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 Wave phenomena in the upper atmosphere can be studied using the high-precision Global Positioning System (GPS) radio navigational field. In this paper, basic principles, accuracy, and vertical resolution of the radioholographic technique for studies of ionospheric wave phenomena are presented for the general case when the orbits of the satellites are arbitrary. Results of testing of the radioholographic method are discussed using orbital station MIR and geostationary satellites (MIR/GEO) and GPS/Meteorology (GPS/MET) radio occultation data. The radioholographic method has high vertical (12–30 m) and angular (4–8 μrad) resolution, which has been validated by directly observing multibeam propagation in the atmosphere and revealing signals, reflected from the sea, in GPS/MET and MIR/GEO radio occultation data. We show that this method allows one to determine the vertical profile of the electron density and monitoring wave structures in the upper atmosphere. As an example of this approach, observations of the summer Antarctic mesosphere on 7 February 1997 are presented. We show, by combining phase and amplitude analysis, that a vertical resolution of 0.3–0.5 km reveals wavelike structures with spatial periods from 1–2 km to 8–10 km in the vertical electron density distribution in the D and E regions. Variations in the gradient of the electron density from ±5 × 103 to ±8 × 103 el/(cm3 km) at altitudes of 72–95 km were observed. The obtained results demonstrate the high-technology level of the radioholography approach and open new perspectives for radio occultation experiments: measurements of the characteristics of the natural processes in the atmosphere, mesosphere, and ionosphere and observations of the state of the sea surface by measuring parameters of reflected signal simultaneously with radio occultation experiments.
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 Radio occultation is an effective high-accuracy, high-vertical-resolution technique for studies of the Earth atmosphere from space. Practical testing began in 1989 using a telecommunication line between the orbital station MIR and geostationary satellites (MIR/GEO) at wavelengths of 2 and 32 cm [Yakovlev et al., 1995]. A satellite known as MicroLab-1 was launched into low Earth orbit (LEO) for radio occultation experiments with high-precision GPS radio navigational signals in the two coherent frequency bands F1 = 1575.6 MHz and F2 = 1227.6 MHz (GPS/MET experiment). Since mid-1995, with more than acceptable accuracy, many thousands of refractive index, temperature, density, and pressure profiles of the atmosphere, distributed over both land and ocean areas, from 1 to 40 km altitude or more, were determined [Ware et al., 1996; Kursinski et al., 1996, 1997; Rocken et al., 1997; Feng and Herman, 1999]. Electron density distribution was also obtained in different areas of the Earth's ionosphere, mainly in the F region [Schreiner et al., 1999; Hajj and Romans, 1998; Vorob'ev et al., 1999]. However, most publications do not concern the upper atmosphere from 45 to 200 km because of the need to account for the influence of the F region when solving inverse radio occultation problems. Influence of horizontal gradients owing to asymmetry of the ionosphere constitutes a problem for inversion radio occultation data also. High precision of radio navigational fields requires more accurate and effective scientific methodology for inferring atmospheric, mesospheric, and ionospheric parameters. Modeling radio propagation in the Earth's atmosphere shows that a multibeam situation may exist for the smooth vertical profiles of the modified refraction index having a monotonically diminishing dependence on height [Pavelyev et al., 1996; Pavelyev, 1998]. Thus the main principle of classical radio occultation theory, the one-ray assumption, is refuted. The radioholographic approach has been proposed to directly reveal multibeam propagation and to accurately measure altitude dependence of physical parameters of atmosphere and ionosphere with high vertical resolution [Pavelyev, 1998; Hocke et al., 1999]. Also, the backward method was suggested [Gorbunov et al., 1996; Gorbunov and Gurvich, 1998; Karayel and Hinson, 1997; Mortensen and Hoeg, 1998; Mortensen et al., 1999] with the aim of removing the influence of multibeam propagation and heightening vertical resolution. The quantum cell approach now is under development for the investigation of extreme spatial resolution and accuracy of radioholographic methodology [Gorbunov et al., 2000]. The aims of this paper are (1) to discuss the results of testing of the radioholographic method using MIR/GEO and GPS/MET radio occultation data and (2) to reveal wave structure in the upper atmosphere using combined radioholographic analysis of the amplitude and phase in the GPS/MET radio occultation data.
