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 definition 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 matrices. However, it is only applicable to those linear or linearized balance constraints. A novel approach 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.