Cycling the Representer Method with Nonlinear Models
Realistic dynamic systems are often strongly nonlinear, particularly those for the ocean and atmosphere. Applying variational data assimilation to these systems requires the linearization of the nonlinear dynamics about a background state for the cost function minimization, except when the gradient of the cost function can be analytically or explicitly computed. Although there is no unique choice of linearization, the tangent linearization is to be preferred if it can be proven to be numerically stable and accurate. For time intervals extending beyond the scales of nonlinear event development, the tangent linearization cannot be expected to be sufficiently accurate. The variational assimilation would, therefore, not be able to yield a reliable and accurate solution. In this paper, the representer method is used to test this hypothesis with four different nonlinear models. The method can be implemented for successive cycles in order to solve the entire nonlinear problem. By cycling the representer method, it is possible to reduce the assimilation problem into intervals in which the linear theory is able to perform accurately. This study demonstrates that by cycling the representer method, the tangent linearization is sufficiently accurate once adequate assimilation accuracy is achieved in the early cycles. The outer loops that are usually required to contend with the linear assimilation of a nonlinear problem are not required beyond the early cycles because the tangent linear model is sufficiently accurate at this point. The combination of cycling the representer method and limiting the outer loops to one significantly lowers the cost of the overall assimilation problem. In addition, this study shows that weak constraint assimilation is capable of extending the assimilation period beyond the time range of the accuracy of the tangent linear model. That is, the weak constraint assimilation can correct the inaccuracies of the tangent linear model and clearly outperform the strong constraint method.
KeywordsData Assimilation Assimilation Experiment Variational Data Assimilation Lorenz Attractor Lorenz Model
Unable to display preview. Download preview PDF.
- Amodei L (1995) Solution approchée pour un problème d’assimilation de données avec prise en compte de l’erreur du modèle. Comptes Rendus de l’Académie des Sciences 321, Série IIa, 1087–1094Google Scholar
- Anderson JL (2001) An ensemble adjustment Kalman filter for data assimilation. Mon Wea Rev 129, 1884–2903Google Scholar
- Bennett AF (1992) Inverse methods in physical oceanography. Cambridge University Press, New York, 347ppGoogle Scholar
- Bennett AF (2002) Inverse modeling of the ocean and atmosphere. Cambridge University Press, Cambridge, 234ppGoogle Scholar
- Evensen G, Fario N (1997) Solving for the generalized inverse of the Lorenz model. J Meteor Soc Japan 75, No. 1B, 229–243Google Scholar
- Gauthier P (1992) Chaos and quadric-dimensional data assimilation: A study based on the Lorenz model. Tellus 44A, 2–17Google Scholar
- Miller RN, Carter EF, Blue ST (1999) Data assimilation into nonlinear stochastic models. Tellus 51A, 167–194Google Scholar
- Xu L, Daley R (2000) Towards a true 4-dimensional data assimilation algorithm: application of a cycling representer algorithm to a simple transport problem. Tellus 52A, 109–128Google Scholar
- Xu L, Daley R (2002) Data assimilation with a barotropically unstable shallow water system using representer algorithms. Tellus 54A, 125–137Google Scholar