GPS radio occultations with CHAMP: A radio ... - Semantic Scholar

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D24, 4802, doi:10.1029/2001JD001402, 2002

GPS radio occultations with CHAMP: A radio holographic analysis of GPS signal propagation in the troposphere and surface reflections G. Beyerle,1 K. Hocke,2 J. Wickert,1 T. Schmidt,1 C. Marquardt,1 and C. Reigber1 Received 15 October 2001; revised 18 February 2002; accepted 11 June 2002; published 28 December 2002.

[1] 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 INDEX TERMS: 1640 Global Change: Remote sensing; layered structures in the refractivity field. 0669 Electromagnetics: Scattering and diffraction; 6994 Radio Science: Instruments and techniques; 6964 Radio Science: Radio wave propagation; KEYWORDS: GPS radio occultation, GPS reflectometry, GPS scatterometry, radio holography, global positioning system, CHAMP Citation: Beyerle, G., K. Hocke, J. Wickert, T. Schmidt, C. Marquardt, and C. Reigber, GPS radio occultations with CHAMP: A radio holographic analysis of GPS signal propagation in the troposphere and surface reflections, J. Geophys. Res., 107(D24), 4802, doi:10.1029/2001JD001402, 2002.

1. Introduction [2] 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-ofconcept 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., 1 2

GeoForschungsZentrum Potsdam, Potsdam, Germany. Communications Research Laboratory, Tokyo, Japan.

Copyright 2002 by the American Geophysical Union. 0148-0227/02/2001JD001402$09.00

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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]. [3] 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(~ r) is spherically symmetric, i.e., n(~ r) = 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]. [4] At mid and low latitudes water vapor contributes significantly to the atmospheric refractive index. Thus, strong gradients of the refractivity profile N(r) (n(r) 1) 106 may cause multipath propagation rendering solutions based on single ray propagation questionable. To resolve problems due to multipath, Fresnel diffraction theory [Mortensen and Høeg, 1999], the back-propagation technique [Gorbunov et al., 1996; Karayel and Hinson, 1997; Gorbunov and Gurvich, 1998], the canonical transformation technique [Gorbunov, 2001], and the radio-optic

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BEYERLE ET AL.: GPS SIGNAL REFLECTIONS

Figure 1. Schematic representation of the radio occultation geometry. Signals transmitted by the GPS satellite are received by the LEO satellite directly (ray ‘‘d’’). Additionally, signals may reach the LEO satellite after being reflected at Earth’s surface (ray ‘‘r’’). R and T mark the reflection point and tangent point, respectively; zT is the tangent point altitude. Distances and angles are not to scale; typical refraction angles a are on the order of 1, distances between T and GPS satellite and between T and LEO satellite are on the order of 26,000 and 2500 km, respectively. method [Lindal et al., 1987; Pavelyev et al., 1996; Hocke et al., 1999] have been discussed in the literature. All these techniques take advantage of the observed GPS signal amplitude variations in addition to the phase observations. [5] 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. [1994]. Pavelyev et al. [1996] 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. [2002] report on the first spaceborne observation of an Earth-reflected GPS signal. Garrison and Katzberg [2000] and Komjathy et al. [2000] 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. [1999]. [6] 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 L2band 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]. [7] Recently, Beyerle and Hocke [2001] 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 [2001] 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 groundlevel refractivity. [8] 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 [9] The German satellite CHAMP was launched at Plesetsk, Russia (62.5N, 40.3E) 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). [10] In occultation mode the ‘‘BlackJack’’ instrument records GPS signal phase and amplitude data with a sampling frequency of 50 Hz. Using single- or doubledifferencing 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 [Ko¨nig 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. [2002] and Wickert et al. [2001].

