With a GPS receiver on board an airplane, the airborne radio occultation (ARO) technique provides dense lower-tropospheric soundings over target regions. Large variations in water vapor in the troposphere cause strong signal multipath, which could lead to systematic errors in RO retrievals with the geometric optics (GO) method. The spaceborne GPS RO community has successfully developed the full-spectrum inversion (FSI) technique to solve the multipath problem. This paper is the first to adapt the FSI technique to retrieve atmospheric properties (bending and refractivity) from ARO signals, where it is necessary to compensate for the receiver traveling on a non-circular trajectory inside the atmosphere, and its use is demonstrated using an end-to-end simulation system.

The forward-simulated GPS L1 (1575.42 MHz) signal amplitude and phase are
used to test the modified FSI algorithm. The ARO FSI method is capable of
reconstructing the fine vertical structure of the moist lower troposphere in
the presence of severe multipath, which otherwise leads to large retrieval
errors in the GO retrieval. The sensitivity of the modified FSI-retrieved
bending angle and refractivity to errors in signal amplitude and errors in
the measured refractivity at the receiver is presented. Accurate bending
angle retrievals can be obtained from the surface up to

Global Positioning System (GPS) satellites transmit radio signals that
undergo refractive bending and a Doppler shift due to variations in the
refractive index of the Earth's atmosphere. With a GPS receiver on board an
aircraft, the airborne radio occultation (ARO) receiver tracks the occulting GPS signals
traversing progressively lower (or higher) atmospheric layers when the GPS
satellite sets below (or rises above) the local horizon of the receiver
(Healy et al., 2002; Xie et al., 2008; Haase et al., 2014). Contrary to the
case for spaceborne RO receivers, the ARO receiver is located inside the
atmosphere, where the non-negligible atmospheric refractive index near the
receiver must be taken into account. Moreover, the RO signals from above the
local horizon also need to be recorded, or otherwise accounted for, to allow
the retrieval of atmospheric properties below the ARO receiver. Similar to
spaceborne RO, the measurements of the ARO carrier wave phase and amplitude
can be inverted to retrieve the bending angle (the cumulative atmospheric
refractive bending along each ray path) as a function of the impact parameter.
In a spherically symmetric atmosphere, the impact parameter, which is the
product of the radius and the refractive index at the tangent point
(Kursinski et al., 2000), is a conservative quantity for each signal ray.
The tangent point is the point along a ray path where the radius vector from
the center of curvature is normal to the ray, and it is the closest point on
the ray path to the Earth surface. The bending angle can then be converted
to refractivity through the inverse Abel transformation (Fjeldbo et al., 1971). The
refractivity (

Various radio holographic methods have been proposed to overcome the limitations of the GO method in the spaceborne RO retrievals (e.g., Gorbunov et al., 1996; Gorbunov and Gurvich, 1998; Sokolovskiy, 2001; Gorbunov, 2002; Jensen et al., 2003, 2004; Gorbunov and Lauritsen, 2004). The full-spectrum inversion (FSI) proposed by Jensen et al. (2003) (hereafter referred to as J03) has been applied to invert spaceborne RO signals and outperforms the GO method in the presence of multipath. However, prior to this work it had not been implemented for airborne RO because of the need to address the unique characteristics of ARO occultation measurements. In particular, due to the the asymmetry of the ray path from GPS to a receiver inside the atmosphere, ARO retrieval requires additional measurement of RO signals above the local horizon. Moreover, the irregular (non-circular) flight path of the airborne platform must also be taken into account.

