Rating curve equation and parameter estimation

Rating curve

A rating curve is the relation between stage (gauge height) and discharge at a river section.

Standard equation

$$Q = C\,(G - a)^{b}$$

$Q$ = discharge (m$^3$/s), $G$ = gauge reading (m), $a$ = gauge reading at zero discharge (m), $C$ and $b$ = constants of the section ($b$ typically lies between about 1 and 3).

Procedure to estimate the parameters

1. Determine the stage for zero discharge $a$

• Estimate $a$ by extending the smooth stage-discharge curve down to $Q=0$, or by surveying the lowest point of the control.
• Refine it by the three-point formula, with $Q_2=\sqrt{Q_1Q_3}$:

$$a=\frac{G_1G_3-G_2^2}{G_1+G_3-2G_2}$$

• Or use trial values on log-log paper: the correct $a$ gives a straight line.

2. Linearise the equation

$$\log Q = \log C + b\log(G-a)$$

that is $y = c + b\,x$, with $y=\log Q$, $x=\log(G-a)$ and $c=\log C$.

3. Tabulate $x$, $y$, $x^2$, $xy$ and $y^2$ for the $n$ observations.

4. Least-squares estimates

$$b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2},\qquad \log C=\frac{\sum y-b\sum x}{n}$$

5. Correlation coefficient

$$r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}$$

A value close to 1 means a good fit.

6. Graphical alternative: plot $Q$ against $(G-a)$ on log-log paper and draw the best straight line; the slope is $b$ and the intercept at $G-a=1$ gives $C$.

7. Validate: plot the fitted curve over the observed points, check it with new discharge measurements, and revise it when the control shifts.