The interpretation of radio occultation data in the lower troposphere is a complicated problem due to strong multipath effects. This problem can be solved on the basis of the wave optics. We analyze Microlab-1 radio occultation using two radio-holographic approaches: the radio-optics method, which employs the analysis of the local spatial spectra of the registered wave field, and the canonical transform method, which is based on the principles of the theory of Fourier integral operators. The radio-optics method is a means of data visualizaion, while the canonical transform method must be used for achieving accurate numerical results in processing measurements in multipath areas. The comparison of the results of the application of these two methods is a useful method of data quality control. We show examples of radio occultations with multipath propagation and reflection from the Earth's surface. The application of the radio-holographic methods also allows for the detection of corrupted data caused by phase lock loop failure.
 The interpretation of GPS meteorology (GPS/MET) radio occultation data in areas of multipath propagation was recently addressed in a series of papers [Gorbunov et al., 1996; Karayel and Hinson, 1997; Gorbunov and Gurvich, 1998a, 1998b; Pavel'ev, 1998; Hinson et al., 1998; Hocke et al., 1999; Gorbunov et al., 2000; Gorbunov, 2001]. Multipath propagation arises in the presence of complicated structures of the atmospheric refractivity, which are typical for the ionosphere and the lower troposphere. All the data processing methods, discussed in the above papers, were designed in order to reconstruct profiles of geometric optical refraction angle from measurements of wave field. These methods use complete records of complex wave fields, or radio holograms. Because of that, these methods can be termed radio-holographic. Some of the basic ideas of radio-holographic data processing were introduced in earlier papers [Marouf et al., 1986; Lindal et al., 1987] discussing the radio occultation sounding of planetary atmospheres.
 The radio-holographic methods are based on the synthesis of the wave optics and geometrical optics. This approach is very effective because of the following reasons: (1) the wavelength of GPS/MET observation is short enough for the short-wave asymptotic solutions of the wave problem [Mishchenko et al., 1990] to be applicable and (2) the inverse problem is formulated much more easily in the geometric optical approximation than in the wave optics.
 In this paper we analyze the GPS/MET data using the radio-optics method and the canonical transform method. The radio-optics, or sliding spectral, method was already used for processing of radio soundings of planetary atmospheres [Lindal et al., 1987]. It uses the spectral analysis of the wave field in small sliding apertures. This method can be very easily numerically implemented, but it is not convenient for the accurate computation of refraction angle profiles. However, the local spatial spectra of wave field can be plotted in the ray coordinates (refraction angle and impact parameter), and the spectral maxima must then trace the refraction angle profile. This is a very simple means of data visualization, which allows for very fast processing of a big number of occultations.
 The canonical transform (CT) method was recently developed as a significant generalization and improvement of the back-propagation technique [Gorbunov, 2001]. It utilizes the theory of Fourier integral operators. In this method the wave field is transformed from the representation of the spatial coordinates to the representation of ray coordinates. The transformed wave function depends on the impact parameter. The derivative of its phase is then the ray direction angle. The amplitude of the transformed wave function is then nearly constant outside the geometric optical shadow zone, and it drops very abruptly at the shadow-light border.
 The application of these two methods for the analysis of the GPS/MET radio occultations allows also for data quality control. Many of the occultations contain corrupted data in their lowest parts. This corruption must be due to the phase lock loop failures [Sokolovskiy, 2001], and it manifests itself in the negative bias of reconstructed refractivities [Rocken et al., 1997]. Corrupted fragments of radio occultations are seen as very chaotic patterns in the visualized local spatial spectra. The application of the canonical transform method to such data also results in strong scintillations of the amplitude and phase.
 We processed the complete Prime Time 4 of the Microlab-1 data. We found a series of occultations, where the spatial spectra in multipath areas behave in a regular way and trace nonmonotonous refraction angle profiles, which are characteristic for multipath propagation. These results are consistent with the results of the canonical transform method. This allows for classifying such data as suitable for further utilization. In many occultations the spectra reveal the ray reflected from the Earth's surface [Beyerle and Hocke, 2001]. We processed such occultations reaching the Earth's surface, and the canonical transform method allowed for the correction of effects of multipath propagation due to reflected rays.
2. Data Analysis
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 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.
 The following remarks must be made about the Fourier integral operator (7):
Although both the operator and wave field ux depend on x, this dependence vanishes in ux because is the vacuum propagator and x() = 0()[Zverev, 1975; Martin, 1992].Thus the position x of the back-propagation plane can be chosen arbitrarily.
