Atmospheric sounding by GNSS radio occultation: An analysis of the negative refractivity bias using CHAMP observations

Transcription

1 1 Atmospheric sounding by GNSS radio occultation: An analysis of the negative refractivity bias using CHAMP observations G. Beyerle 1, S. Sokolovskiy 2, J. Wickert 1, T. Schmidt 1, and Ch. Reigber 1 Short title: ANALYSIS OF NEGATIVE REFRACTIVITY BIAS 1 GeoForschungsZentrum (GFZ) Potsdam, Germany. 2 University Corporation for Atmospheric Research, Boulder, USA.

2 2 Abstract. Validation studies of current GPS radio occultation experiments using meteorological analyses consistently report on a negative refractivity bias in the lower troposphere. It is shown that refractivity profiles obtained from Doppler-inverted bending angle profiles not only deviate significantly within zones of multipath propagation but also depend on the selected end point of the occultation signal in the Earth s radio shadow. End-to-end simulations including the GPS receiver s signal tracking process suggest that receiver-induced phase deviations contribute to this observed bias as well. We propose a heuristic retrieval algorithm based on the canonical transform and the sliding spectral technique that seems less susceptible to tracking phase errors than the canonical transform method. The approach is described using simulated profiles and validated on the basis of 4221 CHAMP occultations recorded between 14 May and 10 June Compared to the canonical transform results the heuristic method results in a significantly smaller tropospheric refractivity bias at low latitudes.

3 3 1. Introduction Remote sensing of Earth s atmospheric refractivity by Global Positioning System (GPS) radio occultation (RO) contributes to numerical weather forecasting and climate change studies [see e.g., Melbourne et al., 1994; Anthes et al., 2000]. Between 1995 and 1997 the proof-of-concept GPS RO experiment GPS/Meteorology (GPS/MET) provided several thousands of globally distributed temperature and water vapor profiles [Ware et al., 1996; Rocken et al., 1997]. Since 2001 RO receivers are operating aboard the CHAMP (CHAllenging Minisatellite Payload) [Reigber et al., 2000, 2002] and SAC-C [Hajj et al., 2003] satellite missions. Between February 2001 and March 2003 CHAMP recorded more than 135,000 occultation events, about 62% of which could be successfully converted to validated profiles of atmospheric temperature [Wickert et al., 2001, 2003a]. Figure 1. In a RO measurement a GPS receiver aboard a low-earth orbiting (LEO) satellite records signal phase and amplitude variations with high temporal resolution [see e.g. Melbourne et al., 1994; Kursinski et al., 1997; Hajj et al., 2002]. Fig. 1 schematically illustrates the occultation geometry within the two-dimensional occultation plane. From these phase shifts bending angles are calculated on the basis of precise GPS and LEO satellite orbit data assuming that the atmospheric refractive index field n( r) is spherically symmetric, i.e. n( r) = n(r), and the geometric optic approximation applies [Vorob ev and Krasil nikova, 1994]. Finally, refractivity profiles are obtained by Abel-inverting the corresponding bending angle profiles [Fjeldbo et al., 1971; Hocke, 1997; Steiner et al., 1999]. At mid and low latitudes the water vapor distribution contributes significantly

4 4 to the atmospheric refractive index field. Strong gradients of the refractivity profile N(r) (n(r) 1) 10 6 may cause multipath signal propagation rendering solutions based on single ray propagation inapplicable. To solve the problem of calculating bending angle profiles within zones of multipath propagation various radioholographic methods have been applied. The most advanced methods are the canonical transform (CT) method developed by M. Gorbunov and described in a series of papers [Gorbunov, 2001, 2002a, b] and the full spectrum inversion (FSI) technique [Jensen et al., 2003]. In this paper we use the CT method. In the upper troposphere and throughout the stratosphere good agreements between RO measurements and meteorological model analyses are found. In the lower troposphere, however, validation studies consistently report a negative refractivity bias of more than several percent. This bias was first noticed and discussed by Rocken et al. [1997] within the GPS/MET data validation study. In the follow-on missions CHAMP and SAC-C negative deviations appear as well [see e.g, Wickert et al., 2003b; Marquardt et al., 2003; Hajj et al., 2003]. The origin of this bias is commonly attributed to superrefraction and receiver tracking errors. Vertical refractivity gradients smaller than about 160 km 1, extending horizontally over a sufficiently large distance, result in superrefraction and negative retrieval errors using Abel inversion (which, formally, is inapplicable in this case) [Kursinski et al., 2000; Sokolovskiy, 2003]. This study is focused on the negative refractivity bias caused by the receiver signal tracking process. The use of Doppler-inverted bending angles within regions of multipath propagation may produce refractivity biases, too. In particular, it is shown that the calculated refractivity bias depends on the selected occultation signal end point within the

