Applied Hydrology
Rainfall-Runoff Modeling (2)
Professor Ke-Sheng Cheng
Dept. of Bioenvironmental Systems Engineering
National Taiwan University
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
1
Peak flow estimation
Rational method
Empirical method
Unit hydrograph technique
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
2
The concept of Isochrones
Let’s assume that
raindrops fall on a
spatial point x within
the watershed require
an amount of travel time
t(x) to move from point
x to the basin outlet.
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
3
Contour lines of the travel time are termed
as isochrones (or runoff isochrones) of the
watershed.
The highest value of isochrones represents
the time of concentration of the watershed.
Assume the effective rainfall has a constant
intensity (in time) and is uniformly
distributed over the whole watershed, i.e.,
ie ( x, t )  ie  constant
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
4
 If the effective
rainfall duration tr =
2 hours, what is the
peak direct runoff at
the basin outlet?
When will the peak
flow occur? When
will the direct runoff
end?
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
5
The time of peak flow occurrence is
dependent on the relative magnitude of
Ai’s.
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
6
 If the effective
rainfall duration tr =
10 hours, what is the
peak direct runoff at
the basin outlet?
When will the peak
flow occur? When
will the direct runoff
end?
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
7
The watershed storage
effect is neglected.
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
8
Under the assumptions of constant intensity
and uniform spatial distribution for effective
rainfall, the peak direct runoff occurs at the
time t = tc, if the duration of the effective
rainfall is longer than or equal to tc.
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
9
tr  tc
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
10
tr  tc
However, for
storms with
duration shorter
than tc, the peak
flow and its time
of occurrence
depend on relative
magnitude of
contributing areas
(Ai’s).
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
11
Rational method for peak flow
estimation
Consider a rainfall of
constant intensity and
very long duration
occurring uniformly
over a basin. The
runoff gradually
increases from zero to
a constant value as
shown in the following
figure.
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
Qp  ie A  ciA
c  runoff coefficient
12
Practical application of the rational
method for peak flow estimation
Calculating the time of concentration, tc.
Determining the average rainfall intensity i
from the IDF curve using tc as the storm
duration and a predetermined return period
T.
Calculating the runoff coefficient, c.
Calculating the peak direct runoff
Q p  ie A  ciA
c  runoff coefficient
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
In engineering practice, it is normally
restricted to catchments with basin
area not exceeding 15 km2.
13
Unit hydrograph of the modified
rational method
 Original rational method only estimates the peak flow. It
does not yield the complete runoff hydrograph.
tr  tc
tr  tc
tr  tc
2tc
tc+tr
Neglecting the watershed storage effect.
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
14
Synthetic unit hydrographs

For areas where rainfall and runoff data are not
available, unit hydrograph can be developed
based on physical characteristics of the
watershed.



Clark’s IUH (time-area method)
SCS unit hydrograph
Linear reservoir model (Nash model)
15
Clark’s IUH (time-area method)

The concept of isochrones

Isochrones are lines of equal travel time. Any point on
a given isochrone takes the same time to reach the
basin outlet. Therefore, for the following basin
isochrone map and assuming constant and uniform
effective rainfall, discharge at the basin outlet can be
decomposed into individual contributing areas and
rainfalls.
16
Contributing area and contributing
rainfall
17
18

Using the basin isochrone map, the cumulative
contributing area curve can be developed. The
derivatives or differences of this curve constitute
the instantaneous unit hydrograph IUH(t).
Time base of the IUH is the time of
concentration of the watershed.
19
If the effect of watershed storage is to be
considered, the unit hydrograph described
above is routed through a hypothetical linear
reservoir with a storage coefficient k located at
the watershed outlet.
 For a linear reservoir with storage coefficient k,
we have

20

Consider the continuity of the hypothetical
reservoir during a time interval t.
21

Example. A watershed of 1000-acre drainage
area has the following 15-minute time-area
curve. The storage coefficient k of the watershed
is 30 minutes. Determine the 15-minute unit
hydrograph UH(15,t).
22
23

