Diagnostics of Es Layer Scintillation Observations Using FS3/COSMIC Data: Dependence on Sampling Spatial Scale

The basic theory and experimental results of amplitude scintillation from GPS/GNSS radio occultation (RO) observations on sporadic E (Es) layers are reported in this study. Considering an Es layer to be not a “thin” irregularity slab on limb viewing, we characterized the corresponding electron density fluctuations as a power-law function and applied the Ryton approximation to simulate spatial spectrum of amplitude fluctuations. The scintillation index S4 and normalized signal amplitude standard deviation S2 are calculated depending on the sampling spatial scale. The theoretical results show that both S4 and S2 values become saturated when the sampling spatial scale is less than the first Fresnel zone (FFZ), and S4 and S2 values could be underestimated and approximately proportional to the logarithm of sampled spatial wave numbers up to the FFZ wave number. This was verified by experimental analyses using the 50 Hz and de-sampled FormoSat-3/Constellation Observing System for Meteorology, Ionosphere and Climate (FS3/COSMIC) GPS RO data in the cases of weak, moderate, and strong scintillations. The results show that the measured S2 and S4 values have a very high correlation coefficient of >0.97 and a ratio of ~0.5 under both complete and undersampling conditions, and complete S4 and S2 values can be derived by dividing the measured undersampling S4 and S2 values by a factor of 0.8 when using 1-Hz RO data.


Introduction
Sporadic E (Es) layers exhibit ionization enhancements in the E region at altitudes usually between 90 and 120 km [1]. A characteristic feature of Es layers is that they are thin layers of a few kilometers' thicknesses and a 10~1000 km horizontal extension. Over the past few decades, since the 1950s, Es layers have been extensively studied using a variety of instruments, including ionosondes, incoherent scatter radars and coherent scatter very high-frequency (VHF) radars operating from the ground and in rocket payloads with in-situ measurements. Several excellent reviews of the theories and observations of ionospheric Es layers have also been published [1][2][3][4]. It is generally believed that Es layer structure is affected by the plasma convergence effect of neutral wind shear through ion-neutral collision processes at mid and low latitudes [1]. Es layers could appear as non-uniform wave layers, a composition of irregular elongated clouds of intense ionization, or multiple layers, occurring simultaneously and separated by several kilometers within the lower E region [5]. Therefore, this gradient plasma instability can give rise to irregularities, with

Methods and Materials
When a radio wave transmitted by a satellite passes through the ionosphere, its wavefront will be distorted by irregular electron densities, i.e., plasma densities. After the radio wave has emerged from an irregularity slab, its phase front is randomly modulated. It is assumed that the deviation and the change in the refractive index over a spatial range of one wavelength and a temporal range of one wave period are slight. Under such a condition, the small-angle scattering of a layer of plasma density turbulence causes spatial and temporal fluctuations of radio wave intensity and phase, which is known as scintillation. In this study, we focus on the diagnostics of amplitude scintillation during GPS/GNSS RO observations carried out on Es layers.
Let us assume the geometry of an Es layer scintillation problem in GPS/GNSS RO observations, as shown in Figure 1. A region of irregular electron density structures is located from x = −L/2 to x = L/2, where L could be the cutoff length of a GPS-LEO limb viewing link through an Es layer. A time-harmonic electromagnetic wave is transmitted by a GNSS satellite located at x = −∞, and it is incident on an Es layer slab at x = −L/2 and received by an LEO satellite receiver at a specific coordinate (x, ρ t ), where ρ t = (y, z) is the transverse coordinate of wave propagation. The small-angle scattering of the scintillation phenomenon reduces the propagation equation of a wave field, where E = u(x, ρ t ) exp(−i k x), to the parabolic wave equation in u(x, ρ t ), as follows [12]. where -L/2 < x < L/2, and ∇ 2 t = ∂ 2 /∂ 2 y + ∂ 2 /∂ 2 z .
