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Michael Ghil and Ricardo Todling

Abstract

Sequential data assimilation schemes approaching true optimality for sizable atmospheric models are becoming a reality. The behavior of the Kalman filter (KF) under difficult conditions needs therefore to be understood. In this two-part paper the authors implemented a KF for a two-dimensional shallow-water model with one or two layers. The model is linearized about a basic flow that depends on latitude; this permits the one-layer (1-L) case to be barotropically unstable. Constant vertical shear in the two-layer (2-L) case induces baroclinic instability.

The stable and unstable 1-L cases were studied in Part I. In the unstable case, even a very small number of observations can keep the forecast and analysis errors from the exponential growth induced by the flow's instability. In Part II, the authors now consider the 2-L, baroclinically stable and unstable cases. Simple experiments show that both cases are, quite similar to their barotropic counterparts. Once again, the KF is shown to keep the estimated flow's error bars bounded, even when a small number of observations—taken with realistic frequency—is utilized.

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Cécile Penland and Michael Ghil

Abstract

Multivariate linear prediction based on single-lag inverse modeling is developed further and critically examined. The method is applied to the National Meteorological Center analyses of Northern Hemisphere 700-mb geopotential height anomalies, which have been filtered to eliminate periods shorter than 10 days. Empirically derived normal modes of the randomly forced linear system are usually correlated, even at zero lag, suggesting that combinations of modes should be used in predictions. Due to nonlinearities in the dynamics and the neglect of interactions with other pressure levels, the lag at which the analysis is performed is crucial; best predictions obtain when the autocovariances involved in the analysis are calculated at a lag comparable to the exponential decay times of the modes. Errors in prediction have a significant seasonal dependence, indicating that the annual cycle affects the higher-order statistics of the field. Optimized linear predictions using this method are useful for about half a day longer than predictions made by persistence.

Conditional probabilities are much more efficiently calculated using normal-mode parameters than from histograms, and yield similar results. Maps of the model's Fourier spectra—integrated over specified frequency intervals and consistent with the assumptions made in a linear analysis—agree with maps obtained from fast Fourier transforms of the data.

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Ricardo Todling and Michael Ghil

Abstract

Sequential data assimilation schemes approaching true optimality for sizable atmospheric models are becoming a reality. The behavior of the Kalman filter (KF) under difficult conditions needs therefore to be understood. In this two-part paper we implement a KF for a two-dimensional shallow-water model, with one or two layers. The model is linearized about a basic flow that depends on latitude; this permits the one-layer (1-L) case to be barotropically unstable. Constant vertical shear in the two-layer (2-L) case induces baroclinic instability.

A model-error covariance matrix for the KF simulations is constructed based on the hypothesis that an ensemble of slow modes dominates the errors. In the 1-L case, the system is stable for a meridionally constant basic flow. Assuming equipartition of energy in the construction of the model-error covariance matrix has a deleterious effect on the process of data assimilation in both the stable and unstable cases. Estimation errors are found to be smaller for a model-error spectrum that decays exponentially with wavenumber than an equipartition spectrum. Then the model-error covariance matrix for the 2-L model is also obtained using a decaying-energy spectrum.

The barotropically unstable 1-L case is studied for a basic velocity profile that has a cosine-square shape. Given this linear instability, forecast errors grow exponentially when no observations are present. The KF keeps the errors bounded, even when very few observations are available. The best placement of a single observation is determined in this simple situation and shown to be where the instability is strongest. The 2-L case and a comparison with the performance of a currently operational data assimilation scheme will appear in Part II.

