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IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 37

SINGULAR VECTOR FOR A SHALLOW WATER MODEL  

CHULARAT SRIKUL AND DUSSADEE SUKAWAT  

Department of Mathematics, Faculty of Science, 

King Mongkut's University of Technology Thonburi, 

Bangkok 10140, Thailand. 

chularat.s@hotmail.com; dusadee.suk@kmutt.ac.th  

ABSTRACT: The atmosphere is a chaotic dynamical system, small errors in the initial 

condition of an atmospheric model can lead to growing errors in the forecast. In order to 

reduce uncertainty in the forecast which occur from using only one initial condition from 

many possible initial conditions, more initial conditions are required. In practice, 

however, there is only single observation available for the initial condition. In this paper 

the singular vector has been applied to a single level shallow water model of the 

atmosphere to generate more initial conditions from one observation.  

ABSTRAK: Atmosfera ialah satu sistem dinamik yang huru-hara. Kesilapan kecil dalam 

kondisi awal model atmosfera boleh membawa kepada kesilapan yang semakin 

meningkat  di dalam ramalan. Dalam usaha untuk mengurangkan ketidakpastian dalam 

ramalan yang berlaku dari menggunakan hanya satu kondisi awal, lebih banyak kondisi 

awal diperlukan. Secara praktikalnya, walau bagaimanapun, hanya terdapat satu 

pemerhatian yang sedia ada untuk kondisi awal. Kajian ini menerangkan vektor tunggal 

telah digunakan untuk model air cetek tahap atmosfera untuk menghasilkan lebih banyak 

kondisi awal dari satu pemerhatian. 

KEYWORDS: shallow water model; tangent linear model; adjoint linear model; singular 

vector 

1. INTRODUCTION  

The results from numerical models of the atmosphere can sometimes contain large 

errors. A reason for this lies in the nature of the atmosphere in which very small errors in 

the initial conditions can lead to large errors in the forecast. Implying a perfect forecast 

system cannot be created because every detail of the initial state of the atmosphere cannot 

be observed. To cope with this uncertainty, ensemble forecasting which is a collection of 

difference forecasts and all valid at the same forecast time is applied. Kalney [1] discussed 

a simple model by using the singular vector technique to create ensemble forecasting. Wei 

and Frederiksen [2] used lyapunov vectors and singular vectors to examine errors growth 

and predictability in atmospheric models. Khade and Hansen [3] found that the singular 

error growth in the most parts of the attractor is rapid, in that errors can grow almost by a 

factor of 3 over one step.  

In this paper the singular vector method is applied to a linearized shallow water model 

of the atmosphere to generate more initial conditions from the initial data. 

2. METHODOLOGY 

2.1 The Shallow Water Model 



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 38

The shallow water model is a simplified version of the equations of motion. The 

momentum, hydrostatic and continuity equations are given below [4]. 

 

      (1) 

 

 

(2) 

 

  

(3) 

 

  

(4) 

 

where r  is the density of the atmosphere, p is the pressure, f is the Coriolis parameter 

which is given by 2 sin= Ω θf  ( Ω  is the angular velocity of the earth, θ is the latitude), g  is 

the acceleration due to gravity and ,u v and w  are the speed of fluid in the ,x y  and z

directions, respectively. 

The shallow water model consists of a set of three equations with three dependent 

variables ( ), , φu v . 

    

(5) 

   

(6) 

   

(7) 

  

where φ  is the geopotential  height.  

