Within the first nine months following the activation of the GPS radio occultation experiment aboard the low Earth orbiting satellite CHAMP, more than 25,000 occultation events have been observed. A radio holographic analysis of 3783 occultation events, recorded between 14 May 2001 and 10 June 2001, reveals that in about 20–30% of these events the received signal contains contributions from components reflected at Earth's surface. On the basis of geometrical ray tracing and multiple phase screen calculations, characteristic frequency shifts in the radio holograms' power spectral densities are analyzed quantitatively. These frequency shifts are found to be dominated by surface elevation at the reflection point location and ground-level refractivity. Using temperature and pressure profiles from European Centre for Medium-Range Weather Forecasting (ECMWF) analyses, ground-level specific humidities are derived in good agreement with ECMWF values. Complex patterns found in radio hologram spectra within a subset of observations at low latitudes are interpreted in terms of multipath propagation caused by layered structures in the refractivity field.
 Atmosphere sounding by Global Positioning System (GPS) radio occultation emerges as a promising new tool for numerical weather forecasting and climate change studies [see, e.g., Melbourne et al., 1994; Kursinski et al., 1996; Anthes et al., 2000]. Between 1995 and 1997 the proof-of-concept GPS radio occultation experiment GPS/Meteorology (GPS/MET) provided several thousand globally distributed temperature and water vapor profiles [Ware et al., 1996; Kursinski et al., 1996; Rocken et al., 1997]. These observations have been successfully validated with rawinsonde data and meteorological analyses [Rocken et al., 1997; Steiner et al., 1999; Marquardt et al., 2001]. Following the successful completion of GPS/MET's mission radio occultation experiments were implemented on several current and future satellite missions (e.g., Ørsted, CHAMP, SAC-C, GRACE, METOP and COSMIC). On 11 February 2001 the occultation experiment aboard the CHAMP (Challenging Minisatellite Payload) satellite [Reigber et al., 2000] was activated. Between 11 February 2001 and 31 December 2001 more than 25,000 occultation events with about 120 temperature soundings per day have been recorded [Wickert et al., 2002].
 In a radio occultation measurement a GPS receiver aboard a low-Earth orbiting (LEO) satellite records signal phase and amplitude variations with high temporal resolution (see Figure 1) [Melbourne et al., 1994]. From these phase shifts bending angles are calculated on the basis of precise GPS and LEO satellite orbit data [Vorob'ev and Krasil'nikova, 1994]. Provided the atmospheric refractive index field n() is spherically symmetric, i.e., n() = n(r), and the geometrical optics approximation applies, temperature profiles are obtained by inverting the bending angle profile by means of the Abel transform [Fjeldbo et al., 1971; Hocke, 1997].
 The study described in this paper is focused on GPS signals reflected at Earth's surface. The detection of GPS signals reflected from the sea surface was first reported by Auber et al. . Pavelyev et al.  describe an experiment in space involving the detection of direct and reflected signals transmitted from the Mir space station to a geostationary satellite as it moved into Earth's radio shadow. Lowe et al.  report on the first spaceborne observation of an Earth-reflected GPS signal. Garrison and Katzberg  and Komjathy et al.  performed a series of air-borne ocean remote sensing measurements using customized GPS receiver instrumentation. An excellent review on experiments involving GPS signal reflections and their theoretical analysis is given by Ruffini et al. .
 Here we are concerned with grazing incidence scattering (incidence angles close to 90°). As incidence angles approach 90° the sea surface appears progressively smooth and the signal reflection process is more and more dominated by specular scattering. However, at grazing incidence contributions from shadowing and diffraction effects, from multiple scattering and trapping by atmospheric ducts may need to be taken into account [Ruffini et al., 1999]. Another important aspect is signal polarization. The L1- and L2-band signals transmitted by the GPS satellites are right-hand circularly polarized (RHCP) [Kaplan, 1996]. CHAMP is equipped with a high-gain helix antenna optimized for RHCP signal detection. Since the reflected signals are essentially linearly polarized at grazing angles ≲1° about 50% signal power is still available in the RHCP components [Ruffini et al., 1999]. Only at grazing angles larger than about 10° left-hand circularly polarized components start to dominate the RHCP components [Katzberg and Garrison, 1996].
 Recently, Beyerle and Hocke  have presented first evidence that GPS signals observed by the GPS/MET radio occultation experiment contain reflected signal components. They employed the radio holographic method to separate direct and reflected signal components [Pavelyev et al., 1996; Hocke et al., 1999; Igarashi et al., 2000]. In the present study we expand upon the Beyerle and Hocke  analysis and show that quantitative information on atmospheric parameters can be extracted from the reflected signal components. Our study is based on the analysis of 3783 occultation events recorded by the CHAMP radio occultation experiment between 14 May 2001 (day of year 134) and 10 June 2001 (day of year 161). During that time period the receiver operated continuously without changes to the on-board tracking and navigation software. By means of the radio holographic technique contributions from reflections are detected and characteristic frequency shifts in the radio holograms' power spectral densities are determined. It is shown that these frequency shifts can be exploited to obtain information on ground elevation at the reflection point and the ground-level refractivity.