Equations (1) and (2) are of the parabolic type, whose solutions are determined uniquely by the boundary conditions at x = −L/2 and x = L/2, respectively. Notably, the irregularity slab can be characterized by its dielectric permittivity, as follows. and where ∆N(x, ρ t ) is the fluctuating part of electron density, and f p0 is the plasma frequency corresponding to the background electron density N 0 . f is the radio wave frequency, i.e., GPS L1-or L2-band frequency, and is much larger than f p0 . The other constants include the electronic charge e, the electronic mass m, and the free space permittivity ε 0 . Notably, ε 1 (x, ρ t ) is proportional to the fluctuating part of electron density ∆N(x, ρ t ) but is in inverse proportion to the square of radio wave frequency f. In this study, we do not consider the effects of wave propagation through an elongated Es layer cloud and/or multiple Es layers. It will be interesting to work on this in future investigations. , ) , where -L/2 < x < L/2, and (2) where x > L/2. Here, ε1(x, ρt) is the fluctuating part of dielectric permittivity ε(x, ρt). Equations (1) and (2) are of the parabolic type, whose solutions are determined uniquely by the boundary conditions at x = −L/2 and x = L/2, respectively. Notably, the irregularity slab can be characterized by its dielectric permittivity, as follows.
where ΔN(x, ρt) is the fluctuating part of electron density, and fp0 is the plasma frequency corresponding to the background electron density N0. f is the radio wave frequency, i.e., GPS L1-or L2-band frequency, and is much larger than fp0. The other constants include the electronic charge e, the electronic mass m, and the free space permittivity ε0. Notably, ε1(x, ρt) is proportional to the fluctuating part of electron density ΔN(x, ρt) but is in inverse proportion to the square of radio wave frequency f. In this study, we do not consider the effects of wave propagation through an elongated Es layer cloud and/or multiple Es layers. It will be interesting to work on this in future investigations. It is now generally accepted that ionospheric electron density irregularities can be characterized by a power-law function [3,[13][14][15][16]. To characterize the general power-law irregularity spectrum with spatial index p, [17] introduced a fairly general correlation spectrum, as follows. It is now generally accepted that ionospheric electron density irregularities can be characterized by a power-law function [3,[13][14][15][16]. To characterize the general power-law irregularity spectrum with spatial index p, [17] introduced a fairly general correlation spectrum, as follows.
where κ 0 is the wave number of the outer scale l 0 (=2π/κ 0 ) in an irregularity slab, and Γ( ) is the Gamma function. The outer scale is defined as an intrinsic property of the instability that causes electron density and plasma irregularities. The power-law continuum encompasses at least three orders of magnitude in the spatial scale size [13]. Notably, the power-law spectrum is expected to be valid when its spatial scale is much less than the outer scale. In practice, it conforms to a GPS-LEO limb viewing an Es layer with a 10~1000 km horizontal extension.
Historically, the first ionospheric scintillation theory was based on the idea of wave diffraction from a phase-changing screen, and named phase screen theory [14,15,18]. This assumes that, as the radio waves propagate through an irregularity slab, to the first order, only the phase is affected by the random fluctuations in refractive index. The phase screen theory can be applied for the weak scintillations usually induced by a "thin" irregularity slab. Ref. [11] derived an Es layer thickness of 1.2 km at a peak distribution based on the 50 Hz "atmPhs" FS3/COSMIC data. Considering such an Es layer distributed at an altitude of 100 km and along the Earth's curvature, the cutoff length of a GPS-LEO limb's viewing link through the Es layer is approximately 160 km, taken as the value of L in Figure 1. Therefore, we consider here that the Es layer in limb viewing is not a thin irregularity slab. The effects of scattering on the field amplitude inside a limb viewing irregularity slab must be included in the treatment of the Es layer scintillation phenomenon. Earlier investigations [13,19,20] derived the Ryton approximation for the parabolic wave Equations (1) and (2), and concluded that the Ryton approximation can be applied to weak and even strong ionospheric scintillations. Using the Ryton approximation, a wave field can be presented in terms of the fluctuation of a complex phase ψ(x, ρ t ), shown as follows: where u 0 is the average field of u(x, ρ t ) and is a reference field with definite amplitude and phase. χ(x, ρ t ) and S 1 (x, ρ t ) are referred to as the log-amplitude and the phase departure of a wave field, respectively. The general Ryton solution of the parabolic wave Equations (1) and (2) was obtained as follows.
