Detection of Superrefraction at the Top of the Atmospheric Boundary Layer from COSMIC-2 Radio Occultations

Sergey Sokolovskiy aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Zhen Zeng aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Douglas C. Hunt aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Jan-Peter Weiss aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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John J. Braun aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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William S. Schreiner aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Richard A. Anthes aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Ying-Hwa Kuo aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Hailing Zhang aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Donald H. Lenschow bNational Center for Atmospheric Research, Boulder, Colorado

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Teresa Vanhove aUniversity Corporation for Atmospheric Research, Boulder, Colorado

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Open access

Abstract

Superrefraction at the top of the atmospheric boundary layer introduces problems for assimilation of radio occultation data in weather models. A method of detection of superrefraction by spectral analysis of deep radio occultation signals introduced earlier has been tested using 2 years of COSMIC-2/FORMOSAT-7 radio occultation data. Our analysis shows a significant dependence of the probability of detection of superrefraction on the signal-to-noise ratio, which results in a certain sampling nonuniformity. Despite this nonuniformity, the results are consistent with the known global distribution of superrefraction (mainly over the subtropical oceans) and show some additional features and seasonal variations. Comparisons to the European Centre for Medium-Range Weather Forecasts analyses and limited set of radiosondes show reasonable agreement. Being an independent measurement, detection of superrefraction from deep radio occultation signals is complementary to its prediction by atmospheric models and thus should be useful for assimilation of radio occultation data in the atmospheric boundary layer.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sergey Sokolovskiy, sergey@ucar.edu

Abstract

Superrefraction at the top of the atmospheric boundary layer introduces problems for assimilation of radio occultation data in weather models. A method of detection of superrefraction by spectral analysis of deep radio occultation signals introduced earlier has been tested using 2 years of COSMIC-2/FORMOSAT-7 radio occultation data. Our analysis shows a significant dependence of the probability of detection of superrefraction on the signal-to-noise ratio, which results in a certain sampling nonuniformity. Despite this nonuniformity, the results are consistent with the known global distribution of superrefraction (mainly over the subtropical oceans) and show some additional features and seasonal variations. Comparisons to the European Centre for Medium-Range Weather Forecasts analyses and limited set of radiosondes show reasonable agreement. Being an independent measurement, detection of superrefraction from deep radio occultation signals is complementary to its prediction by atmospheric models and thus should be useful for assimilation of radio occultation data in the atmospheric boundary layer.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sergey Sokolovskiy, sergey@ucar.edu

1. Introduction

Superrefraction (SR) often occurs at the top of the atmospheric boundary layer (ABL), especially over the subtropical oceans. The SR affects the inversion of radio occultation (RO) data. Below the top of the SR layer, the RO inversion problem becomes ill posed (allows multiple solutions). This presents a challenge for RO data assimilation in the ABL. Several approaches for constrained RO inversions under the SR conditions have been proposed (Xie et al. 2006, 2010; Wang et al. 2017, 2020). Regardless of the approach, it is important to know whether or not a given RO observation is affected by SR. This is not possible from the retrieved bending angle (BA) profile. In RO data assimilation, the model first guess and large retrieved BA are sometimes used as proxies for SR (Cucurull 2015). Earlier it was found that SR produces a weak diffracted signal observed down to the height of straight line (HSL) which is significantly lower than the minimum height common for RO signals (the so-called deep signal) (Sokolovskiy et al. 2014). This signal can be used as a proxy for SR. Detection of weak deep signals requires a high signal-to-noise ratio (SNR), and became routinely possible with Formosa Satellite 7/Constellation Observing System for Meteorology Ionosphere and Climate 2 (FORMOSAT-7/COSMIC-2) mission, hereafter C2, due to RO receivers with high gain antennas developed by NASA’s Jet Propulsion Laboratory (Tien et al. 2012). In this study we introduce a method for detection of SR at the top of the ABL based on the spectral analysis of deep RO signals (Sokolovskiy et al. 2014) and large bending angle lapse in the retrieved BA profiles (Sokolovskiy et al. 2007). We processed 2 years of all C2 deep occultations (tracked down to −350 km HSL) and analyzed the results. We present the global distribution of the SR occurrence for the whole 2-yr period and for the two periods of May–September and November–March separately. We demonstrate that the possibility of detecting SR substantially depends on the SNR. We compared detected cases of SR with those predicted by a high-resolution European Centre for Medium-Range Weather Forecasts (ECMWF) model analyses and found overall reasonable agreement, but for a significant percentage of the cases the occurrence of SR is not present in the model analyses. Additionally, we compared detected cases of SR with a limited number of collocated radiosondes. Overall, our results support feasibility of the detection of SR from deep RO signals, which provides independent information, complementary to standard RO profiles.

2. Superrefraction and radio occultation inversions

We define SR as the condition where the negative vertical gradient of the refractive index dn/dr is strong enough such that the curvature radius of the ray at the tangent point (TP) is smaller than the local Earth’s radius re following (Kursinski et al. 2000; Sokolovskiy 2003; Xie et al. 2006, 2010). As it follows from the Snell’s law (Kursinski et al. 1997) the critical refractivity gradient dN/dr = −106/re = −157 km−1, where N = 106(n − 1). We note that sometimes the SR is defined alternatively as the condition where the vertical refractivity gradient is between the critical value of −157 km−1 and half of that value (Lopez 2009). The layer below the top of the SR layer where external rays (between transmitter and receiver outside the atmosphere) do not have TPs is called the ducting layer. The rays which have TPs inside the ducting layer are internal (trapped) (Ao 2007). It is important to distinguish between elevated and surface ducts. Commonly, for describing ducts a modified refractivity is used which accounts for Earth’s curvature (Lopez 2009; Zhou et al. 2021). However, in this study, we prefer to use the refractional radius x = rn(r) because this variable also is used for the forward modeling of the bending angle α(a), where a is the impact parameter, from the refractive index n(x) in the RO data assimilation:
α(a)=2aadn/dxn(x)a2x2dx.

