ARF Calculator

Open ARF Calculator

Compute Areal Reduction Factors (ARF) that convert point rainfall depths into catchment-averaged design rainfall. This guide covers the two South-African methods provided by HydroDesign — the regionalised Pietersen method (Gericke et al. 2022, WRC K5-2924) and the simplified Alexander formula (2001) — the five homogeneous ARF regions, the two-stage polynomial algorithm, the full set of calibrated coefficients, and practical guidance on region distribution via manual entry, polygon upload, or watershed integration.

Overview

The ARF Calculator estimates Areal Reduction Factors using two methods calibrated for South African rainfall conditions: the Pietersen regionalised approach (Gericke et al. 2022, WRC K5-2924) and the Alexander simplified formula (Alexander 2001).

The tool divides South Africa into five homogeneous ARF regions, each with unique calibrated coefficients derived from 2,053 artificial circular catchments and 1,779 daily rainfall stations with a minimum of 30-year combined record lengths.

What is ARF?

An Areal Reduction Factor (ARF) converts point rainfall estimates to average areal rainfall over a catchment. Point-rainfall measurements from individual rain gauges typically overestimate the average rainfall over an area because storm intensities decrease with distance from the storm centre.

Why ARF matters

Design rainfall estimates are typically derived from point data (see the Design Rainfall tool). When estimating design floods for catchments, the point rainfall must be reduced by the ARF to reflect the actual average rainfall depth over the catchment area. Without ARF correction, flood estimates can be significantly overestimated.

ARF values range from 0% to 100%. An ARF of 85% means that the average areal rainfall over the catchment is 85% of the point rainfall depth. Key relationships:

• ARF decreases with increasing catchment area
• ARF increases with increasing storm duration
• ARF increases with increasing return period
• ARF approaches 100% for very small catchments

When to apply ARF

ARF is rarely relevant for small catchments (< 30 km²) typically handled with the Rational Method. For such small areas the point rainfall is a reasonable proxy for areal rainfall and ARF ≈ 1. ARF becomes important for medium-to-large catchments modelled with the SCS Method, Unit Hydrograph, DRH, or SDF Method, and for reservoir-scale design.

Available methods

The ARF Calculator offers two methods. Select the appropriate method based on your time of concentration ($T_c$) and the level of regional detail required.

Pietersen method (regionalized)

A two-stage polynomial approach calibrated for five homogeneous South African regions. It uses region-specific coefficients and accounts for catchment area, storm duration, and return period. The area input for each region is the overlap area (the portion of the catchment falling within that region), not the total catchment area.

• Recommended $T_c$ range: 24–168 hours
• Area range: 5–40,000 km²
• ARF varies by return period (probabilistically correct)
• Requires region distribution input

Alexander method (simplified)

A single-formula approach that does not require region information. The ARF is computed as:

$$\mathrm{ARF} \;=\; \bigl(90000 - 12800\,\ln A + 9830\,\ln(60\,T_c)\bigr)^{0.4}$$

Alexander (2001) ARF formula

where $A$ is the catchment area in km² and $T_c$ is the time of concentration in hours.

• Recommended $T_c$ range: 0.08–168 hours
• Area range: 5–40,000 km²
• ARF does not vary by return period
• No region distribution required

When to use which

Method selection guide

• $T_c \ge 24$ hours: Use the Pietersen method for the most accurate, regionally-calibrated results.
• $12 \le T_c < 24$ hours: The Pietersen method may produce distorted results. The Alexander method is recommended.
• $T_c < 12$ hours: Use the Alexander method. Pietersen results will be severely distorted at short durations.

ARF regions

South Africa is divided into five homogeneous ARF regions based on a clustering analysis of rainfall characteristics. These regions were derived from an initial 46 clusters that were merged based on similar rainfall-producing mechanisms and spatial correlation patterns.

Region map

The interactive map in the ARF Calculator displays all five regions with distinct colour coding. When using polygon upload or watershed integration, your catchment boundary is overlaid on these regions to calculate the percentage distribution.

Region characteristics

Each region has unique rainfall-producing mechanisms:

• Region 1: Distinct spatial correlation patterns with moderate decay.
• Region 2: Moderate spatial correlation with typical rainfall regimes.
• Region 3: Lower spatial variability, broader storm coverage.
• Region 4: Higher variability, localized storm patterns.
• Region 5: Unique frontal and convective interaction patterns.

Methodology

The ARF is calculated using a two-stage polynomial approach with region-specific calibrated coefficients. This method uses a geographically-centred (fixed-area) approach, which is probabilistically correct as it varies with return period.

Stage 1: Long equation

The intermediate value ($eQ_\mathrm{Long}$) is computed using a second-order polynomial regression on the log-transformed inputs:

$$\begin{aligned} eQ_{\mathrm{Long}} \;=\;& x_1 \bigl[\log_{10}(D/24)\bigr]^2 + x_2 \log_{10}(D/24) \\[2pt] &+ x_3 \bigl[\log_{10} T\bigr]^2 + x_4 \log_{10} T \\[2pt] &+ x_5 \bigl[\log_{10} A\bigr]^2 + x_6 \log_{10} A + x_7 \end{aligned}$$

Stage 1 — Pietersen long equation

where $D$ = duration (hours), $T$ = return period (years), $A$ = area (km²), and $x_1$ through $x_7$ are region-specific calibrated coefficients.

Stage 2: Final ARF

The final ARF percentage is computed from the intermediate value:

$$\mathrm{ARF} \;=\; A\,\bigl(eQ_{\mathrm{Long}}\bigr)^{2} + B\,eQ_{\mathrm{Long}} - C$$

Stage 2 — Final ARF

where $A$, $B$, and $C$ are region-specific constants. If the computed ARF exceeds 100%, it is capped at 100%.