 The paper is organized as follows. First, the GPS radio occultation experiment aboard CHAMP and the data analysis are briefly described. Second, the radio holographic method, geometrical ray tracing and the multiple phase screen technique are reviewed. Simulation studies are presented in support of the data analysis. Finally, the results are discussed and interpreted.
2. Data Analysis
 The German satellite CHAMP was launched at Plesetsk, Russia (62.5°N, 40.3°E) on 15 July 2000, 12:00 UTC. Initially, CHAMP was injected in a near polar orbit with an inclination of 87.2° and an orbit period of about 94 min. Within one year orbit altitude has decreased from 454 km to 434 km; at the end of the 5 year design life time orbit altitude will have decayed to about 350 km. The GPS radio occultation experiment is performed with a state-of-the-art GPS receiver (“BlackJack”) provided by the Jet Propulsion Laboratory (JPL).
 In occultation mode the “BlackJack” instrument records GPS signal phase and amplitude data with a sampling frequency of 50 Hz. Using single- or double-differencing techniques inaccuracies due to satellite clock errors can be corrected for by simultaneous tracking of two GPS satellites, the occulting and the referencing satellite [Wickert et al., 2001; Hajj et al., 2002]. First analysis results indicate similar or better receiver performance in terms of L1 and L2 signal-to-noise ratio (SNR) under conditions of activated antispoofing mode compared to GPS/MET during periods of deactivated antispoofing [Wickert et al., 2001]. Using CHAMP precise orbit data [König et al., Routine CHAMP precise orbit products, manuscript in prepraration, 2002] and GPS ground station observations [Galas et al., 2001] atmospheric excess path delays are derived from the 50 Hz data with a double difference technique. For a detailed discussion see Hajj et al.  and Wickert et al. .
 Our analysis is restricted to the C/A signal at the L1 frequency (1.57542 GHz); high-resolution amplitude data at L2 (1.22760 GHz) are unavailable for the measurement period studied here.
2.1. Radio Holographic Analysis
 In the following we focus on the spectral composition of signals propagated through the troposphere, i.e., the analysis is restricted to tangent point altitudes below 20 km. (The tangent point altitude is denoted by zT in Figure 1.) The temporal evolution of the received signal spectrum is analyzed with the radio holographic method. The method is described in detail by Pavelyev et al.  and Hocke et al. ; here, a brief review of the main aspects suffices.
 Signal amplitude A(t) and phase data ϕ(t) provided by the “BlackJack” GPS receiver are combined to form the complex signal of the electric field
The ratio of the observed field E(t) and a reference field
The reference signal Em(t) is obtained from ray tracing calculations using the actual orbit positions of the LEO and GPS satellites. For reasons that will discussed below only those rays are selected which undergo a reflection at Earth's surface.
 The temporal evolution of the radio hologram frequency spectrum is determined in sliding windows of width w = 64 observations (samples) corresponding to a time period of 1.28 s. The window is shifted in steps of 1/4 of the window width, i.e., 16 samples. The analysis is based on the assumption that the number of frequency components k contained in ΔE(t) is much smaller than w; specifically, we choose k = 5. Our assumption is substantiated by the fact that at grazing incidence the reflected signal undergoes specular scattering at the surface. Thus, the bandwidth of the reflected signal equals the (small) bandwidth of the direct signal.
 Since k ≪ w parametric frequency estimation methods are expected to provide a better frequency resolution of the radio hologram spectrum compared to nonparametric techniques such as the periodogram [Stoica and Moses, 2000]. Following Hocke et al.  we select the multiple signal classification method (MUSIC) [Schmidt, 1986]. MUSIC derives frequency estimates by exploiting the properties of the eigen decomposition of the data covariance matrix. [Stoica and Moses, 2000].
2.2. Geometrical Ray Tracing
 The propagation of an electromagnetic wave can be described within the framework of geometrical optics provided the characteristic length scale of the refractivity field N() is much larger than the Fresnel zone (about 1.3 km above 40 km altitude decreasing to about 400–500 m in the lower troposphere) [Melbourne et al., 1994]. If this condition is not fulfilled wave optical methods are needed to model signal propagating through refractivity gradients; this will be discussed in the next section. In this section geometrical ray tracing is described.
