Radio Science

Application of window functions for full spectrum inversion of cross-link radio occultation data

Authors


Abstract

[1] The full spectrum inversion (FSI) technique allows for effective retrieval of profiles of bending angle and optical depth. These profiles are directly derived from a Fourier transform of a measured radio occulation (RO) signal. Though the entire signal is used in the FSI Fourier transform, only some fraction of the signal contributes significantly to each pair of bending angle and optical depth, whereas noise and disturbances throughout the signal contribute to the errors for all pairs of bending angle and optical depth. The impact from noise and other disturbances may be reduced if window functions are applied in the computation of individual Fourier components. In this study, it is demonstrated how window functions can be applied to the FSI technique. In the approach described here, the window functions are applied in the frequency domain, and it is demonstrated that this technique can be applied in an iterative way to further reduce the impact of noise and disturbances. To assess the merits of using window functions, we apply the windowed FSI to simulated cross-link signals. The results from the simulations show that application of window functions in the FSI technique results in some noise reduction for both white noise and a spike in the signal, whereas iterative use of the window functions was found to significantly reduce the errors from these noise sources, as compared to standard FSI retrieval. It was also found that in terms of relative errors, retrieved derivatives of optical depth are far more sensitive to signal disturbances than retrieved bending angles.

1. Introduction

[2] Temperature, pressure, and humidity profiles of the Earth's atmosphere can be derived through the radio occultation technique. This technique is based on the Doppler shift imposed by the Earth atmosphere on a signal emitted from a GPS satellite and received by a low Earth orbiting (LEO) satellite [see, e.g., Rocken et al., 1997; Kursinski et al., 1997, 2000]. The method is very accurate, with a temperature accuracy of one degree Kelvin, when both frequencies in the GPS system are used.

[3] However, when using GPS frequencies, retrieval of atmospheric water vapor requires a priori information of the atmospheric state to separate contributions from dry air and water vapor to the refractivity. In order to avoid use of external data, more sophisticated satellite systems have been proposed where the atmosphere is sounded through additional LEO-LEO cross links [Kursinski et al., 2002; Herman et al., 2002; Eriksson et al., 2003; Kirchengast and Høeg, 2004; P. Høeg and G. Kirchengast, ACE+: Atmosphere and Climate Explorer, proposal to ESA in response to the second call for proposals for Earth Explorer Opportunity Missions, 2002, hereinafter referred to as Høeg and Kirchengast, proposal, 2002]. The carrier frequencies used in these LEO-LEO links must be located around a water vapor absorption line to enable detection of both phase and amplitude modulations caused by the atmosphere. For instance, the ESA ACE+ Earth Explorer Opportunity mission proposes to use additional LEO-LEO links with three frequencies close to the 23 GHz water vapor line, namely, 9.7, 17.25, and 22.6 GHz [Kirchengast and Høeg, 2004; Høeg and Kirchengast, proposal, 2002].

[4] In order to derive independent profiles of atmospheric water vapor from radio occultations at frequencies around a water vapor absorption line, it is essential, for each frequency, to correctly retrieve the complex refractive index [Lohmann et al., 2003a; Nielsen et al., 2003; Benzon et al., 2004; Kirchengast et al., 2004a, 2004b; S. S. Leroy, Active microwave limb-sounding for tropospheric water vapor: Some theoretical considerations, unpublished manuscript, 2001, hereinafter referred to as Leroy, unpublished manuscript, 2001]. Under the assumption of spherical symmetry, a complex refractivity profile can be computed, through Abelian transforms, from a profile of bending angle and a profile of optical depth measured as functions of ray impact parameter.

[5] Alternatively, geophysical parameters can also be derived from observations of the variations in the absorption coefficient with frequency by using pairs of frequencies known as calibration frequencies [Kursinski et al., 2002; Nielsen et al., 2003; Facheris et al., 2004]. In this differential method, geophysical parameters are derived from a profile of real refractivity and profiles of variations in imaginary refractivity with frequency. These variations are computed by an Abel transform, assuming spherical symmetry, of a profile representing the differences between two absorption profiles computed at two different frequencies.

[6] Gorbunov and Kirchengast [2004, 2005a, 2005b] introduced a new modification of the differential method for the retrieval of the optical depth by combining the differential method with the canonical transform (CT)/full spectrum inversion (FSI) retrieval techniques. They simulated the retrieval of optical depth in a realistic horizontally inhomogeneous, turbulent atmosphere. Their simulations also included a realistic model of receiver noise.

[7] In this study we focus on retrieval of imaginary refractivity profiles; that is, we do not consider calibration frequencies, though it should be noted that the techniques described here are equally applicable for observations with calibration frequencies. For a given frequency, a profile of transmission as a function of ray impact parameter can be derived using either geometrical optics or radio holographic methods. Leroy (unpublished manuscript, 2001) describes how transmission profiles can be computed using geometrical optics, whereas [Gorbunov, 2002a] suggests the use of the canonical amplitude in order to establish the transmission profile. The major advantage of radio holographic methods as compared to geometrical optics is that the former methods can disentangle multipath. So far, a number of high-resolution methods have been introduced in the literature for processing of radio occultation signals in multipath regions: back propagation [Gorbunov et al., 1996; Hinson et al., 1997, 1998; Gorbunov and Gurvich, 1998]; radio optics [Lindal et al., 1987; Pavelyev, 1998; Hocke et al., 1999; Sokolovskiy, 2001; Gorbunov, 2001]; Fresnel diffraction theory [Marouf et al., 1986; Mortensen and Høeg, 1998; Meincke, 1999]; canonical transform (CT) [Gorbunov, 2002a, 2002b]; full spectrum inversion (FSI) [Jensen et al., 2003; Gorbunov et al., 2004]; canonical transform sliding spectral (CTSS) [Beyerle et al., 2004]; phase matching [Jensen et al., 2004]; and CT2 [Gorbunov and Lauritsen, 2004].

