A method of successive corrections of the control subspace in the reduced-order variational data assimilation
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M.Yaremchu, D.Nechaev, G.Panteleev A method of successive corrections of the control subspace in the reduced-order variational data assimilation Submitted to Monthly Wheather Review
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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 highly non-linear 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 due to 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.Introduction
In the past two decades the methods of oceanographic data assimilation into numerical models have undergone a signi_cant progress from the early works of Le Dimet and Talagrand, (1986), Thacker (1988) to solution of the increasingly complex problems, re_ected in a series of monographs by Bennett (1992), Wunsch (1996), Evensen (2006), Talagrand (2008) and others. Most recently, research in data assimilation have got an apparent trend toward the studies of the ensemble-based sequential techniques (Evensen, 2003; Ott et al., 2004; Zupanski, 2005; Uzunoglu et al., 2007). These methods utilize low dimensional ensembles of model states to approximate propagation of error covariances that are vital for improvement of practical weather forecast. At the same time, the classic strong constraint 4dVar methods still remain an important tool in atmospheric and oceanic data analysis in both global (Wenzel et al., 2001; Stammer et al.,, 2003; Blessing et al., 2008) and regional (Zupanski et al., 2005; Yaremchuk, 2006; Di Lorenzo et al., 2007) applications. The strong constraint methods are of particular importance in oceanography where the data coverage is sparse and observations are less accurate. With the ever growing complexity and resolution of the ocean general circulation models (OGCMs), constraining them by 4dVar methods is hampered by the following dif_culties: 1. High computational cost of 4dVar optimization. OGCMs have the typical state vector dimension of 106 ??107 whereas the number of independent observations is only an order in magnitude smaller and growing. On the other hand, optimal estimation of an ocean state with classical 4dVar methods requires the number of model runs (including those of the tangent linear (TL) and/or adjoint models) comparable with the number of observations or model state dimension, that is computationally prohibitive. As a consequence, applications of 4dVar methods are limited to _nding suboptimal solutions, obtained after 10-100 iterations of the minimization procedure. 2. High maintenance cost of the adjoint and tangent linear codes. A signi_cant part of the programming burden related to the maintenance of the adjoint and TL codes cannot be automated at the present state of the adjoint compilers. In addition, keeping the adjoint and TL codes updated in parallel with the perpetually upgraded models is labor-intensive and prone to human errors. 3. Breakdown of the tangent linear approximation (TLA). In the presence of strong physical instabilities of the background state applicability of TLA is restricted to relatively short time intervals (e.g., Oldenborgh et al., 1999). Furthermore, the TL and adjoint codes of the community OGCMs never represent exact TL or adjoint operators, especially when model physics contains parameterized discontinuities (Jiang et al., 2002). To resolve these dif_culties the focus of research has recently shifted toward the development of the R4dVar methods. As a few examples, Robert et al. (2005) utilized EOF analysis of the model trajectory for the de_nition of the reduced control space and parameterization of the background error covariance. Robert et al. (2006) explored preconditioning of the incremental 4dVar assimilation by the R4dVar method. Qiu et al., (2007) studied a possibility to use an ensemble of randomly perturbed model states for generation of the reduced control by singular value decomposition. Another strategy studied by (Cao et al., 2005; Daescu and Navon, 2007) is based on the reduction of the model itself using EOF approach. Although the latter technique improves computational ef_ciency, the issue of _nding an optimal low-dimensional state subspace remains an open question. This paper presents a version of the reduced control space 4dVar data assimilation method. In contrast to previous studies of e.g. Robert et al., (2006), Daescu and Navon (2007), Qiu et al., (2007), which utilize a _xed EOF-generated subspace for optimization, our algorithm employs a sequence of low-dimensional subspaces that are iteratively updated in the process of _nding a minimum of the cost function. The paper is organized as follows: in the next section we _rst present linear considerations, underlying the development of the scheme and outline the algorithm. In Section 3 the setup of the twin-data experiments is described and the issue of the adjoint code instability is considered. The Section 4 we 2 present the results of the twin-data experiments and compare the method with the standard 4dVar technique. Section 5 with conclusions completes the paper.