Four-dimensional variational data assimilation (4DVAR) is frequently used to improve model forecasting skills. This method improves a model consistency with available data by minimizing a cost function measuring the model-data misfit with respect to some model inputs and parameters. Associated with this type of method, however, are difficulties related to the coding of the adjoint model, which is needed to compute the gradient of the 4DVAR cost function. Proper orthogonal decomposition (POD) is a model reduction method that can be used to approximate the gradient calculation in 4DVAR. In this work, two ways of using POD in 4DVAR are presented, namely model-reduced 4DVAR and reduced adjoint 4DVAR (RA-4DVAR). Both techniques employ POD to obtain a reduced-order approximation of the forward linear tangent operator. The difference between the two methods lies in the treatment of the forward model. Model-reduced 4DVAR performs minimization entirely in the POD-reduced space, thereby achieving very low computational costs, but sacrificing accuracy of the end result. On the other hand, the RA-4DVAR uses POD to approximate only the adjoint model. The main contribution of this study is a comparative performance analysis of these 4DVAR methodologies on a nonlinear finite element shallow water model. The sensitivity of the methods to perturbations in observations and the number of observation points is examined. The results from twin experiments suggest that the RA-4DVAR method is easy to implement and computationally efficient and provides a robust approach for achieving reasonable results in the context of variational data assimilation.
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