where x > L/2. Notably, ψ 0 (−L/2, ρ t ) = ln u(−L/2, ρ t ) corresponds to the incident wave of the irregular slab, and ψ(L/2, ρ t ) = ln u(L/2, ρ t ) is its boundary condition at x = L/2. In this study, we develop a plane incident wave with unity amplitude to be incident on an Es layer slab in a limb-viewing direction. The small scattering angle assumption means that the mean values of the log-amplitude and phase departures of the wave field through an Es layer slab are approximately zero, i.e., <χ>~<S 1 >~0. The power spectra for χ and S 1 are given, respectively, by [21], as follows.
As shown in Figure 1, x is the distance from the center of an irregularity slab to the receiver on an LEO satellite, and is equal to D (~3500 km) for a GPS/GNSS RO observation on Es layers. κ t is the wave number of a transverse spatial scale ρ t . Notably, the spatial power spectrum of the log amplitude departure of wave field is controlled by two competing factors: a Fresnel filter function, shown as the expression in the square brackets in Equation (9), and the power-law function of dielectric permittivity or electron density fluctuation. Figure  corresponding to the GPS L1-band signal and a distance D of approximately 3500 km from the Es layer slab to the LEO satellite during a GPS/GNSS RO observation. The oscillatory character of the filter function is known as the Fresnel oscillation [3]. In Figure 2, the spatial power spectrum of the simulated log amplitude departure of wave field from Equation (9) is also shown at a spatial index p of 4. The spatial spectrum is generally of a power-law type with an order of p that decays as κ t increases. It has a maximum around κ t = κ F , corresponding to the first maximum of the Fresnel filter function. This is consistent with an irregularity size in the order of FFZ, which is most effective in causing amplitude scintillation.
Equation (9), and the power-law function of dielectric permittivity or ele fluctuation. Figure 2 shows the Fresnel filter as a function of normalized w κt/κF, where κF (=2π/DF) is the wave number in the first Fresnel zone (FFZ =~0.8 km) corresponding to the GPS L1-band signal and a distance D of a 3500 km from the Es layer slab to the LEO satellite during a GPS/GNSS RO The oscillatory character of the filter function is known as the Fresnel osci Figure 2, the spatial power spectrum of the simulated log amplitude depa field from Equation (9) is also shown at a spatial index p of 4. The spatia generally of a power-law type with an order of p that decays as κt increases. imum around κt = κF, corresponding to the first maximum of the Fresnel f This is consistent with an irregularity size in the order of FFZ, which is mo causing amplitude scintillation.  Applying the Wiener-Khinchin theorem [12] and Equations (3)- (5) and (9), the correlation of the log amplitude departure of the wave field can be derived by its spatial power spectrum, as follows.
As a measure of amplitude scintillation, the scintillation index S 4 is most often used and is defined as a normalized variance of signal power intensity I, as follows.
Applying Ryton approximation, it is natural to suppose that the logarithmic amplitude has a normal distribution with a mean of zero because of the small angle scattering through an irregularity slab. Therefore, <χ> and <χ 3 > are approximately zero, and can be neglected, and <χ 4 > = 3<χ 2 > 2 . Considering the moderate or strong scintillations as radio waves propagate through an Es layer's irregularity slab, we include the first four orders of the Taylor series in our exponential functions and obtain Notably, the square of the S 4 value is a function of the log amplitude departure correlation of wave field, i.e., <χ 2 >. Equation (13) indicates that, for weak scintillations, i.e., χ 2 << 1, S 2 4 ∼ 4 χ 2 , and the S 4 value is approximately equal to double the square root of the integral of the log amplitude departure spectra in the spatial scale.
