The atmospheric propagation of GPS signals under multipath conditions and their detection are simulated. Using the multiple phase screen method, C/A-code modulated L1 signals are propagated through a spherically symmetric refractivity field derived from a high-resolution radio sonde observation. The propagated signals are tracked by a GPS receiver implemented in software and converted to refractivity profiles by the canonical transform technique and the Abel inversion. Ignoring noise and assuming an ideal receiver tracking behavior, the true refractivity profile is reproduced to better than 0.1% at altitudes above 2 km. The nonideal case is simulated by adding between 14 and 24 dB of Gaussian white noise to the signal and tracking the signal with a receiver operating at 50 and 200 Hz sampling frequency using two different carrier phase detectors. In the upper troposphere and stratosphere the fractional refractivity retrieval error is below 0.3% for 50 Hz sampling and below 0.15% for 200 Hz sampling. In the midtroposphere down to altitudes of about 2 km, phase-locked loop tracking induces negative fractional refractivity biases on the order of −1 to −2% at 50 Hz sampling frequency. Modifications to the receiver tracking algorithm significantly improve the retrieval results. In particular, replacing the carrier loop's two-quadrant phase extractor with a four-quadrant discriminator reduces the refractivity biases by a factor of 5; increasing the sampling frequency from 50 to 200 Hz gains another factor of 2.
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 Atmospheric soundings of temperature and water vapor by Global Positioning System (GPS) radio occultation (RO) measurements are increasingly being considered for numerical weather prediction and climate change studies. To date a data set of more than 100,000 temperature profiles has been collected by the proof-of-concept GPS/Meteorology (GPS/MET) mission [Ware et al., 1996] as well as the current CHAMP [Reigber et al., 2000, 2002] and SAC-C missions. First validation studies based on CHAMP observations indicate that the observed temperature bias with respect to European Centre for Medium-Range Weather Forecasts (ECMWF) global analyses is less than 1 K above the tropopause and less than 0.5 K between 12 to 20 km at mid and high latitudes [Wickert et al., 2001; Hajj et al., 2002]. These values are consistent with validation results of GPS/MET data [Rocken et al., 1997].
 However, in the lower troposphere at mid and low latitudes a negative refractivity bias of more than 1% is observed in past and current RO data [see, e.g., Rocken et al., 1997; Marquardt et al., 2001]. The negative bias is commonly attributed to the receivers' inadequate signal tracking behavior within zones of multipath signal propagation [see, e.g., Gorbunov, 2002b]. These multipath zones are caused by complicated structures in the tropospheric refractivity field generated by spatial variations of the water vapor distribution. The analysis of radio occultation data affected by multipath propagation is extensively discussed in the literature [Gorbunov et al., 1996; Karayel and Hinson, 1997; Hocke et al., 1999; Sokolovskiy, 2001b]. Recently, the canonical transform (CT) method was devised to solve the problem of calculating bending angle profiles from phase and amplitude data observed within multipath regions [Gorbunov et al., 2000; Gorbunov, 2002a, 2002b]. CT processing of a large number of GPS/MET and CHAMP observations, however, did not succeed in removing the refractivity bias suggesting that the quality of the lower tropospheric data is degraded (T. Schmidt and J. Wickert, GeoForschungsZentrum, personal communication). This conclusion is corroborated by a sliding spectral analysis [Gorbunov, 2002b].
Sokolovskiy [2001b] performed first studies modeling phase-locked loop tracking of simulated tropospheric radio occultation signals with complicated dynamics. These results indicated a strong sensitivity to loop parameters and motivated to consider open loop tracking techniques. In the present study we follow Sokolovskiy's approach and investigate the observed refractivity bias using end-to-end simulations. The atmospheric propagation of a C/A-code modulated GPS signal is modeled. For simplicity, only the neutral atmosphere is considered in the following, dispersion due to ionospheric propagation is not taken into account. GPS signal demodulation is performed by multiplication with a C/A-code replica. In multipath zones, however, the signal cannot be demodulated unambiguously since multiple interfering rays may have different phase delays and Doppler shifts. The objective of the present study is to investigate the influence of the receiver tracking process on retrieved amplitude and phase data and explore their relation to the negative refractivity bias.