Area between isochrones t = 0 and t = 0.25 hr is
100 acres. Since we are interested in the unit
hydrograph UH(15-min, t), the rainfall intensity in
the time period (0.25 hr) should be 4 inch/hr.
Therefore, the ordinate of UH(15-min, t) at time t
= 0.25 hr is:
4 inch/hr  (1/12 ft/inch)  (1/3600 hr/sec)  100 acres 
(43560 ft2/acre) = 403.33 cfs
24
Synthetic unit hydrographs



Clark’s IUH (time-area method)
SCS unit hydrograph
Linear reservoir model (Nash model)
25
SCS unit hydrograph
Note: tb > tr+tc. The SCS UH takes into
account the storage effect.
26
Misuse of SCS UH – an example
tc=0.5
27
Synthetic unit hydrographs



Clark’s IUH (time-area method)
SCS unit hydrograph
Linear reservoir model (Nash model)
28
Nash’s linear reservoir model

Consider a linear reservoir which has the
following characteristics:
S (t )  KQ(t )
29
30
31
Q(t )  I (1  e
t / k
)
32
33
34
35
Summary of the Conceptual IUH model
Assumption: Linear Reservoir S(t)=kQ(t)
36
Relationship between IUH and the S
curve
RSLAB-NTU
Lab for Remote Sensing Hydrology
and Spatial Modeling
37
38
39
40
41
42
[ A(1  e
t
t
K
)  A(1  e
 ( t  t )
K
1
)]
t
(t  t )
 ( t  t )
t
1
1
t
t
K
K
K
K
K
 A[e  e
]  A[e
e e ]
t
t
t
1
A  t K t K
t
K
K
 Ae (e  1)  e (e  1)
t t
43
44
The Nash Linear Reservoir Model
45
46
The above equation represents a gamma
density function, and thus the integration over 0
to infinity yields 1 (one unit of rainfall excess).
47
Parameter estimation for n and k

The IUH of the n-LR system is characterized by
a function which is identical to the gamma
density.
=M1
=M2
48

The direct runoff can be expressed as a
convolution integral of the rainfall excess and
IUH, i.e.
t
Q(t )   i( )u(t   )d
0
If the direct runoff hydrograph (DRH) is divided
by the total volume of direct runoff, it can be
viewed as a density function. Thus, Q(t ) / VDRH  q(t )
denotes a density function of a random variable t.
 The first moment of this rescaled DRH is


M Q1   tq(t )dt 
0
1
VDRH


0
0
 t
i( )u (t   )ddt
49

The second moment of the rescaled DRH is

M Q 2   t q(t )dt 
0

1
2
VDRH

t
0
2

0
i( )u(t   )ddt
Similarly, the first and second moments of the
rescaled effective rainfall can be respectively
expressed as
M I1 
MI2 
1
VERH
1
VERH


0

ti(t )dt

0
2
t i(t )dt
50

It can be shown that
M Q1  M I 1  nk
MQ2  M I 2  n(n  1)k 2  2nkMI1

Thus, given the ERH and DRH, the parameters
n and k can obtained from the above equations.
51
Calculation of M I 1 and M I 2
52
Calculation of M Q1 and M Q 2
53
坡地開發滯留池之水文設計
Hydrological Analysis for Detention
Pond Design in Hillslope
Development
54
坡地開發水文設計相關規範
55
56
57
58
59
60
61
62
63
64
65
66
現行設計規範之檢討
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
5.75 (60 / 60) / 2  10350
159.14 (15 / 60) 103  (20104 )
 7957
91
92
93
94
95
96
97
98
99
100
101
102
103
逕流量分析
(水土保持技術規範第 十七條 )

洪峰流量之估算，有實測資料時，得採用單位歷
線分析；面積在一千公頃以內者，無實測資料時，
得採用合理化公式(Rational Formula)計算。合理
化公式如下：
1
Qp 
CIA
360
式中，
Q ：洪峰流量
(立方公尺/ 秒),
p
C：逕流係數(無單位),
I：降雨強度(公釐/ 小時),
A：集水區面積(公頃。
)
104
滯洪設施規劃設計原則
(水土保持技術規範第 九十五 條 )
105
滯洪量之估算
(水土保持技術規範第 九十六 條 )
106