Research Article Some Aspects of Sensitivity Analysis in Variational Data Assimilation for Coupled Dynamical Systems Sergei Soldatenko, 1 Peter Steinle, 1 Chris Tingwell, 1 and Denis Chichkine 2 1 Centre for Australian Climate and Weather Research, 700 Collins Street, Melbourne, VIC 3008, Australia 2 University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1 Correspondence should be addressed to Sergei Soldatenko; s.soldatenko@bom.gov.au Received 18 December 2014; Revised 24 February 2015; Accepted 11 March 2015 Academic Editor: Guijun Han Copyright © 2015 Sergei Soldatenko et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Variational data assimilation (VDA) remains one of the key issues arising in many felds of geosciences including the numerical weather prediction. While the theory of VDA is well established, there are a number of issues with practical implementation that require additional consideration and study. However, the exploration of VDA requires considerable computational resources. For simple enough low-order models, the computational cost is minor and therefore models of this class are used as simple test instruments to emulate more complex systems. In this paper, the sensitivity with respect to variations in the parameters of one of the main components of VDA, the nonlinear forecasting model, is considered. For chaotic atmospheric dynamics, conventional methods of sensitivity analysis provide uninformative results since the envelopes of sensitivity functions grow with time and sensitivity functions themselves demonstrate the oscillating behaviour. Te use of sensitivity analysis method, developed on the basis of the theory of shadowing pseudoorbits in dynamical systems, allows us to calculate sensitivity functions correctly. Sensitivity estimates for a simple coupled dynamical system are calculated and presented in the paper. To estimate the infuence of model parameter uncertainties on the forecast, the relative error in the energy norm is applied. 1. Introduction Te earth system consists of several interactive dynamical subsystems and each of them covers a broad temporal and spatial spectrum of motions and physical processes. Te components of the earth system have many diferences in their physical properties, structure, and behavior but are linked together by fuxes of momentum and mass as well as sensible and latent heat. All of these subsystems interact with each other in diferent ways and can be strongly or weakly coupled. Prediction of future state of the earth system and its components is one of the most important problems of mod- ern science. Te most signifcant progress has been achieved in the forecasting of the atmosphere via numerical models, which describe the dynamical and physical processes in the earth’s gaseous envelope. It is clear that further improvement of forecasts can be pursued via the development of coupled modeling systems that primarily combine the atmosphere and the ocean and describe the interactions between these two systems. Since numerical weather prediction systems calculate a future state of the atmosphere and ocean by integrating a set of partial diferential equations that describe the fuid dynamics and thermodynamics, initial conditions that accurately represent the state of the atmosphere and ocean at a certain initial time must be formulated. Numerical weather prediction systems use data assimilation procedures to estimate initial conditions for forecasting models from observations. Data assimilation remains one of the key issues not only in the numerical weather prediction (NWP) but also in other geophysical sciences. One of the most advanced and efective data assimilation techniques is four-dimensional variational data assimilation (4D-Var). In particular, the weather forecasts produced by the ACCESS (Australian Community Climate and Earth System Simulator) at the Bureau of Meteorology use 4D-Var in the incremental formulation developed at the Met Ofce [1, 2]. Hindawi Publishing Corporation Advances in Meteorology Volume 2015, Article ID 753031, 22 pages http://dx.doi.org/10.1155/2015/753031