  The vertical velocity does not appear explicitly. The total time derivative is given by 

   

(8) 

 

2.2  The Singular Vector Method 

Singular vector (SV) method is explained here according to Kalney [1]. The initial 

conditions for the ensemble prediction system (EPS) are created to represent the 

uncertainties in the operational analysis. They are made by adding to the operation 

analysis perturbations which produce the fastest growth during the first stage of the 

forecast period, defined using the singular vector technique. The purpose is to find those 

perturbations to a given initial state which grow most rapidly [5]. The calculation of the 

singular vectors is the tangent - linear version of the full (non -linear) operational forecast 

1
0

∂
− + =

ρ ∂

du p
fv

dt x

1
0

∂
+ + =

ρ ∂

dv p
fu

dt y

0
∂

+ ρ =
∂

p
g

z

0
∂ ∂ ∂

+ + =
∂ ∂ ∂

u v w

x y z

0
∂φ

− + =
∂

du
fv g

dt x

0
∂φ

+ + =
∂

dv
fu g

dt y

0
φ

+ φ∇ ⋅ =
H

d
V

dt

.
d

u v
dt t x y

∂ ∂ ∂
= + +

∂ ∂ ∂



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 39

model. A second model, the adjoint of the tangent linear model is also needed in the 

singular vector calculation. 

a)  The Tangent Linear Model Method (TLM) 

Consider a non linear model 

(9) 

where 
0

( )x t is the state at the initial time and ( )x t is the state at time t  and M is the time 

integration of the numerical scheme from the initial condition to time t . 

The tangent linear model (TLM) is defined such that 

 

(10) 

where L is the tangent linear model of M . 

So the TLM evolves a perturbation in time; 

(11) 

where 
0

( )y t is the perturbation state at the initial time 0=t  and ( )y t is the perturbation state 

at time t  [3].  

b)  The Adjoint of the Tangent Linear Model 

The adjoint tangent linear model is the transpose of the tangent linear model. 

c)  The Singular Vector 

Singular value decomposition theory indicates that for any matrix L there exist two 

orthogonal matrices ,U V such that  

 (12) 

where

 

1

2

0 0

0 0

0 0

σ 
 

σ
 =
 
 

σ 

K

K

M M M

K
n

S  and   =TUU I , =TVV I  

S  is a diagonal matrix whose elements are the singular values of L and I  is the identity 

matrix. 

So that 

(13) 

The vector 
1

( , , )K
n

v v are the right singular vectors of L .  

Consequently 

(14) 

 

0
( ) [ ( )]=x t M x t

∂
=

∂

M
L

x

0 0
( ) ( , ) ( )=y t L t t y t

=
T

U LV S

.= σ
i i i

Lv u

= σ
T

i i i
L u v



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 40

The vector 
1

( , , )K
n

u u are the left singular vectors of L . 

Therefore, the initial singular vectors can be obtained as the eigenvectors of TL L , the 

final singular vectors can be obtained as the eigenvectors of TLL and eigenvalues are the 

squares of the singular values. 

Consider a basic trajectory  

 

  

(15) 

 

  

(16) 

Applying the tangent linear model L in (11) and (15) and rearrange in the form 

(17) 

Applying the adjoint of the tangent linear model TL  in (11) and (16)  

 
(18) 

Applying TL L by running tangent linear model forward in time, and then the adjoint 

backward in time. 

 
(19) 

 

2.3  The Numerical Method of the Shallow Water Model 

The forward in time center in space finite difference method  is used to solve the 

shallow water equations on Arakawa - A grid. 

a)  Discretized the Shallow Water Model 

 

 

 

                                                                                                                                                                                   (20) 

where 
2

∆
α =

∆

t

x
,

2

∆
β =

∆

t

y
 

 

b)  Initial Condition 

For this research the initial conditions are  

                    
  

(21) 

 
                   

  

(22) 

 

0 0

1

( ) ( ),
=

= < >∑
n

i i

i

y t y t v v

1 1

1

( ) ( ), .
n

i i

i

y t y t u u
=

= < >∑

1 0
( ), ( ),

i i i
y t u y t vσ< >=< >

0 1
( ), ( ),< σ >=< >

i i i
y t v y t u

2

1 0
( ), ( ),< σ >= σ < >

i i i i
y t u y t v

( ) ( ) ( )1 1 1 1 1 1 1
+

+ − + − + −
= − α − − β − + ∆ − α φ − φ

j j j j j j j j j j j

i i i i i i i i i i i
u u u u u v u u tfv g

( ) ( ) ( )1 1 1 1 1 1 1
+

+ − + − + −
= − α − − β − − ∆ − β φ − φ

j j j j j j j j j j j

i i i i i i i i i i i
v v u v v v v v tfu g

( ) ( ) ( ) ( )j 1 j j j j j j j j j j j j ji i i i 1 i 1 i i 1 i 1 i i 1 i 1 i i 1 i 1u v u u v v
+