5 5 Earth s shadow zone. First attempts to model phase-locked loop tracking of the simulated tropospheric RO signals with complicated dynamics indicated strong sensitivity of the results to loop parameters and motivated to consider the tracking technique without a feedback (open loop) [Sokolovskiy, 2001b]. More recently, two independent end-to-end simulation studies confirmed that receiver tracking errors can lead to an underestimation of refractivity in the lower troposphere [Ao et al., 2003b; Beyerle et al., 2003]. In addition, further simulation runs could reproduce the true refractivity profiles with high accuracy if the receiver model was excluded from the simulation chain. Since retroactive correction of the receiver-induced phase deviations seems impossible we investigate in this study a heuristic retrieval method to derive reliable bending angle profiles from observations that are affected by these phase errors. The study s objective is to separate receiver-induced refractivity biases from deviations caused by superrefraction. The structure of the paper is as follows: First, we briefly review the canonical transform technique and describe the heuristic canonical transform / sliding spectral (CTss) retrieval method. An application of the method is illustrated using simulated data from end-to-end simulation studies. Second, simulation results are presented showing refractivity profiles derived from a single-valued approximation of Doppler bending angle. Finally, the heuristic CTss retrieval method is applied to a data set of about 4,200 CHAMP observations recorded between 14 May and 10 June The results are compared with the corresponding refractivity profiles obtained by the canonical transform retrieval.

6 6 2. Retrieval methods In order to better understand the factors contributing to the observed negative refractivity bias we performed end-to-end simulations starting with the atmospheric propagation, the tracking of the propagated signal, and finally the conversion from amplitudes and phases to vertical profiles of atmospheric refractivity [Beyerle et al., 2003]. Multiple phase screen calculations using a high-resolution tropical radio sonde profile determine the propagated electromagnetic field which is tracked by a simplified model of a GPS receiver. From the receiver output the bending angle and refractivity profiles are calculated using the CT method and the Abel transform. Alternatively, the receiver model may be bypassed allowing for direct computation of bending angles from the multiple phase screen data. A schematic diagram of the end-to-end simulation chain is shown in Fig. 2 (taken from Beyerle et al. [2003]). Figure 2. The simulation study showed that the signal tracking process may induce significant phase errors in regions of multipath propagation; the signal amplitude (expressed as signal-to-noise ratio of the tracked signal) is less affected. It turns out that the CT method is susceptible to these phase deviations induced by the tracking process. For simulation runs with bypassed tracking receiver bending angle profiles obtained by the CT technique reproduce the true profiles with excellent accuracy; inserting the receiver in the simulation chain, however, introduces significant phase deviations leading to bending angle errors exceeding 10 20%. In an independent simulation study, which used tracking algorithms essentially identical to those currently employed by the BlackJack receiver aboard CHAMP and SAC-C, Ao et al. [2003b] have found similar receiver-induced deviations.

7 7 Thus, receiver-induced contributions to the observed negative refractivity bias are likely to be present in the observed CHAMP and SAC-C data set and have to be accounted for. We developed a heuristic method which (partially) compensates the receiver-induced refractive bias. Our heuristic algorithm rests on two foundations: the canonical transform method [Gorbunov, 2001, 2002a, b] and the sliding spectral technique [Sokolovskiy, 2001a; Gorbunov, 2002c]. Before describing the details of the heuristic canonical transform / sliding spectral (CTss) method we briefly review the CT procedure Canonical transform method Recently, the CT method was developed to solve the problem of calculating bending angle α(p) as a function of impact parameter p within regions of multipath signal propagation [Gorbunov, 2001, 2002a, b]. Within the CT approach the solution u(x, y) of the Helmholtz equation is transformed from geometric space characterized by vertical coordinate y (see Fig. 1) and direction cosine 1 η 2 to impact parameter space characterized by impact parameter p and its generalized momentum ξ. The transformation equations relating geometric and impact parameter space are p = y 1 η 2 η x (1) ξ = arcsin η. In the following we consider solutions on straight lines, x = const, motivating the notation u x (y). The mathematical problem of finding the solution of the Helmholtz equation