Figure 1 shows functions y(z), where y = xre and z = rre, for elevated (left) and surface (right) ducts (z is measured from the surface). Normally, y decreases with decreasing z, but in the SR layer, y increases. The top of the SR layer is the top of the duct. The height where y decreases again and equals the value at the top of the duct is the bottom of the duct. If the bottom of the duct is above the surface it is an elevated duct, otherwise it is a surface duct. For the elevated duct, there are external rays below the bottom of the duct, while there are no external rays below the top of the surface duct. In other words, for the elevated duct, α(a) (which tends to infinity at the duct) can be obtained from RO observations below the bottom of the duct, while α(a) does not exist below the top of the surface duct. Thus, information about elevated ducts should be useful for assimilation of RO data below the bottom of the duct. Elevated ducts most commonly occur below the top of the inversion layer capping the ABL, especially over the subtropical ocean.

Fig. 1.
Fig. 1.

Examples of the profiles y(z) for (left) elevated and (right) surface ducts. White and gray areas correspond to TP heights of the external and internal (trapped) rays; i.e., the gray areas represent the ducting layers.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

Figure 2 (left panels) shows two profiles N(z) for strong inversion layers with dN/dz > −157 km−1 (top panel, no SR) and dN/dz < −157 km−1 (bottom panel, SR). Figure 2 (right panels) shows the corresponding profiles α(h), where h = are, obtained by forward modeling using Eq. (1). Without SR, the maximum BA may be very large but it is finite. With SR, the maximum BA formally equals infinity and the corresponding h equals y at the top and bottom of the duct (see Fig. 1). In the case of SR, calculation of n(x) from α(a) using the equation known as the Abel inversion (Kursinski et al. 1997, 2000),
n(x)=exp[1πxα(a)a2x2da],
returns the profile NA(z), which is negatively biased below the top of the SR layer. The original N(z) and the Abel-retrieved NA(z) profiles are shown in Fig. 2 (lower-left panel). They both correspond to the same BA profile shown in Fig. 2 (lower-right panel). Furthermore, forward modeling of α(a) from a linear combination cN(z) + (1 − c)NA(z), where 0 < c < 1 yields the profile α(a), which is not strictly equal [due to nonlinearity of Eq. (1)] but is very close to the original α(a). Several such N(z) profiles are shown in the lower-left panel. The corresponding α(a) profiles, which are indistinguishable from the original, are shown in the lower-right panel. This illustrates that the inverse RO problem is underdetermined in the case of SR. Analytical expressions for multiple profiles N(z) obtained by parameterization of the SR layer can be found in Xie et al. (2006), Ao (2007), and Wang et al. (2017, 2020).
Fig. 2.
Fig. 2.

Examples of the (left) N(z) and (right) α(h) profiles (top) without SR and (bottom) with SR.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

We note that assimilation of α(h) in the region of large BA or singularity has no physical meaning. Due to large defocusing and observational noise resulting in low SNR, α(h) is estimated with large uncertainty or is not measurable. However, below the elevated duct, the SNR increases and α(h) again becomes measurable. Data assimilation is an optimal estimation that uses forward modeling. The possibility to model the same BA profile from quite different N profiles introduces an uncertainty in data assimilation. Since the uncertainty introduced by an SR layer extends from the top of the layer down to the surface, the importance of the information about SR increases with the height of the SR layer. To know whether α(a) can be assimilated directly, by applying additional constraints, or discarded, it is necessary to know whether a certain RO observation is affected by SR. Since SR is not detectable from the RO-retrieved BA profile, sometimes in RO data assimilation a large BA from the RO observation and/or large |dN/dz| from the model first guess are used as the proxies for SR (Cucurull 2015). In this study, SR is detected based on only the observation itself.

3. Superrefraction and deep radio occultation signals

Commonly, RO signals used for atmospheric profiling are distinguishable from noise down to about −150 km HSL. Earlier, by wave optics (WO) modeling and analysis of COSMIC RO signals it was found that elevated ducts result in weak diffracted signals observed much deeper, as low as −350 km HSL (Sokolovskiy et al. 2014). These deep signals do not contribute to atmospheric profiling, but their existence in the spectrograms indicates the existence of SR. They found that deep signals below −200 km HSL appear when |dN/dz| exceeds the critical value of −157 km−1 (Sokolovskiy et al. 2014). Here we perform additional WO modeling aimed at evaluating the effects of height and strength of the elevated and surface ducts on the existence, magnitude, and structure of deep RO signals.

Figure 3 (top panel) shows a set of y(z) profiles (similar to those in Fig. 1) with the SR layers corresponding to surface (1, 2, 7) and elevated (3–6, 8–12) ducts (the two profiles 13 and 14 without SR are used as the references). The numbers on the left show N gradients calculated between top and bottom of the corresponding SR layers. The bottom panel shows amplitudes of the corresponding deep signals obtained by WO modeling (amplitudes are normalized to 1 above the atmosphere). For the elevated ducts, the magnitude of the amplitude variations increases with the increasing strength of the SR layer, while the mean amplitude is not significantly different. There is no significant dependence of the mean amplitude on the height of the SR layer. However, the HSL of the maxima and minima depend on the height of the SR layer. For strong SR layers, the deep signal may fade at some HSL but then reemerge at lower HSL. For the spherically symmetric refractivity used in this modeling, there are no deep RO signals for the surface ducts. We note that strong horizontal gradients potentially may create short deep signals for surface ducts (Sokolovskiy et al. 2010); however, this effect requires additional investigation and is not considered in this study.

Fig. 3.
Fig. 3.

