Radio Science

In situ irregularity identification and scintillation estimation using wavelets and CINDI on C/NOFS

Authors


Abstract

[1] Wavelets are a time-domain method that is able to extract both time and frequency information from a signal. The Morlet wavelet is used here to characterize the magnitude of ionospheric irregularities using measurements of the total ion density from the Coupled Ion Neutral Dynamics Investigation package onboard the Communications/Navigation Outage Forecasting System spacecraft. The power in ionospheric irregularities at scale sizes less than 128 km is used to generate an irregularity amplitude index. This index is used with a phase screen analysis to form an estimate of scintillation at the satellite location. The temporal information retained in a wavelet analysis also allows for an accurate power spectrum calculation even when used on short segments of data which is useful for real-time processing of irregularity detection onboard a satellite or for analyzing the long data sets produced by a satellite. To validate the process, a comparison of the in situ scintillation estimate and Scintillation Network and Decision Aid measurements of the S4 index is presented.

1 Introduction

[2] The formation of equatorial irregularities in the ionosphere can cause communications outages by disrupting waveforms as a signal travels through the irregularity. Sharp changes in ion density within and around an irregularity cause communication signals to refract as they travel through the ionosphere, distorting the waveform recorded by an observer. The Communications/Navigation Outage Forecasting System (C/NOFS) satellite carries a suite of instruments to measure a number of expected drivers for irregularity formation.

[3] The purpose of this paper is to establish a methodology for detecting plasma irregularities from in situ satellite data which will reveal instability regions that coincide with the observations of scintillation on the ground. For this purpose, we have calculated a “quasi S4 index” for which the appearance above a given threshold can be compared with the true S4 index measured by a ground station. The S4 index [Briggs and Parkin, 1963] measures the root mean square fluctuations in signal intensity, normalized by the average signal intensity:

display math(1)

It is not our intent here to compare the absolute magnitude of the derived S4 index with true values observed on the ground. Rather, we seek to establish that irregularity occurrence regions identified with an elevated “quasi S4 index” coincide with those observed on the ground.

[4] Perturbations to in situ measurements of total ion density by the Coupled Ion and Neutral Dynamics Investigation (CINDI) along the satellite track are extracted using wavelets. Wavelets are a time-domain technique useful for signal analysis and many other applications. Wavelets have been used to analyze amplitude scintillation measurements by Wernik [1997] and Materassi and Mitchell [2007] as well as to study turbulence in the high latitude ionosphere [Lagoutte et al., 1992]. A review of wavelets for geophysical applications can be found in Kumar and Foufoula-Georgiou.

[5] The Morlet wavelet is chosen here due to its construction using the sines and cosines present in Fourier analysis. This construction gives the Morlet wavelet a limited bandwidth that can be used to isolate variations at a particular scale size. Further, the power spectrum as well as the waveforms isolated for a particular scale size may be interpreted similarly to those produced by Fourier analysis. Thus, the results from the Morlet wavelet may be compatible with analyses and intuition based upon Fourier analysis of ionospheric irregularities.

[6] An advantage of wavelets over the Fourier method is that time information is retained, providing greater specificity on when irregularities occur. The retained time information also offers some practical benefits in producing an accurate characterization of density variations in a long satellite data set. Using wavelets, density measurements may be analyzed using very short segments of the data set in real time but produce the same result as if the whole data set was analyzed at once. While the frequency resolution of the Fourier transform is dependent upon the overall length of the data set analyzed, this does not apply to wavelets. Wavelets are time-limited signals thus the convolution of a wavelet and a signal that comprises the wavelet transform is only nonzero within the wavelet. Data sets longer than the wavelet thus have no impact on the quality of the results. While the use of a short data segment will introduce errors in the wavelet transform near the edges of the segment, these locations are known and may be ignored. By using a series of overlapping segments along the satellite track, a complete analysis of the data set may be constructed using only wavelet transforms unaffected by edge effects, producing an accurate power spectrum for all density measurements.