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[11] 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 [12] 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. [1996] and Hocke et al. [1999]; here, a brief review of the main aspects suffices. [13] Signal amplitude A(t) and phase data f(t) provided by the ‘‘BlackJack’’ GPS receiver are combined to form the complex signal of the electric field E ðt Þ ¼ Aðt Þ expðifðt ÞÞ:

ð1Þ

The ratio of the observed field E(t) and a reference field Em ðt Þ ¼ expði fm ðt ÞÞ

ð2Þ

is denoted as radio hologram [Pavelyev et al., 1996; Hocke et al., 1999], E ðt Þ E ðt Þ m ¼ Aðt Þ expðifðt Þ ifm ðtÞÞ: E ðt Þ

ð3Þ

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. [14] 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 E(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. [15] 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. [1999] 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 [16] 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(~ r) is much larger than the Fresnel zone (about 1.3 km above 40 km altitude decreasing to about 400– 500 m in the

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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. [17] 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 104 m of the LEO satellite position after 3 – 4 iterations. Numerical integration of the differential equations [Haselgrove, 1963] d~ r ~ k ¼ dt n2

d~ k rn ¼ dt n

ð4Þ

is performed by an implementation of Adams-BashforthMoulton methods within the Matlab programming environment [Shampine and Gordon, 1975]. Here, t denotes the optical path length and ~ k the ray direction (not the wave vector) with j~ kj = n. [18] Refractivity is parameterized in terms of pressure p, temperature T and water vapor partial pressure pw by p pw T pw pw N ¼ Nd þ k2 þ k3 2 : T T Nd ¼ k1

ð5Þ

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 Pa1, k2 = 0.648 K Pa1 and k3 = 3.776 103 K2 Pa1 [Thayer, 1974]. In the following Nd will be denoted as dry refractivity. [19] The refractive index field n(r) = 1 + 106 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. [20] 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 [21] As will be discussed in section 3.3 humidity distributions observed typically at low latitudes are likely

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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. [22] We performed simulation studies using the multiple phase screen (MPS) technique [Knepp, 1983; Martin and Flatte´, 1988]. Within the MPS approach the refractive index field n(~ r) 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.) [23] At each individual screen the electromagnetic (EM) wave experiences a phase shift xþx=2 Z

2p l ð zÞ ¼ l

dx½nð x; zÞ 1

ð6Þ

Figure 2. Schematic representation of the multiple phase screen method. The atmosphere is represented by a series of phase screens. An electromagnetic wave propagates through the screens from left to right. Observation screen O and central screen are separated by the distance D. For details, see section 2.3.

xx=2

whereas the amplitude remains unchanged. Here, l = 1,. . ., L with L denoting the total number of screens. [24] Excellent descriptions of the MPS method can be found in Karayel and Hinson [1997] and Sokolovskiy [2001]. Here, we briefly review the key equations following the notation introduced by Karayel and Hinson [1997], for consistency with the remainder of the paper, however, the xand z-axis are interchanged (see Figure 2). [25] The complex field u(xl+, z) emerging from phase screen at position xl is obtained from the incoming complex field u(xl, z), u xþ l ; z ¼ u xl ; z expðil ð zÞÞ

ð7Þ

where l(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]

[26] In our simulation we follow the approach presented by Sokolovskiy [2001]. Each of the 2001 phase screen consists of 218 = 262,144 grid points with a separation of z = 1 m, vertically the screens extend over 262.143 km. Horizontally, the 2001 phase screens are separated by a distance of x = 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. [27] Using discrete Fourier transforms in the calculations implicitly assumes that the screens are periodic in the zdirection. 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,

wð zÞ ¼ 1 u x lþ1 ; z ¼ 2p

Z1

dkz U kz ; x lþ1 expði kz zÞ

ð8Þ

1

where þ U kz ; x lþ1 ¼ U kz ; xl expðikx ðxlþ1 xl ÞÞ Z1 þ dz u xþ U kz ; xl ¼ l ; z expðikz zÞ:

ð9Þ

1

Here, ~ k ðkx ; kz Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 kz2 ; kz

ð10Þ

denotes the wave vector with k = 2p/l. Starting with u(xl=1, z) = 1 the field u(x, z) is stepped through all L screens by repeated application of equations (7) and (8). For details, see Karayel and Hinson [1997] and Goodman [1968].

8 >

:

:

0

z < zt

: zt < z < zt þ dz :

ð11Þ

z > zt þ dz:

Here, we choose dz = 2 km and zt = 252qkm. Toffi account for ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Earth’s surface we set u(xl, z) = 0 for z < rE2 x2l rE where rE denotes the radius of the Earth. [28] Phase = arctan(=(u)/ 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

ð12Þ

where the subscripts r and s denote reflected and direct components, respectively. Taylor expansion of the radio hologram using the notation Es(t) Es(t)/Em(t) yields Er ð t Þ þ Es ðt Þ E m ðt Þ ¼ Ar ðt Þ expðifr ðt Þ ifm ðt ÞÞ þ Es ðt Þ Ar ðt Þ expðif þ i2pft þ . . .Þ þ Es ðt Þ:

EðtÞ ¼

ð13Þ

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 f (cf. Figures 3 and Figure 14

Figure 6. Histogram distribution of component number Nc. Events with Nc exceeding a threshold value of 30 (dashed line) are regarded as reflection events.