 The two point boundary value problem of finding the ray originating from the GPS satellite and reaching the position of the LEO satellite within a predefined radius is solved numerically by Newton-Raphson iteration [Press et al., 1992]. With an appropriate choice of initial values the ray end point lies within 10−4 m of the LEO satellite position after 3–4 iterations. Numerical integration of the differential equations [Haselgrove, 1963]
is performed by an implementation of Adams-Bashforth-Moulton methods within the Matlab programming environment [Shampine and Gordon, 1975]. Here, τ denotes the optical path length and the ray direction (not the wave vector) with ∣∣ = n.
 Refractivity is parameterized in terms of pressure p, temperature T and water vapor partial pressure pw by
p, T and pw profiles are obtained from European Centre for Medium-Range Weather Forecasts (ECMWF) analyses. Pressure and temperature values are calculated by linear interpolation between grid points (2.5° × 2.5° resolution) on 21 pressure levels. Then, linear interpolation in time is performed between 6 h ECMWF analyses fields. In equation (5) the parameters are given by k1 = 0.7760 K Pa−1, k2 = 0.648 K Pa−1 and k3 = 3.776 · 103 K2 Pa−1 [Thayer, 1974]. In the following Nd will be denoted as dry refractivity.
 The refractive index field n(r) = 1 + 10−6 · N(r) is assumed to be spherically symmetric with respect to the center of surface curvature. The reflected ray is determined by taking advantage of the trace's symmetry with respect to the reflection point R (see Figure 1). Integration of equation (4) is terminated at R; the ray segment from R onwards is obtained by mirroring the ray trace with respect to R. Again, Newton-Raphson iteration is applied to reach the LEO satellite position. Signal amplitudes are obtained from ray bundle cross-sections.
 The Matlab implementation requires several seconds of CPU time per ray on a PC-class computer. In order to process the 3783 occultation events within a reasonable amount of time for each occultation only 10 out of 1200–1500 observations below 20 km tangent point altitude are modelled and their optical path lengths are determined. The full optical path profile is obtained by cubic spline interpolation. Comparisons between interpolated profiles and results obtained by tracing all rays yield path length differences below 0.05 mm justifying our approach.
2.3. Multiple Phase Screen Method
 As will be discussed in section 3.3 humidity distributions observed typically at low latitudes are likely to produce strong vertical gradients in the refractivity field. Furthermore, it will be shown that these refractivity gradients give rise to diffraction effects in GPS signals observed by the LEO satellite. Analysis and interpretation of these diffraction effects requires the utilization of wave optical techniques.
 We performed simulation studies using the multiple phase screen (MPS) technique [Knepp, 1983; Martin and Flatté, 1988]. Within the MPS approach the refractive index field n() is modelled as a series of infinitesimally thin phase screens (see Figure 2). (Since the following discussion of the MPS method is restricted to the two dimensional occultation plane the phase screens are de facto one-dimensional objects.)
 At each individual screen the electromagnetic (EM) wave experiences a phase shift
whereas the amplitude remains unchanged. Here, l = 1,…, L with L denoting the total number of screens.
 The complex field u(xl+, z) emerging from phase screen at position xl is obtained from the incoming complex field u(xl−, z),
where Φl(z) is given by equation (6). Between the screens the EM wave is propagated through vacuum. Thus, the incoming field at screen xl+1 is obtained by the Fourier transforms [e.g., Born and Wolf, 1980]
 In our simulation we follow the approach presented by Sokolovskiy . Each of the 2001 phase screen consists of 218 = 262,144 grid points with a separation of Δz = 1 m, vertically the screens extend over 262.143 km. Horizontally, the 2001 phase screens are separated by a distance of Δx = 1 km; thus, the screens cover a horizontal range of 2000 km. The distance between the last screen and the observation screen (marked by the letter O in Figure 2) is 1500 km. The distance between central and observation screen is therefore 2500 km corresponding to a satellite orbit altitude of 450 km.
 Using discrete Fourier transforms in the calculations implicitly assumes that the screens are periodic in the z-direction. In order to prevent energy from leaving at one end of the screen and entering at the other a windowing function w(z) is applied at each phase screen,
Here, we choose δz = 2 km and zt = 252 km. To account for Earth's surface we set u(xl, z) = 0 for where rE denotes the radius of the Earth.
 Phase Φ = arctan and amplitude are extracted from the complex signal u(x−, z) at the observation screen; here, ℜ(u) ((u)) denotes the real (imaginary) part of u. We assume a perpendicular satellite velocity v⟂ = 2.7 km/s corresponding to a LEO satellite orbit altitude of 450 km. The simulated phase and amplitude values are downsampled from 2700 Hz to 50 Hz by taking the mean over 54 adjacent samples.
3. Results and Discussion
 An example for the temporal evolution of power spectral densities (PSD) calculated by the method described in the previous section is shown in Figure 3, top panel. CHAMP occultation event 2001.134#0008 (occultation number 8 on day 134 of the year 2001) was observed in the Arctic at 83.9°N, 162.9°W on 14 May 2001 between 1h15m25s and 1h16m43s UTC. The vertical coordinate is occultation time, i.e., the time elapsed since the beginning of the occultation. Circles in the top panel of Figure 3 mark the dominant frequency components of the radio hologram; circle diameter indicates the component's SNR.