[7] To validate the wavelet-extracted density perturbations along the satellite track, the perturbations are used to estimate S4 following the in situ analysis performed by Basu et al. [1976] for the equatorial OGO-6 satellite. Basu et al. [1976] implements the phase screen analysis by Rufenach[1975, 1976] which presumes that the irregularity follows a power law with scale size. This power law is used to relate the magnitude of irregularities at sizes measurable by the satellite to scale sizes that impact radio waves of interest. Wernik et al. [2007] use Rino [1979a] and incorporates the International Reference Ionosphere (IRI) model to account for the location of the peak in ion density relative to the satellite to identify polar scintillation using Dynamics Explorer data. The work by Rino [1979a] requires a turbulence parameter which may be determined using the measured spectral index as well as the power within the irregularity at one scale size. These parameters may also be determined using the presented method.

[8] Scaling the observed densities to those observed at the F peak is not attempted here and is reserved for future work. Without knowledge of the full density altitude profile, the presented in situ-estimated S4 index is expected to under report the scintillation intensity and should not be used in an absolute sense. Further, in situ measurements reported by Singh and Szuszczewicz [1984] also demonstrate deviations from a power law in the spectral index based upon scale size, which are not included here. Using a single power law scaling may overestimate scintillation at smaller length scales [Retterer, 2010]. For this paper, a constant spectral index is assumed.

[9] The in situ estimate is compared against measurements of S4 performed by the Scintillation Network and Decision Aid (SCINDA) [Groves et al., 1997; Carrano and Groves, 2006], a network of ground-based receivers that monitor scintillations at the UHF and L-Band frequencies caused by electron density irregularities in the equatorial ionosphere. A comparison of the in situ S4 and SCINDA 5 min averages of S4 at 250 MHz is presented and used to establish that wavelets are effective at identifying irregularities from total ion density measurements and that the wavelet-determined variance associated with the irregularity may be used as input for scintillation estimation methods.

2 Data Analysis Technique

[10] The Morlet wavelet is used to extract changes in the CINDI measurements of total ion density onboard C/NOFS. The C/NOFS satellite is in a 13° inclination elliptical orbit with perigee near 400 km and apogee near 850 km. Measurements of total ion density, the ion drift velocity, ion temperature, and the constituent ion composition [Heelis and Hanson, 1998] are performed every 1/2 s or about every 4 km along the satellite track. The wavelet-derived characterization of irregularities from the in situ density measurements will be used to drive a scintillation estimate that is compared to SCINDA scintillation observations.

2.1 Morlet Wavelet

[11] The Morlet wavelet

display math(2)

is the combination of a Fourier term windowed by a Gaussian function, localizing the wavelet in a nondimensional “time” parameter η [Torrence and Compo, 1997]. The nondimensional frequency ω is taken to be six to satisfy the admissibility condition for wavelets [Farge, 1992].

[12] This wavelet shares a similar interpretation with the standard Fourier transform but provides greater specificity on where oscillations occur in a time series. A Fourier decomposition of a time series characterizes frequencies present throughout the entire time series. The continuous wavelet transform is calculated by performing a convolution of the wavelet with a time series for each point in the series. Though we are dealing with discrete signals, this particular wavelet transform is called continuous as it allows for wavelets of any scale size. At each point, the Morlet wavelet is centered upon the sample and a convolution is performed. The size of the Morlet wavelet is varied to investigate the presence of oscillations at different sizes while retaining the same functional form. Small-scale sizes involve a small number of samples around the desired time while larger scales necessarily involve a larger range of samples. The Morlet wavelet has a total length inline image [Torrence and Compo, 1997], where s is the scale size of the wavelet.

[13] The trade-off between frequency resolution and the temporal resolution can be chosen by varying ω. Increasing ω produces more sinusoidal periods within the Gaussian time window (decreasing time resolution), narrowing the bandwidth of the Morlet wavelet (providing greater frequency resolution). The relationship between the Morlet wavelet scale size s and the equivalent Fourier wavelength λ is thus ω dependent [Torrence and Compo, 1997],

display math(3)

For ω=6 used here, λ=1.03s. The trade-off between frequency and temporal resolution for the Morlet wavelet implies that it will not resolve pure sinusoids as sharply as the Fourier transform [Torrence and Compo, 1997]. Since irregularities contain sharp changes in density, this results in a broad spatial spectrum; thus, the increased bandwidth of a Morlet wavelet at a particular scale is not a significant disadvantage for irregularity identification or scintillation estimation.