In practical situations of GPS/GNSS RO observations, the radio signals are transmitted from a GPS/GNSS satellite and received by an LEO satellite at an altitude of approximately 100 km. The projected speed v z of the LEO satellite in the azimuth direction, i.e., the z direction in Figure 1, is approximately 3.2 km/s, which is much faster than the projected speed of the GPS/GNSS satellite and also the drift speed of the Es layer irregularities along the z direction. The temporal variations in GPS/GNSS signals received by an LEO receiver can be considered as the result of radio beam scanning over the z-direction variations of frozen Es layer irregularities. Therefore, the sampling frequency f s on GPS/GNSS RO observations can be transferred to the wave number κ s of the sampling spatial scale as Figure 3 shows three profiles of simulated S 4 values, where the saturated S 4 values are equal to 0.8, 0.5, and 0.2 separately, as functions of a normalized sampling wave number κ s /κ F . Three simulated S 4 profiles present results for different cases of strong, moderate, and weak scintillations, separately. Notably, when the wave number κ s of the sampling spatial scale is larger than the wave number κ F at the FFZ, saturation of the scintillation index S 4 becomes apparent because of the power-law spectrum of electron density fluctuation, in the order of −p. We could conclude that the S 4 values are "complete" when κ s is larger than κ F , i.e., the sampling spatial scale is less than the FFZ. However, for values of κ s less than κ F , the S 4 values decrease as the logarithm of κ s decreases. This means that the derived S 4 values could be underestimated when the sampling spatial scale is larger than the FFZ, i.e., in "undersampling" conditions.
Another parameter used to identify the Es layer from GPS/GNSS RO observations [8][9][10][11] is the normalized signal amplitude standard deviation σ A , which would be equivalent to the S 2 index [22]. We apply Ryton approximation to derive a wave amplitude u a = u 0 exp(χ). Assuming that the logarithmic amplitude χ of the wave field through an Es layer slab has a normal distribution with a mean of zero, we define S 2 as follows: Being similar to the scintillation index S 4 , the square of the normalized signal amplitude standard deviation S 2 is a function of the log amplitude departure correlation of wave field, i.e., <χ 2 >. For weak scintillations, i.e., χ 2 << 1, S 2 2 ∼ χ 2 , and the S 2 value is approximately half that of S 4 and equal to the square root of the integral of the log amplitude departure spectra in the spatial scale. Figure 3 shows another three profiles of simulated S 2 values, corresponding to three complete S 4 values equal to 0.8, 0.5, and 0.2 separately, as functions of the normalized sampling wave number κ s /κ F . Notably, being similar to S 4 , the normalized signal amplitude standard deviation S 2 values are saturated, and are complete when κ s is larger than κ F . However, for the values of κ s less than κ F , the S 2 values decrease as the logarithm of κ s is decreasing. This also means that the derived S 2 values are underestimated when the sampling spatial scale is larger than the FFZ. Figure 3 also shows three profiles of simulated S 2 /S 4 values, corresponding to three complete S 4 values equal to 0.8, 0.5, and 0.2 separately, as functions of the normalized sampling wave number κ s /κ F . Notably, the three profiles of simulated S 2 /S 4 ratios are mostly overlapped, and all simulated S 2 values are approximately half of the simulated S 4 values for different complete S 4 values, whenever κ s is larger or less than κ F . Another parameter used to identify the Es layer from GPS/GNSS RO [8][9][10][11] is the normalized signal amplitude standard deviation σA, which wou lent to the S2 index [22]. We apply Ryton approximation to derive a wave a u0 exp(χ). Assuming that the logarithmic amplitude χ of the wave field th layer slab has a normal distribution with a mean of zero, we define S2 as fol ( )

