QRT Calculator

Open QRT Calculator

Estimate peak discharge for ungauged rural catchments in South Africa using the Quantile Regression Technique (QRT) — a regional flood-frequency method developed by W.J.R. Alexander and adopted in the SANRAL Drainage Manual. This guide covers the theoretical basis, the veld-type regions used as homogeneous zones, the regression form and its coefficients, required inputs, a fully worked example, and the method’s limitations.

Overview

The Quantile Regression Technique is a regional flood-frequency method that estimates peak discharge directly as a function of catchment descriptors. Rather than first computing a rainfall input and then converting it to runoff (the rainfall-runoff paradigm of the Rational, SCS, and Unit Hydrograph methods), QRT fits multiple linear regressions in log-space between observed flood quantiles and catchment attributes at gauged sites within a homogeneous region. The resulting equations can then be applied to ungauged catchments in the same region.

The technique was developed for South Africa by W.J.R. Alexander in the late 1970s and progressively refined through the 1980s, 1990s, and 2000s. It is documented in Alexander (2003) and codified in the SANRAL Drainage Manual (Chapter 3, Section 3.6) as one of the preferred methods for peak-flow estimation in rural catchments where at-site gauged data are not available.

Regional flood frequency, not rainfall-runoff

QRT belongs to the family of regional flood frequency methods — it bypasses the rainfall-runoff transformation and regresses flood peaks directly on catchment descriptors. This makes it conceptually similar to the Index Flood Calculator (which uses a two-step index-flood / growth-curve approach) but different from the Rational Method, SCS Method, and Unit Hydrograph, which all compute peak flow from a design rainfall input.

When to use

QRT is appropriate — and often preferred per the SANRAL Drainage Manual — in the following situations:

• Rural catchments without reliable at-site gauged data. QRT is the workhorse method for ungauged rural hydrology in South Africa.
• Medium to large catchments (broadly 10 km² to several thousand km²) where the single-cell, uniform-rainfall assumptions of the Rational Method start to break down.
• Catchments that fall clearly within a single QRT region, so that the regional homogeneity assumption is defensible.
• Cross-check of another method. Even when a Rational, SCS, or Unit Hydrograph estimate is the primary result, a QRT cross-check helps triangulate the magnitude of the design flood.

The method is not suitable when:

• The catchment straddles two QRT regions with substantially different regression coefficients.
• Reliable gauged data exist at or very near the site — at-site Flood Frequency Analysis will outperform regional regression.
• The catchment is heavily urbanised or has significant anthropogenic storage (reservoirs, floodplains) not represented in the gauged calibration sample.
• The catchment is substantially outside the size range of the calibration catchments for the region.

Theoretical foundation

Quantile regression vs mean regression

Most elementary regression methods in hydrology fit a relation between the mean of the response and the predictors. QRT fits a separate regression for each quantile of interest — that is, one equation for $Q_2$, another for $Q_5$, another for $Q_{10}$, and so on up to $Q_{200}$. The advantage is that the tail behaviour of the flood distribution (rare events) can evolve with catchment size and rainfall in a different way to the central tendency — a feature that single mean-regression approaches miss.

The method does not explicitly fit a parametric frequency distribution (such as the GEV or Log-Pearson III) at each gauge and interpolate its parameters. Instead, it uses the empirical quantiles at each gauge as the response variable and lets the regional regression absorb both the growth-curve shape and the between-site variability in a single step.

Regional homogeneity assumption

The central assumption is that catchments within a given region share a similar flood-generating mechanism and hence a similar response to the chosen descriptors. Homogeneity is established by partitioning South Africa into regions based on observable physiographic and climatic boundaries — chiefly the veld-type classification of Acocks / Mucina & Rutherford — and by verifying at the calibration stage that within-region residual variance is small relative to between-region variance.

Region boundaries are approximate

QRT regions are drawn from vegetation and climate criteria that are continuous in reality but rendered as sharp polygon boundaries for computation. Catchments close to a boundary should be evaluated against both adjacent regions; a substantial change in the result signals that regional homogeneity is not well satisfied at that location and the engineer should apply additional judgement.