[14] Note that the continuous wavelet transform allows any scale size to be specified. Though constructed using sinusoids, the Morlet wavelet is limited in time by a Gaussian; thus, the Morlet wavelet is overcomplete and generally nonorthogonal. A dyadic set of scale sizes should be used if an orthonormal set is desired. By applying a range of scale size wavelets, a power spectrum of the signal similar to that produced by Fourier analysis is generated.

[15] In practical terms, by retaining time information, wavelet transforms can be applied to short time segments of a longer signal and avoid the accuracy penalty from using short time signals. Consider a moving centered window of length n viewing a long time signal of length L, subdividing the signal into smaller segments that are analyzed with wavelets. When each segment is analyzed, the convolution of samples near the segment edges will be distorted. Since the wavelet transform is centered upon the time to be investigated, for samples near the segment edge, a portion of the wavelet will extend past the edge of the data segment, preventing an accurate characterization of the signal at that scale size and time. The choice of wavelet and scale size determines the number of samples from the edge before an accurate wavelet transform is possible. In general, m samples will have accurate convolutions over the scale size range of interest where m<n. The longer the data segment, the larger the maximum scale size that may be investigated. Using the total length of the Morlet wavelet of inline image, the largest wavelet free from edge effects has scale size smaxnδt/2.82, where δt is the distance between samples, or roughly three periods in the data segment of length n. For oscillations at smax, only one sample per data segment will get an accurate characterization. For a fixed scale size, an increase in n leads to a larger fraction of samples that may be characterized accurately.

Since the influence of the segment edge on a wavelet is well specified, portions of a given data segment that are impacted by the edge may be ignored. If the moving window of length n only moves by m samples each iteration, then the data segments characterized by wavelets will overlap. The m samples calculated without error and extracted each iteration are sufficient to characterize the whole time signal with sufficient iterations. This method is particularly useful for real-time applications such as onboard satellite processing or analyzing very long time signals. Wavelet software provided by Torrence and Compo [1997] is integrated into this method to produce the results here.

The properties of the Morlet wavelet can be used to determine scale-limited variances in ion density, reconstruct perturbation waveforms in density as well as investigate the scaling properties of irregularities with size. Here we concentrate on determining the variance in ion density as a means of identifying irregularities and use that variance as an input to a scintillation model.

2.2 Characterizing Irregularities

[16] An example wavelet decomposition is shown in Figure 1. The black line in the top panel is the total ion density measured by the Retarding Potential Analyzer (RPA) on C/NOFS as part of CINDI. This density is first normalized by 104 cm−3, and the mean value of the time series is subtracted; thus, the waveform shown is the deviation from the mean. The scale size of the subtracted mean is larger than the scale sizes of density perturbations to be investigated. The Morlet wavelet decomposition of this signal is shown in Figure 1b. Measurements of density by CINDI equally spaced in time are converted to measurements along the satellite track presuming a constant satellite velocity of 7.5 km/s. Though C/NOFS' elliptical orbit does lead to a variation in orbit velocity, the variation is less than 10%. For each sample, the complex decomposition coefficients for each scale size can be computed. The vertical axis is the scale size of the Morlet wavelet along the satellite track where the wavelet scale size (s) is related to the equivalent Fourier wavelength (λ) by λ=1.03s [Torrence and Compo, 1997]. Power levels are indicated by colored contours (blue, green, orange, and red) with magnitudes of 1,100,1500,2500×1E8(N/cc)2, respectively. Contour lines are drawn around regions with a 90% confidence level, estimated using the global wavelet power spectrum (not shown) [Torrence and Compo, 1997]. Wavelet decompositions for scale sizes that extend beyond the time series are influenced by the edges of the finite sample size and are discarded as indicated by the hatched area. The line that separates these regions is known as the cone of influence.

Figure 1.

