One of the most significant error sources in radio occultation soundings are residual errors of the ionospheric correction originating from small scale ionospheric turbulence and receiver noise. These errors degrade the quality of retrievals above 35–40 km. We describe a combined algorithm of the ionospheric correction and noise reduction including the following important details: (1) fitting of climatological model, (2) dynamical estimation of signal and noise covariances using radio occultation signals above 50 km, and (3) statistically optimized retrieval of neutral atmospheric and ionospheric refraction angles from L1 and L2 measurements using background refraction angles and signal and noise covariances (optimal linear combination). This method allows for the estimation of the error covariances of the retrieved neutral atmospheric and ionospheric refraction angles. We show some examples of GPS/MET data processing and perform a statistical comparison with the retrievals obtained by UCAR and DMI/IGAM algorithms. Our algorithm was previously used in the statistical validation of GPS/MET data on the basis of ECMWF analyses.
 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.
 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).
 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.
 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 ε1,2 are L1 and L2 refraction angles, p is impact parameter, f1,2 are L1 and L2 frequencies, εLC is the ionospheric-free linear combination. It must be noticed that this linear combination does not require spherical symmetry of the refractivity field.
 Refraction angles obtained after linear ionospheric correction cannot be immediately used for Abelian inversion. Profiles of εLC(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 εLC(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 f1,22/(f12 – f22) ≈ 2.5, 1.5.
 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.
 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.
 In the work of Gorbunov and Sokolovskiy  the refraction angle profile εLC(p) was replaced with the background profile εBG(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.
where σS is the covariance of neutral atmospheric signal εLC(p) − εBG(p), and σN is the covariance of the residual errors of the ionospheric correction estimated from εLC(p) – εBG(p) at big heights. The covariance of the signal is typically estimated as the square of a fixed fraction of the background profile: σS = γ2εBG2(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  analyzed the propagation of measurement errors into retrievals.
Hocke  suggested a modification of this method, where the signal and error covariances were replaced with the absolute values of signal and error estimates: σS = γεBG(p), σN = |εLC(p) − εBG(p)|. At big heights, where σN ≫ σS this gives:
and a noise with a constant relative magnitude γ is added to the background profile. In the standard statistical optimization, σS and σN are square functionals of εBG(p) and εLC(p) – εBG(p), and at big heights noise is suppressed.
 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.
 More general approach to the statistical optimization was suggested by Healy . 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 , noise with a fixed magnitude of 5 × 10−6 is assumed, and the signal (background error) magnitude is estimated as 0.2εBG(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.
 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.
2. Ionospheric Correction Algorithm
 The basic a priori information about the neutral atmosphere can be provided by a climatological model. We use the MSISE-90 climatological model [Hedin, 1991]. For this model and given observation geometry we compute the refraction angle profile εMSIS(p). Because GPS/MET refraction angle profiles often indicate systematic deviations from MSIS profiles, we multiple this profile with the fitting coefficient, α, which is calculated from regression of refraction angles εLC(p) with respect to εMSIS(p) in the height interval 40–60 km. In order to stabilize the estimate of the fitting coefficient, εLC(p)(p) was computed using a heavy smoothing with a sliding window of 3 km. Further we work with the deviations of L1 and L2 refraction angles from the fitted model refraction angle profile:
We assume the following model of the measurement data:
where ΔεN(p) is the deviation of the neutral atmospheric refraction angles from fitted model profile, εI(p) is the ionospheric refraction angle profile for L1 frequency channel, and ξ1,2(p) are stationary uncorrelated L1 and L2 noises. As shown in [Vorob'ev and Kan, 1999], covariances of L1 and L2 fluctuations conform the dispersion relation 〈ξ12〉 f14 = 〈ξ22〉 f24, which follows from the theory of weak fluctuations [Tatarskii, 1961]. ΔεN(p) is the neutral atmospheric useful signal.
 We also compute large-scale component of ionospheric refraction angle profile, for which linear ionospheric correction is expected to work well. Applying a strong smoothing with a sliding window width of 2 km to Δε1,2(p), we calculate their low-frequency components and the low-frequency component of the ionospheric refraction for the L1 channel:
 Now we can apply the statistical optimization solution for the useful signal consisting of two components: neutral atmospheric ΔεN and ionospheric . For this purpose we need the estimates of the signal and noise covariances.
 Above 50 km, where the ionospheric signal dominates, the covariance of ionospheric high-frequency noise in L1 channel that cannot be corrected for by the linear combination is estimated as follows:
where a is the Earth's local curvature radius, and p-a is approximately ray perigee height.
