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Roger Daley

Abstract

The goal of atmospheric data assimilation is to determine the most accurate representation of the signal from the available observations. The optimality of a data assimilation scheme measures how much information has been extracted from the observations. It is possible to quantify the optimality of the scheme using on-line performance diagnostics. Such a diagnostic is the proposed lagged innovation covariance procedure. This diagnostic has been developed from Kalman filter theory. Its characteristics are examined using a simple scalar model, a univariate one-dimensional linear advection model, and a linear quasigeostrophic model. The model results are compared with actual lagged innovation covariances derived from the innovation sequences of an operational data assimilation system.

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Roger Daley

Abstract

In the design of a rational global observing system, one of the most important questions is “What are the best variables (wind, temperatures, etc.) to observe for the optimal specification of the initial conditions for numerical forecast models?” The slow manifold theory of Leith (1980), together with geostrophic adjustment theory, can be used to analyze the information content of different variables under different conditions. This type of analysis was applied to the shallow-water equations to demonstrate that under most conditions the rotational wind components optimally specify the initial state.

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Roger Daley

Abstract

Objective analysis procedures such as statistical interpolation require reliable estimates of forecast-error statistics in order to optimize the analysis weights. Reasonably good estimates of the forecast-error statistics can be obtained from radiosonde networks by the zero lag innovation covariance technique. However, over the data-sparse regions of the tropics, Southern Hemisphere, and oceans, these techniques cannot he applied and much more ad hoe procedures must be used.

This study uses a simple Kalman filter system to actually generate forecast-error statistics for a hierarchy of wind-height observation networks-from uniform, time-invariant networks to inhomogeneous, time-dependent networks. The forecast-error statistics are characterized by their variance and measures of their spatial scale and anisotropy. Several methods of generating forecast-error statistics in data-sparse regions are compared with the optimal results.

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Roger Daley

Abstract

Recent work by Gent and McWilliams (1982) suggests that of all models intermediate in complexity between quasi-geostrophic (QG) and primitive equation (PE), the balance equations (BE) have the best performance when compared with the standard of the PE. The BE system [and the simpler linear balance equations (LBE)] have been infrequently integrated during the past 30 years because of problems with solvability conditions and the apparent necessity for iteration each timestep.

The present article proposes a non-iterative procedure for the time integration of the BE and LBE systems. The procedure was tested on the barotropic (vertically integrated) equations on the sphere. High-resolution BE, LBE and QG integrations were compared with a control PE integration initialized with observed data. The growth of error in the various models as a function of time, latitude and scale was carefully monitored. The high accuracy of the BE equations was confirmed.

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Roger Daley

Abstract

The development of nonlinear normal mode initialization for shallow water models by Machenhauer (1977) and Baer (1977) and its subsequent successful extension to baroclinic models has provided the impetus for further exploitation of model normal modes. The present work is a straightforward application of model normal mode expansions to the problem of taking long timesteps in primitive equation models. A methodology is developed which treats the “fast” gravity modes of the model in a special manner while treating the ensemble of “slow” gravity modes and Rossby modes by explicit leapfrog techniques.

The schemes developed were tested experimentally using the GCM of Bourke et al. (1977) and found to be numerically stable, efficient and accurate.

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Roger Daley

Abstract

Forecast-error statistics have traditionally been used to investigate model performance and to calculate analysis weights for atmospheric data assimilation. Forecast error has two components: the model error, caused by model imperfections, and the predictability error, which is due to the model generation of instabilities from an imperfectly defined initial state. Traditionally, these two error sources have been difficult to separate.

The Kalman filter theory assumes that the model error is additive white (in time) noise, which permits the separation of the model and predictability error. Progress can be made by assuming that the model-error statistics are homogeneous and stationary, an assumption that is more justifiable for the model-error statistics than for the forcast-error statistics. A methodology for estimating the homogeneous, stationary component of the model- error covariance is discussed and tested in a simple data-assimilation system.

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Roger Daley

Abstract

Modern data assimilation algorithms such as the four-dimensional variational algorithm or the extended Kalman filter (EKF) can, in theory, estimate the wind field from chemical constituent observations. This seems to be possible because of the wind-constituent coupling in the chemical transport equation. This paper examines this possibility by applying an EKF to the one-dimensional constituent transport equation and to a prognostic, linear wind model. Generally, both transport and wind models are assumed to be perfect. Tangent linear (TLM) and adjoint models for the chemical transport model are developed and examined.

A set of preliminary experiments was performed assuming perfect winds and examining the propagation of constituent errors. For this case, it was shown that the analysis of the constituent in zones of strong convergence can only be determined from nearby observations inside the zone; but the situation is much more favorable in divergent zones. This was shown to be consistent with observability theory.

Experiments with imperfect wind fields focused on the validity of the TLM and on wind and constituent error propagation. EKF experiments with constituent observations only showed that the wind field can indeed be recovered from these observations provided there is sufficient structure in the constituent field, the observations are sufficiently frequent and accurate (particularly for low constituent concentrations), and data voids are small.

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Roger Daley

Abstract

The direct analysis of the divergent component of the wind has traditionally been difficult. The higher quality datasets now available offer an opportunity to produce meaningful analyses of the divergent wind. The present work considers how the statistical interpolation objective analysis formalism can be generalized to improve the analysis of the divergent wind.

The formulation of the prediction error correlation is modified to be weakly divergent instead of completely nondivergent. This formulation is tested on several lower-order analysis systems. The spectral characteristics of the analysis operators are determined and a simple case study is performed with real data. The results suggest that the present formulation is likely to improve the analysis of the divergent wind field.

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Roger Daley

Abstract

Observation error statistics are required in most atmospheric data assimilation systems. While observation errors are often assumed to be spatially correlated, serial correlations have received virtually no attention. In this article, the effect of serially correlated observation error is examined in the context of Kalman filter theory. It is shown that for spatially uncorrelated observation errors, serial correlations will only be detrimental for rapid-sampling instruments or low-flow regimes.

In standard Kalman filter theory, it is assumed that the model error is not serially correlated. This assumption has been questioned in the past. In this article, certain types of serially correlated model errors are shown to have a serious detrimental effect on atmospheric data assimilation. It is also suggested that certain performance diagnostics may be capable of detecting serial correlations.

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Roger Daley

Abstract

Effective data assimilation algorithms require the estimation and specification of second-moment forecast error statistics. The imposition of multivariate constraints has been found to be a particularly effective way of extracting the maximum amount of information from the observations. In the optimal interpolation (OI) algorithm, geostrophic constraints were imposed, and these have been extended to the global case in the newer three-dimensional variational (3DVAR) algorithm by the application of the linear balance equation or a Rossby–Hough expansion.

This study shows that the imposition of the linear balance equation (or the Rossby–Hough expansion) in the Tropics (although mathematically attractive) is not correct and may have deleterious effects on the assimilated products.

A procedure is developed, based on the singular-value decomposition (svd) of the linear balance equation, for generating global forecast error covariances in which the multivariate coupling between wind and mass is completely user controlled. Thus, two modal spaces are defined: a spectral (spherical harmonic) space for controlling the redness of the spectrum and a balance (svd) space for controlling the coupling. In this way it is possible for the rotational wind and mass error covariances to be closely coupled in line extratropics and essentially univariate at low latitudes.

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