Example Morelet Wavelet decomposition for a portion of a satellite pass on 19 April 2009. (a) The total ion density measured using CINDI is in black, the wavelet reconstruction of the perturbation in density for scale sizes above the cone of influence is in green. (b) The Morlet wavelet power as a function of satellite track over a range of scale sizes is shown. Power levels are indicated by colored contours (blue, green, orange, and red) with magnitudes of 1,100,1500,2500×108cm−6, respectively. The central region outlined in yellow allows investigation over a range of scale sizes free from edge effects. The perturbation waveform that corresponds to this region is denoted by the vertical lines in Figure 1a.

[17] The yellow box in the center of Figure 1b outlines the central quarter of the time series. In this region, the behavior over a constant range of scale sizes can be investigated free from edge effects that arise from the use of a finite sample size. The green line in Figure 1a is the reconstructed waveform using the wavelet transforms for scale sizes above the cone of influence. Since the largest scale sizes have been removed, this waveform may be interpreted as the perturbation in ionospheric density. Outside the central quarter of elements, the perturbation waveform is constructed using a changing range of scale sizes and is not retained. To extract a continuous perturbation waveform over a constant range of scale sizes, the buffer is advanced by a quarter of the total elements as shown in Figure 2 and the waveform in the central quarter is again extracted. The calculated wavelet power is only used when the wavelet transform is free from edge effects and thus characterizes a signal roughly three Fourier periods long. Statistical significance tests are sometimes used to filter both Fourier and wavelet analyses [Torrence and Compo, 1997]. The test provides information on whether a detected oscillation is larger than an assumed background. As care has been taken to ensure a robust wavelet transform and RPA measurements of density have low noise, significance tests are not utilized here.

Figure 2.

This figure follows the same format as Figure 1, demonstrating the advancement of the finite buffer used in the wavelet decomposition by a quarter of the buffer size to extract information over a range of scale sizes free of edge effects.

[18] The particulars of the C/NOFS orbit can lead to measured density changes that are not due to irregularities. In Figure 3, changes in the ion density occur for scale sizes greater than 512 km. To reduce the chance of similar waveforms, which are not related to irregularities, from being used in a scintillation estimation only scale sizes less than 128 km are considered, equivalent to approximately 17 s of observations on CINDI at an orbital velocity near 7.5 km/s. The spacing between irregularities has been observed and modeled between 100 to 200 km [Aveiro et al., 2012], thus choosing 128 km as a maximum gives a density perturbation of similar magnitude to the total density depletion. A full reconstruction of the density signal requires all spatial scales. This figure also emphasizes the interpretation of the reconstruction waveform as a density perturbation. Though the measured density is changing and is away from the mean, these changes occur at scale sizes that are larger than the retained wavelet coefficients, thus the perturbation waveform is near zero.

Figure 3.

This figure follows the same format as Figure 1. The density variation observed by CINDI on 19 April 2009 is not due to bubbles in the equatorial ionosphere. Only scale sizes less than or equal to 128 km are used in the presented scintillation estimation to limit the false prediction of scintillation for similar nonirregularity density variations.

[19] The perturbation waveforms in Figures 13 demonstrate the ability of wavelets to isolate scale-limited changes; however, the waveforms are not the best input for generating a scintillation estimate. The change in ion density can be positive or negative, and in general, the signal oscillates between these values, generating zero crossings that would falsely indicate no scintillation.

[20] Parseval's Theorem for the Morlet wavelet [Torrence and Compo, 1997] analysis allows us to determine the variance of the measured irregularities. The Morlet wavelet power for scale sizes less than 128 km is summed, weighted by scale size, and normalized appropriately [Torrence and Compo, 1997] to produce an equivalent perturbation amplitude at each measurement location,

display math(4)

where |A|2 is the variance of the ionospheric irregularities and Ps is the wavelet power for scale size s. The standard deviation |A| provides a measure of the strength of the irregularities at each measurement location. The sampling rate δt is the distance between measurements for the RPA (δt=3.75), δj=0.25 relates to the interval between the scale lengths over which the wavelet analysis is performed (four per octave) and C=0.776 [Torrence and Compo, 1997] is a wavelet-dependent constant. The Morlet wavelet analysis yields the amplitude of scale-limited perturbations in the ionosphere. The maximum density change associated with this oscillation is given by the peak-to-peak perturbation amplitude,

display math(5)

