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The main points i the stochastic hydrology are listed below:Design Precipitation Hyetographs, Alternating Block Method, Precipitation Depth, Successive Time Intervals, Rainfall Intensity, Multiple Linear Regression, Simple Linear Regression, Sum of Square Errors
Typology: Study notes
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Design precipitation Hyetographs from IDF relationships:
Alternating block method :
successive time intervals of duration Δt over a total
duration T
d
3
Duration
Rainfall intensity
Procedure
return period and duration(t
d
d
additional unit of time Δt.
P
Δt
= P
td
td
4
t
d
t
d
Δt
Obtain the design precipitation hyetograph for a 2-
hour storm in 10 minute increments in Bangalore with
a 10 year return period.
Solution:
The 10 year return period design rainfall intensity for
a given duration is calculated using IDF formula by
Rambabu et. al. (1979)
6
a
n
For Bangalore, the constants are
a = 0.
b = 0.
n = 1.
For T = 10 Year and duration, t = 10 min = 0.167 hr,
7
( )
9
Duration
(min)
Intensity
(cm/hr)
Cumulative
depth (cm)
Incremental
depth (cm)
Time (min)
Precipitation
(cm)
10 13.251 2.208 2.208 0 - 10 0.
20 10.302 3.434 1.226 10 - 20 0.
30 8.387 4.194 0.760 20 - 30 0.
40 7.049 4.699 0.505 30 - 40 0.
50 6.063 5.052 0.353 40 - 50 0.
60 5.309 5.309 0.256 50 - 60 2.
70 4.714 5.499 0.191 60 - 70 1.
80 4.233 5.644 0.145 70 - 80 0.
90 3.838 5.756 0.112 80 - 90 0.
100 3.506 5.844 0.087 90 - 100 0.
110 3.225 5.913 0.069 100 - 110 0.
120 2.984 5.967 0.055 110 - 120 0.
10
10 20 30 40 50 60 70 80 90 100 110 120
Precipita)on (cm)
Time (min)
other independent variables, x
1
, x
2
x
3
, x
4
and so on.
water shed depends on many
factors like rainfall, slope of
catchment, area of catchment,
moisture characteristics etc.
12
y
x
1
x
2
x
3
x
4
these variables
(x
i
, y
i
) are observed values
is predicted value of x
i
Error,
Sum of square errors
13
ˆ
i
y
x
y
Best fit line
ˆ
i i
y = a + bx
ˆ
i i i
e = y − y
( )
( ) { }
2
2
1 1
2
1
n n
i i i
i i
n
i i
i
= =
=
∑ ∑
∑
Estimate the parameters a, b such that
the square error is minimum
i
y
i
y
A general linear model of the form is
y = β
1
x
1
2
x
2
3
x
3
+…….. + β
p
x
p
y is dependent variable,
x
1
, x
2
, x
3
,……,x
p
are independent variables and
β
1
, β
2
, β
3
,……, β
p
are unknown parameters
corresponding n observations on each of the p
independent variables.
15
y
1
= β
1
x
1,
2
x
1,
p
x
1,p
y
2
= β
1
x
2,
2
x
2,
p
x
2,p
y
n
= β
1
x
n,
2
x
n,
p
x
n,p
parameters.
practice n must be at least 3 to 4 times large as
p.
16
1,1 1,2 1,3 1, 1 1
2,1 2,2 2,3 2, 2
2
3,1 3 3
,1 ,1 ,
p
p
n n n p p
n
x x x x
y
x x x x y
x y
x x x
y
β
β
β
β
18
Y is an nx1 vector of observations on the dependent
variable, X is an nxp matrix with n observations on
each p independent variables, Β is a px1 vector of
unknown parameters.
nx
nxp px
i,
=1 for ∀ i, β
1
is the intercept
j
, j = 1….p are estimated by
minimizing the sum of square errors (e
i
19
ˆ
i i i
e = y − y
,
1
ˆ
p
i j i j
j
=
=
∑
inverted.
21
1 1
' ' ' '
1
' '
− −
−
( )
'
X X
( )
1
' '
ˆ
X X X Y
−
Β =
or
( )
1
'
−
matrix is as follows
22
( )
2
,1 ,2 ,1 ,3 ,
1 1 1
' 2
,1 ,2 ,2 ,3 ,
1 1 1
2
,1 ,3 ,2 ,3 ,
1 1 1
n n n
i i i i i
i i i
n n n
i i i i i
i i i
n n n
i i i i i
i i i
x x x x x
X X x x x x x
x x x x x
= = =
= = =
= = =
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
=
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
∑ ∑ ∑
∑ ∑ ∑
∑ ∑ ∑
( )
'
X X
