## 1. Introduction

[2] The radio occultation technique of sounding Earth's atmosphere is based on measuring signals from the satellites of Global Positioning System (GPS) [*Melbourne et al.*, 1994; *Ware et al.*, 1996; *Kursinski et al.*, 2000]. During a radio occultation experiment, a low–Earth orbiter (LEO) implemented with a receiver of GPS signals moves in such a way that the ray linking it to a GPS satellite immerses into the atmosphere. Amplitude and phase of the signal as functions of time are measured. The Doppler frequency shift of the signal is a function of the ray directions and satellite positions and velocities. Given precise orbit data, the refraction (bending) angle and impact parameter (leveling height with respect to Earth' local curvature center) of the ray can be found [*Vorob'ev and Krasil'nokova*, 1994]. Using Abel integral inversion [*Phinney and Anderson*, 1968; *Fjeldbo et al.*, 1971; *Gorbunov and Sokolovskiy*, 1993; *Gorbunov et al.*, 1996b], we can compute the vertical profile of temperature. Reconstruction of temperatures at heights up to 40 km with an accuracy not worse than 1 K requires measurements of neutral atmospheric refraction angles for rays with perigee heights up to 100 km.

[3] However, in radio occultations, GPS signals received by LEO propagate through the ionosphere, and the effect of ionospheric refractivity integrated over a long ray path cannot be neglected. Ionospheric and neutral atmospheric components of refraction angle become comparable for rays with a perigee height of about 45 km, and above 50 km the ionospheric component dominates. For the reconstruction of neutral refractivity, ionospheric component of refraction angle profile must be removed. Separation of the ionospheric and neutral atmospheric component is referred to as the ionospheric correction (calibration).

[4] All methods of the ionospheric correction use the frequency dependence of refractivity. GPS has two channels (L1-1.57542 GHz, L2-1.22763 GHz). In this band, ionospheric refractivity is proportional to the inverse square of the frequency, whereas the frequency dependence of neutrospheric refractivity is negligible. Radio occultation data measured in the two channels are used to solve for neutral atmospheric and ionospheric components of refraction angles. Although two unknown variables must be found from two measured quantities, the problem is nontrivial, and it has no general solution. The reason is that measured L1 and L2 signals as well as neutral atmospheric and ionospheric refraction angles are complicated nonlinear functionals of the unknown 3D neutral atmospheric and ionospheric refractivity fields. Neutral atmospheric and ionospheric refraction angles, generally, cannot be expressed as (nonlinear) functionals of L1 and L2 phases. Thus the problem can only be solved under some approximations.

[5] The simplest approach to the ionospheric correction is based on the linear approximation. The main term of the ionospheric effect in observables such as phase excess or refraction angle is a linear functional of the ionospheric refractivity, which is inversely proportional to the square of the frequency. This allows for writing the ionospheric-free linear combination of L1 and L2 signals. Discussed in the literature were the following: (1) the linear combination of the phase paths at the same sample time [*Spilker*, 1980], (2) the linear combination of the phase modulations at the common ray perigee height [*Melbourne et al.*, 1994], applied when using the Fresnel inversion technique[*Mortensen and Høeg*, 1998], (3) the linear combination of the Doppler shifts at the same sample time [*Ladreiter and Kirchengast*, 1996], which is equivalent to the linear combination of phase paths, and (4) the linear combination of the refraction angles at the common impact parameter [*Vorob'ev and Krasil'nikova*, 1994]. For example, the linear combination of refraction angles takes the following form:

where ε_{1,2} are L1 and L2 refraction angles, *p* is impact parameter, *f*_{1,2} are L1 and L2 frequencies, ε_{LC} is the ionospheric-free linear combination. It must be noticed that this linear combination does not require spherical symmetry of the refractivity field.

[6] Refraction angles obtained after linear ionospheric correction cannot be immediately used for Abelian inversion. Profiles of ε_{LC}(*p*) computed for GPS/MET or CHAMP radio occultation data in height region 70–100 km look like a homogeneous high-frequency noise. They differ significantly from neutral atmospheric refraction angles, which are approximately proportional to refractivity and in this height range must decrease exponentially from 10^{−6} to 10^{−8} rad. If we compute Doppler shift by differentiating phase after smoothing with a sliding window of 0.3 km, the characteristic magnitude of ε_{LC}(*p*) varies from 10^{−6} to 10^{−5} rad. These refraction angles are the sum of ionospheric residuals and receiver noise [*Vorob'ev and Kan*, 1999]. Their high variability from occultation to occultation suggests that ionospheric noise prevails, because the ionosphere is very volatile. The ionospheric residuals result from small-scale ionospheric turbulence, whose effects in L1 and L2 channels are approximately uncorrelated. Linear combination (LinComb) amplifies uncorrelated L1/L2 noise by a ratio of *f*_{1,2}^{2}/(*f*_{1}^{2} – *f*_{2}^{2}) ≈ 2.5, 1.5.