2.3 Scintillation Estimation

[21] The spatial changes in ion density obtained using the Morlet wavelet can be used in the scintillation estimation by Rufenach [1975] and used by Basu et al. [1976]. The S4 index is estimated using,

display math(6)

where F is a Fresnel filter function, f(β) is a geometric factor for anisotropic irregularities, and Φis the phase deviation. The phase deviation is given by

display math(7)

where ΔN is the magnitude of the irregularity, re is the classical electron radius, λ is the free space wavelength of the trans-ionospheric signal, Le is the thickness of the scattering layer, α is the axial ratio of field-aligned irregularities (assumed to be greater than 5), and K is the outer scale wave number of irregularities. The zenith angle between the wavefront and the plane of the ionosphere is χ and is taken to be 0. The remaining term, β, is the axial ratio transverse to α, defined in terms of the angle between the wavefront and the magnetic field ψ, β2= cos2ψ+α2 sin2ψ. Given the equatorial orbit of C/NOFS, it is assumed that ψ=π/2.

[22] The Fresnel filter function is given by

display math(8)

where

display math(9)

is defined in terms of the mean distance between observer and irregularities (z) as well as the outer scale wave number. The Fresnel wave number is

display math(10)

and the geometric factor is

display math(11)

Though the largest impact to a radio signal occurs at the F peak density maximum generally located below the satellite, the height of the irregularities (z) is taken as 450 km, reflecting the average altitude of CINDI when both ion drifts and densities are available during the recent solar minimum [Stoneback et al., 2011]. The layer thickness (Le) is assumed to be 200 km.

[23] The ΔN used by Rufenach [1975] covers all wave numbers and presumes a power law spectrum for the irregularity. Here we use the scale-limited wavelet-derived ΔN(5) between 8 and 128 km. While this underestimates the true ΔN, the altitude variation of C/NOFS limits the largest scale size that may be investigated before density variations due to the orbit contaminate the desired irregularity density change. Irregularity power decreases with decreasing scale size thus the minimum sample size from CINDI near 8 km has less impact on scintillation levels than the choice of maximum scale size of 128 km.

[24] The outer scale size of the wavelet-derived inline image determines the outer scale wave number K. The scaling of irregularities measurable by the satellite to sizes that impact the desired communications signal was performed by Rufenach [1975] which relates the total density perturbation in an irregularity to the scintillation at a desired frequency using an assumed three-dimensional power law scaling for ionospheric irregularities with spectral index p=4. Using the specified parameters for the phase screen analysis, the estimated S4 is the product of a scaling constant and the magnitude of the density perturbation associated with the irregularity.

[25] The presented S4 estimation only accounts for weak scintillation (S4<0.4) of incident electromagnetic waves due to Fresnel scattering. In reality, multiple scattering events may occur for a given radio signal and these events may also be strong. Thus, the scintillation estimation used here is unable to accurately characterize all scintillation events. Further, when the actual power spectrum of an irregularity deviates from the presumed spectrum by Rufenach [1975] than an additional error in the scintillation estimate will be present.

3 Results

[26] CINDI measurements of ion density from 20 February 2009 along with the perturbation amplitude for scale sizes less than or equal to 128 km is in Figure 4. In Figure 4a, the ion density measured by CINDI is in black along with the inline image obtained using equation (5) in blue. Near 2000 magnetic local time (MLT), an isolated irregularity is seen in density measurements which has a corresponding spike in inline image. Later near 2030 MLT, a long train of irregularities are seen in the density measurements. inline image rises to 1×105 cm3 at the start of the irregularities in a background ionosphere with densities near 2×105 cm3. The inline image remains positive over the region of irregularities observed and varies as the strength of the irregularities changes. Near 2230 MLT, a weaker set of irregularities are observed with a correspondingly weaker inline image. The estimated scintillation using the wavelet-derived inline image is shown for VHF frequencies of 140 and 250 MHz in Figure 4b. Before the start of irregularities, this value remains close to zero. During the train of irregularities, the S4 estimation is distinctly elevated, identifying the whole region as disturbed.

Figure 4.

(a) Example of a C/NOFS pass on 20 February 2009. The total measured ion density is in black while the effective irregularity amplitude inline image is in blue and uses the right-hand side y axis. Both density curves are displayed on a log scale. (b) The scintillation estimation obtained using inline image and the method by Basu et al. [1976].

