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:

- Short durations

Durations less than 12 hours; a single equation for the whole of Australia

- Long durations

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.

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 11 separate cases to consider which I list in 5 groups below.

1. Very small catchments:

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

2. Very large catchments:

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

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

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

4.3. Between long duration and short duration: interpolate

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

5.2. Long duration: interpolate for the area between an ARF of 1 at 1 km^{2} and the long duration ARF for 10 km^{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.

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 smallest duration considered in the derivation of the ARFs was 60 min so anything shorter than that is an extrapolation (Stensmyr et al., 2014). If the critical case ends up being less than 60 min, check that the ARFs 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.

### 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

Stensmyr, P., Babister, M. and Retallick, M. (2014) Spatial patterns of rainfall. Australian Rainfall and Runoff Revision Project 2. http://arr.ga.gov.au/__data/assets/pdf_file/0020/40556/ARR_Project2_Report_Short_ARF.pdf