Context. The wind-driven halo is a feature that is observed in images that were delivered by the latest generation of ground-based instruments that are equipped with an extreme adaptive optics system and a coronagraphic device, such as SPHERE at the Very Large Telescope (VLT). This signature appears when the atmospheric turbulence conditions vary faster than the adaptive optics loop can correct for. The wind-driven halo is observed as a radial extension of the point spread function along a distinct direction (this is sometimes referred to as the butterfly pattern). When this is present, it significantly limits the contrast capabilities of the instrument and prevents the extraction of signals at close separation or extended signals such as circumstellar disks. This limitation is consequential because it contaminates the data for a substantial fraction of the time: about 30% of the data produced by the VLT/SPHERE instrument are affected by the wind-driven halo. Aims. This paper reviews the causes of the wind-driven halo and presents a method for analyzing its contribution directly from the scientific images. Its effect on the raw contrast and on the final contrast after post-processing is demonstrated. Methods. We used simulations and on-sky SPHERE data to verify that the parameters extracted with our method can describe the wind-driven halo in the images. We studied the temporal, spatial, and spectral variation of these parameters to point out its deleterious effect on the final contrast. Results. The data-driven analysis we propose provides information to accurately describe the wind-driven halo contribution in the images. This analysis confirms that this is a fundamental limitation of the finally reached contrast performance. Conclusions. With the established procedure, we will analyze a large sample of data delivered by SPHERE in order to propose post-processing techniques that are tailored to removing the wind-driven halo.