# Design Rainfall Depth

The design rainfall depth is estimated as:

P = DDF \cdot ARF \cdot SCF

Where $DDF$ is the estimate of rainfall depth for a given duration and required frequency. The $ARF$ and $SCF$ are the areal reduction and seasonal correction factors that reflect the fact that the initial DDF model estimate is a point estimate rather than a catchment estimate and that the depth estimate is based on an analysis of annual maximum frequency rather than seasonal maximum frequency.

# Areal Reduction Factors

The estimates of design rainfall calculated using the DDF model are point values as the model is based on data from individual gauges. To allow the estimation of catchment average design rainfall, the concept of the areal reduction factor ($ARF$) has been adopted from the existing FSR/FEH method. The FSR (NERC, 1975) originally defined and analysed the $ARF$ as “the ratio of rainfall depth over an $AREA$ to the rainfall depth of the same duration and return period at a representative point in the $AREA$”. The $FSR$ values of $ARF$ were adopted in the FEH.

The values used and those used in ReFH are those expressed mathematically by Keers and Wescott (1977) as:

ARF - 1-bD^{-a}

where $D$ is the duration of the design rainfall and $a$ and $b$ are the parameters presented on Table 3.

Table 3. Areal reduction factor parameters (Keers and Wescott, 1977)

AREA A (km²) A b
A ≤ 20 0.40 – 0.0208 ln[4.6–ln[A]] 0.0394 A$^{0.354}$
20 ≤ A < 100 0.40 – 0.00382 (4.6–ln[A])² 0.0394 A$^{0.354}$
100 ≤A < 500 0.40 – 0.00382 (4.6–ln[A])² 0.0627 A$^{0.254}$
500 ≤A < 1000 0.40 – 0.0208 ln[ln(A]–4.6) 0.0627 A$^{0.254}$
1000 ≤A 0.40 – 0.0208 ln(ln[A]–4.6) 0.1050 A$^{0.180}$

# Seasonal Correction Factors (SCF)

A seasonal correction factor was introduced within the ReFH method to support the use of summer and winter design inputs. The $SCF$ converts the $DDF$ estimate of design rainfall depth based on annual maximum rainfall into an estimate of seasonal design rainfall through simple multiplication. The $SCF$ is a correction factor depending on location, season, duration and selected return period. A detailed description of the development of the $SCF$ is given by Kjeldsen et al. (2006) with a shorter summary provided in the FEH Supplementary Report No1 (Kjeldsen, 2007).

The SCF was derived by fitting a GEV distribution to a series of annual and seasonal maximum rainfall obtained from 523 daily raingauges and 172 subdaily recording raingauges located throughout the UK.

The $SCF$ for a given duration, $D$, are estimated using the following functional relationships:

SCF_D=\left\{\begin{matrix} \alpha SAAR + \beta & summer\\ (1-e^{[-\Phi SAAR]})^{\Psi} & winter \end{matrix}\right.

Where $SAAR$ is the catchment average value of the Standard period Average Annual Rainfall for the Meteorological Office 1961-1990 standard period. Please note the formulation is incorrect in the FEH Supplementary Report No1 (Kjeldsen, 2007).

Summer is defined as May to October and Winter as November to April.

For the summer relationship, a constraint was included in the parameter estimation that for $SAAR$ = 500 mm the seasonal correction factor equals one, i.e. 1 = $\alpha$ 500 mm + $\beta$. The parameters for duration of 1, 2, 6 and 24 hours are presented on Table 4. Values for other durations in the range [1,24] hrs are estimated using linear interpolation between adjacent values.

Table 4. Seasonal correction factor parameters

Summer Summer Winter Winter
Duration $\alpha$ $\beta$ $\Phi$ $\Psi$
≤1 hour –8.03x10$^{-5}$ 1.04 0.0004 0.4000
2 hours –6.87x10$^{-5}$ 1.03 0.0006 0.4454
6 hours –4.93x10$^{-5}$ 1.02 0.0009 0.4672
≥ 24 hours –10.26x10$^{-5}$ 1.05 0.0011 0.5333

Other than at the extremes of $SAAR$, where the tendency may be for the majority of annual maxima to lie within a particular season, annual maxima can occur in the other season. The seasonal rainfall will therefore tend be lower than the annual counterpart. The SCF is an integral part of the ReFH method. The application of the SCF in other models will depend on the context within which the rainfall data is used.