 The paper is organized into two parts: first, the methods employed in the simulation chain are introduced and described. Briefly reviewed are the multiple phase screen method which is used to simulate GPS signal propagation through the atmosphere and the CT method which transforms the amplitude and phase data into bending angle profiles. Second, the simulation studies are described and the results are discussed.
 In our simulation study the refractivities retrieved at altitude ranges between 2 and 6 km differ characteristically from values above 6 km and below about 2 km. For brevity, we denote the altitude range between 2 and 6 km as “midtroposphere,” whereas the altitude range below 2 km is termed “lower troposphere.”
 The atmospheric propagation of GPS signals, the tracking process by a receiver instrument and the data analysis is studied using a sequence of simulation elements. Each element processes the output of the previous element and generates the input of the following element. This simulation “chain” is schematically illustrated in Figure 1.
 Using the multiple phase screen (MPS) method, the atmospheric propagation of a C/A-code modulated GPS signal based on a high-resolution refractivity profile N() = (n() − 1) · 106 is modeled. Here, n() denotes the real part of the refractive index. The refractivity field is assumed to be spherically symmetric, i.e., N() = N(r). The the refractivity profile N(r) is derived from a high-resolution radio sonde observation. The generated electromagnetic (EM) field serves as input to a single channel GPS software receiver. The receiver's output data, carrier phases and signal-to-noise ratios (SNRs), are converted to bending angle profiles using the CT method. Finally, refractivity profiles are obtained by Abel-transforming the bending angle profiles thereby closing the simulation loop. Statistical analyses of original (true) and retrieved refractivity profiles provide insight into systematic deviations introduced by the individual simulation steps. In the following subsections the individual simulation elements are described.
2.1. Multiple Phase Screen Simulation
 Atmospheric propagation of GPS signals is numerically simulated using the multiple phase screen (MPS) technique [Knepp, 1983; Martin and Flatté, 1988]. Several studies simulating radio occultation observations successfully employed the MPS technique [see, e.g., Karayel and Hinson, 1997; Gorbunov and Gurvich, 1998; Sokolovskiy, 2001b]. Since the spatial scales of the refractive index field variations are much larger than the GPS signal wavelength (19.03 cm at L1) backscattering can be ignored and the propagated signal can be determined by representing the continuous medium by a sequence of phase screens. For simplicity, we model an infinitely remote GPS satellite immovable with respect to the Earth's atmosphere. The incident wave is a plane wave with a unit amplitude.
 The temperature and humidity measurement, the atmospheric refractivity profile N(r) is calculated from, took place on the Atlantic ocean at 23.1°S, 26.0°W on 29 October 1996 between 12:00 and 14:00 UTC during the ALBATROS field measurement campaign aboard the research vessel “POLARSTERN.” 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 GmbH, 1989]. In order to reduce the humidity 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 sonde profile ends at an altitude of 26.3 km. Above that altitude the profile is continued by an exponential function using a scale height of 7.9 km. The profile, the tropospheric part of which is plotted in Figure 2, was selected because of several layered humidity structures in the midtroposphere which translate into corresponding layers in the vertical refractivity gradient. As will be shown below, these layers cause multipath beam propagation in the lower and midtroposphere.
 Within the MPS approach the refractivity field N(r) is modeled as a series of phase screens; Figure 3 illustrates the concept and defines the coordinate system. At each screen the incident EM wave suffers a phase shift whereas the wave's amplitude remains unchanged. Between the screens the EM wave is propagated through vacuum. The result of the MPS calculation is the signal amplitude A(y) and phase ϕ(y) on the observation screen O.
