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Max Yaremchuk
and
Paul Martin

Abstract

The sensitivity of model forecasts to uncertainties in control variables is evaluated using the adjoint technique and the ensemble generated by the reduced-order four-dimensional variational data assimilation (R4DVAR) algorithm within the framework of twin-data experiments with a quasigeostrophic model. To simulate real applications where the true state is unknown, the sensitivities were estimated using model solutions that were obtained after assimilating sparse observations extracted from the true solutions. The numerical experiments were conducted in the linear, weakly nonlinear, and strongly nonlinear (NL) regimes with special emphasis on the NL case characterized by the instability of the tangent linear model. It is shown that the ensemble-based R4DVAR method provides better sensitivity estimates in the NL case, primarily due to the better accuracy of the optimized solutions. The concept of sensitivity in the NL case is also considered within the statistical framework. Using analytical arguments and numerical experimentation, averaging the adjoint sensitivity estimates over an ensemble of model trajectories generated by finite perturbations of the optimal control is shown to provide an estimate similar to that obtained with the adjoint model stabilized by enhanced dissipation. This observation allows for evaluation of the sensitivities of strongly nonlinear optimal solutions by using both the adjoint (4DVAR) and ensemble (R4DVAR) optimization algorithms.

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Max Yaremchuk
and
Matthew Carrier

Abstract

Many background error correlation (BEC) models in data assimilation are formulated in terms of a smoothing operator , which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of ) is a computationally expensive procedure, which needs an efficient numerical approximation.

In this study approximate renormalization techniques based on the Monte Carlo (MC) and Hadamard matrix (HM) methods and on the analytic approximations derived under the assumption of the local homogeneity (LHA) of are compared using realistic BEC models designed for oceanographic applications. It is shown that although the accuracy of the MC and HM methods can be improved by additional smoothing, their computational cost remains significantly higher than the LHA method, which is shown to be effective even in the zeroth-order approximation. The next approximation improves the accuracy 1.5–2 times at a moderate increase of CPU time. A heuristic relationship for the smoothing scale in two and three dimensions is proposed for the first-order LHA approximation.

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Max Yaremchuk
and
Dmitry Nechaev

Abstract

Improving the performance of ensemble filters applied to models with many state variables requires regularization of the covariance estimates by localizing the impact of observations on state variables. A covariance localization technique based on modeling of the sample covariance with polynomial functions of the diffusion operator (DL method) is presented. Performance of the technique is compared with the nonadaptive (NAL) and adaptive (AL) ensemble localization schemes in the framework of numerical experiments with synthetic covariance matrices in a realistically inhomogeneous setting. It is shown that the DL approach is comparable in accuracy with the AL method when the ensemble size is less than 100. With larger ensembles, the accuracy of the DL approach is limited by the local homogeneity assumption underlying the technique. Computationally, the DL method is comparable with the NAL technique if the ratio of the local decorrelation scale to the grid step is not too large.

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Dmitri Nechaev
and
Max Yaremchuk

Abstract

A new representation of the Coriolis terms on the Arakawa C grid is proposed. The approximation dumps the grid-scale noise that arises because of spatial averaging of the Coriolis terms when the grid spacing is larger than the deformation radius. The proposed approximation can also be applied in C-grid schemes with semi-implicit treatment of the Coriolis terms. The new scheme is analyzed in the context of the linear inertial–gravity waves and its advantageous behavior is demonstrated with respect to the conventional technique.

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Max Yaremchuk
,
Dmitri Nechaev
, and
Gleb Panteleev

Abstract

A version of the reduced control space four-dimensional variational method (R4DVAR) of data assimilation into numerical models is proposed. In contrast to the conventional 4DVAR schemes, the method does not require development of the tangent linear and adjoint codes for implementation. The proposed R4DVAR technique is based on minimization of the cost function in a sequence of low-dimensional subspaces of the control space. Performance of the method is demonstrated in a series of twin-data assimilation experiments into a nonlinear quasigeostrophic model utilized as a strong constraint. When the adjoint code is stable, R4DVAR’s convergence rate is comparable to that of the standard 4DVAR algorithm. In the presence of strong instabilities in the direct model, R4DVAR works better than 4DVAR whose performance is deteriorated because of the breakdown of the tangent linear approximation. Comparison of the 4DVAR and R4DVAR also shows that R4DVAR becomes advantageous when observations are sparse and noisy.

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Max Yaremchuk
,
Dmitri Nechaev
, and
Chudong Pan

Abstract

A hybrid background error covariance (BEC) model for three-dimensional variational data assimilation of glider data into the Navy Coastal Ocean Model (NCOM) is introduced. Similar to existing atmospheric hybrid BEC models, the proposed model combines low-rank ensemble covariances with the heuristic Gaussian-shaped covariances to estimate forecast error statistics. The distinctive features of the proposed BEC model are the following: (i) formulation in terms of inverse error covariances, (ii) adaptive determination of the rank m of with information criterion based on the innovation error statistics, (iii) restriction of the heuristic covariance operator to the null space of , and (iv) definition of the BEC magnitudes through separate analyses of the innovation error statistics in the state space and the null space of .

The BEC model is validated by assimilation experiments with simulated and real data obtained during a glider survey of the Monterey Bay in August 2003. It is shown that the proposed hybrid scheme substantially improves the forecast skill of the heuristic covariance model.

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Douglas R. Allen
,
Sergey Frolov
,
Rolf Langland
,
Craig H. Bishop
,
Karl W. Hoppel
,
David D. Kuhl
, and
Max Yaremchuk

Abstract

An ensemble-based linearized forecast model has been developed for data assimilation applications for numerical weather prediction. Previous studies applied this local ensemble tangent linear model (LETLM) to various models, from simple one-dimensional models to a low-resolution (~2.5°) version of the Navy Global Environmental Model (NAVGEM) atmospheric forecast model. This paper applies the LETLM to NAVGEM at higher resolution (~1°), which required overcoming challenges including 1) balancing the computational stencil size with the ensemble size, and 2) propagating fast-moving gravity modes in the upper atmosphere. The first challenge is addressed by introducing a modified local influence volume, introducing computations on a thin grid, and using smaller time steps. The second challenge is addressed by applying nonlinear normal mode initialization, which damps spurious fast-moving modes and improves the LETLM errors above ~100 hPa. Compared to a semi-Lagrangian tangent linear model (TLM), the LETLM has superior skill in the lower troposphere (below 700 hPa), which is attributed to better representation of moist physics in the LETLM. The LETLM skill slightly lags in the upper troposphere and stratosphere (700–2 hPa), which is attributed to nonlocal aspects of the TLM including spectral operators converting from winds to vorticity and divergence. Several ways forward are suggested, including integrating the LETLM in a hybrid 4D variational solver for a realistic atmosphere, combining a physics LETLM with a conventional TLM for the dynamics, and separating the LETLM into a sequence of local and nonlocal operators.

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