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- Author or Editor: S. E. Cohn x
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Abstract
The fixed-lag Kalman smoother was proposed recently by S. E. Cohn et al. as a framework for providing retrospective data assimilation capability in atmospheric reanalysis projects. Retrospective data assimilation refers to the dynamically consistent incorporation of data observed well past each analysis time into each analysis. Like the Kalman filter, the fixed-lag Kalman smoother requires statistical information that is not available in practice and involves an excessive amount of computation if implemented by brute force, and must therefore be approximated sensibly to become feasible for operational use.
In this article the performance of suboptimal retrospective data assimilation systems (RDASs) based on a variety of approximations to the optimal fixed-lag Kalman smoother is evaluated. Since the fixed-lag Kalman smoother formulation employed in this work separates naturally into a (Kalman) filter portion and an optimal retrospective analysis portion, two suboptimal strategies are considered: (i) viable approximations to the Kalman filter portion coupled with the optimal retrospective analysis portion, and (ii) viable approximations to both portions. These two strategies are studied in the context of a linear dynamical model and observing system, since it is only under these circumstances that performance can be evaluated exactly. A shallow water model, linearized about an unstable basic flow, is used for this purpose.
Results indicate that retrospective data assimilation can be successful even when simple filtering schemes are used, such as one resembling current operational statistical analysis schemes. In this case, however, online adaptive tuning of the forecast error covariance matrix is necessary. The performance of this RDAS is similar to that of the Kalman filter itself. More sophisticated approximate filtering algorithms, such as ones employing singular values/vectors of the propagator or eigenvalues/vectors of the error covariances, as a way to account for error covariance propagation, lead to even better RDAS performance. Approximating both the filter and retrospective analysis portions of the RDAS is also shown to be an acceptable approach in some cases.
Abstract
The fixed-lag Kalman smoother was proposed recently by S. E. Cohn et al. as a framework for providing retrospective data assimilation capability in atmospheric reanalysis projects. Retrospective data assimilation refers to the dynamically consistent incorporation of data observed well past each analysis time into each analysis. Like the Kalman filter, the fixed-lag Kalman smoother requires statistical information that is not available in practice and involves an excessive amount of computation if implemented by brute force, and must therefore be approximated sensibly to become feasible for operational use.
In this article the performance of suboptimal retrospective data assimilation systems (RDASs) based on a variety of approximations to the optimal fixed-lag Kalman smoother is evaluated. Since the fixed-lag Kalman smoother formulation employed in this work separates naturally into a (Kalman) filter portion and an optimal retrospective analysis portion, two suboptimal strategies are considered: (i) viable approximations to the Kalman filter portion coupled with the optimal retrospective analysis portion, and (ii) viable approximations to both portions. These two strategies are studied in the context of a linear dynamical model and observing system, since it is only under these circumstances that performance can be evaluated exactly. A shallow water model, linearized about an unstable basic flow, is used for this purpose.
Results indicate that retrospective data assimilation can be successful even when simple filtering schemes are used, such as one resembling current operational statistical analysis schemes. In this case, however, online adaptive tuning of the forecast error covariance matrix is necessary. The performance of this RDAS is similar to that of the Kalman filter itself. More sophisticated approximate filtering algorithms, such as ones employing singular values/vectors of the propagator or eigenvalues/vectors of the error covariances, as a way to account for error covariance propagation, lead to even better RDAS performance. Approximating both the filter and retrospective analysis portions of the RDAS is also shown to be an acceptable approach in some cases.
Abstract
An efficient implicit finite-difference method is developed and tested for a global barotropic model. The scheme requires at each time step the solution of only one-dimensional block-tridiagonal linear systems. This additional computation is offset by the use of a time step chosen independently of the mesh spacing. The method is second-order accurate in time and fourth-order accurate in space. Our experience indicates that this implicit method is practical for numerical simulation on fine meshes.
Abstract
An efficient implicit finite-difference method is developed and tested for a global barotropic model. The scheme requires at each time step the solution of only one-dimensional block-tridiagonal linear systems. This additional computation is offset by the use of a time step chosen independently of the mesh spacing. The method is second-order accurate in time and fourth-order accurate in space. Our experience indicates that this implicit method is practical for numerical simulation on fine meshes.
Abstract
Data assimilation has traditionally been employed to provide initial conditions for numerical weather prediction (NWP). A multiyear time sequence of objective analyses produced by data assimilation can also be used as an archival record from which to carry out a variety of atmospheric process studies. For this latter propose, NWP analyses are not as accurate as they could be, for each analysis is based only on current and past observed data, and not on any future data. Analyses incorporating future data, as well as current and past data, are termed retrospective analyses. The problem of retrospective objective analysis has not yet received attention in the meteorological literature.
