STOCHASTIC HYDROLOGY Lecture -34

Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.

INDIAN  INSTITUTE  OF  SCIENCE  

2  

Summary  of  the  previous  lecture  

•  Multivariate stochastic models –  Cross correlation

–  Two site Markov model

( ) ( ) ( )2, , , ,0 1 0hh t h j h j t j t h j h

j

sX x r X x u s rs

= + − + −

( )( )( )

( )

, ,1

,

n

j i j h i k hi

j hj h

x x x xr k

n k s s

+=

− −=

−

∑

Multisite Markov model: •  Multisite generation requires simultaneous

generation of data at several sites while preserving the correlation between the data at various sites.

•  Consider xj,t

Multivariate Stochastic models

3  

( ),,

dj t j

j tj

x xx

s−

=

•  The first order Markov model for site h is

•  The first order Markov model for site j is

•  The equations are written in matrix form

Multivariate Stochastic models

4  

( ) ( )2, 1 , , 11 1 1h t h h t h t hx xρ ε ρ+ += + −

( ) ( )2, 1 , , 11 1 1j t j j t j t jx xρ ε ρ+ += + −

( ) 21 1 1 11t x t x t xX X uµ ρ µ σ ρ+ += + − + −

µ= 0 and σ =1 because it is standardized data

where Xt is a p x 1 vector of standardized values of the

variable generated at time t, E is a p x p diagonal matrix whose jth diagonal element

is ρj(1), G is a p x p diagonal matrix whose jth diagonal element

is

ε is a p x 1 vector of random variates

Multivariate Stochastic models

5  

1t tX EX Gε+ = +

( )21 1jρ−

•  ε is defined to preserve the first order serial correlation (auto correlation) of the xj‘s and the lag zero cross-correlation between xj and xh .

•  ε is made of elements that are εj,t+1; each εj,t+1 is independent of xj,t ; εj is N(0,1)

•  The cross correlation between εj and εh is ρ*j,h(0),

•  ρ*j,h(0) reproduces the desired ρj,h(0), which is the lag zero cross correlation between xj and xh .

Multivariate Stochastic models

6  

( )( ) ( ){ } ( )

( ){ } ( ){ },*

, 2 2

1 1 1 00

1 1 1 1

j h j hj h

j h

ρ ρ ρρ

ρ ρ

−=

− −

where is a p x p diagonal matrix whose jth diagonal

element is the square root of the jth largest eigenvalue of the p x p correlation matrix whose elements are ρ*j,h(0)

A is a p x p matrix consisting of eigenvectors of correlation matrix,

e is p x 1 vector of independent observations from N(0,1)

Multivariate Stochastic models

7  

12AD eλε =

12Dλ

•  Matalas (1967) has given a multisite normal generation model that preserves the mean, variance, lag one serial correlation, lag one cross-correlation and lag zero cross-correlation.

where Xt and Xt+1 are p x 1 vectors representing standardized

data corresponding to p sites at time steps t and t+1 resp.

Multivariate Stochastic models

8  

1 1t t tX AX Bε+ += +

Ref.: Matalas, N.C. (1967) Mathematical assessment of synthetic hydrology, Water Resources Research 3(4):937-945

Assumption is that the model is multivariate normal.

εt+1 is N(0,1); p x 1 vector with εt+1 independent of Xt. A and B are coefficient matrices of size p x p. B is

assumed to be lower triangular matrix

Multivariate Stochastic models

9  

( )( )

( )

( )

1

1, 12, 1.., 1.., 1

t

x tx t

Xx i t

x p t

+

⎡ ⎤+⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

+⎢ ⎥⎣ ⎦

( )( )

( )

( )

1,2,..,..,

t

x tx t

Xx i t

x p t

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

( )( )

( )

( )

1

1, 12, 1.., 1.., 1

t

tt

i t

p t

ε

ε

ε

ε

ε +

⎡ ⎤+⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

+⎢ ⎥⎣ ⎦

•  The scalar form is

where ai,j and bi,j denote the (i , j)th elements of the matrices

A and B.

Multivariate Stochastic models

10  

( ) ( ), 1 , ,1 1

, , 1p i

i t i j i jj j

x a x j t b i tε+= =

= + +∑ ∑

Coefficient matrices A and B: •  The expectation of Xt Xt

’ is denoted by M0

If m0(i , j) is a element of M0 matrix (size p x p) in the ith row and jth column,

Multivariate Stochastic models

11  

'0 t tM E X X⎡ ⎤= ⎣ ⎦

( ) ( ) ( )01

1, , ,n

tm i j x i t x j t

n =

= ∑

( ) ( ) ( )0 , , ,m i j E x i t x j t⎡ ⎤= ⎣ ⎦

Expected value of a matrix is matrix of expected values of individual elements

n is no. of time periods

Jayarami Reddy, P. Stochastic Hydrology, Laxmi publications, New Delhi, 1997

i.e., m0(i , j) is correlation coefficient between the data at sites i and j at time t. Therefore M0 is the cross-correlation matrix of lag zero