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Figure 7. Geographical distribution of 3783 occultation events observed between 14 May and 10 June 2001. Blue dots indicate 2571 observations without reflection signatures; 1212 reflection events are marked as red circles. Circle diameter is proportional to the reflected intensity. 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). [38] The meridional distribution of the reflection percentage is plotted in Figure 8. The decrease in reflection events at 40N–60N compared to 40S – 60S is explained by a corresponding decrease in available ocean surface (full line). Between 40S and 40N 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. [39] For reasons that will become clear in the following discussion we correlate the observed frequency shift f 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 f 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 f 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. [40] In order to interpret the relationship between f and refractivity (Figure 9) as well as between f 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 ~ ~ m(t), (3)), are replaced by simulated fields E(t) and E m ~ and E ~ (t) are obtained from ray tracing respectively. E(t) calculations using the model refractivity profile z z þ N0w exp : N ð zÞ ¼ N0d exp hd hw

ð14Þ

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regression analysis of f obtained from the observations and corresponding elevations from GTOPO-30 yields a slope of 0.192 km Hz1 (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. [44] Figure 11 shows, however, that the observed correlation between zR and f 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 t is then given by Figure 8. Meridional distribution of reflection event abundance. The fraction of sea-covered areas in the sealand distribution between 60S, and 60N is shown as full line. N0d

Ground-level dry refractivity and dry scale height hd are taken from ECMWF analyses; wet scale height is taken to ~ is calculated for several values of N0w be hw = 2.5 km. E(t) ~ m(t) is calculated using and reflection point altitudes zR; E w N0 = 0 and zR = 0 m. Finally, the radio hologram spectra are analyzed analogous to the procedure discussed before and a simulated frequency shift ~f is obtained. In general, we find that the relationship between ~f and N0w as well as zR is almost linear. Examples will be shown below. [41] 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 [42] Radio holographic analyses of ray tracing simulations show that f~ depends almost linearly on the reflection ~ is point elevation zR. In these simulations the field E(t) determined by adding fields obtained from direct and ~ is repeated whereby reflected rays. The calculation of E(t) the reflection point altitude zR is varied between 0 and 4 km ~ m(t), however, is in steps of 500 m; the reference field E 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 Hz1. [43] The CHAMP observations (open symbols in Figure 11) exhibit a correlation between zR and f as well. For each of the 234 occultation events at zR > 0 m the frequency shift f 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

tanðaÞ tanða 2tÞ tanðtÞ þ tanða 2tÞ 2t xL at

d ¼ xL

ð15Þ

since a 1 and t 1. Here, xL and zL denotes the LEO satellite’s position in an coordinate system centered at reflection point R and a is the reflection angle (cf. Figure 12). Assuming an average slope angle of t = 1 mrad, a = 5 mrad and xL = 2400 km the reflection point moves by d 1200 km horizontally and by d t 1.2 km in elevation. With 0.192 km Hz1 the corresponding frequency shift is 6.25 Hz in approximate agreement with typical deviations

Figure 9. Correlation of frequency shift f with groundlevel refractivity. 1212 events with reflection signature are plotted. Three classes of events can be distinguished: reflection point at nonzero elevation (circles, 235 cases), sea-level reflections following a correlation between ground-level refractivity and f (crosses within grey area, 963 cases), and reflections apparently inconsistent with the correlation (crosses outside of grey area, 14 cases). For a discussion, see sections 3.1, 3.2, and 3.3.

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84 ellipsoid. A height offset of 100 m translates into a frequency shift of 100 m/0.22 km Hz1 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]. [48] Simulation studies show that the frequency shift f depends linearly on ground-level refractivity N0 and its wet contribution N0w = N0 N0d (assuming N0d = const, equation (14)); the ratio N0w/f, however, varies from occultation event to occultation event. Therefore, the relation between f 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 ~ m(t) the 977 sea-level reflection events the reference field E w,8 is then calculated using N0 50 N-units yielding a frequency shift ~f 8 . The calibration factor relating wet refractivity and frequency shift is X

N0w;? : ~f ?

