## 1. Introduction

[2] The method of the wave front inversion consists of determining the radio field inside the inhomogeneous media using a radio hologram registered on some interface or curve in space. The name of the method has been introduced by *Zeldovich et al.* [1985]. They described its application for the optical sounding of the inhomogeneous media. Unlike the optical range, the digital methods are applied for the radio holographic remote sensing. The digital methods for remote sensing use the diffractive integrals connecting the electromagnetic fields on some interface or curve in the space (e.g., the orbital trajectory of a low Earth orbit (LEO) satellite with a radio holographic receiver) with the field in the space between the transmitter and receiver. *Zverev* [1975] obtained the three-dimensional (3-D) scalar equation, which links the angular spectrum of the field with the angular spectrum of the back-propagated wave in the free space. *Marouf and Tyler* [1982] described the inversion method for obtaining the spatial structure of Saturn's rings using the radio holograms registered onboard Voyager spacecraft. They constructed a reference signal using the known form of the rings and diffraction theory and obtained spatial resolution about of 1/10–1/100 of the Fresnel's zone size. *Kunitsyn and Tereshchenko* [1991] and *Kunitsyn et al.* [1994] considered the application of the tomographic method for the remote sensing of Earth's ionosphere using the radio emission of the LEO satellites. *Gorbunov et al.* [1996] introduced the back-propagation (BP) method on the basis of the scalar diffractive 2-D integral, to heighten the vertical resolution in the radio occultation (RO) experiments. The BP method has a significant difference in comparison with the radio holographic approach suggested by *Marouf and Tyler* [1982]. Back propagation is performed using the 2-D free space Green function rather than the Green function obtained as a solution of a boundary diffraction problem in a 3-D medium. *Pavelyev* [1998], *Hocke et al.* [1999], and *Igarashi et al.* [2000, 2001] derived a radio holographic–focused synthetic aperture (RFSA) principle for RO data analysis. The Fourier analysis in the finite time intervals is applied to the product of the RO and reference signals with the aims (1) to obtain 1-D radio images of the atmosphere and terrestrial surface and (2) to retrieve the vertical profiles of the physical parameters in the atmosphere and mesosphere. Using their method, one can directly determine the dependence of the refraction angle on the impact parameter without application of complex BP technology. *Hocke et al.* [1999] and *Igarashi et al.* [2000, 2001] determined by the RFSA method the electron density *N*_{e}(*h*) and its vertical gradient *dN*_{e}(*h*)/*dh* in the mesosphere and temperature *T*(*h*) in the atmosphere. *Beyerle and Hocke* [2001] and *Igarashi et al.* [2001] applied the RFSA method to visualize signals reflected from the terrestrial surface. They were the first to reveal surface reflections and obtained 1-D radio images of the troposphere and the surface by analyzing the Global Positioning System/Meteorology Experiment (GPS/MET) RO data. *Igarashi et al.* [2001] and *Pavelyev et al.* [2002a] provided preliminary analysis of radio images and estimated the vertical resolution of the RFSA method as about of 70 m. *Beyerle et al.* [2002] applied the RFSA method for a radio holographic analysis of the GPS signal propagation in the troposphere and surface reflections. They obtained important information on a global scale on the humidity concentration in the boundary layer of the atmosphere using the CHAMP RO data. It may be noted that the RFSA method is distinctive in comparison with the unfocused synthetic aperture (Doppler selection) method applied early by *Lindal et al.* [1987] for the spectral analysis of the RO data to obtain the radio images of Uranus' atmosphere. *Gorbunov* [2002a, 2002b] used also the representation of the wave field as a sum of spherical waves to correct approximately for the wave front curvature and considered examples of radio images with multiple direct and reflected rays for the GPS/MET RO data. In the case of unfocused synthesis, the size of the synthetic aperture and the vertical resolution are limited by an uncertainty condition between resolution in the impact parameter and refraction angle [*Gorbunov et al.*, 2000]. The RFSA method, in principle, does not obey the uncertainty condition because it accounts for the curvature of the wave fronts corresponding to the physical rays after propagation in Earth's atmosphere and can use the large size of the synthetic aperture for effective compression of the angular plane wave spectrum of the RO signal. According to this advantage, the RFSA method can discern the surface reflections near the powerful tropospheric RO signal and can realize in practice the high values of the vertical resolution ∼100 m as expected early for the BP method [*Hocke et al.*, 1999; *Igarashi et al.*, 2000, 2001]. Note, however, that to achieve high resolution in the multipath areas, one must use an accurate model of the refractivity in the RO region to construct the reference signal, which has high level of coherence with the RO signal. Recently, *Gorbunov* [2002c] introduced the canonical transform (CT) method for processing the GPS RO data in lower troposphere. The main idea of the CT method consists of using Fourier integral operators (FIO) to find directly the dependence of the refraction angle on the impact parameter for each physical ray in multipath conditions. *Jensen et al.* [2003] introduced the full spectrum inversion (FSI) method to process the RO signals. They established a novel connection between the derivative of the phase of a physical ray on the instantaneous frequency in the full Fourier spectrum of the RO signal and the time of intersection of the physical ray with orbital trajectory of LEO satellite. This feature of the FSI method can be used to obtain under an assumption of spherical symmetry the refraction angle and impact parameter for each physical ray.

[3] The progress in developing the radio holographic investigations is connected, in particular, with existence of the radio navigational satellite systems GPS/Global Navigation Satellite System (GLONASS), which are emitting high-precision, coherent, and stable radio signals. The diffractive integrals can be used to realize the high precision and stability of the radio signals of the radio navigational systems and to obtain extreme values of the spatial resolution and accuracy in the remote sensing of the atmosphere and surface of Earth from space. The aim of this paper consists of the presentation of the diffractive vector integrals for bistatic radio holographic remote sensing of the terrestrial surface and atmosphere, substantiating the RFSA method, introducing a simple way to obtain the CT and BP transforms and establishing their limitations using a GIO. In section 2 we derive the diffractive 3-D vector integrals for describing the direct propagation of the radio waves in the 3-D inhomogeneous media. To accomplish this end, we modernize the Stratton-Chu vector theory developed early for the case of 3-D homogeneous media [*Stratton*, 1941]. The derived vector equations include Green function, which is a solution of the 3-D scalar wave equation for an inhomogeneous medium. This is significantly different from results obtained early by *Müller* [1969] and *Ström* [1991] for an inhomogeneous medium because their vector equations contain the free space Green function. In section 3 we derive 3-D vector equations for back-propagating radio waves and then show that the scalar 2-D diffractive integral can be derived from the 3-D vector diffractive integrals for a 2-D medium. A connection between the solution of 3-D diffraction problem and the reference signal for the RFSA method is established in section 4. In section 4 we derived the basic equations of the RFSA method, illustrated its capability to compress the angular spectrum of the RO signal, and achieved high vertical resolution. In section 5 we compared the RFSA, CT, and BP methods and established their limitations using the GIO transform.