# On the calculation of equal area slope

As noted in the previous post, the equal area slope was adopted for use with the Bransby Williams time of concentration formula in the 1987 version of Australian Rainfall and Runoff to: “…give a better indication of flow response times, especially where there are large variations of slope within a catchment” (ARR 1987 Boo IV, Section 1.3.2(d)).  The equal area slope is also used as part of flood estimation in New Zealand (NZ, 1980; Auckland Regional Council, 1999), in the Papua New Guinea Flood Estimation Manual (SMEC, 1990) – where it is used in the estimation of overland flow times and runoff coefficients for use in the rational method – and is discussed in the Handbook of Hydrology (Pilgrim and Cordery, 1993).

The equal area slope is the slope of a straight line drawn on a profile of a stream such that the line passes through the outlet and has the same area under and above the stream profile.

An alternative to the equal area slope, as used by Bransby Williams, is the average slope.  The differences between the equal area and average slopes are highlighted in the Figure 1 below.

Figure 1: Average slope and equal area slope (McDermott and Pilgrim, 1982, page 28)

I haven’t been able to find the history of the equal area slope.  I’m guessing that the average slope was found to be too steep for use in hydrologic calculations.  The equal area slope may have been considered more representative and was easy to calculate in pre-computer times.  The procedure, perhaps to be undertaken by a draftsman, is  specified in NZ (1980) (see Figure 2).

The method involves the calculation of the slope of the hypothetical line AC, which is so positioned that the enclosed areas above and below it, i.e. areas X and Y, are equal. The procedure is to planimeter the total area under the longitudinal profile.  This area Ad, equals the area of the triangle ABC.

Figure 2: Definition diagram for the calculation of the equal area slope (NZ, 1980)

$A_d = \frac{1}{2}AB \times BC$

$A_d = \frac{1}{2}L \times h$

$\therefore h= \frac{2A_d}{L}$

(Point C is known at the ‘equal area slope ordinate’).

Hence the equal area slope $S_a$ is given by

$S_{ea} = \frac{h}{L} = \frac{2A_d}{L^2}$

When the units for the elevation and length in the diagram above are used:

$S_{ea} = \frac{2A_d}{1000L^2} \;\; \mathrm{m/m}$

Notice that nothing iterative is required, we are just calculating the area under the profile and then working out the triangular area to match.

This simple calculation procedure seems to have been forgotten, as some modern approaches suggest an iterative procedure is necessary, for example, in this excel equal area slope tool.

An R function to calculate the equal area slope is available as a gist.

### References

Auckland Regional Council (1999) Guidelines for stormwater runoff modelling in the Auckland Region. (link)

McDermott, G. E. and Pilgrim, D. H. (1982) Design flood estimation for small catchments in New South Wales.  Department of National Development and Energy.  Australian Water Resources Council.  Research Project No. 78/104. Australian Government Publishing Service, Canberra (link)

Pilgrim, D. H. and Cordery, I. (1993) Flood runoff. In: Maidment, D. R. (ed) Handbook of Hydrology.  McGraw Hill.

SMEC (1990) Papua New Guinea Flood Estimation Manual. Department of Environment and Conservation, Bureau of Water Resources. (link)

NZ (1980)A method for estimating design peak discharge: Technical Memorandum No. 61. New Zealand. Ministry of Works and Development. Water and Soil Division. Planning and Technical Services; National Water and Soil Conservation Organisation (N.Z). (1980)  (link to catalog entry) (link to document)

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