ð16Þ

Mean value and standard deviation of X is 49.6 Hz1 and 7.1 Hz1, respectively. [49] Specific humidity q at ground-level then follows from Figure 10. Schematic overview of the simulation procedure. The procedure determines the relationship between ~f and ground-level refractivity as well as between ~f and ~ m(t) and the reflection point altitude. The reference field E ~ field E(t) are derived from geometrical ray tracing calculation using refractivity profiles for dry and wet ~ m(t) is calculated atmospheric conditions, respectively. E ~ altitude zR is with reflection point altitude zR = 0 m, for E(t) ~ yields varied. Spectral analysis of the radio hologram E the simulated frequency shift ~f . For details, see section 3.

q¼

X f T02 k2 p0 T0 þ k3 p0 ð1 ÞX f T02

ð17Þ

where = 0.62197 denotes the ratio of water vapor and dry air molar mass, T0 and p0 are the ECMWF temperature

found in Figure 11. However, ground-level refractivity affects the frequency shift f as well; this is the topic of the next section. 3.2. Ground-Level Refractivity [45] 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 f 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. [46] In the previous section it was shown that variations in reflection point elevation zR represent the dominating contribution to f with about 0.22 km Hz1. Here, we focus on sea-level reflections, i.e., events with reflection point altitude at mean sea level. [47] 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

Figure 11. Dependence of observed frequency shift f and simulated frequency shift ~f on reflection point elevation zR. Open symbols indicate results from CHAMP observations. For comparison, results from ray tracing simulations are plotted as well (full symbols). Linear regression of the observations (dashed line) yields a slope of 0.192 km Hz1 in good agreement with the simulation result of 0.221 km Hz1 (full line).

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Figure 12. Reflection point shift caused by tilting the reflecting surface. The reflection geometry for the horizontal (dashed lines) and tilted (full lines) surface is shown. Tilting the reflecting surface by an angle t shifts the reflection point from R to R0 and causes a corresponding change in reflection point altitude of about d t. 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 Pa1 and k3 = 3776 K2 Pa1 [Thayer, 1974]. We note that the calibration factor X is derived from ray tracing simulations; f, 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. [50] 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 f [see, e.g., Budden, 1985]. In order to investigate effects on f 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 [51] 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 f. For example, the PSD’s temporal evolution of event 2001.149#0150 (37.2N, 144.6W), 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 f = 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. [52] Patterns in radio hologram’s spectra such as those shown in Figure 14 are characteristic for low-latitude

Figure 13. Comparison of ground-level specific humidities derived from the radio holographic analysis of CHAMP data (vertical axis) and from ECMWF data (horizontal axis). A subset containing only events with solar angles below 20 (nighttime occultations) and above 20 (daytime occultations) is indicated by full symbols. The linear regressions of the complete data set and the subset are given as broken and full line, respectively. For clarity, not all error bars are drawn. occultation events: the frequency components do not line up vertically at constant frequency shifts as in the highlatitude 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.

Figure 14. Same as Figure 3 (top panel) however for event 2001.149#0150 observed at 37.2N, 144.6W on 29 May 2001.


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Figure 15. Left: vertical refractivity gradient of model refractivity profile used for MPS simulations. The threshold value for super-refractivity is shown as dotted line. Right: vertical refractivity gradient derived from an aerological sounding at 24.73S, 27.15W on 29 October 1996, 23:00 UTC. [53] 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

z h

ð1 þ A f1 ðzÞ f2 ðzÞÞ 2 z zt0 f1 ðzÞ ¼ 1 arctan zh p b 2 z z f2 ð zÞ ¼ 1 arctan 0 p zh

N ðzÞ ¼ N0 exp

ð18Þ

with a scale height h = 8 km, a ground-level refractivity of N0 = 400, and A = 0.04, zb0 = 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 24.73S, 27.15W 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.

