On optimal solution error in the variational data assimilation problem for the ocean thermodynamics model
Victor Shutyaev, Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia. E-mail: [email protected]
Auxiliary minimization problem
Thermodynamics equations
 T  L(T ) T
t


T


S ( Q)


Tt  (U  Grad )T  Div ( aˆ T  Grad T )  fT in D  (t0  t1 )
T  T0 for t  t0 on D
T
 T
 Q on  S  (t0  t1 )
z
T
 0 on  wc  (t0  t1 )
NT
  T
Un
inf S (Q)
Q
 t  L(T )  Bv in D  (t0  t1 )
  0 for t  t0 

 


( )t  ( L (T ))   B m0 in D  (t0  t1 )
   0 for t  t1

Hv   v  
on   (t0  t1 )
t  (t0  t1 )
for t  t0 
Error control equation
H  Q  R11  R2 2 
R1   E  R2 2   z 0 

and L, F, B, are defined by
( LT  Tˆ )   (TDiv(UTˆ )) 
D

ˆ
TTd

U
()
n
wop
ˆ )dD
ˆ
a
Grad
(
T
)

Grad
(
T
 T
 


( )t  ( L (T ))   B m0 2 in D  (t0  t1 )

D
ˆ  ( BQ Tˆ )  QTˆ  d 
ˆ  f TdD
ˆ  (Tt  Tˆ )  TTdD
(QT  U d )TdT
t
 z 0
T
( )
n T
wop

for t  t0 
Hessian of the functional S
1
ˆ
ˆ
ˆ
ˆ
ˆ
(Tt  T )  ( LT  T )  F (T )  ( BQ T ) T W2 ( D)

0
t1
in a week sense:
F (Tˆ ) 

in D  (t0  t1 )
1
1
2
S ( Q)       Q  1  d dt    m0   T z 0  2 2 d dt
2 t0 
2 t0 
T
 0 on  H  (t0  t1 )
NT
T  T0 
B Q
t1
T

 U n dT  QT on  wop  (t0  t1 )
NT
Tt  LT  F  BQ

D
D

  0 for t  t1

The optimal solution error:
 Q  T11  T22 
1
1
T1  H R1  T2  H R2 
Sensitivity coefficients
T  LT
t


 T
 J (Q)


 F  BQ in D  (t0  t1 )

T0

r1 
for t  t0 
inf J (Q)

r1 

 min
Q
t1
t1
1
1
(0) 2
2
J (Q)      Q  Q  d dt    m0  T z 0 Tobs  d dt
2 t0 
2 t0 
r2 
T1*T1 ,
r2 

(T )t  L T  B m0 (T  Tobs ) in D  (t0  t1 )

T  0 for t  t1
 (Q  Q
(0)

) T  0
on   (t0  t1 )
k  0 for t  t0 
 
k

k

  0 for t  t1
 vk  k  k vk on   (t0  t1 )
T  T0 for t  t0 

in D  (t0  t1 )
( )  L   B m0k in D  (t0  t1 )

k t
Tt  LT  F  BQ in D  (t0  t1 )

( H   E ) H 2 
Fundamental control functions
(k )t  Lk  Bvk

T2*T2
Singular vectors:
1
T T w   w  wk 
k z 0 
k  

2 2 k
2
k k
k  1 2… 
2
k
Numerical examples (Indian Ocean)
Q (0)  Q  1  Tobs  T  z  0  2 
where
T  T  T   Q  Q  Q
and
T t  LT  F  BQ in D  (t0  t1 )
T  T0 for t  t0 
 Tt  L(T ) T  B Q in D  (t0  t1 )
 T  0 for t  t0 
t=24h
t=240h
(T )t  ( L(T )) T  B m0 ( T   2 ) in D  (t0  t1 )



T  0

for t  t1 
 ( Q  1 )  T   0
on   (t0  t1 )
1. Dimet, F.-X., Shutyaev, V.P. On deterministic error analysis in variational data assimilation. Nonlinear Processes in Geophysics (2005).
2. Gejadze I., Le Dimet F.-X., Shutyaev V. On analysis error covariances in variational data assimilation. SIAM J. Sci. Computing (2008).
Hessian eigenvalues (t=1month)
k  

.
2
k