2.1. Radio-Optics Method
 The radio-optics method uses the spectral analysis of radio holograms in small sliding apertures. Each radio hologram is a set of coordinates rGPS(t), rLEO(t) of the GPS satellite and low-Earth orbiter (LEO) and corresponding measurements of the complex wave field u(t). In each sliding aperture centered at time t we compute the local spatial spectrum (t, ω) as the Fourier spectrum of the function u(t) in the time interval from t − T / 2 to t + T / 2, where T is the aperture size, which corresponds to the Fresnel zone [Gorbunov et al., 2000]. The maxima of its absolute value must correspond to the Doppler frequencies of multiple interfering rays. For given satellite positions and Doppler frequency ω we can compute refraction angle ϵ (rGPS(t), rLEO(t), ω) and impact parameter p(rGPS(t), rLEO(t), ω) [Vorob'ev and Krasil'nikova, 1994] (Figure 1. For satellite coordinates at a fixed moment of time t, these functions specify a curve in the plane of the ray coordinates (ϵ,p), each point of the curve corresponding to some Doppler frequency ω, or a virtual ray direction. In the occultation plane we may introduce the Cartesian coordinates (x, y), where the x-axis points in the direction from the GPS satellite to the planet limb, and the y axis is transversal to it (Figure 1). In the approximation of the infinitely remote transmitter and small refraction angles the equation of this curve can be written as
from which it follows that the slope of this curve is approximately equal to the observation distance. The spectral amplitude |(t,ω)| can then be plotted along curve (1). For different moments of time these curves cover the complete (,p) plane. This specifies the mapping of (t,ω) coordinates to (ϵ,p) coordinates. All the spectral amplitudes |(t, ω)| can then be plotted in ray coordinates (ϵ,p), and their maxima must then trace the profile ϵ(p).
Figure 1. Radio occultation geometry. Each observation point at the LEO orbit is characterized by the range of virtual rays passing through the point. A ray reflected from the Earth is shown as a dashed line. In the (ϵ, p) plane each observation point can be represented by a curve of virtual rays. Real rays are defined by the intersection of these curves with the dependence ϵ(p). The branch of the dependence ϵ(p) corresponding to reflected rays is shown as a dashed line.)
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 Figure 1 shows that multipath propagation can arise both owing to complicated atmospheric structures resulting in nonmonotonous profiles ϵ(p) and owing to reflection from the Earth's surface. The complete profile ϵ(p) consists of two branches: direct rays and reflected rays. These two branches have one common point corresponding to the lowest direct ray tangent to the Earth's surface. This ray has the impact parameter pE, and it forms the border of the geometric optical shadow. Rays with p ≥ pE are direct, while reflected rays have impact parameters p < pE. As we show below, the application of the radio-optics method allows for the visualization of both branches of the profile ϵ(p).
 This method has two basic limitations [Gorbunov et al., 2000]: (1) it has a limited resolution and (2) in subcaustic areas the spectra cannot be interpreted in terms of rays. Owing to that, this method has a limited accuracy and is not fit for data processing in the operational mode. Notwithstanding, this method is a very convenient means of the data visualization.
2.2. Canonical Transform Method
 Like in the back-propagation method, we assume a vertical radio occultation with an immovable GPS satellite. The real occultation geometry can be reduced to this case by means of the appropriate coordinate transform [Gorbunov et al., 1996; Kursinski et al., 2000]. The measured wave field uLEO(t) is then given along some curve x(t), y(t). Using the back-propagation algorithm, we transform it to the vertical line at some position x. For the back-propagated field we assume the notation ux(y) because the propagation coordinate x can be looked at as the independent “temporal” coordinate and will be mostly considered as a fixed parameter. We study the ray structure of the back-propagated field because it is equivalent to that of the wave field registered along the LEO orbit.
 In the geometrical optics we introduce the momentum η corresponding to the spatial coordinate y, and rays are described by the canonical Hamilton system [Kravtsov and Orlov, 1990]. In single-ray areas the momentum is equal to ∂Ψ/∂y, where Ψ is the eikonal of the wave field. As a consequence of the Helmholtz equation in a vacuum, |∇Ψ| = 1, and thus . In the geometrical optics, ∇Ψ is the ray direction vector [Kravtsov and Orlov, 1990].
 We may consider a canonical transform from the old coordinate and momentum (y,η) to the new ones (z,ξ). The new coordinate z must be chosen in such a way that the there is not more than one ray for a given z coordinate. This means that in the new coordinates we have single-ray propagation and the momentum ξ is a single-valued function of the coordinate z and can be defined as the derivative of the eikonal.