 As discussed in Beyerle and Hocke  the strong frequency component increasing from zero frequency shift at 0 km to 25 Hz frequency shift at about 6 km is due to the direct ray. Above 6 km the frequency shift exceeds 25 Hz. Owing to the signal sampling frequency of 50 Hz the frequency interval [25 Hz, 75 Hz] is aliased back to the [−25 Hz, +25 Hz] range. The weaker frequency component which appears between 0 and 10 km altitude close to zero Hz is caused by a signal reflected at Earth's surface [Beyerle and Hocke, 2001]. We note that the orientation of the direct and the reflected components differs from Figure 3 in Beyerle and Hocke  since they derived the reference field Em(t) from direct rays whereas here the reflected rays are used.
 Each of the 3783 power spectral densities (PSD) are analyzed with the methodology described in the previous section. The reference field Em(t) is obtained from geometrical ray tracing calculations taking into account precise orbit data of the CHAMP and the occulting GPS satellite as well as refractivity profiles derived from ECMWF analyses. From the ray tracing solutions only the reflected rays are selected. We note that the ray traces yield the location of reflection point as well. For the calculations of Em(t) contribution from water vapor are ignored, i.e., pw is taken to be zero in equation (5) and the reflection point altitude zR is assumed to be at sea level, i.e., zR = 0 m. The reference field Em(t), together with the observed phase and amplitude data, form the radio hologram ΔE(t) according to equation (3). Then, the location of the k = 5 dominant frequency components are determined by the MUSIC algorithm. Finally, the frequency shift Δf of the reflected component within the radio hologram's estimated spectra (if it is present) is determined; the algorithm is described below. A schematic overview of the analysis procedure is shown in Figure 4.
 Typically, the reflected signal is detected only during the last 10–20 s of an occultation corresponding to tangent point altitudes below 5–10 km. From the observed 4678 occultations 895 (19.1%) continue for less than 20 s from the time the 20 km tangent point altitude is reached. These data are excluded from the following analysis. Optical path lengths derived from geometrical ray tracing calculations indicate that at about 15 km tangent point altitude the path length difference between direct and reflected ray is about 150 m. Figure 5 shows the relationship between path length difference and tangent point altitude derived from a ray trace calculation based on event 2001.134#0008.
 An optical path length difference of 150 m corresponds to a time delay of 0.5 μs or half a chip length of the C/A code modulated onto the L1 carrier [Kaplan, 1996]. Owing to the pseudorandom noise properties of the C/A code the correlation between a direct and reflected signal separated by more than one chip length is close to zero and therefore the reflected component can no longer be tracked. However, the reflected component might be tracked at tangent point altitudes exceeding 15 km using a GPS receiver equipped with additional correlators added to the primary correlators. Each additional correlator is controlled by the corresponding primary correlator whereby specific time delays are inserted. Details can be found in [Garrison and Katzberg, 2000].
 In order to comprehend the patterns of frequency components found in the radio hologram spectra (see Figure 3, top panel) we assume that the observed fields E(t) can be separated into two components, a direct and a reflected signal,
where the subscripts r and s denote reflected and direct components, respectively. Taylor expansion of the radio hologram using the notation ΔEs(t) ≡ Es(t)/Em(t) yields
We recollect that E is obtained from the GPS measurements whereas Em denotes the reference field derived from ray tracing calculations. In the following analysis it is assumed that the reference field Em(t) is a sufficiently close approximation of the observed reflected component Er(t) such that higher terms in the Taylor expansion (equation (13)) need not be taken into account. Thus, the reflected signal components in E(t) will appear as vertical structures located at a frequency shift Δf (cf. Figures 3 and Figure 14 in section 3.3). We note, however, that under conditions of strong diffraction the assumption leading to equation (13) may not be fulfilled and Δf may exhibit significant temporal variability (cf. Figure 14 in section 3.3).
 We begin the quantitative analysis by describing a heuristic algorithm for the calculation of Δf. First, the number of signal components N in a sliding window of width 2 Hz is determined (see Figure 3, top panel). In order to remove contributions from the direct ray, frequency components whose SNR exceed half the median of all SNR values are ignored. Figure 3 (bottom panel) shows the number of components as a function of window position Δfi for event 2001.134#0008. In the following the maximum value of N(Δfi) will be denoted by Nc ≡ max(N(Δfi)). The frequency shift Δf corresponding to Nc is taken to be the median of all frequency components within the corresponding 2 Hz window. The error of Δf is approximated by the standard deviation of all Δf values within the 2 Hz window contributing to Nc; typically, they are on the order of 0.5 Hz.
