2076 Asoj · 10 marks

S-curve: 4-hr UH to 3-hr UH

Short answer

Construct S-curve by adding 4-h UH lagged every4 h, lag S-curve by3 h, multiply difference by4/3. The3-h UH peaks at113.3 m³/s at4 h; ordinates at1-h intervals:0,8.0,48.0,88.0,113.3,101.3,84.0,72.0,62.7,54.7,44.0,36.0,30.7,25.3,20.0,13.3,12.0,6.7,5.3,0,2.0 m³/s.

The ordinates of a 4 hr UH of a basin area of 300 km $^2$ measured at 1-hr intervals are 6, 36, 66, 91, 106, 93, 79, 68, 58, 49, 41, 34, 27, 23, 17, 13, 9, 6, 3 and 1.5 m $^3$ /s respectively. Obtain the ordinates of a 3 hr UH using the S-curve technique.

The 4-h UH is given at 1-h intervals (the ordinate is 0 at t = 0 and 6 m $^3$ /s at t = 1 h). Check: the sum of the ordinates is 826.5 m $^3$ /s, so the volume under the UH is $826.5\times3600 = 2.975\times10^6$ m $^3$ . This is 0.99 cm over 300 km $^2$ , so the UH is a 1 cm UH.

Method

Build the S-curve: $S(t)=U_4(t)+S(t-4)$ (add the 4-h UH repeatedly, each lagged by 4 h).
Lag the S-curve by the new duration of 3 h: $S(t-3)$ .
The difference is the response to 3 h of rain at the 4-h rate (1 cm per 4 h). Scale it by $4/3$ :

U_3(t)=\frac{4}{3}\left[S(t)-S(t-3)\right]

t (h)	$U_4$	$S(t)$	$S(t-3)$	$S(t)-S(t-3)$	$U_3$ (m $^3$ /s)
0	0	0	0	0	0.00
1	6	6	0	6	8.00
2	36	36	0	36	48.00
3	66	66	0	66	88.00
4	91	91	6	85	113.33
5	106	112	36	76	101.33
6	93	129	66	63	84.00
7	79	145	91	54	72.00
8	68	159	112	47	62.67
9	58	170	129	41	54.67
10	49	178	145	33	44.00
11	41	186	159	27	36.00
12	34	193	170	23	30.67
13	27	197	178	19	25.33
14	23	201	186	15	20.00
15	17	203	193	10	13.33
16	13	206	197	9	12.00
17	9	206	201	5	6.67
18	6	207	203	4	5.33
19	3	206	206	0	0.00
20	1.5	207.5	206	1.5	2.00

The S-curve reaches an equilibrium value of about 207 m $^3$ /s, in agreement with $Q_e=2.78A/T_r = 2.78\times300/4 = 208.5$ m $^3$ /s.

Answer: The 3-h UH ordinates at 1-h intervals are 0, 8.0, 48.0, 88.0, 113.3, 101.3, 84.0, 72.0, 62.7, 54.7, 44.0, 36.0, 30.7, 25.3, 20.0, 13.3, 12.0, 6.7, 5.3, 0, 2.0 m $^3$ /s (t = 0 to 20 h), with a peak of 113.3 m $^3$ /s at 4 h. The small ripples at the tail come from rounding of the given ordinates.

✓ Answer checked by Claude Sonnet 5.5

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S-curve: 4-hr UH to 3-hr UH

Method

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