[8] In terms of computational simplicity, resolution, and ability to handle multipath, the most efficient radio holographic methods are currently the Fourier integral operator (FIO) and phase-matching-based methods: canonical transforms CT2 and FSI. In these techniques, the entire RO signal, or the part of the signal which transverses the lower part of the atmosphere, is used to compute each set of impact parameter, bending angle, and optical depth. However, the major contribution to each of these sets comes from a smaller segment of the signal located around some stationary phase point. This means that noise and disturbances in the signal located outside that region contribute to the errors in the corresponding bending angle and optical depth, though, for a given impact parameter, this part of the signal only contributes negligibly to the true values. For that reason, it seems intuitively clear that the sensitivity to noise and disturbances of FIO-based methods can be reduced if window functions are applied in the processing so that different segments of the signal are used for each impact parameter.

[9] The first attempt to use window functions when transforming RO signals was proposed by Marouf et al. [1986] in a study of Saturn's rings by Voyager 1 radio occultations. In that study it was found that a significant reduction in sidelobe level can be achieved if a window function is applied in the inverse Fresnel transform of the observed complex signal. Lohmann et al. [2003b] used window functions of varying lengths when applying FSI to invert simulated cross-link radio occultation signals to reduce the impact of background noise. Beyerle et al. [2004] introduced a heuristic method combining the canonical transform and the radio optic techniques. The aim of that method is to reduce negative refractivity bias introduced by tracking errors using a square window with constant length. To reduce the impact from noise, Gorbunov and Lauritsen [2004] introduced radio holographic filtering in the frequency (impact parameter) domain by first multiplying the transformed wave field with a reference signal (to compress the spectrum) and then convolving it with a window function. These processing steps are very similar to signal processing in a digital phase-locked loop [see, e.g., Stephens and Thomas, 1995]. Filtering of the transformed field is a particularly advantageous, as compared to separate filtering of the phase and amplitude, field for low signal-to-noise ratios (SNRs). For low SNR, the nonlinear relations between a complex field and its phase and amplitude result in larger errors, relative to the SNR, as the relation between phase/amplitude errors and SNR becomes nonlinear. For that reason, it is desirable to apply noise reduction before the phase and log amplitude variations are computed from the transformed field.

[10] The method presented in this study is also based on filtering in the frequency (impact parameter) domain after compressing of the spectrum, but here a more sophisticated algorithm is introduced where variable window lengths are used adaptively. The advantage of using adaptive variable window lengths is that the filter bandwidth of the noise reduction filter depends on the bandwidths of the transformed field. This means that for impact parameters where the bending angle profile is smooth, more smoothing is applied than for impact parameters where the bending angle profile contains many small-scale structures and where the application of strong smoothing would otherwise distort these structures. As the window length must be computed iteratively, the advantage of using a variable window length comes at the expense of reduced computational efficiency.

[11] Though this study focuses on the application of the FSI technique, it should be noted that the method described here can be applied also to other FIO-based techniques. The study is organized in the following structure: Section 2 describes the principles of deriving complex refractivity. In section 3 we briefly review the FSI method and discuss advantages and disadvantages of this technique in relation to absorption measurements. In section 4 the windowed FSI technique is described, and section 5 discusses how this technique can be applied iteratively. Results from numerical simulations are presented in section 6, where we apply the FSI, the windowed FSI (WFSI), and iterative use of WFSI (IWFSI) to simulated signals with both absorption and multipath.

2. Retrieval of Complex Refractivity

[12] Radio waves traversing the Earth's atmosphere from a LEO or a GPS satellite to a receiving LEO satellite are subject to refraction, absorption, and diffraction effects. For frequencies in the L, X, and K band, the importance of these phenomena depends upon the complex refractive index field, which is a function of temperature, pressure, water vapor, liquid water, frequency, and the electromagnetic field [Liebe, 1989]. When diffraction effects are small compared to the effects of focusing/defocusing (i.e., refraction) and absorption, the measured signal can be interpreted using geometrical optics (GO). This is generally the case for Earth atmospheric RO observations at GPS or higher frequencies, excluding regions with strong focusing where diffraction effects may become significant [Kursinski et al., 1997, 2000; Gorbunov and Gurvich, 1998].

[13] When the GO approximation (small wavelength) is valid, the ray trajectories through the atmosphere are determined by the real part of the complex refractive index field, nr, whereas the imaginary part of the refractive index, ni, determines the absorption along these ray paths. For a spherically symmetric atmosphere, where the GO approximation is valid, the total ray bending angle, α, and optical depth, τ, can readily be computed using the following Abelian transforms [Kursinski et al., 2002; Leroy, unpublished manuscript, 2001]; see also definition sketch in Figure 1.

equation image
equation image

where rt is the geometrical radius of the tangent point and ARX and ATX are the ray amplitude at the receiver and transmitter, respectively. The impact parameter, a, is related to rt through a = nr(rt)rt. The Abelian inverses of (1) and (2) are [Kursinski et al., 2002; Leroy, unpublished manuscript, 2001]

equation image

and

equation image

where x is the so-called refractional radius defined as x = nr(r)r.

Figure 1.

