## 1. Introduction

[2] The interpretation of GPS radio occultation data in the lower troposphere is of a great importance for the assimilation of this type of data into numerical weather prediction models. The standard algorithm of processing radio occultation data involves the derivation of the dependence of refraction (bending) angle versus impact parameter, which is used for the formulation of the inverse problem in the geometric optical approximation [*Ware et al.*, 1996; *Rocken et al.*, 1997; *Kursinski et al.*, 1997]. The computation of refraction angles is straightforward in single-ray areas, where the derivative of the phase, or the Doppler frequency, can be related to the satellite velocities and ray directions at GPS and Low Earth Orbiter (LEO) satellites [*Vorob'ev and Krasil'nikova*, 1994]. Complementing this relation with the formula of Bouger, one can derive the refraction angle and impact parameter from the Doppler frequency.

[3] Radio signals propagated through the lower troposphere can have a very complicated structure due to the effects of multipath propagation and diffraction [*Gorbunov et al.*, 1996b; *Gorbunov and Gurvich*, 1998a, 1998b; *Sokolovskiy*, 2001a]. Refraction angle cannot be directly derived from the phase of wave field in multipath zones, because at each point the phase is defined by amplitudes and phases of multiple interfering rays.

[4] Two methods were suggested for processing radio occultation signals in multipath zones: (1) back propagation method [*Gorbunov et al.*, 1996b; *Karayel and Hinson*, 1997; *Hinson et al.*, 1997, 1998; *Gorbunov and Gurvich*, 1998a, 1998b] and (2) radio-optic (or radio-holographic) method [*Lindal et al.*, 1987; *Pavelyev*, 1998; *Hocke et al.*, 1999]. The comparative analysis of the two methods performed by *Gorbunov et al.* [2000] indicates that they have very strong restrictions of their applicability.

[5] In the first method, the wave field measured along the LEO trajectory is used as the boundary condition for the back-propagating solution of the Helmholtz equation in a vacuum. The underlying ray structure of the back-propagated field may have two types of caustics: real and imaginary. There may be a single-ray area between them, where the back-propagated field can be processed in the standard way. However, the position of the single-ray area is unknown a priori, because the caustic structure depends on the profile of refraction angle, which must yet be determined. Numerical simulations with global field of atmospheric parameters from analyses of European Centre for Medium-Range Weather Forecasts (ECMWF) [*Gorbunov et al.*, 2000] show that complicated caustic structures with overlapping real and imaginary caustics may occur, which causes the back propagation method to fail. The real atmosphere contains small-scale inhomogeneities, which are not represented in numerical weather prediction models and which will result in still more complicated structures of wave fields [*Sokolovskiy*, 2001].

[6] The radio-optic method utilizes the analysis of spatial spectra of the wave field in small sliding apertures. Rays are locally associated with plane waves and are visualized as the maxima of local spatial spectra. This method has the following disadvantages: (1) it cannot be applied in sub-caustic zones, where the wave field cannot be interpreted in terms of rays and (2) it has limited resolution [*Gorbunov et al.*, 2000].

[7] In this paper, we suggest a new approach to the problem, based on the theory of Fourier integral operators. A wave problem can be associated with a canonical Hamilton system, which describes the geometric optical ray structure of the wave field [*Kravtsov and Orlov*, 1990; *Mishchenko et al.*, 1990]. The Hamilton system is written in terms of spatial coordinates and momentum related to the ray direction. We can consider the manifold of all rays in the phase space. Multipath propagation arises in areas where multiple points of this manifold have the same spatial coordinate. A canonical transform in the phase space can be used in order to introduce new spatial coordinate and momentum in such a way that the rays have unique spatial coordinates. Using Egorov's theorem [*Egorov*, 1985; *Treves*, 1982], it is possible to write a Fourier integral operator associated with the canonical transform. The operator transforms the wave function to the new representation with single-ray propagation. The ray structure can then be reconstructed from the derivative of the phase of the transformed wave function in the standard way. We show that the canonical transform to the ray coordinates (impact parameter, ray direction angle) can be used over a wide range of conditions.

[8] We designed a fast numerical implementation of the method based on the FFT. For the validation of the method we process simulated radio occultations, where the application of the back propagation method results in big errors. The simulations show that the method has a much wider applicability than the back propagation.