PurposeSusceptibility maps are usually derived from local magnetic field estimations by minimizing a functional composed of a data consistency term and a regularization term. The data‐consistency term measures the difference between the desired solution and the measured data using typically the L2‐norm. It has been proposed to replace this L2‐norm with the L1‐norm, due to its robustness to outliers and reduction of streaking artifacts arising from highly noisy or strongly perturbed regions. However, in regions with high SNR, the L1‐norm yields a suboptimal denoising performance. In this work, we present a hybrid data fidelity approach that uses the L1‐norm and subsequently the L2‐norm to exploit the strengths of both norms.MethodsWe developed a hybrid data fidelity term approach for QSM (HD‐QSM) based on linear susceptibility inversion methods, with total variation regularization. Each functional is solved with ADMM. The HD‐QSM approach is a two‐stage method that first finds a fast solution of the L1‐norm functional and then uses this solution to initialize the L2‐norm functional. In both norms we included spatially variable weights that improve the quality of the reconstructions.ResultsThe HD‐QSM approach produced good quantitative reconstructions in terms of structural definition, noise reduction, and avoiding streaking artifacts comparable with nonlinear methods, but with higher computational efficiency. Reconstructions performed with this method achieved first place at the lowest RMS error category in stage 1 of the 2019 QSM Reconstruction Challenge.ConclusionsThe proposed method allows robust and accurate QSM reconstructions, obtaining superior performance to state‐of‐the‐art methods.