Nonlinear balance constraints in 3DVAR data assimilation

In many applications of 3DVAR, the balance constraints can be considered via two main approaches: weak constraint method which adds penalty terms to the cost function; and proper defi- nition of the background error covariance matrix with non-zero cross-correlation sub-matrices. The weak constraint approach requires determining the weighting matrices of the penalty terms. The background error covariance approach does not require determining those additional weighting ma- trices. However, it is only applicable to those linear or linearized balance constraints. A novel ap- proach is proposed based on the background error covariance approach by generalizing the so-called Derber-Bouttier formulation. An assimilation experiment of estimating temperature and salinity from the sea surface dynamic height observation is given to illustrate the proposed treatments of nonlinear balance constraints.