(top) Profiles y(z) used for WO modeling of deep RO signals. Each profile y(z) is shifted in y with respect to the previous profile by 0.5 km for display purposes. (bottom) Amplitudes of deep RO signals modeled by using profiles y(z) from the top panel.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

4. Detection of deep signals from COSMIC-2 radio occultations

We use representation of the complex RO signals by spectrograms (Sokolovskiy et al. 2009; Wang et al. 2017). This allows us to visualize very weak signals such as those in Fig. 4 which are not easily distinguishable from the observational noise in the amplitude and Doppler representations. For down-conversion (reduction of the frequency) of the RO signal, we use the model described by Sokolovskiy (2001). This model, which is based on transmitter and receiver positions and velocities and N climatology, predicts the frequency of RO signals with 5–10 Hz accuracy (Sokolovskiy 2001; Sokolovskiy et al. 2009) and is routinely used at the COSMIC Data Analysis and Archive Center (CDAAC) for processing RO signals acquired in the open-loop (OL) mode. For visualization, we perform spectral analysis in a sliding window of 1.28 s by representing the normalized power spectrum as a function of the HSL corresponding to the center of the window.

Fig. 4.
Fig. 4.

Spectrogram of the (top) C2 RO and (bottom) power spectrum calculated in the window between −250 and −350 km HSL without (black) and with (red) smoothing using 1-Hz Hann’s window.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

Figure 4 (top panel) shows an example of such a spectrogram for a C2 occultation with deep signal visible down to −350 km HSL. The color scale denotes the normalized spectral power (the same color scale is used for the spectrograms in other figures). For automated detection of deep signals, we calculate the power spectrum S(f), where f is the frequency, in a large window between −350 and −250 km. This makes it possible to reduce the effects of noise, interfering signals (see discussion below) and variations of the amplitude of deep signals with the HSL (see Fig. 3). The possibility to use a large window is based on the fact that the frequency model used for down-conversion substantially eliminates the dependence of the RO signal frequency on the HSL. Next, we smooth the spectrum using Hann’s window of 1 Hz width and calculate the spectral signal-to-noise ratio (SSNR), SSNR = Smax/Sbgr, where Smax and Sbgr are maximum and background spectral powers. The Sbgr is estimated by averaging S(f) in the frequency intervals (−50, −25) Hz and (25, 50) Hz. The raw and smoothed power spectra are shown in Fig. 4 (bottom panel) by black and red lines.

The possibility to detect deep signals depends on the SNR above the atmosphere (hereafter we refer to the SNR averaged between 40 and 80 km HSL) and on the accuracy of the frequency and range models used by the receiver firmware for the OL signal tracking. Errors of the receiver models reduce the SNR in the lower atmosphere in addition to the atmospheric defocusing (Sokolovskiy et al. 2009). Generally, receiver models are less accurate for rising occultations, especially the range model, which is affected by the ionospheric delay. In this study, we do not use the SNR and error estimates of the receiver models as constraints. We calculate the SSNR for all occultations and select those where the SR is identified by using the metrics discussed below. It is obvious that not all SR cases are detected and thus the number of detected SR cases represents a lower estimate.

Figure 5 shows examples of spectrograms of 12 C2 RO signals. Occultations 1–4 do not show any evidence of signals below about −150 km HSL where only noise is seen on the spectrograms. Occultations 5–8 show deep signals extending to −350 km HSL which are thought to be caused by elevated ducts. Occultations 9–11 show signals which we cannot explain by propagation effects. One type of such interfering signals was investigated and explained by Sokolovskiy et al. (2014) as the global positioning system (GPS) cross-satellite interference. Cross-satellite interference was not expected for the Global Navigation Satellite System (GLONASS) due to the use of the frequency division multiple access (Kaplan 1996). In C2 occultations, some types of the interfering signals are seen in the GPS spectrograms (occultation 9) while other types are seen both in the GPS and GLONASS spectrograms (occultation 10). Our testing shows that these signals may reduce the SSNR but they do not result in false detection of deep RO signals, because either their frequency is rapidly changing with respect to the frequency of the RO signal (occultation 9) or their spectrum is spread (occultation 10). The interfering signals with narrow spectra staying close in frequency to the RO signal for an extended time (occultation 11) are seen less frequently, but they may result in a large SSNR and false detection of deep RO signals. We note that our analysis of the interfering signals is based on the visual analysis of a limited number of spectrograms. An automatic detection of the interfering signals with different structure is a complicated task and is not considered in this study.

Fig. 5.
Fig. 5.

Spectrograms of C2 RO signals.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

Commonly, navigation data modulation (NDM) is removed from RO signals recorded in the OL mode by applying the NDM replica in the ground processing. However, for some occultations, when the replica is not available, the NDM is removed by 2-quadrant phase extraction. While such RO signals are suitable for standard inversions, they are not suitable for detection of deep signals because the two-quadrant phase extraction “despreads” the spectrum of noise by creating an artificial deep signal (Sokolovskiy et al. 2009). This is seen in the spectrogram of the occultation 12 (Fig. 5) processed with the 2-quadrant phase extraction. The percentage of such occultations at CDAAC is small, on average about 0.2%. But since in a significant number of such occultations, deep signals are identified using the metric discussed below, their percentage increases to about 6%. Such RO signals were removed from the analysis based on the information about OL signal processing available in the RO data files at CDAAC.

5. Detection of superrefraction from COSMIC-2 deep occultations

The main metric for detection of SR is the existence of a deep RO signal with sufficient SSNR. In this study, we consider occultations with SSNR > 8. Currently, this threshold is set ad hoc; i.e., it does not have clear physical justification and is based on the analysis of spectrograms. Figure 6 shows examples of the spectrograms with different SSNRs which support the choice of the threshold. This threshold may be adjusted in the future based on a more detailed analysis.

Fig. 6.
Fig. 6.

