---
title: "Rating curve equation and parameter estimation"
subject: "Engineering Hydrology"
chapter: 4
key: "semester_5/engineering_hydrology:rating-curve-equation-parameters"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/rating-curve-equation-parameters
site: BCT Engineering
---

# Rating curve equation and parameter estimation

Engineering Hydrology · Chapter 4: Surface Runoff

- Asked in: 2067 Mangsir (old course), 1+1+8 marks
- Video: [Developing stream Rating Curve Equation and curve Using EXCEL [ Step By Step ]](https://www.youtube.com/watch?v=CSPLLGA4c3M) — Hydraulics and Hydrology Tutorials
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/rating-curve-equation-parameters
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-4

## Question

What is a rating curve? Write down a standard equation for a rating curve. Explain in detail the procedure to estimate the parameters of that rating equation.

## Short answer

A rating curve is a stage-discharge relation expressed as Q = C (G - a)^b. Parameters are estimated by determining the zero-discharge stage a via extrapolation or three-point method, then linearising to log Q = log C + b log(G - a) and applying least-squares regression to find b and C, validated by plotting.

## Answer

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is a rating curve?
   A. The relation between rainfall and runoff
   B. The relation between velocity and depth
   C. A curve of annual flood against return period
   D. The relation between stage (gauge height) and discharge at a river section
   Answer: D. It lets stage records be converted to discharge. (Key point: Definition)

2. In Q = C(G-a)^b, what does a represent?
   A. The gauge reading at zero discharge
   B. The cross-sectional area
   C. The slope of the river
   D. The correlation coefficient
   Answer: A. a is the stage at which flow stops, the level of the control. (Key point: Equation)

3. After finding a, how are b and C estimated?
   A. Read them directly from a rainfall chart
   B. Linearise to log Q = log C + b log(G-a) and use least squares for b and log C
   C. Choose b = 1 and fit C by eye
   D. Use Manning's equation
   Answer: B. The slope of the log-log line is b and the intercept gives log C. (Key point: Parameter estimation)

4. What does the correlation coefficient r tell you?
   A. r near 0 means a good fit
   B. r gives the value of b
   C. A value close to 1 means a good fit
   D. r gives the zero-flow stage
   Answer: C. r measures how well the straight line fits the log-log data. (Key point: Goodness of fit)

## Related in Engineering Hydrology

- [Rating curve equation, zero-discharge stage 161.0 m](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Arating-curve-equation-161-0) — Engineering Hydrology, chapter 4
- [Rating curve equation and daily discharge hydrograph](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Arating-curve-equation-daily-hydrograph) — Engineering Hydrology, chapter 4
- [DAD curve procedure and isohyetal area data](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Adad-curve-procedure-isohyetal) — Engineering Hydrology, chapter 2
- [Rating curve definition and uses](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Arating-curve-define-uses) — Engineering Hydrology, chapter 4, asked 2×
- [Zero-discharge stage of a smooth rating curve](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Astage-zero-discharge-23-37) — Engineering Hydrology, chapter 4, asked 2×
- [Current meter discharge, rating equation](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Acurrent-meter-0-53n) — Engineering Hydrology, chapter 4

## Related in other subjects

- [Components of detail estimate report (road)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Adetail-estimate-report-road) — Estimating and Costing, chapter 5
- [Detail estimate report for sewerage project](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Aestimate-report-sewerage) — Estimating and Costing, chapter 5, asked 2×
- [Quadrilateral speed-time curve on down gradient](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Aquadrilateral-curve-down-gradient-energy) — Utilization of Electrical Energy, chapter 4
- [Hyperbola: locus and standard equation](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ahyperbola-standard-equation-locus) — Mathematics I, chapter 3, asked 6×