3.1 SCINDA Comparison

[27] A comparison of the scintillation estimation using the wavelet-extracted inline image to measurements by the SCINDA ground station network is shown in Figure 5. On 4 November 2009, a quiet night was observed by the Kwajelein SCINDA station as shown by the blue trace in the top panel. The ground station is monitoring signals at 250 MHz with a noise floor near 0.1 on the S4 index. The 5 min averages are shown. The in situ scintillation estimation using C/NOFS is in black. The scintillation estimates displayed are restricted to within 12° apex longitude of the magnetic flux tubes monitored by the SCINDA station. The longitudinal width was chosen so that the 15 passes made by C/NOFS over the longitude sector centered upon the SCINDA measurement location form a quasi-continuous signal in local time. Though the satellite results convolve altitude, longitude, and local time changes, only the local time is reported. Perigee is located near 2100 LT, with apogee 12 h later.

Figure 5.

A comparison of the S4 index at 250 MHz at (a) Kwajelein (blue) and the estimated scintillation using in situ density measurements (black) from CINDI on 5 November 2009. Similar results are shown for (b) Christmas Island and (c) Cape Verde, both on 6 November 2009. The C/NOFS scintillation estimation over local time is comprised of 15 passes through each respective SCINDA longitude sector.

[28] The in situ estimation shows a small amount of scintillation near midnight, reaching values near 0.1 on the estimated S4 index. To obtain a good correspondence between the scintillation estimation and SCINDA observations, a constant empirical multiplicative factor of 3 was included with the in situ estimate. The factor was chosen to make the in situ estimate close to but generally below 0.1, consistent with this SCINDA station for this quiet night. Since C/NOFS is not measuring density perturbations at the peak in ion density and scintillation depends upon the absolute change in density, scintillation estimates are expected to underestimate the value measured on the ground. Thus, this factor is expected to depend upon the altitude of the F peak in ion density relative to the altitude of the spacecraft. The empirical scaling may also account for other limitations of the process used to estimate S4. This factor of 3 is included only for the passes in Figure 5.

[29] The next day similarly quiet conditions were observed at Kwajalein (not shown), though significant irregularities were observed at other SCINDA stations. In the middle panel, both C/NOFS and the Christmas Island station agree that significant scintillation at 250 MHz started at 2100 LT. The in situ estimation only accounts for weak-scattering [Basu et al., 1976] while values above 0.4 likely involve multiple scattering events. Thus, values above 0.4 for the in situ estimated S4 index will not accurately describe the extent of scintillation.

[30] The difference in the Christmas Island SCINDA measurement location and the position of C/NOFS for Figure 5 is shown in Figure 6 for both apex longitude and geographic latitude. C/NOFS also varies in altitude, not shown. In general, C/NOFS will measure the same apex longitude sector 15 times per day, though the satellite will only be near the SCINDA measurement location 1–2 times per day in both apex longitude and latitude. The large field-aligned conductivity of the equatorial ionosphere will map plasma irregularities along the field line, increasing the latitudinal range that C/NOFS could detect the presence of an irregularity that is simultaneously observed by a SCINDA station. However, the field line actually measured by C/NOFS will generally map to a different altitude at the SCINDA station than is actually measured. The difference in position between the satellite and SCINDA could be a significant contribution to the differences between the in situ estimated and ground-based S4.

Figure 6.

The difference in the position of C/NOFS and the SCINDA ionospheric measurement location for Christmas Island is shown for both apex longitude (black) and geographic latitude (blue) for the passes shown in Figure 5.

[31] While the scintillation estimate from C/NOFS, as it repeatedly moves in and out of the longitude sector, does not account for every detail in the SCINDA measurements, both the SCINDA and C/NOFS observations agree that scintillation occurred between 20 and 25 LT. On the same night at the Cape Verde station, shown in the lower panel, scintillation was observed though not as strong as observed at Christmas Island. The lower scintillation levels are also reported by C/NOFS. Thus, relative changes in the wavelet-derived in situ irregularity amplitude are consistent with changes observed on the ground.