Spectrograms of C2 RO signals with different SSNR.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

However, the SSNR metric is necessary but not sufficient for detection of SR. First, because of the interfering signals discussed above (Fig. 6, occultation 11), which may result in false detection. Second, numerical modeling proves the existence of deep RO signals in the case of horizontally homogeneous elevated ducts, but it does not prove that deep signals exist only in those cases. In particular, effects of horizontal gradients have not been sufficiently investigated. Because this study is focused on the elevated ducts caused by the SR at the top of the ABL, we apply an additional metric for detection of a sharp ABL top. Different metrics based on RO signals and retrieved profiles have been used as the proxy for the ABL top (von Engeln et al. 2005; Sokolovskiy et al. 2007; Ratnam and Basha 2010; Guo et al. 2011; Ao et al. 2012; Ho et al. 2015; Kalmus et al. 2022). We use the metric based on the maximum bending angle lapse (max. BAL) introduced by Sokolovskiy et al. (2007) and used in subsequent studies. Figure 7 shows histograms of max. BAL for all setting occultations (blue) and for those with SSNR > 8 (red). It is obvious that max. BAL values on the left side of the histogram (blue) are more related to random structures of BA profiles, while the values on the right side are related to inversion layers. It is also obvious that these two mechanisms overlap. In this study we use the max. BAL = 0.007 rad corresponding to the maximum of the histogram (blue) in Fig. 7 as the conventional boundary between these mechanisms and as the threshold for detection of the ABL top. Thus, in this study detection of SR at the top of the ABL is based on 1) SSNR > 8 and 2) max. BAL > 0.007 rad. The max. BAL constraint removes about 5% of the occultations with SSNR > 8 as this can be seen from the histogram (red) in Fig. 7. Both the SSNR and max. BAL metrics may be revised in the future based on a more detailed analysis.

Fig. 7.
Fig. 7.

Histograms of max. BAL for all C2 setting occultations (blue) and those with SSNR > 8 (red).

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

6. Analysis of the superrefraction detected from 2 years of COSMIC-2 data

Deep tracking was tested on some of the C2 receivers after launch in 2019. Preliminary results are available (Schreiner et al. 2020). After mid-2020 all C2 receivers were configured to track RO signals down to HSL = −350 km for setting occultations. This provided sufficient data for analysis. Below we present results for the 2-yr period 2020.181–2022.180. Detection of SR was based on the two combined metrics discussed above. This resulted in about 81 000 occultations with detected SR (the SSNR metric alone resulted in about 85 500 occultations), i.e., about 4% of all setting occultations, which corresponds to the global SR detection rate in this study. We also consider the local SR detection rate, which is the ratio of the number of detected SR cases to the total number of setting occultations in a latitude–longitude bin as well as the SR detection rate for a given SNR, i.e., defined in an SNR bin. We note that SR detection rate depends on RO mission and data processing which may use different metrics. Thus, our estimation of the SR detection rate is specific to C2 mission and data processing used in this study.

Figure 8 (top panel) shows the global distribution of C2 occultations with detected SR. Different colors denote different height intervals estimated from the heights of max. BAL: black, < 1 km; green, 1–2 km; blue, 2–3 km; and red, > 3 km. The bottom panel shows gridded map of the local SR detection rate. The main regions of SR at the top of the ABL are over the subtropical ocean, west of the American and African continents in the Northern and Southern Hemispheres where local SR detection rate is highest, up to 36%. Two other noticeable regions are over the north and south Indian Ocean. SR also occurs over the continents, though less frequently. Three noticeable regions are over South America and southern Africa (where the heights of SR layers are often above 3 km) and Australia. Ducting climatology has been studied based on radiosondes (Ao 2007), global atmospheric models (von Engeln and Teixeira 2004; Lopez 2009; Feng et al. 2020), and profiles obtained by one-dimensional variational data assimilation of RO data (Zhou et al. 2021). The global distribution of detected SR in Fig. 8 is consistent with the known ducting climatology, such as the one derived from the ECMWF global analyses fields (von Engeln and Teixeira 2004). One noticeable difference is that von Engeln and Teixeira (2004) found that the maximum ducting probability is adjacent to the west coasts of continents, while there are noticeable “gaps” in Fig. 8. These gaps (which exist without applying max. BAL constraint) are explained by low heights of the SR layers resulting in surface ducts which do not produce deep signals (see Fig. 3) and thus are not detectable by RO.

Fig. 8.
Fig. 8.

(top) Global distribution of C2 occultations with detected SR for 2020.181–2022.180. Colors indicate different height intervals: <1 km (black); 1–2 km (green); 2–3 km (blue); >3 km (red). (bottom) Global distribution of the local SR detection rate.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

The distribution of C2 occultations with detected SR in Fig. 8 shows quasi-periodic structures in the longitudinal direction with a period of about 10°. Analysis shows that these structures are not related to the atmospheric properties. They reflect the dependency of the SR detection rate on the SNR. Figure 9 (left panel) shows the SNR histograms for all setting occultations (blue) and for those with detected SR (red). The right panel shows the global SR detection rate (which is the ratio of histograms in left panel normalized by total numbers of occultations) as a function of the SNR. It is seen that SR detection rate is rather low for SNR < 1000 V/V. Fifty percent of all SR cases are detected with SNR > 1750 V/V, which is available in 17.5% of all setting occultations. Thus, Fig. 9 supports the importance of high SNR for detection of SR and the statement made earlier that detection of SR in this study provides a lower estimate. We note that the SR detection rate in Fig. 9 is specific for C2 mission and used SR metrics and should be different for different RO missions and signal processing.

Fig. 9.
Fig. 9.

(left) Histograms of the SNR for all C2 setting occultations (blue) and for those with detected SR (red). (right) Global SR detection rate as function of the SNR.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

Figure 10 shows the global distribution of occultations with the SNR > 1750 V/V, which shows a pattern similar to that in Fig. 8. This pattern is the result of C2 orbits, GPS and GLONASS orbits and transmitted signal strengths, and the C2 antenna gain pattern. Signals with the same strengths received at different azimuth angles with respect to C2 orbit plane have different SNRs. Distribution of the azimuth angle depends on the orbits. Thus, the pattern in Fig. 10 is specific to C2 mission and should be different for different RO missions.

