We introduce new vector diffractive integrals, which can be used for the radio holographic remote sensing of the atmosphere and terrestrial surfaces. These integrals are exact relationships connecting the electromagnetic fields known at some interface or curve in space with radio fields on the terrestrial surface or inside the atmosphere. They allow one to restore the radio image of the atmosphere or Earth surface in the investigated regions using a radio hologram registered in space by a small instrument installed on the low Earth orbit satellite. The high-precision radio signals of the Global Positioning System (GPS) navigational satellites can be used as a source of the radio emission for radio holograms. We indicated a connection between the vector diffractive integrals and scalar diffractive integral, which is now applied for the GPS occultation investigation of Earth's atmosphere under an assumption of the spherical symmetry. For the atmosphere itself the accuracy of the scalar theory corresponds to the accuracy of the GPS occultation measurements. The most significant factor that affects the polarization is the reflection from the surface. The use of vector theory can thus be useful for the investigation of Earth's atmosphere by detecting the reflected rays. We show that the reference signal needed for restoration of the radio field from the registered radio hologram is coinciding with the Green function of the scalar wave equation corresponding to a three-dimensional inhomogeneous medium. This substantiates the radio holographic--focused synthetic aperture principle (RFSA) in its application to the atmosphere and surface research. We validated the high vertical resolution of the RFSA method by obtaining radio image of the atmosphere and Earth's surface. Zverev's diffractive integral is used to compare the canonical transform (CT), back propagation (BP), and RFSA methods. For comparison, a general inverse operator (GIO) is introduced. The CT and BP transforms can be obtained by application of the GIO transform to Zverev's diffractive integral. The CT method can resolve physical rays in multipath situations under an assumption of the global spherical symmetry of the atmosphere and ionosphere. The RFSA method can account for the multipath in the case when the global spherical symmetry is absent by using the appropriate model of the refractivity and has a promise to be effective for operational data analysis.