In this paper, the development and implementation of the FSI retrieval method for ARO measurements are presented. An end-to-end simulation system is developed to carry out a sensitivity analysis of the modified FSI algorithm, and it is compared with the GO method. The occultation geometry of ARO events used in this simulation study is based on characteristics of the flights in the PRE-Depression Investigation of Cloud-systems in the Tropics (PREDICT) field campaign over the tropical Atlantic during August–September of 2010 (Montgomery et al., 2012; Haase et al., 2014; Murphy et al., 2015). During the PREDICT campaign, the Global Navigation Satellite System (GNSS) Instrument System for Multistatic and Occultation Sensing (GISMOS) was deployed on the Gulfstream-V High-performance Instrumented Airborne Platform for Environmental Research (GV HIAPER) aircraft (Garrison et al., 2007). Haase et al. (2014) and Murphy et al. (2015) presented very promising initial results of the airborne RO observations for the upper troposphere above 7 km based on the GO retrieval. However, significant refractivity retrieval errors in the middle and lower troposphere were detected in the GO retrievals for open-loop tracking data. These retrieval errors in the lower troposphere are likely caused by the multipath problem that plagues the GO method. The FSI retrieval method presented in this paper is expected to solve the multipath problem and offer high-quality lower-tropospheric ARO soundings with high vertical resolution to enable more comprehensive studies near tropical storms of the hurricane genesis process by including near-surface moisture observations. In this study, high-vertical-resolution ERA-Interim reanalysis profiles (60 vertical levels) from the European Centre for Medium Range Weather Forecasts (ECMWF) over the campaign area are used to represent the atmospheric conditions.

This paper is organized as follows: Sect. 2 describes the key implementation steps for the FSI method for ARO. An end-to-end simulation system is presented in Sect. 3. Section 4 presents simulated ARO observations under severe multipath conditions, where the FSI is expected to provide major improvement. The sensitivity of the FSI retrievals to the ARO measurement errors in signal amplitude and the refractivity at the receiver are explored in Sect. 5. The conclusions are summarized in Sect. 6.

The FSI method operates directly on the measured signal along the receiver
trajectory, recognizes the recorded RO signal as radio waves of different
frequencies determined by the refractive index of the media through which
they pass, and accounts for interference of waves with different
frequencies. In contrast, the GO retrieval does not account for the possible
superposition in time of multiple waves. Jensen et al. (2003) demonstrated
that the derivative of the phase of the Fourier transform of the measured
signal as a function of open angle distinguishes the different frequency
components by using the method of stationary phase (MSP). The open angle (

The GPS signal composed of several narrowband subsignals resampled with
respect to

The MSP assumes that at an instantaneous pseudo frequency in the Fourier
integral in Eq. (3) is dominated by the contribution from a single subsignal
near the stationary point (Born and Wolf, 1999). Equation (3) can then be
simplified to Eq. (4):

The last two terms in Eq. (7) are frequency changes caused by the radial
variations due to the non-circular trajectories of the GPS and the receiver,
which will be removed through a correction described in the following
section. When both trajectories are circular, the equation can be simplified to

After identifying the contribution from individual subsignals, these can be
attributed to distinct ray paths, and subsequently the impact parameter (

The open angle can be computed using Eq. (6). The bending angle (

The FSI retrieval equations in the preceding section are derived assuming a circular orbit of the GPS and the receiver relative to the center. When the GPS and the receiver trajectories become non-circular, the radial velocity terms in Eq. (7) are nonzero.

Schematic plot of airborne radio occultation geometry. Projection of
the receiver from its original position

In real airborne occultation measurements, the perfectly circular trajectory assumption is not valid in part because of the oblateness of the Earth, as well as any height variations of the receiver. In addition, the asymmetry of the ray path from the source to a receiver inside the atmosphere requires additional measurement of RO signals above the local horizon and correction for the irregular (non-circular) flight path of the airborne platform. To take into account the oblateness of the Earth, Syndergaard (1998) showed that the inversion of the RO data should be performed assuming local spherical symmetry tangential to the Earth's ellipsoid.

