Numerical simulations of radio occultation experiments indicate that the effect of multipath propagation and diffraction must play a very significant role in the lower troposphere. Two methods of processing radio occultation data in the multipath zone were previously discussed: the radioholographic (radio-optics) method and the back propagation method. Both methods, however, have their restriction. The radio-optics method cannot accurately take into account the effects of diffraction in subcaustic zones. The back propagation method does not work for complicated caustic structures. In this paper we describe a combined method. The radio-optics analysis is applied both for the radio occultation data measured along the low Earth orbiter orbit and for the back propagated data. The computed spatial spectra are then averaged. This allows one to get rid of diffraction effects in subcaustic zones to a significant extent. The method is validated in numerical simulations on the material of global fields of atmospheric variables from analyses of the European Centre for Medium-Range Weather Forecasts. An example of processing a real GPS/MET occultation is given.
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 Radio occultations using the signals of the Global Positioning System (GPS) can play a very important role in the sounding of the Earth's atmosphere for the purposes of numerical weather prediction [Ware et al., 1996; Rocken et al., 1997; Kursinski et al., 1997]. It is also clear that the reconstruction of the structure of the lower troposphere is of primary importance for these purposes. The Microlab-1 experiment performed in the frame of the GPS Meteorology Program (GPS/MET) provided a big array of occultation data. However, the quality of these data was degraded in the lower troposphere due to technical problems with the receiver [Rocken et al., 1997; Sokolovskiy, 2001b]. To a significant extent, this is explained by the results of numerical simulations of GPS radio occultation experiments, which show that the effects of multipath propagation must play a significant role in the lower troposphere [Gorbunov and Gurvich, 1998b; Gorbunov et al., 2000; Sokolovskiy, 2001a]. Radio signals in multipath areas have a complicated structure, which impedes both their tracking and interpretation. In this paper we shall discuss a new method of the interpretation of lower tropospheric radio occultation signals.
 Two techniques for processing radio occultation data in multipath regions were discussed in the literature: (1) the back propagation method [Gorbunov et al., 1996; Hinson et al., 1997, 1998; Gorbunov and Gurvich, 1998a] and (2) the radio-optics (or radioholographic, or sliding spectral) method [Lindal et al., 1987; Pavelyev, 1998; Hocke et al., 1999; Sokolovskiy, 2001a]. The former method consists in the numerical back propagation of the wave field recorded during a radio occultation experiment to a single-ray region. The latter method is based on the analysis of local spatial spectra of the wave field, where the spectral maxima are interpreted as rays. It can also be observed that the radio-optics method is very similar to the wavelet analysis.
 The comparative analysis of both methods [Gorbunov et al., 2000] indicates that they have their restrictions. The back propagation method does not work for complicated caustic structures. Numerical simulations with high-resolution radiosonde data indicate that for the real atmosphere this restriction may be critical [Sokolovskiy, 2001a]. The radio-optics method cannot locate rays in subcaustic zones, where the wave field cannot be described in terms of geometric optics. The radio-optics method also has a limited resolution. This restriction, however, is not very significant in the lower troposphere. Another problem was that the radio-optics method in its standard variant was inconvenient for the automated data processing.
 In this paper we discuss two simple modifications of the radio-optics method, the first of which allows for the automated data processing, and the second modification reduces the effects of subcaustic zones. For a record of complex field uLEO(t) measured at the low Earth orbiter (LEO) during a radio occultation experiment, we can compute a two-dimensional (2-D) array of sliding spectra ũLEO(t, ω). For each moment of time t the spectrum ũLEO(t, ω) is defined as the local Fourier spectrum of the complex field uLEO(t) for time interval t − T/2 … t + T/2, where T is the size of the sliding aperture. Rays can be identified as the maxima of these spectra. Given the satellite positions for time t, the refraction angle ϵ and ray impact parameter p can be computed from the Doppler frequency ω. This means that the coordinates (t, ω) can be mapped to ray coordinates (ϵ, p), and we can now compute the 2-D spectrum as a function of ray coordinates ũLEO(ϵ, p). Plotting the absolute values of sliding spectra |ũLEO(ϵ, p)| is a very convenient means of data visualization, because the spectral maxima trace the dependence ϵ(p). For automated data processing the dependence ϵ(p) is defined as the position ϵ of the main maximum of the cross section of the spectrum ũLEO(ϵ, p) for given p. This method has an advantage as compared to the standard approach, where multiple spectral maxima ωi are detected for given time t and then corresponding refraction angles ϵi and impact parameters pi are computed. This is because (1) the detection of the main maximum is algorithmically very simple, unlike the detection of multiple maxima and (2) not all of the multiple spectral maxima can be really attributed to rays, while the main maximum for given impact parameter p is much more likely to correspond to the only ray with this p.
