Shape factor in regional flood estimation

[Edit 17 Oct 2016.  Some web links have changed.  See Where is ARR?]

The new Australian Rainfall and Runoff includes a regional flood frequency estimation (RFFE) approach (Book 2, Section 3).  In summary, parameters of the Pearson III distribution for a particular location can be determined from catchment characteristics combined with estimation equations that are based on flood data from nearby gauges.

The catchment characteristics are:

• Catchment area in km2 (area)
• Design rainfall intensity (in mm/h) at the catchment centroid for the 6 hour and 50% AEP event (50%I6h)
• The ratio of design rainfall intensities (2%I6h/50%I6h  )
• A shape factor (Sf) – the shortest distance between the catchment outlet and centroid divided by the square root of the catchment area.

A web-app has been developed to undertake the required calculations and is available here.

This post looks at the shape factor.

Table 3.3.2 in ARR (Book 2, Section 3.4.1) provides a range of shape factors that were found in the 798 catchments considered by the ARR team.  This table is reproduced below.

Sf % of catchments
less than
0.32 1
0.51 10
0.76 50
1.06 90
1.51 99

The table shows that 98% of shape factors are in the range 0.32 to 1.51.  Shape factors are roughly log-normally distributed i.e. the log(Sf) v standard normal deviate plots approximately as a straight line (Figure 1).

Figure 1: Log(shape factor) v standard normal deviate.

So, what does a catchment with a shape factor of 0.32 or 1.51, or any other value, look like?  The RFFE web app uses ellipses to approximate catchment shapes.

Figure 2: Catchment shape as approximated by an ellipse.  The distance from the centroid to the outlet is ‘a’.

For an ellipse, the area is:

$\mathrm{Area} = \pi a b$

From the definition of the shape factor (Sf):

$S_f = \frac{a}{\mathrm{\sqrt{Area}}} = \frac{a}{\sqrt{\pi a b}} = \sqrt{\frac{a}{\pi b}}$

If we set the area to one, it is straightforward to construct a series of ellipses the correspond to various shape factors.  See examples in the table and figures below.

If Area = 1:

$S_f = a$
$\mathrm{Area} = \pi S_f b$
$b = \frac{1}{\pi S_f}$

Sf % of catchments
less than
a b
0.32 1 0.32 1.0
0.51 10 0.51 0.62
0.76 50 0.76 0.42
1.06 90 1.06 0.30
1.51 99 4.51 0.21

Figure 3:  These catchments have the same area but different shapes as characterised by the shape factor.

In the regional flood frequency approach the shape factor  is used in the estimation of the mean of the Pearson III distribution.  The is equivalent to the mean of the natural logs of the peak discharges.  The estimating equation is:

$M = b_0 + b_1(log_e(area)) + b_2(log_e(^{50\%} I _{6h})) + b_3(log_e(S_f))$    (1)

There was a comment at the ARR launch in December 2015 that flood estimates are sensitive to the shape factor.  We can test this by looking at a specific example.

The idea is to use the RFFE tool to generate flood estimates for a range of shape factors while holding all the other factors constant.  Steps are as follows.  Choose somewhere in region 1 or 2. In these regions the estimating equations for skew and standard deviation do not include rainfall intensity or catchment area.  To hold the factors in equation (1) constant (i.e. the area and rainfall intensity), on the Input Data screen – Basic tab, use a constant area and on the Advanced tab, specify a 50% AEP 6 Hour Rainfall Intensity.  On the Basic tab, maintain a consistent outlet location so the bias correct factor remains constant.  Then, on the map, move the catchment centroid around to generate different shape factors while ensuring the estimated skew and standard deviation don’t change.

An example is shown below (Figure 4).  Clearly, when the shape factor is less than about 0.5, the effect on the flood estimate is dramatic.  Small errors in shape factor, when the shape factor is small, will cause very large errors in flood estimates.  (To be precise, this only applies to per unit errors e.g. an error of 0.1 in the shape factor, not percent errors).

Figure 4: 1% flood estimate as a function of the catchment shape factor (when all other factors are held constant)

The explanation for this non-linear relationship is that the log of the shape factor is linearly related to the log of the flood estimate as specified in equation (1) (as long as all other factors are held constant).  A log-log plot is shown in Figure 5.

Figure 5: 1% flood estimate as a function of the catchment shape factor (log-log plot)

So is this effect real?  Do catchments with small shape factors have much larger floods mutatis mutandis.  There must be some influence because the estimating equations (e.g. equation 1) are based on analysis of data, but we need to be careful.  The RFFE tool provides a warning when the shape factor is less than 0.51 that results may not be reliable.

Lets look at an example.  The Moe River at Darnum (gauge 226209) drains a catchment with a shape factor of 0.281.  This is in the smallest 1% of shape factors considered in the development of the estimating equations.   Details, from the RFFE tool are:

• Outlet latitude = -38.21
• Outlet longitude = 146
• Centroid latitude = -38.1861
• Centroid longitude = 145.966
• Catchment area = 214

It is  possible to undertake a regional estimate for the Moe River at Darnum using this data.  We can compare these estimates with the at-site frequency values which can be obtained from the RFFE tool by downloading the ‘nearby’ file.  Look for the following toolbar in the RFFE web app.

Figure 6: The ‘Nearby’ file contains information on at-site flood frequency results at gauges surrounding the location of the regional estimate

These two approaches are compared in the figure below. The regional estimates are much larger than the at-site estimates for the same location.  Clearly the small shape factor is contributing to these large errors.  The upshot is, take the warning provided by the RFFE tool seriously, regional flood estimates for catchments with unusual shape factors may not be reliable.  So, what range of shape factors will produce reasonable results?  The RFFE tool will warn if the shape factor is approximately within the largest or smallest 10% (Table 1).  A shape factor between 0.50 and 1.10 will not result in a warning.

Figure 7: Comparison of at-site and regional estimates for a catchment with a small shape factor

Code is available as a gist.