 The MPS integration step and resolution are taken from Sokolovskiy [2001a]: each of the L = 2001 phase screens consists of 219 = 524, 288 grid points with a separation of Δy = 0.54 m, vertically the screens extend over about 283 km. Horizontally, the 2001 phase screens are separated by a distance of Δx = 1 km covering a horizontal range of 2000 km. The distance between the last phase screen and the observation screen is 1500 km, the distance between central and observation screen therefore is 2500 km corresponding to a satellite orbit altitude of 450 km.
 The GPS signal's spread spectrum modulation implies that the transmitter signal uI(t) arriving at the first phase screen is not a pure tone but exhibits a finite bandwidth [Kaplan, 1996]. As an illustration, the main lobe and the first few ancillary lobes of a normalized power spectral density calculated from a L1 signal modulated with C/A-code PRN 17 is shown in Figure 4. The main lobe extends from about fc − 1.1 MHz to fc + 1.1 MHz; here, fc denotes the L1 carrier frequency of 1575.42 MHz.
 IN the present study the propagation of the spread spectrum signal is simulated by propagating a number of discrete frequencies covering the main lobe. An alternative approach would be the propagation of a pure tone adding the signal's modulation afterwards [see, e.g., Ao et al., 2003]. For simplicity, we treat only C/A-code modulation and ignore the effects from P-code and the 50 Hz navigation data modulation [Kaplan, 1996].
 The C/A-code consists of a pseudo-random sequence of 1023 bits (chips), CAk = ±1 with k = 1, …, 1023 and is periodic with a repeat frequency of 1 kHz (ignoring Doppler shifts due to relative motion between transmitter and receiver). Thus, uI(t) exhibits a discrete power spectrum with individual spectral lines separated by Δf = 1 kHz,
Here, Nc = fc/Δf = 1,575,420, fc is the carrier frequency and N★ is defined below. The coefficients cn are obtained by evaluating the Fourier integral
where ca(t) ≡ CA⌊1023 (t/T+1/2)⌋+1 is a piecewise constant function given by the chips CAk. (⌊·⌋ denotes the floor function and we set ca(T/2) ≡ ca(−T/2) = CA1.) The cn are found to be
The propagation of the C/A-code modulated signal is simulated by performing a series of MPS calculations using plane waves with frequencies varying between fc - 1.1 MHz and fc +1.1 MHz roughly covering the first main lobe, i.e., we choose N★ ≡ 1100.
 MPS simulations for monochromatic signals are very fast. However, the propagation of 2 N★ + 1 = 2201 components is computationally expensive. Therefore, the number of frequency components is reduced to 129 and the MPS amplitudes and phases, An(y) and ϕn(y), n = (Nc − N★, …, Nc + N★), are obtained by linear interpolation. Finally, the signal uO(y, t) at observation screen position y is assembled from the individual spectral components according to
In the following, we assume that the receiver moves with a constant velocity v = y/t ≡ 2.7 km/s along the observation screen corresponding to a LEO satellite orbit altitude of about 450 km. The observed GPS signal, therefore, is uR(y) ≡ uO(y, t = y/v).
2.2. Receiver Simulation
 The simulated GPS signals uR(t) are processed by a receiver module implemented in software. For simplicity it is designed as a single channel receiver and is restricted to C/A-code tracking only; decoding of navigation data bits is not implemented. Here we focus on specific aspects of the simulation receiver relevant to this study; for detailed discussions of GPS receiver technology we refer to the literature [Kaplan, 1996; Parkinson and Spilker, 1996; Tsui, 2000; Misra and Enge, 2002].
 Typical GPS signal levels found in antenna phase centers of ground-based or LEO-based receivers are on the order of −130 dBm, about 19 dB below the noise floor of −111 dBm [Kaplan, 1996; Tsui, 2000; Misra and Enge, 2002]. We model the receiver's input uA(t) as the sum of simulated signal uR(t) and white Gaussian noise
where SNR denotes the signal-to-noise in dB and N(t) is a Gaussian distributed random variable with unit standard deviation, i.e., σ(N(t)) = 1. In our simulation runs signal-to-noise ratios of SNR = −14 dB (strong signal) and SNR = −24 dB (weak signal) are used. These SNR values correspond to post-correlation carrier-to-noise density ratios of C/N0 = 44 dB-Hz and C/N0 = 34 dB-Hz, respectively. The sum uA(t) is three-level quantized and tracked by the software receiver.
