# Probabilistic Rational Method

## Table of Contents

4. Chapter 4 of 6. PROBABILISTIC RATIONAL METHOD

### 4.1 General

Australian Rainfall and Runoff Book IV Section 1.4.7 has probabilistic Rational Method for all of Western Australia. Discretion should be exercised in using this methods for the Goldfields. The Rational Method equations recommended in ARR for the Goldfields is to use the Wheatbelt method (loamy soils) with Goldfields frequency factors. This was derived from only one gauged catchment located near Kambalda. Flood information gathered from larger flood events in the last 5 years has highlighted that this method may be reasonably suited to the Goldfields Woodlands (Norseman to Kalgoorlie and to Southern Cross) however appears to overestimate the flow to some degree especially when the catchment is fully wooded (i.e. no natural clear area or rock).When the catchment is fully wooded a better guide appears to be the Wheatbelt Loamy/Lateritic Soil with wheatbelt frequency factors. Some catchments contain rock outcrops and when this is reasonably significant the flow is higher than the method predicts. In the Northern Goldfields (north of Kalgoorlie -especially north of Menzies) there is more scrub like vegetation and less pervious red earth) the recommended Goldfields method greatly underestimates the design flow by a factor of around three.

### 4.2 Theoretical Considerations

The amount of catchment losses in a particular rainfall-runoff event depends very much on the "antecedent wetness" of the catchment. Consequently, runoff coefficients do vary widely from event to event, depending upon the initial or antecedent wetness. To calculate the Y year flood peak from the Y year rainfall intensity, which is the usual application of the Rational Method, it is necessary to use some kind of average value of C_{Q}as the runoff coefficient. This average value is called a statistical or probabilistic runoff coefficient (C

_{Y}). The runoff coefficients traditionally used on the other hand, are called "deterministic" runoff coefficients.

The traditional deterministic Rational Method is no longer featured in ARR and is replaced by the probabilistic approach to the application of this method. As discussed in ARR,

Q_{Y} = 0.278 C_{Y} ItC,_{Y} A (4.1)

where Q_{Y} = peak flow rate (m^{3}/s) of average recurrence interval (ARI) of Y years;

C_{Y} = runoff coefficient (dimensionless) for ARI of Y years;

A = area of catchment (km^{2});

It_{C,Y} = average rainfall intensity (mm/h) for design duration t_{C} hours and ARI of Y years.

The probabilistic application of the Rational Method promotes the concept that the Rational Method formula can be used to estimate a design flood of a selected probability from a design rainfall of the same probability. To do so would require the evaluation of probabilistic runoff coefficients from analysis of flood frequency characteristics of rivers. This has become the basis of all subsequent development of Regional Probabilistic Rational Methods around Australia.

- From analysis of flood frequency in rivers, estimates of peak flows for a range of probability can be determined;
- Design rainfall intensities of corresponding probabilities for the catchment being analysed are determined from the procedure discussed in Chapter 2.
- The Rational Method formula is then applied to compute the probabilistic runoff coefficient;
- The probabilistic runoff coefficients are then plotted on maps and contours of runoff coefficients drawn to facilitate regional applications.

Thus it can be seen that the Probabilistic Rational Method is an approach which pools regional information on flood characteristics for use on ungauged catchments.

In all the methods discussed in ARR, a reference probability (the 10 year ARI in most cases except for the North West region) is used. Book IV of ARR Volume 1(page 16-21) lists the formulae for the time of concentration (t_{c}) and the reference runoff coefficients (C_{2} or C_{10} ) for the various regions in Western Australia.

This method is similar to the Index Flood Method discussed in Chapter 5 as both these methods are largely based on flood frequency analysis of the same set of flow data. Frequency factors are assigned to convert the reference design discharge to design discharge corresponding to the required probability. These are listed in Table 4.2. It can be seen here that the probabilistic runoff coefficient is merely a ratio of peak flow to rainfall intensity and has no physical interpretation. It is quite possible and not uncommon (especially in NSW) to have very high runoff coefficient (even > 1.0).

The rainfall intensity depends on knowing t_{c} which is equal to the design rainfall duration. In the deterministic use of the method, the critical rainfall duration was the time taken for the total catchment to contribute to the flow, i.e. the time of concentration. In the probabilistic approach, use of the physical time of concentration is not necessary. All that is needed is a representative rainfall duration, which has to be determined using the same techniques as that used in developing the reference probabilistic runoff coefficient and frequency factors.

- the method for each region assumes that all variations in the catchment characteristics which are known to have an effect on the catchment's flood hydrology are accounted for in the computation of the design rainfall duration. In most methods, only the catchment area is included in the formula used for computing t
_{c} - the method for each region uses an average slope of the flood frequency curve to compute the frequency factors. This inherently assumes that the skew of the flood frequency curves in the region are similar. Recent research work has shown this to be incorrect and that geographical boundaries do not necessary imply catchment homogeneity;

Refer to Book IV Section 1 of ARR Volume 1 for detailed worked examples using Rational Method in Western Australia.

## References

Australian Rainfall And Runoff; A Guide to Flood Estimation, Volume 1, 2001 .Australian Rainfall And Runoff; A Guide to Flood Estimation, Volume 2, 1987