Fig. 10.
Fig. 10.

Global distribution of C2 setting occultations with SNR > 1750 V/V.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

To see the seasonal variability of SR, we use two “expanded seasons” (variations between four seasons are less pronounced) November–March and May–September, by removing the “transition” months of April and October after the equinoxes. Hereafter we refer to these expanded seasons “winter” and “summer.” In Figure 11 the top and middle panels show global distributions of SR for the winter and summer seasons, respectively. The bottom panel shows a gridded map of the difference (winter–summer) of the local SR detection rates. In the Northern Hemisphere, in the east Pacific and Atlantic regions the SR is more pronounced in the summer season while in the west Pacific and Indian Ocean regions it is more pronounced in the winter season. In the Southern Hemisphere, the differences in SR between summer and winter seasons are less pronounced in the east Pacific region. The SR west of southern Africa and Australia is more pronounced in winter season. Over the continents, South America, southern Africa, and Australia, SR occurs mainly in the summer season. Though Fig. 11 and von Engeln and Teixeira (2004) use different representations, some seasonal features such as increased probability of SR in the north Indian Ocean and west of Australia and a more pronounced South Pacific convergence zone in winter season are seen in both representations.

Fig. 11.
Fig. 11.

(top),(middle) Global distributions of the C2 occultations with detected SR for winter and summer seasons, respectively. (bottom) Distribution of the difference of SR detection rates between winter and summer seasons.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

Figure 12 (top panel) shows the global distribution of SR detected from C2 (blue) and SR predicted by the ECMWF model (red) for those occultations with SR detected from C2. The occultations with SR predicted by model but not detected by C2 are not considered as explained in section 8. We use the operational ECMWF analysis (137 levels up to 80 km, 0.25° horizontal resolution, 6-h update time) used by CDAAC. For the model, we use the physically defined SR metric dN/dz < −157 km−1 calculated on the model grid. We note that in data assimilation empirically defined metrics may be used (Cucurull 2015). The bottom panel shows gridded map of the ECMWF local prediction rate calculated for the set of occultations with SR detected from C2. The SR was predicted by the ECMWF model in 51% cases, though the local prediction rate is larger and smaller in different regions (close to 100% near west coasts of continents and low over the continents). It is reasonable to expect that for the models with lower vertical resolution, the prediction rate may be lower. Thus, the information about the SR detected from C2 should be useful for RO data assimilation.

Fig. 12.
Fig. 12.

(top) Global distribution of the C2 occultations with SR detected by C2 (blue) and predicted by ECMWF (red). (bottom) Gridded map of the ECMWF SR prediction rate for occultations with SR detected from C2.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

7. Comparison to radiosondes

Comparison of RO to radiosondes in the ABL is challenging due to horizontal and vertical representation differences (Sjoberg et al. 2021; Rieckh et al. 2021, and references therein). First, radiosondes represent local profiles so that dN/dz < −157 km−1 does not necessarily indicate SR on the larger horizontal scales represented by the RO observations. Second, SR mainly occurs over the oceans where radiosondes commonly are launched from islands, which may locally change the ABL structure. Nevertheless, as examples, we compared the SR cases detected from C2 near the Hawaiian Islands to local radiosondes from two stations because this region has a high SR detection rate from RO, but not a high prediction rate from the ECMWF model, as can be seen in Fig. 12. We used radiosonde data from the Integrated Global Radiosonde Archive (IGRA) version 2 (Durre and Yin 2008). These data are collected from different sources (Durre et al. 2006); their vertical sampling is highly nonuniform and is different for different stations.

We used the collocation criteria: <300-km distance, <3-h time difference for occultations and radiosondes and interpolated ECMWF data to the times and locations of the occultations. Most radiosonde, C2, and ECMWF N profiles show pronounced break points characterizing a sharp ABL top (Guo et al. 2011), but sometimes at different heights. In many cases, a sharp ABL top is better pronounced in radiosonde N profiles than in C2 and ECMWF profiles. This is consistent with the radiosonde being a local measurement (point value) while both RO and ECMWF values represent the atmosphere with considerable horizontal averaging (footprint). Figure 13 shows several examples of the radiosonde, C2, and ECMWF N profiles (left panel) along with the spectrograms of deep RO signals (right panels). The SR cannot be determined from RO N profiles due to negative bias (section 2), but it is clearly detected from deep signals.

Fig. 13.
Fig. 13.

(left) Examples of radiosonde profiles N(z) from Hawaii (black), collocated C2 RO (red), and ECMWF (green) profiles N(z). Each set of the profiles is shifted in z with respect to previous profile by 2 km for display purposes. (right) Spectrograms of deep signals for the corresponding occultations.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

For the two Hawaiian stations used in this study the averaged vertical sampling interval between surface and 500 hPa is about 145 m. To detect the SR, the raw grid profiles N(z) were linearly interpolated onto a uniform 10-m grid and smoothed by sliding averaging with a 150-m window. The SR layers were identified by requiring Δyz < 0 throughout the layer (see Fig. 1). The N gradient and the middle height between the top and bottom of the layer were calculated. In the case of multiple layers, the layer with minimal N gradient was selected for comparison (SR layers corresponding to surface ducts were not considered). For ECMWF, the SR layers were identified by using the original model grid without smoothing.

Analysis of 72 radiosonde profiles collocated with C2 occultations with detected SR showed that SR was detected by radiosondes in 62 cases (86%) and predicted by ECMWF in 16 cases (22%). This supports the frequent occurrence of SR in this region, which seems to be underestimated by the ECMWF model. Figure 14 shows comparison of the heights of max. BAL for C2 occultations with detected SR and the heights of SR determined from the collocated radiosonde profiles as discussed above. Overall, the heights are in reasonable agreement. For the two outliers there is some evidence of multiple layers in the C2 BA profiles, which may be a subject of the future investigation.