 The receiver itself consists of two parts, the front-end and the code/carrier tracking loops [Kaplan, 1996; Tsui, 2000; Thomas, 1995]. The front-end performs digitization and down-conversion of the received signal from L1-band at fc = 1575.42 MHz to an intermediate frequency. Down-conversion is achieved by direct sampling, i.e., by sampling at a smaller rate than the carrier frequency fc [Tsui, 2000]. We choose fs = 5.045 MHz; thus, the L1 signal centered at fc is aliased to a frequency range centered at the intermediate frequency of fd = 1.38 MHz.
 The samples are digitized with a three-level discriminator, quantization thresholds are ±0.61 times the noise's standard deviation [Thomas, 1995]. The digitized samples are tracked with two phase-locked loops (PLL), the code and the carrier tracking loop. The PLLs compensate changes in carrier and code frequency due to delays induced by the atmospheric refractivity field (and possible relative motions between transmitter and receiver). The modulation-free carrier, required for PLL carrier tracking, is obtained by multiplying the received signal with a C/A-code model produced by the code tracking loop. Similarly, the code tracking loop inputs a carrier-free signal. In the carrier loop the demodulated signal is multiplied with sine and cosine values generated by a numerically controlled oscillator (NCO) and the result is low-pass filtered by summing over a large number of samples. Typical summation periods (predetection integration times) are 10 or 20 ms; thus the sum extends over 10 ms · fs = 50,450 or 20 ms · fs = 100,900 samples. Similar to the carrier tracking loop, the code tracking PLL reads carrier-free signals which are obtained by multiplying the received signal with a cosine wave produced by the carrier tracking loop. The carrier-free signal then is multiplied with C/A-code replicas shifted by one half-chip (corresponding to a time offset of about ±0.5 μs).
Table 1. Carrier and Code Tracking Loop Parameters Used in the Simulation Study
Damping ratio ζ
 Within multipath propagation zones the signal phase exhibits strong fluctuations. These phase fluctuations are amplified by forming the time derivative. An example is plotted in Figure 5, bottom panel; strong Doppler frequency variations, e.g., between −35 km and −25 km, indicate the occurrence of multipath. Thus, significant deviations between observed and model signal, denoted as phase residual , occur and need to be detected reliably by the carrier tracking loop. The phase residual is related to the in-phase component I and quadrature-phase component Q, the outputs of the two carrier PLL low-pass filters, according to [Kaplan, 1996]
(We note that in the literature [e.g., Kaplan, 1996] approximations to equation (6) are given which are computationally less demanding. However, these approximations suffer from large deviations already at moderate values of and therefore will not be considered here.)
 The characteristic feature of PLLs with two-quadrant phase extraction is their insensitivity with respect to 180° phase changes; in particular, the phase residual remains unchanged if I and Q change their sign, i.e., (I, Q) = (−I,−Q). As a consequence, the 50 Hz navigation message modulation does not impair operation of the PLL provided the sample summation does not straddle data bit boundaries. However, two-quadrant phase extraction is less attractive if the carrier exhibits phase fluctuation exceeding 90°. Phase residuals larger than 90° are aliased into the [−90°, 90°] interval; e.g., a true phase residual of 135° produces a value of = −45° causing the NCO to decrease its frequency instead of increasing it.
 In order to improve the carrier loop's tracking behavior within multipath zones we tentatively replaced the two-quadrant arctangent phase detector arctan(Q/I) (equation (6)) by a four-quadrant arctangent extractor arg(I, Q) defined as
For phase residuals between −90° and 90° the arg phase extractor gives the same result as the two-quadrant arctangent extractor. However, it also yields the correct values for phase angles between −180° and −90° and between 90° and 180° where the two-quadrant arctangent extractor deviates by ±180°. Thus, the arg phase extractor extends the valid phase range from ±90° to ±180°. The drawback is that a carrier loop with an arg phase extractor no longer is insensitive with respect to phase reversals due to the navigation message modulation. The navigation message thus needs to be removed in advance from the incoming signal by a process called “data wipeoff” [Kaplan, 1996].