Radio occultation geometry. Radio waves propagate from a transmitter, TX, to a receiver, RX, through the Earth's atmosphere.

[14] From (3) and (4) it follows that both the bending angle profile and the real refractivity profile must be retrieved before the imaginary refractive index profile can be computed. The physics behind this is that the bending angle and the real refractive index are related to the ray paths, whereas the imaginary refractive index is related to the absorption along these ray paths and, obviously, the ray path must be known in order to invert the ray amplitude profile.

[15] It is also worth noting that the measured ray amplitude, ARX, may be scaled with any arbitrary constant without affecting the results of the integral in (4). The reason for this is that the optical depth is related to the relative variations in the signal intensity and not to the absolute variations. Consequently, there will be no need for calibration of the measured amplitudes, as long-term drifts in the instrument will not affect the observations. This makes the cross-link technique very suitable for climate monitoring.

[16] From the discussion above it follows that for a spherically symmetric atmosphere, a complex refractive index profile can be derived from a profile of bending angles and a profile of ray amplitudes. However, the amplitude of a measured RO signal will not only be subject to modulations caused by absorption but will also be affected by amplitude modulations caused by defocusing and spreading of the rays [Kursinski et al., 1997, 2002; Sokolovskiy, 2000; Jensen et al., 2003, 2004; Gorbunov and Lauritsen, 2004; Leroy, unpublished manuscript, 2001]. Consequently, the measured amplitudes must be corrected for amplitude modulations caused by spreading and focusing/defocusing before the ray amplitude profile can be computed. For a given frequency, computation of the complex refractive index profile follows five different processing steps:

[17] 1. Computation of the bending angle profile as a function of ray impact parameter.

[18] 2. Correction for unwanted amplitude modulations (i.e., focusing/defocusing and spreading).

[19] 3. Computation of the absorption along each ray trajectory, which is equivalent to computing the ray amplitude as a function of impact parameter.

[20] 4. Inversion of the bending angle profile into a real refractivity profile.

[21] 5. Inversion of the ray amplitude profile into an imaginary refractivity profile.

[22] The specific implementation of the different steps depends on the retrieval technique used to compute the bending angle and the ray amplitude profiles. The reader should consult Nielsen et al. [2003] for an overview of how the most common retrieval techniques can be used to retrieve complex refractive index profiles.

[23] In the following, it will be described how the FSI technique can be used to compute profiles of bending angles and optical depths from a LEO-LEO link. As will be clear from the next sections, the FSI technique has a number of advantages, which it shares with the CT2 technique [Gorbunov and Lauritsen, 2004], that make these techniques very suitable for LEO-LEO measurements: (1) it works in multipath zones, (2) defocusing is automatically accounted for, (3) it has high vertical resolution, and (4) computationally expensive back propagation to a straight line is not required.

3. Application of FSI in a LEO-LEO Link

[24] When the FSI technique is applied to radio occultation signals, both bending angles and signal attenuations are automatically assigned to the corresponding impact parameters. This property makes the FSI technique particularly suitable for inversion of cross-link measurements, also when calibration frequencies are used. Generally, the Fourier amplitude represents the distribution of energy in a given signal as a function of frequency. For circular orbits, the impact parameter is proportional to the instantaneous frequency. Consequently, the FSI amplitude describes the distribution of energy in the measured wave field with respect to impact parameter and is, in that sense, identical to the CT and CT2 amplitudes [see, e.g., Gorbunov et al., 2004; Gorbunov and Lauritsen, 2004].

[25] When using FSI to invert radio occultation data, a distinction between ideal occultations and realistic occultations must be made. Here ideal occultations are defined as occultations with perfect circular orbits, whereas realistic occultations are defined as occultations with approximately circular orbits. In the former case, a global Fourier transform can be applied directly to the measured signal, and pairs of opening angle, θ0 (see Figure 1), and ray Doppler angular frequency, ω0, are related through [Jensen et al., 2003]

equation image

where F(ω) represents the Fourier transform of the measured signal resampled with respect to θ. Subsequently, the corresponding bending angle profile is readily computed from the geometry of the occultations by simultaneously solving the following set of equations (for details, see, e.g., Jensen et al. [2003]):

equation image
equation image
equation image

where

ω

derivative of the signal phase with respect to θ;

rTX

distance from the center of curvature to the transmitter satellite;

rRX

distance from the center of curvature to the receiver satellite;

ϕTX

angles between the ray path and the satellite radius vector at the transmitter satellite;

ϕRX

angles between the ray path and the satellite radius vector at the receiver satellite.

[26] Equation (6) describes the Doppler shifts in the transmitter frequency measured at the receiver produced by atmospheric refraction and by the projection of the spacecraft velocity onto the ray path, whereas (7) and (8) are geometrical relations (see Figure 1). The amplitude of the Fourier transformed wave field is given by

equation image

where D is the damping of the ray amplitude for a ray with instantaneous frequency, ω, caused by absorption; ATX is the amplitude of the transmitted signal; and S represents amplitude modulations caused by spreading of the signal. S may be expressed as [Jensen et al., 2003]

equation image

[27] In (10), the first square bracket accounts for amplitude modulations caused by spreading within the occultation plane, whereas the last term accounts for spreading in the transverse direction. From (9), the ray amplitude versus impact parameter profile is readily computed using ephemeris data and the direct relation between ray Doppler frequency and impact parameter.