4 Discussion and Conclusion

[32] The wavelet transform is an effective tool for characterizing irregularities in measurements of ion density. The wavelet decomposition presented only uses wavelet coefficients when they are unaffected by edge effects, providing an accurate description of the strength of irregularities at different scale sizes for each measurement location. The technique can be used to process very long time spans of data as well as process irregularity detection onboard a satellite in real time. The total power in irregularities below and including 128 km are summed and converted into an equivalent density perturbation amplitude using Parseval's Theorem. To validate the calculated density perturbation, the results are used with the scintillation estimation method by Rufenach [1975] to obtain S4 values, which are compared against SCINDA measurements.

[33] The convolution of longitude and local time inherent in CINDI observations of the ionosphere complicates comparisons with ground-based stations. To limit the influence of this convolution, CINDI measurements were restricted to a 24° longitude sector centered upon SCINDA measurement locations. However, there can be significant differences in the state of the ionosphere over 24° longitude, thus even a perfect calculation of scintillation from in situ measurements will in general only approximate what is seen on the ground at a particular location.

[34] An intrinsic limitation to the interpretation and use of these data lie in the altitude limits of the CINDI data available. In cases where ionospheric irregularities remain at or below the F peak and do not have strong signatures that map to the altitudes being sampled, then the correspondence between satellite and ground scintillation estimates may be weak. The altitude range of C/NOFS also ensures that when irregularities are observed, it will generally be at altitudes away from the peak in F region ion density. Scintillation intensity is dependent on the magnitude of the density perturbation which decreases with distance from the peak; thus, the in situ scintillation estimation will generally underestimate the true value. No attempt to account for the altitude variation is made here.

[35] Additionally, the phase screen approach by Rufenach [1975] used here calculates a single Fresnel scattering of a wave through a weak irregularity. In reality, multiple scattering events occur and the strength of these individual events may be stronger than can be accounted for by Rufenach [1975]. Finally, the existence of irregularities confined to flux tubes that lie completely below the satellite location can be largely undetected.

[36] Given these complications, a one-to-one correspondence between SCINDA and C/NOFS was not expected. However, the general behavior of the in situ estimation of scintillation displays a good correspondence with observations from the SCINDA ground station network when including an empirically determined constant multiplicative factor. The correspondence with SCINDA only establishes that relative changes in the estimation of scintillation using CINDI are consistent with ground measurements. This is sufficient, however, to investigate climatological variations of irregularities measured in situ with geophysical parameters.

[37] The empirical scaling applied to the scintillation estimate for the SCINDA comparison is not expected to remain constant. The comparison shown demonstrates the empirical constant was suitable for scintillation estimation over 24 h and at multiple locations around the equator. Daily updates to this scaling may be sufficient to provide a general estimate of the equatorial distribution of scintillation for the next 24 h or more.

[38] The scintillation estimation could be improved by using an alternate method by Rino [1979a]. Used by Wernik et al. [2007], the one-dimensional spectral power index as well as the power at a given scale size inside an irregularity is used to estimate scintillation. The spectral power index has a large effect upon the scintillation levels, and the index can vary between irregularities. Using the actual power spectrum in an irregularity as done by Rino [1979a] and Wernik et al. [2007] is thus expected to improve the accuracy of the in situ scintillation estimate. As the altitude variation of C/NOFS has not been accounted for, this improved in situ estimate would still require an empirical scaling factor to coincide with ground-based measurements. Wernik et al. [2007] also uses a density model to estimate peak ion density levels in the F region. Adopting a similar process in future work could reduce the impact of the altitude variations in the C/NOFS orbit and eliminate the need for an empirical scaling factor. Without the large-scale changes due to altitude, then the details of the scintillation estimate may be better appreciated. However, the focus of this paper is on establishing the use of wavelets in isolating variations in density from total ion density measurements; thus, we reserve improvements to the scintillation estimation for future work.

Acknowledgments

[39] Work supported at UT Dallas by NASA grant NNX10AT029. AFRL would like to acknowledge the contribution of the U.S. Air Force Space Test Program in providing support for the C/NOFS program. Wavelet software was provided by C. Torrence and G. Compo, and is available at http://atoc.colorado.edu/research/wavelets/.

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