In our current approach, we account for the oblateness of the Earth by
calculating the local center of curvature for each occultation event and
transforming the coordinates of receiver and transmitter to that center of
curvature. After the oblateness correction, a correction is applied to
account for the non-spherical trajectories of the receiver and the
transmitter. In the current algorithm, the correction for non-spherical
trajectories has been performed by projecting the position of both the
receiver and the transmitter at each epoch to a circular trajectory. Figure 1
shows the schematic diagram of the projection of the GPS signal from a
non-circular receiver trajectory onto a fixed radius circular trajectory
inside the atmosphere. Similarly, the method is applied to the GPS orbit to
project its position (in a vacuum) onto a circular orbit. The figure shows
the receiver at position

In practice, the additional phase resulting from the bending from

After the projection at each sample time, the new trajectories of both GPS
and receiver are circular relative to the local center of curvature, and both

End-to-end simulation system for airborne RO data processing. (Note: derivation of bending from refractivity in forward Abel does not need the occultation geometry information but only the receiver height and refractivity at the receiver.)

In the case of airborne RO measurements, the bending angle is not a unique
function of the impact parameter (

An end-to-end simulation system (Fig. 2) was developed to investigate the performance of the modified FSI algorithm for airborne RO retrievals. The simulation system consists of two major components: (i) a forward simulator and (ii) an inverse simulator, i.e., FSI retrieval. The forward simulator is used to simulate the phase and the amplitude for an ARO signal given an atmospheric refractivity model and occultation geometry. The inverse simulator (or retrieval component) processes the simulated ARO signal phase and amplitude to retrieve the atmospheric bending angle and refractivity profiles.

Two different types of forward simulators were used in the study. The first option is a ray tracer that simulates the ARO signals as geometric optics rays (e.g., Xie et al., 2008). The initial ray direction from the transmitter is iteratively perturbed until the ray is found that reaches the receiver. Forward simulation using the ray-tracing technique becomes problematic when the atmosphere has sharp refractivity gradients. If the phase fluctuates rapidly for small perturbations in initial ray direction, the technique will not converge and cannot find a ray path connecting the GPS and receiver. Often this occurs when there are multiple solutions (rays) for a given GPS-to-receiver geometry, i.e., atmospheric multipath.

The second type of forward simulator that is used to simulate the GPS signal
in the presence of sharp refractivity gradients is a two-step procedure
that combines an Abel integral forward model followed by a FSF
simulator based on J03. The refractivity profile is input to
the forward Abel integral (e.g., Fjeldbo et al., 1971; Xie et al., 2008), which
calculates the bending as an integral function of radius from the surface to
the aircraft height and GPS transmitter height. The resulting bending angle
profile,

The complex signal after Fourier transform in Eq. (4) can be expressed as a
function of

Each pseudo frequency corresponds to a single ray path having a unique impact
parameter (

In this FSF forward model, the input atmospheric condition is represented by
a bending angle profile. Given

Therefore, the complex occultation signal phase and amplitude can be
constructed through the inverse Fourier transform of

The phase,

Subsequently, the excess phase as a function of the open angle

The inverse simulators in the end-to-end simulation comprise both the GO retrieval (e.g., Xie et al., 2008) and the newly developed FSI. Both the GO and FSI retrievals derive the bending angle profiles as a function of the impact parameter from the input excess Doppler as a function of time (for GO) or the combination of both the total phase and amplitude as a function of open angle (for FSI). The inverse Abel transform is then applied to retrieve refractivity from the bending angle.

In the following section, the input atmospheric refractivity and/or bending angle profiles to the forward simulator are directly compared to the output from the inverse simulators to assess the performance of the inversion technique, and to quantify the sensitivities of the ARO FSI to the potential errors presented in several key input variables such as the SNR and refractivity at the receiver.

To assess the performance of the ARO FSI retrieval algorithm, we used the
occultation characteristics from an actual aircraft flight and the
atmospheric profile of temperature and water vapor from the ERA-Interim
reanalysis at the flight location and time. One specific occultation
involves the GPS satellite PRN24 (the pseudo-random number identifies the
satellite) and the airborne receiver during the PREDICT flight from 18:20 to 19:00 UTC
on 14 September 2010 (research flight no. 19). In the simulation,
the radius of the curvature of the Earth was found at the occultation
location; then the aircraft height was set to a constant 14 km to produce an
occultation geometry with a circular orbit for the receiver at the time when
PRN24 was setting. The radius of the GPS orbit was set to a constant 26 000 km
above the center of curvature. The grid profiles of ERA-I temperature and
water vapor mixing ratio and the calculated refractivity profiles from the
ARO sounding region are shown in Fig. 3a and e, respectively. Very
moist atmospheric conditions with high mixing ratio (