[28] The amplitude of the received signal, ARX, is related to the amplitude of the transmitted signal and can be expressed as [Eshleman et al., 1980; Sokolovskiy, 2000; Jensen et al., 2003, 2004; Gorbunov and Lauritsen, 2004]

equation image

where the so-called defocusing term, ∣dθ/da∣, accounts for amplitude variations caused by defocusing. This expression is similar to the expression for the Fourier amplitude, as (11) is proportional to (9) except for the defocusing term.

[29] In realistic occultations, unwanted frequency variations are introduced in the RO signal because of variations in satellite radii. These frequency variations must be removed prior to the application of a Fourier transform [Jensen et al., 2003]. This can be done by removing frequency variations caused by radial variations in the radius vectors as described by Jensen et al. [2003] or by applying CT2. Alternatively, the measured signal can be directly propagated from the real orbit to some nearby artificial ideal circular orbit [Jensen et al., 2003] using either wave optic propagation or geometrical optic propagation [Sokolovskiy, 2002]. For realistic orbits, it may also be necessary to account for variations in the satellite radii when correcting for signal spreading, as described by Gorbunov and Lauritsen [2004].

[30] The noncircular correction techniques described above all work well outside multipath regions; on the other hand, if the signal experiences strong multipath, some errors may be introduced in the FSI amplitude. However, in regions with strong multipath, a more significant error source in the FSI amplitude than the propagation from real to close circular orbit is the diffraction in the atmosphere, not accounted for by the FSI (and by other radio-holographic (RH) methods). These errors are especially large when the refractivity gradient is close to critical and the caustics are located deep inside the atmosphere. According to Sokolovskiy [2003], strong multipath may result in errors of the RH-transformed amplitude close to 100%. Also, the nonspherically symmetric irregularities of refractivity (turbulence), regardless of whether they cause multipath or not, may cause significant fluctuations of the RH amplitude [Sokolovskiy, 2003; Gorbunov and Kirchengast, 2005a]. Additionally, in case of strong multipath, insufficiently low truncation of the RO signal may cause both bending angle and RH amplitude errors above the cutoff impact height [Sokolovskiy, 2002, 2003]. To suppress the errors caused by turbulence, Gorbunov and Kirchengast [2005a] proposed a differential method which shall suppress errors caused by corrections for noncircular orbits as well. Hence, under realistic conditions, errors caused by correction for noncircular orbits are not considered to be the dominant error source. As RO observations from real orbits can be transformed into a form where the equations for ideal orbits can be applied, we have restricted this study to ideal orbits only where the FSI technique can be implemented directly as a Fourier transform.

4. Windowed FSI

[31] As discussed earlier, only a segment of a RO signal contributes significantly to a given set of impact parameter, bending angle, and optical depth (a0, α0, τ0) when these parameters are computed using FSI. This segment is different for each impact parameter and is centered on the point in the time series, t0, when the ray with impact parameter a0 arrived at the receiver. The extent of each segment depends on how fast the Doppler frequency varies around t0; t0 is a stationary phase point in the Fourier integral for the frequency component, ω0, corresponding to impact parameter a0. For slowly varying Doppler frequencies, these segments are long compared to situations with rapid variations in the Doppler frequency.

[32] To reduce the impact of noise and disturbances located away from the stationary phase points, a window function can be applied in the computation of each Fourier component as suggested by Lohmann et al. [2003b]. In this case we can write the windowed Fourier transform as

equation image

where w is an arbitrary window function, s is the measured RO signal, tc is the time around which w is centered, and Δt is the duration of w.

[33] In order to apply (12), Δt and tc must be determined for each frequency component/ray. Clearly, tc should be equal to or close to the ray arrival time, t0. The choice of window length is more complicated; if the window is chosen to be too short, significant contributions to the Fourier integral may be excluded, which might result in significant distortion, whereas a too long window will include more noise and will increase the likelihood that a local disturbance in the signal, for example, a tracking error, could corrupt the given Fourier component. The problem of choosing the right window length is further complicated in caustic regions where the signal has a complicated structure and where diffraction is not negligible [see, e.g., Gorbunov and Gurvich, 1998]. For that reason, it is convenient to express (12) in the frequency domain where there is no interference caused by multipath [Jensen et al., 2003] and where diffraction due to big propagation distances is reduced [Gorbunov et al., 2004]. As multiplication in the time domain corresponds to convolution in the frequency domain, (12) can be rewritten as a double integral. By doing so, interchanging the integration order, and employing t0 = tc = −ϕ′(ω), this integral can be expressed as

equation image

where W is the Fourier transform of the window function, ϕ is the phase of the Fourier-transformed RO signal, that is, ϕ(ω) = arg{F(ω)}, and the apostrophe means differentiation of Fourier phase with respect to angular frequency, ω.

[34] As a long window in the frequency domain corresponds to a short window in the time domain, the frequency window length, Δω, should be as long as possible to reduce the impact of noise yet short enough to avoid distortion of the original Fourier transform. Expanding the phase term in (13) into power series yields

equation image

This shows that the integrand in (13) has a parabolic phase variation if the phase of W(ω) is constant, which is the case for any symmetric time window function, and when the third-order or higher phase derivatives are small compared to the second derivative of ϕ. Assuming that this is the case and that the amplitude variations are small within the duration of the window, then if a boxcar window function

equation image

with length Δω is applied in the frequency domain, the integral in (13) can be approximated as

equation image

The last integral in (16) describes the so-called Cornu spiral [see, e.g., Born and Wolf, 1999]. By expanding (16) into power series, the windowed Fourier transform can be approximated as

equation image

This expression reveals that the following criterion must be fulfilled to ensure that the application of window functions only result in negligible distortion of the Fourier transform:

equation image

Here ϕ2 is basically a tunable parameter and must be chosen as a trade-off between noise reduction and distortion caused by the use of a window function as both noise damping and distortion increase with increasing ϕ2.

equation image

where the plus signs apply to integration from 0 to Δω/2 in (16), and the minus signs apply to integration from 0 to −Δω/2.