 The histogram distribution of Nc derived from 3783 events is plotted in Figure 6. The tail of the distribution extending to large values of Nc indicates the significant fraction of reflection events within the observed data set. The distribution at low values we attribute to random noise. Heuristically, a threshold value of 30 is introduced to separate the two sets. Events with Nc > 30 are regarded as events containing reflection signatures, Nc ≤ 30 are considered as events without reflected signal components.
 1212 out of 3783 events (32.0%) obey Nc > 30. The geographical locations of the corresponding reflection points is shown in Figure 7. Occultations without reflection signatures (Nc ≤ 30) are plotted as blue dots, reflection events (Nc > 30) are marked in red. The circles' diameter are drawn proportional to the mean SNR of the reflected components. The geographical distribution indicates that a large fraction of events over the Atlantic and Pacific oceans at mid and high latitude contain contributions from GPS signal reflections. Ice- and snow-covered surfaces reflect as well as can be seen from the large percentage of reflections over the Arctic, Antarctica and Greenland. Few reflections occur over lakes or wetlands within the continents (e.g., the Great Lakes or the Caspian Depression north of the Caspian Sea).
 The meridional distribution of the reflection percentage is plotted in Figure 8. The decrease in reflection events at 40°N–60°N compared to 40°S–60°S is explained by a corresponding decrease in available ocean surface (full line). Between 40°S and 40°N an additional decrease in the number of reflection events is observed. This decrease at tropical latitudes is, at present, not understood; it might be related to ionospheric disturbances at low latitudes or a corresponding decrease in the number of occultations which penetrate the tropical lower troposphere caused by water vapor induced inhomogeneities in the refractivity field.
 For reasons that will become clear in the following discussion we correlate the observed frequency shift Δf with ground-level refractivity. The result is plotted in Figure 9. Three different event classes can be distinguished. First, reflections at elevated altitudes (marked as circles in Figure 9) exhibit large frequency shifts of up to 25 Hz. These events are mostly due to reflections from the polar regions (Antarctica, Greenland) with reflection altitudes reaching almost 4 km. Zero-elevation (sea-level) events (marked as crosses) are distributed differently. Second, most sea-level reflections (crosses within the grey area) follow an approximate correlation between Δf and ground-level refractivity. Third, 14 out of 977 events (1.4%) deviate significantly from this approximate correlation (crosses outside of the grey area). These three classes will be discussed separately in the following subsections. However, we note that Δf not only depends on ground-level refractivity and reflection point elevation but on the occultation geometry as well, i.e., the positions of the LEO satellite, the occulting GPS satellite and the reflection point and their relative motions.
 In order to interpret the relationship between Δf and refractivity (Figure 9) as well as between Δf and the reflection point altitude zR (see Figure 11 in section 3.1) a large number of ray tracing simulations were performed. Figure 10 shows a schematic overview of the procedure. Here, the observed field E(t) and reference field Em(t), required for the calculation of the radio hologram (equation (3)), are replaced by simulated fields (t) and m(t), respectively. (t) and m(t) are obtained from ray tracing calculations using the model refractivity profile
Ground-level dry refractivity N0d and dry scale height hd are taken from ECMWF analyses; wet scale height is taken to be hw = 2.5 km. (t) is calculated for several values of N0w and reflection point altitudes zR; m(t) is calculated using N0w = 0 and zR = 0 m. Finally, the radio hologram spectra are analyzed analogous to the procedure discussed before and a simulated frequency shift Δ is obtained. In general, we find that the relationship between Δ and N0w as well as zR is almost linear. Examples will be shown below.
 We end this section by summarizing the number of events within the various subsets discussed above. From the analyzed 4678 events 895 measurements continue less than 20 s below 20 km tangent altitude and are removed from the data set. From the remaining 3783 events 2571 observations are without reflection signature (Nc ≤ 30). The rest, 1212 events, is divided into 977 observations with reflection point at zero-elevation and 235 cases with zR > 0.
3.1. Reflection Point Elevation
 Radio holographic analyses of ray tracing simulations show that Δ depends almost linearly on the reflection point elevation zR. In these simulations the field (t) is determined by adding fields obtained from direct and reflected rays. The calculation of (t) is repeated whereby the reflection point altitude zR is varied between 0 and 4 km in steps of 500 m; the reference field m(t), however, is computed with zero reflection point elevation. The frequency shifts derived from these simulations are plotted in Figure 11 as a function of zR (filled symbols). Linear regression yields a slope of 0.221 km Hz−1.
 The CHAMP observations (open symbols in Figure 11) exhibit a correlation between zR and Δf as well. For each of the 234 occultation events at zR > 0 m the frequency shift Δf is determined according to the heuristic algorithm described in sections 2.1 and 3. Reflection point elevations are taken from the GTOPO-30 digital elevation data set (http://edcdaac.usgs.gov/gtopo30/gtopo30.html). Linear regression analysis of Δf obtained from the observations and corresponding elevations from GTOPO-30 yields a slope of 0.192 km Hz−1 (dashed line in Figure 11) in good agreement with the simulation result. A simple theoretical model estimate is discussed in the appendix and yields a slope value which is consistent with the observed value within a factor of two.