Multivariate Stochastic models

12  

( ) ,,0

1

1,n

j t ji t i

t i j

Q QQ Qm i j

n s s=

⎛ ⎞⎛ ⎞ −−= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠∑

Q is the original random variable before standardization e.g., stream flow

•  The expectation of Xt Xt-1’ is denoted by M1

If m1(i , j) is a element of M1 matrix (size p x p) in the ith row and jth column,

Multivariate Stochastic models

13  

'1 1t tM E X X −⎡ ⎤= ⎣ ⎦

( ) ( ) ( )12

1, , , 11

n

tm i j x i t x j t

n =

= −− ∑

( ) ( ) ( )1 , , , 1m i j E x i t x j t⎡ ⎤= −⎣ ⎦Expected value of a matrix is matrix of expected values of individual elements

n is no. of time periods

i.e., m1(i , j) represents lag one cross correlation between the data at sites i and j . Therefore M1 is the cross-correlation matrix of lag one.

Multivariate Stochastic models

14  

( ) , 1,1

2

1,1

nj t ji t i

t i j

Q QQ Qm i j

n s s−

=

⎛ ⎞⎛ ⎞ −−= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠

∑

Q is the original random variable before standardization e.g., stream flow

Considering the model,

Post multiplying with Xt’ on both sides and taking the

expectation,.

Multivariate Stochastic models

15  

1 1t t tX AX Bε+ += +

' ' '1 1t t t t t tE X X AE X X BE Xε+ +⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1 01

1 0

0M AMA M M −

= +

=

0

Post multiplying with Xt+1’ on both sides and taking the

expectation,.

Multivariate Stochastic models

16  

1 1t t tX AX Bε+ += +

' ' '1 1 1 1 1t t t t t tE X X AE X X BE Xε+ + + + +⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0M

Multivariate Stochastic models

17  

'1 1t tM E X X −⎡ ⎤= ⎣ ⎦

{ }{ }

'' '1 1

''1

'1

t t

t t

t t

M E X X

E X X

E X X

−

−

−

⎡ ⎤= ⎣ ⎦

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎡ ⎤= ⎣ ⎦

' '1 1t tM E X X +⎡ ⎤= ⎣ ⎦

or

Multivariate Stochastic models

18  

Taking expectation on both sides,

{ }''1 1 1 1

' ' ' '1 1 1

t t t t t

t t t t

X AX B

X A B

ε ε εε ε ε

+ + + +

+ + +

= +

= +

' ' ' ' '1 1 1 1 1

' ' ' '1 1 1

'

'

0

t t t t t t

t t t t

E X E X A B

E X A E B

IBB

ε ε ε ε

ε ε ε+ + + + +

+ + +

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦

= +

=

Multivariate Stochastic models

19  

Substituting in the equation,

' ' '1 1 1 1 1t t t t t tE X X AE X X BE Xε+ + + + +⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

' '0 1M AM BB= +

11 0A M M −=1 ' '

0 1 0 1' 1 '

0 1 0 1

M M M M BBBB M M M M

−

−

= +

= −

If C = BB’ 1 '

0 1 0 1C M M M M−= −

•  B does not have a unique solution. •  One method is to assume B is a lower triangular

matrix.

Multivariate Stochastic models

20  

( )( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )

'

1,1 0 0 . . 0 1,1 2,1 . . . ,12,1 2,2 0 . . 0 0 2,2 . . . , 2. . . . . . . .. . . . . . . .,1 ,2 . . . , 0 0 . . . ,

b b b b pb b b b p

BB

b p b p b p p b p p

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )

1,1 1,2 1,3 . . 1,2,1 2,2 2,3 . . 2,. . . . . .. . . . . .,1 ,2 . . . ,

c c c c pc c c c p

C

c p c p c p p

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

•  The diagonal elements of the B matrix are obtained as,

Multivariate Stochastic models

21  

( ) ( )

( ) ( ) ( ){ }

12

12 2

1,1 1,1

2,2 2,2 2,1

b c

b c b

=

= −

.

.

.

( ) ( ) ( ) ( ) ( ){ }1

2 2 2 2, , , 1 , 2 ... ,1b k k c k k b k k b k k b k= − − − − − −

•  The elements in the kth row are obtained as,

Multivariate Stochastic models

22  

( ) ( )( )

( ) ( ) ( ) ( )( )

,1,1

1,1

,2 2,1 ,1,2

2,2

c kb k

b

c k b b kb k

b

=

−=

.

.

. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ), ,1 ,1 ,2 ,2 ...... , 1 , 1

,,

c k j b j b k b j b k b j j b k jb k j

b j j− − − −

=

•  Assumption is that the model is multivariate normal.

Multivariate Stochastic models

23  

The annual flow in MCM at two sites X and Y is given below. Generate the first two values of data from these two sites.