Parent distribution

When a frequency distribution is fitted for comparison or for extrapolation beyond the gauged range, the parent distributions most commonly adopted in South African QRT work are the General Extreme Value (GEV) and the Log-Pearson Type III (LP3) distributions. These are the same distributions used elsewhere in HydroDesign’s Flood Frequency Analysis and Index Flood Calculator tools, so their properties and parameter-fitting methods are treated in depth there.

Methodology

Regression form

QRT fits a log-linear multiple regression between the peak discharge $Q_T$ and a set of catchment descriptors. The working form is:

$$Q_T \;=\; K_T \cdot A^{\alpha} \cdot \mathrm{MAP}^{\beta} \cdot S^{\gamma} \cdot \mathrm{LCOEF}^{\delta}$$

QRT regression — general multiplicative form

where $Q_T$ is the peak discharge (m³/s) for return period $T$, $K_T$ is the region-specific and return-period-specific coefficient, $A$ is catchment area (km²), $\mathrm{MAP}$ is mean annual precipitation (mm), $S$ is an average slope metric, $\mathrm{LCOEF}$ is a land-cover / vegetation coefficient, and the exponents $(\alpha, \beta, \gamma, \delta)$ are fitted per region by least squares in log-space.

Taking logarithms converts the multiplicative form to an additive linear regression, which is how the coefficients are actually estimated:

$$\ln Q_T \;=\; \ln K_T + \alpha \ln A + \beta \ln \mathrm{MAP} + \gamma \ln S + \delta \ln \mathrm{LCOEF}$$

QRT regression — additive log-space form

Only the predictors that are statistically significant within a given region are retained; it is therefore common for a particular region to use a reduced form with, for example, only area and MAP as predictors.

Regional coefficient tables

The SANRAL Drainage Manual (2013), Chapter 3 tabulates the fitted coefficients $(K_T, \alpha, \beta, \gamma, \delta)$ for each homogeneous region and each of the seven standard return periods. The HydroDesign QRT Calculator hard-codes the most recent revision of these tables — users do not interact with the coefficients directly, but the coefficient set used in the calculation is always reported in the result panel for auditability.

Auditability

Every QRT calculation returns not only the peak-flow result but also the region name, the coefficient set used, and the values of $A$, MAP, and slope that were substituted into the equation. Include this block in the project report — it makes the calculation reproducible by a reviewer without access to HydroDesign.

QRT regions

South Africa is divided into a set of homogeneous veld-type regions for QRT application. The regions follow the broad vegetation biomes of the country, which in turn reflect the underlying rainfall, temperature, and runoff generation characteristics. The regional split used by QRT is broadly consistent with the regions used in the SDF Method and Index Flood Calculator, although the specific boundaries and coefficients differ between methods.

Regional summary

Region	Typical biome / climate	Character
Coastal Fynbos	Winter-rainfall Western Cape	Moderate floods; steep coastal catchments
Forest and Thicket	Southern Cape, eastern escarpment	High MAP; steep; rapid response
Grassland — Highveld	Interior plateau, summer rainfall	Large summer floods; moderate slope
Grassland — Drakensberg	Escarpment and upper reaches	High rainfall; steep; flash-flood prone
Karoo (Nama and Succulent)	Arid to semi-arid interior	Episodic flash floods; low MAP
Savanna — Bushveld	Northern summer-rainfall areas	Intense convective storms; moderate runoff
Savanna — Lowveld	Eastern lowveld, Limpopo	Large subtropical storm systems
Desert and Nama Karoo fringes	North-west arid zone	Extreme episodic events; very low baseline

Region map

The QRT Calculator displays the active region boundaries on the map; clicking a point automatically determines the region assignment by point-in-polygon lookup. River segments that are known to behave anomalously relative to the regional envelope are flagged for the engineer’s attention, mirroring the treatment used in the RMF Calculator.

Boundary catchments

Catchments that straddle two regions are an operational problem for any regional method. Two strategies are commonly used:

• Area-dominant rule — assign the catchment to whichever region contains the largest share of its area. Simple and auditable.
• Dual-region cross-check — compute $Q_T$ with both regions’ coefficients and report the range. The engineer then selects a value (often the higher) with explicit justification.

The HydroDesign tool supports both: the default is the area-dominant rule, with a toggle to display the dual-region comparison.

Input parameters

The QRT Calculator requires the following inputs for the catchment:

Catchment area (A)

The contributing area in km². This is the single most important variable in the regression. Obtain it from the Watershed Delineation tool, from a GIS catchment layer, or by manual polygon entry.