+ − + − + − + −φ = φ − α φ − φ −β φ − φ − αφ − −βφ −

0

φ
= −

d
u g f

dy

0

φ
=

d
v g f

dx



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 41

                    
 (23) 

 

where    

 

2
( ) * ( 0 * ( 0) 0.5* * ( 0) )= − −φ + −
uBar

Bar f Y y beta Y y
g

5500 φ =Ref is the constant reference 

geopotential height,  = 500φAmp is the constant amplitude geopotential height. 1=mx and 

1=my are wave numbers in x and y directions, respectively. Lx and Ly are channel lengths 

in x and y directions, respectively. 

c)  Boundary Condition 

The boundary condition in the east-west direction is a cyclic boundary and the north-south 

direction is an updated boundary.  

d)  Experiment  

The domain of the model is 632 10× meters in x-direction and 68 10×  meters in y-direction, 

grid size in x-direction is 58 10×  meters and grid size in y-direction is 58 10×  meters, time 

step is 600 seconds. The output interval from the forecast model is 6 hours.  

2.4  Step of Calculation the Singular Vector 

Step1: The tangent linear model 

From (10) for the shallow water model is calculated as  

 

  

                                                                                                                                          (24) 

 

 

The discretized form of (24) is then used to calculate the tangent linear model at each of 

grid point. 

Step2: The adjoint linear tangent model is the transpose of the  linear tangent model. 
Step3: Calculating the singular vector and singular value from (13) and (14) have to be 

repeated many times to generate several perturbations. 

The maximum norm of perturbations and minimum norm  of perturbations are then added 

to the initial condition to generate the new initial conditions. 

In this paper the norm of vector is defined as 

(25)                        

For example, at a point where the initial pertubations  
0

0.1871

( ) 0.4344

277.2987

− 
 

=  
  

y t  a single cycle of L and 

T
L transformations result in 

1

-10

1.7712

( ) 0.0629

3.8762e

 
 

=  
 
 

y t . 

0
 =   + +φ φ φ φWave Bar Ref

* sin(2 * * * / ) * cos( * * ( 0) / )φ = φ −Wave Amp pi mx X Lx my pi Y y Ly

2 2 2 2

2 2 2 2

2 2 2 2 2

 ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + ⋅ − 

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 
 ∂ ∂ ∂ ∂ ∂ ∂ ∂

= = + + + + 
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 

 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + + + + 

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂  

i

u u u u u v z
u v f g

u t u x x u y y y z x

M v v v v v z
L f u v g

U x v t v x v y y z y

z u z v z z z u v
z z u v

x u x y v y z t z x z y x y

2 2 2
= + + φ

i
y u v



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 42

3. RESULTS AND DISCUSSION 

3.1   Result of  the shallow water model 

Figure 1 shows the velocity and geopotential field following (21) and (22). Figure 2 

shows the result of forecast at 6 hours. 

 

Fig. 1: The initial state of the velocity field (vectors) 

and geopotential heights (contour lines). 

 

Fig. 2:  The forecast state of the velocity field (vectors)  

and geopotential heights (contour lines) at 6 hours. 

Table 1: 

Examples of singular vectors and singular values of perturbation at grid point (2,3). 