 In order to reduce the effect of subcaustic zones, where the spectral maxima cannot be interpreted in terms of rays, we proceed as follows. We compute the back-propagated field uBP(t), and its local spectra transformed the ray coordinates ũBP(ϵ, p). Simple geometrical considerations show that “false” spectral maxima arising in subcaustic zones will have different positions for back-propagated field as compared to the field measured along the LEO trajectory, while the positions of “true” maxima corresponding to rays remain the same. The computation of the spectral maxima of the averaged logarithmic spectrum [ln uLEO(ϵ, p) + ln uBP(ϵ, p)]/2 proves then more stable to the effects of subcaustic zones.
 The method is validated on the material of global fields of atmospheric variables from analyses of the European Centre for Medium-Range Weather Forecasts (ECMWF).
2. Data Analysis
 The back propagation method is used in order to correct for the effects of multipath propagation [Gorbunov et al., 1996; Hinson et al., 1997, 1998; Gorbunov and Gurvich, 1998a]. It uses the solution of the boundary problem of the Helmholtz equation in a vacuum. In this technique the complex wave field recorded along the LEO orbit during a radio occultation experiment is numerically back-propagated to a virtual observation plane (the back propagation plane) located inside the atmosphere. From the geometric optical viewpoint this corresponds to the back continuation of the rays as straight lines, which often allows for disentangling the multipath structure.
 The dependence of refraction angle ϵ versus impact parameter p defines a configuration of straight back-propagated rays and corresponding caustics. There are two classes of caustics: real and imaginary ones [Gorbunov et al., 2000], and the back propagation plane must be positioned in the single-ray area between them. If this is possible, then the back propagation method provides a high accuracy in the reconstruction of refraction angle profiles. In the presence of strong refraction or superrefraction, however, structures with overlapping real and imaginary caustics can arise. In this case, no positioning of the back propagation plane can get rid of multipath propagation.
 Another problem with this method is the choice of the back propagation plane position. This position depends on the caustic structure, which is a priori unknown.
 An alternative technique of processing radio occultation data in multipath areas is the radio-optics method [Lindal et al., 1987; Pavelyev, 1998; Hocke et al., 1999; Sokolovskiy, 2001a]. It is based on the analysis of the local spatial spectra of the recorded complex field in small sliding apertures. Rays correspond to the maxima of the spectra. This method, however, does not work in subcaustic zones, where the wave field cannot be described in terms of geometric optics.
 In this paper we describe an improved approach to the radio-optics analysis based on combining it with the back propagation. By means of the appropriate coordinate transform, the occultation geometry can be reduced to the case of a vertical occultation with the stationary GPS satellite [Gorbunov et al., 1996; Kursinski et al., 2000]. The coordinate transform uses the assumption that the orbital motion of the GPS satellite with respect to the Earth is negligible, and it is occulted by the atmosphere along the vertical direction imposed by the LEO orbit. In the occultation plane we introduce Cartesian coordinates x, z defined as shown in Figure 1. During a radio occultation experiment the complex wave field uLEO(t) along the LEO orbit is measured. The number of geometric optical rays connecting the GPS and LEO satellites is different in different moments of time t. An example of such a situation is shown in Figure 1, where in the moment t1 there is only one ray, while in the moment t2 there are as many as three interfering rays. In the radio-optics method the complex wave field uLEO(t) is subjected to the Fourier analysis in small sliding apertures. For each moment of time t we specify the time interval t− T/2 … t+ T/2, where T is the aperture size, and compute the local Fourier spectrum ũLEO(t, ω). In Figure 1, for the moments of time t1 and t2 the corresponding apertures are schematically shown as thick fragments of the LEO trajectory. For given time t the frequencies ωi of the spectral maxima are then interpreted as the Doppler frequencies of interfering rays at the corresponding observation point.