Figure 11 shows, however, that the observed correlation between zR and Δf is only approximate. Individual data points exhibit strong deviations on the order of several Hz. These deviations are caused by nonzero surface slope angles. For simplicity we ignore atmospheric refraction and assume that the GPS satellite is located at an infinite distance, i.e., the incoming rays are parallel. Figure 12 illustrates the reflection geometry for a horizontal and a tilted surface (dashed and full lines, respectively). The horizontal displacement d of the reflection point caused by tilting the surface by an angle τ is then given by
since α ≪ 1 and τ ≪ 1. Here, xL and zL denotes the LEO satellite's position in an coordinate system centered at reflection point R and α is the reflection angle (cf. Figure 12). Assuming an average slope angle of τ = 1 mrad, α = 5 mrad and xL = 2400 km the reflection point moves by d ≈ 1200 km horizontally and by d · τ ≈ 1.2 km in elevation. With 0.192 km Hz−1 the corresponding frequency shift is 6.25 Hz in approximate agreement with typical deviations found in Figure 11. However, ground-level refractivity affects the frequency shift Δf as well; this is the topic of the next section.
3.2. Ground-Level Refractivity
 In the following it will be shown that some information about ground-level refractivity can be extracted from the observed frequency shifts. In particular, specific humidity at the ground is determined from observed Δf values and the results are compared with ECMWF analyses. Our analysis is based on the assumption that the horizontal motion of the reflection point on the sea surface does not induce frequency shifts in the reflected signal. The justification for this assumption is discussed in the appendix.
 In the previous section it was shown that variations in reflection point elevation zR represent the dominating contribution to Δf with about 0.22 km Hz−1. Here, we focus on sea-level reflections, i.e., events with reflection point altitude at mean sea level.
 The reference field Em(t) is calculated using zR = 0 m (cf. section 3) in a coordinate system based on the WGS 84 reference ellipsoid. The ocean surface, however, follows Earth's geoid and deviates by almost ±100 m from the WGS 84 ellipsoid. A height offset of 100 m translates into a frequency shift of 100 m/0.22 km Hz−1 ≈ 0.45 Hz. In order to remove this non-negligible effect the offsets between the WGS 84 ellipsoid and the true ocean surface at the locations of the reflections points are corrected for using the EGM96 geopotential model [Lemoine et al., 1998].
 Simulation studies show that the frequency shift Δf depends linearly on ground-level refractivity N0 and its wet contribution N0w = N0 − N0d (assuming N0d = const, equation (14)); the ratio N0w/Δf, however, varies from occultation event to occultation event. Therefore, the relation between Δf and N0w is determined quantitatively from ray tracing calculations for each occultation individually. Ground-level dry refractivity N0d and dry scale height hd are taken from ECMWF analyses; hw is chosen to be 2.5 km. For each of the 977 sea-level reflection events the reference field m(t) is then calculated using N0w,★ ≡ 50 N-units yielding a frequency shift Δ★. The calibration factor relating wet refractivity and frequency shift is
Mean value and standard deviation of X is 49.6 Hz−1 and 7.1 Hz−1, respectively.
 Specific humidity q at ground-level then follows from
where ϵ = 0.62197 denotes the ratio of water vapor and dry air molar mass, T0 and p0 are the ECMWF temperature and pressure values at ground-level, respectively, and the dependence of refractivity on water vapor partial pressure and temperature is described by the parameters k2 = 0.648 K Pa−1 and k3 = 3776 K2 Pa−1 [Thayer, 1974]. We note that the calibration factor X is derived from ray tracing simulations; Δf, however, are the observed frequency shifts. A comparison between specific humidities derived from equation (17) and the corresponding ECMWF specific humidities is shown in Figure 13. A linear regression (dashed line) yields an intercept of −0.29 g/kg and slope of 1.23.
 Within the geometrical ray tracing calculations the ionosphere and its effect on ray propagation is not taken into account. Since the ionosphere is most active and variable at post-sunset and dawn, deviations from spherical symmetry are likely to occur causing deviations from the calculated ray traces and, thereby, variations in Δf [see, e.g., Budden, 1985]. In order to investigate effects on Δf due to ionospheric disturbances around sunrise and sunset the analysis is repeated with a subset of the data. This subset contains events observed at sun elevation angles below −20° (nighttime occultations) and above 20° (daytime occultations). The subset results are plotted in Figure 13 as full symbols. Linear regression of the restricted data set (full line) yields an intercept of 1.00 g/kg and slope of 1.12. In particular, we note that in the restricted data set only 2 events are found with negative values of q compared to 54 events found in the complete data set.
