Areal reduction factor calculator

What an areal reduction factor does

Design rainfall is a point value: the depth one gauge can expect for a given duration and return period. A storm never covers a large catchment at its peak point intensity, so the catchment-average depth is smaller:

The ARF falls as catchment area grows, rises as storm duration grows (long storms are more uniform), and in some methods varies with return period. It approaches 1 for small catchments.

Which method the calculator uses

ARF formulas

South Africa: Pietersen method

The Pietersen method, developed for the Water Research Commission (project K5-2924), works in two stages. The first is a second-order polynomial in the logarithms of duration $D$ (hours), return period $T$ (years) and area $A$ (km²):

The second converts it to the ARF in percent, capped at 100:

Where a catchment spans more than one region, each region’s ARF is evaluated at the part of the catchment inside that region, and the results are weighted by that area. The method is calibrated for storms of 1 to 7 days; below 24 hours its results distort, and below 12 hours the calculator steers you to the Alexander formula.

South Africa: Alexander formula

with $A$ in km² and $T_c$, the critical storm duration, in hours. The result is in percent, capped at 100, and the same for every return period.

United States: Leclerc and Schaake fit to TP-29

with $t$ in hours and $A$ in km². It is one national curve, valid up to 1 000 km² and 0.5 to 24 hours.

Australia: ARR 2019

ARR 2019 gives a national equation for short storms ($D \le$ 12 h, $D$ in minutes):

with $a$ = 0.287, $b$ = 0.265, $c$ = 0.439, $d$ = 0.36, $e$ = 0.00226, $f$ = 0.226, $g$ = 0.125, $h$ = 0.0141, $i$ = −0.021 and $j$ = 0.213. Storms of 24 hours or longer use a similar equation with nine coefficients for each of ten regions; the calculator looks the region and its coefficients up from the ARR Data Hub for your location, or you pick the region. Between 12 and 24 hours the two are interpolated linearly. Catchments of 1 km² or less take ARF = 1; between 1 and 10 km² the value is scaled from the 10 km² ARF by $1 - 0.6614(1 - \text{ARF}_{10})(A^{0.4} - 1)$. ARR 2019 does not define an ARF for catchments over 1 000 km² with storms of 12 hours or less, and the calculator says so instead of returning a number.

ARF vs area

ARFs from the calculator for four storm durations. The ARR values are for the SE Coast region at 1% AEP:

The two methods agree for long storms and large areas, and differ most for short storms on small catchments: at 10 km² and one hour, ARR 2019 reduces the point depth by about 10% and TP-29 by about 1%. For 250 km² and 6 hours in the SE Coast region, the ARR 2019 ARF falls from 0.890 at 50% AEP to 0.836 at 1% AEP and 0.826 at 0.5% AEP.

Worked example: South Africa, 1 500 km²

A 1 500 km² catchment lies entirely in ARF Region 2. The design storm duration is 48 hours and the return period 1:100 years.

1. Logarithms: $\log_{10}(48/24)$ = 0.3010, $\log_{10}100$ = 2, $\log_{10}1500$ = 3.1761.
2. Stage 1 with the Region 2 coefficients: $eQ_{Long}$ = 92.402.
3. Stage 2: ARF = −0.037 × 92.402² + 7.896 × 92.402 − 319.770 = 93.9%.

Across return periods the Region 2 ARF rises from 83.9% at 1:2 years to 89.5% at 1:10, 93.0% at 1:50 and 94.7% at 1:200. The Alexander formula gives 89.0% for the same area and duration at every return period. If the 48-hour 1:100-year point depth were 200 mm, the catchment depth would be 0.939 × 200 = 188 mm with the Pietersen method and 178 mm with Alexander.

Inputs and where they come from

• Catchment area. From the catchment boundary, for example from watershed delineation.
• Storm duration. The duration of the design storm, usually the critical duration of the flood method, which for peak-flow methods is close to the time of concentration.
• Return period or AEP. The same as the point design rainfall you are reducing, from design rainfall.
• Region (South Africa and Australia). For South Africa, enter the percentage of the catchment in each of the five ARF regions, upload the catchment polygon (GeoJSON or shapefile, WGS84) or pick a saved watershed, and the calculator intersects it with the region map. For Australia, the region is looked up from the catchment location.

Limitations

• An ARF from one method cannot be paired with design rainfall or a flood method calibrated with another without care. Use the ARF that goes with your national design rainfall.
• The South African Pietersen method is for storms of 24 to 168 hours. For shorter storms use Alexander.
• Leclerc–Schaake stops at 1 000 km² and 24 hours.
• The generic log-area curve has no regional calibration and is clipped between 0.5 and 1.0. It is for preliminary work and teaching, not final design.
• An ARF is a fixed-area, catchment-average ratio. It does not describe how rainfall is distributed in space within the catchment.

What the full Studio tool adds

The Studio tool shows the ARF for every standard return period or AEP as a table and chart, the ARF of each South African region the catchment touches, and the regional map with your catchment overlaid. It flags inputs outside each method’s calibrated range. Calculations save to a project and export to CSV, Word and Excel. The ARF can be sent straight to the rational method calculator, and South African design flood estimation uses the same five ARF regions.