Image restoration under multiple adverse weather conditions aims to develop a single model to recover the underlying scene with high visibility. Weather-related artifacts vary with the particle's distance to the camera according to the established scene visibility analysis, where close and faraway regions are more affected by falling drops and fog effects, respectively. Existing methods fail to consider this weather-specific physical visual process; thus, the restoration performance is limited. In this work, we analyze the common visual factors in adverse weather conditions and present a unified imaging model that considers the individually visible particles and fog-like aggregate scattering effects. Further, we design a novel weather-prior-based network, which leverages the weather-related prior information to help recover the scene by enhancing the features using the estimated occlusion and transmission. Experimental results in multiple adverse scenarios show the superiority of our method against state-of-the-art methods.