# kc prediction equations

This page bring together some prediction equations that have been proposed for the RORB routing parameter, kc.

# kc as a function of catchment area

## 1. RORB manual, equation 2.5

$k_c = 2.2A^{0.5} \left( \frac{Q_p}{2} \right )^{0.8 - m}$

Where

• $A$ is the catchment area (km2)
• $Q_p$ is the maximum discharge of the hydrograph (m3s-1)

This can be simplified when $m$ is equal to 0.8.

$k_c = 2.2 \sqrt A$

All the $k_c$ prediction equations below apply when m = 0.8.  If is different value of m is used the following conversion may be necessary (ARR1987 Book 5, Equation 3.19):

$k_c (m) =k_c (m = 0.8) \left( \frac{Q_p}{2} \right )^{0.8 - m}$

## 2. MMBW (Melbourne and Metropolitan Board of Works)

Yarra and Maribyrnong areas.

$k_c = 1.19A^{0.56}$

## 3. DVA (Dandenong Valley Authority)

Dandenong Valley and South East Melbourne.

$k_c = 1.53A^{0.55}$

## 4. Australian Rainfall and Runoff (1987)

• NSW  – Eastern(ARR Book V, Eqn 3.20; Kleemola, 1987)

$k_c = 1.22A^{0.46}$

• Victoria, areas with mean annual rainfall > 800 mm (ARR Book V, Eqn 3.21, Hansen et al. (1986a,b))

$k_c = 2.57A^{0.45}$

• Victoria,  areas with mean annual rainfall < 800 mm (ARR Book V, Eqn 3.22, Hansen et al. (1986a,b))

$k_c = 0.49A^{0.65}$

• Queensland (ARR Book V, Eqn 3.23; Weeks (1986) )

$k_c = 0.88A^{0.53}$

• South Australia – south east (ARR Book V, Eqn 3.25-3.26; Maguire et al., (1986))

$k_c = 0.60A^{0.67}, \, A < 100$

$k_c = 1.09A^{0.51}, \, A \geq 100$

• South Australia – northern and western (ARR Book V, Eqn 3.27; Lipp (1983))

$k_c = f(slope) \, A^{0.51} \, slope < 1\%$

See Lipp (1983) for further details of the relationship with slope

• South Australia (D. Kemp pers. comm.)

$k_c = 0.89A^{0.55}$

• Tasmania – western (ARR Book V, Eqn 3.32)

$k_c = 0.86A^{0.57}$

• Western Australia

See ARR 1987 Book V equations 3.28 and 3.29.

• Northern Territory

See ARR 1987 Book V equations 3.30 and 3.31.

## 5. Australian Rainfall and Runoff 2016

Regional relationships for RORB are provided in ARR 2016 Book  7 Chapter 6.2.1

• Queensland

Several equations are listed for different areas of Queensland with the equation from Weeks (1986) considered as a ‘good average’ and is recommended.

$k_c = 0.88A^{0.53}$

• NSW

The recommended equation for catchments east and west of the Great Dividing Range is:

$k_c = 1.18A^{0.46}$

• Victoria

For Victoria the recommended equations are a function of location and mean annual rainfall.

For regions with mean annual rainfall greater than 800 mm, mainly the eastern part of Victoria:

$k_c = 2.57A^{0.45}$

For regions with mean annual rainfall less than 800 mm, mainly the western part of Victoria:

$k_c = 0.49A^{0.65}$

(more to come)

# kc as a function of dav

Australian Rainfall and Runoff also provides estimation equations as a function of $d_{av}$.  $d_{av}$ is the average flow distance in the channel network of all the sub-area inflows i.e. the average distance from the sub-area inputs to the model outlet.

5.  $k_c = c_{0.8} \times d_{av}$

Where $c_{0.8}$ is approximately:

•  1.2 for coastal south-east Queensland (ARR Book V, Equation 3.24; McMahon and Muller, 1986)
• 1.25 for Victorian catchments (Pearse et al., 2002)
• 1.14 for all Australian catchments from Dyer (1994; Pearse et al., 2002)
• 0.96 for all Australian catchments from Yu (1989; Pearse et al., 2002)

The 0.8 subscript means that the c value applies when the routing parameter m = 0.8.

Table 1 and 2 in Pearse et al. (2002) provides information on the uncertainty in $c_{0.8}$.  These are reproduced below.

