A PV-related leading control variable in Var.
Watch this space for details about the current status of this project.

A (currently) 29-page ps document giving background, motivation and technical details is available here in ps (gzipped) and in pdf formats. It is under development.

A 4-page document giving a derivation of PV for the vertical modes is available here in ps (gzipped) and in pdf formats.

A presentation of this work at a MetO seminar (Sept 04) is available here for download as a powerpoint file.

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Current problems and results
Pictures of PV of normal modes - try 1 (simple projection)

Pictures of PV of normal modes - try 2 (derivation of PV-like quantity in grid space)

Pictures of PV of normal modes - try 3 (derivation of PV-like quantity via spectral space)

Pictures of vorticity projected onto normal modes

Introduction to the project
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.

Page last updated 22/09/03. Ross Bannister.