When GPS radio occultation (RO) signals are used for monitoring the neutral atmosphere, the ionospheric effect has to be removed. While the neutral atmospheric effect on the RO signals exponentially decreases with the height, the ionospheric effect on the average slowly increases with the height. Thus, removal of the ionospheric effect (i.e., ionospheric correction) is most important in the stratosphere and above. Nevertheless, neglecting the ionospheric correction in the troposphere results in a small but statistically significant inversion bias. One of the common methods of dual-frequency, model-independent ionospheric correction is a linear combination of the L1 and L2 GPS RO observables and, in particular, the linear combination of the L1 and L2 bending angles taken at the same impact parameter (Vorob'ev and Krasil'nikova, 1994) (hereafter the “standard ionospheric correction”). However, a disadvantage of the standard ionospheric correction is the amplification of uncorrelated noises (errors) on L1 and L2.

When the first GPS RO data became available in 1995 (Ware et al., 1996), it became clear that the standard linear combination of the L1 and L2 bending angles is useless in the troposphere, mainly due to noise and tracking errors on the encrypted L2P signal. In order to approximately remove the mean ionospheric effect, it was proposed that the L1 bending angle be corrected by using the L1–L2 bending angle extrapolated from above the troposphere (Kursinski et al., 1997, 2000; Rocken et al., 1997; Steiner et al., 1999; Hajj et al., 2002). As a result of the open-loop (OL) tracking of the L1 in the troposphere (Sokolovskiy et al., 2009b; Ao et al., 2009), the L2P is unavailable and only the ionospheric correction by extrapolation of the L1–L2 can be applied. With the OL tracking of the un-encrypted L2C (Sokolovskiy et al., 2014), both the L1CA and L2C signals, free of the tracking errors, are available in the troposphere. Theoretically, this may allow an extension of the standard ionospheric correction into the troposphere. However, in practice, as noted by Steiner et al. (1999), not only the noise but also the small-scale refractivity irregularities in the troposphere may introduce additional errors into the standard ionospheric correction. As follows from the analysis of the occultations with the L1CA and L2C available in the OL mode (examples shown below in Sect. 1), the standard ionospheric correction down to the surface is feasible for only smooth RO signals under low-moisture conditions (e.g., high latitudes, local winter). In the presence of moisture, especially in the tropics, fluctuations of the RO signals result in an increase of noise (errors) after the standard ionospheric correction. Also, as shown in this study, the horizontal inhomogeneity of the ionosphere may result in additional errors of the standard ionospheric correction around strong inversion layers. The ionospheric correction by extrapolation of the L1–L2 reduces the errors of these types.

For an optimal application of the ionospheric correction in the troposphere, it is necessary to define the transition height for replacement of the standard linear combination by extrapolation of the L1–L2. It is also necessary to define the extrapolation function and the height interval in which this function is fitted to the L1–L2 and then used to model the L1–L2 below the transition height. In previous studies (Rocken et al., 1997; Steiner et al., 1999; Hajj et al., 2002), the transition height and the fitting interval were not always defined and simple extrapolation functions (constant and linear) were used primarily. In this study, the extrapolation function, which models the effects of the ionospheric F and E layers, is introduced, and a special case of this function is used for processing and statistical analysis of the GPS RO data. The selection of the transition height is based on the noisiness of the ionosphere-free bending angle. A transition too high results in an increase of noise due to the uncorrected, small-scale ionospheric effects. A transition too low results in an increase of noise due to the small-scale tropospheric irregularities and L2P tracking errors. An optimal transition height minimizes the noise of the ionosphere-free bending angle at all heights. As follows from the results of this study, the optimal transition height depends on the latitude and is different for different GPS RO receivers and firmware, for L2P and L2C signals, and for setting and rising occultations.

Section 2 presents examples of the occultations with L1CA and L2C available in the OL mode and with the standard ionospheric correction applied down to the surface. Section 3 introduces a model of the L1–L2 bending angle for extrapolation of the ionospheric correction in the troposphere. Section 4 presents statistical distributions of the dynamic transition heights (as determined individually for each occultation). Section 5 presents the statistical comparison of the GPS RO bending angles to those from the European Centre for Medium-range Weather Forecasting (ECMWF) forecast for different static transition heights. This allows us to determine optimal static transition heights. Section 6 concludes the study.