3.3. Diffraction Effects
 In 14 out of 977 sea-level events (1.4%) the observed frequency shifts are found to be inconsistent with the linear relationship between N0w and Δf. For example, the PSD's temporal evolution of event 2001.149#0150 (37.2°N, 144.6°W), which is one of these 14 events, is shown in Figure 14. The heuristic algorithm described in sections 2.1 and 3 determines a frequency shift of Δf = 6.6 Hz. However, additional patterns resembling signal reflections appear at about 0–4 Hz, 10 Hz, 13 Hz and 16 Hz. The appearance of more than one pattern and their significant variability with respect to frequency suggests that these spectral components are not due to ground-level reflections.
 Patterns in radio hologram's spectra such as those shown in Figure 14 are characteristic for low-latitude occultation events: the frequency components do not line up vertically at constant frequency shifts as in the high-latitude events (cf. Figure 3) but tend to vary by several Hz during the last 10–15 s of the occultation. Furthermore, two or three patterns spaced apart by several Hz are a common feature of this event class. We associate these pattern with layered structures in the refractivity field.
 We substantiate our hypothesis by comparing results from multiple phase screen (MPS) simulation studies with the observed frequency spectra. Two refractivity profiles are used: one describes a layer of enhanced refractivity between 2.5 and 3.0 km altitude and is given by the following expression
with a scale height h = 8 km, a ground-level refractivity of N0 = 400, and A = 0.04, z0b = 2.5 km, z0t = 3.0 km and zh = 50 m. The other profile is derived from an aerological sounding during the ALBATROSS field measurement campaign aboard the research vessel “POLARSTERN” at 24.73°S, 27.15°W on 29 October 1996, 23:00 UTC. The dominating source of error with respect to refractivity measured by the rawinsonde is the relative humidity (RH) sensor's accuracy of 2% RH and the sensor's precision of 1% RH [Vaisala, 1989]. In order to reduce this measurement noise the refractivity profiles are smoothed with an 8 point running mean filter thereby reducing the vertical resolution from about 15 m to about 120 m. The vertical refractivity gradients derived from the model and rawinsonde profile are plotted in Figure 15, left and right panel, respectively.
 For both refractivity profiles MPS simulations are performed using the parameters detailed in section 2.3. From the complex signal at the observation screen the received field E(t) is determined; the reference signal Em(t) is calculated from geometrical ray tracing simulation. The LEO spacecraft is placed on a straight-line orbit following the observation screen; similarly, the occulting GPS satellite is assumed to move along a straight line. It is placed at an (unrealistically large) distance of 107 km since the MPS technique is based on the assumption of incoming plane waves.
 The temporal evolution of the PSDs derived from the corresponding radio holograms are shown in Figure 16. The top panel in Figure 16 shows the result for the model N-profile (equation (18)), the result for the radiosonde profile is plotted in the bottom panel. In both cases a component at or close to zero Hz is evident corresponding to a ground-level reflection. For the rawinsonde simulation the main signal vanishes at about 45 s occultation time and disintegrates into several vertical patterns corresponding to the layers in the refractivity gradient profile (cf. Figure 15 right). In the PSD derived from the model profile at about 50 s occultation time two vertical structures related to the enhancements in refractivity gradient (cf. Figure 15 left) appear within the main ray.
4. Conclusions and Outlook
 Radio holographic analyses of CHAMP occultation data reveal the presence of reflected signal components in about 1212 of 3783 events (32.0%) recorded between 14 May 2001 and 10 June 2001. This percentage number increases to about 70–80% at high latitudes. At low latitudes inhomogeneities in the refractivity field caused by the distribution of water vapor often prevent single-path propagation of the direct ray in the lower troposphere.
 Frequency shifts Δf of spectral patterns in radio hologram PSDs are found to contain information on ground elevation at the reflection point and ground-level refractivity. By correlating these frequency shifts with ground-level refractivity the events can be separated in three subsets. First, Δf depends almost linearly on reflection point elevation following about 0.22 km Hz−1. Second, within the subset of zero-elevation reflections an approximate correlation with specific humidity at ground-level is found. Thereby, specific humidities at the ground are derived on the basis of the GPS phase and amplitude observations, precise orbit data and ECMWF dry refractivities. The results are consistent with corresponding ECMWF values of specific humidity. Finally, in about 1.4% of the zero-elevation events the observed frequency shifts are inconsistent with the linear relationship between ground-level refractivity and Δf. Comparisons with results from multiple phase screen simulations suggest that in these cases signatures caused by diffraction effects of layered structures in the refractivity field are present.