Results
The FS3/COSMIC is a joint Taiwan-U.S. mission consisting of six identical LEO microsatellites from mid-2006 to 2019. Each spacecraft carried a GPS receiver developed by the Jet Propulsion Laboratory (JPL) and utilized four GPS antennas: two limb viewing occultation antennas remotely sensing the atmosphere and low ionosphere at 50 Hz from the Earth surface up to approximately 125 km, and two slant observing antennas for precise orbit determination (POD) and remotely sensing the ionosphere at 1 Hz [23]. Therefore, the use of both limb viewing and POD antennas could produce simultaneous occultation observations consisting of two sets of 50 Hz and 1 Hz limb viewing links separately with different altitude resolutions of approximately 0.064 and 3.2 km at E-region altitudes. Both the GPS RO measurements include GPS and LEO satellite orbits, carrier excess phase and carrier signal-noise ratio (SNR) amplitudes for dual GPS L-band signals (f 1 = 1575.42 MHz and f 2 = 1227.60 MHz) at 50 Hz and 1 Hz, i.e., the "atmPhs" and "ionPhs" data, respectively. Notably, the FS3/COSMIC also provides 1 Hz amplitude scintillation S 4 data at the L1 band as the "scnLv1" data. The amplitude scintillation measurements are performed by an on-board algorithm under a Gaussian distribution assumption applied to the raw 50 Hz L1-band SNR amplitude data (Syndergaard 2006, COSMIC S4 data, from the website: http://cdaac-www.cosmic.ucar.edu/cdaac/doc/documents/s4_description.pdf, accessed on 22 August 2021). The on-board S 4 measurements are not derived by the scintillation index definition shown in Equation (12), and are thus not discussed in this study.
As described in the last section, the FFZ D F of a GPS RO observation on the Es layer is approximately 0.8 km. This means that a 50 Hz limb viewing setting can be used to determine the complete scintillation index S 4 and S 2 values upon Es layer observation, but a 1 Hz limb viewing cannot. Figure 4 shows one example of FS3/COSMIC "atmPhs" (50-Hz) RO observation with an Es event recorded on 1 January 2007. As shown, significant amplitude fluctuations from L1-band signals were observed at E-region altitudes, and the corresponding S 4 and S 2 altitudinal profiles can be derived by Equations (12) and (14) separately applied to a sliding window of 4 s amplitude data. However, the L2-band signals are much weaker and have insufficient sensitivity to derive reliable S 4 and S 2 values to be shown. Notably, there is strong scintillation in the Es event, and the peak S 4 (S 2 ) value is 0.82 (0.39) for the GPS L1-band signals.
tation observations consisting of two sets of 50 Hz and 1 Hz limb viewing links separately with different altitude resolutions of approximately 0.064 and 3.2 km at E-region altitudes. Both the GPS RO measurements include GPS and LEO satellite orbits, carrier excess phase and carrier signal-noise ratio (SNR) amplitudes for dual GPS L-band signals (f1 = 1575.42 MHz and f2 = 1227.60 MHz) at 50 Hz and 1 Hz, i.e., the "atmPhs" and "ionPhs" data, respectively. Notably, the FS3/COSMIC also provides 1 Hz amplitude scintillation S4 data at the L1 band as the "scnLv1" data. The amplitude scintillation measurements are performed by an on-board algorithm under a Gaussian distribution assumption applied to the raw 50 Hz L1-band SNR amplitude data (Syndergaard 2006, COSMIC S4 data, from the website: http://cdaac-www.cosmic.ucar.edu/cdaac/doc/documents/s4_description.pdf, accessed on 22 August 2021). The on-board S4 measurements are not derived by the scintillation index definition shown in Equation (12), and are thus not discussed in this study.
