Probabilistic rational method

The equation

The probabilistic rational method keeps the Rational form and replaces the runoff coefficient with two calibrated terms:

The effective runoff coefficient for each return period is $C_T \times SF_{C10}$. South Africa is divided into 41 homogeneous clusters (numbered 1 to 42; there is no cluster 7), each with its own regression coefficients and seven growth factors from 1:2 to 1:200 years. The distance to the coast stays in decimal degrees, the unit the regressions were calibrated in, so do not convert it to kilometres. The calculator also reports the companion mean annual flood scaling factor from the same cluster, which is the index flood the Index Flood method scales.

How it differs from the Rational Method

The SANRAL Drainage Manual Rational Method builds a rural coefficient from slope, soil permeability and vegetation, adjusts it for urban cover, lakes and return period, and applies an areal reduction factor to the rainfall. Each component is a judgement from a table, so two engineers can defensibly arrive at different coefficients for the same catchment. The PRM removes that choice: the coefficient comes from a regional calibration against observed floods, and the rainfall is a point value with no areal reduction because the calibration used point rainfall.

The cost is that the PRM sees only area, MAP and distance to the coast. It cannot tell a grassland catchment from a suburb with the same three numbers. Run it next to the Rational Method, not instead of it.

Inputs and where they come from

• Location: the catchment centroid, from a map click, typed coordinates, an uploaded polygon or a saved watershed delineation. It sets the cluster, the MAP from the national design rainfall grid and the coast distance.
• Area: from the delineation or entered by hand.
• Time of concentration: from the time of concentration calculator or your own flowpath analysis.
• Rainfall intensity: by default derived for every return period from the South African design rainfall grid (Smithers and Schulze, median point depth) at duration $T_c$, then divided by $T_c$. You can switch to entering one intensity per return period from your own design rainfall analysis.
• DR10%, the 1:10-year 24-hour depth, is recorded with the result for provenance. It does not enter the equation, and the calculator warns if it differs from the grid by more than 20 %.

Worked example

A 26.45 km² catchment east of Pretoria has its centroid in PRM cluster 31, with MAP = 724.9 mm, $T_c$ = 2.137 h and $D_c$ = 4.196 decimal degrees. The cluster’s regression gives $SF_{C10}$ = 0.1427 (and a mean annual flood scaling factor of 8.65 m³/s). The intensities are the point design rainfall depths for a storm lasting $T_c$, one per return period, divided by $T_c$.

At 1:100:

The effective coefficient rises from 0.07 at 1:2 to 0.27 at 1:200, which is the PRM’s version of the Drainage Manual’s return-period adjustment.

Why one intensity per return period matters

If the 1:10-year intensity of 33.03 mm/h were used for every return period, the 1:2-year peak would come out at 16.3 m³/s, 67 % too high, and the 1:200-year peak at 65.1 m³/s, 49 % too low. The intensity at duration $T_c$ grows with return period just as the coefficient does, and both have to move together.

Against the other regional methods

On the same catchment the Index Flood method gives 47.6 m³/s at 1:100 and the QRT gives 24.2 m³/s. The catchment is about two-thirds urban, which none of the regional methods sees, and that spread is the reason to compare them in a design flood estimation with the deterministic methods alongside.

Reference table: growth factors $C_T$ by centre

Growth factors for the cluster that contains each city centre. Your catchment’s centroid, not the nearest town, decides the cluster.

The spread in the 1:200 column, from 1.24 at Polokwane to 3.05 at George, is the regional flood signature the method encodes.

Limitations

• The method applies to South Africa only. The Studio calculator runs for South African sites, and a centroid outside the cluster coverage returns an error.
• It was calibrated on gauged, mostly rural catchments. Urban cover, large dams and dolomite are not represented.
• Reliability falls for very small catchments and above about 10 000 km²; the calculator warns above 50 000 km².
• Cluster boundaries are sharp lines through gradual change. Near a boundary, run the neighbouring cluster as well.
• The calibration assumes the historical flood record is stationary.
• Where a long, reliable record exists at or near the site, at-site flood frequency analysis should govern.

What the full Studio tool adds

The calculator shows the cluster map, finds the cluster, MAP and coast distance from the centroid, and derives the seven intensities from the design rainfall grid at your $T_c$. The result lists the cluster, both scaling factors, $C_T$, the point depth and intensity for every return period, and the peaks. Calculations save to your projects and export to CSV, Word and Excel. The PRM can also be ticked on a design flood estimation request, where it runs on the delineated catchment’s own area, MAP, $T_c$ and rainfall next to Index Flood, QRT and the deterministic methods.