For inversions of the GPS radio occultation (RO) data in the neutral atmosphere, this study investigates an optimal transition height for replacing the standard ionospheric correction using the linear combination of the L1 and L2 bending angles with the correction of the L1 bending angle by the L1–L2 bending angle extrapolated from above. The optimal transition height depends on the RO mission (i.e., the receiver and firmware) and is different between rising and setting occultations and between L2P and L2C GPS signals. This height is within the range of approximately 10–20 km. One fixed transition height, which can be used for the processing of currently available GPS RO data, can be set to 20 km. Analysis of the L1CA and the L2C bending angles shows that in some occultations the errors of standard ionospheric correction substantially increase around the strong inversion layers (such as the top of the boundary layer). This error increase is modeled and explained by the horizontal inhomogeneity of the ionosphere.

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

For the uncorrelated, random errors

Figure 1 shows

Examples of COSMIC L1CA (black), L2C (red), and ionosphere-free (green) bending angles 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,

At heights where errors of

Left panel: examples of electron density profiles modeled by Chapman layer. Blue lines: F2 layer at 300 km with vertical scale 50 km (dashed line) and 75 km (solid line). Green lines: E layer at 100 km with vertical scale 5 km (dashed line) and 7.5 km (solid line). Right panel: L1–L2 bending angles obtained by ray tracing for the profiles in left panel (blue and green lines); approximations by bending angles calculated for refractivities modeled by delta functions at 300 and 100 km (red lines). Numbers (%) show fractional rms deviations (averaged below 20 km) between L1–L2 bending angles and their approximations.

Examples of COSMIC L1–L2 bending angle profiles (thin lines) and
fitting functions (thick lines).

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

Latitudinal distributions of the dynamic L2 drop heights for COSMIC and Metop 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

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.

Mean differences (solid) and standard deviations (dashed) of
retrieved COSMIC L2P bending angles (left) and refractivities (right) for
different extrapolation (transition) heights (colors indicated in panel

Similar to Fig. 5, except for the COSMIC L2P rising

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 (

Similar to Fig. 5, except for the Metop rising

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

Similar to Fig. 5, except for the COSMIC L2C setting occultations,
for all latitudes. Panels

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.

The ionospheric correction Eq. (

A layout of ray tracing is shown in Fig. A1. A transmitter (GPS) and
receiver (in low Earth orbit: LEO) are in co-planar circular orbits. This simplifies
calculations and is sufficient to model and explain the errors of the
ionospheric correction. The Earth radius is

Layout of ray tracing.

We start each ray from

We calculate a set of rays with the increment

In our modeling of refractivity we use the following function (continuous
with derivatives):

Atmospheric refractivity

The ionospheric model is decomposed into radial and horizontal factors as
follows:

First, we calculate

Next we model the effect of horizontal inhomogeneity of electron density in
the ionosphere. Figure A4 (lower panel) shows a set of traced rays, in the
coordinates

Bending angle profiles obtained from modeling by ray tracing. Black: bending angle obtained with the atmosphere only. Red and green: L1 and L2 bending angles obtained with the atmosphere and ionosphere. Blue: ionospheric-free bending angles.

Lower panel: set of ray trajectories between GPS and LEO spanning the troposphere. Upper panel: functions modeling horizontal inhomogeneity of electron density in the ionosphere.

To further justify the statements made at the beginning of this section,
Fig. A5 shows

The

Alternatively, the structure of the ionospheric effects on

The effect of the horizontal inhomogeneity of the ionosphere on the accuracy of the ionospheric correction was also investigated in VK94. By applying ray tracing, it was concluded that the error is negligible. The difference with our conclusions can be explained by two reasons. First, VK94 used the model of horizontal ionospheric inhomogeneity which is similar to our case E in Fig. A4 for which we also did not find errors. Second, and most important, is that VK94 applied the ionospheric correction for the true bending angle as a function of the height of the ray tangent point. According to VK94, this observable can be decomposed into the atmospheric and ionospheric terms, similarly to Eq. (A1), even when the ionospheric refractivity is non-spherically symmetric. We verified the results of VK94 with ray tracing and confirmed that for this observable the errors of the ionospheric correction are negligible not only in case E but also in cases C and D. But this observable is not realistic because it cannot be derived from RO Doppler in a general case of non-spherically symmetric refractivity. For the realistic RO observable given by Eqs. (A4, A5), the errors of the ionospheric correction can be significant, as demonstrated in this section.

This work is supported by the National Science Foundation under Cooperative Agreement no. AGS-1033112. The authors acknowledge the contributions to this work from members of the COSMIC team at UCAR. The authors are grateful to Stig Syndergaard (DMI) for useful discussions and to the anonymous referees for reading the paper and providing valuable comments. Edited by: M. Nicolls