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conditional distribution of truth given a previous estimate of the true state; this last
density will hereafter be referred to as the “prior.” The density
describes
the conditional distribution of the truth given a particular observation and previous
estimate; this density will hereafter be referred to as the “posterior.”
.
x
j
y
;
x
f
/
7.2.2
Data Assimilation as a Problem in Nonlinear Regression
The presentation below relates the estimation of the posterior mean to the
well-known methods of nonlinear regression and is presented as a review of
Hodyss
(
2011
). A standard estimation technique for the true state given the posterior
density is to find its mean, i.e.
Z
x
y
x
f
D
x
j
y
x
f
d
;
x
;
x
:
(7.2)
1
This estimate of the true state has the property that it is unbiased and that it min-
imizes the posterior error variance (
Jazwinski 1998
).In ensemble-based estimation
techniques it is often assumed that our previous estimate of the truth is the mean
of the prior distribution,
x
f
. Note that a random draw from the prior distribution
behaves as
x
©
f
is a random variable with zero mean. Adding
and subtracting the prior mean allows (
7.2
) to be written as
D
x
f
C ©
f
,where
Z
x
y
;
x
f
D
x
f
C
x
x
f
x
j
y
;
x
f
d
x
:
(7.3)
1
Equation (
7.3
) shows that the “correction” to the mean of the prior that produces
the mean of the posterior is the expected error given the distribution of errors
conditioned on today's observation and prior mean. Without loss of generality
we may consider the right-hand side of (
7.3
) as simply an unknown function of
today's observation and prior mean. By taking this perspective (
7.3
) may be written
concisely as
x
x
f
D
f
y
;
x
f
:
(7.4)
The vector-function
f
is assumed smooth and is the object of central interest.
One way to understand the structure of
f
is through an expansion in terms of the
observation about the prior estimate of that observation (
Jazwinski 1998
, pp. 340-
346), i.e.
x
x
f
D
f
0
C
M
1
v
C
M
2
v
2
C
:::
(7.5)
where the innovation,
v
D
y
H x
f
, the matrix
H
is the observation operator that
takes the
N
-dimensional state vector,
x
f
, into the
p
-dimensional observation space,
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