Enhancement of the ensemble nonlinear least-squares algorithm for i4DVar

The integral-correcting 4DVar method (i4DVar) is an evolution of the traditional strongly constrained 4DVar method (s4DVar), which performs much better than s4DVar without increasing computational cost and complexity, and is solved using an ensemble, adjoint-free algorithm (i.e., NLS-i4DVar). However, like most previous ensemble-based methods, NLS-i4DVar also faces the paradox of using the same set of perturbed samples to approximate the background error covariance matrix and the joint tangent linear (TL) operator, making it difficult to guarantee their accuracy. To solve this problem, we divide ensemble anomalies with a (very large) shrinkage factor omega$$ \omega $$ to ensure the validity of the TL operator approximation, while still using the original (unshrunk) samples to approximate the background error covariance matrix, which ultimately solves the above difficulty. Finally, the advantages of the newly developed algorithm are verified by numerical evaluation experiments.