7/10/2023 0 Comments Subspace definition sexual![]() Adding a single adaptive observation assimilated by 3DVar (3DVar-BDAS) stabilizes the prediction-assimilation cycle further, the only positive exponent being slightly greater than zero ( λ max = 0.002 day − 1, Kaplan–Yorke dimension 1.1), and reduces the normalized RMS analysis error to around 0.16. When fixed observations only are assimilated (3DVar), the number of positive exponents is reduced to three, with the leading exponent ( λ max = 0.088 day − 1 ) corresponding to a doubling time of 7.9 days, while the Kaplan–Yorke dimension is reduced to 6.9 and the normalized RMS error to 0.321. In all three experiments, the fixed observations are assimilated by a least-squares fit, according to the three-dimensional variational (3DVar) algorithm 5,9 in wide operational use, while the adaptive observations are assimilated either by 3DVar (3DVar-BDAS) or in the unstable subspace (AUS-BDAS) according to Eq. 5 In this paper, we examine the long-term stability of the set of modified equations that are referred to as the prediction-assimilation system, in the case in which the original physical system is fully nonlinear and chaotic. In practice, a data assimilation algorithm has to provide the best possible estimate of the evolving state of the system, using the observations available and the equations governing the system’s time evolution. 1,2 In numerical weather and ocean prediction, this classical estimation problem goes under the name of data assimilation 3,4 as data assimilation is spreading rapidly to other fields of the geosciences and of continuum physics, it is important to better grasp its fundamental theoretical aspects. ![]() Estimating the state of a nonlinear dynamical system from partial and noisy observations is therefore crucial in applied physics and engineering. Physical systems-in nature, the laboratory, or industry-can only be measured at a limited number of points in space and time.
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