With the publication of the 2019 edition of Australian Rainfall and Runoff most, but not all, of the ambiguities around the calculation of Areal Reduction Factors have been fixed.

Equations are provided that allow calculation of Areal Reduction Factors (ARFs) for:

Durations less than 12 hours; a single equation for the whole of Australia based on data rich regions around Sydney, Brisbane and Melbourne.

Durations between 24 hours and 168 hours, with different coefficients required for each of 10 regions (Figure 1).

These equations, and the long duration coefficients, are available from the data hub and from from Australian Rainfall and Runoff (ARR) (Book 2, Chapter 4.3). The reference is to ‘coefficients’ in ARR and to ‘parameters’ on the data hub but they are the same thing.

Figure 1: ARF regions

What the data hub doesn’t make clear is that procedures used to calculate an ARF depend on both duration and area. Details are in ARR Book 2, Table 2.4.1.

There are 9 separate cases to consider which I list in 5 groups below. These are summarised on Figure 2.

Figure 2. Application of ARF equations depends on area and duration (hi-res)

1. Very small catchments:

1.1. Catchment area ≤ 1 km^{2}, ARF = 1 for any duration

2. Very large catchments and very long durations

2.1. Catchment area > 30,000 km^{2}, ARF can not be calculated using the generalised equations

2.2. ARF equations are only available for durations less than 168 hours

3. Catchments between 1000 km^{2} and 30,000 km^{2}

3.1. Short durations: for duration ≤ 12 hours, ARF can not be calculated using the generalised equations

3.2. Long durations: for duration ≥ 24 hours calculate ARF using the long duration equation.

3.3. Between long and short durations (between 12 and 24 hours); interpolate between the long duration and short duration ARFs. So, although it is not valid to use the short duration ARFs in catchments of this size, the guidance suggests the 12 hour short duration ARF can be used as one terminal in the required interpolation.

4. Catchments between 10 km^{2} and 1000 km^{2}

4.1. Long durations: use the long duration equation for durations ≥ 24 hours

4.1. Short durations: use the short duration equation for durations ≤ 12 hours

4.3. Between long duration and short duration: interpolate

5. Catchments between 1 km^{2 } and 10 km^{2}

5.1. Long duration: interpolate for the area between an ARF of 1 at 1 km^{2} and the long duration ARF for 10 km^{2}. This is the ‘Interpolate 1’ region on Figure 2.

5.1. Short duration: interpolate for the area between an ARF of 1 at 1 km^{2} and the short duration ARF for 10 km^{2}. This is the ‘Interpolate 3’ region on Figure 2.

5.3. Between long duration and short duration: Interpolate for the duration between the long duration ARF and short duration ARF for a catchment of 10 km^{2}. Then interpolate for the area between an ARF of 1 at 1 km^{2} and the value for a 10 km^{2} catchment. This is the ‘Interpolate 2’ region on Figure 2.

Note, that for the area-based interpolations in 5.1 and 5.2, equation 2.4.4 is required (below). For the duration based interpolations, equation 2.4.3 should be used. There is an error in Table 2.4.1 in ARR where the wrong interpolation formula is referred to.

(2.4.3)

(2.4.4)

Another thing to be careful of is that the unrealistic negative values are calculated for large catchments and short durations. For example, the ARF for a 1000 km^{2} catchment, 1 minute duration and AEP of 1% in the Southern Temperate zone is -0.79. Of course, most practitioners are not interested in situations like this but if a Monte Carlo approach is used, these odd results may come up unless the parameter bounds are set carefully.

When setting up an ARF spreadsheet or script it is probably worth setting any values less than zero, to zero. But also check that your hydrologic model won’t crash if ARF is zero.

The shortest duration considered in the derivation of the ARFs was 30 min so anything shorter than that is an extrapolation (Stensmyr and Babister, 2015; Podger et al., 2015). If you use ARFs where the duration is less than 30 min, check that the values are realistic.

### Example

There is a worked example in ARR Book 2, Chapter 6.5.3.

Region = East Coast North, Area = 245.07, AEP = 1%, Duration = 24 hour (1440 min)

**ARF = 0.929**

### ARF calculator

I’ve developed a simple ARF calculator as a web app here. The code is available as a gist. If you just need the functions to calculate ARFs look here. Code to produce Figure 2 is here.

### Test cases

The test cases I used when developing the ARF calculator are here. This gives ARFs for a range of AEPs, durations and areas that correspond to the 11 cases listed above, along with other checks. I calculated these manually. Please let me know if you think any are incorrect.

### References

Podger, S., Green, J., Stensmyr, P. and Babister, M. (2015) Combining long and short duration areal reduction factors. Hydrology and Water Resources Symposium. Hobart, Tasmania.

Stensmyr, P. and Babister, M. (2015) Australian Rainfall and Runoff Revision Project 2: Spatial Patterns of Rainfall. Stage 3 Report: Short Duration Areal Reduction Factors.