---
title: "Procedure of deriving Clark UH"
subject: "Engineering Hydrology"
chapter: 7
key: "semester_5/engineering_hydrology:clark-uh-procedure"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/clark-uh-procedure
site: BCT Engineering
---

# Procedure of deriving Clark UH

Engineering Hydrology · Chapter 7: Flow Routing

- Asked in: 2082 Bhadra, 6 marks; 2070 Asar, 8 marks (asked 2 times)
- Video: [L 7 b Unit Hydrograph Derivation](https://www.youtube.com/watch?v=AdamfBo2Mco) — HEC-HMS
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/clark-uh-procedure
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-7

## Question

Explain the procedure of deriving Clark UH.

## Short answer

Divide basin by isochrones to get time-area diagram; inflow from each strip I = 2.78 A_i/Δt. Route through linear reservoir (S=KQ) using O2 = C0(I1+I2) + C2 O1 with C0=Δt/(2K+Δt), C2=(2K-Δt)/(2K+Δt). Average consecutive ordinates to get Δt-h UH.

## Answer

Clark's method derives a unit hydrograph by passing the excess rainfall through two effects separately: **translation** (movement of water to the outlet, represented by a time-area curve) and **attenuation/storage** (represented by a linear reservoir at the outlet).

#### Data needed

Catchment area $A$, time of concentration $t_c$, storage constant $K$, and the isochrone map (areas between lines of equal travel time to the outlet).

#### Procedure

1. **Delineate isochrones.** Divide the basin into zones by lines of equal travel time to the outlet at interval $\Delta t$ (up to $t_c$). Find the area $A_i$ between successive isochrones by planimeter.
2. **Time-area curve.** Plot the cumulative area against travel time (or the area histogram).
3. **Translation hydrograph.** For 1 cm of excess rain, the inflow from each strip to the outlet reservoir is
$$
I_i = \frac{2.78\,A_i}{\Delta t}\ \text{m}^3/\text{s}\quad (A_i \text{ in km}^2,\ \Delta t \text{ in h})
$$
4. **Find $K$.** Estimate the storage constant $K$ from the recession limb of an observed hydrograph, $K = -Q/(dQ/dt)$ at the inflexion point, or from regional relations.
5. **Route through a linear reservoir** $S = KO$:
$$
O_2 = C_0 (I_1 + I_2) + C_2 O_1,\quad C_0 = \frac{\Delta t}{2K + \Delta t},\quad C_2 = \frac{2K - \Delta t}{2K + \Delta t}
$$
The outflow ordinates form the **instantaneous unit hydrograph (IUH)** of the catchment.
6. **Unit hydrograph of duration $\Delta t$.** Take the average of two successive IUH ordinates:
$$
U_n = \frac{O_n + O_{n-1}}{2}
$$
This is the $\Delta t$-hour unit hydrograph, i.e. the response to 1 cm of excess rain in $\Delta t$ hours.
7. **Use.** Multiply by excess rainfall depths and superpose to get the flood hydrograph for a storm.

#### Flow chart

```text
 Isochrone map -> area per strip
        |
        v
 Inflow I = 2.78 A / dt
        |
        v
 Route through linear reservoir (K)
        |
        v
 IUH ordinates O
        |
        v
 dt-hour UH = average of two IUH ordinates
```

## Check yourself

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

1. Which two effects does Clark's method separate?
   A. Infiltration and evaporation
   B. Translation (time-area) and attenuation (linear reservoir storage)
   C. Interception and depression storage
   D. Rainfall and runoff coefficient
   Answer: B. Translation is represented by the time-area curve and storage by a linear reservoir. (Key point: Two effects)

2. What is done with the isochrone map?
   A. Areas between successive equal-travel-time lines up to t_c give the strip areas A_i
   B. It gives the rainfall at each gauge
   C. It gives K directly
   D. It gives the infiltration rate
   Answer: A. Strip areas between isochrones form the time-area histogram. (Key point: Isochrones)

3. How can the storage constant K be estimated?
   A. From the catchment area alone
   B. From the rainfall depth
   C. From the isochrone interval dt
   D. From the recession limb of an observed hydrograph, K = -Q/(dQ/dt), or regional relations
   Answer: D. K is obtained from recession data or regional relations. (Key point: Estimating K)

4. What is the final step to get the dt-hour unit hydrograph?
   A. Add successive IUH ordinates
   B. Divide IUH by dt squared
   C. Average two successive IUH ordinates
   D. Multiply IUH by C2
   Answer: C. U_n = (O_n + O_n-1)/2 gives the dt-hour UH. (Key point: IUH to UH)

## Related in Engineering Hydrology

- [Linear reservoir and Clark UH from time-area](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Alinear-reservoir-clark-uh) — Engineering Hydrology, chapter 7
- [UH derivation from complex storms](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Auh-complex-storms-procedure) — Engineering Hydrology, chapter 5