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We describe two theoretical bases for an algorithm for back-projection. The first is (1) Fourier inversion of the mathematical expression for the far electric field components in terms of the aperture electric field. The second is (2) Fourier inversion of the complete vectorial transmitting characteristic of Kerns' scattering matrix. It is this characteristic that results from the standard process of planar near-field (PNF) scanning and the ensuing reduction of the PNF transmission equation. We demonstrate that the theoretical approaches (1) and (2) yield identical back-projection algorithms. We report on back-projection measurements of an 18 inch X-band flat plate phased array using the far-field obtained from spherical near-field scanning. The spherical measurements were made on a large arch range.