As described in the last section, the FFZ DF of a GPS RO observation on the Es layer is approximately 0.8 km. This means that a 50 Hz limb viewing setting can be used to determine the complete scintillation index S4 and S2 values upon Es layer observation, but a 1 Hz limb viewing cannot. Figure 4 shows one example of FS3/COSMIC "atmPhs" (50-Hz) RO observation with an Es event recorded on 1 January 2007. As shown, significant amplitude fluctuations from L1-band signals were observed at E-region altitudes, and the corresponding S4 and S2 altitudinal profiles can be derived by Equations (12) and (14) separately applied to a sliding window of 4 s amplitude data. However, the L2-band signals are much weaker and have insufficient sensitivity to derive reliable S4 and S2 values to be shown. Notably, there is strong scintillation in the Es event, and the peak S4 (S2) value is 0.82 (0.39) for the GPS L1-band signals.  As mentioned, 50 Hz GPS-LEO limb viewing has a vertical sampling spatial scale of approximately 0.064 km, which is over one order less than the FFZ (~0.8 km) of the Es layer RO observation. We can perform de-sampling to obtain more sets of amplitude data at different sampling frequencies of 50/n Hz, where n = 1, 2, . . . , 100. The resulting one hundred sets of amplitude data can be used to derive different altitudinal S 4 and S 4 profiles using a 4 s sliding window, and obtain corresponding peak S 4 and S 2 values at sampling frequencies from 0.5 to 50 Hz, which are approximately equal to normalized sampling spatial scales κ s /κ F of 0.2 to 20. Figure 5 shows the measured peaks S 4 , S 2 , and their ratio S 2 /S 4 values as a function of κ s /κ F based on the FS3/COSMIC RO amplitude data, as used in Figure 4. Notably, the measured peak S 4 and S 2 profiles in κ s /κ F fluctuate but are mainly similar to the simulated S 4 and S 2 profiles in the strong scintillation case of a complete S 4 value of 0.8, as shown in Figure 3. The S 4 and S 2 values are saturated tõ 0.8 and~0.4, respectively, when κ s is larger than κ F , and, when κ s is less than κ F , the S 4 Remote Sens. 2021, 13, 3732 9 of 14 and S 2 values decrease in general as the logarithm of κ s decreases. Figure 5 also shows the measured S 2 /S 4 values to be more or less than 0.5 at different normalized sampling wave numbers κ s /κ F from 0.2 to 20, i.e., the measured S 2 values are also approximately half of the measured S 4 values whenever κ s is larger or less than κ F . using a 4 s sliding window, and obtain corresponding peak S4 and S2 values at sampling frequencies from 0.5 to 50 Hz, which are approximately equal to normalized sampling spatial scales κs/κF of 0.2 to 20. Figure 5 shows the measured peaks S4, S2, and their ratio S2/S4 values as a function of κs/κF based on the FS3/COSMIC RO amplitude data, as used in Figure 4. Notably, the measured peak S4 and S2 profiles in κs/κF fluctuate but are mainly similar to the simulated S4 and S2 profiles in the strong scintillation case of a complete S4 value of 0.8, as shown in Figure 3. The S4 and S2 values are saturated to ~0.8 and ~0.4, respectively, when κs is larger than κF, and, when κs is less than κF, the S4 and S2 values decrease in general as the logarithm of κs decreases. Figure 5 also shows the measured S2/S4 values to be more or less than 0.5 at different normalized sampling wave numbers κs/κF from 0.2 to 20, i.e., the measured S2 values are also approximately half of the measured S4 values whenever κs is larger or less than κF. Ref. [11] determined that around 30% of those observations were Es events, based on the criteria of its peak S2 value (>0.2) and altitude (>80 km and less than the top altitude of each 50 Hz GPS RO observation). The local Es event occurrence rate could approach 70% Ref. [11] determined that around 30% of those observations were Es events, based on the criteria of its peak S 2 value (>0.2) and altitude (>80 km and less than the top altitude of each 50 Hz GPS RO observation). The local Es event occurrence rate could approach 