F-region electron density and Te / Ti measurements using incoherent scatter power data collected at ALTAIR

The ALTAIR UHF radar was used in an incoherent scatter experiment to observe the low-latitude ionosphere during the Equis 2 rocket campaign. The measurements provided the first high-resolution electron density maps of the low-latitude Dand E-region in the Pacific sector and also extended into the F-region and topside ionosphere. Although the sampling frequency was well below the Nyquist frequency of F-region returns, we were able to estimate Te/Ti ratio and infer unbiased electron density estimates using a regularized inversion technique described here. The technique exploits magnetic aspect angle dependence of ISR cross-section for Te>Ti .


Introduction
During the 2004 Equis 2 NASA Rocket Campaign, incoherent scatter radar (ISR) measurements of the low-latitude daytime ionosphere were conducted using the ALTAIR UHF radar. The measurements were made for common volume comparisons with rocket based D-and E-region electron density estimates described in Friedrich et al. (2006). During the observations, ALTAIR beam was scanned along a northsouth oriented plane that coincides with the rocket trajectory, and ISR signal returns were sampled from D-region up to the topside F-region heights. The scan geometry and radar data collected in a single scan (lasting about 5 min 20 s) are illustrated in Fig. 1 where the color map corresponds to the logarithm of backscattered power. A detailed analysis of the power data shown in this figure and how it can be used to obtain F-region density maps corrected for T e /T i variations Correspondence to: E. Kudeki (erhan@uiuc.edu) throughout the bottom-side F-region are the main concerns of this paper.
As the ALTAIR beam was scanned, it probed the ionosphere at different magnetic aspect angles, going through a perpendicular orientation with respect to the geomagnetic field at an elevation of ∼80 • . That orientation coincides with the spike-like intrusion seen in Fig. 1. This is better illustrated in Fig. 2 where power is now displayed as function of elevation angle and height. The superposed white arrows depict the relative orientation of the geomagnetic field B with respect to the radar beam direction over a sample grid. The enhanced power corresponding to the spike observed in both figures is characteristic of incoherent scatter (IS) process for probing directions perpendicular to B and for heights where electron temperature exceeds the ion temperature (i.e., T e >T i ). The theory presented by Farley (1966) quantifies this enhancement in terms of IS radar cross-section (RCS) that depends on electron density N e , T e /T i ratio, and magnetic aspect angle α. The RCS function constitutes the core of a forward model for the power data displayed in Fig. 1 which will be used to infer the height distributions of N e and T e /T i as described in Sect. 4.
In practice, the standard RCS function of Farley (1966) is inaccurate -and, in fact, unspecified -for small aspect angles α where electron and ion Coulomb collisions are becoming important as shown by Sulzer and González (1999). Since the ALTAIR beam goes through α = 0 • , we had to extend the RCS model to include these collisional effects. This was done by using the collisional incoherent scatter spectrum model described by Woodman (1967) and Kudeki et al. (1999), and an empirical collision frequency model proposed by Woodman (2004).
Another detail of the measurements concerns the linear frequency modulation (LFM) used to generate the transmitted radar pulses. The response of the resultant "chirped" pulse was modeled in terms of the corresponding radar ambiguity function (AF) and its influence on the collected power  Backscattered power data displayed as function of beam elevation angle and height. The white arrows depict the relative orientation of the geomagnetic field with respect to the radar beam direction over a sample grid. Note that the power enhancement at ∼80 • elevation is from altitudes with T e >T i and a radar line-ofsight perpendicular to B. data was included in the forward model used to estimate N e and T e /T i . The paper is organized as follows: after a brief description of the radar configuration in Sect. 2, we present, in Sect. 3, the derivation of the forward theory model for backscattered ISR power, including the AF analysis of transmitted pulses and our extended ISR RCS model, as outlined above. Section 4 describes our inversion method to estimate the N e and T e /T i profiles from measured power data, namely, a regularized least-squares minimization algorithm relying on a graphic criterion for optimal selection of the regularization parameter (L-curve). Finally, Sect. 6 is devoted to the presentation of our results, N e and T e /T i , and their discussion.

