285
4.
Simplified schemes based on static
background errors
In the KF algorithm, the forecast error covariance is updated before
every analysis step by using the model dynamics. Equation (18) can be
rewritten as
P
f
i+1 = M(MP
a
i )
T
+ Q
(26)
showing that the model code has to be used to propagate the n columns
of P
a
i (in addition to propagate the model state itself). Given the huge
size of realistic ocean models, this leads to computing requirements that
even the largest computers in the world will not meet in a foreseeable future. In addition, the many imperfections inherent in the representation
of the error covariance matrices P
a
i and Q make the explicit computation
of this equation questionable.
If the flow of assimilated observations is fairly regular over time, it
makes sense to assume that the errors in an assimilation sequence tend
to fluctuate around some average level after a couple of cycles. This asymptotic behaviour reflects a balance between the increase of uncertainty
during the forecast step and the error reduction during the analysis step.
The existence of such an asymptotic limit for P
f
i+1 provides justification
for simply using a static error covariance, noted B for “background error” (a term usually adopted when the error statistics are not propagated
from one assimilation cycle to the next), instead of explicitly computing the forecast error according to (18). Dierent techniques have been
developed to prescribe the background error covariances in ocean and
atmospheric assimilation schemes. These are discussed below.
4.1
Optimal Interpolation
Optimal Interpolation (OI) designates a wide range of statistical assimilation schemes in which the B matrix is pre-determined empirically.
The main advantages of OI methods are their cost, their ease of use and
the possibility of conducting sensitivity studies to test many dierent
models of error covariances. Today, the vast majority of operational systems involved in GODAE (e.g., FOAM, MFS, HYCOM, MERCATOR)
are based on OI schemes. In contradiction with its name however, OI is
a sub-optimal assimilation process and it would actually be more correct
to designate this class of methods as “statistical interpolation” [Daley,
1991].
Any covariance matrix can be normalized into a correlation matrix C,
dividing each component by the product of the error standard deviations.
OCEAN DATA ASSIMILATION
4.
Simplified schemes based on static
background errors
In the KF algorithm, the forecast error covariance is updated before
every analysis step by using the model dynamics. Equation (18) can be
rewritten as
P
f
i+1 = M(MP
a
i )
T
+ Q
(26)
showing that the model code has to be used to propagate the n columns
of P
a
i (in addition to propagate the model state itself). Given the huge
size of realistic ocean models, this leads to computing requirements that
even the largest computers in the world will not meet in a foreseeable future. In addition, the many imperfections inherent in the representation
of the error covariance matrices P
a
i and Q make the explicit computation
of this equation questionable.
If the flow of assimilated observations is fairly regular over time, it
makes sense to assume that the errors in an assimilation sequence tend
to fluctuate around some average level after a couple of cycles. This asymptotic behaviour reflects a balance between the increase of uncertainty
during the forecast step and the error reduction during the analysis step.
The existence of such an asymptotic limit for P
f
i+1 provides justification
for simply using a static error covariance, noted B for “background error” (a term usually adopted when the error statistics are not propagated
from one assimilation cycle to the next), instead of explicitly computing the forecast error according to (18). Dierent techniques have been
developed to prescribe the background error covariances in ocean and
atmospheric assimilation schemes. These are discussed below.
4.1
Optimal Interpolation
Optimal Interpolation (OI) designates a wide range of statistical assimilation schemes in which the B matrix is pre-determined empirically.
The main advantages of OI methods are their cost, their ease of use and
the possibility of conducting sensitivity studies to test many dierent
models of error covariances. Today, the vast majority of operational systems involved in GODAE (e.g., FOAM, MFS, HYCOM, MERCATOR)
are based on OI schemes. In contradiction with its name however, OI is
a sub-optimal assimilation process and it would actually be more correct
to designate this class of methods as “statistical interpolation” [Daley,
1991].
Any covariance matrix can be normalized into a correlation matrix C,
dividing each component by the product of the error standard deviations.
OCEAN DATA ASSIMILATION