Kalman Filter Method with Iterative Sparse Regularization and Its Application to the Retrieval of the Initial Field for the Convection–Diffusion Equation

Sparse regularization methods play an important role in inverse problems for extracting key features of underlying parameters and have attracted increasing attention in meteorological data assimilation. However, when the condition number of the background error covariance matrix is extremely large (e.g., 10 12 ), the instability of the inverse problem makes accurate reconstruction difficult. To address this issue, a gradient operator is incorporated into the sparse regularization term of the cost function, and a Kalman filter (KF) algorithm is developed within a majorization–minimization (MM) framework to solve the resulting optimization problem. The problem is reformulated as a weighted least-squares problem via the MM strategy and further decomposed into two subproblems in the null space and its oblique complementary space through oblique projection, which are then solved using the KF method. This approach avoids the use of an adjoint model typically required in four-dimensional variational data assimilation (4D-Var). In addition, a modified f-slope strategy with a constrained search interval is introduced to adaptively select the regularization parameter during computation. Numerical experiments on the initial condition inversion of the Convection–Diffusion equation demonstrate that the proposed method achieves more accurate reconstruction of key features than the l 1 -norm regularized 4D-Var method, particularly in capturing sharp gradients and sparse structures. The adaptive regularization strategy automatically balances sparsity and smoothness without manual tuning. The inversion errors remain low even when the condition number ranges from 10 8 to 10 14 , with relative MSE and MAE below 0.01 and relative bias below 0.005, indicating improved robustness and reconstruction accuracy under severely ill-conditioned settings.