8 8 in impact parameter space, u x (p), that corresponds to u x (y), the solution in geometrical space, is solved within the framework of the Fourier integral operator theory [Egorov et al., 1999; Mishchenko et al., 1990]. A detailed discussion of the CT method is given by Gorbunov [2001, 2002a, b]. The CT method is based on the assumption that every ray is uniquely determined by its impact parameter. In the first step the signal observed by the GPS receiver along the LEO orbit is backpropagated to an auxiliary line x = const using the Kirchhoff diffraction integral [Gorbunov et al., 1996; Ao et al., 2003a]. We note that in the simulation case the processing procedure misses this first step out since the simulated signal is already determined for a straight line receiver trajectory. Fig. 1 shows the occultation geometry and defines the coordinate system. Interfering rays are then disentangled (multipath is unfolded) by transforming the backpropagated field u x (y) to impact parameter space. According to the theory of Fourier integral operators the solution is u x (p) = k ( )) dη (1 η 2 ) 1 4 exp (ik p arcsin η x 1 η 2π 2 dy exp( i k y η) u x (y). (2) Here, k = 2π/λ denotes the wave vector corresponding to GPS carrier wavelength λ. In this study we interchange the order of the two integrations in Eqn. 2 and approximate the integral over η using the method of stationary phase [Born and Wolf, 1980]. We obtain u x (p) k ( i 2π exp π ) dy 1 η2 0(y, p) 4 x + p η 0 (y, p) u x(y) (3) ( ( )) exp i k p arcsin η 0 (y, p) x 1 η0(y, 2 p) y η 0 (y, p)

9 9 with the stationary point η 0 (y, p) given by η 0 (y, p) = x p + y x 2 + y 2 p 2 x 2 + y 2. (4) In the following the magnitude of u x (p), u x (p), is denoted as CT amplitude. The bending angle α(p) follows from the transformed signal s phase α(p) = 1 k d dp arg(u x(p)) + α 0 (p) (5) where the second term, α 0 (p), corrects for the non-zero ray direction angle at the GPS satellite, α 0 (p) = arcsin x G p + y G x 2 G + yg 2 p 2. (6) x 2 G + yg 2 (x G, y G ) denote the position of the GPS satellite Canonical transform / sliding spectral (CTss) method Simulation studies ignoring the receiver tracking process confirm that the Abel inversion of bending angles obtained by the CT method precisely reproduces the true refractivity profile even under severe multipath conditions provided the vertical refractivity gradient remains above about 160 km 1. However, phase deviations introduced by the receiver tracking loops significantly affect the performance of the CT analysis. An example is shown in Figs. 3 and 4. The bending angle profile (blue line) derived from a tropical radio sonde observation is plotted as a function of the height of the ray asymptote (ray height) h p r C, where r C denotes the radius of local curvature at the occultation s location [Syndergaard, 1998]. The temperature and humidity sounding took place over the Atlantic Ocean at 23.1 S, 26.0 W on 29 October 1996 between 12:00 and 14:00 UTC during the ALBATROS

10 10 field measurement campaign aboard the research vessel POLARSTERN. Superimposed are plotted the CT-derived bending angle profiles (green) obtained from simulations including (Fig. 4) and excluding (Fig. 3) the receiver tracking process. In Fig. 4 the CT-derived bending angles deviate significantly from the true bending angles (blue) between 4.5 and 5.5 km and below 3 km ray height. Without receiver modelling the CT solution (green broken line) agrees precisely with the true profile (blue) as shown in Fig. 3. In the heuristic CTss method a RO signal is subjected to a CT in a sliding window (aperture) of limited width (extension). Thus, the transformed signal depends on impact parameter as well as position and width of the aperture. That aperture which provides maximal signal amplitude for a given impact parameter is used for estimating the bending angle. In contrast to the CT method the bending angle is not calculated from the derivative of the transformed signal s phase, which is susceptible to tracking errors, but is estimated by assuming that the corresponding ray arrives at the center of the aperture. This ad hoc approach is justified by the results. In the CTss method we determine the CT amplitude of a signal u x (y) modified by a windowing function. For simplicity, a square windowing function w(ỹ, y) of width y centered at y is used, w(ỹ, y) = 1 : if ỹ y < y/2 0 : else. (7) Inserting w(ỹ, y) in Eqn. 3 yields u x (p, y) = k ( i 2π exp π ) 4 dỹ 1 η2 0(ỹ, p) x + p η 0 (ỹ, p) w(ỹ, y) u x(ỹ) (8)