There is sufficient information here to calculate prediction intervals for $c_{0.8}$.  Table 1 provides information for logarithmically transformed data (using base 10 logs).  We need to back-transform to get the mean value and prediction intervals for $c_{0.8}$.  Following the approach here and and in Zhou and Gao (1997):

the mean value of $c_{0.8}$ is

$10^{\left ( \mu +\frac{ \sigma^2}{2} \right )}$

The confidence intervals are:

$10^{\left ( \mu + \frac{ \sigma^2}{2} \pm \, s \cdot t_{\alpha (n)} \sqrt{1 + \frac{1}{n} + \frac{s^2}{2(n-1)}} \right )}$

Where $t_{\alpha (n)}$ is the quantile of the t distribution, with n degrees of freedom, corresponding to the required confidence level.

$\alpha = 1 - \frac{1-level}{2}$

The level is commonly set at 90% or 95%.

Given a value of $d_{av}$ we can use these formulas to estimate $k_c$ and its confidence intervals.

A function to perform these calculations is as follows:

Calc_kc <- function(dav, method = c("Victorian", "Yu", "CRCCH"), level = 0.90){
method <- match.arg(method)

if(method == "Victorian") {

mu <- 0.096
s <- 0.219
n <- 39
}

if(method == "Yu") {

mu <- -0.018
s <- 0.306
n <- 119
}

if(method == "CRCCH") {

mu <- 0.057
s <- 0.271
n <- 72
}

# back transformed
est <- dav * 10^(mu + s^2/2)
alpha <- 1 - (1-level)/2
t_quantile <- qt(alpha, df = n)

LCL <- dav * 10^(mu - t_quantile * s * sqrt(1 + 1/n + s^2/(2*(n-1))))
UCL <- dav * 10^(mu + t_quantile * s * sqrt(1 + 1/n + s^2/(2*(n-1))))

return(data.frame(est = est, LCL = LCL, UCL = UCL))

}
Calc_kc(dav = 1, method = "CRCCH", level = 0.95)

#      est    LCL      UCL
#     1.24   0.326     3.99


Therefore, for a $d_{av}$ of 1, using the CRCCH parameters from Pearse et al (2002), the estimated $k_c$ value is 1.27 and the 95% confidence limits are (0.33, 3.99).

The confidence intervals provided in Pearse et al. (2002) (Table 2 above) are approximately 68% intervals (i.e. plus or minus 1 standard deviation) but are biassed because the of the way the back-transformation has been done although any differences are small so it probably doesn’t matter much.

Using an improved method (Cox’s method, see Zhou and Gao, 1997) the expected value of $c_{0.8}$ along with 95% confidence levels is shown in the table below.  These confidence levels are for individual values rather than for the mean value.

Group Mean Lower confidence level Upper confidence level
Victorian 1.32 0.44 3.51
Yu 1.07 0.24 3.90
CRCCH 1.24 0.33 3.99

# Transposing kc between models

The prediction equations above show that $k_c$ is a function of area or $d_{av}$ so if the area or the $d_{av}$ changes, the $k_c$ also needs to be changed.  The scaling will be approximately linear with  $d_{av}$ and roughly proportional to the square root of the area.  Other work shows that $k_c$ is also a function of the number of sub-catchments in the model.

The upshot is that we need to take care when transposing $k_c$ values between models of different sizes.   As noted in the RORB manual (Section 2.2.4.2):

It is important to note that $k_c$ depends on the size of the catchment area.  Consequently, when a $k_c$ value determined in a FIT run is used in a DESIGN run, the same catchment and channel network must be used for both runs, even though the main point of interest might be different in the two runs.

This may be important when using a model with interstation areas.  The $k_c$ estimated for the whole model may not be appropriate for an interstation area.  If it is important to get good results for a smaller area internal to an existing model, it may be necessary to build another RORB model that specifically focuses on this area.  Getting a good calibration at the catchment outlet does not mean that all the internal results are fit for purpose.  There is a good, general discussion of this issue by Grayson et al. (1992) (behind a paywall unfortunately).

Recently RORB has been used in regional flood modelling (Lett and Wilkinson, 2006). We need to be careful about expecting a single model and a single design storm to provide good flow estimates at internal points of a large model especially if calibration data are limited to a single point at the catchment outlet.