70% during hemispheric summer seasons. To determine the dependence on the sampling spatial scale of Es layer scintillation index determinations, the top, middle, and bottom panels of Figure 6 show the peak S 4 and S 2 , and their S 2 /S 4 ratio values, as a function of the normalized sampling wave number for three cases of strong scintillations (complete S 4~0 .8(±0.01) and 35 observations), moderate scintillations (complete S 4~0 .5(±0.01) and 57 observations), and weak scintillations (complete S 4~0 .2(±0.01) and 143 observations), respectively. The experimental 50 Hz FS3/COSMIC data were obtained on 1 January and 1 July 2007, and comprise 4750 GPS RO observations totally. Each RO observation was de-sampled into one hundred sets of amplitude profiles used to derived corresponding peak S 4 , S 2 , and S 2 /S 4 values with different sampling frequencies from 0.5 to 50 Hz, i.e., normalized sampling wave numbers κ s /κ F from approximately 0.2 to 20. Generally, as shown in Figure 6, there was a greater spread of peak S 4 , S 2 , and S 2 /S 4 values when sampling wave number was less. This is reasonable, because the mean error increases as its sampled number decreases, i.e., the sampling frequency decreases, and the same 4 s sampling window is used. However, the dependence on sampling spatial scale of the S 4 , S 2 , and S 2 /S 4 values is similar to that of the simulated S 4 , S 2 , and S 2 /S 4 profiles, shown in Figure 3, regardless of whether the scintillation cases are strong (complete S 4~0 .8), moderate (complete S 4~0 .5), or weak (complete S 4~0 .2). S2/S4 values with different sampling frequencies from 0.5 to 50 Hz, i.e., normalized sampling wave numbers κs/κF from approximately 0.2 to 20. Generally, as shown in Figure 6, there was a greater spread of peak S4, S2, and S2/S4 values when sampling wave number was less. This is reasonable, because the mean error increases as its sampled number decreases, i.e., the sampling frequency decreases, and the same 4 s sampling window is used. However, the dependence on sampling spatial scale of the S4, S2, and S2/S4 values is similar to that of the simulated S4, S2, and S2/S4 profiles, shown in Figure 3, regardless of whether the scintillation cases are strong (complete S4~0. 8

Discussions
Following the FS3/COSMIC mission, a follow-on program called FormoSat-7 (FS7)/COSMIC2 is in progress, with satellites launched on 25 June 2019. Similar to the FS3/COSMIC, the FS7/COSMIC2 is a six-LEO-satellite constellation mission orbiting at 24°

Discussions
Following the FS3/COSMIC mission, a follow-on program called FormoSat-7 (FS7)/ COSMIC2 is in progress, with satellites launched on 25 June 2019. Similar to the FS3/COSMIC, the FS7/COSMIC2 is a six-LEO-satellite constellation mission orbiting at 24 • inclination and 520~720-km altitude. It enhances GNSS receiver capability to accommodate signals from GPS and GLONASS satellites, and can provide more than 5000 RO observations per day within the region between geographic latitudes of ±40 • . The large dataset from the FS7/COSMIC2 program enables precise specifications and statistical studies on equatorial and low-latitude ionospheric electron densities and irregularities. Notably, each LEO satellite of FS7/COSMIC2 also carries two limb viewing occultation antennas remotely sensing the atmosphere at 50 Hz, and another two slant observing antennas for POD and remotely sensing the ionosphere at 1 Hz. However, the altitude data of GPS/GNSS RO 50 Hz "atmPhs" are distributed between the Earth's surface and a 60 km altitude, and do not include the E region range because of mission data storage constraints and satellite downlink bandwidth limitations [24]. This means that Es layer observations by FS7/COSMIC2 can be performed using 1 Hz "ionPhs" data only, and the S 4 and S 2 values derived by Equations (12) and (14) are underestimated under undersampling conditions and are not complete. Therefore, it is necessary to find a method for solving the underestimated scintillation indexes (S 4 and S 2 ).