 If we consider a spherically stratified atmosphere, then the ray impact parameter p, which is equal to , always defines an unique ray. This means that p can be chosen as the new canonical coordinate. For the definition of the new momentum ξ we have the condition [Arnold, 1978]
If we assume that ξ = ξ(η), then we can derive the canonical transform as
which indicates that the new momentum ξ is the ray direction angle with respect to the x axis.
 Egorov's theorem [Egorov, 1985; Hörmander, 1985; Taylor, 1981; Treves, 1982; Mishchenko et al., 1990] states that with this canonical transform we can associate a Fourier integral operator , which will transform the wave function to the new representation. The transformed wave function ux depends on the impact parameter p. This transformation reduces the wave function to the single-ray representation. The refraction angle can then be computed from the derivative of the phase of the transformed wave function.
 The Fourier integral operator associated with a canonical transform can be written in the form [Egorov, 1985]
where S(p,η) is the generating function [Arnold, 1978] of this transform, defined by the equation dS = ξdp + ydη:
and x(η) is the k-Fourier transform of ux(y):
This allows for writing the Fourier integral operator as
The transformed wave function ux(p) has the form A(p)exp(ik∫ξ(p)dp), where A(p) is the amplitude. This allows for the computation of the momentum as
Finally, the refraction angle is computed as
where a is the Earth's radius and xGPS is the x coordinate of the GPS satellite. The second term corrects for the initial direction angle of the incident ray at the GPS satellite.
 This method has the following advantages: (1) it allows for the computation of the refraction angle ϵ for a given ray impact parameter p, unlike the other methods, where ϵ and p are computed simultaneously for given point in the LEO trajectory or in the back-propagation plane, and noise may result in a nonunique function ϵ (p); and (2) the method has no tuning parameters, unlike the back-propagation, where the choice of the back-propagation plane position x is very critical.
 For an atmosphere with strong horizontal gradients, the impact parameter is not a ray invariant. Strong systematic horizontal gradients of the refractivity can result in multivalued profiles ϵ(p) [Gorbunov and Kornblueh, 2001]. In this case, the CT method cannot be applied. Such gradients can arise in the presence of intensive atmospheric fronts. However, the simulations with European Centre for Medium-Range Weather Forecasts (ECMWF) analyses used by Gorbunov  indicate that such strong systematic horizontal gradients occur rarely (we expect the ECMWF analyses to reproduce large-scale atmospheric fronts well). It must also be noticed that the statistical comparisons of GPS/MET and ECMWF data [Gorbunov and Kornblueh, 2001] show their agreement.
 Although the refraction angle is computed from the phase of the transformed wave function ux, its amplitude can also be utilized in data processing algorithms. The amplitude of the measured wave field in single-ray areas can be represented as the vacuum amplitude multiplied by the defocusing and absorption factors [Sokolovskiy, 2000]. In multipath zones the amplitude is also influenced by the effects of interference and diffraction. The integral transform corrects for the effects of defocusing (in the approximation of spherically stratified atmosphere) and multipath propagation. This allows for the following use of the amplitude:
- In the absence of atmospheric absorption, which is the case for the GPS wavelength, the geometric optical shadow border is marked by a very abrupt drop of the amplitude of the transformed wave function. This was found in the numerical simulations with ECMWF global fields used by Gorbunov .
- In the presence of absorption the amplitude of the transformed wave function is proportional to the absorption factor for the ray with given impact parameter, exp (−k∫n″ds), where n″ is the imaginary part of the refractive index and s is the ray arc length. The specific absorption kn″ can then be reconstructed by means of the Abel inversion. The standard technique of the retrieval of absorption uses the computation of the absorption factor by removing the defocusing related to the refraction angle profile, which is computed from the phase [Sokolovskiy, 2000; S. Leroy, Amplitude of an occultation signal in three dimensions, submitted to Radio Science, 2001]. This approach, however, can only be applied in single-ray areas.
- The amplitude can also be used for the data quality control, which will be shown in section 3 in the discussion of the results of processing of real observational data.
 A very important remark must be made about the resolution of this method. As shown in section 2.2, the reconstructed profile ϵ(p) is independent of the back-propagation plane position x. In the same way, it can be seen that it is also independent of the observation distance L. From this we conclude that the resolution is independent of the Fresnel zone size, which is equal to . In an idealized situation, where no restrictions are imposed on the sampling rate and aperture of the measurement data, the only resolution limit is the wavelength. Given a finite measurement aperture A, the resolution d can be estimated as
For GPS/MET observations the observation distance L is ∼3000 km, and the synthesized aperture A can be made as big as 20 km. Then for a wavelength of 20 cm we have the estimated resolution d = 30 m.