
72
*
48
Percentage commonality of storm events for a station can be calculated by dividing
the number of repetitions (i.e. the number of asterisk) occurred divided by the maximum
possible number of repetitions over the available periods of record. The maximum
possible number of repetition among storms of different standard durations is 28 in any
year, as shown by shaded cells in Table 1. For Station 82042, the percentage
commonality is 7/28 = 25% for the year 1996.
For the 91 pluviograph stations, percentage commonality values range from 33 to
67%, with an average value of 51%. The distribution of parentage commonality values
for the 91 stations is presented in Figure 2. Some 65% of the pluviograph stations show
45 to 60 percent commonality value.
This commonality indicates that same rainfall spells are shared by many events. This
Distribution of percentage dependency
No of Occurence
35
30
25
20
15
10
5
0
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
Dependency class (%)
Figure 2. Distribution of percentage commonality values in Victoria
is not the case for the Joint Probability Approach in that a rainfall spell can be included
5
only once in the data series for constructing IFD curves. This makes rainfall intensity
data points in the partial duration series in the Joint Probability Approach more
independent.
Relationship between ARR IFD and JP IFD curves
IFD curves by the ARR method for durations 1, 2, 6, 24, 48 and 72 hours and ARIs of 1,
2, 5, 10, 20, 50 and 100 years were developed for 88 of the selected pluviograph stations
using a computer program developed for this study. Figure 3 shows a typical ARR IFD
curves.
IFD Curves in ARR Method
Station: 79079 Tottington
Rainfall Intensity mm/hr
100
100 Yr
50 Yr
20 Yr
10 Yr
5 Yr
2 Yr
1 Yr
10
1
1 hr
2 hr
3 hr
6 hr
12 hr
24 hr
48 hr
72 hr
Duration (hr)
Figure 3. IFD curves by ARR method for the Station Tottington (79079)
A FORTRAN programme was developed to generate Joint Probability IFD curves as
per the method described before. Figure 4 presents a typical IFD curve in the Joint
Probability Approach. Results from the 88 stations in the study area indicated a high
degree of consistency in the derived Joint Probability IFD curves.
IFD Curves in Joint Probability Approach
Station: 79079 Tottington
Rainfall Intensity mm/hr
100
100 Yr
50 Yr
20 Yr
10 Yr
5 Yr
2 Yr
1 Yr
10
1
1hr
2hr
6hr
24hr
48hr
72hr 100hr
Duration (hr)
Figure 4. IFD curves by Joint Probability approach for the Station Tottington (79079)
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Relationship between ARR and JPA IFD Curves
A key issue for the practical application of the Joint Probability approach to design flood
estimation is whether the IFD curves of this method and ARR are related. Any
meaningful relationship between these two types of IFD curves can be used to estimate
Intensity-Frequency-Duration Curve - 100 Yr ARI
Station: 86142 Mt St Leonard
100
Joint Probability
Rainfall Intensity
(mm/h)
ARR
10
1
1
10
100
Duration (h)
Figure 5. Comparison plots of ARR IFD and JP IFD for 100 years ARI of the Station
Mt St Leonard (86142)
Intensity-Frequency-Duration Curve - 100 Yr ARI
Station: 85237 Noojee Eng. HMSD
Rainfall Intensity (mm/h)
100
ARR
Joint Probability
10
1
1
10
100
Duration (h)
Intensity-Frequency-Duration Curve - 2 Yr ARI
Station: 85237 Noojee Eng. HMSD
100
ARR
Joint Probability
10
1
0.1
1
10
100
Duration (h)
Figure 6. Comparison plots of ARR IFD and JP IFD curves for the station Noojee Eng.
HMSD for ARI 100 and 2 years
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Joint Probability IFD curves from readily available ARR IFD curves. To examine the
relationship between the two types of IFD curves, comparison plots were constructed for
durations of 1, 2, 3, 6, 12, 24, 48, 72 hours and ARIs of 1, 2, 5, 10, 20, 50, 100 years for
20 pluviograph stations. Examination of these plots revealed a fairly regular relationship
between the Joint Probability and ARR IFD curves. The ARR estimates of design rainfall
intensities are generally slightly higher than that of the Joint Probability approach. This is
as expected because Joint Probability rainfall events are more independent as mentioned
before. However there were a few exceptions where ARR IFD values were little higher,
these were mainly limited to stations with short period of data.
