## 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2004b; S. S. Leroy, Active microwave limb-sounding for tropospheric water vapor: Some theoretical considerations, unpublished manuscript, 2001, hereinafter referred to as Leroy, unpublished manuscript, 2001].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.