The effect of uncorrelated L1 and L2 random errors on the ionospheric correction is known. Let α1 and α2 be the L1 and L2 bending angles and β1 and β2 their observational noises (errors), whereas f1=1.57542 GHz and f2=1.2276 GHz are the GPS frequencies. The 1st-order ionosphere-free bending angle is equal to α=c1α1(h)-c2α2(h), where h=a-rc-Δhg is the impact height (a is the impact parameter, rc is the local curvature radius of reference ellipsoid, and Δhg is the geoid undulation), c1=f12/(f12-f22)≅2.5457, and c2=f22/(f12-f22)≅1.5457.

For the uncorrelated, random errors β1 and β2, the root mean square (rms) error of the ionosphere-free bending angle equals <β2>1/2=(c12<β12>+c22<β22>)1/2. If the uncorrelated L1 and L2 random errors have the same rms magnitudes (<β12>=<β22>), then, after the ionospheric correction, the rms error is amplified by a factor of (c12+c22)1/2≅3. If the L2 random error is much larger than the L1 error, then it is feasible to correct α1 by an additionally smoothed α1-α2: α(h)=α1(h)+c2[α1(h)‾-α2(h)‾]. This reduces the effect of L2 errors but increases the errors due to un-corrected small-scale ionospheric effects on L1. The optimal smoothing interval for α1-α2 is determined by minimizing the combined effect of both errors individually for each occultation (Sokolovskiy et al., 2009a). This approach is routinely used in the COSMIC Data Analysis and Archive Center (CDAAC) processing (Schreiner et al., 2011).

Figure 1 shows α1(h), α2(h), and α(h) for four Constellation Observation System for Meteorology, Ionosphere, and Climate (COSMIC) occultations with the L1CA and L2C acquired in the OL mode in the troposphere. Bending angles were calculated from the complex RO signals with the use of a wave optics (WO) transform (phase matching) (Jensen et al., 2004) and then smoothed with a window of 0.1 km. Figure 1a shows a high-latitude wintertime occultation. Both α1(h) and α2(h) are smooth functions due to low humidity resulting in small fluctuations of refractivity. The fluctuation of α(h) is not substantially different from those of α1(h) and α2(h). The standard ionospheric correction performs satisfactorily in the troposphere.

Figure 1b shows a tropical occultation (from the center of the Pacific Ocean). Commonly, most of the tropical occultations are affected by strong random refractivity irregularities caused by moist convection in the troposphere, which broaden the spectra of RO signals both in time and impact height representations (Gorbunov et al., 2006; Sokolovskiy et al., 2010). Correspondingly, α1(h) and α2(h) have strong fluctuations that are partially uncorrelated (Sokolovskiy et al., 2014). After applying the standard ionospheric correction, these fluctuations are amplified as seen in Fig. 1b. Thus, in the moist convective troposphere, the standard ionospheric correction, generally, results in substantial increase of noise. Figure 1c and d show two occultations affected by strong refractivity gradients on top of the boundary layer, resulting in the large bending angle gradients and lapses around h ∼ 2.8 km and h ∼ 3.3 km. In these regions, the errors of the standard ionospheric correction are clearly increased. Such increased errors can be often (but not always) observed around the inversion layers in L2C occultations. They can be viewed as resulting from different shifts of the impact heights at L1 and L2 frequencies. An explanation and modeling of these errors is discussed in Appendix A. Extrapolation of the ionospheric correction (discussed in the next section) reduces the errors such as those below ∼ 7 km in Fig. 1b and below ∼ 3 and ∼ 3.5 km in Fig. 1c and d.

