## 1. Introduction

[2] In this paper we discuss the application of Fourier integral operators (FIOs) for processing radio occultation data. The FIO is a technique for constructing short-wave asymptotic solutions of wave problems. The simplest short-wave asymptotic solution is geometric optics (GO) [*Kravtsov and Orlov*, 1990]. It has the following limitations of the applicability: (1) It cannot describe details of wave fields with small characteristic scales below the Fresnel zone size, (2) it does not work on caustics, where the GO amplitude goes to infinity. A generalization of asymptotic GO solution for quantum mechanics and theory of waves in inhomogeneous media was introduced in two equivalent forms: (1) the technique of Maslov operators and (2) FIO [*Maslov and Fedoriuk*, 1981; *Mishchenko et al.*, 1990; *Hörmander*, 1985a, 1985b]. This technique significantly reduces the limitation due to Fresnel diffraction. In particular, the FIO asymptotic solution in a vacuum coincides with the exact solution.

[3] Processing radio occultation data posed the inverse problem: reconstruction of the GO rays from measurements of wave fields, especially in multipath zones. The simplest GO approximation, which can only be applied in single-ray areas, uses the connection between ray direction and Doppler frequency [*Vorob'ev and Krasil'nikova*, 1994]. The resolution of this approximation is limited by the Fresnel zone. Analysis of local spatial spectra is a simple technique that can separate multiple rays. It was applied for processing planetary occultations [*Lindal et al.*, 1987], and it was also used for the analysis of GPS radio occultation data for the Earth's atmosphere, obtained by Microlab-1 and CHAMP satellites [e.g., *Pavelyev*, 1998; *Igarashi et al.*, 2000; *Pavelyev et al.*, 2002; *Gorbunov*, 2002b]. Another direction took origin from the early work by *Marouf et al.* [1986], who suggested the back propagation (BP) technique for the reconstruction of Saturn rings. This technique reduced the effects of diffraction at a large propagation distance. A very similar Fresnel inversion, based on the thin screen approximation, was introduced by *Melbourne et al.* [1994] and applied by *Mortensen and Høeg* [1998]. In addition, the combination of BP preprocessing with the standard GO inversion, which does not use the thin screen model, was suggested by *Gorbunov and Gurvich* [1998a, 1998b].

[4] The application of Fourier integral operators for processing radio occultations was suggested by *Gorbunov* [2002a, 2002b] in the framework of the canonical transform (CT) method. The CT method uses back propagation as the preprocessing tool, which reduces the radio occultation geometry to a vertical observation line. The CT method uses the phase space with coordinate *y* along the vertical line and conjugated momentum η (vertical projection of ray direction vector). In single-ray areas the momentum is equal to the derivative of the eikonal (optical path) of the wave field. The back-propagated complex wave field *u* as a function of vertical coordinate *y* is subjected to the Fourier integral operator associated with a canonical transform from (*y*, η) to (*p*, ξ), where *p* is the ray impact parameter, and the momentum, ξ, conjugated with it is equal to the ray direction angle. Under the assumption that each value of the impact parameter occurs not more than once, this transform allows for disentangling multiple rays. The phase of the transformed wave field can then be differentiated with respect to the impact parameter *p* giving the ray direction angle ξ, which is linked to refraction (bending) angle by simple geometrical relationships. The CT method provides high accuracy and resolution in the reconstruction of refraction angle profiles. The FIO used in its formulation allows for a fast numerical implementation on the basis of FFT. The disadvantage of this CT method is the necessity of BP, which is the most time-consuming part of the numerical algorithm. However, a very fast implementation of BP can be designed using a simple geometric optical approximation (S. Sokolovskiy, private communication, 2003). Also, the best implementations of BP using numerical computation of diffractive integral are reasonably fast.

[5] The full spectrum inversion (FSI) method was recently introduced by *Jensen et al.* [2003]. The simplest formulation of this method applies to a radio occultation with a circular geometry (i.e., spherical satellite orbits in the same vertical plane, spherical Earth and spherically symmetrical atmosphere). The Fourier transform is applied to the complete record of the complex field *u*(*t*) as function of observation time *t* or another parameterization of the observation trajectory such as satellite-to-satellite angle. For circular occultation geometry, the derivative of the phase, or the Doppler frequency ω, of the wave field *u*(*t*) is proportional to ray impact parameter *p*. We can introduce the (multivalued) dependence ω(*t*), which is by definition equal to Doppler frequency (or frequencies) of the ray(s) received at time moment *t*. In multipath zones, where the dependence ω(*t*) is multivalued, it cannot be found by the differentiation of the phase of the wave field *u*(*t*). Unlike ω(*t*), the inverse dependence *t*(ω) is single-valued, if we assume that each impact parameter and therefore frequency ω occurs not more than once. Using a stationary phase derivation, it can be easily shown that the derivative of the phase of the Fourier spectrum (ω) equals −*t*(ω). Then, for each impact parameter *p* we can find the time *t* and therefore the positions of the GPS satellite and low-Earth orbiter (LEO) for which this impact parameter occurred. This allows for the computation of the corresponding refraction angle. The advantage of this method is its numerical simplicity. Its disadvantage is that it is formulated for circular radio occultation geometry and its generalization for realistic radio occultations required some approximation.

[6] *Gorbunov and Lauritsen* [2002] suggested a synthesis of the CT and FSI methods. They derived a general formula for the FIO applied directly to radio occultation data measured along a generic LEO trajectory without the BP procedure. The phase function of the FIO is described by a differential equation. The equation was approximately solved for a generic occultation geometry, to the first order in a small parameter measuring the deviation from a circular orbit. It was shown that for a circular occultation geometry this operator reduces to a Fourier transform. This method was thus a generalization of the FSI technique for generic observation geometry. Recently, *Jensen et al.* [2004] obtained the exact expression for the phase function of the FIO introduced by *Gorbunov and Lauritsen* [2002]. *Jensen et al.* [2004] also considered the amplitude function and below we shall use the synthesis of the approaches used by *Jensen et al.* [2004] and by *Gorbunov* [2002a, 2002b].

[7] The paper is organized as follows: In section 2 we give a survey of the FIO technique. We discuss the method of the Maslov operator and show that its application for the computation of the Green function results in the FIO. We also discuss the similarity between canonical transforms in the geometric optical approximation and FIOs in asymptotic approximation in wave optics. We introduce a new class of operators, FIOs of the second type (FIO2). In section 3 we discuss the application of the new FIO2 technique for processing radio occultation data. We show that the FSI method can also be classified as a specialization of the FIO2 technique, which explains the link between the FSI and CT methods. We describe a generalization of FSI for generic observation geometry and derive a new simple inversion algorithm, which can be implemented using FFT. The algorithm combines the numerical efficiency of the FSI and improved accuracy of the computation of the impact parameter. In section 4 we make a further development of the FIO technique for direct modeling of radio occultation data, which was discussed by *Gorbunov* [2003]. We construct a fast new algorithm for direct modeling, which can be used for moving GPS and LEO satellites (or for LEO-LEO occultations [*Kursinski et al.*, 2002]). Section 5 presents the results of numerical modeling. In section 6 we make our conclusions.