A set of new, potential vorticity (PV)-based control variables used
within an atmospheric variational data assimilation (Var.) scheme has
advantages over sets that are currently used operationally by some
leading meteorological centres. A choice of new variables, formulated
by Mike Cullen, of which a PV-related field is the leading variable, is
described together with the strategy for its implementation within the
Met Office's Var. scheme. Detailed is the transformation from the
PV-based set to model variables, its adjoint, its inverse and the
boundary conditions that must be considered when solving the
transformation equations.
The process of variational data assimilation can be described as the
task of adjusting a model state vector in view of gaining optimal
consistency simultaneously with (i) a background state and (ii) a set
of observations, relevent to some time window. Other contraints are
sometimes also imposed that encourage the state vector, e.g., to obey
balance conditions or to discourage model error. The whole process is
achieved by minimizing a cost function that penalizes misfit between
the state vector variable and the background, and the state vector's
'prediction' of the observations and the observations themselves (plus
costs that penalize departure from the other conditions imposed). The
state vector that achieves this best fit within the characterised
errors of the background and observations is called the
analysis. The cost function is minimized at the anlysis.
Atmospheric assimilation schemes make extensive use of numerical
weather prediction (NWP) models to provide a background state (a
forecast from a previous analysis) and, in four-dimensional variational
data assimilation (4d-var.), the time evolution part of the forward
model and its adjoint. The state vectors used by these models describe
the atmosphere typically by the fields u, v, T, q, etc. These are
represented on a set of model levels in the vertical and either a real-
or spectral-space representation in the horizontal. It is helpful to
refer to this representation of the state vector as in model
space.
All information that goes into the Var. scheme has uncertainties, and
it is very important to take uncertainties into account. The
background error covariance matrix characterizes the uncertainties
within the background state by describing variances of and covariances
between the model variables (in a Gaussian context). The model state
space is of high rank (10e6 to 10e7 and so we cannot represent the
background error covariance matrix explicitly.
Most leading assimilation schemes do not perform the minimization
process in model space, but instead use a transformed or
control space. This new space is chosen to have a special and
desirable property - when the background field is represented in this
space, its errors are uncorellated and variances are of unit size (the
problem is said to be preconditioned). It is very convenient
to express state vectors in this form in the minimization process as
the background error covariance matrix becomes the identity matrix.
The remaining problem is determine the transformation that (at least
approximately) achieves this.
The transformation between model and control variables is practically a
multi-step process. The first stage involves a change of parameters
(the parameter transform). This is designed to shift from the
model variables - whose background errors are strongly correlated
(multivariate) - to an alternative set of parameters - whose background
errors are uncorrelated (univariate) (or at least weakly correlated).
There however remains non-local correlations within each of the
parameter's fields. The role of the remaining (vertical and
horizontal) parts of the transformation is to project the parameters
onto sets of vertical and horizontal modes that have no background
error correlations.
This paper is about the first step in the transformation. It describes
a change from model variables to a proposed set of pseudo-uncorrelated
parameters which are partitioned according to whether they are
balanced or unbalanced. The choice of parameters is
discussed, together with the mathematical details of the
transformations that need to be solved.
The key advantages of using a set of pseudo-uncorrelated parameters as
part of the transformation include the following (in no particular
order).
- The parameter transform is an essential stage in the
preconditioning process. This procedure block-diagonalizes the
background error covariance matrix thus limiting the amount of
information needed to describe it. This simplifies the process of
determinging the approximate eigenmodes (EOFs) of the background error
covariance matrix whose variances (eigenvalues) are required for the
preconditioning to work. A preconditioned cost function helps to
control better the iterations of the minimization procedure, resulting
in a well behaved algorithm that should converge quickly. With
parameters that have a minimum of correlation between them, the full
variances of the real problem is preserved during the transformation.
- The atmospheric state can be partitioned into balanced (
slow manifold) and unbalanced components, which often have
separate spatio-time scales, and evolve in a quasi-independent manner.
This is a useful property in data assimilation, not only for item 1,
but also so that each component can treated according to its own error
characteristics. Balanced modes of variability often have greater
variance than that of unbalanced modes. This has two consequences in
data assimilation. Firstly, Var. will implicitly weight its analysis
increment to the variances of each mode, resulting in a largely
balanced increment. Secondly, unbalanced modes will be tightly
constrained automatically in Var., which will lessen the need for
initialization of the analysis.
- The atmosphere is dominated by balanced flow, and the residual
weight is made up of unbalanced components. Thus most of the flow
should be represented by one (leading) control parameter). The
other parameters represent the residual flow.
- A set of PV-based balanced/unbalanced partitioned parameters is
expected to satisfy better the assumption of non-correlation between
parameters than for existing control parameters. Thus the true
background errors are expected to be better approximated with the
proposed method. The new parameters are thus expected to lie close to
the true principal axes of the background error covariance matrix, and
so we expect to capture more of the variance of the background errors.
As a consequence, the problem posed in terms of the PV-based parameters
will be worse conditioned than for the existing parameters, but this
will be compensated for in the vertical and horizontal transforms.
- PV is a non-linear parameter. In formulating the transforms, it is
linearized about a non-zero and synoptic dependent reference state.
This introduces some flow-dependence to the errors.
- Many of the transforms that arise from the proposed PV-based scheme
involve solving three-dimensional elliptic equations. Sets of only
two-dimensional equations are involved in the current scheme. The
vertical coupling may improve vertical consistency.
This is the philosophy behind many leading atmospheric data
assimilation schemes, such as those used by the United Kingdon
Meteorological Office (Met Office) and the European Centre for Medium
Range Weather Forecasts (E.C.M.W.F.). Their choice of control
varaibles are, for practical reasons, not properly partioned into
balanced and imbalanced components and so their schemes cannot exploit
to the full the advantages outlined above. The focus of this report is
to describe a new set of parameters that is alternative to the existing
set used in the Met Office's operational 3d-var scheme. The new set is
designed around potential vorticity (PV), and is hoped to be
advantagous in the scope of the points outlined above.
In the present scheme used by the Met Office, the control parameters
are (i) streamfunction, (ii) velocity potential, (iii) geostrophically
unbalanced pressure and (iv) relative humidity. Streamfunction is the
leading parameter that is meant to represent the balanced component of
the flow (but only approximately over some flow regimes). Our new set
will involve parameters that will be related to (i) PV, (ii) departure
from linear balance, (iii) divergence and (iv) relative humidity. Our
leading control variable related to PV is better suited to bescribe the
balanced part of the flow than is the streamfunction.
Although the proposed scheme is expected to improve the representation
of the background error covariance matrix, we also point out what it
will not do. According to Kalman filter theory, the background error
covariance matrix is a projection, forward in time, of the previous
cycle's analysis error covariance matrix (the inverse hessian). This
covariance matrix is influenced by the observations that are used in
the previous analysis. A projection onto dynamically pseudo-decoupled
parameters will not take into account the covariances intoduced by the
observing system.
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