Fig. 14.
Fig. 14.

Comparison of the heights of SR layers determined from C2 and radiosondes.

Citation: Journal of Atmospheric and Oceanic Technology 41, 1; 10.1175/JTECH-D-22-0100.1

8. Discussion and conclusions

SR at the top of the ABL detected from deep RO signals is an additional RO product complementary to the standard profiles, which should be useful for RO data assimilation in the ABL. This product is based on direct measurement as opposed to proxies based on the magnitude of retrieved BA and modeled (predicted) N. As it was mentioned above, the tested method underestimates the occurrence of SR. When SR is not detected by RO, this does not mean that SR does not occur. The SR may not be detectable due to insufficient SNR (estimated at the top of an occultation), additional loss of the SNR at the bottom of the occultation due to receiver open-loop model errors (Sokolovskiy et al. 2009), or interfering signals. Also, there is lack of justification for some settable parameters which may be overly conservative. Thus, currently for comparison to ECMWF we used only the occultations with SR detected by C2 RO. For this set of occultations the ECMWF model predicted SR in about half of the cases with the local prediction rate mostly varying between about 20% and 80% in different regions. An additional comparison of C2 occultations with detected SR to collocated radiosondes from the Hawaiian Islands showed that SR was detected by radiosondes in about 80% of the cases and predicted by ECMWF in about 20% of the cases. Thus, the tested method detects the SR in some cases that are not predicted by the model, and can be useful in addition to the check based on the vertical model N gradient. When RO observations are available below the top of the ABL, information about the existence of SR at the top indicates whether to assimilate or discard RO data below, or apply any of the possible constrained inversions when they are implemented in RO data assimilation. However, there are multiple questions which need further investigation. They are discussed below.

The effect of SNR on the quality of RO profiling is currently under investigation. Ho et al. (2020) and Gorbunov et al. (2022), using different approaches, found weak dependence of the retrieval statistics below 10 km on the SNR. Multiple studies confirm that high SNR results in deeper penetration of retrieved profiles obtained with the same retrieval algorithm (Schreiner et al. 2020; Ho et al. 2020; Rieckh et al. 2021; Gorbunov et al. 2022; Anthes et al. 2022). As it follows from results of this study, the probability of SR detection using deep RO signals substantially depends on the SNR (Figs. 810). Results of WO modeling in Sokolovskiy et al. (2014) and in this study suggest that the SNR of about 2000 V/V should be sufficient for detection of deep RO signals caused by SR. However, in our analysis we did not find clear evidence of the saturation of dependence of the probability of SR detection on the SNR up to the maximum observed C2 SNR of about 2500 V/V (Fig. 9, right panel). This can be explained by additional loss of the SNR due to errors of the receiver OL models (mentioned above) and possibly by loss of the SSNR due to interfering signals like those in Fig. 6 (occultations 9 and 10). In other words, some deep RO signals, in order to be detectable, may require SNR greater than 2000 V/V to compensate for additional losses. This may be a subject of a future investigation.

Strong dependence of the SR detection rate on the SNR and nonuniformity of the distribution of SNR for the C2 mission result in nonuniformity of the SR sampling. Generally, sampling nonuniformity is common to most (if not all) remote and in situ observing systems. For example, radiosondes have strong spatial sampling nonuniformity while satellites in solar-synchronous orbits have strong sampling nonuniformity in local time. The nonuniformity of the distribution of the SNR (Fig. 10) is related to C2 orbits, GPS and GLONASS orbits and transmitted signal strengths, and on the C2 antenna pattern. Thus, the pattern in Fig. 10 is specific to C2 and should be different for different RO missions. For the future RO missions, increase of the antenna gain would increase the SR detection rate and flattening of the antenna gain pattern would reduce the SNR-related nonuniformity of the SR sampling.

Most commonly, SR occurs at the top of the ABL. This is one of the reasons for using an additional metric: existence of an elevated inversion layer which helps to discriminate between deep signals induced by SR and other signals. The height of the strongest inversion layer currently is used as the proxy for the height of the SR layer. However, potentially there may be more than one inversion layer. In this case the SR will be assigned to the layer with the largest BAL, which may result in error. It is possible to estimate the impact height of deep signals from the signal itself. One such approach based on wave optics was tested for individual occultations in Sokolovskiy et al. (2014). Another approach based on geometric optics is currently under testing. Comparison of both approaches is the subject of an ongoing study. It should be noted that independent estimation of the height of SR layer is useful as it mitigates the errors in case of multiple layers, but it does not eliminate the need to check the BA profile for existence of strong inversion layer characterized by a large BAL.

The effects of horizontal inhomogeneity of the SR layers on deep signals also need further investigation. So far, the effect of bumpiness of the SR layer at the top of an elevated duct and the effect of a surface duct with a finite horizontal extent on the existence of deep signals have been modeled (Sokolovskiy et al. 2010, 2014). Effects of the tilted SR layer and of the N gradient gradually changing from below to above critical also need investigation by modeling. This is also the subject of an ongoing study.

In this study, we set a threshold of 8 for the SSNR and 1 Hz value for the spectral filtering window based on our analysis, without clear physical justification. Though setting parameters based on analysis is not uncommon in remote sensing, it is desirable to better justify these parameters in the future.

The possibility to detect SR by spectral analysis of deep RO signals was first demonstrated in 2014. Currently, RO seems to be the only space remote sensing technique capable of detecting SR. While a number of questions require further investigation, the results of this study on detection of SR from 2 years of C2 deep occultations are promising based on the global distribution of detected SR events, seasonal variations, and comparison to ECMWF analyses and radiosondes. Being an independent measurement, SR detection should be useful for RO data assimilation in the ABL. Currently, CDAAC is implementing an SR flag in the files containing RO retrieval products available to users.

Acknowledgments.