 Refraction angles and impact parameters can be computed as a function of satellite positions rLEO(t) and rGPS(t) and Doppler frequency ω [Vorob'ev and Krasil'nikova, 1994]:
This specifies the mapping of the coordinates (t, ω) to the ray coordinates (ϵ, p). An observation point for a given moment of time t can then be represented as a curve in the ray coordinates described by (1) with fixed time t and variable frequency ω. An approximate equation for this curve can be derived in the approximation of an infinitely remote GPS satellite. In this case, rays coming to a fixed observation point (x, z) must satisfy the geometric relationship:
Each spatial frequency of the complex field corresponds to a ray direction at the observation point and specifies a ray with ϵ, p coordinates satisfying relationship (1). The intersections of the refraction angle profile ϵ(p) for given atmospheric state with the curve defined by (1) for given observation point (x, z) define the interfering rays at this point. The absolute values of the spectrum |ũLEO(t, ω)| for the corresponding moment of time t can then be plotted along this curve in the ray coordinates, and it must have maxima corresponding to rays. This is schematically shown in Figure 1, where the spectrum at the moment of time t1 has one maximum, while the spectrum for the moment of time t2 has three maxima. The maxima of the spectra for all the moment of time must then trace the dependence ϵ(p).
 The local spatial spectra of the recorded complex field can thus be represented as a function of the ray coordinates ũLEO(ϵ, p) [Gorbunov et al., 2000]. For its computation the spatial spectra ũLEO(t, ω) for different moments of time t corresponding to different curves (1) are interpolated to a standard rectangular ϵ − p grid.
 The standard application of the radio-optics methods consists in the detection of (a variable number of) spectral maxima ωi for different moments of time tk and the computation of corresponding pairs of refraction angles and impact parameters (ϵk, i, pk, i). These pairs are then ordered with respect to p, in order to produce a profile ϵ(p). The disadvantage of this approach is that the spectra can contain a number of local maxima not necessarily corresponding to rays. This is especially important in subcaustic zones, where the spectral maxima cannot be interpreted in terms of rays.
 Our method consists in the use of the spectra transformed to the ray coordinates ũLEO(ϵ, p). The refraction angle profile ϵ(p) is defined as the position ϵ of the main maximum of |ũLEO(ϵ, p)| for given p. This method is based on the assumption that there is only one ray with a given impact parameter. This allows one to get rid of the necessity of detecting and reordering multiple spectral maxima.
 It must be noticed that a similar approach was introduced by Sokolovskiy [2001a]. In this approach, the refraction angle ϵ is computed as a weighted average over the complete cross section of the spectrum |ũLEO(ϵ, p)| for given p:
The disadvantage of this approach is that this average is influenced by the effects of subcaustic zones, and its application results in a significant smearing-out of retrieved profiles of refraction angles.
 Another improvement of the method allows for the suppression of the effect of caustics. For an observation point located in a subcaustic zone the spectral maxima cannot be interpreted in terms of rays. The spectrum will then contain “false” maxima. If we represent the observation point with a corresponding curve in the ray coordinates, then the false maxima will be located along this curve. Using the approximation of small refraction angles, (1) can be rewritten as follows:
which indicates that the slope of this curve is approximately equal to the observation distance x. This means that when the back-propagated wave field is subjected to the radio-optics analysis, the locations of “false” maxima will be different as compared to the analysis of the field recorded at the LEO orbit. This situation is schematically shown in Figure 2. For a given profile ϵ (p) we show dashed thick curves representing points of the LEO trajectory located at caustics separating the areas with one and three rays. The spectra in the vicinities of these curves cannot be interpreted in terms of rays. If we perform the spectral analysis of the back-propagated field uBP(t), whose effective observation distance is smaller, then the caustics will be represented by thick solid curves with a different slope. The quality of local spectra of the back-propagated field ũBP(ϵ, p) will be degraded along these lines. If we use the averaged logarithmic spectrum
the effects of subcaustic zones will be suppressed in it.
3. Numerical Simulations and Data Processing
 In the numerical simulations we processed a few artificial occultation data sets. The artificial data were generated using the geometry of real GPS/MET occultation events and global fields of atmospheric variables from analyses of the ECMWF. We used the wave optics model [Gorbunov and Gurvich, 1998b] for generating the artificial data. We also computed the refraction angles using the geometric optical ray equations, which were treated as the exact solution, which had to be reconstructed from the wave field. We chose the occultation events with complicated caustic structures, where the back propagation method is inapplicable.