The occultation phase and amplitude time series are divided into positive
and negative elevation angle parts for the FSI retrieval. The time epoch of
the local horizon (zero elevation angle) is estimated by a ray-tracing
simulation with the CIRA

The bending angle retrievals from GO (blue) and FSI (red) are plotted in
Fig. 3d along with the “true” bending angle profile (black) from the Abel
integral forward simulation. Two distinct and important features of the ARO
retrievals are shown. The first is the large error in the retrieved bending
angle near zero elevation when the tangent point is near the receiver. This
feature is present for both the GO and FSI methods. There is a singularity
in both the GO and FSI retrievals near zero elevation angle where small
errors in ray tangent angle (near

For both the GO- and FSI-retrieved bending angle, the refractivity below the aircraft is obtained through the inverse Abel transform, by integrating the partial bending angle (defined as the difference in bending angle between the negative and positive elevation at each impact parameter) from the tangent point height up to the receiver height (Fig. 3e) (X08). The retrieved refractivity has high errors for both GO and FSI immediately below the aircraft due to higher bending angle errors discussed above. In the inverse Abel integral, the effect of bending angle errors at the receiver height propagates downward to lower levels; however the bending angle increases exponentially downwards, so the refractivity errors also decrease exponentially downward (Fig. 3f, solid lines). This is consistent with the GO simulation study in X08.

In the lower troposphere, the large refractivity errors in the GO retrieval in the lowest 1 km are due to the bending angle retrieval error in the presence of multipath. The FSI retrieval, on the other hand, successfully resolves the fine vertical structure of both bending angle and refractivity in the presence of the multipath in the moist atmosphere near the surface without introducing retrieval biases.

The accuracy of the FSI retrieval depends on the accuracy of the signal phase and amplitude measurements, the occultation geometry, and the refractivity observation at the receiver. The sensitivity of the ARO retrieval to the excess phase or Doppler error in the geometrical determination of the aircraft position and velocity has been explored in the GO retrieval system in Xie et al. (2008) and Muradyan et al. (2010), and we expect the same sensitivity for the FSI retrievals. In this section, we will quantify the sensitivity of FSI retrievals to the errors in signal amplitude (not used in the GO retrieval), amplitude-dependent phase error, and the refractivity error at the receiver.

In the ARO measurements, the amplitude is affected by the aircraft heading
and attitude relative to the line of sight when using a directionally
focused antenna gain pattern. Sharp amplitude variations are introduced by
aircraft turns during the ARO measurements (Murphy et al., 2015). Therefore,
it is important to know how sensitive the FSI retrieval is to the
uncharacterized changes in signal amplitude. In addition, under low-SNR conditions, phase measurements have greater
uncertainties. To test the sensitivity of the FSI retrieval algorithm to the
variations in signal amplitude, we need to account for (1) the effect of
sharp fluctuations in the SNR and (2) the possible phase errors that may
arise under low-SNR conditions. To accomplish this task, a baseline signal
phase and amplitude were first simulated using the ray-tracing method in a
smooth refractivity model that accounts for atmospheric losses due to
attenuation. Then a sinusoidal amplitude function (Eq. 24) was added to
simulate the sharp amplitude jumps produced by the changing aircraft
direction, and finally Gaussian noise was added to the amplitude to
represent the variations in the ARO amplitude measurements.

