We propose an entanglement purification protocol for high-dimensional multipartite systems. In the protocol we can select a subensemble in a pure generalized Greenberger-Horne-Zeilinger (GHZ) state. This post-selection can be made by detecting the noise which contaminated the initial pure ensemble when the systems past through a noisy environment. For the detection of noise we investigate correlation properties of GHZ states and analyze their possible errors due to a noisy environment. We show that the presented protocol is more efficient than a simple generalization of the purification protocol for a bipartite state in high dimensions.