11 11 = ( ( )) exp i k p arcsin(η 0 (ỹ, p)) x 1 η0(ỹ, 2 p) ỹ η 0 (ỹ, p) k ( i 2π exp π ) y+ y/2 dỹ 4 y y/2 ( ( exp i k p arcsin(η 0 (ỹ, p)) x 1 η2 0(ỹ, p) x + p η 0 (ỹ, p) u x(ỹ) )) 1 η0(ỹ, 2 p) ỹ η 0 (ỹ, p) with the stationary point η 0 (y, p) given by Eqn. 4. We note that the CTss signal u x (p, y) depends on impact parameter p as well as window position y. For convenience y is expressed in terms of bending angle ɛ according to ɛ = ( ) x p y x2 + y arcsin p 2 + α x 2 + y 2 0 (p) (9) = arcsin(η 0 (y, p)) + α 0 (p) where α 0 (p) is given by Eqn. 6. (The symbol ɛ rather than α(p) is used to emphasize that ɛ is a free variable whereas α(p) is a single-valued function of p.) Thus, we can write Eqn. 8 in terms of bending angle ɛ instead of window position y and define A(p, ɛ) u x (p, ɛ). (10) Figs. 3 and 4 show A(p, ɛ) coded in shades of gray as a function of ray height Figure 3. Figure 4. and bending angle for two simulation runs. In Fig. 3 the result without receiver simulation is plotted, in Fig. 4 the receiver tracking is included. Superimposed are shown the true geometric optical bending angle profile (blue line) and the CT solution (green line). Whilst in the case of a bypassed receiver (Fig. 3) the CT solution agrees precisely with the true bending angle profile, in the simulation with activated receiver tracking the CT solution deviates significantly due to receiver-induced phase errors. In Fig. 4 the largest values of A(p, ɛ) (plotted in dark colors) seem to follow the true bending angle profile more closely than the CT solution. We assume that

12 12 this phenomenon applies generally; it is the key assumption of the heuristic CTss method. Thus, for each impact parameter p the value of ɛ where A(p, ɛ) assumes its maximum in the following this value of ɛ is denoted by α (p) in order to distinguish it from α(p) given by Eqn. 5 is regarded as the bending angle that corresponds to p, i.e. max ɛ (A(p, ɛ)) = A(p, α (p)) A(p). (11) where the subscript indicates that the maximum is determined in ɛ space for a given p. In the following A(p) will be denoted as CTss amplitude. The resulting bending angle profiles α (p) are superimposed as red lines in Figs. 3 and 4. In Fig. 4 the CTss profile is less affected by the receiver-induced deviations between 4 and 6 km and at around 3 km ray height compared to the CT bending angle profile. On the other hand, the CTss solution completely misses the layer at 5.2 km and exhibits in general a lower vertical resolution. Figure 5. The CTss amplitude profiles A(p) corresponding to Figs. 3 and 4 are plotted in Fig. 5 (left panel); both profiles are almost identical. The CT amplitudes, on the other hand, differ significantly as shown in the right panel suggesting that the CTss amplitude is less affected by receiver tracking errors than the CT amplitude. The CTss bending angle profiles are truncated at the smallest impact parameter p where p is defined by A(p ) = 0.5 A. Furthermore, profiles with σ(a) > 0.2 A are removed as well. Here, A is the mean of A(p) and σ(a) the standard deviation of A(p) between 15 and 20 km ray height. 0.5 A and A are marked as dotted lines in Fig. 5. Figure 6. Fig. 6 shows the fractional retrieval error (N N 0 )/N 0 where N 0 is the true

13 13 refractivity. Solid and dashed lines correspond to the N profiles inverted by use of the CT and CTss methods (green and red lines in Fig. 4, respectively). For reference the CT solution obtained from the receiver-free simulation is plotted as well (dotted line). While the CTss solution shows somewhat larger deviations from N 0 than the CT solution above 6 km, the CTss solution performs significantly better within the multipath region between 1 and 4 km altitude. 3. Simulations of Doppler inversion It is known, that in the presence of multipath propagation the function α(p), derived from phase (Doppler), is multi-valued and can only be used as a coarse approximation for true α(p). Here we demonstrate that the Doppler-derived α(p) can substantially depend on the extension of acquired RO signal. For this purpose we use the RO signal simulated for one of high resolution tropical radiosonde N profiles (can be seen in Fig. 8B) as discussed by Sokolovskiy [2001b]. Figure 7. The simulated signal was sampled at 50 Hz, downconverted by cubic spline least square fit of the accumulated phase with 1 s increment, and subjected to spectral analysis in sliding apertures of 0.64 s. Each frequency was associated with ray arrival angle and, with account for the position of the center of the aperture, with impact parameter p. Fig. 7A shows the simulated RO signal in the form of sliding spectrogram, as introduced by Beyerle and Hocke [2001] (the density of gray color represents the spectral amplitude A), but in different coordinates: z = y r C, h = p r C. The horizontal stripes can be associated with rays (signals) diffracted by refractivity layers and by Earth s surface. The darkest part of the spectrogram traces the mean dependence z(h). The signals positioned at the left