At heights where errors of α2(h) become too large (such as in the troposphere), it makes sense to approximate α1(h)-α2(h) by a smooth function αext(h), which is fitted to α1(h)-α2(h) at some interval (hext,htop) where the errors of α2 are smaller: ∫hexthtop[α1(h)-α2(h)-αext(h)]2dh=min⁡ and, at h<hext, replace Eq. () with α(h)=α1(h)+c2αext(h). The ionospheric correction by extrapolation eliminates the errors of α2 by instead introducing the errors due to uncorrected small-scale ionospheric effects on α1. If the α2 errors are larger than the small-scale ionospheric effects, the ionospheric correction by extrapolation results in the overall reduction of the magnitude of bending angle error. However, such replacement of the errors may also change the vertical error correlation, as noted by Syndergaard et al. (2013). Though detailed investigation of this effect is outside the scope of this study, it is important to note that increase of the error correlation radius increases the error of retrieved refractivity for a given rms bending angle error (see, for example, frequency response of the Abel inversion in Lohmann (2005), Fig. 1). Thus reduction of the rms magnitude of the bending angle error, by itself, does not warrant reduction of the refractivity error. Since, on average, α1-α2 is a smooth function, in previous studies, αext was either a constant or linear function (Kursinski et al., 1997, 2000; Rocken et al., 1997; Steiner et al., 1999; Schreiner et al., 2011) fitted to the observational α1-α2 in some interval above the extrapolation height. However, α1-α2 is also affected by the ionospheric irregularities and L2 tracking errors. To reduce these effects on αext, it makes sense to increase the fitting interval and to apply a more complicated model of the ionospheric effect on bending angle. For this purpose, we use an approximate expression for the bending angle response δα from an infinitely thin refractivity layer at a height z0: δα(h)∼∫h∞dδ(z-z0)/dz(z-h)1/2dz∼δ(z-z0)(z-h)h∞+∫h∞δ(z-z0)(z-h)3/2dz∼(z0-h)-3/2, where δ(z) is a delta function. The ionospheric layer, such as F2, is not thin. Figure 2 shows electron density profiles modeled by Chapman layer (left panel) and α1(h)-α2(h) calculated by ray tracing and approximated by the Eq. () fitted in the interval 20 km < h < 80 km (right panel). Details are provided in the figure caption. It is seen that the model (Eq. ) satisfactorily represents α1(h)-α2(h) for modeling both F2 layer at height 300 km and E layer at height 100 km. The accuracy of the approximation reduces with the increase of the thickness of the layer, which is to be expected. We model the ionospheric bending angle by the function αext(h)=A+B⋅h+C(zE-h)-3/2+D(zF2-h)-3/2, where zE=100 km and zF2=300 km are approximate heights of the E and F2 layers and search for the coefficients A,B,C, and D by least-squares fitting αext(h) to the observational α1(h)-α2(h) in the interval hext<h<80 km. Figure 3a shows α1(h)-α2(h) and αext(h) for six COSMIC occultations for hext=20 km. Figure 3b shows the same but with an excluded fourth term in Eq. (). It is seen from comparison of Fig. 3a and b that keeping the fourth term in αext(h) results in an overfitting. Our statistical analysis (details are omitted) also shows strong cross-correlation between the coefficients B and D. Thus it is sufficient to keep the third term, which models the response from E layer, while the response from F2 layer, as well as the effects of horizontal inhomogeneity of the ionosphere (see Appendix A), are modeled by the first two terms: αext(h)=A+B⋅h+C(zE-h)-3/2. We found that the use of the fitting function (Eq. ), compared to the fitting functions with only the first term (constant) or first two terms (linear fit) (fitted in a shorter interval, 10 km above the transition height), results in slightly smaller standard deviation of RO bending angles from those for the collocated ECMWF analyses. In the following, we use the function (Eq. ) for extrapolation of the ionospheric correction.

In one of the options used in the CDAAC inversion algorithm, the L2 quality is checked from top to bottom based on a simple criterion |Δs1-Δs2| < 6 cm, where Δs1 and Δs2 are raw (unsmoothed) excess phase changes between 50 Hz samples (Kuo et al., 2004). The standard ionospheric correction is replaced by extrapolation below the height (called the L2 drop height, same as the transition height), defined as the maximal height below 40 km where this criterion is not satisfied or the height where L2 becomes unavailable. Occultations with an L2 drop height above 20 km are not processed. The most common reason for a large |Δs1-Δs2| is a strong fluctuation of the RO signal resulting in an L2P tracking instability. The threshold of 6 cm (approximately 1/4 of an L2 wavelength) has no clear physical justification and was found empirically based on the processing of a large amount of RO data, as a tradeoff between the quality (i.e., noisiness) and quantity (i.e., number of processed occultations).

