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. Return to Data Assimilation menu |
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).
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Page last updated 22/09/03. Ross Bannister. |