14 14 and right sides from z(h) can be treated as refracted (dh/dz = 0) and reflected (dh/dz < 0) signals. The signals marked by RS and RL are reflected from the surface and from the refractivity layer at altitude 2 km (see Fig. 8B). It is important that at z < 140 km, the amplitude of the main signal, diffracted at the Earth s surface, at h 3 km, substantially decreases and becomes comparable to the amplitude of the signals refracted by elevated layers, at h 4, 5 and 6.5 km. This means that a receiver, which descends deep into the radio shadow zone, may effectively receive radiowaves from above the Earth s limb, due to diffraction by elevated refractivity layers. Fig. 7B shows the mean height of ray asymptote for the signals arriving at receiver, h(z) = ha(z, h)dh/ A(z, h)dh. As seen, h(z) noticeably increases at z < 100 km. Figure 8. Bold line in Fig. 8A shows the multi-valued function α(h) derived from the least square fit of phase used for the downconversion of RO signal (discussed above). For comparison, thin line shows the single-valued function α(h) obtained by CT method (strong positive spikes in the CT α(h) trace supercritical or close to critical refractivity gradients). For the Abel inversion, the multi-valued Doppler α(h) must be approximated by a single-valued function. The result of such approximation may substantially depend on the extension of acquired RO signal. As an example, dotted line shows the single-valued approximation α(h) when the RO signal was used down to z = 175 km. Fig. 8B shows the original N profile used for the simulation (solid line), the N profiles retrieved from the CT α(h) (dashed line), and from the single-valued approximation of Doppler α(h) when the signal was used down to different z, as indicated in the figure (thick solid lines). As seen, the Doppler inversion results

15 15 in negative N bias at altitudes between 2.5 and 6 km. However, below 2 km the Doppler inversion may result in both negative and positive N bias, depending on the extension of the used RO signal, and thus it may not be used as the reference for comparison to other methods. The magnitude of this effect depends on amplitude of radiowaves diffracted by elevated N layers. We note that the negative N bias of the CT inversion, which is due to only the effect of superrefraction, converges when RO signal extends down to low enough observation altitude z [Sokolovskiy, 2003]. 4. Discussion of CHAMP observations The heuristic CTss algorithm is validated on the basis of CHAMP RO observations [Wickert et al., 2001, 2003a]. Refractivity profiles derived from CTss bending angle profiles are intercompared with meteorological analysis results provided by the European Centre for Medium-Range Weather Forecasts (ECMWF). ECMWF pressure and temperature values are calculated by linear interpolation between grid points ( resolution) on 21 pressure levels. Then, linear interpolation in time is performed between 6 h ECMWF analyses fields. In order to illustrate the improvements gained by the CTss method the corresponding refractivity profiles obtained by the CT method are compared with ECMWF analyses as well. The simulations described above focus on signal propagation in the neutral atmosphere; contributions from the ionosphere are not taken into account. The CHAMP measurements, however, are affected by ionospheric contributions that need to be corrected for. We approximate the ionospherically corrected signal by S(t) = A ic (t) exp(i Φ(t)) (12)

16 16 with signal phase Φ(t) given by Φ(t) = k (l ic (t) + d(t)). (13) The amplitude A ic (t) A L1 (t) is assumed to be frequency independent, d is the geometrical distance between GPS and LEO and the excess path l ic is calculated from the L1 and L2 excess paths l L1 and l L2 by the standard method l ic = (f L1) 2 l L1 (f L2 ) 2 l L2 (f L1 ) 2 (f L2 ) 2. (14) f L1 = GHz and f L2 = GHz are the L1/L2 carrier frequencies, respectively. Since L2 tracking errors in the lower troposphere start to occur earlier compared to L1 tracking due to low L2 signal-to-noise ratios and losses due to codeless tracking techniques [Kaplan, 1996] Eqn. 14 should not be applied at low altitudes. We determine the range of valid L2 tracking by analyzing the ratio of the first excess path differences, R(t i ) l L1(t i+1 ) l L1 (t i ) l L2 (t i+1 ) l L2 (t i ). (15) Within the neutral atmosphere the refractive index is frequency independent. Thus, R(t) = 1 if the excess path is dominated by contributions from the neutral atmosphere. If, on the other hand, the excess path is controlled by ionospheric propagation the increase in excess path from one time step to next, l l(t i+1 ) l(t i ) is proportional to square of the inverse carrier frequency. We obtain R(t) = (f L2 /f L1 ) Figure 9. Two example profiles of R(t) are plotted in Fig. 9. The top panel shows the temporal evolution of R(t) for occultation number 9 recorded on 14 May 2001 at 1:15 h (UTC) and W, 83.9 N, (occultation identification string