Figure 4 shows the latitudinal distributions of the L2 drop heights for April 2012. Figure 4a and b show the L2P drop heights for the COSMIC setting and rising occultations. The upper part of the “donut-shaped” structure in Fig. 4b is correlated with the height of the tropopause. In many occultations, the sharp structure of the tropopause is sufficient to cause a strong enough fluctuation of the RO signal resulting in a tracking instability of the L2P. For those setting occultations where the L2P tracking remains sufficiently stable at the tropopause, it becomes unstable in the presence of fluctuations caused by the tropospheric moisture, or the L2 becomes unavailable below the transition height from the phase-locked loop (PLL) to the OL mode; this explains the lower part of the donut-shaped structure. For rising COSMIC occultations, the structure of the distribution of the L2P drop heights is, in general, similar to that of the setting occultations, except that the lower part is higher because the OL–PLL transition generally needs extra time for locking on the L2P and this, on average, happens higher than the PLL–OL transition for setting occultations. The L2C drop heights for COSMIC setting occultations (Fig. 4c) are primarily related to the PLL–OL transition; this confirms that the L2C PLL tracking is stable and not susceptible to signal fluctuations (Sokolovskiy et al., 2014). Although there are some differences, the structures of the distributions of the L2P drop heights for the Meteorological Operational satellite program (Metop) occultations prior to the 2013 firmware update (Figs. 4d and e) are, in general, similar to those for COSMIC. For setting occultations (Fig. 4e), fewer occultations are affected by the L2P tracking instability induced by the tropopause, and, due to the dynamic PLL–OL transition, the L2P can, on average, be more stably tracked down to the lower height in the troposphere than for COSMIC (Fig. 4b). For rising occultations (Fig. 4d), the lock on the L2P happens, on average, at lower heights than for COSMIC (Fig. 3a). Overall, the comparison of Fig. 4a and b to d and e suggests less noisy (more stable) L2P tracking for Metop than for COSMIC.

As follows from Fig. 4, the dynamic approach for extrapolation of the ionospheric correction results in a rather large spread of the L2P drop heights in low latitudes as well as systematic differences of the L2 drop heights for different missions, rising and setting occultations, and L2P and L2C signals. Generally, this approach may be optimal for the processing of RO data for general purposes and weather applications. An alternative (static) approach applies extrapolation of the ionospheric correction below a fixed transition height that is predetermined based on statistical evaluation and minimization of the inversion errors. We believe that the static approach, when all occultations are processed in the same way, may be superior for climate applications.

In this section, we investigate the effect of different fixed transition heights for extrapolation of the ionospheric correction on the retrieved ionosphere-free bending angle profiles. For retrieval of the L1 bending angle, the phase matching method (Jensen et al., 2004) is applied up to 20 km or the transition height hext (whichever is higher), and geometric optics is applied above. For retrieval of the L2 bending angle, the geometric optics is applied. To evaluate the retrieved ionosphere-free bending angle profiles, they are statistically compared to the bending angles obtained from the ECMWF global forecasts, by calculating the mean (bias) and standard deviation. A transition too high results in the increase of random errors due to uncorrected, small-scale ionospheric effects, while a transition too low results in the increase of random errors due to noise and tracking errors on the L2. Although the ECMWF model, which is used as the reference, also has errors, it is important that both the errors related to uncorrected ionospheric effects and L2 noise are uncorrelated with the model forecast errors. Consequently, the minimal standard deviation (where all independent errors are summed with squares) can be used as the criterion for finding an optimal transition height. The mean deviation (where all biases are summed with their signs), generally, cannot be used as such a criterion unless the bias becomes too large and can be clearly attributed to RO rather than to the model.

Figures 5 through 8, in left panels, show statistical comparisons of the RO-retrieved ionosphere-free bending angles to the bending angles forward-modeled from the ECMWF forecasts for April 2012. Because extrapolation of the ionospheric correction may change the bending angle error correlation which affects the refractivity error (mentioned in Sect. 3), we also perform statistical comparison of refractivities. These results are shown in right panels of Figs. 5–8. Solid and dashed lines show the mean and standard deviations, respectively. Different colors correspond to different transition heights applied for extrapolation of the ionospheric correction. In each case, an optimal transition height is located between those fixed heights which result in smaller standard deviations at all heights. More accurately, the optimal transition height can be estimated as the height where the difference between these standard deviations for the fixed heights changes the sign.

