# Cycling the Representer Method with Nonlinear Models

## Abstract

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.

## Keywords

Data Assimilation Assimilation Experiment Variational Data Assimilation Lorenz Attractor Lorenz Model## Preview

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## Reference

- 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
- Barron CN, Kara AB, Martin PJ, Rhodes RC, Smedstad LF (2006) Formulation, implementation and examination of vertical coordinate choices in the Global Navy Coastal Ocean Model (NCOM), Ocean Model 11, pp 347–375CrossRefGoogle 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
- Bennett AF, Chua BS, and Leslie LM (1996) Generalized inversion of a global numerical weather prediction model. Meteor Atmos Phys 60, 165–178CrossRefGoogle Scholar
- Chua BS, Bennett AF (2001) An inverse ocean modeling system. Ocean Model 3, 137–165CrossRefGoogle Scholar
- Egbert GD, Bennett AF, Foreman MGG (1994) TOPEX/POSEIDON tides estimated using a global inverse method. J Geophys Res 99, 24821–24852CrossRefGoogle Scholar
- Evensen G (1997) Advanced data assimilation for strongly nonlinear dynamics. Mon Wea Rev 125, 1342–1354CrossRefGoogle 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
- Evensen G, Van Leeuwen PJ (2000) An Ensemble Kalman Smoother for nonlinear dynamics. Mon Wea Rev 128, 1852–1867CrossRefGoogle Scholar
- Gauthier P (1992) Chaos and quadric-dimensional data assimilation: A study based on the Lorenz model. Tellus 44A, 2–17Google Scholar
- Hogan TF, Rosmond TE (1991) The description of the navy operational global atmospheric prediction system. Mon Wea Rev 119 (8), pp 1786–1815CrossRefGoogle Scholar
- Hurlburt HE, Thompson JD (1980) A numerical study of the loop current intrusions and eddy shedding. J Phys Oceanogr 10(10), 1611–1651CrossRefGoogle Scholar
- Jacobs GA, Ngodock HE (2003): The maintenance of conservative physical laws within data assimilation systems. Mon Wea Rev 131, pp 2595–2607CrossRefGoogle Scholar
- Lawson WG, Hansen JA (2004) Implications of stochastic and deterministic filters as ensemble-based data assimilation methods in varyingregimes of error growth. Mon Wea Rev 132, 1966–1981CrossRefGoogle Scholar
- Lorenz E N (1963) Deterministic nonperiodic flow. J Atmos Sci 20, 130–141CrossRefGoogle Scholar
- Lorenz EN, Emanuel KA (1998) Optimal sites for supplementary weather observations: simulation with a small model. J Atmos Sci 55, 399–414CrossRefGoogle Scholar
- Miller RN, Ghil M, Gauthiez F (1994) Advanced data assimilation in strongly nonlinear dynamical systems. J Atmos Sci 51, 1037–1056CrossRefGoogle Scholar
- Miller RN, Carter EF, Blue ST (1999) Data assimilation into nonlinear stochastic models. Tellus 51A, 167–194Google Scholar
- Muccino JC, Bennett AF (2002) Generalized inversion of the Korteweg-De Vries equation. Dyn Atmos Oceans 35, 3,227–263CrossRefGoogle Scholar
- Ngodock HE, Chua BS, Bennett AF (2000) Generalized inversion of a reduced gravity primitive equation ocean model and tropical atmosphere ocean data. Mon Wea Rev 128, 1757–1777CrossRefGoogle Scholar
- Ngodock HE, Jacobs GA, Chen M (2006) The representer method, the ensemble Kalman filter and the ensemble Kalman smoother: a comparison study using a nonlinear reduced gravity ocean model. Ocean Model 12, pp 378–400CrossRefGoogle Scholar
- Ngodock HE, Smith SR, Jacobs GA (2007a) Cycling the representer algorithm for variational data assimilation with the Lorenz attractor. Mon Wea Rev 135, 373–386CrossRefGoogle Scholar
- Ngodock HE, Smith SR, Jacobs GA (2007b) Cycling the representer algorithm for variational data assimilation with a nonlinear reduced gravity ocean model. Ocean Model 19, 3–4, pp 101–111CrossRefGoogle Scholar
- Teague WJ, Jarosz E, Carnes MR, Mitchell DA, Hogan PJ (2006) Low-frequency current variability observed at the shelfbreak in the northeastern Gulf of Mexico: May–October, 2004, Continental Shelf Res 26, pp 2559–2582CrossRefGoogle Scholar
- Whitaker JS, Hamill TM (2002) Ensemble data assimilation without perturbed observations. Mon Wea Rev 130, 1913–1924CrossRefGoogle 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