17 #0009) in the bottom panel R(t) is plotted for occultation number 10 recorded on the same day at 1:27 h (UTC) at 22.9 W, 43.6 N ( #0010). To remove high frequency noise of R a 2-second (100 samples) running mean filter is applied. Occultation #0009 occurred during quiet ionospheric conditions characterized by a smooth transition from R = 0.61 to R = 1. L2 tracking in the lower troposphere is lost at about 66 s when R starts to deviated from unity. Occultation #0010, by contrast, observed 15 minutes later suffers from reduced signal-to-noise ratios due to low gain far from the antenna boresight possibly in combination with ionospheric disturbances causing significant fluctuations of R within the first 20 s of the observation. Observations corrupted by ionospheric disturbances are removed from the data set using the ratio R. Specifically, profiles have to fulfill the following two requirements in order to be included in the analysis: < R < 1.03 for at least 750 samples; i.e. if 0.97 < R < 1.03 holds true for a time period [t A, t B ] then t B t A 15 s must be fulfilled. 2. For t < t A at least 50 samples fall within the 0.41 < R < Following Rocken et al. [1997] the range of the ionospheric correction formula (Eqn. 14) is extended to the lower troposphere by fitting δl l ic l L1. (16) within the range t A < t < t B to a second order polynomial. Then, the ionospheric excess paths for t > t B are determined from the extrapolated value of δl, l ic = δl + l L1. (17)

18 18 I.e., for t > t B the observed L2 phase data is not taken into account. Multipath ray propagation is unlikely to occur within the stratosphere; under singlepath conditions, however, the standard Doppler retrieval is expected to yield the most precise bending angle profiles. Thus, the CTss and CT bending angles are replaced by the corresponding Doppler-retrieved values above 20 km ray height (corresponding to an altitude of approximately 18 km). We discuss the CTss results in terms of refractivity profiles calculated from the ionospherically corrected signals (Eqn. 12) and compare the results with ECMWF analyses. In the CHAMP data analysis u x (p, ɛ) (Eqn. 8) is calculated for ray heights between 0 and 20 km corresponding to an impact parameters range from r C to r C + 20 km with a resolution of 20 m. The backpropagation plane is placed at a distance of x = 500 km. Window position y ranges from r C 17 km to r C + 23 km with a spacing of 2 m, window width y is taken to be 200 samples corresponding to 400 m. The following intercomparison of CT- and CTss-processed data is based on 5079 excess path profiles derived from CHAMP RO observations. In 94 cases the standard Doppler retrieval did not produce refractivity profiles covering the altitude range between 20 and 40 km. From the remaining 4985 profiles 647 are removed since they fail to meet the two conditions with respect to the ratio R described above. In 86 out of the remaining 4338 observations the standard deviation of the CTss amplitude between 15 and 20 km ray height exceeds 20% of the mean amplitude A (see subsection 2.2). Finally, 31 refractivity profiles deviating from the corresponding ECMWF profile on average by more than 10% between 10 and 20 km altitude are removed from the data set in post-processing

19 19 leaving 4221 refractivity profiles available for the intercomparison. Figure 10. Fig. 10 shows the mean fractional refractivity error (N N 0 )/N 0 as a function of altitude based on these 4221 observations. N 0 now denotes the ECMWF refractivity. The CTss result is plotted as solid line, the corresponding canonical transform result as dashed line. Below about 8 km altitude the CTss method performs significantly better than the CT algorithm with a bias compared to ECMWF reduced by more than a factor of 2. The right panel in Fig. 10 shows the relative number of data points used in the intercomparison. I.e., 50% of the profiles reach an altitude of 1.1 km, only 10% pass below 300 m. Figure 11. The corresponding root-mean-square (rms) deviation of the fractional refractivity (N N 0 )/N 0 is plotted in Fig. 11. Again, below about 10 km altitude the CTss rms deviations are significantly less than the CT values. Figure 12. The meridional distribution of the refractivity bias derived with the heuristic CTss method is plotted in Fig. 12, the corresponding CT result is shown in Fig. 13. The comparison is restricted to the common subset of CTss and CT profiles consisting of 4221 observations (83.1% of the available excess phase measurements). Within the lower stratosphere at polar latitudes the CT refractivities (Fig. 13) are less biased than the corresponding CTss profiles. On the other hand, the CTss result (Fig. 12) exhibits a significantly reduced refractivity bias in the tropics below about 6 km altitude. A negative bias persists, though, that is most likely due to occurrences of superrefraction. At high latitudes the CTss results are almost bias-free. Figure 13.