Mean annual precipitation (MAP)

MAP in mm is used as the climatic descriptor. It is extracted from the national rainfall grid by the tool automatically using the catchment centroid — the value used is displayed in the result panel. For a cross-check, compare the tool’s value against the Daily Rainfall Data or Design Rainfall station-by-station summaries.

Average slope (S)

The average catchment slope — typically the 10-85 watercourse slope. Obtain this automatically from the Watercourse Profile tool, which computes the standard 10-85, equal-area, and average slope metrics in a single pass.

Land-cover / vegetation coefficient

A dimensionless coefficient summarising the dominant land-cover class of the catchment. HydroDesign derives this automatically from the national land-cover classification; the coefficient is shown in the result panel for transparency and can be manually overridden for catchments where the classification is known to be out of date.

Parameter extraction tools

Where possible, use the dedicated HydroDesign tools to populate QRT inputs: Watershed Delineation for area, Watercourse Profile for slope, Design Rainfall for MAP. Consistent parameter extraction is one of the main advantages of running the full HydroDesign pipeline rather than assembling inputs from disparate sources.

Return period handling

The QRT Calculator produces peak-discharge estimates for seven standard return periods. Each return period has its own coefficient set $K_T$ within each region, which is why the method is called quantile regression — every quantile is fitted as its own regression rather than by interpolation on a fitted frequency curve.

Return Period (yr)	AEP (%)	QRT coefficient
2	50	$K_2$
5	20	$K_5$
10	10	$K_{10}$
20	5	$K_{20}$
50	2	$K_{50}$
100	1	$K_{100}$
200	0.5	$K_{200}$

Extrapolation beyond 1:200

The regression coefficients were fitted using gauged records of typically 30-60 years of annual-maximum data. Extrapolation to return periods beyond 1:200 — such as for dam safety applications — is not supported within QRT and should be handled by the RMF Calculator envelope or, for the Probable Maximum Flood, the PMP Calculator plus a rainfall-runoff model.

Comparison with other methods

Method	Best for	Key input
Rational Method	Small urban catchments (< 25 km²)	Design rainfall IDF
Alternative Rational	Small rural catchments (SA practice)	Design rainfall IDF + return-period correction
SCS Method	Small to medium catchments with CN data	Curve Number, design rainfall
Unit Hydrograph	Medium rural catchments, full hydrograph	Catchment lag, design rainfall
SDF Method	Medium to large rural SA catchments	Regional K, drainage basin ID
QRT	Medium to large rural SA catchments, ungauged	Area, MAP, slope, land cover
Index Flood Calculator	Ungauged catchments with gauged neighbours	Index flood + growth curve
Flood Frequency Analysis	Gauged catchments	Annual-maximum series
RMF Calculator	Upper-bound envelope for critical infrastructure	Area, K-value region

QRT compares to these other methods as follows:

• vs Rational Method. QRT is generally preferred for larger rural catchments where the Rational Method’s uniform-rainfall and small-catchment assumptions become strained.
• vs SDF Method. Both are deterministic regional approaches for South African catchments, but QRT uses regression on multiple descriptors while SDF uses a single regional K-value applied to an envelope form. SDF is typically more conservative; QRT typically produces a central estimate consistent with the observed record.
• vs Flood Frequency Analysis. QRT is designed for ungauged catchments. Where a long reliable at-site record is available, classical flood-frequency analysis will almost always outperform QRT.
• vs Index Flood. Both are regional methods; Index Flood separates the index (mean annual flood) from the growth curve, while QRT fits one regression per quantile.

Worked example

The following example computes the 1:100 year peak flow for a hypothetical rural catchment in the Highveld grassland region.

Given information

• Catchment area: $A = 200$ km²
• Region: Grassland — Highveld (hypothetical coefficient set shown below for illustration)
• MAP: 680 mm/yr
• Average slope: $S = 0.018$ m/m
• Land-cover coefficient: $\mathrm{LCOEF} = 0.80$ (predominantly grassland / cultivated)
• Target return period: 1:100 year

Step 1 — Region lookup. The catchment centroid falls inside the Highveld grassland region. For illustration, assume the fitted coefficients for this region at the 100-year return period are:

$$K_{100} = 1.5, \quad \alpha = 0.70, \quad \beta = 1.10, \quad \gamma = 0.25, \quad \delta = 0.50$$

These are illustrative values used for the worked example only; the actual coefficients used by the tool are those from the latest SANRAL Drainage Manual revision.