[54] 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. [55] 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 [56] Radio holographic analyses of CHAMP occultation data reveal the presence of reflected signal components in

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Figure 16. Temporal evolutions of radio hologram power spectra calculated from corresponding MPS simulations. Circles mark the location of the dominant frequency components, their diameter correspond to the component’s intensity. Top: result derived from the model refractivity profile (cf. equation (18)); its vertical gradient is plotted in Figure 15, left panel. Bottom: result derived from rawinsonde observation. Its vertical gradient is shown in the right panel of Figure 15. 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. [57] Frequency shifts f 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, f depends almost linearly on reflection point elevation following about 0.22 km Hz1. 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 zeroelevation events the observed frequency shifts are inconsistent with the linear relationship between ground-level refractivity and f. 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. [58] However, we note that the value of f is not uniquely determined by ground-level refractivity and reflection point elevation. Our simulation studies show that f 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. [59] Clearly, with an uncertainty of about 0.5 Hz in the determination of f (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 f 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. [60] 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 [61] 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 which has its origin at the reflection point. The x- and zaxes define the occultation plane (see Figure 17). v = (vx, vy, vz). [62] In the surface moves with velocity ~ Thus, the transformation between the surface’s rest frame

Figure 17. Reflection at a plane surface. The reflecting surface is moving with velocity ~ v = (vx, vy, vz). The schematic is not to scale, reflection angle a is typically less than 1.


~ and frame is described by the Lorentz transformation matrix [Møller, 1983] 0

g

g

B B g vx B c ¼B B vy B g c @ g vcz

vx c v2x v2

1þh h

g

vx vy v2

h vvx 2vz

vy c

vx vy v2 v2 þ h vy2

h 1

h

vy vz v2

g vcz

1

C h vvx 2vz C C C: vy vz C h v2 C A v2 1 þ h vz2

ð19Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

with v v2x þ v2y þ v2z , g 1= 1 v2 =c2 , h g 1 and velocity of light c. The inverse transformation 1 follows from by reversing the signs of vx, vy, and vz. The wave fourvector of the incoming ray is kmi = (2pf i/c, kxi, 0, kzi ) with q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 i i i kxi þ kzi ¼ 2p=l ; f and l denote frequency and wave length of the incoming ray in , respectively. [63] The wave four-vector of the reflected ray, kmr, is calculated by Lorentz-transforming kmi to the surface’s rest frame, i.e., ~k mi = kmi . According to Snell’s law the ~ is given by reflected ray in ~ kmr 2p~f r =c; ~kxr ; ~kyr ; ~kzr ¼ 2p~f i =c; ~kxi ; ~kyi ; ~kzi

c2 v2 þ 2v2z i f c2 v2 g 2 ðg 1Þv2z vx kx þ v2x þ v2y þ gv2z vz kz : pv

ð21Þ

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 105 and equation (21) can be expanded with respect to v/c. Disregarding terms of order v2/c2 and higher yields fr fi

vz kz vz ¼ f i 2 sin a: p l

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(within a factor of two) consistent with the slope value of 0.192 km Hz1 (5.2 Hz km1) derived in section 3.1.

[66] Acknowledgments. Helpful discussions with M. Gorbunov, A. G. Pavelyev, Ph. Hartl, G. Kirchengast and H. Lu¨hr 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. Fo¨rste and W. Ko¨hler for data handling and preprocessing, R. Ko¨nig 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 Universita¨t, Berlin. We thank K. Labitzke and K. Schulz-Scho¨llhammer (Freie Universita¨t 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).

References ð20Þ

Finally, kmr follows from the inverse transformation, kmr = 1~k mr. [64] The first component of kmr contains the frequency of the reflected ray, f r, in . We obtain fr ¼

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ð22Þ

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. [65] Within a time interval dT the reflection angle a changes by da dz/D and the frequency shift f f r f i changes accordingly by dðf Þ vz dz ¼ 2 ; lD dT dT

ð23Þ

dðf Þ vz : ¼ 2 lD dz

ð24Þ

thus,

With a distance D between reflection point and LEO satellite of about 2500 km (see Figure 17), vz 2.7 km/s and l = 0.19 m we obtain d(f )/dz 11 Hz km1. This result is

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G. Beyerle, C. Marquardt, C. Reigber, T. Schmidt, and J. Wickert, Division 1, Kinematics and Dynamics of the Earth, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg, D-14473 Potsdam, Germany. ([email protected]; [email protected]; reigber@ gfz-potsdam.de; [email protected]; [email protected]) K. Hocke, Space Science Division, Upper Atmosphere Section, 4-2-1, Communications Research Laboratory, Nukui Kita-machi, Koganei-shi, Tokyo 184-8795, Japan. ([email protected])