20 20 5. Conclusions Refractivity profiles calculated from Doppler-inverted bending angles depend on the chosen end point of the occultation signal within the shadow zone. Thus, geometrical optics solutions having no physical meaning in the presence of multipath propagation should not be used as a reference for intercomparisons with other radioholographic techniques. In order to compensate for phase errors induced by the receiver tracking process during a RO event we propose a heuristic signal analysis method for the derivation of bending angles that is based on the canonical transform technique and the sliding spectral approach. The CTss method significantly reduces the negative refractivity bias and the rms deviation in the lower troposphere. The meridional distribution of the CTss refractivity bias shows improvements at high and mid latitudes compared to the CT results almost completely eliminating the refractivity bias with respect to ECMWF. The residual bias still found in the tropical lower troposphere is most likely due to superrefraction. Acknowledgments. Helpful discussions with C. Ao, P. Hartl and T. Meehan are gratefully acknowledged. We thank R. Weller, Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany for high-resolution rawinsonde data. The European Centre for Medium-Range Weather Forecasts provided meteorological analysis fields. This study was carried out within the HGF project GPS Atmospheric Sounding (grant no. FKZ 01SF9922/2).

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24 24 Sokolovskiy, S. V., Modeling and inverting radio occultation signals in the moist troposphere, Radio Sci., 36, (3), , 2001a. Sokolovskiy, S. V., Tracking tropospheric radio occultation signals from low Earth orbit, Radio Sci., 36, (3), , 2001b. Sokolovskiy, S., Effect of superrefraction on inversions of radio occultation signals in the lower troposphere, Radio Sci., in print, Steiner, A. K., G. Kirchengast, and H. P. Ladreiter, Inversion, error analysis, and validation of GPS/MET occultation data, Ann. Geophysicae, 17, , Syndergaard, S., Modeling the impact of Earth s oblateness on the retrieval of temperature and pressure profiles from limb sounding, J. Atmos. Solar-Terr. Phys., 60, (2), , Vorob ev, V. V., and T. G. Krasil nikova, Estimation of the accuracy of the refractive index recovery from Doppler shift measurements at frequencies used in the NAVSTAR system, Phys. Atmos. Ocean, 29, , Ware, R., M. Exner, D. Feng, M. Gorbunov, K. Hardy, B. Herman, Y. Kuo, T. Meehan, W. Melbourne, C. Rocken, W. Schreiner, S. Sokolovskiy, F. Solheim, X. Zou, R. Anthes, S. Businger, and K. Trenberth, GPS sounding of the atmosphere from low Earth orbit: Preliminary results, Bull. Am. Meteorol. Soc., 77, (1), 19 40, Wickert, J., C. Reigber, G. Beyerle, R. König, C. Marquardt, T. Schmidt, L. Grunwaldt, R. Galas, T. K. Meehan, W. G. Melbourne, and K. Hocke, Atmosphere sounding by GPS radio occultation: First results from CHAMP, Geophys. Res. Lett., 28, (17), , Wickert, J., G. Beyerle, T. Schmidt, C. Marquardt, R. König, L. Grunwaldt, and

25 25 C. Reigber, GPS radio occulation with CHAMP, in First CHAMP mission results for gravity, magnetic and atmospheric studies, edited by C. Reigber, H. Lühr, and P. Schwintzer, , Springer Verlag, Berlin, 2003a. Wickert, J., T. Schmidt, G. Beyerle, R. König, C. Reigber, and N. Jakowski, The radio occultation experiment aboard CHAMP: Operational data processing and validation of atmospheric parameters, J. Meteorol. Soc. Jpn., submitted, 2003b. G. Beyerle, GeoForschungsZentrum Potsdam (GFZ), Department 1, Geodesy and Remote Sensing, Telegrafenberg, D Potsdam, Germany ( Ch. Reigber, GeoForschungsZentrum Potsdam (GFZ), Department 1, Geodesy and Remote Sensing, Telegrafenberg, D Potsdam, Germany ( T. Schmidt, GeoForschungsZentrum Potsdam (GFZ), Department 1, Geodesy and Remote Sensing, Telegrafenberg, D Potsdam, Germany ( S. Sokolovskiy, University Corporation for Atmospheric Research P.O. Box 3000 Boulder CO , USA, ( J. Wickert, GeoForschungsZentrum Potsdam (GFZ), Department 1, Geodesy and Remote Sensing, Telegrafenberg, D Potsdam, Germany ( Received

26 26 To appear in the Journal of Geophysical Research, May, 2003 This manuscript was prepared with AGU s L A TEX macros v5, with the extension package AGU ++ by P. W. Daly, version 1.6b from 1999/08/19.

