---
title: "Rating curve equation, zero-discharge stage 161.0 m"
subject: "Engineering Hydrology"
chapter: 4
key: "semester_5/engineering_hydrology:rating-curve-equation-161-0"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/rating-curve-equation-161-0
site: BCT Engineering
---

# Rating curve equation, zero-discharge stage 161.0 m

Engineering Hydrology · Chapter 4: Surface Runoff

- Asked in: 2069 Chaitra, 8 marks
- Video: [Stage Discharge Curve in Excel](https://www.youtube.com/watch?v=4HliZ7c1xn4) — NyaNew
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/rating-curve-equation-161-0
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-4

## Question

The stage and discharge data of a river are given below. Derive the equation of rating curve (stage-discharge relationship) to predict the discharge for a given stage. Assume the value of stage for zero discharge as 161.0 m.

| Stage (m) | 161.3 | 161.7 | 161.9 | 162.8 | 163.4 | 163.8 | 164.5 | 165.4 | 165.7 |
|---|---|---|---|---|---|---|---|---|---|
| Discharge (m$^3$/s) | 30 | 120 | 210 | 450 | 650 | 825 | 900 | 1000 | 1050 |

## Short answer

Assuming Q = C(G-161.0)^b, log transformation and linear regression give C = 185.0 and b = 1.277. The rating equation is Q = 185 (G - 161.0)^1.28 m³/s.

## Answer

Use the power form $Q = C(G-a)^b$ with $a = 161.0$ m. In logarithms: $\log Q = \log C + b\log(G-a)$, a straight line $y = c + b\,x$.

| G (m) | G - a | Q | $x=\log(G-a)$ | $y=\log Q$ | $x^2$ | $xy$ |
|---|---|---|---|---|---|---|
| 161.3 | 0.3 | 30 | -0.5229 | 1.4771 | 0.2734 | -0.7724 |
| 161.7 | 0.7 | 120 | -0.1549 | 2.0792 | 0.0240 | -0.3221 |
| 161.9 | 0.9 | 210 | -0.0458 | 2.3222 | 0.0021 | -0.1063 |
| 162.8 | 1.8 | 450 | 0.2553 | 2.6532 | 0.0652 | 0.6773 |
| 163.4 | 2.4 | 650 | 0.3802 | 2.8129 | 0.1446 | 1.0695 |
| 163.8 | 2.8 | 825 | 0.4472 | 2.9165 | 0.2000 | 1.3041 |
| 164.5 | 3.5 | 900 | 0.5441 | 2.9542 | 0.2960 | 1.6073 |
| 165.4 | 4.4 | 1000 | 0.6435 | 3.0000 | 0.4140 | 1.9304 |
| 165.7 | 4.7 | 1050 | 0.6721 | 3.0212 | 0.4517 | 2.0305 |
| **Sum** | | | 2.2187 | 23.2365 | 1.8709 | 7.4184 |

With $n = 9$ and $\sum y^2 = 62.2104$:
$$
b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}=\frac{9(7.4184)-(2.2187)(23.2365)}{9(1.8709)-(2.2187)^2}=1.277
$$
$$
\log C=\frac{\sum y-b\sum x}{n}=2.2671\;\Rightarrow\;C=185.0
$$

**Rating equation:** $Q = 185.0\,(G-161.0)^{1.277}$ (m$^3$/s, $G$ in m)

The correlation coefficient is $r = 0.986$, so the fit is good.

Check: at $G=163.4$ m, $Q = 185.0\times2.4^{1.277} = 566$ m$^3$/s (observed 650).

**Answer: $Q = 185(G-161.0)^{1.28}$.**

## Check yourself

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

1. Which equation form is fitted to the stage-discharge data?
   A. Q = C(G-a)^b
   B. Q = C + b(G-a)
   C. Q = C e^(bG)
   D. Q = C(G+a)^b
   Answer: A. The rating curve is the power form Q = C(G-a)^b, with a the stage at zero discharge. (Key point: Power form)

2. Why are logarithms taken before fitting?
   A. It removes the need for the zero-discharge stage a
   B. log Q = log C + b log(G-a) is a straight line, so least squares can be used
   C. It makes all the discharge values equal
   D. It converts stage into velocity
   Answer: B. Taking logs turns the power law into y = c + b x, a straight line that can be fitted by linear regression. (Key point: Linearising)

3. In the line y = c + bx, what are x and y?
   A. x = log Q, y = log(G-a)
   B. x = G-a, y = Q
   C. x = log(G-a), y = log Q
   D. x = G, y = log Q
   Answer: C. x is log of the stage above zero-flow stage, y is log of discharge; b is the slope. (Key point: Regression variables)

4. What rating equation does the answer obtain?
   A. Q = 161(G-185)^1.28
   B. Q = 1.28(G-161.0)^185
   C. Q = 185(G-161.0)^2.27
   D. Q = 185(G-161.0)^1.28
   Answer: D. Least squares gave b = 1.277 and log C = 2.2671, so C = 185 and Q = 185(G-161.0)^1.28. (Key point: Result)

## Related in Engineering Hydrology

- [Stage at zero discharge from rating data](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Astage-zero-discharge-rating-20-80) — 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×
- [Rating curve and stage for zero discharge](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Arating-curve-zero-stage) — 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
- [How stage-discharge relationship is established](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Astage-discharge-established) — Engineering Hydrology, chapter 4
- [Stage for zero discharge with sketch](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Astage-zero-discharge-sketch) — Engineering Hydrology, chapter 4

## Related in other subjects

- [Rankine's table, IP 2+450 km, 30°, 3° curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Arankine-ip-2450-deflection-30-3deg) — Surveying II, chapter 6
- [Stage discharge curve and its application](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Astage-discharge-curve) — Hydro Power, chapter 2
- [Source P and Q, Ean = -120∠210° V](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Asource-power-ean-120-210) — Power System Analysis I, chapter 3
- [River stage and water table, three grids](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Ariver-stage-water-table-numerical) — Computational Techniques in Civil Engineering, chapter 7