2. Radioholographic Method
 The scheme of the radio occultation experiments is shown in Figure 1. The MIR/GEO team used a transmitter installed on the orbital station MIR and receivers installed on GEO satellites to test the radio occultation technique in decimeter and centimeter wavelength ranges at λ = 32 and 2 cm [Yakovlev et al., 1995]. The GPS/MET team used a small satellite called MicroLab-1 and the radio navigational field emitted by a GPS satellite at λ = 19 and 24 cm. For analysis of the basics of the radioholographic method the terrestrial atmosphere is modeled locally as spherically symmetric, with a local center of curvature O. One of the “ray paths” corresponding to signals propagating from GPS to the low orbital satellite (LEO) (points G and P, respectively) lies in the plane of Figure 1, which contains point O. This ray intersects the line PO at the angle β (Figure 1) and has impact parameter p and corresponding refraction angle ξ(p). A signal reflected from the Earth may also be observed (the path GDP in Figure 1). The next connection exists between ξ(p) and p for the case of spherical symmetry:
where R1 and R2 are the distances PO and GO, respectively. Since the functions θ(t), R1(t), and R2(t) are known from trajectory data, equations (1) give impact parameter p if the refraction angle ξ(p) or the angle β is measured. The radio field E(r, t) may be presented as a superposition of the waves having complex amplitude Asj(β) propagating at different angles βj relative to the line PO:
where ωo = 2πfo; fo is the carrier frequency of the radio field, nj(l) is the refraction index distribution along the jth ray trajectory, M is the number of the ray trajectories connecting points P and G (M may be a function of time depending on physical conditions in the atmosphere), and Φ(βj, t) is the eikonal function [Kravtsov and Orlov, 1990]. The function Φ(βj, t) includes the free space Sb(t) and combined ionospheric and atmospheric contributions S(βj) to the eikonal function. The eikonal functions S(βj) that arise owing to refraction and reflection mechanisms have a common property derived under the assumption of spherical symmetry of the atmosphere [Pavelyev et al., 1986]:
where n(R1) and n(R2) are the refraction indexes at the points P and G, respectively, and Fdj is the Doppler frequency. The record of complex radio signals along the LEO trajectory is the radiohologram's envelope that contains the amplitude A(t) and phase path excess ψ(t) = kSe(t) of the radio field as functions of time. Temporal dependencies of the amplitude A(t) and eikonal function Se(t) are given in the GPS/MET radio occultation data. The functions A(t) and ψ(t) may be combined in the complex form
A reference wave field Em(t) = exp [iψm(t)], ψm(t) = kSm(t) may be used to reveal the angular spectra Asj(βj) from the radiohologram, where ψm(t) and Sm(t) are the expected path excess and eikonal function for the radio occultation region. The ray corresponding to the phase function ψm(t) begins at the point G and intersects the direction PO at the angle βm(t) at the point P (Figure 1). The function Sm(t) may be evaluated using models of the atmosphere and ionosphere [Pavelyev et al., 1996]. The next equation for Asj(βj) may be found after multiplying both sides of equation (2) by the reference field Em(t):
Using relation (3), the difference S(βj) − Sm(t) may be presented inside the time interval −T/2 < t < T/2 in the form
where all derivatives in equation (6) correspond to the time instant t = 0, which determines the spatial position of the phase center of the focused synthetic aperture, and T is the time interval of coherent data handling. It is convenient to apply a spectral method for solution (5). For this aim, it is necessary to multiply both parts of equation (5) by factor exp (−iωt) and integrate on time in the interval − T/2 ≤ t ≤ T/2. In the result the main power will correspond to the rays that are matched with function Sm(t) in the sense that only two terms on the right side of equation (5) may be retained. For such rays, after integration the next equation describing the radiohologram's spectrum W(ω) may be obtained:
where M1 is the number of rays that are in coherence with the reference beam and index “o” refers to the time instant t = 0. According to equations (8) the module of function f(ω, ωj ) has a sharp maximum at the value of the angle βj equal to
A similar connection exists between the impact parameter pj and ω:
The measured values of the angle βj and pj correspond to the time instant t = 0. If the orbits of LEO and GPS satellites are circular, then Q(pj, pm)o = dθ/dt and v is equal to R1dθ/dt. Equations (9) and (10) may be used for estimating the angular Δβ and vertical Δh resolution of the radioholography approach. This estimation may be obtained using the form of the function f(ω, ωj) (equation (8)). The amplitude value of f(ω, ωj) at ω – ωj = π/T is equal to 2/π, which is nearly one half (on the power) relative to the maximal value f(ω, ωj) = 1. To obtain an estimation of the resolution, one may set ω − ωj = π/T in equations (9) and (10). Then the next equations for Δβ and Δh may be revealed:
Equations (11) describe the angular and vertical resolution in the general case when the orbits of the satellites are arbitrary. Both parameters Δβ and Δh depend on the effective velocity v, which is determined from equations (7) and (9), if the trajectories of the satellites in an inertial coordinate system with center at the point O (Figure 1) are known. The vertical resolution Δh depends on the distance R1 and does not include dependence on the angle βm. The angular resolution Δβ diminishes when the angle βm increases. The accuracy of the radioholographic method increases when v and T are growing and the wavelength is diminishing. The wavelength dependence of angular and vertical resolution, equations (11), is distinct from the Fresnel one: Δh, Δβ ∼ λ (radioholographic), and Δh, Δβ ∼ λ1/2 (Fresnel). The position of the main peak in the radiohologram spectrum according to equations (7) and (8) determines the impact parameter p and then, from equations (1), the bending angle ξ(p). After estimating dependence ξ(p) the standard Abel inversion procedure may be used for determining the refraction index altitude profile [Gorbunov et al., 1996]. The angular spectrum may be prolonged along the rays up to any plane disposed near the atmosphere. In this case the angular spectrum may be interpreted as a “radio image” or “radio brightness distribution” of the atmosphere and ionosphere as seen from LEO. The brightness of the radio image may be found using estimation of the module of the amplitude Asj(βj) of the angular spectrum using the following equation:
After prolongation along the rays, the radioholography approach can give the distribution of the radio field in the space between the LEO satellite and the atmosphere. Thus the radioholography approach can realize, in principle, the main task of the backward method described by Karayel and Hinson .
 It may be noted that the spectral radioholographic method reveals only the rays in the angular spectrum that are in coherence (or matched in the sense of high-resolution radar imaging technology) with the reference beam. The influence of turbulence, horizontal irregularity of the atmosphere, and other effects that cannot be predicted by the reference model diminishes the vertical resolution. The resolution may be lower owing to the finite time of the ray's existence (in full analogy with the time interval of reflection existence from the radar target). This time depends on the physical mechanism of the ray's arising. The amplitude of each ray may depend on time also. In this case the radioholographic method may account for the regular time dependence of the ray's amplitude evaluated from knowledge of its physical nature.
3. Tests of Radioholography: Direct Observation of Multibeam Propagation and Reflections From the Sea
 The application of radioholography to the analysis of radio occultation data (MIR/GEO, wavelength 32 cm) led to direct observations of multibeam propagation in the troposphere, including signals reflected from the sea [Pavelyev et al., 1996, 1997]. An example of multibeam propagation in the troposphere (at level H = 3 km), obtained from a MIR/GEO radio occultation experiment, is shown in Figure 2c. The time interval between neighboring spectra was 1/32 s, which corresponds to a distance of nearly 250 m along the LEO trajectory. The time increased from the lower left corner toward the upper right corner (along the vertical direction). The width of the main beam in the angular spectrum at half power was about 40 μrad, which corresponds to a vertical resolution in the troposphere of the order of 80 m. Usually only one maximum is seen for a ray trajectory connecting a transmitter on the MIR orbital station and a receiver on a geostationary satellite. An example is the spectrum in the lower left corner in Figure 2c. However, a gradual transition from one- to three-beam propagation is noticeable in Figure 2c. The power and vertical size of the secondary beam were sometimes of the order of the corresponding values of the main beam. Changes in the angular spectrum were connected with the spatial distribution of the radio field. This effect is associated with the intersection of the caustic boundary. Details of this process are important for interpretation of diffraction phenomena in the frame of the fundamental theory of radio wave propagation in a layered atmosphere. The results obtained by radioholography (Figure 2) give new experimental evidence of the caustic surface intersection and associated diffraction phenomena as observed from space.