[36] This is a more convenient criterion, as it requires computation only of the first derivative of the Fourier phase, which reduces the sensitivity to noise. Additionally, this criterion can easily be evaluated simultaneously with the computation of the windowed Fourier transform, as given by the first integral in (16), to determine when to stop the integration.

[37] The last term in (19) represents the difference between the tangent to the Fourier phase curve at the center, ωc, of the window function and the Fourier phase curve. This principle is illustrated in Figure 2.

Figure 2.

Principle of finding the maximum allowable window length, Δω, from the Fourier phase when evaluating the windowed Fourier transform. Integration starts at ωc and continues until the difference between the tangent and phase curve exceeds a certain threshold, ϕ2.

[38] A drawback of the frequency boxcar window function is that the corresponding time domain window is a sinc function, which has indefinite duration [Proakis and Manolakis, 1996]. For that reason, a windowed Fourier transform based on a frequency boxcar window will be more sensitive to local disturbances in the time domain than a windowed Fourier transform based on a window with a final duration in the time domain. However, it should be noted that for the boxcar frequency window, the main contribution to each frequency component will arise from a time interval of length 2π/Δω.

[39] For other window functions than the boxcar, it is straightforward to derive expressions analogous to (18) and (19) in order to determine the maximum allowable window length in the frequency domain. However, only the boxcar window allows for determining Δω simultaneously with the evaluation of (13) in a simple manner. In the following, only the frequency boxcar window will be applied because of the simplification and computational efficiency which can be achieved by using this window function.

[40] For a frequency boxcar window function, the procedure for applying a windowed Fourier transform to a radio occultation signal can be summarized as follows:

[41] 1. Apply a standard FSI to compute the Fourier transform of the RO signal.

[42] 2. Compute the Fourier phase derivative for each frequency component.

[43] 3. Perform integration for each frequency component using the first integral in (16), starting at the center of the frequency window and using (19) to decide when to stop the integrations.

[44] This procedure is similar to the processing in a digital phase-locked loop where a complex signal is down converted to (close to) zero mean frequency by use of a phase model derived from the signal itself and is integrated over some fixed time interval [see, e.g. Stephens and Thomas, 1995]. However, the technique presented in this study does not use fixed intervals.

[45] For real signals, the higher-order phase derivatives may not always be negligible, and the Fourier phase may not vary symmetrically around the stationary phase points. In this case, the length of the integration interval on one side of ωc could be different from the length of the integration interval on the opposite side of ωc. Alternatively, a specific criterion can be introduced to ensure that the integrations are symmetric around ωc, which means that ϕ2 could be different for the two integration intervals. The authors have tested both approaches, but no notable differences were found. Thus we prefer to use a symmetric integration interval, as it centers the integration around ωc.

[46] Finally, we can establish the following relations between wave number and the extent of the window functions in time, frequency, and impact parameter domain (Δt, Δf, Δa):

equation image

where it has been exploited that

equation image

[47] The first two relations above can be derived from (6) and (18), respectively. Equation (20) shows that the lengths of the window functions in both time and impact parameter domain are inversely proportional to the square root of the wave number. That is, the impact of using a windowed Fourier transform will increase with increasing wave number. This makes WFSI particularly suitable for cross-link radio occultations operating at X and K band, or higher, frequencies.

[48] From (21) it follows that the required window length is inversely proportional to the square root of the second derivative of the Fourier phase. If the received signal is affected by caustics (strong focusing), there will be frequencies for which the second derivative is zero; on the other hand, if the received signal is affected by superrefraction (strong defocusing), there will be frequencies for which the second derivative of the phase will be infinite. Radio occultation measurements of the lower moist troposphere are frequently affected by strong focusing and defocusing. Consequently, the required window length may vary significantly between different frequencies/rays for the same occultation and particularly between different occultations.

5. Iterative Use of the Windowed Fourier Transform

[49] For noisy data, there is a tendency for the procedure described above to underestimate the window lengths, as fluctuations in the Fourier phase caused by the noise may result in larger differences between the Fourier phase curve and its tangent. For that reason it is desirable to repeat the procedure described above, applying the windowed Fourier transform in an iterative way; in the following, we will refer to this approach as IWFSI. Iterations do not only improve the noise filtering by improving the estimates of the required window lengths; through each iteration, small structures in the phase are suppressed. By removing these structures, longer windows can be applied, which allows for stronger noise filtering.

[50] By iteratively applying window functions, additional distortion is introduced after each iteration. However, as long as the higher-order phase derivatives are small, the phase distortion will be approximately the same for all frequencies and will therefore only result in minor errors in the phase derivatives and thereby in the retrieved bending angle and refractivity profiles. Similarly, the relative distortion in the Fourier amplitudes will also be approximately identical for all frequencies and for that reason only leads to minor errors in the retrieved derivatives of optical depths and imaginary refractivities; see (4). However, when the receiving satellite passes through a caustic ϕ″ = 0 for the affected frequencies/rays, the higher-order phase derivatives become dominant, which may increase the distortion introduced through each iteration. This is particularly a problem for the Fourier phase derivative when ϕ″ changes sign at such a point since this will also change the sign of the phase distortion as follows from (17). In this case, additional smoothing of the Fourier phase and hence of the bending angle profile will be introduced around the points where ϕ″ = 0. The impact on the Fourier amplitude will be smaller, as the relative distortion is approximately the same on both sides of these points. We will elaborate more on this in section 6 when interpreting the results from our simulations.