27 27 Figure Captions ray from GPS y backpropagation plane p Earth α p x LEO orbit Figure 1. Occultation geometry. A ray characterized by impact parameter p propagates through the atmosphere and is refracted with bending angle α. The signal is observed by a GPS receiver aboard a low-earth orbiting satellite. For further processing the observed signal is propagated to a line x = const (backpropagation plane) and transformed to impact parameter space. (Figure is not to scale.)

28 28 Figure 2. A schematic diagram of the end-to-end simulation. Using the multiple phase screen (MPS) method the atmospheric propagation of a GPS signal based on refractivity profiles is modelled. The generated signal amplitude and phase serves as input to a GPS software receiver. The receiver s output are converted to bending angle profiles using the canonical transform (CT) method. The simulation loop is closed by Abel-transforming the bending angle profiles into refractivity profiles. (Schematic taken from Beyerle et al. [2003].)

29 29 height of ray asymptote (km) true CT CTss bending angle (rad) Figure 3. CTss amplitude A(h, ɛ) (coded in different shades of gray) as a function of bending angle ɛ and ray height h. In this simulation run the receiver tracking process is not taken into account. Dark gray indicate enhanced values of A(h, ɛ), values close to zero are plotted in light gray or white. Diffraction patterns are caused by the rectangular window shape (Eqn. 7). Profiles of the true bending angle (blue), the CT result (dashed green) and the CTss processing result (red) are superimposed. For details see text.

30 30 height of ray asymptote (km) true CT CTss bending angle (rad) Figure 4. Same as Fig. 3 however with the receiver tracking model included in the end-to-end simulation chain. The large deviations of the CT bending angle profile are caused by receiver-induced phase errors.

31 31 20 height of ray asymptote (km) norm. CTss amplitude norm. CT amplitude Figure 5. Normalized CTss (left panel) and CT (right panel) amplitude profiles derived from the simulation study. Results including the receiver tracking simulation are plotted as thick lines, the amplitudes obtained from the bypassed receiver simulation are shown as thin lines. The CT amplitude (right) is extremely sensitive with respect to receiver tracking errors producing large deviations between the two cases. In comparison the CTss amplitude differences (left) are significantly smaller. The profiles are normalized to unity between 15 and 20 km ray height.

32 altitude [km] (N N 0 )/N 0 [%] Figure 6. Fractional refractivity deviations as a function of altitude obtained from CTss (solid line) and CT (dashed line) processing. In both cases the receiver tracking is included in the simulation. For reference, the CT result obtained from the bypassed receiver simulation is marked as dotted line.

33 33 Figure 7. (A) sliding spectrogram of the simulated RO signal (for details see text). (B) mean height of ray asymptote derived from the spectrogram.

34 34 10 A B 10 height of ray asymptote (km) km 90km 150km 175km altitude (km) bending angle (rad) refractivity (N units) 0 Figure 8. (A) bending angle as the function of height of ray asymptote derived by CT method (thin line) and from Doppler (bold line) with a single-valued approximation (bold dotted line). (B) refractivity profile used for the simulation (thin line), Abel inversion of the CT bending angle (thin dotted line) and of the single-valued approximation of the Doppler bending angle for different length of RO signal (bold lines).

35 L1 / L L1 / L2 1 plot time [s] Figure 9. The ratio of the first forward differences of the L1/L2 excess paths. The top panel shows the ratio R for occultation number 9 recorded on 14 May 2001 at 1:15 h (UTC) and W, 83.9 N; the next occultation on that day 12 minutes later (at 22.9 W, 43.6 N) is plotted in the bottom panel.

36 altitude (km) (N N 0 )/N 0 (%) # prf. [%] Figure 10. Mean fractional refractivity error derived from 4221 CHAMP observations recorded between 14 May and 10 June 2001 and the corresponding ECWMF meteorological analyses (left). The CTss and CT results are plotted as solid and dashed lines, respectively. The right panel shows the fractional number of data points available for the intercomparison.

37 h [km] σ((n N 0 )/N 0 ) [%] # prf. [%] Figure 11. Same as Fig. 10 however showing the root-mean-square deviation of the fractional refractivity.

38 38 15 altitude (km) latitude (deg) N/N [%] Figure 12. Meridional distribution of the fractional refractivity deviation between CHAMP observations and ECMWF fields as a function of altitude. Results of 4221 CHAMP occultations obtained the heuristic CTss method are shown. Note that isopleth lines are non-equidistant.

39 39 15 altitude (km) latitude (deg) N/N [%] Figure 13. Same as Fig. 12, however derived from refractivity profiles processed by the CT method.