This study presents a variational approach for optimized estimation of specific differential phase ( K_DP) for polarimetric radars using a linear forward operator. A cubic B-spline interpolating filter is included to mitigate the impact of measurement error in the total differential phase and ensure the spatial continuity of K_DP. For rain, non-negative constraints are introduced to ensure that the K_DP estimates are within the physical bounds. The variational approach is flexible to incorporate the background information constructed from the measurements of horizontal reflectivity factor ( Z_H) and differential reflectivity ( Z_DR) based on the self-consistent relationship of polarimetric variables. The variational approach is evaluated using simulated experiments, as well as real observations from an S-band operational weather radar. Without including background information, the variational approach has slightly better performance compared to the approach based on linear programming (LP), and the background information helps to further improve the performance. In addition, the linear forward operator makes this variational approach computationally efficient. It needs less than 3% computational power required by the approach based on LP, making it more suitable for real-time operational applications.