Radar measurements
ALTAIR, located on Roi-Namur island (9.39 • N, 167.47 • E) of the Kwajalein Atoll, is a dual-band (VHF and UHF) highsensitivity radar designed for missile detection and tracking. Its 150-ft parabolic dish antenna is fully steerable with angular rates of up to 10 • /s (Lemnios and Grometstein, 2002). Its transmitter can deliver up to 4 MW of peak power in the UHF band with a maximum duty cycle of 5%. In transmission, circularly polarized pulses are used while in reception both right-and left-circularly polarized returns are monitored and detected independently. Table 1 presents a summary of the primary ALTAIR parameters used in ionospheric measurements conducted on 20 September 2004. As indicated, the radar was operated using a LFM waveform (chirped pulse) of 400 µ s duration, 422 MHz center frequency, 250 kHz bandwidth, and a 120 Hz pulse repetition frequency (PRF) corresponding to a 8.33 ms inter-pulse period (IPP) greater than the correlation time of ISR signal returns. The pulse compression effect which results upon a matched filter detection of the pulse returns will be described and formulated in the next section. The matched-filter receiver outputs were sampled at a rate corresponding to 240 m radar range, and 2874 gates were sampled to cover from 65 to 755 km of radar range. During each scan the radar beam was swept in the −13.5 • azimuth plane from an elevation angle of 50 • to zenith position in about 5 min and 20 s (at 8 s/deg rate).
In Fig. 3, we display power versus height calculated from matched-filter receiver outputs of the right-and leftcircularly polarized antenna channels. The individual profiles in Fig. 3 correspond to different elevation angles of the ALTAIR beam and represent an integration over 1000 radar pulses (i.e., 8.33 s) during which the beam moves about 1 • . Notice that only the channel labeled as "principal polarization" contains backscattered radar power, indicating that possible depolarization effects can be neglected in our analysis. This is an expected result, not only because circularly polarized transmissions correspond to a normal mode of propagation for most scanning directions involved, but also because generalized Faraday effects (e.g. Yeh et al., 1999) are very weak at 422 MHz even for propagation perpendicular to the geomagnetic field (in which case the normal modes are linearly polarized).
In the next section we develop a quantitative model for the enhanced power data recorded in the principal polarization channel, which is subsequently used in Sect. 4 to estimate N e and T e /T i profiles in the probed regions.

Radar equation and ambiguity function (AF)
In soft-target Bragg scattering each infinitesimal volume dv≡r 2 drd of a radar field-of-view behaves like a hard- target with a radar cross-section (RCS) dv dω 2π σ (k, ω), where σ (k, ω), by definition, is soft-target RCS per unit volume per unit Doppler frequency ω 2π , and k≡−2k or denotes the relevant Bragg vector for a radar with a carrier wavenumber k o . As a consequence, hard-target radar equation generalizes for soft-target Bragg-scatter radars as where P t (t) ∝ |f (t)| 2 is the transmitted power carried by a pulse waveform f (t), G(r) and A(r) are the gain and effective area of the radar antenna as a function of radial unit vectorr, r is the radar range, and c the speed of light. Furthermore, Eq. (1), which assumes an open radar bandwidth, can be re-cast for a match-filtered receiver output as where E t is the total energy of the transmitted radar pulse, K s denotes a system calibration constant including loss factors ignored in Eq. (1), g(r) is the self-normalized version of G(r), and has a magnitude known as radar ambiguity function (e.g., Levanon, 1988). In Eq.
(3) the normalization constant T denotes the duration of pulse waveform f (t) and Eq. (3) itself is effectively the normalized cross-correlation of Doppler-shifted pulse f (t)e j ωt with a delayed pulse echo ∝ f (t−τ ) that would be expected from a point target located at a radar range cτ/2. Hence, variable τ in Eq.
(3) not only represents a time delay, but also a corresponding radar range r, and we will refer to τ as delay or range as we find convenient in the following discussion. In our experiment, we used a chirped pulse waveform which, in effect, causes a slow linear variation of the carrier frequency over a bandwidth f and at a rate β 2π = f T . The corresponding ambiguity function, is plotted in Fig. 4 for T =400 µs and f =250 kHz (parameters used in the experiment). According to Eq. (5), the overall width of |χ(τ, ω)| along the delay axis τ over the frequency bandwidth B of the RCS spectrum corresponds to 2π B/β (given B f ), which implies a range resolution r -defined as the range separation between independent measurements -described by For instance, for a plausible ionospheric bandwidth of B=10 kHz, the range resolution is r=2.4 km. In addition, any mean Doppler shift of the RCS spectrum will imply a displacement of the actual position of the backscatter signal; such a displacement can be estimated as where v d is the Doppler shift measured in m/s and f o the radar carrier frequency in Hz. In ionospheric observations, possible Doppler shifts are around 100 m/s, with a corresponding displacement of about 67 m, a small fraction of the range resolution r that can be safely neglected.
Assuming that σ (k, ω) varies slowly with the radar range r, and that the AF is almost flat within the bandwidth of the RCS spectrum (which is the case of this application), we can simplify Eq. (2) to obtain where R= ct 2 is the measured radar range, is the effective range depth of the radar scattering volume, and is the total volumetric RCS where dependence on range R is implied.

