## 1. Introduction

[2] Until recently, deciphering the geometric optical ray structure of wave fields in multipath zones was considered one of the main problems in processing radio occultation (RO) data. The following radio holographic techniques for solving this problem were developed: (1) back propagation (BP) [*Marouf et al.*, 1986; *Gorbunov and Gurvich*, 1998a, 1998b; *Mortensen et al.*, 1999], (2) Fresnel inversion [*Melbourne et al.*, 1994; *Mortensen and Høeg*, 1998], (3) sliding spectral, focused synthetic aperture methods [*Lindal et al.*, 1987; *Pavelyev*, 1998; *Igarashi et al.*, 2000, 2001; *Pavelyev et al.*, 2002]. An attempt was also made to combine the back propagation and radio holographic analyses [*Gorbunov*, 2002c]. However, it was recognized that all of these approaches have strong restrictions [*Gorbunov et al.*, 2000], which impeded their application for processing lower tropospheric data with severe multipath [*Sokolovskiy*, 2001a, 2001b].

[3] The introduction of the canonical transform (CT) method [*Gorbunov*, 2001, 2002a, 2002b] practically solved the problem. The CT method has the following advantages: (1) it allows for the achievement of very high resolution (30–60 m), which is not limited by the Fresnel zone size, even under very severe multipath conditions, (2) it has no tuning parameters (such as back propagation distance in the BP approach), and (3) its only restriction is the requirement that the refraction angle is a single-valued function of impact parameter. This made the CT algorithm a reference method indicating the highest possible accuracy and resolution, and this algorithm is already very effectively used in operational applications.

[4] The CT method uses BP as a preprocessing tool, which reduces the observation trajectory to a vertical line. A Fourier integral operator maps the back-propagated wave field from the representation of the spatial vertical coordinate to the representation of the ray impact parameter. Under the assumption that impact parameter is a unique ray coordinate (i.e., there is no more than one ray for a given value of impact parameter), the refraction angle can be found in geometric optical approximation from the derivative of the phase of the transformed wave field. This is similar to the standard derivation of impact parameter from the derivative of the phase (Doppler frequency) of the measured wave field under no multipath conditions [*Vorob'ev and Krasil'nikova*, 1994]. The Fourier integral operator allows for a very effective FFT-based numerical implementation.

[5] The main drawback of this approach is the necessity of the BP preprocessing, which is the most time-consuming part of the numerical algorithm. Accurate BP preprocessing uses diffractive integrals, and for an arbitrary occultation geometry it defies a fast FFT-like numerical implementation. However, the best numerical implementations of the accurate BP algorithm are reasonably fast. For processing one radio occultation with a standard 50 Hz sampling rate, the BP part of the complete numerical algorithm may take as little as 10 s on a Pentium-based system. A very fast approximate numerical implementation of BP can be based on geometrical optics (S. Sokolovskiy, personal communication, 2003).

[6] The use of a global Fourier transform in the retrieval of radio occultation signals with a circular geometry (i.e., circular satellite orbits in the same vertical plane, spherical Earth and spherically symmetrical atmosphere) was introduced in the work of *Høeg et al.* [2001] and *Jensen et al.* [2002]. A complete development of the full spectrum inversion (FSI) method, including realistic orbits, has been presented in the work of *Jensen et al.* [2003]. This method uses the Fourier transform of the complete radio hologram, i.e., the record of the complex field *u*(*t*) as function of observation time *t*. In the case of a circular geometry, the derivative of the phase, or Doppler frequency, ω of the wave field *u*(*t*) is proportional to ray impact parameter. We can introduce the (multivalued) dependence ω(*t*), which is by definition equal to the 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 stationary phase derivation, it can be easily shown that the derivative of the phase of the Fourier spectrum (ω) is equal to −*t*(ω). Thus for each impact parameter 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 finding the corresponding refraction angle. This method is much more numerically effective as compared to CT. FSI does not require a computationally expensive BP, it uses only one FT instead of two and does not need interpolation of the spectrum on another grid.

[7] For a generic satellite orbit, which deviates from a circle, Doppler frequency ω may not be an unique ray coordinate, and then some modification of this approach is necessary. *Jensen et al.* [2003] showed that the parameterization of the observation trajectory by the satellite-to-satellite angle θ instead of time *t* together with subtracting a correction model from the phase is a very effective solution. The phase model corrects for the radial velocities of the satellites, and it is an approximate and numerically very efficient form of short distance BP from the real orbit to a circle. *Lauritsen and Lohmann* [2002] investigated an integral linear transformation of the phase followed by Fourier transform over time *t*. This procedure also results in unique frequencies of different rays. However, this transform is nonlinear with respect to complex field *u*(*t*), which impairs the quality of amplitude reconstruction.

[8] In this paper, we compare the CT and FSI methods. In section 2 we show that the two methods are closely related. They both process the wave field by means of a Fourier integral operator. The transformed wave function is a function of ray impact parameter for the CT method, and a function of Doppler frequency, approximately proportional to the impact parameter, for the FSI method. The derivation of refraction angles uses the differentiation of the phase of the transformed wave function.

[9] The stationary phase method (which was mainly used when introducing the FSI method) is also the basic computational means in theory of Fourier integral operators [*Egorov et al.*, 1999]. This can be used to show that the CT and FSI methods are different modifications of the same approach.

[10] Section 3 contains an analysis of the resolution of the two methods. The Fresnel zone size, which is related to diffraction on a big propagation distance from the planet limb to the LEO satellite, does not limit the resolution of these methods because they both transform the wave function to the representation of impact parameter and in this representation diffraction effects are reduced. The basic limitation factors are the synthesized aperture and the diffraction inside the atmosphere. These conclusions are confirmed by numerical simulations on the material of simple analytical profiles and high-resolution radiosonde profiles, presented in section 4. We also investigate the sensitivity of both methods to random noise. Section 5 contains the conclusions.