Figure 5 shows the statistics for the COSMIC L2P occultations with transition heights at 25, 20, 15, and 10 km over three latitude bands. It is apparent that if the transition is too high, such as at 25 km, the standard deviation in some interval (∼ 5 km) below the transition height increases, which indicates the noise from uncorrected small-scale ionospheric structures. If the transition is too low, such as at 15 or 10 km, the standard deviation in some interval above the transition height (∼ 5–10 km) substantially increases. Clearly, involving the noisy L2P in the ionospheric correction significantly deteriorates the statistical results. The optimal transition heights for different latitudes are not very different, ranging from ∼ 17 km at high latitudes to ∼ 19 km at low latitudes. Though the vertical error correlation might change due to the ionospheric correction by extrapolation, the statistical results for refractivity (right panels) affected by error propagation through the Abel transform, are overall consistent with those for bending angle (left panels). Figure 6 shows the statistics for the COSMIC L2P rising and setting occultations. Again, the optimal transition heights are not very different: slightly lower for setting (∼ 17 km) than for rising (∼ 19 km) occultations. This difference may suggest, on average, lower quality of the L2P signal right after lock for rising occultations than after tracking for an extended time for setting occultations. We also examine the statistics for the Metop L2P rising and setting occultations before the 2013 firmware update in Fig. 7. The optimal transition heights are ∼ 15 and ∼ 20 km for setting and rising occultations, respectively. A comparison of Figs. 6 and 7 shows that Metop has a smaller standard deviation (i.e., better L2P quality) than COSMIC, especially for setting occultations. Figure 8a and b show statistics for the COSMIC L2C setting occultations (rising L2C occultations currently are not available). The optimal transition height is about 10 km. Because the L2C is tracked in the OL mode down to the bottom of the occultations, Fig. 8c and d also show the standard deviation in the lower troposphere for the transition height at 10 km and for the standard ionospheric correction without extrapolation. It is apparent that application of the standard ionospheric correction down to the bottom of the occultations makes the ionosphere-free bending angles in the lower troposphere, on average, noisier than the correction by extrapolation below 10 km.

Application of the standard ionospheric correction for the L1 and L2 GPS RO bending angles results in an increase of random errors in the troposphere due to noise and tracking errors (mainly on the L2P signal) and tropospheric irregularities, and is also limited by availability of the L2 signal. Correction of the L1 bending angle by the L1–L2 bending angle extrapolated from above reduces these errors by introducing other errors due to uncorrected small-scale ionospheric effects. The extrapolation height, which minimizes the combination of both errors, is considered optimal and depends mainly on the quality of the L2 signal, which, in turn, depends on the receiver, tracking firmware, signal strength, and structure, and may depend on the tropospheric and ionospheric irregularities.

Results of this study show that the optimal transition height for the COSMIC L2P occultations is about 20 km (the height differs slightly for different latitudes and for rising and setting occultations). For Metop occultations before the 2013 firmware update, the optimal transition height is about 20 km for rising and ∼ 15 km for setting occultations, indicating a less noisy L2P than for COSMIC.

For the COSMIC L2C setting occultations (tracked in the OL mode in the lower troposphere), the optimal transition height is about 10 km. Extension of the standard ionospheric correction without extrapolation down to the bottom of the occultations, generally, increases the noisiness of the ionosphere-free bending angles in the troposphere. In particular, most significant increase is observed (i) in the presence of non-spherically symmetric refractivity irregularities (such as the moist convection) and (ii), sometimes, around strong inversion layers (such as the top of boundary layer). The latter effect is modeled and explained by horizontal inhomogeneity of the ionosphere. The results of the modeling suggest the ionospheric correction of the boundary layer depth determined from RO bending angles (application of this correction is a subject of a separate study). To minimize the noise of the ionosphere-free bending angles, occultations from different missions, rising and setting occultations, and L2P and L2C occultations can all be processed with different extrapolation heights. Alternatively, an optimal extrapolation height can be determined for each occultation individually. This may be better for weather applications. However, when GPS RO data are used for climate applications, different or dynamically determined extrapolation heights may result in different biases, which may propagate into climate signals. Thus, for climate applications, a constant extrapolation height may be superior. Based on the results of this study, for currently available GPS RO data, a transition height of 20 km (when the L2 is available down to that height) may be considered reasonable.