 At big heights, where neutral atmospheric refraction angles are negligible, the L1 ionospheric refraction angles can be estimated as follows:
This is optimal estimation, which suppresses L1 and L2 uncorrelated noise. The covariance of the ionospheric signal is then estimated as follows:
where we subtracted the residual of uncorrelated L1 and L2 noises.
 The covariance of the neutral signal relative to the model can be estimated using the height interval from 12 to 35 km, where the neutral signal dominates and the atmosphere is dry. We exclude troposphere, where strong variations of humidity may result in overestimation of the signal covariance.
 Now we can write a system of equations for the signal components, ΔεN and ΔεI:
where the matrix of the system is defined as follows:
This matrix allows for expressing the statistically optimized solution (optimal linear combination) as follows:
The diagonal elements of the matrix (KTCN−1K + CS−1)−1 are equal to error covariances of the components of the solution vector, εN(p) and εI(p).
 This solution cannot be used in the lowest 2–3 km, where L2 signal becomes too noisy. This area is usually very sharply marked [Steiner et al., 1999], and its upper limit pL can be easily estimated using a simple heuristic criterion, such as pL = max[p:Δε2I > 100σIS] + 1km. For this area we assume εI(p) = εI(pL) and εN(p) = ε1(p) − εI(pL).
3. Processing GPS/MET Data
 Our algorithm of ionospheric correction and noise reduction was used for the statistical validation of GPS/MET data [Gorbunov and Kornblueh, 2001]. Figures 1 and 2 show examples of processing a GPS/MET radio occultations events. We also perform a comparison with two other algorithms:
We compare with the algorithm designed by University Corporation for Atmospheric Research (UCAR) [Rocken et al., 1997]. Retrievals obtained by this algorithm are published at the GPS Science and Technology Web site (http://www.gst.ucar.edu/).
We compare with the algorithm designed at the Danish Meteorological Institute and Institute for Geopysics, Astrophysics, and Meteorology. This is an enhanced algoirthm using global search for MSIS initialization profile and inverse covariance weighting for the statistical optimization [Gobiet and Kirchengast, 2002; Healy, 2001]. It is implemented in End-to-end GNSS Occultation Performance Simulator (EGOPS) software [Kirchengast et al., 2002].
Figures 1a and 2a show L1 and L2 refraction angles and their linear combination. For the computation of refraction angles we differentiated the L1 and L2 phase excesses using smoothing with a sliding window of 0.3 km. These plots illustrate the influence of small scale L1 and L2 noises resulting in big residual errors in the linear combination. The small scale perturbations in L1 and L2 channels do not conform the ionospheric dispersion relation. This can be attributed to the following two factors: (1) the signal contains some noise uncorrelated in L1 and L2 channels; (2) small-scale ionospheric perturbations result in nonlinear effects. This results in the amplification of the noise in the linear combination. Very often the ionospheric perturbations in L1 and L2 channels have amplitude ratio very close to f12/f22, but they are shifted with respect to each other, which also results in a significant residual error. This shift depends on the location of ionospheric inhomogeneities, and differs singificantly and may have different signs. Linear correction does not work well for ionospheric perturbations with scales less or equal 1 km [Vorov'ev and Kan, 1999].
Figures 1b and 2b show the refraction angle profiles obtained after the application of the optimal linear combination (OLC) algorithm described above. Also shown are error bars. These plots indicate that the small-scale variations of the refraction angle profiles above 40 km are smaller than the estimate of the error of refraction angle retrieval. Relative error is plotted in Figures 1c and 2c. At a height of 50 km, error level corresponds to the relative magnitude of the useful signal, i.e. variations of refraction angles with respect to the climatology. This value is very different for different occultations and it may vary from 3% to 20%.
Figures 1d and 2d show temperatures reconstructed by the three algorithms using the statistical optimization. These profiles are also compared with the reanalysis of European Centre for Medium-Range Weather Forecasts (ECMWF), which were used for statistical validation of GPS/MET data [Gorbunov and Kornblueh, 2001]. Because ECMWF data were given up to a height of about 28 km, we combine them with MSIS climatology. In order to obtain continuous profiles of refractivity and temperature we multiply MSIS data by a fitting coefficient, which is found from the hydrostatic equation in the ECMWF-MSIS transfer area [Gorbunov and Kornblueh, 2001].