 The receiver outputs are I and Q components at the C/A-code period of nominally 1 msec. From these values receiver signal amplitudes AR are obtained by
at 50 Hz (N = 20) or 200 Hz (N = 5); the carrier phase is obtained from the carrier loop's NCO phase.
 The digitization and tracking process described above is repeated 100 times, in each case using the same signal uR(t) only replacing the noise component N(t) in equation (5). In each iteration the complete retrieval leading from signal amplitude and phase data to refractivity profiles are performed. Mean and standard deviations of bending angles and refractivities are calculated from the profile ensemble.
 In addition, the MPS signal is processed directly omitting the receiver simulation step in order to quantify to what extend the receiver tracking influences the derived refractivity (“ideal receiver case”). For this purpose the MPS result at the center frequency, ANc(y) · exp(), is used and no noise is added.
 The effect of multipath propagation on the C/A-code correlation function, i.e., the cross-correlation between observed and replica signal as a function of code offset, is shown in Figure 6. In the top panel the single path region from −14 km to −16 km is shown, the multipath zone (screen range from −24 km to −26 km) is plotted in the bottom panel. The occurrence of multipath leaves the shape of the correlation function relatively unchanged since the optical path length differences between individual rays are much less than the C/A-code chip length of 300 m (see discussion below). The correlation functions' magnitudes, however, start to fluctuate strongly induced by signal amplitude variations within multipath regions. We note, that the shape of the correlation functions deviates from the theoretically expected triangular form [see, e.g., Kaplan, 1996] since the bandwidth of the simulated signal is limited to 2.2 MHz.
 Not only phase fluctuations but also strong amplitudes variations within multipath regions contribute to receiver tracking errors. Incidences of low amplitude values push the antenna signal below the closed-loop's tracking detection limit and increase the probability of phase tracking errors. This correlation of low signal amplitudes with enhanced probability of phase errors is illustrated in Figure 7. It shows the difference between retrieved and true phase ΔΦ as a function of retrieved signal amplitude AR. Cycle slips, i.e., changes in ΔΦ, are more likely if AR falls below a certain threshold value.
2.3. Canonical Transform Technique
 The next step in the simulation procedure involves the derivation of bending angle profiles from phase and amplitude data produced by the software GPS receiver. The refractivity profile considered in this study (see Figure 2) induces strong multipath beam propagation. As a result, the EM field's phases and amplitudes exhibit strong fluctuation at the observation screen as is illustrated in Figure 5.
 A schematic representation of multipath beam propagation is shown in Figure 8. The projection of the ray manifold onto the (x, y)-plane (occultation plane in geometrical space) illustrates a ray structure caused by a strongly refracting atmospheric layer. Between t1 and t2 a receiver following the LEO orbit detects signal contributions from several interfering rays.
 The canonical transform (CT) method solves the problem of calculating the bending angle ϵ(p) as a function of ray impact parameter p within multipath regions. A detailed and motivated description of the method is given by Gorbunov [2001, 2002a, 2002b].