6. Simulations

[51] In this section, the WFSI and the IWFSI techniques are applied to simulated LEO-LEO radio occultation signals, and their performance is compared to the performance of the traditional FSI. Three different scenarios are considered: (1) no noise, (2) white noise, and (3) signal spike. The first scenario corresponds to ideal conditions, and the purpose here is to investigate what distortions are introduced when applying WFSI and IWFSI. In the second scenario we assess how much additional noise damping can be achieved by using WFSI and IWFSI. The purpose of the third scenario is to investigate to what extent a local error in the time domain is spread out in the impact parameter domain when applying FSI, WFSI, and IWFSI. For all three scenarios we compute the differences between the true profiles and the retrieved profiles of bending angles and derivatives of optical depth. We have chosen to present errors in the retrieved derivatives of optical depth rather than errors in optical depth, as it is the derivative of optical depth that is inverted to imaginary refractivity (see (4)).

[52] The simulated satellite system consists of two counterrotating satellites orbiting the Earth in the same plane with the transmitter and receiver at heights of 850 and 650 km, respectively. The transmitter frequency was set to 10 GHz.

[53] The simulated signal was generated by using asymptotic direct modeling [Gorbunov, 2003; Gorbunov and Lauritsen, 2004] in the form of an inverse FSI and sampled at 1092 Hz. For circular orbits, the inverse FSI can easily be derived from the equations presented in section 3.

[54] Before inversion, the total signal phase was reconstructed in order to restore the “original” signal. This was done by frequency shifting the signal to base band, followed by up-sampling of the signal through interpolation of amplitude and phase to a sampling rate of 5074 Hz to satisfy the sampling theorem.

[55] The simulated RO signal was based on an atmosphere defined by a real refractivity profile, Nr(h) = 106(nr(h) − 1), and a imaginary refractivity, Ni = 106ni (h), expressed as

equation image

and

equation image

which are exponential profiles with Gaussian-shaped bumps and correspond to an atmosphere with multipath propagation and strong absorption at 5 km height. In the simulations, these profiles are transformed into profiles of bending angles and optical depth, using (1) and (2), which are used as input to the inverse FSI to generate the simulated RO test signal.

[56] In the inversion we apply moderate smoothing of both phase and log amplitude of the Fourier-transformed signal corresponding to a smoothing window of 25 m in the impact parameter domain in agreement with the diffraction-limited resolution for a 10 GHz signal [Gorbunov et al., 2004]. Obviously, the results could be compared without any smoothing. However, we prefer to compare the results smoothed to a resolution comparable to the diffraction limit rather than comparing the results at a somehow arbitrary resolution related to the specific retrieval scheme. Furthermore, by applying a reasonable amount of smoothing in the retrievals, the performance, in terms of noise reduction, of the IWFSI technique can be compared to simple smoothing of phase and log amplitude.

[57] Smoothing is applied to the log amplitude rather than to the amplitude, as the log amplitude is directly related to the optical depth, which can be inverted into a profile of imaginary refractivity. The smoothing of the log amplitude was implemented as a moving average, whereas smoothing of the phase was performed through differentiation of the phase by applying the vector phase change scheme [Sirmans and Bumgarner, 1975]:

equation image

where δω is the spectral resolution and the over bar represents complex conjugate. The advantage of this approach is that cycle ambiguities are automatically accounted for, and unwrapping of the phase is therefore not needed. The ϕ2 stop criterion used to decide when to stop the integration in (16) was set to 0.04π on the basis of experience with inversion of simulated RO signals.

6.1. Scenario 1: No Noise

[58] The amplitude variations of the signal derived directly from (22) and (23), corresponding to ideal conditions without any background noise or disturbances, are depicted in Figure 3. Figures 4 and 5 show the errors in the retrieved bending angles and derivatives of optical depth, dτ/da, respectively, for retrievals based on FSI, WFSI, and IWFSI. For the IWFSI, 0, 1, 2, 5, and 10 iterations were considered, where zero and one iteration correspond to applying FSI and WFSI, in that order.

Figure 3.

Scenario 1 (no noise): amplitude variations of simulated LEO-LEO signal.

Figure 4.

Bending angle retrieval, scenario 1 (no noise): performance of FSI (0 iterations), WFSI (1 iteration), and IWFSI (2, 5, and 10 iterations). The first plot from the left shows the true bending angle profile, while the other plots depict the corresponding differences between retrieved bending angles and true bending angles as a function of impact height.

Figure 5.

Retrieval of derivatives of optical depth, scenario 1 (no noise): performance of FSI (0 iterations), WFSI (1 iteration), and IWFSI (2, 5, and 10 iterations). The first plot from the left shows the true profile of derivatives of optical depth, while the other plots depict the corresponding differences between the retrieved profiles and the true profile.

[59] In this ideal case all the retrievals result in very small bending angle errors. The most notable difference between the retrieved bending angle profiles and the true profile occur at the peak in the bending angle profile at an impact height of 6.75 km. At this height, the errors are between 750 and 900 μrad, with the smallest errors for the FSI retrieval and the largest errors for the IWFSI with 10 iterations. The main contribution to these errors comes from the smoothing applied in the processing. As discussed in the previous section, the IWFSI introduces additional smoothing of the bending angles at heights where ϕ″ = 0, resulting in slightly larger errors at those heights. Similarly, ϕ″ is also zero at an impact height of 5.8 km, which also leads to additional smoothing around this height. The effect of the additional smoothing at heights where ϕ″ = 0 increases after each iteration. However, even after 10 iterations, the maximum relative errors caused by additional smoothing are less than 0.5%. Additionally, it is worth noting that on both sides of the heights where ϕ″ = 0, hardly any distortion is introduced, even after 10 iterations, though the bending angle profile has large gradients in these regions.