To illustrate and compare the complete and undersampled scintillation observations, Figure 7 shows two scatter plots and the corresponding least-squares fitting lines for the measured peak S 4 and S 2 values using 1 Hz FS3/COSMIC data versus the peak S 4 and S 2 values measured using 50 Hz FS3/COSMIC data, respectively, on E and Es layer observations. The experimental FS3/COSMIC data were recorded on 1 January and 1 July 2007, and the 1 Hz amplitudes have been de-sampled from the 50 Hz amplitudes of "atmPhs" data. The results show that the peak S 4 (S 2 ) measurements of E and Es layer observations using 50 Hz and de-sampled 1 Hz amplitude data have very high correlation coefficients of 0.95 (0.97). Meanwhile, from the least-squares fitting line equations shown in Figure 7, we see that the underestimated S 4 (S 2 ) values derived from the 1 Hz data have a ratio of 0.77 (0.84) to the complete S 4 (S 2 ) values derived from 50 Hz data. The strong correlations prove the reliability of complete S 4 and S 2 determinations using undersampled S 4 and S 2 measurements and corresponding sampling frequency or spatial scale information. Notably, the normalized spatial wave number κ s /κ F is approximately 0.4 at a limb-viewing sampling rate of 1 Hz on the E or Es layer RO observations. The resulting ratio of 0.77 (0.84) is close to the ratio of~0.8 for the simulated S 4 (S 2 ) value (at a κ s /κ F of 0.4, as shown in Figure 3) to the complete S 4 (S 2 ) value, regardless of whether the scintillation cases are strong (complete S 4~0 .8), moderate (complete S 4~0 .5), or weak (complete S 4~0 .2). Furthermore, as shown in Figure 7, the measured complete S 2 values are distributed from 0.07 to 0.5, approximately half of the measured complete S 4 distribution range from 0.15 to 1.0. Figure 8 shows another two scatter plots and the corresponding least-squares fitting lines for the peak S 2 values versus peak S 4 values measured using 50 Hz and de-sampled 1 Hz FS3/COSMIC "atmPhs" data separately in E and Es layer observations. The experimental data are the same as those used in Figure 7. The results show that the peak S 2 versus S 4 measurements for the E and Es layers have very high correlation coefficients of 0.98 and 0.97 using 50 Hz and de-sampled 1 Hz amplitude data, respectively. Meanwhile, from the least-squares fitting line equations shown in Figure 8, we see that the slopes of both fitting lines, i.e., the mean ratios of peak S 2 to S 4 values, are approximately equal to 0.5, which is the same as the simulated S 2 /S 4 values shown in Figure 3. This means that the scintillation index S 4 and the normalized signal amplitude standard deviation S 2 measurements have high correlations with the determined ionospheric scintillation level.    Scatter plot for the peak S4 (S2) values measured using 1 Hz data versus the peak S4 (S2) values measured using 50 Hz data is shown in blue (red). Two corresponding least-squares fitting lines, line equations, and the obtained correlative coefficients are also shown. The experimental FS3/COSMIC data were obtained on 1 January and 1 July 2007. Figure 8. Scatter plot for the peak S2 values measured using 50 Hz (1-Hz) data versus the peak S4 values measured using 50 Hz (1-Hz) data is shown in blue (red). Two corresponding least-squares Figure 8. Scatter plot for the peak S 2 values measured using 50 Hz (1-Hz) data versus the peak S 4 values measured using 50 Hz (1-Hz) data is shown in blue (red). Two corresponding least-squares fitting lines, line equations, and the obtained correlative coefficients are also shown. As in Figure 7, the experimental FS3/COSMIC data were obtained on 1 January and 1 July 2007.

Conclusions
The diffraction theory of wave propagation in a stationary random medium establishes a connection between the spatial spectrum of ionospheric electron density fluctuations and the power spectrum of observed signal intensities, and therefore the amplitude scintillations. We have reported in this paper on the relevant theories and FS3/COSMIC experiments of radio wave scintillations of Es layer RO observations. The theoretical results show that applying Ryton approximation to radio wave propagations through an Es layer irregularity slab can provide reasonable descriptions of weak-, moderate-, and strongamplitude scintillations. We conclude that a GPS RO scintillation observation is complete when the sampling spatial scale is less than the FFZ, otherwise it is a case of undersampling. It is shown that the simulated results are consistent with interpretations of FS3/COSMIC RO scintillation index estimates in terms of the different sampling frequencies. It is useful to clarify the dependence of the determination of the scintillation index on the sampling spatial scale, and find a method for solving complete scintillation indices (S 4 and S 2 ) under undersampling conditions. Institutional Review Board Statement: Not applicable.