Integral (7) can be rewritten using the coordinate ξ = arcsin and functions with shifted argument vx(Δy) = ux(a + Δy), and vx(Δp) = ux(a + Δp), which turns it into the inverse k-Fourier transform of the function
This shows that the algorithm allows for a fast numerical implementation on the basis of fast Fourier transform.
 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.
 In this section we describe the results of processing real radio occultation data by the radio-optics and canonical transform methods. We processed the Prime Time 4 of Microlab-1 data (2–16 February 1997), which includes a total of 1947 radio occultations. All the occultations were processed in batch mode without tuning any inversion parameters. We selected 85 occultations indicating multipath propagation and 35 occultations reaching the Earth's surface, where the reflected ray was detected. Here we discuss a few of the most interesting occultation events. We also chose a few examples of data, which we classify as corrupted in order to demonstrate the data quality control possibilities.
Figure 2 shows the most interesting and unique example of strong multipath propagation. We show the local spatial spectra plotted in gray scale in ray coordinates and the refraction angle profile computed by the canonical transform method (Figure 2a), the amplitude of the transformed wave function (Figure 2b), the hybrid humidity computed from the reconstructed refractivity, and temperature from analyses of the European Centre for Medium-Range Weather Forecast (ECMWF) (Figure 2c), and the hybrid temperature computed from the reconstructed refractivity and humidity from the ECMWF analyses and local temperature profile from ECMWF analyses (Figure 2d).
 The computation of the hybrid humidity uses the reconstructed refractivity profile, hydrostatic equation, and the prescribed temperature profile taken from the ECMWF analyses [Kursinski et al., 2000]. The hybrid temperature is computed from the reconstructed refractivity profile, hydrostatic equation, and the prescribed humidity profile taken from the ECMWF analyses [Kursinski et al., 2000]. We utilized the interpolated ECMWF temperature and humidity profiles computed by Gorbunov and Kornblueh .
 In the refraction angle plots the vertical coordinate is the ray leveling height Δp = p − a, i.e., the ray impact parameter p shifted by the Earth's local curvature radius a. We use the maximum of the correlation of amplitude with the step function as the cutoff height. Reconstructed hybrid humidity is not always positive, which is due to the inaccuracy of the ECMWF temperature and the errors of the Abel inversion resulting from horizontal gradients. We did not correct for this effect because we use these inversions for illustrative purposes only. The differences between GPS and ECMWF humidities and temperatures are consistent with retrieval error estimates given by Kursinski et al. .
 In this occultation the refraction angle profile is characteristic for an atmospheric waveguide. We assume that the waveguide is created by a humidity layer at altitudes of 3–4 km because humidity field usually has a complicated structure. This layer is seen in the reconstructed humidity. The ECMWF data also indicate a humidity layer at the same height. It is an example of a situation, where the back-propagation method cannot be applied. The amplitude of the transformed wave function indicates here some oscillations, which shows that the data quality in the multipath region may be degraded. On the other hand, the results of the canonical transform are consistent with the local spatial spectra. This allows for the conclusion that these data are suitable for inversion.
Figures 3 and 4 show further examples of occultations, where humidity layers result in multipath propagation. In these examples the spatial spectra maxima do not show as good a correlation to the refraction angles derived from the CT method. This must be explained by the limited resolution of the radio-optics method and its inapplicability in subcaustic zones [Gorbunov et al., 2000]. In Figure 3 we see extra spectral maxima in the subcaustic or shadow areas (similar examples of the spectra of artificial occultation data were given by Gorbunov et al. . However, the CT curve almost always follow the spectral maxima, which can be strong for the primary ray or weak for the secondary rays. In Figure 4 the radio-optics method proves incapable of resolving small-scale structures due to its limited resolution.
 As shown by Beyerle and Hocke , GPS/MET data contain examples of rays reflected from the Earth's surface. Figure 5 presents an example of an occultation event reaching the Earth's surface, where a reflected ray was detected. In this example, the height of the lowest direct ray is estimated as 2.04 km. The spectral maxima of the reflected rays are visible as a white stripe below this height (compare Figure 1). In this occultation the last measurements were performed in the zone of the geometric optical shadow. The border between light and shadow zones is marked by a very sharp drop of the amplitude of the transformed wave function. This allows for the location of the impact parameter of the lowest ray that touches the Earth's surface (shadow border), within ∼30 m. The difference between GPS and ECMWF humidities is here ∼1 g/kg, and the temperature difference is ∼5 K. A possible explanation is that this occultation event was observed above the Pacific Ocean, where the quality of analysis data is expected to be degraded because of the lack of observational data.