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Dmitri Kondrashov, Chaojiao Sun, and Michael Ghil

Abstract

The parameter estimation problem for the coupled ocean–atmosphere system in the tropical Pacific Ocean is investigated using an advanced sequential estimator [i.e., the extended Kalman filter (EKF)]. The intermediate coupled model (ICM) used in this paper consists of a prognostic upper-ocean model and a diagnostic atmospheric model. Model errors arise from the uncertainty in atmospheric wind stress. First, the state and parameters are estimated in an identical-twin framework, based on incomplete and inaccurate observations of the model state. Two parameters are estimated by including them into an augmented state vector. Model-generated oceanic datasets are assimilated to produce a time-continuous, dynamically consistent description of the model’s El Niño–Southern Oscillation (ENSO). State estimation without correcting erroneous parameter values still permits recovering the true state to a certain extent, depending on the quality and accuracy of the observations and the size of the discrepancy in the parameters. Estimating both state and parameter values simultaneously, though, produces much better results. Next, real sea surface temperatures observations from the tropical Pacific are assimilated for a 30-yr period (1975–2004). Estimating both the state and parameters by the EKF method helps to track the observations better, even when the ICM is not capable of simulating all the details of the observed state. Furthermore, unobserved ocean variables, such as zonal currents, are improved when model parameters are estimated. A key advantage of using this augmented-state approach is that the incremental cost of applying the EKF to joint state and parameter estimation is small relative to the cost of state estimation alone. A similar approach generalizes various reduced-state approximations of the EKF and could improve simulations and forecasts using large, realistic models.

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Michael Ghil, Boris Shkoller, and Victor Yangarber

Abstract

We derive a system of diagnostic equations for the velocity field, or “wind laws,” for a barotropic primitive-equation model of large-scale atmospheric flow. The derivation is mathematically exact and does not involve any physical assumptions, such as nondivergence or vanishing of derivatives of the divergence, which are not already present in the prognostic equations. Therefore, initial states computed by solving these diagnostic equations should be compatible with the type of motion described by the prognostic equations of the model, and should not generate initialization shocks when inserted into the prognostic model.

Based on the diagnostic system obtained, we are able to give precise meaning to the question whether the wind field is determined by the mass field and by its time history. The answer to this important question is affirmative, in the precise formulation we provide.

The diagnostic system corresponding to the chosen barotropic model is a generalization of the classical balance equation. The ellipticity condition for this system is derived and given a physical interpretation. Numerical solutions of the diagnostic system are exhibited, including cases in-which the system is of mixed elliptic-hyperbolic type.

Such diagnostic systems can be obtained for other primitive equation models. They are valid for all atmospheric scales and regions for which the prognostic models from which they are derived hold. Some problems concerning the possibility of implementing such a system in operational numerical weather prediction are discussed.

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Chaojiao Sun, Zheng Hao, Michael Ghil, and J. David Neelin

Abstract

The assimilation problem for the coupled ocean–atmosphere system in the tropical Pacific is investigated using an advanced sequential estimator, the extended Kalman filter (EKF). The intermediate coupled model used in this study consists of an upper-ocean model and a steady-state atmospheric response to it. Model errors arise from the uncertainty in atmospheric wind stress. Data assimilation is applied in this idealized context to produce a time-continuous, dynamically consistent description of the model's El Niño–Southern Oscillation, based on incomplete and inaccurate observations. This study has two parts: Part I (the present paper) deals with state estimation for the coupled system, assuming that model parameters are correct, while Part II will deal with simultaneous state and parameter estimation.

The dynamical structure of forecast errors is estimated sequentially using a linearized Kalman filter and compared with that of an uncoupled ocean model. The coupling produces large changes in the structure of the error-correlation field. For example, error correlations with opposite signs in the western and eastern part of the model basin are caused by wind stress feedbacks.

The full EKF method is used to assimilate various model-generated synthetic oceanic datasets into the coupled model in an identical-twin framework. The assimilated datasets include the sea surface temperature and a combination of wave velocities and thermocline depth anomaly. With the EKF, the model's forecast-assimilation cycle is able to estimate correctly the phase and amplitude of the basic ENSO oscillation while using very few observations. This includes a set of observations that only cover a single meridional section of the ocean, preferably in the eastern basin.

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