Step 2 — Substitute into the regression.

$$\begin{aligned} Q_{100} &= K_{100} \cdot A^{0.70} \cdot \mathrm{MAP}^{1.10} \cdot S^{0.25} \cdot \mathrm{LCOEF}^{0.50} \\[4pt] &= 1.5 \cdot 200^{0.70} \cdot 680^{1.10} \cdot 0.018^{0.25} \cdot 0.80^{0.50} \end{aligned}$$

QRT worked example — substitution

Step 3 — Evaluate term by term.

Term	Value
$200^{0.70}$	42.9
$680^{1.10}$	1 240
$0.018^{0.25}$	0.366
$0.80^{0.50}$	0.894

$$Q_{100} \;=\; 1.5 \cdot 42.9 \cdot 1240 \cdot 0.366 \cdot 0.894 \;\approx\; \boxed{26{,}100 \ \mathrm{m^3/s}}$$

QRT worked example — final evaluation

Illustrative numbers only

The coefficient values used above are illustrative — they are chosen to demonstrate the arithmetic and will not match a real calculation. In the HydroDesign tool the actual fitted coefficients are used and the per-term evaluation is shown in the result panel for verification.

Step 4 — Cross-check. Compare the QRT result against at least one other method — typically SDF, Unit Hydrograph, and/or RMF. If the QRT value sits outside the envelope of the other estimates by more than ±50%, investigate the inputs (MAP, slope, area) before adopting the QRT value as the design flood.

Limitations

• Regional boundaries are approximate. QRT regions are drawn from continuous vegetation and climate criteria rendered as polygon boundaries. Catchments near a boundary should be evaluated against both adjacent regions.
• Sub-regional heterogeneity. The regression fits a single relationship to each region, implicitly assuming that all catchments within that region respond similarly. Sub-regional variability — for example between the eastern and western Highveld — is not captured.
• Stationarity. The fitted coefficients rely on gauged records that reflect historical hydroclimate. Climate-change-induced shifts in rainfall intensity and frequency are not explicitly modelled.
• Urbanisation. QRT is calibrated against predominantly rural catchments. Heavily urbanised catchments should be treated with Rational or SCS methods that explicitly represent impervious cover.
• Catchment size. Regression equations are fitted over a finite range of catchment sizes. Extrapolation to very small (< ~10 km²) or very large (> ~10 000 km²) catchments substantially reduces reliability.
• Single peak-flow output. QRT returns a peak flow, not a hydrograph. When the full hydrograph is required for routing through a storage element (see Flood Routing) or for a dam-safety analysis (see Dam Safety Evaluation), QRT can provide the peak but the hydrograph shape must come from another method (Unit Hydrograph, SCS, or DRH).

Adopt with engineering judgement

The QRT result is a statistically-grounded central estimate for an ungauged rural catchment. It should be one input — not the sole input — to the engineer’s choice of design flood. For any consequential design, present QRT alongside at least two independent methods and document the rationale for the adopted value.

References

• Alexander, W.J.R. (2003). Flood Risk Reduction Measures — Incorporating Flood Hydrology for Southern Africa. University of Pretoria, Department of Civil and Biosystems Engineering.
• SANRAL. (2013). Drainage Manual (6th ed.). South African National Roads Agency, Pretoria. Chapter 3 — Hydrology; Section 3.6 — Quantile Regression Technique.
• Van der Spuy, D. & Rademeyer, P.F. (2018). Flood Frequency Estimation Methods as Applied in the Department of Water and Sanitation. Department of Water and Sanitation, South Africa.
• Smithers, J.C. & Schulze, R.E. (2003). Design Rainfall Estimation in South Africa. Water Research Commission Report K5/1060. Pretoria.
• Görgens, A.H.M. (2007). Joint Peak-Volume Design Flood Hydrographs for South Africa. Water Research Commission Report 1420/3/07. Pretoria.
• Acocks, J.P.H. (1988). Veld Types of South Africa. Memoirs of the Botanical Survey of South Africa No. 57. Botanical Research Institute, Pretoria. (Underlying vegetation classification used for region delineation.)

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