IFD Curve -100 Yr ARI
IFD Curve - 5 Yr ARI
Station: 89016 Lake Bolac
Station: 89016 Lake Bolac PO
Rainfall Intensity (mm/h)
100
100
ARR
Joint Probability
ARR
Joint Probability
10
10
1
1
1
10
Duration (h)
100
0.1
1
10
100
Duration (h)
Figure 7. Comparison plots of ARR IFD and Joint Probability IFD curves for the station
Lake Bolac PO for ARI 100 and 5 years
Three remarkable trends in the relationships of ARR IFD and Joint Probability IFD
curves were noted. Firstly, the ARR IFD curve is higher than the Joint Probability IFD
curve for the short storm durations and the difference diminishes with the increase in
duration and decrease in ARI, as shown in Figure 5. This trend was generally noticed
when the percentage commonality value of rainfall spells across different durations was
around 50%. Secondly, both the IFD curves are very similar, with ARR IFD curve
slightly higher than Joint Probability IFD curve for higher ARIs, as shown in Figure 6.
This trend was generally noticed when the percentage commonality value of rainfall
spells across different durations was around 64%. Lastly, both the IFD curves are very
close at the shorter storm durations but the Joint Probability IFD curve is smaller than
ARR IFD curve with the increase in storm duration and decrease in ARI, as illustrated in
Figure 7. This trend was generally noticed when the percentage commonality value of
rainfall spells across different durations was smaller than 50%.
CONCLUSIONS
This paper examines the sampling properties of rainfall events for constructing intensityfrequency-duration (IFD) curves in the Australian Rainfall and Runoff method and Joint
Probability Approach. Following conclusions can be made from this study.

The method of selecting storm burst events in Australian Rainfall and Runoff allows
repetition of same rainfall spell in different durations, which implies that many data
8


points across various durations are not independent. For 91 stations in Victoria, it
was found that about 50% storm burst events share common rainfall spells in
Australian Rainfall Runoff Method.
In general, the Australian Rainfall Runoff IFD curves are higher than that of the Joint
Probability approach.
A relationship between Australian Rainfall Runoff and Joint Probability IFD curves
exists as a function of storm durations, average recurrence intervals and locations.
Further analysis is required to identify the precise nature of this relationship.
ACKNOWLEDGEMENT
The authors would like to thank Bureau of Meteorology, Australia for supplying the
pluviograph data for the study, Mr Faruk Kader for his assistance in data collation. and
the Cooperative Research Centre for Catchment Hydrology, Monash University,
Australia, where the Second Author (Dr Rahman) developed the FORTRAN programs
used in this analysis.
REFERENCES
[1] I. E. Aust. Australian Rainfall and Runoff – A guide to flood estimation. Institution
of Engineers Australia, (1997)
[2] Eagleson, P. S. Dynamics of flood frequency. Water Resources Research, 8, 4,
(1972), 878-898.
[3] Rahman, A., Weinmann, P. E., Hoang, T.M.T, Laurenson, E. M. Monte Carlo
Simulation of flood frequency curves from rainfall. Journal of Hydrology, 256 (3-4),
(2002), 196-210.
[4] Rahman, A., Weinmann, P. E. and Mein, R.G. The use of probability-distributed
initial losses in design flood estimation. Australian Journal of Water Resources. 6(1),
(2002), 17-30.
[5] Kuczera, G., Lambert, M., Heneker, T., Jennings, S., Frost, A. and Coombes, P.
2003. Joint Probability and Design Storms at the Crossroads. Keynote paper. 28th
International Hydrology and Water Resources Symposium, I. E. Aust, 11 – 13 Nov,
Wollongong, Australia, (2003).