This study was supported by the National Science Foundation (NSF) Grant 2054356, National Aeronautics and Space Administration (NASA) Grant C22K0658, and National Oceanic and Atmospheric Administration (NOAA) Cooperative Agreement R4310383. The authors acknowledge NASA’s Jet Propulsion Laboratory for design and support of the C2 RO receivers. The authors thank two anonymous referees for their thoughtful comments, which helped to improve the paper.

Data availability statement.

C2 data used in this study are profiles of RO observables and retrieved profiles as well as the profiles extracted from the ECMWF grid data. All C2 data are openly available from the University Corporation for Atmospheric Research, COSMIC Data Analysis and Archive Center (https://doi.org/10.5065/T353-C093). Radiosonde profiles used in this study are openly available from the National Oceanic and Atmospheric Administration National Centers for Environmental Information (https://doi.org/10.7289/V5X63K0Q).

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
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Save
  • Anthes, R., J. Sjoberg, X. Feng, and S. Syndergaard, 2022: Comparison of COSMIC and COSMIC-2 radio occultation refractivity and bending angle uncertainties in August 2006 and 2021. Atmosphere, 13, 790, https://doi.org/10.3390/atmos13050790.

    • Search Google Scholar
    • Export Citation
  • Ao, C. O., 2007: Effect of ducting on radio occultation measurements: An assessment based on high-resolution radiosonde soundings. Radio Sci., 42, RS2008, https://doi.org/10.1029/2006RS003485.

    • Search Google Scholar
    • Export Citation
  • Ao, C. O., D. E. Waliser, S. K. Chan, J.-L. Li, B. Tian, F. Xie, and A. J. Mannucci, 2012: Planetary boundary layer heights from GPS radio occultation refractivity and humidity profiles. J. Geophys. Res., 117, D161117, https://doi.org/10.1029/2012JD017598.

    • Search Google Scholar
    • Export Citation
  • Cucurull, L., 2015: Implementation of a quality control for radio occultation observations in the presence of large gradients of atmospheric refractivity. Atmos. Meas. Tech., 8, 12751285, https://doi.org/10.5194/amt-8-1275-2015.

    • Search Google Scholar
    • Export Citation
  • Durre, I., and X. Yin, 2008: Enhanced radiosonde data for studies of vertical structure. Bull. Amer. Meteor. Soc., 89, 12571262, https://doi.org/10.1175/1520-0477-89.9.1251.

    • Search Google Scholar
    • Export Citation
  • Durre, I., R. S. Vose, and D. B. Wuertz, 2006: Overview of the Integrated Global Radiosonde Archive. J. Climate, 19, 5368, https://doi.org/10.1175/JCLI3594.1.

    • Search Google Scholar
    • Export Citation
  • Feng, X., F. Xie, C. O. Ao, and R. A. Anthes, 2020: Ducting and biases of GPS radio occultation bending angle and refractivity in the moist lower troposphere. J. Atmos. Oceanic Technol., 37, 10131025, https://doi.org/10.1175/JTECH-D-19-0206.1.

    • Search Google Scholar
    • Export Citation
  • Gorbunov, M., V. Irisov, and C. Rocken, 2022: The influence of the signal-to-noise ratio upon radio occultation retrievals. Remote Sens., 14, 2742, https://doi.org/10.3390/rs14122742.

    • Search Google Scholar
    • Export Citation
  • Guo, P., Y.-H. Kuo, S. Sokolovskiy, and D. H. Lenschow, 2011: Estimating atmospheric boundary layer depth using COSMIC radio occultation data. J. Atmos. Sci., 68, 17031713, https://doi.org/10.1175/2011JAS3612.1.

    • Search Google Scholar
    • Export Citation
  • Ho, S.-P., L. Peng, R. A. Anthes, Y.-H. Kuo, and H.-C. Lin, 2015: Marine boundary layer heights and their longitudinal, diurnal, and interseasonal variability in the southeastern Pacific using COSMIC, CALIOP, and radiosonde data. J. Climate, 28, 28562872, https://doi.org/10.1175/JCLI-D-14-00238.1.

    • Search Google Scholar
    • Export Citation
  • Ho, S.-P., and Coauthors, 2020: Initial assessment of the COSMIC-2/FORMOSAT-7 neutral atmosphere data quality in NESDIS/STAR using in situ and satellite data. Remote Sens., 12, 4099, https://doi.org/10.3390/rs12244099.

    • Search Google Scholar
    • Export Citation
  • Kalmus, P., C. O. Ao, K.-N. Wang, M. P. Manzi, and J. Teixeira, 2022: A high-resolution planetary boundary layer height seasonal climatology from GNSS radio occultations. Remote Sens. Environ., 276, 113037, https://doi.org/10.1016/j.rse.2022.113037.

    • Search Google Scholar
    • Export Citation
  • Kaplan, E. D., 1996: Understanding GPS Principles and Applications. Artech House, 554 pp.

  • Kursinski, E. R., G. A. Hajj, J. T. Schofield, R. P. Linfield, and K. R. Hardy, 1997: Observing Earth’s atmosphere with radio occultation measurements using the global positioning system. J. Geophys. Res., 102, 23 42923 465, https://doi.org/10.1029/97JD01569.

    • Search Google Scholar
    • Export Citation
  • Kursinski, E. R., G. A. Hajj, S. S. Leroy, and B. Herman, 2000: The GPS radio occultation technique. Terr. Atmos. Oceanic Sci., 11, 53114, https://doi.org/10.3319/TAO.2000.11.1.53(COSMIC).

    • Search Google Scholar
    • Export Citation
  • Lopez, P., 2009: A 5-yr 40-km-resolution global climatology of superrefraction for ground-based weather radars. J. Appl. Meteor. Climatol., 48, 89110, https://doi.org/10.1175/2008JAMC1961.1.

    • Search Google Scholar
    • Export Citation
  • Ratnam, M. V., and S. G. Basha, 2010: A robust method to determine global distribution of atmospheric boundary layer top from COSMIC GPS RO measurements. Atmos. Sci. Lett., 11, 216222, https://doi.org/10.1002/asl.277.