 Different retrievals become very close to each other and to ECMWF data below 25 km. At heights 45–60 km the UCAR and DMI profile indicates strong temperature fluctuations. Because in this area radio occultation data are not reliable, this means that the UCAR and DMI statistical optimization algorithms overestimate the signal-to-noise ratio in this area.
Figure 3 shows the results of the statistical intercomparison of the three algorithms for the complete day (February 2, 1997) including about 80 radio occultation events. This plot shows that above 30 km discrepancies between different algorithms are very big due to different initializations. At 30 km the characteristic temperature difference is about 2 K at 30 km, at 20 km it is 1 K, and at 10 km it decreases to 0.3 K. The discrepancies in the lower troposphere are due to different handling of multipath propagation, which we do not discuss here. The systematic difference between OLC and UCAR data is negligible, while DMI initialization results also in a visible systematic deviation from OLC and UCAR algorithms.
 The importance of ionospheric correction and noise reduction algorithms for processing radio occultation data cannot be overestimated. Ionospheric residuals define the upper bound of the usability of radio occultation data. Typically, the residual error of the ionospheric correction above 35–40 km becomes comparable with the natural variations of refraction angles, which means that measurement data do not give much new information about the atmosphere above these heights.
 The linear correction in combination with the statistical optimization is widely used in retrieval algorithms. The statistical optimization is mostly employed in its simplest one-sample variant. The reason is that it is difficult to obtain reliable covariance matrices for ionospheric refraction angles, because the ionosphere is very volatile and it contains a mixture of turbulence and regular structures, such as sporadic layers. On the other hand, above 50 km none ionospheric correction scheme can be expected to give good results. Below 30 km, where the neutral atmospheric signal is strong, it is not important which noise reduction scheme is used. Really important is what climatological data are used for the initialization at big heights, and what estimations of signal and noise covariances are used in the height range 30–50 km. Different algorithms use different initializations and different estimations of signal-to-noise ratio, which often results in significant differences in retrievals.
 We describe our approach to the statistical optimization and ionospheric correction. Critical parts of the algorithm are the following:
We use a fitting coefficient for the standard climatology, which is important, because climatological models such as MSIS or CIRA often indicate significant biases with respect to radio occultation data. Such biases can result in the overestimation of the useful signal magnitude, which degrades the quality of noise reduction at heights 30–50 km.
We use dynamical estimates of signal and noise covariances. Noise is estimated from the discrepancy of dispersion relation of L1 and L2 data in the upper parts of occultations where ionospheric signals prevail. Signal is estimated from the deviation of the refraction angle from climatology in the height interval, where neutral atmospheric signals prevail. In other algorithms, the signal covariance is assumed to be a fixed fraction (from 5% to 20% in different algorithms) of the climatological refraction angles. Sometimes also a fixed estimation for the noise covariance is used. Our solution is, on our view, preferable, because these estimations depend on the state of the ionosphere, which is very volatile. They also depend on filtering used for the differentiation of phase exess.
Using the signal and noise covariances, we solve for neutral atmospheric and ionospheric refraction angles from L1 and L2 refraction angles using 2–D statistical optimization. This generalizes the standard linear correction, where linear combination of L1 and L2 data is first computed, and 1-D statistical optimization is then used to solve for neutral atmospheric data. It must, however, be noticed that the variations of the results due to different initialization and different signal and noise estimation can significantly exceed the difference between 1-D and 2-D statistical optimization. Nevertheless, the consequent use of the statistical optimization is achieved at a very low computational cost (it only requires inversion of 2 × 2 matrices. This approach is also convenient, because it allows for the estimation of the accuracy of the retrieved neutral atmospheric and ionospheric refraction angles.
 This algorithm constitutes an important part of our data processing software, which we used to process GPS/MET data acquired during Prime Time 4. The retrieved temperatures were validated by means of statistical comparison with ECMWF analyses [Gorbunov and Kornblueh, 2001]. This algorithm is also implemented in EGOPS. Our method of the estimation of signal and noise covariances can also be incorporated into other algorithms, such as optimal smoothing [Healy, 2001].
 This work was performed with the financial support of the Institute for Geophysics, Astrophysics, and Meteorology/University of Graz (IGAM/UG, Graz, Austria), Max-Planck Institute for Meteorology (Hamburg, Germany), and Russian Foundation for Basic Research (grant 01-05-64269). The author is grateful to G. Kirchengast (IGAM/UG) and to S. Sokolovskiy (Institute for Atmospheric Physics, Moscow) for useful scientific discussions. The author is also grateful to G. Kirchengast and J. Ramsauer (IGAM/UG) for their help in the computations using EGOPS.