Hyperspectral remote sensing images are affected by different types of noise. In addition to typical random noise, nonperiodic partially deterministic disturbance patterns generally appear in the data. These patterns, which are intrinsic to the image formation process, are characterized by a high degree of spatial and spectral coherence. We present a new technique that faces the problem of removing the spatially coherent noise known as vertical striping, usually found in images acquired by push-broom sensors. The developed methodology is tested on data acquired by the Compact High Resolution Imaging Spectrometer (CHRIS) onboard the Project for On-board Autonomy (PROBA) orbital platform, which is a typical example of a push-broom instrument exhibiting a relatively high noise component. The proposed correction method is based on the hypothesis that the vertical disturbance presents higher spatial frequencies than the surface radiance. A technique to exclude the contribution of the spatial high frequencies of the surface from the destriping process is introduced. First, the performance of the proposed algorithm is tested on a set of realistic synthetic images with added modeled noise in order to quantify the noise reduction and the noise estimation accuracy. Then, algorithm robustness is tested on more than 350 real CHRIS images from different sites, several acquisition modes (different spatial and spectral resolutions), and covering the full range of possible sensor temperatures. The proposed algorithm is benchmarked against the CHRIS reference algorithm. Results show excellent rejection of the noise pattern with respect to the original CHRIS images, especially improving the removal in those scenes with a natural high contrast. However, some low-frequency components still remain. In addition, the developed correction model captures and corrects the dependency of the noise patterns on sensor temperature, which confirms the robustness of the presented approach.