[60] For the errors in dτ/da, we see the same trend as for the bending angle errors with one significant difference: In agreement with the theoretical findings in the previous section, no notable additional smoothing is introduced at the heights where ϕ″ = 0 for the IWFSI retrievals. For the FSI retrieval (0 iterations), the effect of ringing (also known as Gibb's phenomenon) results in small fluctuations in the dτ/da profile, which disappear after a single iteration because of the application of window functions.

6.2. Scenario 2: White Noise

[61] In this scenario we add white Gaussian noise to our test signal and repeat the retrievals. The noise power was chosen so that the signal-to-noise density above the atmosphere was 60 dBHz, which is 7 dB less than the 67 dBHz noise density specified for the ACE+ mission.

[62] Figure 6 shows the corresponding signal amplitude, and Figures 7 and 8 show profiles of errors in bending angles and optical depth derivatives, respectively. Again we notice that the bending angle distortion at the heights where ϕ″ = 0, which results in additional smoothing, increases for each iteration, but more important, the noise level is significantly reduced by iteratively applying the window functions. After 5 iterations, most of the noise is removed, and the additional improvements achieved by applying 10 iterations are minor and hardly compensate for the added distortion. However, it should be noted that even with zero iterations, the relative bending angle errors caused by noise is relatively small even for this very noisy signal.

Figure 6.

Scenario 2 (white noise): amplitude variations of simulated LEO-LEO signal.

Figure 7.

Bending angle retrieval, scenario 2 (white noise): performance of FSI (0 iterations), WFSI (1 iteration), and IWFSI (2, 5, and 10 iterations). The first plot from the left shows the true bending angle profile, while the other plots depict the corresponding differences between retrieved bending angles and true bending angles as a function of impact height.

Figure 8.

Retrieval of derivatives of optical depth, scenario 2 (white noise): performance of FSI (0 iterations), WFSI (1 iteration), and IWFSI (2, 5, and 10 iterations). The first plot from the left shows the true profile of derivatives of optical depth, while the other plots depict the corresponding differences between the retrieved profiles and the true profile.

[63] This is not the case for the optical depth derivatives for which the relative errors are very large for the FSI retrieval. On the other hand, the IWFSI greatly reduces the impact from noise; outside the multipath region, the magnitude of the noise fluctuations is reduced by approximately an order of magnitude after 5 iterations, with some further improvements after 10 iterations. The noise reduction in the multipath region, around the bump, are smaller because of the larger phase gradient in that region, which leads to shorter windows and thus less noise damping. Nevertheless, after 5 and 10 iterations, the magnitude of the noise fluctuations is damped by more than a factor of 2 in that region. The higher noise level observed below 6.8 km is a result of the strong absorbing layer at that height, which reduces the SNR for rays traversing that layer.

6.3. Scenario 3: Signal Spike

[64] To test the impact of local errors in the time domain, a spike is now added to the test signal. The spike is introduced by multiplying the signal amplitude by −10 at 30 s. In this way, a significant spike in amplitude and sudden phase shift of 180° is introduced right in the middle of the multipath zone, thus affecting three rays with impact heights of 4.3, 6.1, and 6.3 km.

[65] Figures 9 and 10 depict the error profiles for the retrieved bending angles and derivatives of optical depths. As for the white noise scenario, only minor errors are introduced in the bending angle profile by introducing an amplitude and a phase spike. Generally, the errors from the spike are spread out in the impact parameter domain for the FSI retrieval, as expected, with almost constant magnitude. The resulting errors are smaller than the error caused by smoothing, even at the impact heights of the rays affected by the spike. It is also found that by applying window functions, the errors are greatly reduced even after a single iteration, corresponding to WFSI retrieval, and with further improvements through additional iterations. After 5 and 10 iterations, the impact from the spike is nearly completely removed except for some minor errors at the impact heights of the rays affected by the spike.

Figure 9.

Bending angle retrieval, scenario 3 (signal spike): performance of FSI (0 iterations), WFSI (1 iteration), and IWFSI (2, 5, and 10 iterations). The first plot from the left shows the true bending angle profile, while the other plots depict the corresponding differences between retrieved bending angles and true bending angles as a function of impact height.

Figure 10.

Retrieval of derivatives of optical depth, scenario 3 (signal spike): performance of FSI (0 iterations), WFSI (1 iteration), and IWFSI (2, 5, and 10 iterations). The first plot from the left shows the true profile of derivatives of optical depth, while the other plots depict the corresponding differences between the retrieved profiles and the true profile.

[66] The impact on the retrieved derivatives of optical depths is more significant, but as for the retrieved bending angles, the application of a windowed Fourier transform significantly reduces the magnitude of the errors. Additional noise reduction is achieved by applying iterations, though most improvements are achieved during the first two iterations. In contrast to the bending angle retrieval, some notable errors, around the impact heights of the rays affected by the spike, still remain after 10 iterations. Furthermore, the errors in the derivative of optical depth are more concentrated around these heights.