 The artificial data were processed by means of the techniques described above. The vertical aperture size used for the radio-optical processing of LEO data was 10 km. The back propagation plane position was x= 400 km. The vertical aperture size for the radio-optical processing of back-propagated (BP) wave field was 1 km. The spectra were interpolated to the standard rectangular homogeneous grid of ray coordinates with a mesh size Δp× Δϵ = 20 m × 0.0005 rad.
 For the simulations we chose two artificial occultations. In one of them the back propagation method results in big errors in the reconstruction of the refraction angle profile. This is the case if the refraction angle profiles have sharp spikes, which results in structures with overlapping real and imaginary caustics [Gorbunov et al., 2000].
Figures 3 and 4 show the results of the processing of two artificial occultations. Shown are the local spatial spectra of the LEO wave field and the back-propagated wave field, the averaged spectrum, and refraction angles. Instead of the impact parameter pwe use the ray leveling height defined as p − rE, where rE is the local curvature radius of the reference ellipsoid. We compare refraction angles computed using the geometric optics (GO), refraction angles computed using the back propagation (BP) method, and refraction angles computed using the combination of back propagation and radio-optics methods described above (BP + RO).
 These plots show that the local spatial spectra plotted in the ray coordinates are a very convenient means of data visualization. The comparison of LEO and BP spectra shows that they consist of the same basic structure, corresponding to the true refraction angle profile, on which caustic effects are superimposed. The caustic effects are different for LEO and BP spectra. The averaging allows for some improvement of the reproduction of the basic structure. The reconstructed refraction angle profiles show a good agreement with the accurate geometric optical solution.
 In Figure 3a we indicated three cross sections of the 2-D spectra corresponding to local spectra in three selected observation points. The local spectra in the observation points are shown in Figure 5. The spectra in one-ray and multiray areas indicate very distinct maxima corresponding to the rays. However, they also have some additional maxima, shown by arrowheads. The spectrum in the subcaustic zone indicates a lot of local maxima. When using the standard approach to the radio-optics analysis, these maxima will produce many false (ϵ, p) pairs, which will distort the retrieved profile ϵ(p).
Figure 6 shows an example of processing a real GPS/MET occultation event. This is a very interesting example of an occultation with a refraction angle profile containing a very sharp spike at ray heights of 4–5 km, which must have resulted from a humidity layer. This occultation was processed with the same parameters of the inversion algorithm as those used for processing the artificial occultations. The application of the back propagation method results in an ambiguous refraction angle profile with very strong oscillations. This is characteristic for the situation, where the back propagation procedure cannot disentangle the multipath structure. On the other hand, the spectral analysis allows for revealing the refraction angle profile, and the algorithm described above gives sensible results.
 Two methods of processing radio occultation data in multipath regions were previously suggested, which are (1) the back propagation method and (2) the radio-optics method. It was also shown that both methods have their restrictions. The radio-optics method cannot be applied in subcaustic zones, where the diffraction effects are significant and the wave field cannot be interpreted in terms of rays. On the other hand, the back propagation method cannot be applied for complicated caustic structures, which can arise if the refraction angle profile has sharp spikes.
 We suggest two improvements for the radio-optics method. The first improvement consists in the computation of the local spatial spectra of the wave field in the ray coordinates ϵ, p. The dependence ϵ(p) is then computed as the position of the main maximum of the cross section of the spectrum for given p. This simplifies the automated use of the radio-optics method.
 The second improvement utilizes the combination of the radio-optics and back propagation methods. For this purpose, we compute the local spatial spectra of the back-propagated wave field. For the determination of the profile ϵ(p) we use the averaged spectrum. This allows for the suppression of the effects of caustics.
 The simulations showed that the combined algorithm can significantly improve the accuracy of the reconstruction of the refraction angles for the cases where the back propagation method does not work. Another advantage of this algorithm is that it can be used with fixed values of tuning parameters unlike the back propagation method, where the correct choice of the back propagation plane position is critical.
 This work was performed with the financial support of the Max-Planck Institute for Meteorology (Hamburg, Germany), Institute for Geophysics, Astrophysics, and Meteorology, University of Graz (IGAM/UG), and the Russian Foundation for Basic Research (grant 98-05-64717). The author is grateful to S. V. Sokolovskiy, for his valuable comments on the paper.