There are two equations provided for areal reduction factors (ARFs) in Australian Rainfall and Runoff 2019. One for long durations, greater than 1440 minutes (24 hours), and one for short durations, less than 720 minutes (12 hours).
To estimate ARFs between 12 hours and 24 hours we need to interpolate. The recommendation in Australian Rainfall and Runoff 2019 is to do the required interpolation linearly (Podger et al., 2015) Complicating matters, the short duration ARF equations are only suitable for catchment areas less than 1000 km2 while long duration ARFs are available for catchments out to 30,000 km2. The situation is summarised on Figure 1.
Surprisingly, ARR2019, recommends that interpolation can be used for catchment areas all the way out to 30,000 km2. That is, we can use the short duration equations as end points for interpolation for catchment areas larger than 1000 km2. This involves applying the short duration equations well beyond the range of the data that was used in their derivation (Stensmyr and Babister, 2015).
Let’s start with the equations:
Short duration:
(equation 1)
Long duration:
(equation 2)
where, is area, is duration and is AEP.
The constants a, b, c, d, e, f, g, h, i depend on the region (Figure 2). The values of these constants is shown below in a screen shot of Table 2.4.2 from ARR Book 2, Chapter 4.3.1.
Here, we are interested in the case where area is larger than 1000 km2 and duration falls between 12 and 24 hours. Figure 2 shows an example for a catchment area of 30,000 km2. Between 12 and 24 hours, the ARFs are linearly interpolated between the long duration value at 24 hours and the short duration value calculated at 12 hours (solid line in Figure 2). The short duration 12 hour value is subject to considerable uncertainty because the prediction equation was developed using data from catchments smaller than 1000 km2 , much smaller than the 30,000 km2 catchment area used here.
My first though was, if we require ARF estimates for durations between 12 and 24 hours, it would be better to just extend the long duration equations (the dashed line in Figure 2). This would result in more conservative ARFs in the interpolation region. And it is not obvious why extending the long duration equations is any worse than extrapolating short duration values and then interpolating from these values. However, I now realise there are problems with this approach.
Extrapolating the long duration ARFs does not work for catchment areas close to 1000 km2 This can be seen on Figure 3. For a catchment area of 1000 km2 , both short and long duration ARFs can be calculated and the extrapolation between 12 and 24 hours is reasonable because the end points will be accurate. Both equations are being used where there is data to support their derivation. This is the red line on Figure 3.
Now consider the case for a catchment of 1100 km2. The Short duration ARFs can no longer be calculated (because the limit of 1000 km2 is exceeded). So, to calculate the ARF between 12 and 24 hours we could, either
1) Extend the long duration equation (dashed green line in Figure 3), or
2) Interpolate between the long duration value at 24 hours and the short duration value at 12 hours (the green solid line in Figure 3).
In this case, only approach 2 will work. If we adopt approach 1, the ARFs for a larger 1100 km2 catchment will exceed those for the smaller 1000 km2 catchment (the dashed green line exceeds the solid red line). This is not physically realistic. As area increases, ARF is expected to decrease.
The upshot is, if you are working on catchments with areas only a bit larger than 1000 km2 it would be best to do the ARF calculations using the recommended method, to avoid discontinuities in ARF values.
If you are working on catchments much larger than 1000 km2 , and need to calculate an ARF between 12 and 24 hours, it may be worth checking the ARF values that result from extending the long-duration equations. They will likely to be conservative, in that they will produce higher flood flows than the ARR recommended approach. It is also in this region, i.e. where catchments area are much larger than 1000 km2, that there is not strong justification for using the short duration equations as end points for the interpolation. The support for these short duration equations reduces as catchment areas increase beyond the 1000 km2 limit that was used in their derivation.
Code to produce graphs is available as a Gist.
Reference
Stensmyr, P. and Babister, M. (2015) Short duration areal reduction factors. Australian Rainfall and Runoff Revision Project 2: Spatial Patterns of Rainfall. Stage 3 Report. September 2015) (link).
Podger, S., Green, J., Stensmyr, P. and Babister, M. (2015). Combining long and short duration areal reduction factors. Hydrology Water Resources Symposium (link to paper at Informit).