 The high vertical resolution of radioholography is also evident from analysis of the GPS/MET radio occultation data. The resulting vertical radio brightness distributions are shown in Figures 2a, 2b, and 2d. These data are related to radio occultation event 0392 (5 February 1997, 1354:42 UT; 55.6°N, 139.2°E). The curve in Figure 2a shows the experimental angular spectrum at a height of 64 km. The angular width of the maximum was about 20 μrad, and the corresponding vertical size of the maximum was near 70 m. The broadening of the angular spectrum in the upper atmosphere may be related to turbulence effects. The radio brightness distribution in the troposphere at 2.8 km is shown in Figures 2b and 2d. One pixel in the angular spectrum corresponds to a 0.004 mrad variation in the arrival angle and a 12 m change in the minimum height of the ray above the terrestrial surface. The main peak in Figures 2b–2d corresponds to the radio occultation signal propagated along the path GP (Figure 1). The peak has two components because of multibeam propagation in the troposphere. A signal reflected from the Okhotskoe Sea may also be seen (two small peaks in Figures 2b and 2d). According to prolongation along the ray in the angular spectrum these signals are projected on the line DD' (Figure 1) and have, as a result, negative height values. The intensity of a reflected signal is about 15–20 dB below the level of a radio occultation signal. Because of this obstacle, reflected signals are not seen directly in the phase data. It follows from Figure 2 that the radioholographic method may resolve details in one-dimensional vertical radio images of the atmosphere with a scale of 30–50 m, which corresponds to a spatial resolution of about 1/10 of the Fresnel zone size. This value is somewhat higher than the magnitude of the expected spatial resolution, ∼100 m, based on the modified version of the backward propagation method [Karayel and Hinson, 1997; Mortensen and Hoeg, 1998].
4. Application of Radioholograpy to Observations of Waves in the Upper Atmosphere
 Radioholography can be used to determine electron density altitude profiles and reveal various kinds of wave motion in the upper atmosphere. In the radioholography approach the upper ionosphere influence may be accounted for directly in the phase of the reference beam Ψm(t). This may be done by means of using, for example, the international reference ionosphere 1995 (IRI-95) model for the region and time of a radio occultation experiment. The amplitude and phase components of the radiohologram envelope of the D and E regions of the ionosphere are presented in Figure 3 for GPS/MET occultation event 79. The radiohologram envelope has been obtained after subtracting the phase of the reference beam accounting for the F region contribution. Event 79, of 9 February 1997, corresponds to winter daytime in the north part of the Pacific Ocean near Japan. The time-spatial coordinates of ray minimal height H varied from 48.1°N, 226.8°W, H = 112 km, and 0719:53 UT to 47.5°N, 226.74°W, H = 60 km, and 0720:25 UT. The two curves at the bottom of Figure 3 correspond to the experimental phase excess variations at the first frequency F1 (upper curve) and at the second frequency F2, as functions of height. Both curves are multiplied by a factor of 2. The variations in amplitude (top pair of curves in Figure 3) are also strongly correlated. The level of variations in the phase path excess and in amplitudes at the two frequencies is proportional to the ratio f22/f12, and this demonstrates that the variations originate because of fluctuations in the electron density. The phase excess that is connected with wave structures changes in the interval ±2 cm, with a random noise contribution of about ±1 mm. Spatial periods in the 1–3 km range can be seen in the phase excess data of Figure 3. The spatial periods in the 1–2 km range can be clearly observed in the amplitude data. A feature at the height interval 72–78 km is seen in both the phase excess and amplitude data. These variations correspond to wave structures in the electron density distribution in the D region of the ionosphere. A perturbation method may be applied to find the wave part of the electron density altitude profile. In this case the phase and amplitude part of the radiohologram envelope may be considered separately as independent phase and amplitude information channels. Correspondingly different inversion formulas may be applied to find the vertical distribution of electron density and its gradient. From a radar technology point of view, the principal spatial resolution in the two independent channels, the phase and the amplitude, is dependent on the length of the aperture (or time of observation) as was shown in section 2 for the spectral method.
