Geoscience Reference
In-Depth Information
19.2.1
Data Assimilation Overview
We assume that reader is at least partially familiar with data assimilation, thus we
will only describe data assimilation methodologies in general. This section will also
serve to introduce the notation.
Data assimilation (DA) may be referred to as a mathematical algorithm that
provides optimal combination of observations and model prediction. DA produces
optimal estimates of the model state vector (e.g., analysis), as well as its uncer-
tainty, typically represented by the analysis error covariance matrix. “Optimal”
is commonly defined as a minimum variance or a maximum likelihood estimate
(e.g.,
Jazwinski 1970
). Current data assimilation methodologies generally rely on
the use of Bayes formula for describing conditional probability density function
(pdf). A detailed overview of data assimilation methods can be found in topics by
Daley
(
1993
),
Kalnay
(
2003
),
Lewis et al.
(
2006
), and
Evensen
(
2009
).
We briefly describe two major DA methodologies, variational and ensemble.
Variational DA is commonly used in operational weather centers, while ensemble
DA is mostly used in research and is making progress towards operational use.
Also, there are a variety of sub-methods and hybrid methods that combine the two
methodologies.
Following
Lorenc
(
1986
), for Gaussian probability distribution one can derive
a cost function by taking a negative logarithm of the Bayes formula for posterior
probability
J.x/
D
1
2
Œx
x
f
T
P
f
Œx
x
f
C
1
2
Œy
h.x/
T
R
1
Œy
h.x/
(19.1)
Where
x
denotes the state vector,
y
is the observation vector, the superscript
f
denotes forecast,
are the
forecast and observation error covariances, respectively. Although this cost function
is typically mentioned in variational methods, it is important to note that this
function is also relevant for Kalman filter based methodologies, including ensemble
Kalman filter (
Li and Navon 2001
). As shown in
Jazwinski
(
1970
), minimization of
the cost function (
19.1
) defined for linear observation operator produces the Kalman
filter analysis solution formally obtained using Newton method for minimization of
quadratic function (e.g.,
Luenberger 1989
). This apparent equivalence between the
minimum variance and maximum likelihood estimates is ultimately a consequence
of using Gaussian pdfs for which the mean and the mode are identical.
Variational and ensemble data assimilation methods can be applied sequentially,
in which case they are referred to as filters, or in batch mode, in which case they are
referred to as smoothers. Variational filtering method is called the three-dimensional
variational (3d-Var) DA, while when it is used as a smoother it is referred
to as the four-dimensional variational (4d-Var) DA. Ensemble data assimilation
methodologies are mostly applied sequentially, although there is a possibility to
apply them in a smoother framework (e.g.,
Evensen and van Leeuwen 2000
).
h
is a nonlinear observation operator and
P
f
and
R
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