    • Search Google Scholar
    • Export Citation
  • Rieckh, T., J. P. Sjoberg, and R. A. Anthes, 2021: The three-cornered hat method for estimating error variances of three or more atmospheric datasets. Part II: Evaluating radio occultation and radiosonde observations, global model forecasts, and reanalyses. J. Atmos. Oceanic Technol., 35, 17771796, https://doi.org/10.1175/JTECH-D-20-0209.1.

    • Search Google Scholar
    • Export Citation
  • Schreiner, W. S., and Coauthors, 2020: COSMIC-2 radio occultation constellation: First results. Geophys. Res. Lett., 47, e2019GL086841, https://doi.org/10.1029/2019GL086841.

    • Search Google Scholar
    • Export Citation
  • Sjoberg, J. P., R. A. Anthes, and T. Rieckh, 2021: The three-cornered hat method for estimating error variances of three or more datasets. Part I: Overview and evaluation. J. Atmos. Oceanic Technol., 38, 555572, https://doi.org/10.1175/JTECH-D-19-0217.1.

    • Search Google Scholar
    • Export Citation
  • Sokolovskiy, S. V., 2001: Tracking tropospheric radio occultation signals from low Earth orbit. Radio Sci., 36, 483498, https://doi.org/10.1029/1999RS002305.

    • Search Google Scholar
    • Export Citation
  • Sokolovskiy, S. V., 2003: Effect of superrefraction on inversions of radio occultation signals in the lower troposphere. Radio Sci., 38, 1058, https://doi.org/10.1029/2002RS002728.

    • Search Google Scholar
    • Export Citation
  • Sokolovskiy, S. V., C. Rocken, D. H. Lenschow, Y.-H. Kuo, R. A. Anthes, W. S. Schreiner, and D. C. Hunt, 2007: Observing the moist troposphere with radio occultation signals from COSMIC. Geophys. Res. Lett., 34, L18802, https://doi.org/10.1029/2007GL030458.

    • Search Google Scholar
    • Export Citation
  • Sokolovskiy, S. V., C. Rocken, W. Schreiner, D. Hunt, and J. Johnson, 2009: Postprocessing of L1 GPS radio occultation signals recorded in open-loop mode. Radio Sci., 44, RS2002, https://doi.org/10.1029/2008RS003907.

    • Search Google Scholar
    • Export Citation
  • Sokolovskiy, S. V., C. Rocken, W. Schreiner, and D. Hunt, 2010: On the uncertainty of radio occultation inversions in the lower troposphere. J. Geophys. Res., 115, D22111, https://doi.org/10.1029/2010JD014058.

    • Search Google Scholar
    • Export Citation
  • Sokolovskiy, S. V., W. Schreiner, Z. Zeng, D. Hunt, Y.-C. Lin, and Y.-H. Kuo, 2014: Observations, analysis, and modeling of deep radio occultation signals: Effects of tropospheric ducts and interfering signals. Radio Sci., 49, 954970, https://doi.org/10.1002/2014RS005436.

    • Search Google Scholar
    • Export Citation
  • Tien, J., and Coauthors, 2012: TriG: Next generation scalable spaceborne GNSS receiver. Proc. 2012 Int. Technical Meeting, Newport Beach, CA, Institute of Navigation, 882–914, https://www.ion.org/publications/abstract.cfm?articleID=9999.

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  • Fig. 1.

    Examples of the profiles y(z) for (left) elevated and (right) surface ducts. White and gray areas correspond to TP heights of the external and internal (trapped) rays; i.e., the gray areas represent the ducting layers.

  • Fig. 2.

    Examples of the (left) N(z) and (right) α(h) profiles (top) without SR and (bottom) with SR.

  • Fig. 3.

    (top) Profiles y(z) used for WO modeling of deep RO signals. Each profile y(z) is shifted in y with respect to the previous profile by 0.5 km for display purposes. (bottom) Amplitudes of deep RO signals modeled by using profiles y(z) from the top panel.

  • Fig. 4.

    Spectrogram of the (top) C2 RO and (bottom) power spectrum calculated in the window between −250 and −350 km HSL without (black) and with (red) smoothing using 1-Hz Hann’s window.

  • Fig. 5.

    Spectrograms of C2 RO signals.

  • Fig. 6.

    Spectrograms of C2 RO signals with different SSNR.

  • Fig. 7.

    Histograms of max. BAL for all C2 setting occultations (blue) and those with SSNR > 8 (red).

  • Fig. 8.

    (top) Global distribution of C2 occultations with detected SR for 2020.181–2022.180. Colors indicate different height intervals: <1 km (black); 1–2 km (green); 2–3 km (blue); >3 km (red). (bottom) Global distribution of the local SR detection rate.

  • Fig. 9.

    (left) Histograms of the SNR for all C2 setting occultations (blue) and for those with detected SR (red). (right) Global SR detection rate as function of the SNR.

  • Fig. 10.

    Global distribution of C2 setting occultations with SNR > 1750 V/V.

  • Fig. 11.

    (top),(middle) Global distributions of the C2 occultations with detected SR for winter and summer seasons, respectively. (bottom) Distribution of the difference of SR detection rates between winter and summer seasons.

  • Fig. 12.

    (top) Global distribution of the C2 occultations with SR detected by C2 (blue) and predicted by ECMWF (red). (bottom) Gridded map of the ECMWF SR prediction rate for occultations with SR detected from C2.

  • Fig. 13.

    (left) Examples of radiosonde profiles N(z) from Hawaii (black), collocated C2 RO (red), and ECMWF (green) profiles N(z). Each set of the profiles is shifted in z with respect to previous profile by 2 km for display purposes. (right) Spectrograms of deep signals for the corresponding occultations.

  • Fig. 14.

    Comparison of the heights of SR layers determined from C2 and radiosondes.

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