 For the phase channel, Abel's inversion formula may be applied in the form
where p is an impact parameter corresponding to the free-space propagation, h is the height above the Earth's surface, and N(h) is the perturbation part of the electron density altitude distribution (el/m3). The function κ(p) in equation (13) may be used in different forms corresponding to the way chosen for inversion of radio occultation data. Below, the following three formulas will be used for κ(p) for description of wave structures in the D and E regions of the ionosphere:
where F1r and F2r are the phase residuals at the frequencies F1 and F2, respectively, after subtracting the F region's contribution using the reference phase ψm(t); C is the scaling factor; and f1 and f2 are the frequencies L1 and L2 (hertz), respectively. To obtain the N(h) profile in the D region, the first two formulas (14) have been used. The frequency difference F1 – F2 has been applied for finding the vertical profile of the electron density distribution in the E region of the ionosphere.
 The amplitude variations connected with ionosphere influence may be used separately for obtaining the wave part of the electron density distribution and its gradient. The refraction angle may be restored from the amplitude measurements using the equations developed by Pavelyev et al. :
where pb(t) is the impact parameter corresponding to free space and X(t) is the power attenuation of the radio occultation signal owing to refraction relative to free space.
Equations (1) and (15) give the temporal dependence of the impact parameter p(t) and the refraction angle ξ(p). To reveal the gradient of the electron density distribution from the amplitude data, the following equations [Pavelyev et al., 1986] may be used:
where a is Earth's radius, A1,2 is the amplitude of the radio occultation signal, A1,2o is the amplitude before radio occultation, and Ca1,2 is a scaling factor dependent on frequency. Expressions (16) and (17) give the dependence dN(h)/dh1,2 on the height h separately for each frequency f1, f2.
 The results of the restoration dNe(H)/dH for event 79 are shown in Figure 4 (left curve). This profile reveals wavelike structures in the D region plasma with a spatial period of 1–2 km and variations in the electron density gradient from ±3 × 103 to ±6 × 103 el/(cm3 km). The main peak in the D region gradient is observed at H = 75 km with a value of 8 × 103 el/(cm3 km). This peak is also clearly seen in the amplitude part of the radiohologram (Figure 3). The sharp spike in the vertical distribution of the electron density at height 106 km corresponding to the sporadic E region is seen in the right curve of Figure 4 (displaced for comparing at 30 × 103 el/(cm3 km)). The strong feature that is seen at 72–79 km may correspond to the breaking of gravity waves in a region near the temperature inversion. This temperature inversion region has been observed sometimes in different geographic positions by Earth-based radar and lidar tools [Hauchecorne et al., 1987]. It follows from analysis of the data shown in Figures 3 and 4 that combined phase and amplitude radioholography gives high vertical resolution of 0.4–0.6 km (more than the Fresnel zone size of 1.5 km) for revealing wavelike structure in electron density distribution altitude profiles in the D and E regions of the ionosphere. Radioholographic analysis permits one to use high-precision GPS/MET radio occultation measurements to identify detailed structures in the electron density profile in the upper atmosphere.