 However, we note that the value of Δf is not uniquely determined by ground-level refractivity and reflection point elevation. Our simulation studies show that Δf depends to some degree on the refractivity profile, the ionospheric electron density distribution, deviations from spherical symmetry, occultation geometry and, in the case of reflections on ice and snow, slope tilt angle as well.
 Clearly, with an uncertainty of about 0.5 Hz in the determination of Δf (corresponding to about 15–20 N-units in refractivity and about 100 m reflection point elevation) the analysis of reflection data as described in this study is insufficient to extract useful information on topography or surface humidity which is not observed by other remote sensors. We hope that in the future improvements in GPS receiver tracking algorithms in combination with more elaborate data analysis techniques, e.g., the inclusion of Δf observations in data assimilation schemes, will allow us to extract the maximum amount of information from the reflection measurements. Thereby, the monitoring of GPS signal reflections at mid and high latitudes might develop into a useful tool for meteorological and climatological observations.
 In this study we have focused on the analysis of frequency shifts related to reflected signal components; we have not addressed the question concerning the information content of the SNR values in the reflected components. Possible relations of these SNR values to ocean surface characteristics such as significant wave heights or wave slope statistics [Katzberg and Garrison, 1996; Garrison and Katzberg, 2000; Ruffini et al., 1999] will be the topic of future research.
Appendix: Doppler Shift Due to Surface Reflection
 In this section it will be shown that in the nonrelativistic limit only relative vertical motions of the reflecting surface give rise to frequency shifts between incoming and reflected rays; Doppler shifts due to horizontal velocities can be neglected. For simplicity we consider a plane reflecting surface and ignore atmospheric refraction, i.e., N = 0. The GPS receiver aboard the LEO satellite is assumed to be at a fixed position in a frame denoted by Γ which has its origin at the reflection point. The x- and z-axes define the occultation plane (see Figure 17).
 In Γ the surface moves with velocity = (vx, vy, vz). Thus, the transformation between the surface's rest frame and frame Γ is described by the Lorentz transformation matrix [Møller, 1983]
with , and velocity of light c. The inverse transformation Λ−1 follows from Λ by reversing the signs of vx, vy, and vz. The wave four-vector of the incoming ray is kμi = (2πfi/c, kxi, 0, kzi) with ; fi and λi denote frequency and wave length of the incoming ray in Γ, respectively.
 The wave four-vector of the reflected ray, kμr, is calculated by Lorentz-transforming kμi to the surface's rest frame, i.e., μi = Λ kμi. According to Snell's law the reflected ray in is given by
Finally, kμr follows from the inverse transformation, kμr = Λ−1μr.
 The first component of kμr contains the frequency of the reflected ray, fr, in Γ. We obtain
Analysis of tangent and reflection point velocities in CHAMP occultation events suggests vx,y ≲ 5 km/s and vz ≲ 2.7 km/s. Thus, v/c ≈ 10−5 and equation (21) can be expanded with respect to v/c. Disregarding terms of order v2/c2 and higher yields
We conclude that in the nonrelativistic regime horizontal motions do not lead to Doppler shifts in the reflected GPS signals. Contributions to the observed frequency shifts can only arise from vertical velocity components.
 Within a time interval δT the reflection angle α changes by δα ≡ δz/D and the frequency shift Δf ≡ fr − fi changes accordingly by
With a distance D between reflection point and LEO satellite of about 2500 km (see Figure 17), vz ≈ 2.7 km/s and λ = 0.19 m we obtain δ(Δf)/δz ≈ 11 Hz km−1. This result is (within a factor of two) consistent with the slope value of 0.192 km Hz−1 (5.2 Hz km−1) derived in section 3.1.
 Helpful discussions with M. Gorbunov, A. G. Pavelyev, Ph. Hartl, G. Kirchengast and H. Lühr are gratefully acknowledged. We are also grateful to two anonymous reviewers, whose comments and corrections strengthened and improved the manuscript significantly. We thank our colleagues at GFZ and JPL: T. Meehan and L. Grunwaldt for receiver operation and control, C. Förste and W. Köhler for data handling and preprocessing, R. König and K.-H. Neumayer for supplying CHAMP precise orbit, and R. Galas for GPS ground station data processing. We are grateful to R. Weller, Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany, for high-resolution rawinsonde data. The German Weather Service (DWD) provided ECMWF analyses via the Stratospheric Research Group, Freie Universität, Berlin. We thank K. Labitzke and K. Schulz-Schöllhammer (Freie Universität Berlin) for ECMWF data preparation. The GTOPO-30 digital elevation data are distributed by the EROS Data Center Distributed Active Archive Center (EDC DAAC), located at the U.S. Geological Survey's EROS Data Center in Sioux Falls, South Dakota (http://edcdaac.usgs.gov/gtopo30/gtopo30.html). Trademarks are the property of their respective owners. This study was carried out within the HGF project “GPS Atmospheric Sounding” (grant FKZ 01SF9922/2).