 The CT method uses the connection between geometrical optics and wave optics. In geometrical optics rays are described by the canonical Hamilton system which defines the evolution of coordinate y and corresponding momentum η. The momentum η is equal to the sine of the angle between the ray direction and the x-axis. The propagation distance x is looked at as the temporal coordinate. Multipath propagation is characterized by the fact that multiple rays may have the same coordinate y. In geometrical optics we can introduce a new coordinate and momentum by means of a canonical transform. If we take the ray impact parameter p as the new coordinate, then the canonical transform can be written in the form
This transform from the old canonical coordinates (y, η) to the new ones (p, ξ) resolves multipath propagation, because, for a spherically symmetrical medium, ray impact parameters p are different for different rays. In wave optics we consider the wave field as a function of x and y, which we denote ux(y). The corresponding transform of the wave function to the new representation is given by the following Fourier integral operator
where [ux](η) denotes the Fourier transform of ux(y)
and k = 2π/λ is the vacuum wave number. Because in this representation we have single-ray propagation, the momentum ξ is equal to the derivative of the optical path of the transformed wave function. The bending angle ϵ(p) follows from
In the present study the signal transmitter is placed at infinity. We note that for a finite transmitter distance a correction term has to be added to equation (12) [Gorbunov et al., 2000; Gorbunov, 2002a, 2002b]. The receiver trajectory follows a straight line at x = 2500 km (observation screen O in Figure 3). Consequently, backpropagation of the signal ux(y) is unneeded. Equation (11) is implemented as a Fast Fourier Transform [Press et al., 1992] for −120 km < y < 60 km yielding [ux](η) covering the range from η ≈ −0.176 to η ≈ 0.176.
Figure 8 shows a graphical illustration of the ray manifold in the three-dimensional (x, y, p)-space. The orbit segment in the (p, x)-plane deviates significantly from the near-circular shape in geometrical space.
 The squared magnitude of the transformed field in impact parameter space, or the CT amplitude, describes the energy distribution in the impact parameter space. For a spherically symmetrical atmosphere without absorption this function is a constant in the light zone and it drops abruptly to very small values in the shadow zone. In order to illustrate this property CT amplitudes are computed for a simulated EM field obtained by MPS calculations. Spatial filters are applied to the field resulting in signals w(y); for simplicity the filter shape is taken to be a rectangular window, i.e.,
Two signals are calculated using y★ = −15 km and y★ = −25 km, the corresponding CT amplitudes ∣x(p)∣ are shown in Figure 9. In both cases the window width Δy is 2 km. At y★ = −15 km single ray propagation dominates and the observed signal is mainly determined by rays which have probed the atmosphere at ray heights of about 8 km (about 6 km altitude). However, the signals observed around y★ = −25 km contain contributions from two height ranges, one at 6.5 km ray height (4.5 km altitude) and the other in the lower troposphere at 4.5 km ray height (2.5 km altitude). Thus, at y★ = −25 km the receiver resides within a multipath zone. The implications will be discussed below. Here, ray height is defined as h = p − rE. (Typically, ray heights are about 2 km larger than the corresponding tangent altitudes.)
2.4. Abel Transform
 Finally, the bending angle profiles ϵ(p) are inverted back into refractivity profiles N(r) using the Abel transform [Fjeldbo et al., 1971; Melbourne et al., 1994]. The Abel transform is the last step in the simulation chain.
3. Discussion of Simulation Results
 A large number of simulation runs have been performed; for brevity we restrict the discussion to four cases: in the baseline receiver experiment sampling frequency is taken to be 50 Hz and carrier tracking with two-quadrant arctangent phase extraction is used, the SNR level is −14 dB. The results are compared to simulations with a four-quadrant arctangent phase detector, 200 Hz sampling and/or a larger SNR level of −24 dB. The results are compiled in Figures 10–13. Figure 10 shows the result for the baseline experiment, Figure 11 shows the result for a four-quadrant arctangent phase detector, Figure 12 shows the result for a four-quadrant arctangent phase detector/200 Hz sampling, and Figure 13 shows the result for a four-quadrant arctangent phase detector/200 Hz sampling/−24 dB SNR.
 In all four figures the relative deviation between derived and true refractivity profile is plotted, 1-sigma standard deviations are marked in grey. For reference the simulation excluding the receiver tracking is plotted as a dashed line. The enhancement at altitude below 2 km is due to limitations of the MPS calculation. Extending the screen range from 2000 km to 3000 km and reducing the screen separation from 1 km to 0.5 km reduces the residual error from 0.25% to 0.1%.
 Above 6 km altitude the retrievals including the receiver tracking agree to better than 0.3% at 50 Hz sampling frequency (Figures 10, 11, and 13); at 200 Hz the agreement improves and reaches 0.1% in the low noise case (Figure 12). The zero standard deviations above 6 km indicate that the retrieval results are insensitive to the noise added (see equation (5)).