[7] Improvements of the linear ionospheric calibration by taking into account higher-order terms were also suggested [*Gu and Brunner*, 1990; *Brunner and Gu*, 1991; *Hardy et al.*, 1994; *Syndergaard*, 2000]. They are based on some estimations of the nonlinear terms, which are only valid for large-scale ionospheric inhomogeneities. These correction schemes cannot effectively remove small-scale perturbations, whose effect completely overwhelms any possible improvement of the linear correction. However, these algorithms can be useful for the investigations of large-scale ionospheric inhomogeneities.

[8] The refraction angle for a ray with a perigee height of 35 km is about 10^{−4} rad. Comparison of GPS/MET data with MSIS climatological model [*Hedin*, 1991] allows for estimating natural variations of refractivity as 5–20%, which means that variations of refraction angles are about 5 × 10^{–6}−2 × 10^{−5}. This indicates that above 35–40 km the errors due to the imperfectly calibrated ionosphere become very significant. Above 50 km they always exceed natural variations of refraction angles, and at these heights, measurements do not give any new information about the neutral atmosphere.

[9] In the work of *Gorbunov and Sokolovskiy* [1993] the refraction angle profile ε_{LC}(*p*) was replaced with the background profile ε_{BG}(*p*), which is computed for a climatological model, above the height at which observational errors exceed natural variations of refraction angles. This combined profile was then inverted. The use of background estimate of refraction angles instead of measurements at big heights, where measurements are too noisy, is referred to as the initialization of inversions.

[10] A better approach to noise reduction is based on the application of the statistical optimization introduced by *Turchin and Nozik* [1969]. Application of the statistical optimization to ε_{LC}(*p*) neglecting sample-to-sample correlations results in the following estimation for the refraction angle [*Sokolovskiy and Hunt*, 1996; *Gorbunov et al.*, 1996a]:

where σ^{S} is the covariance of neutral atmospheric signal ε_{LC}(*p*) − ε_{BG}(*p*), and σ^{N} is the covariance of the residual errors of the ionospheric correction estimated from ε_{LC}(*p*) – ε_{BG}(*p*) at big heights. The covariance of the signal is typically estimated as the square of a fixed fraction of the background profile: σ^{S} = γ^{2}ε_{BG}^{2}(*p*), where γ varies from 0.05 to 0.2 in different algorithms. This approach is widely used [*Rocken et al.*, 1997; *Gorbunov and Gurvich*, 1998; *Steiner et al.*, 1999; *Hajj et al.*, 2002]. *Rieder and Kirchengast* [2001] analyzed the propagation of measurement errors into retrievals.

[11] *Hocke* [1997] suggested a modification of this method, where the signal and error covariances were replaced with the absolute values of signal and error estimates: σ^{S} = γε_{BG}(*p*), σ^{N} = |ε_{LC}(*p*) − ε_{BG}(*p*)|. At big heights, where σ^{N} ≫ σ^{S} this gives:

and a noise with a constant relative magnitude γ is added to the background profile. In the standard statistical optimization, σ^{S} and σ^{N} are square functionals of ε_{BG}(*p*) and ε_{LC}(*p*) – ε_{BG}(*p*), and at big heights noise is suppressed.

[12] GPS system uses Anti-Spoofing (A/S), i.e. pseudo-random encoding of L2 data. For processing data with A/S turned on, a strong smoothing for the L2 data before the linear combination was used [*Steiner et al.*, 1999; *Hajj et al.*, 2002]. We only process data acquired during so-called Prime Time 4 (February 2–16, 1997) period, i.e. one of short periods of time, when A/S was turned off. Due to that, we will not use this approach.

[13] More general approach to the statistical optimization was suggested by *Healy* [2001]. Using signal and noise covariance matrices and the estimation of the background refraction angles it is solved for the signal. The difficulty of this approach consists in obtaining reliable signal and noise covariance matrices. In the work of *Healy* [2001], noise with a fixed magnitude of 5 × 10^{−6} is assumed, and the signal (background error) magnitude is estimated as 0.2ε_{BG}(*p*). Signal covariances are assumed to have a correlation distance of 6 km above 30 km and to be uncorrelated below this height. Correlation distance practically represents the smoothing scale. It must, however, be noticed that the ionosphere is very volatile, and it is a mixture of regular structures like sporadic E-layers and small-scale turbulence. The ionospheric noise can vary significantly from occultation to occultation. Its magnitude also depends on the filter used for the differentiation of the phase excess. This is also the case for the signal magnitude. In addition, smoothing with a scale of 6 km may be too strong.

[14] In this paper we describe a simple combined algorithm of ionospheric correction and noise reduction based on statistical optimization. Our algorithm uses dynamic estimates of the background refraction angle profile, as well as signal and noise covariance. The background profile is computed from MSIS model and multiplied by a fitting coefficient. The ionospheric noise covariance is estimated using residuals of the linear combination at heights 70–100 km. The signal covariance is estimated from the difference of measurements and background refraction angles at heights 12–35 km. Using statistical optimization we write the optimal estimations of both neutral atmospheric and ionospheric components of refraction angles from L1 and L2 data (optimal linear combination). Particularly, for zero noise this optimal solution will transfer to the standard linear combination. This algorithm does not contain parameters like γ, which are often assigned in more or less arbitrary way, and it also reduces the dependence of the quality of the climatological model. It also allows for the estimation of error covariances for both neutral atmospheric and ionospheric refraction angles.