A variational approach for parameter estimation based on balanced proper orthogonal decomposition

The adjoint method has been used significantly for parameter estimation. This requires a significant programming effort to build adjoint model code and is computationally expensive as cost of one adjoint simulation often exceeds several original model runs. The work proposed here is variational data assimilation based on balanced proper orthogonal decomposition (BPOD) to identify uncertain parameters in numerical models and avoids the implementation of the adjoint with respect to model input parameters. An ensemble of model simulations (forward and backward) is used to determine the model subspace while considering both inputs and outputs of the system. By projecting the original model onto this subspace an approximate linear reduced model is obtained. The adjoint of the tangent linear model is replaced by the adjoint of linear reduced model and the minimization problem is then solved in the reduced space at very low computational cost. The performance of the method are illustrated with a number of data assimilation experiments in a 2D-advection diffusion model. The results demonstrate that the BPOD based estimation approach successfully estimates the diffusion coefficient for both advection and diffusion dominated problems. The paper also proposes an efficient method for computing the observable subspace when the number of observations is large is also proposed.