 At altitudes between 2.5 km and 6 km the two-quadrant phase detector tracking produces refractivity biases exceeding −1.5% with 50 Hz sampling frequency (Figure 10). The bias is reduced by about a factor of five if the two-quadrant phase detector is replaced by the four-quadrant phase detector (Figure 11); further improvement is obtained by increasing the sampling frequency from 50 to 200 Hz (Figure 12). Even under weak signal conditions with SNR of −24 dB the four-quadrant phase detector outperforms the baseline receiver by about a factor of three in terms of the refractivity bias (Figure 13). However, the standard deviation increases significantly indicating that loss of lock occurs more frequently below 6 km under low SNR conditions.
 Within the planetary boundary layer below 2 km altitude the retrievals exhibit strong negative biases (Figures 10–13). In all simulation experiments the bias increases strongly with decreasing altitude and reaches −25% at the surface (not shown). Ducting (confinement of the signal to a narrow region of the troposphere) is excluded as possible explanation since the vertical refractivity gradient never becomes smaller than the critical value of about −160 km−1 (compare Figure 2). An analysis similar to the one discussed in Figure 9 shows that rays passing below 2 km altitude strike the observation screen at locations below −40 km. However, below −40 km, in particular between −60 km and −50 km, the GPS signal is characterized by low amplitudes as shown in the top panel of Figure 5. As a consequence, below about 2 km altitude the signal amplitude approaches the threshold value for PLL tracking and loss of lock occurs. The simulation results shown in Figures 10–13 indicate that loss of lock consistently produces a negative bias with respect to the true refractivity profile. Thus, the observed refractivity bias below 2 km altitude is induced by early failure of closed-loop tracking which in turn is due to low signal amplitudes caused by defocusing. It is expected that a transition from closed-loop to open-loop tracking at low altitudes will remove this bias provided the vertical refractivity does not fall below the critical value [Sokolovskiy, 2001b].
 The CT amplitudes corresponding to Figures 10–13 are plotted in Figure 14. The CT amplitude obtained from the ideal receiver simulation is shown as well (left panel, dashed line); the latter follows closely the ideal step function shape indicating the beginning of the shadow zone at about 2.3 km ray height, whereas the receiver tracking results (Figure 10) expose frequent drop-offs of the CT amplitude by more than 50%.
 It was shown in Figure 9 that the measurement taken around −25 km on the observation screen is largely determined by signals which probed the atmosphere at ray heights between 4–5 km (about 2–3 km altitude). Thus, differences between the two-quadrant phase extractor retrieval (Figure 10) and the four–quadrant cases (Figures 11 and 12) within the 2–3 km altitude region are expected to correspond to deviations in retrieved amplitudes and phases around y = −25 km. These frequency and amplitude deviations are indeed observed: in Figures 15 and 16 the retrieved amplitude and Doppler profiles are compared with the true profiles as obtained from the MPS calculations. The Doppler and amplitude profiles show strong fluctuations which are characteristic for multipath signal propagation (compare Figure 17). From Figure 15 it is evident that the four-quadrant phase detector performs significantly better than the two-quadrant phase extractor in resolving the fine structure of the Doppler profile, in particular between −24 km and −25 km.
 The deviations in Doppler frequency between the two tracking simulations (Figure 15) are not matched by corresponding amplitude differences (Figure 16). Despite their strong variability both receivers are able to follow the amplitude signatures closely with only small differences between the two tracking results. We note that the baseline's receiver normalized signal amplitude rarely drops below 0.1 within the screen range considered here despite its poor Doppler tracking performance.
