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.
Figure 2. Temperature (solid line) and humidity (dashed line) are plotted in the left panel, and the vertical refractivity gradient is shown in the right panel. The profiles are derived from a tropical radiosonde sounding.
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 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.
Figure 3. Schematic representation of the multiple phase screen method. The atmosphere is represented by a series of phase screens. A plane electromagnetic wave propagates through the screens from left to right and is detected on observation screen O.
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 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.
Figure 4. Power spectral density of a C/A-code modulated L1 signal (PRN 17). The carrier frequency of 1.57542 GHz is marked by a dashed line. Note that the spectrum is carrier-free.
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 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
|Parameter||Carrier PLL||Code PLL|
|Bandwidth Bn||20 Hz||1 Hz|
|Gain G||2π 200||50|
|Damping ratio ζ||1/||1/|
 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.)
Figure 5. Amplitude (top panel) and Doppler frequency profile (bottom panel) at the carrier frequency of 1.57542 GHz obtained by MPS calculations. The profiles are obtained from the refractivity field plotted in Figure 2. The observation screen ranges analyzed in Figure 9 (single path propagation between −14 and −16 km, multipath propagation between −24 and −26 km) are marked in grey.
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 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.
Figure 6. C/A-code correlation as a function of code chip delay calculated for two observation screen ranges. The grey area marks the 1-sigma standard deviation. The correlation for single path propagation (screen range from −14 to −16 km) is shown in the top panel, the corresponding result for multipath propagation (range from −24 to −26 km) is shown below.
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 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.
Figure 7. Relation between Δϕ, the differences between retrieved and true phase, and the retrieved signal amplitude. Data are taken from the screen range between −35 km and −25 km.
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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.
Figure 8. Ray manifold in three-dimensional (x, y, p) space. x and y span the occultation plane in geometrical space, and p and x form the impact parameter space with p denoting the impact parameter. In this example, the receiver encounters a multipath region between t1 and t2: within this region rays probing the atmosphere on different paths superimpose and interfere with each other. The individual contributions can be separated by transforming the measured signal from geometrical space to impact parameter space (projection onto (p, x)-plane). For details, see text. (Figure is not to scale.)
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 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.)
Figure 9. CT amplitudes for two spatially filtered signals showing the contributions from rays observed between −24 and −26 km on the observation screen (solid line) and between −14 and −16 km (dashed line). The −24 to −26 km range lies within a multipath region where rays arriving from of about 6.5 and 4.5 km ray height interfere. For comparison the CT amplitude for the nonfiltered signal is plotted as well (dash-dotted line).
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