The work is devoted to the consideration of the issues of eliminating information redundancy of raster data of remote sensing of the Earth (RDRSE), including the latest hyperspectral data (HSD). The characteristic properties of raster hyperspectral images (RHSI) are listed, a brief description of the existing methods of RDRSE compression is given. The possibility of using local, homogeneous "well-adapted" basic functions (LHWABF) to eliminate information redundancy and adaptive compression of RDRSE is considered. An algorithm for constructing a LHWABF system for the RHSI based on the Chebyshev approximation is proposed. The results of computational experiments are given. The effectiveness of the proposed method of adaptive compression RHSI is shown.