Incoherent scatter RCS spectrum (σ )
The IS theory establishes that the volumetric RCS spectrum of a quiescent ionosphere can be written as (e.g. Dougherty and Farley, 1960;Farley et al., 1961) σ (k, ω) = 4πr 2 e |n e (k, ω)| 2 where r e is the classical electron radius and |n e (k, ω)| 2 = 2N e ω |j k 2 λ 2 e + µy i (k, ω)| 2 Re{y e (k, ω)} |j k 2 λ 2 e + y e (k, ω) + µy i (k, ω)| 2 + |y e (k, ω)| 2 Re{y i (k, ω)} |j k 2 λ 2 e + y e (k, ω) + µy i (k, ω)| 2 denotes the space-time spectrum of electron density fluctuations. Above, N e is the mean electron density, λ e is the electron Debye length, µ the temperature ratio T e /T i , and y e and y i are normalized electron and ion admittance functions proportional to conductivities in the medium. These admittances can be expressed as where s denotes each species (electrons or ions) and θ s ≡ ω √ 2kC s is the frequency normalized by wavenumber k and thermal speed C s . The function J s , that will be specified later, is a Gordeyev integral that can be interpreted as one-sided Fourier transform of normalized auto-correlation of echoes from charged particles in the absence of collective interactions (e.g. Hagfors and Brockelman, 1971).
Replacing Eq. (11) in Eq. (8), we have where is an efficiency factor that is 1 2 in a plasma in thermodynamic equilibrium, or, more generally, (1+ T e T i ) −1 in a nonmagnetized plasma with an arbitrary T e /T i and negligible Debye length (e.g. Farley, 1966). But, in our forward model, we will need the efficiency η for a magnetized plasma with an arbitrary T e /T i and for arbitrarily small aspect angles. Here, we define the aspect angle α as the complement of the angle between probing vector k and geomagnetic field B.
In general, non-collisional models for (12) become singular as α→0 • and make numerical evaluation of η impossible in the same limit. This difficulty can be circumvented by using a collisional model for Eq. (12). In our computations of η, we used the collisional model developed by Woodman (1967), which is based on a simplified Fokker-Planck operator for Coulomb collisions, the relevant collision process for F-region altitudes (e.g. Sulzer and González, 1999). The model leads to a Gordeyev integral (e.g. Kudeki et al., 1999;) where ψ s ≡ ν s √ 2kC s and φ s ≡ s √ 2kC s are normalized collisionand gyro-frequencies, respectively, and γ s ≡ tan −1 ( ν s s ). For ν i and ν e we make use of the collision frequency models of Woodman (1967) and Woodman (2004), respectively, which are and (Eq. (14) in the original paper) ν e /ν e0 = 1.06 + 7.55 sin α/ sin α c − 2.00(sin α/ sin α c ) 2 +0.27(sin α/ sin α c ) 3 , where α c ≡ sin −1 ( 2π ν e0 kC e ) is a critical aspect angle and ν s0 ≡ e 4 N e is the so-called Spitzer collision frequency for species s (electrons or ions). In numerical evaluations of η, Gordeyev integrals J s were computed using a chirp z-transform algorithm suggested by Li et al. (1991). Each integral was evaluated for a total of 2 15 sample frequencies equally spaced from 0 to 125 kHz. The algorithm samples the integrand of J s in sections of the same length, and adds iteratively the contribution of each section as it is needed for convergence of the integral. Despite the accuracy that this algorithm provides, it became computationally demanding to obtain J s at angles close to perpendicular to B because of the narrowness of the IS spectrum in this regime. Nevertheless, computations were possible even at α=0 • as a consequence of the collisional model used.
We have calculated the efficiency factor η for different values of aspect angle α, temperature ratio T e /T i , and electron density N e . As an example, we display in Fig. 5a the values of η computed for a constant density of 10 12 m −3 and T e =1000 K. Here, we can notice that the dependence of η on the magnetic aspect angle becomes relevant for small α (smaller than ∼3 • ). In addition, Fig. 5b shows the variation of the efficiency factor as function of N e for α=0 • and T e =1000 K. In this case, η increases as T e /T i rises from unity. Note that the rate of change is not uniform for different density values. Overall, we have found that η is not highly sensitive to variations of the collision frequency parameter. For instance, increasing ν e by factor of 3 or decreasing it by a factor of 10 did not change the computed η values by more than ∼1%. On the other hand, drastic changes, such as a factor of 30 increase in ν e , cause noticeable changes in η.

