In this paper we apply a recently developed class of method known as the forward sensitivity method (FSM) which is based on the evolution of the forward sensitivity of the states of the model with respect to the control variables . This method can be shown to be a dual of the classical adjoint based 4D Var method and does not need the development of the adjoint. FSM directly computes the corrections to the control as a solution to linear inverse problems that exploits the forward sensitivity information and the forecast error.
We demonstrate the power of this approach by applying it to the spectral version of the linearized shallow-water model.
 A. Apte, C. K. R.T. Jones and A. M. Stuart (2008) "A Bayesian approach to Lagrangian data assimilation", Tellus, 60A, 336-347
 S. Lakshmivarahan and John M. Lewis (2010) Forward Sensitivity Based Approach to Dynamic Data Assimilation, Advances in Meteorology, Volume 2010, Article ID 375615, doi 1155/2010/375615