 Although we do not use the information about reflected rays in the inversion procedure, this example shows that the radio-holographic techniques allow for the correction of effects of multipath propagation due to both direct and reflected rays. Processing occultations with reflected rays do not require any modifications of the algorithms of the derivation of refraction angles. In the radio-optics method we can see spectral maxima of both direct and reflected rays. In the CT method the restored profile of refraction angle may also contain the branch corresponding to reflected rays. In the inversion procedure this branch is cutoff using the amplitude of the transformed wave function as described above. Because the amplitude of reflected rays is significantly lower than that of direct rays, this allows for the effective removal of reflected rays.
Figures 6 and 7 show examples of occultation events, where the spatial spectra of the wave field indicate a sudden shift at some moment of time. The shifts of the spectra are extremely big, and they may have positive as well as negative sign. They cannot be attributed to any atmospheric structures like superrefraction layers because for this interpretation we must assume atmospheric irregularities with characteristic vertical scales of 6–8 km. Because of that, we consider these spectral patterns to be signatures of phase lock loop failures.
Figure 8 presents an occultation where the spectra indicate a complicated pattern at altitudes, where the effects of atmospheric inhomogeneities (mostly due to humidity) cannot be as significant. The CT amplitude indicates strong scintillations, which also corroborates the conclusion about the bad quality of these data. Strong amplitude oscillations can also be seen in Figure 7. Figure 6 shows a different situation. In the interval of ray heights from 6 to 12 km the wave field computed in the representation of the ray coordinates (p,ξ) can be approximately represented as the superposition of the strong regular component and the weak shifted component. Because of this, the amplitude oscillations of the shifted component are suppressed. This example shows that the CT amplitude cannot be always used as the indicator of bad data quality.
 In this paper we discussed the application of two radio-holographic methods for the analysis and quality control of GPS/MET data. The radio-optics method is a very convenient means of data visualization; the canonical transform method can be used for accurate data processing in operational applications. The comparison of the results of the two methods allows for data quality control. This is very important at the stage of primary data analysis preceding their operational use in systems of numerical weather prediction. We believe that at this stage data must be analyzed by different techniques, which will help to check that the data are usable and to understand the limits of their usability.
 The canonical transform method must be looked at as a generalization of the back-propagation method because it is also based on the transformation of the wave field to single-ray situation. However, the canonical transform method has a much wider limits of the application than back propagation. Algorithmically, it is a composition of back propagation and a Fourier integral operator. It can always be applied when the refraction angle profile is single-valued. This condition can be broken by strong horizontal refractivity gradients in the lower troposphere. However, the results of the numerical simulations with ECMWF data show that such strong gradients occur rarely. On the other hand, there are no algorithms of inversion of such multivalued refraction angle profiles.
 The advantages of the canonical transform method can be classified as follows: (1) it has no tuning parameters, which allows for its application for processing big arrays of measurement data operationally; and (2) it allows for the computation of refraction angle for a prescribed impact parameter and the measurement noise does not result in multivalued refraction angle profiles. In the computation of refraction angles, only phase of the transformed wave function is used. However, its amplitude can also be used (1) for the identification of the geometric optical shadow border, (2) for the reconstruction of the atmospheric absorption, and (3) for data quality control.
 We processed the complete Prime Time 4 of the Microlab-1 data and chose occultations with multipath propagation and reflections from the Earth's surface. In this paper, we show a few such examples of occultation events with multipath propagation due to humidity layers. We analyze them using both the radio-optics and canonical transform methods and show that they produce consistent results. This indicates that these data are usable. We also show that the reflected ray can be separated from the direct ray by means of the canonical transform method.
 Many occultations contain fragments corrupted because of phase lock loop failures. We show examples of processing such data. The corrupted fragments can be identified by chaotic patterns in the visualized spatial spectra as well as by strong scintillations of the amplitude computed in the canonical transform method.
 This work was performed with the financial support of the Institute for Geophysics, Astrophysics, and Meteorology (Graz, Austria) and the Russian Foundation for Fundamental research (grant 01-05-64269). The author is grateful to A. S. Gurvich (Institute for Atmospheric Physics, Russian Academy of Sciences), Y. A. Kravtsov (Space Research Institute, Russian Academy of Sciences), and G. Kirchengast (Institute for Geophysics, Astrophysics, and Meteorology) for useful scientific discussions.