 Finally, the optical path lengths obtained from a ray tracing calculation are plotted in Figure 17 (thin line) for comparison with the corresponding MPS profiles (dashed and thick solid lines). Multivalued optical paths in the geometrical optics solution indicate the extent of multipath regions; e.g., between y = −35 km and y = −24 km the GPS receiver measures the interference pattern of three, between y = −31.5 km and y = −30.5 km even the pattern of five individual rays. The largest optical path difference between interfering rays are found to be about 5 m at y = −24.5 km corresponding to about 1/60 of the 300 m C/A-code chip length. In addition, the optical path differences obtained by MPS calculations including and excluding the receiver tracking are shown in Figure 17 as well (dashed and thick solid lines, respectively). Whereas the receiver reproduces the true MPS values up to a constant factor within regions of single ray propagation (y > −24 km), the tracking results start to deviate significantly within the multipath region (−35 km < y < −24 km).
 Simulation studies of atmospheric signal propagation, receiver tracking and canonical transform data analysis were performed for a high-resolution refractivity profile derived from a tropical radiosonde sounding. The study is based on the parabolic approximation of the electromagnetic wave equations, the assumption of spherical symmetry in the atmospheric refractivity field and the absence of absorption effects.
 Within regions of multipath signal propagation interference between individual rays cause enhanced amplitude and Doppler frequency fluctuations. However, these fluctuations need to be mapped accurately in order to be able to reconstruct the bending angle profile within the framework of the canonical transform method.
 From the simulation results we arrive at the following conclusions:
 1. Simulations with a noise-free signal and an ideal receiver reproduce the true refractivity profile to better than 0.1% at altitude above 2 km.
 2. Tracking with a baseline receiver model at 50 Hz sampling and using a two-quadrant PLL phase detector induces negative refractivity biases on the order of −1 to −2% in the midtroposphere. These values are of the same order of magnitude as biases observed in current satellite missions.
 3. In the midtroposphere the retrieved refractivity profiles depend on receiver tracking algorithms and parameters. By modifying receiver tracking algorithms the simulated retrievals could be improved significantly at altitudes between 2 and 6 km. In particular, replacing the two quadrant phase extractor with a four-quadrant discriminator reduced the refractivity biases by a factor of 5. Increasing the sampling frequency from 50 to 200 Hz gained another factor of 2.
 4. At altitudes below 2 km an even larger negative refractivity bias is observed in all tracking experiments reaching −25% at the surface. Most likely this bias is caused by early occurrence of loss of lock when signal amplitudes fall below the closed-loop detection limit.
 5. While the retrieved Doppler profiles differ significantly within multipath regions between different receiver models there is quite close agreement in the amplitude data. Thus, in selecting tracking algorithms emphasis should be put on recovering the phase information; tracking of signal amplitude is less critical.
 6. We identify two main reasons for the failure of the simulation receiver to track the signal correctly within multipath zones. First, the carrier PLL is unable to follow strong phase fluctuations in particular if two-quadrant phase extraction is used. Second, occurrences of low signal amplitudes within zones of large amplitude variations may push the signal below the closed-loop's tracking detection limit increasing the probability of tracking errors.
 In our study the atmosphere is assumed to be free from ducting layers, i.e., the vertical refractivity gradient dN/dz exceeds about −160 km−1 at all altitude levels. If, on the other hand, dN/dz ≲ −160 km−1 the corresponding trapping layers cause a negative refractivity bias [Sokolovskiy, 2003; C. O. Ao et al., manuscript in preparation, 2003].
 While the influence of the receiver algorithms on amplitude and phase data probably cannot be removed completely, our simulation results suggest that improvements can be obtained by using a four-quadrant phase detector and by increasing the sampling frequency. These improvements come at the price of an increase in data volume and the added complexity of a “data wipeoff” implementation required for removing the 50 Hz navigation message modulation.
 Helpful discussions with P. Hartl, C. Marquardt, T. Meehan, T. Schmidt, and J. Wickert are gratefully acknowledged. We are grateful to two anonymous reviewers whose comments and corrections strengthened and improved the manuscript significantly. We thank R. Weller, Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany, for high-resolution rawinsonde data. This study was carried out within the HGF project “GPS Atmospheric Sounding” (grant FKZ 01SF9922/2) lead by C. Reigber.