In the simulation, the Gaussian noise power is assumed to be a constant
percentage of the peak signal power during the occultation. The observed
peak SNR prior to occultation is typically 200 V/V, and the observed noise
floor is typically

Wang et al. (2016) have shown that, at low SNR, increased phase variance
results in larger errors in the unwrapped phase of the signal. Therefore, to
test the impact of signal amplitude errors on the FSI retrievals, it is
important to assess its impact on the measured signal phase. Wang et al. (2016)
developed and tested a realistic model that relates the phase error
from ARO open-loop signal processing to SNR, which we use here to estimate
phase error and add to the simulated excess phase. Two different model
atmospheric profiles were used in the simulations. One ERA-Interim profile
(12:00 UTC, 13 September 2010 at 15

Figure 4c shows the difference between the FSI-retrieved bending angle of
the noisy signal and the true bending angle, calculated by forward Abel
integration of the ERA-I refractivity profile. Similarly, Fig. 4d shows
the percentage error of the FSI-retrieved refractivity compared to the input
refractivity profile. Both bending and refractivity errors show near-zero
mean with small variations, which indicate that large variations in the
amplitude measurement do not introduce systematic bias in the FSI bending
and refractivity retrievals when the SNR is high. It is worth noting that in
very low SNR conditions the amplitude error could potentially lead to
integer cycle unwrapping errors manifesting as cycle slips, which could lead
to a systematic bias in the signal phase or Doppler observation if a
climatological profile is not used (e.g., Wang et al., 2016). Such biased
phase or Doppler will lead to biases in both the bending and refractivity
retrievals. In a simulation using noise variance of 10 % of the peak
signal power (not shown), the reconstructed residual phase deviates from the
original residual phase below the 3.5–4 km height range due to the large
unwrapping errors. The retrieved bending angle errors increase below this
height, and the refractivity errors exceed

We now assess the sensitivity to uncertainties in the in situ refractivity
at the receiver. The refractivity at the receiver can be obtained from the
in situ temperature, pressure, and water vapor mixing ratio measurements at
the aircraft flight level. It is also one of the key parameters required in
both the GO and FSI retrievals. With an ARO receiver flying at

Figure 5a and b show the absolute bending angle error and fractional
refractivity error of the FSI retrieval considering the phase noise combined
with the in situ refractivity error. The bending angle error is dominated by
the phase error of about 0.02

The refractivity at the receiver is also used to reduce the refractivity
errors in the Abel inverse. Figure 3d shows that only the top

In this study, a FSI algorithm is developed and successfully applied to simulated airborne GNSS RO (ARO) measurements for the first time. The simulation study demonstrates the capability of the FSI method to retrieve the atmospheric vertical structure in the lower moist troposphere where frequent multipath occurs.

In the FSI retrieval process, the oblateness correction is applied to
transform the original occultation geometry to a local spherical radius of
curvature to fulfill the local spherical symmetric assumption. Then the
non-spherical trajectories of both the ARO receiver and the GPS satellite
are projected onto circular trajectories relative to local center of
curvature. Additional phase correction terms as a result of projection are
then added to the measured phase. Afterward, the occultation phase and
amplitude time series are divided into positive and negative elevation angle
sections. The separation point at the local horizon can be estimated by
ray tracing through the CIRA

The end-to-end simulation system was used to quantify the sensitivity of the
FSI bending and refractivity retrievals to the noise in two key parameters:
the signal amplitude (which induces phase errors as well as amplitude
errors) and the refractivity at the receiver. The FSI retrieval showed a
weak sensitivity to signal amplitude errors. Even the abrupt changes in
signal amplitude due, for example, to aircraft turns do not introduce any
systematic bias to the retrieval, as long as the SNR is high. The
sensitivity to refractivity errors at the receiver is greater. The 1 %
in situ refractivity errors at the receiver height could introduce a maximum
refractivity retrieval error of 0.5 % (1 K) near the receiver, but that
error decreases gradually to

Funding for this research was provided by NSF grant AGS 1262041. Jennifer S. Haase was supported by NSF grant AGS 1015904. Special thanks to Sergey Sokolovskiy at UCAR for helping develop the prototype FSI retrieval model. Brian Murphy, Kuo-nung Wang, and James Garrison at Purdue University are thanked for useful discussions over the course of the project. We would also like to acknowledge the continued support of NSF program managers Anjuli S. Bamzai and Eric DeWeaver. ERA-Interim reanalysis profiles were provided by the European Centre for Medium Range Forecasts (ECMWF). Edited by: I. Moradi Reviewed by: two anonymous referees