State 

 

Singular vector 

 Perturbation 

v1 v2 v3 

I 
5

0.859

0.511

6.907e
−

 
 

− 
 − 

 

 

5

0.512

0.859

4.979e
−

 
 
 
 
 

 

 

5

5

3.386e

7.813e

0.999

−

 
 
− 
 
 

 

0.187

0.434

277.298

− 
 
 
  

 

N 1 1 1 277.2992 

F 

 

5

0.221

0.131

1.778e
−

 
 
− 
 − 

 

 

 

1.598

2.683

0.0001

 
 
 
  

 

 

 

2.422 7

5.589 7

7.154 7

e

e

e

+ 
 
− + 
 + 

 

 

10

1.771

0.063

3.877e
−

 
 
 
 
 

 

N 0.2574 3.12273 7.154e+11 1.77234 

 

U

 

 

0 0.5 1 1.5 2 2.5 3

x 10
7

0

1

2

3

4

5

6

7

8
x 10

6

5100

5200

5300

5400

5500

5600

5700

5800

5900

6000

49.55 m/s 

U

 

 

0 0.5 1 1.5 2 2.5 3

x 10
7

0

1

2

3

4

5

6

7

8
x 10

6

5100

5200

5300

5400

5500

5600

5700

5800

5900

49.55 m/s 



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 43

In Table 1, I denotes the initial state of singular vector, F is the final state of singular 

vector and N is the norm of singular vector. The singular vector at the grid point in Table 1 

can be represented as vectors as shown in Fig. 3–5. 

 

Fig. 3: 
i

V are the initial time singular vectors; x, y and z axes 

represent u, v, and φ , respectively. 

 

 

Fig. 4: 
i

U are the final time singular vectors with 
1

0.2574σ = ,
2

3.1227σ = and 
3

7.1537e11σ = ; 

x, y and z axes represent u, v, and φ , respectively. 

The tangent linear model stretches (or shrinks) and rotates each of the perturbation 

vectors. At each step the initial state can be transformed to the final state as shown in Fig. 

3. Figure 4 shows the adjoint tangent linear model stretches (or shrinks) and rotats for each 

of the perturbation vectors. 

 

Fig. 5: 
i

V  are the final time singular vectors with 
2

1
0.066σ = , 2

2
9.7512σ = and 2

3
5.1175e 23σ = + ; 

x, y and z axes represent u, v, and φ , respectively. 

0

0.2

0.4

0.6

0.8

1

-0.5

0

0.5

1

-0.5

0

0.5

x
y

z

v
3

v
1

v
2

-1

-0.5

0

0.5

-0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1

xy

z u2

u
1

u
3

0

5

10

15

20

-40

-20

0

20

-20

0

20

40

60

xy

z

v
3

1

v
3

3

v
3

2



IIUM Engineering Journal, Vol. 12, No. 5, 2011: Special Issue -1 on Science and Ethics in Engineering  

Srikul and Sukawat  

 44

4.   CONCLUSION 

In this paper a singular vector is applied at a grid point in linearized shallow water 

wave model. A single vector transformation results in a significant the change in the norm 

of the initial perturbation. The method can be used to generate initial states to an ensemble 

shallow water model for ensemble forecasts.  

ACKNOWLEDGEMENTS 

The first author would like to thank Rajamangala University of Technology Tawan-ok 

(RMUTTO) and Mathematics Department of King Mongkut's University of Technology 

Thonburi  (KMUTT) for the financial support.  

REFERENCES  

[1] Kalnay E., 2003, Atmospheric Modeling Data Assimilation and Predictability, Cambridge 

University Press, UK., vol. 341, pp.1-2.  

[2] Mozheng Wei and Jorgen S. Frederiksen., 2000, “Iterative methods for Lyapunov vectors 

and singular vectors and atmospheric predictability,” ANZIAM J., vol. 42, pp. 1501-1519.  

[3] V. M. Khade and J. A. Hansen., 2004, “State dependent predictability: Impact of uncertainty 

dynamics, uncertainty structure and model inadequacies,” Nonlinear Processes in 

Geophysics, vol. 11, pp. 351-362.  

[4] Holton J. R., 2004, Introduction to Dynamic Meteorology, 4
th
 ed., Elsevier Inc., UK., pp. 

188-219  

[5] L. Magnusson, E. Kallen, and J. Nycander, 2008, “Initial state perturbations in ensemble 

forecasting,” Nonlinear Processes in Geophysics, vol. 15, pp. 751-759.