7. Discussion

[67] In section 6 all retrievals were based on a fixed number of iterations and a fixed value of ϕ2, and no attempt was made to estimate the optimal number of iterations or the optimal value of ϕ2, though for all three scenarios, five iterations seem to be a reasonable choice. However, it was also found that distortion of the bending angle profile was introduced through each iteration at heights where ϕ″ = 0. Therefore it may be desirable to use more iterations when retrieving profiles of derivatives of optical depth than when retrieving bending angle profiles. This would also agree with our findings that in terms of relative errors, retrieval of derivatives of optical depth is far more sensitive to noise and disturbances than retrieval of bending angles.

[68] The authors have also experimented with different values of ϕ2; values between 0.03π and 0.16π have all been found to work well and to give results similar to the results presented in this study. The effect of increasing ϕ2 is similar to the effect of increasing the window length. For ϕ2 ≲ 0.03π, the noise reduction effect is greatly reduced, whereas for ϕ2 ≳ 0.16π, the additional smoothing at the heights where ϕ″ = 0 becomes significant.

[69] The optimal number of iterations and the optimal ϕ2 value depend on the atmosphere and the SNR; for an atmosphere with multiple small-scale irregularities it would be desirable to use fewer iterations and a small value of ϕ2 than for a smooth atmosphere, whereas a low SNR favors the use of more iterations and a large ϕ2. Generally, the optimal number of iterations and the optimal ϕ2 value must be chosen as a trade-off between noise damping, computational efficiency, and distortion.

8. Summary and Conclusions

[70] In order to derive independent profiles of atmospheric water vapor and temperature from cross-link radio occultations, it is essential to correctly retrieve the complex refractive index. Under the assumption of spherical symmetry, a complex refractivity profile can be computed, through Abelian transforms, from a profile of bending angles and a profile of derivatives optical depth measured as functions of ray impact parameter.

[71] The FSI technique allows for effective retrieval of profiles of bending angles and optical depths as functions of impact parameter from radio occultation measurements. These profiles are directly derived from a Fourier transform of the measured RO signal. Though the entire signal is used in the Fourier transform, only some fraction of the signal contributes significantly to each pair of bending angle and optical depth, whereas noise and disturbances throughout the entire signal contribute to the errors for all pairs of bending angle and optical depth. Accordingly, the impact from noise and other disturbances may be reduced if window functions are applied in the processing so that different segments of the signal are used for computing different Fourier components. The challenge when using window functions is to choose the window lengths; if the time window is chosen to be too short, significant contributions to the Fourier integral may be excluded, which may result in distortion of the Fourier transform, whereas a too long window will include more noise and will increase the likelihood that a local disturbance in the signal, for example, a tracking error, could corrupt the given Fourier component. The problem of choosing the right window length is further complicated in caustic regions where RO signals have a complicated structure and where diffraction is not negligible. For these reasons, it is more convenient to apply the window functions in the frequency domain, where diffraction effects are greatly reduced and multipath interference is eliminated.

[72] This study presents a method where both the application of the window functions and the estimation of the individual window lengths are performed in the frequency domain. We refer to this approach as the windowed FSI (WFSI). In this method, rectangular window functions are applied in the frequency domain, which is equivalent to applying sinc function windows to the Fourier transform in the time domain. Estimation of the window lengths is based on the phase variations caused by the second derivative of the phase of the Fourier-transformed RO signal, which is assumed to be constant within the duration of each window function. This procedure is similar to the processing in a digital phase-locked loop where a complex signal is down converted to (close to) zero mean frequency and is integrated over some fixed time interval.

[73] As the window lengths depend on the phase of the transformed signal, the lengths of the windows are affected by noise and disturbances. We therefore suggest applying WFSI iteratively to further reduce the impact of these error sources; we refer to this approach as iterative WFSI (IWFSI).

[74] The IWFSI technique includes two tunable parameters: a stop criterion used to determine the window lengths and the number of iterations. Generally, the optimal stop criterion and the optimal number of iterations must be chosen as a trade-off between noise damping, computational efficiency, and distortion caused by the windows.

[75] To assess the performance of FSI, WFSI, and IWFSI, these techniques were applied to simulated LEO-LEO radio occultation signals with both multipath and strong absorption. Three different scenarios were considered: (1) no noise, (2) white noise, and (3) signal with amplitude and phase spike. For the IWFSI, 2, 5, and 10 iterations were considered. For each retrieval, profiles of bending angle errors and errors in the derivative of optical depth were computed.

[76] The results from the simulations showed that application of window functions in the FSI technique results in some noise reduction for both global disturbances (white noise) and local disturbances (signal with a amplitude spike). Iterative use of the window functions was found to further reduce the errors through each iteration, and after five iterations, a significant reduction in the errors was found as compared to the FSI retrieval. It was also found that in terms of relative errors, both local errors and global errors have a far more significant impact on the retrieved derivatives of optical depth than on the retrieved bending angles.

[77] Although the simulations were based on a transmitter frequency of 10 GHz, we expect similar results for GPS-LEO occultations; however, the impact of using WFSI and IWFSI will be smaller for GPS signals because of the longer wavelength, which results in longer windows in the time domain.

[78] In this study, window functions were applied only for the FSI technique; nevertheless, it is expected that the findings and methods in this study can also be applied to other retrieval techniques based on Fourier integral operators.

Acknowledgments

[79] The authors are grateful to Michael Gorbunov and Sergey Sokolovskiy for valuable suggestions to this study. This work has been supported by ESA/ESTEC contract 16743/02/NL/FF and by the National Environmental Satellite, Data and Information Service (NESDIS).

Ancillary