Multi-region weighting

When a catchment spans multiple regions, the final ARF is a weighted average:

$$\mathrm{ARF}_{\mathrm{weighted}} \;=\; \frac{\sum_i \mathrm{ARF}_i \cdot p_i}{100}$$

Multi-region ARF weighting

where $p_i$ is the percentage of the catchment falling in region $i$.

Region distribution

The region percentages must sum to 100%. For catchments entirely within one region, set that region to 100% and leave the others at 0%.

Regional coefficients

The following tables list the calibrated coefficients for each of the five ARF regions. These were derived from non-linear 2nd-order polynomial regressions on log-transformed variables, fitted using Linear Moments (LM) to Annual Maximum Series (AMS) data from the DREU rainfall database.

Stage 1 — long-equation coefficients ($x_1$ to $x_7$)

Region	x₁	x₂	x₃	x₄	x₅	x₆	x₇
Region 1	-9.415	19.494	-1.164	7.666	-0.754	-1.081	97.000
Region 2	-9.527	18.229	-1.042	6.816	-0.629	-1.058	88.019
Region 3	-7.608	15.724	-0.330	4.562	-0.330	-1.216	89.190
Region 4	-12.363	24.372	-0.817	7.660	-0.540	-2.436	85.056
Region 5	-11.957	23.453	-0.896	7.037	-0.953	-0.129	84.444

Stage 2 — final ARF coefficients ($A$, $B$, $C$)

Region	A	B	C
Region 1	-0.034	7.286	287.648
Region 2	-0.037	7.896	319.770
Region 3	-0.055	11.395	487.770
Region 4	-0.024	5.391	196.710
Region 5	-0.025	5.502	200.890

Calibration dataset

Coefficients were calibrated from 2,053 artificial circular catchments using daily rainfall data from 1,779 stations (DREU database) with a minimum 30-year combined record length. An initial 46 clusters were merged to 5 final regions where Goodness-of-Fit statistics did not decrease by more than 15%. The GEV distribution was fitted to Annual Maximum Series using Linear Moments.

Input parameters

Parameter	Unit	Calibration range	Description
Catchment area (A)	km²	0–30,000	Total catchment area
Storm duration (D)	hours	24–168	Duration of the design storm (1–7 days)
Return period (T)	years	2–200	Design return period (1:2 to 1:200)
Region distribution	%	Sum = 100%	Percentage of catchment in each of 5 regions

Extrapolation warning

Values outside the calibration ranges will generate warnings. The calculator will still produce results, but they should be used with caution as they represent extrapolation beyond the calibrated dataset.

Region distribution modes

The calculator offers three ways to determine what percentage of your catchment falls within each ARF region.

Manual entry

Enter the percentage of your catchment within each region manually. This is useful when you already know the distribution from GIS analysis or from consulting the ARF region map. The percentages must sum to exactly 100%.

Polygon upload

Upload a GeoJSON or Shapefile containing your catchment-boundary polygon. The tool automatically intersects your polygon with the five ARF regions and calculates the percentage distribution. The intersection uses the Turf.js geospatial library for client-side polygon intersection.

Supported formats

GeoJSON files (.geojson, .json) and Shapefiles (.shp) are supported. For Shapefiles, upload the .shp file directly. The polygon must be in WGS84 (EPSG:4326) coordinates.

From watershed

If you have previously saved a watershed delineation using the Watershed Delineation tool, you can select it directly. The catchment-boundary polygon will be loaded and intersected with the ARF regions automatically. The catchment area will also be pre-filled from the watershed data.

Interpreting results

The calculator produces several outputs:

• Primary ARF value: The weighted ARF for your selected return period, displayed prominently. This is the value to apply to your point rainfall estimate.
• Region breakdown: Individual ARF values for each region with non-zero distribution, showing how each region contributes to the weighted result.
• Return-period table: ARF values for all seven standard return periods (1:2, 1:5, 1:10, 1:20, 1:50, 1:100, 1:200) with your selected period highlighted.
• Chart: Visual bar chart showing ARF variation across return periods.

Worked example

For a 1,500 km² catchment entirely within Region 2, a storm duration of 48 hours, and a 1:100 year return period:

1. Set Region 2 = 100% in the region distribution panel.
2. Enter $A = 1500$ km², $D = 48$ h, $T = 100$ years.
3. The Stage 1 equation computes $eQ_\mathrm{Long}$ from the Region 2 coefficients.
4. Stage 2 yields an ARF close to 90% (exact value depends on the Stage 2 coefficients for Region 2).
5. Apply the ARF to the point rainfall depth from the Design Rainfall tool to obtain the catchment-averaged design depth, then feed this depth into your flood-estimation method.

Saving & exporting

Results can be saved to your account for future reference and downloaded as CSV files for use in spreadsheets or reports. The CSV includes all input parameters, region distributions, and the full return-period table.

References

• Gericke, O.J., Pietersen, W.H. & Du Plessis, J.A. (2022). Development of a Regionalised Approach to Estimate Areal Reduction Factors in South Africa. WRC Report No. TT 891/22. Water Research Commission, Pretoria. (WRC Project K5-2924)
• Pietersen, W.H. (2023). Development of a Geographically-centred Approach to Estimate Areal Reduction Factors in South Africa. Central University of Technology, Free State.
• Alexander, W.J.R. (2001). Flood Risk Reduction Measures. University of Pretoria. (Source for the Alexander ARF method.)
• Bell, F.C. (1976). The Areal Reduction Factor in Rainfall Frequency Estimation. Institute of Hydrology, Report No. 35. Wallingford, UK.
• NERC (1975). Flood Studies Report, Vol. II. Natural Environment Research Council, London. (Original UK FSR areal-reduction-factor framework referenced in generic workflows.)

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Open ARF Calculator