Full waveform inversion (FWI) gradient inherently contains both tomographic and migration components, which are responsible for updating large-scale background velocity and small-scale structural details, respectively. A key challenge in FWI is to decouple these components to ensure a robust, multi-scale inversion strategy. To overcome this, we propose a novel gradient decomposition method based on bidimensional ensemble empirical mode decomposition (BEEMD). In contrast to conventional bidimensional empirical mode decomposition (BEMD), the proposed method employs a noise-assisted ensemble averaging scheme to effectively mitigate mode-mixing artifacts. In this framework, the migration and tomographic components are respectively derived from fine-scale bidimensional intrinsic mode functions (BIMFs) and the large-scale residual. This decoupling allows the tomographic component mitigates cycle-skipping in the presence of inaccurate initial models, whereas the migration component accelerates convergence to a high-resolution solution once the background is well-defined. Sensitivity kernel analyses and numerical experiments on both synthetic and the Marmousi models demonstrate that the proposed method possesses superior stability and robustness compared to conventional approaches.