 Radioholography was also applied to restoration wave structure in a sporadic E region GPS/MET occultation event (7 February 1997, event 0158). The event of 7 February 1997, event 0158, corresponds to summer daytime in the Antarctic region. The time-spatial coordinates of ray minimal height H varied from 71.2°S, 18.2°W, H = 95 km, 1451:05 UT to 70.5°S, 16.4°W, H = 60 km, 1451:25 UT. The amplitude corresponding to frequencies F1 and F2 and phase path difference F1 – F2 are given in Figure 5 (left plots). A general coincidence between phase and amplitude changes in the altitude interval 102–108 km may be seen. After removing trends from the phase path data caused by the regular part of the E region and interference of the upper part of the ionosphere, the influence of the wave part of the electron density distribution on the phase data may be seen (Figure 5, right plots). For comparison, the ratio A2/A1 is also shown. The value of the phase path connected with wave structures changes in the interval ±1 cm; this is lower than the whole contribution of the E region by a factor of 10. The amplitude ratio also followed phase path changes with some delay. In Figure 6 the results of restoration of the electron density altitude profile Ne(H) (from phase data) and gradient dNe(H)/dH (amplitude data) are shown (left plots). The maximum in electron density distribution, which is equal to 61 × 103 el/cm3, corresponds to a height of 104 km. The maximum gradient of the electron density distribution (dNe(H)/dH = 68 × 103 el/(cm3 km)) corresponds to a height of 103 km. The forms of features seen in the electron density distribution may be revealed in the perturbed part of the phase path and amplitude using Abel inversion. The wave structure in the electron density distribution Ne(H) (from phase data) and gradient dNe(H)/dH (amplitude data) is shown in Figure 6 (right plots). The vertical period of the wave changed from 2 km to 0.5 km. The amplitude of the wave structure is 2–3 × 103 el/(cm3). The origin of these wave structures may be internal waves propagating through the ionospheric E region.
 In its current state, the radioholography approach combines the radar focused synthetic aperture principle and the perturbation method. The focused synthetic aperture principle is applied for constructing the reference beam using some knowledge of the average parameters of the atmosphere and ionosphere in the radio occultation region. The perturbation method is applied to analysis of residuals in the phase and amplitude to find the deflections in the ionospheric and atmospheric parameters from expected values. The effectiveness of such an approach may be seen from consideration of the phase path residuals described in sections 3 and 4. The statistical error in the phase residuals is about ±1 mm for a distance between LEO and GPS satellites equal to 30,000 km. The useful effect connected with a variation in electron density distribution of ±2–3 × 103 el/cm3 is equal to ±1 cm. Thus the radioholography may correspond in accuracy to a high-precision level of the GPS radio navigational field. The necessity to account for horizontal gradients and deflections of different kinds from spherical symmetry existing in the ionosphere may be noted. The influence of these gradients may be diminished by using the subtracting of the phase excess corresponding to frequencies F1 and F2 as has been done in the case of the E region in section 4. However, these gradients may introduce some systematic errors, as described by Ahmad . The possibility of including the known horizontal gradient in the ionosphere (for example, day-night asymmetry) might be considered. Also, measurement of these kinds of effects may be possible by further development of the radioholography approach. Nowadays, the accuracy of the radioholographic approach seems to be higher than usually supposed for radio occultation data. As was shown in section 2, the possibility exists to combine radioholography and bward approaches to achieve extreme vertical resolution and accuracy in estimating parameters of natural processes in the atmosphere and ionosphere.
 The equations for angular and height resolutions obtained for the general case of arbitrary satellite orbits indicated the effectiveness of the radioholography approach. The radioholographic method combines phase and amplitude data to achieve high resolution, better than Fresnel's zone, and thereby reveals fine structures in electron density altitude profiles with short spatial periods (0.2–0.5 km). The efficiency of radioholography has been confirmed by direct observation of multibeam propagation and reflected from the sea signals using MIR/GEO and GPS/MET radio occultation data. This demonstrates the high-technology level of the radioholography approach and opens new perspectives for radio occultation experiments: observation of natural processes in the atmosphere, mesosphere, and ionosphere and measurements of the parameters of the sea surface by means of analysis of the reflected signal. Radioholography may also be used for evaluation of electromagnetic fields along the trajectory of a satellite with the goal of revealing diffraction and multibeam propagation effects to determine the location of caustic zones and to investigate various scattering phenomena (turbulence, influence of rain and convection regions, observation of wind shear regions in the mesosphere, etc.). Experimental observation of these effects is also important for advances in the theory of radio wave propagation in irregular media.
 We are grateful to UCAR for access to the GPS/MET data. This contribution was prepared partly under the sponsorship of one of the authors, A. Pavelyev, by the Russian Foundation for Basic Research (RFBR), grant 01-02-17649, and an STA fellowship, host organization Communications Research Laboratory, Japan. A. Pavelyev expresses his gratitude to RFBR, STA, and CRL.