Radar calibration and backscattered power calculations
The use of Eq. (14) to model the measured power data requires the availability of transmitted pulse energy E t and system constant K s . Since the transmitted peak power P t is recorded in ALTAIR data headers on a per pulse basis, the pulse energy is trivially obtained as E t ≡P t T . In addition, the K s value that accounts for all the losses and gains of the entire system is available from power measurements conducted with a hard target of a known RCS. The effective range depth δR needed in Eq. (14) is obtained by using Eq. (9) with radar returns detected from a suitable point target.
The computation of the solid angle integral in Eq. (14) was simplified assuming that ionospheric parameters do not vary locally in longitude, therefore, only changes along the direction of the scan were considered. This assumption is justified by the narrowness of the ALTAIR UHF antenna beam with a beam-width of less than one degree. As a result, we collapsed the solid integral into a 1-D angular integral along the direction of the scan. Moreover, because only a few points of the antenna pattern were available from calibration measurements, we used a Fourier interpolation method to improve the angular grid of the normalized antenna gain.
We considered two different approaches to evaluate the angular integral just described: The first one corresponds to assuming a fixed beam direction during the transmission of the 1000 pulses used to obtain a single averaged power profile. This approach is justified if the efficiency factor η varies linearly across the beam, in which case, it can be replaced by its central value and reduce the integral to the calculation of an equivalent solid angle. In the second approach, we accounted for the effect of the angular displacement of the beam from pulse to pulse and averaged the 1000 profiles corresponding to each beam position to obtain a model profile for the measured power data. Comparison of the two methods revealed negligible differences in test cases, and, as a consequence, the bulk of the results to be presented in the next section were obtained using the less costly first approach.

Inversion technique
A regularized least-squares fitting algorithm (e.g. Tikhonov and Arsenin, 1977) was used to perform electron density and T e /T i estimations from the modeled power data. In general, a regularization technique allows us to introduce some prior information about the characteristics of the parameters to be estimated in order to obtain better results. Next, we will describe our algorithm and the assumptions we made in this work.
First, we envisioned the ionosphere as a stratified medium composed of layers of a fixed width that extend along longitude and latitude following the curvature of the Earth, such that, in each layer, the density and temperature ratio can be considered constants. Below, the vectors N e and T r denote respectively the electron density and T e /T i profiles that were estimated after minimizing the following cost function Regularization-term (20) In this expression, p m is the measured power collected by the radar, and P r and σ p denote its model and standard deviation. Our power model also accounts for the noise level detected by the radar receiver, level that was estimated from samples taken below 80 km altitude. Above, the variable λ denotes the regularization parameter, and its significance will be discussed later in this section. In Eq. (20), the index i implies summation over a set of radar ranges and scanning angles that correspond to the region where the power enhancement was observed. Thus, we are considering seven data profiles whose magnetic aspect angles are between −2.5 • and 2.5 • (and corresponding scanning directions between 77 • and 83 • ). In range, samples taken at each 6 km (which is also the width of each j -th layer) provide a good enough resolution to characterize the F-region.
The simple form of the data-term in Eq. (20) is justified by the independence of our measurements. Based on our AF analysis, we know that samples with a separation of 6 km are range independent. In addition, given that the inter-pulse period of the experiment (8.33 ms) was longer than the correlation time of ISR returns, there is no angular or pulseto-pulse correlation between power profiles. Therefore, we can define the standard deviation of the measured power as σ p ≡ p m √ n i , n i being the number of incoherent integrations (averages) performed to obtain an individual profile. Here, it is useful to indicate that at 422 MHz, the correlation time of beam weighted ISR returns coming from the region around perpendicular to B is of the order of 1 ms.
From the analysis of our power model, it turns out that the dependence of the measurements on the plasma density is stronger than its dependence on T e /T i . Therefore, at ranges Ann. Geophys., 24, 1333-1342, 2006 www.ann-geophys.net/24/1333/2006/ with relatively low SNR, the inverted T e /T i values may be strongly biased. This fact motivated the inclusion of the regularization-term in the cost function (20) which is the discrete gradient of the temperature ratio profile weighted by λ. Our goal in applying regularization is to obtain smooth and unbiased T e /T i estimates.
The outcome of the inversion procedure has some dependence on the right choice of the regularization parameter. An optimal value of λ reduces the noise variance in the inversion results without biasing the model appreciably away from the measured data. Hansen (1992) describes a graphical method for choosing the regularization parameter which is named Lcurve and consists in plotting the residuals of the data-and regularization-terms for different realizations of λ. The shape of this curve usually resembles an L and its "corner" or point of maximum curvature provides the optimal regularization parameter. The method is based on the criterion of equalizing the contributions of each term in the cost function (20). Figure 6 displays one of the L-curves we computed for the data collected on 20 September 2004. In this case, a value of λ=1.58 provided the desired quality in our estimated profiles. Furthermore, we noticed that the L shape is not very sharp around the corner, and regularization parameters between 1 and 2 produced almost the same quality in the results. For other scans, we found optimal parameters within the same range, therefore, we chose a constant value of λ = 1.5 for all the estimations we performed.
In Fig. 7, we show some of the outputs of the fitting procedure corresponding to the same case of Fig. 6. The first two columns display the measured and modeled power profiles, the third one is the difference between data and model weighted by their uncertainties, and the last one is the regularization term. We are judging the quality of our fitting results following the goodness-of-fit criterion, which states that good estimates are obtained when the data-term normalized by the degrees of freedom of the data, i.e. the number of data points minus the number of estimating parameters, is around unity. Much larger or smaller values may imply a poor knowledge of σ p and/or an inappropriate data model. In our estimations, we found values for the normalized dataterm typically between 0.9 and 1.1, giving us confidence about the quality of our results. Figure 8 presents N e and T e /T i maps for 20 September 2004, constructed from profile estimates obtained with the data model and inversion technique described in the previous sections. The density map shows a typical low-latitude F-region ionosphere exhibiting a characteristic rise during the morning hours. Also, T e /T i is elevated above unity mainly between 200 and 300 km as expected in daytime measurements.

