ICHIRO FUKUMORI
324
asaki, 1970) that assumes that models have no errors except in
init
Although smoothed solutions satisfy model equations (Eq 14),
smoothing should not be confused with the so-called “strong constraint”
estimation (S
ial condition. In fact, the model solution by itself does not satisfy the
model; viz.,
0
1
1
ˆ
ˆ
s
s
t
t
t
z
x Ax
Gu . Smoothing is generally a “weak constraint”
inversion that allows for model errors, but one that explicitly provides
estimates of these inaccuracies. The explicit estimation of these model error
sources as opposed to leaving them unknown ( ˆ
s
t
u in Eq 14 instead of
0
ˆ t
u ), is
what allows for the temporal evolution of the smoothed solution to be
physically consistent.
While the discussion above has focused on sequential smoothing, there
are other equally effective smoothing algorithms. In particular, when model
error sources are made part of the estimate, the so-called adjoint method or
4dVAR (Chapter 10) is equivalent to the RTS smoother (Eq 11). The
adjoint estimation directly solves for the smoothed solution ( ˆ
s
x and ˆ
s
u )
without deriving intermediate filter estimates.
4. What are data errors and model errors?
” error covariance R and “model” error covariance Q in effect
ther
Data” and “model” errors can be best understood by considering the
vis-à-vis that of the observations and the ocean.
The following discussion follows that of Cohn (1997). For instance, the
mod
“Data
is
efore, fundamental to assimilation and in utilizing their results. In fact,
as described below, a part of what is commonly regarded as “model” error
should in fact be considered “data” error. R and Q are better considered
error covariances of the “data constraint” and the “model constraint”,
respectively.
4.1 “Data” Error
“
true nature of the model
el’s true state t
x (overbar denotes true solution) can be recognized as
representing the ocean in finite dimension,
t
t
{
x Ȇw
(15)
Function Ȇ defines the model state given the complete state of the ocean
t
w (which has infinite degrees of freedom). Observations t
y are samples of
this ocea
that could be written as,
n t
w
t
t
H
y Ew
(16)
define the solution to the data assimilation problem (e.g., Eqs 5 and 11). (P
a function of R and Q.) Their understanding and specification are,
324
asaki, 1970) that assumes that models have no errors except in
init
Although smoothed solutions satisfy model equations (Eq 14),
smoothing should not be confused with the so-called “strong constraint”
estimation (S
ial condition. In fact, the model solution by itself does not satisfy the
model; viz.,
0
1
1
ˆ
ˆ
s
s
t
t
t
z
x Ax
Gu . Smoothing is generally a “weak constraint”
inversion that allows for model errors, but one that explicitly provides
estimates of these inaccuracies. The explicit estimation of these model error
sources as opposed to leaving them unknown ( ˆ
s
t
u in Eq 14 instead of
0
ˆ t
u ), is
what allows for the temporal evolution of the smoothed solution to be
physically consistent.
While the discussion above has focused on sequential smoothing, there
are other equally effective smoothing algorithms. In particular, when model
error sources are made part of the estimate, the so-called adjoint method or
4dVAR (Chapter 10) is equivalent to the RTS smoother (Eq 11). The
adjoint estimation directly solves for the smoothed solution ( ˆ
s
x and ˆ
s
u )
without deriving intermediate filter estimates.
4. What are data errors and model errors?
” error covariance R and “model” error covariance Q in effect
ther
Data” and “model” errors can be best understood by considering the
vis-à-vis that of the observations and the ocean.
The following discussion follows that of Cohn (1997). For instance, the
mod
“Data
is
efore, fundamental to assimilation and in utilizing their results. In fact,
as described below, a part of what is commonly regarded as “model” error
should in fact be considered “data” error. R and Q are better considered
error covariances of the “data constraint” and the “model constraint”,
respectively.
4.1 “Data” Error
“
true nature of the model
el’s true state t
x (overbar denotes true solution) can be recognized as
representing the ocean in finite dimension,
t
t
{
x Ȇw
(15)
Function Ȇ defines the model state given the complete state of the ocean
t
w (which has infinite degrees of freedom). Observations t
y are samples of
this ocea
that could be written as,
n t
w
t
t
H
y Ew
(16)
define the solution to the data assimilation problem (e.g., Eqs 5 and 11). (P
a function of R and Q.) Their understanding and specification are,