In this paper, the fixed-lag Kalman smoother (FLKS) is proposed as a means of providing retrospective analysis capability in data assimilation. The FLKS is a direct generalization of the Kalman filter. It incorporates all data observed up to and including some fixed amount of time past each analysis time. A computationally efficient form of the FLKS is derived. A simple scalar examination of the FLKS demonstrates that incorporating future data improves analyses the most in the presence of dynamical instabilities, for accurate models and for accurate observations. An implementation of the FLKS for a two-dimensional linear shallow-water model corroborates the scalar analysis. The numerical experiments also demonstrate the ability of the FLKS to propagate information upstream as well as downstream, thus improving analysis quality substantially in data voids.
Abstract
Data assimilation has traditionally been employed to provide initial conditions for numerical weather prediction (NWP). A multiyear time sequence of objective analyses produced by data assimilation can also be used as an archival record from which to carry out a variety of atmospheric process studies. For this latter propose, NWP analyses are not as accurate as they could be, for each analysis is based only on current and past observed data, and not on any future data. Analyses incorporating future data, as well as current and past data, are termed retrospective analyses. The problem of retrospective objective analysis has not yet received attention in the meteorological literature.
In this paper, the fixed-lag Kalman smoother (FLKS) is proposed as a means of providing retrospective analysis capability in data assimilation. The FLKS is a direct generalization of the Kalman filter. It incorporates all data observed up to and including some fixed amount of time past each analysis time. A computationally efficient form of the FLKS is derived. A simple scalar examination of the FLKS demonstrates that incorporating future data improves analyses the most in the presence of dynamical instabilities, for accurate models and for accurate observations. An implementation of the FLKS for a two-dimensional linear shallow-water model corroborates the scalar analysis. The numerical experiments also demonstrate the ability of the FLKS to propagate information upstream as well as downstream, thus improving analysis quality substantially in data voids.
Abstract
The accuracy of interpolating semi-Lagrangian (SL) discretization methods depends on the choice of the interpolating function. Results from barotropic transport simulations on the sphere are presented, using either bicubic Lagrangian or bicubic spline SL discretization. The spline-based scheme is shown to generate excessively noisy fields in these simulations. The two methods are then tested in a one-dimensional advection problem. The damping and dispersion relations for the schemes are examined. The analysis and numerical experiments suggest that the excessive noise found in the spline-based simulations is a consequence of insufficient damping of the small scales for small and near-integer values of the Courant number. Inspection of the local Courant number for the two-dimensional spline-based simulation confirms this hypothesis. This noise can be controlled by adding a scale-selective diffusion term to the spline-based scheme, while retaining its excellent dispersion characteristics.
Abstract
The accuracy of interpolating semi-Lagrangian (SL) discretization methods depends on the choice of the interpolating function. Results from barotropic transport simulations on the sphere are presented, using either bicubic Lagrangian or bicubic spline SL discretization. The spline-based scheme is shown to generate excessively noisy fields in these simulations. The two methods are then tested in a one-dimensional advection problem. The damping and dispersion relations for the schemes are examined. The analysis and numerical experiments suggest that the excessive noise found in the spline-based simulations is a consequence of insufficient damping of the small scales for small and near-integer values of the Courant number. Inspection of the local Courant number for the two-dimensional spline-based simulation confirms this hypothesis. This noise can be controlled by adding a scale-selective diffusion term to the spline-based scheme, while retaining its excellent dispersion characteristics.
Abstract
A Kalman filter for the assimilation of long-lived atmospheric chemical constituents was developed for two-dimensional transport models on isentropic surfaces over the globe. Since the Kalman filter calculates the error covariances of the estimated constituent field, there are five dimensions to this problem, x 1, x 2, and time, where x 1 and x 2 are the positions of two points on an isentropic surface. Only computers with large memory capacity and high floating point speed can handle problems of this magnitude.
This article describes an implementation of the Kalman filter for distributed-memory, message-passing parallel computers. To evolve the forecast error covariance matrix, an operator decomposition and a covariance decomposition were studied. The latter was found to be scalable and has the general property, of considerable practical advantage, that the dynamical model does not need to be parallelized. Tests of the Kalman filter code examined variance transport and observability properties. This code is being used currently to assimilate constituent data retrieved by limb sounders on the Upper Atmosphere Research Satellite.
Abstract
A Kalman filter for the assimilation of long-lived atmospheric chemical constituents was developed for two-dimensional transport models on isentropic surfaces over the globe. Since the Kalman filter calculates the error covariances of the estimated constituent field, there are five dimensions to this problem, x 1, x 2, and time, where x 1 and x 2 are the positions of two points on an isentropic surface. Only computers with large memory capacity and high floating point speed can handle problems of this magnitude.
This article describes an implementation of the Kalman filter for distributed-memory, message-passing parallel computers. To evolve the forecast error covariance matrix, an operator decomposition and a covariance decomposition were studied. The latter was found to be scalable and has the general property, of considerable practical advantage, that the dynamical model does not need to be parallelized. Tests of the Kalman filter code examined variance transport and observability properties. This code is being used currently to assimilate constituent data retrieved by limb sounders on the Upper Atmosphere Research Satellite.