Results and discussion
In order to validate our results, we performed comparisons with electron density profiles obtained from the Roi-Namur digisonde system. As shown in Fig. 9, the agreement between fitted-ISR and digisonde estimates are excellent, specifically at the peak and bottom-side of the F-region. The agreement at the peak effectively verifies the correctness of the ALTAIR calibration constant K s , whereas the agreement in the bottom-side verifies the forward model underlying the data inversion given that the values and altitudes of enhanced T e /T i are plausible in view of our understanding of the physics of the region. This was also verified by the good comparison of our results with temperature values obtained from the International Reference Ionosphere (IRI) model (e.g. Bilitza, 2001).
In Fig. 9, we are also comparing our fitted results with a set of power profiles from the vicinity of α=0 • .  . 9. Electron density comparison between fitted-ISR and digisonde measurements obtained (a) before and (b) after noon on 20 September 2004. We also display density-like profiles calculated by scaling the power data collected at directions around α=0 • and assuming η= 1 2 as if the ionosphere were in thermal equilibrium. been scaled into equivalent density profiles by proper K s and assuming η= 1 2 as if the ionosphere were in thermal equilibrium. Somewhat surprisingly, we note that the profile corresponding to the scan angle closest to α=0 • (about 80.5 • elevation) agrees very well with the fitted-ISR and digisonde profiles. This may be the smooth result of having a moving beam with a finite width that averages the scattering contributions from different aspect angles around perpendicular to B. Finally, note that the effect of T e /T i >1 disappears at altitudes below 130 km, where density estimates can be obtained at all elevation angles assuming η= 1 2 as discussed in  which is focused on E-region results of the same ALTAIR data set.
In summary, the technique described in this paper requires only a few scan directions across magnetic perpendicularity in order to obtain unbiased N e and T e /T i estimates with a calibrated IS radar operating in the UHF band. The time resolution of the profile estimates can be improved by restricting the overall scan angle to a few degrees (2 • or 3 • ) centered about α=0 • and increasing the frequency of the scans. This procedure can also be applied with IS radars in VHF range provided that ionospheric depolarization effects, more important at VHF, are taken into account in a proper manner (e.g. Kudeki et al., 2003).

Conclusions
We have reported on N e and T e /T i estimation with ALTAIR ISR power data at UHF obtained in meridional scans through the F-region ionosphere that included a radar probing direction orthogonal to the geomagnetic field. The estimation method makes use of a forward theory model of magnetic aspect angle as well as T e /T i dependence of the the total RCS of ionospheric incoherent scatter, including the proper physics of electron and ion Coulomb collisions formulated using a Fokker-Planck type collision model.
The physically meaningful results obtained in the experiment effectively verifies the forward theory model used in the inversion technique, including a new collision frequency dependent model for the total RCS of incoherent scattering at small aspect angles formulated in this paper. Indirectly, the electron collision frequency model proposed by Woodman (2004) has also been verified; however, the test of the frequency model was not very stringent given that the collision frequency dependence of the total RCS was found to be quite weak. Finally, we also regard the success of our data inversion procedure to be an indication that Woodman's collisional Gordeyev integral expressions, derived specifically for the T e =T i case (Woodman, 1967), can also be used for T e >T i as argued in