24  

Example – 1

Year 1 2 3 4 5 6 7 8 9 10 Annual flow at site X (MCM) 4946 7017 6653 6355 5908 5327 4548 3556 3852 5319

Annual flow at site Y (MCM) 5142 6240 5648 5977 6008 5045 4630 4604 4250 6182

Year 11 12 13 14 15 16 17 18 19 Annual flow at site X (MCM) 4631 5746 5111 5419 6060 7336 3736 3780 6034

Annual flow at site Y (MCM) 4703 6582 5461 5288 5440 7546 4634 4823 5577

M0 matrix is cross covariance matrix of lag zero

25  

Example – 1 (Contd.)

Site X Y

Mean 5333 5462

Std.dev. 1125.1 823.5

( ) ( )( ) ( )

, ,0

, ,

0 00 0

X X X Y

Y X Y Y

r rM

r r⎡ ⎤

= ⎢ ⎥⎣ ⎦

X

Y

X Y

M1 matrix is lag one cross covariance matrix

26  

Example – 1 (Contd.)

( )( )( )

( )

, ,1

, 0

n

X i X Y i Yi

X YX Y

x x x xr

n s s=

− −=∑

0

1 0.7960.796 1

M ⎡ ⎤= ⎢ ⎥⎣ ⎦

( ) ( )( ) ( )

, ,1

, ,

1 11 1

X X X Y

Y X Y Y

r rM

r r⎡ ⎤

= ⎢ ⎥⎣ ⎦

X

Y

X Y

27  

Example – 1 (Contd.)

( )( )( )

( )

, , 11

, 1 1

n

X i X Y i Yi

X YX Y

x x x xr

n s s

+=

− −=

−

∑

1

0.302 0.1640.02 0.118

M ⎡ ⎤= ⎢ ⎥−⎣ ⎦

11 0A M M −=

10

2.73 2.172.17 2.73

M − −⎡ ⎤= ⎢ ⎥−⎣ ⎦

28  

Example – 1 (Contd.)

0.47 0.210.31 0.37

A−⎡ ⎤

= ⎢ ⎥−⎣ ⎦

1 '0 1 0 1C M M M M−= −

0.89 0.760.76 0.95

C ⎡ ⎤= ⎢ ⎥⎣ ⎦

29  

Example – 1 (Contd.)

( ) ( ) ( )1 12 21,1 1,1 0.89 0.94b c= = =

( ) ( ) ( )( )1,1 0.89, 1,2 2,1 0.76,

2,2 0.95

c c c

c

= = =

=

( ) ( )( )2,1 0.762,1 0.811,1 0.94

cb

b= = =

( ) ( ) ( ){ }

{ }

12 2

12 2

2,2 2,2 2,1

0.95 0.81 0.54

b c b= −

= − =

30  

Example – 1 (Contd.)

( ) ( ) ( )1,1 0.94, 2,1 0.81, 2,2 0.54b b b= = =

0.94 00.81 0.54

B ⎡ ⎤= ⎢ ⎥⎣ ⎦

1 1t t tX AX Bε+ += +

, 1 , , 1

, 1 , , 1

0.47 0.21 0.94 00.31 0.37 0.81 0.54

X t X t X t

Y t Y t Y t

x xx x

ε

ε+ +

+ +

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

31  

Example – 1 (Contd.)

The initial value of xX,0 and xY,0 are considered as zero εX,1 = -0.134 and εY,1 = -0.268

,1 ,0 ,1

,1 ,0 ,1

0.47 0.21 0.94 00.31 0.37 0.81 0.54

X X X

Y Y Y

x xx x

ε

ε−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

,1

,1

0.94 0 0.1340.81 0.54 0.268

0.1260.254

X

Y

xx

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

32  

Example – 1 (Contd.)

εX,2 = 1.639 and εY,2 = 0.134

,2 ,1 ,2

,1 ,1 ,2

0.47 0.21 0.94 00.31 0.37 0.81 0.54

X X X

Y Y Y

x xx x

ε

ε−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1.5431.449⎡ ⎤

= ⎢ ⎥⎣ ⎦

,2

,1

0.47 0.21 0.126 0.94 0 1.6390.31 0.37 0.254 0.81 0.54 0.134

X

Y

xx

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

33  

Example – 1 (Contd.)

Generated annual flow at Site X : QX,1 = 5333 – 0.126*1125.1 = 5191.2 MCM QX,2 = 5333 + 1.543*1125.1 = 7069 MCM Similarly at Site Y : QY,1 = 5462 – 0.254*823.5 = 5252.8 MCM QY,2 = 5462 + 1.449*823.5 = 6655.3 MCM

Site X Y Mean 5333 5462

Std.dev. 1125.1 823.5

( ),1 ,1X X XQ x x X s= +

( ),2 ,2X X XQ x x X s= +

CASE STUDIES

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