In theory K aim an filters can be used to solve many on-line data assimilation problems. However, for models resulting from the discretization of partial differential equations the number of state variables is usually very large, leading to a huge computational burden. Therefore approximation of the Kalman filter equations is in general necessary. In this paper two new algorithms are proposed, that extend the idea of the Reduced Rank Square Root filter [15] for use with non-linear models. The algorithms are based on a low rank approximation of the error covariance matrix and use a square root representation of the error covariance. For both algorithms the tangent linear model is not needed. The first algorithm proposed is accurate up to first order terms, which is comparable to the extended Kalman filter. The second, at the cost of twice the number of computations, is second order accurate, which may be important for strongly nonlinear models. Several experiments were performed on a model of the southern part of the North Sea to measure the performance of both algorithms. Both algorithms perform well when the the number of modes, i.e. the rank of the approximation, is set to 30. This corresponds to a computation time of approximately 30 model runs for the first order algorithm and 60 for the second order algorithm.