# The distribution of burst initial loss

Recent posts have looked at the distribution of storm initial loss and pre-burst rainfall.  Here I extend the analysis to burst initial loss.  The definitions of storm and burst initial loss are described here.

According the the Australian Rainfall and Runoff (ARR) data hub:

Burst initial loss = storm initial loss – pre-burst rainfall.                  (1)

This seems clear, but the text in ARR is confusing.  For example, Book 5, Chapter 3.3.2:

However, if design bursts, rather than complete storms, are used in design then the burst initial loss needs to be reduced to account for the pre-burst rainfall. (emphasis added)

Shouldn’t the underlined word be ‘storm’?  That would make the statement consistent with equation 1.  There is a similar problem in Chapter 3.5.1.   I’ve sent these corrections into the editors of ARR so perhaps they will be addressed in the next edition.

In applying equation 1, it is assumed that median values of storm initial loss and pre-burst rainfall should be used to determine the median burst initial loss.  This assumption was tested by WMA Water for NSW. They found that this approach resulted in an over-estimation of burst initial loss (See Section 7 of WMA Water 2019).

We can test the WMA Water approach using the data for Toomuc Creek in Victoria (-38.064520N, 145.463277E). The distributions of storm initial loss and pre-burst rainfall were discussed here and here.  From the data hub, the median storm initial loss is 25 mm and the median burst initial loss is 1.5 mm for the 2 hour 1% AEP storm.  Therefore, using equation 1, the median burst initial loss for the 2 hour 1% AEP storm is estimated as 25 – 1.5 = 23.5 mm.

A histogram of burst initial loss, based on 10,000 randomly generated values of pre-burst rainfall and storm initial loss is shown in Figure 1.  Note that many of the values are less than zero, which occurs when the pre-burst rainfall is larger than the storm initial loss.  The median burst initial loss is 17.6 mm.  If we set all the negative burst initial loss values to zero, the histogram becomes as shown in Figure 2.  The median is unchanged at 17.6 mm.

Figure 1: Histogram of 10,000 burst initial loss values

Figure 2: histogram of 10,000 burst initial loss values restricted to be zero or greater

The upshot is that equation 1, when based on median storm and pre-burst values does not provide an accurate estimate of the burst initial loss.  The loss is over estimated which means modelled flood peaks will be biassed low.   Although only one example has been used in this analysis, the result confirms that reported by WMA Water.  There analysis was based on a large number of catchments, durations and AEPs.  WMA Water address this problem by calculating ‘probability neutral’ burst initial loss values based on the distributions of storm initial loss and pre-burst rainfall.  These probability neutral burst initial loss value are included in the data hub for sites in NSW.  There has been no similar work in Victoria or other states.  The probability neutral loss for Toomuc Creek would be 17.6 mm for the 2 hour 1% AEP event.  These losses could be calculated for other durations and AEPs by extending the methods shown here.  It would be reasonably straight forward to create a web app to read in the file from the data hub and output all the required losses.

One key assumption with this approach is that storm initial loss and pre-burst rainfall are independent.  This is probably reasonable but is something to keep in mind.

Code to reproduce the figures and calculations is available as a gist.

I’d like to acknowledge the assistance of Scott Podger but any mistakes are mine.

### References

WMA Water (2019) Review of ARR design inputs for NSW.  Report for the NSW, Office of Environment and Heritage.  Authors: Podger, S., Babister, M., Trim, A., Retallick, M. and Adam, M. (link)