F region plasma density estimation at Jicamarca using the complex cross-correlation of orthogonal polarized backscatter fields

Authors


Abstract

[1] The differential-phase method for Jicamarca F region density measurements has been modified to fit the real and imaginary components of the cross-correlation of orthogonally polarized radar returns. The original method [Kudeki et al., 2003; Feng et al., 2003] based on fitting the cross-correlation phase could only make use of correlation data from lower F region heights since the phase noise in low-SNR upper F region returns is typically non-Gaussian. However, upper F region correlation data become useful in the modified method since the fluctuations in real and imaginary parts of cross-correlation remain Gaussian even under low-SNR conditions.

1. Introduction

[2] Recently, Kudeki et al. [2003] and Feng et al. [2003] described a new incoherent scatter radar technique for measuring F region plasma densities with the Jicamarca radar. The new technique makes use of a pair of orthogonal polarized antenna beams pointed perpendicular to the geomagnetic field equation image. F region density estimates Ne(z) are obtained by least squares fitting the differential phase and average power of southward and eastward polarized antenna outputs Vθ and Vϕ to density dependent data models ∠〈VθV*ϕ〉 and 〈∣Vθ2〉 + 〈∣Vϕ2〉. Detailed descriptions of 〈VθV*ϕ〉 and 〈∣Vθ2〉 + 〈∣Vϕ2〉 models are given in Feng et al. [2003]. As described in Kudeki et al. [2003] and Feng et al. [2003], the new technique was developed to obtain density measurements during Jicamarca drifts experiments which rely on antenna beams pointed perpendicular to equation image.

[3] In this paper we will describe a recently introduced feature of the new technique that makes use of the real and imaginary parts of the complex cross-correlation 〈VθV*ϕ〉 of Vθ and Vϕ data instead of their phase difference ∠〈VθV*ϕ〉. The advantage of using 〈VθV*ϕ〉 instead of ∠〈VθV*ϕ〉 is as follows: When signal-to-noise ratio (SNR) is low the standard deviation of phase errors approaches equation image and the error distribution takes a flat non-Gaussian character. Phase estimates with flat error distributions are not usable in data inversions. Differential-phase based inversion procedures described in Kudeki et al. [2003] and Feng et al. [2003] have therefore avoided using cross-correlation data from upper F region heights with low SNR values. However, as it is well known, real and imaginary components of cross-correlation 〈VθV*ϕ〉 remain Gaussian random variables even under low SNR conditions. Therefore, by reformulating the inversion procedure in terms of real and imaginary parts of 〈VθV*ϕ〉 the use of cross-correlation data from upper F region heights becomes possible and the overall quality of inversion results can be improved.

[4] The present paper describes the details of the revisions outlined above. It is organized as follows: In section 2 we review the data models 〈VθV*ϕ〉 and 〈∣Vθ2〉 + 〈∣Vϕ2〉 from Feng et al. [2003] and describe their joint covariance matrix needed in a revised formulation of a χ2 expression representing the data/model misfit. Density estimation is carried out by minimizing χ2 over the model parameters as described earlier in Feng et al. [2003]. Section 3 describes the modified χ2 in detail and experimental results are shown and discussed in section 4.

2. Data Models and Covariances

[5] Let vectors equation imageθ and equation imageθ represent time- and frequency-domain outputs of a southward polarized radar antenna such that equation imageθ contains a sequence of complex numbers corresponding to discrete-time Fourier transform (DTFT) of a complex time-series stored in equation imageθ. The vectors have equal lengths and represent the backscatter radar response of some ionospheric range gate z collected after N pulse transmissions. Likewise, equation imageϕ and equation imageϕ contain time- and frequency-domain data from the same ionospheric range gate obtained using an eastward polarized antenna.

[6] The expected values

equation image

and

equation image

where a summation over N scalar products is implied in each case, are defined to be the average power and cross-correlation of the antenna outputs. Clearly, these power and cross-correlation parameters are frequency-domain averages of self- and cross-spectra

equation image

and

equation image

defined over N frequency bins. In these spectral definitions scalars Vθ and Vϕ stand for the elements of equation imageθ and equation imageϕ vectors, respectively, and are functions of frequency even though our notation does not show this dependence explicitly for the sake of simplicity. The estimators of these spectra,

equation image

and

equation image

computed using I independent measurements of Vθ and Vϕ data, include zero-mean estimation errors δPθ,ϕ, δR and δS. It can be shown that the covariance matrix of spectral estimation errors is

equation image

[7] Furthermore, if the spectra are white, i.e., frequency independent, then the covariance matrix of the estimators for power 〈equation imageθ,ϕ · equation image*θ,ϕ〉 and cross-correlation 〈equation imageθ · equation image*ϕ〉 — frequency domain averages of spectral estimators — can be obtained by replacing I in (7) with IN. This is an acceptable procedure for incoherent scatter returns with nearly white frequency spectra (and amounts to neglecting the small pulse-to-pulse correlation of the incoherent scatter signal returns).

[8] To proceed, we next recall from Feng et al. [2003] that

equation image

and

equation image

where

equation image
equation image

and

equation image

Above, K is an unknown system constant that depends on radar parameters (transmitted power, antenna aperture, etc.), Ne is the electron density at radar range z, μ is an unknown that depends on electron and ion temperature ratio Te/Ti at the same radar range, b(ɛ) is a known function that describes the angular variation of effective beam pattern of the radar system in the magnetic meridian plane, and D(ɛ) as well as C(ɛ) are density dependent functions describing magneto-ionic effects on the propagation of radar signals. These definitions from Feng et al. [2003] constitute the fundamentals of idealized data models of concern here. In practice the data models should also account for additive noise and channel gain differences and therefore take the forms

equation image
equation image

and

equation image

where noise powers Aθ,ϕ and relative gain g = ∣g∣ ∠ {g} are additional system unknowns. Also, Ac is a complex unknown representing the cross-correlation of the noise in θ- and ϕ-channels. Thus, spectral data Pθ,ϕ and R + jS obtained with noisy channel outputs have the expected values

equation image
equation image
equation image

and

equation image

and carry zero-mean estimation errors

equation image

Since the estimation errors are jointly Gaussian, the joint pdf of Pθ, Pϕ, R and S data ∝ eequation image, where

equation image

and, evaluating ((7)) with data values, we find

equation image

[9] Alternatively, using P ≡ ∣g2Pθ + Pϕ with the expectation

equation image

the relevant error vector is

equation image

and the corresponding error covariance matrix is

equation image

3. Data Misfit Model

[10] In the previous section, we described the models and covariance matrices for spectral and cross-spectral data Pθ,ϕ, R, and S obtained in transverse-beam F region radar experiments. The covariance matrix (22) describing Pθ, Pϕ, R and S data is crucial for the revised data inversion method that will be described here and in the next section.

[11] First, we note that if the off-diagonal elements of matrix (22) can be neglected, then χ2 defined in (21) can be calculated as

equation image

In that case a global misfit measure of spectral and cross-spectral data available over data range gates zm, m ∈ [1, M = 58], would be obtained by summing terms of the form (26) over all m. However, the assumption of negligible off-diagonal elements does not hold at all data range gates and therefore our global misfit expression takes the following hybrid form:

equation image

[12] In this hybrid expression four different quadratic terms are summed over four different sets of data range gates. First two sums are carried over lower F region range gates and contain differential phase ψm and total power data as described in our earlier papers [e.g., Feng et al., 2003]. The last two sums are carried over the upper F region and are compatible with (26). Mrs in the last sum is the index m of the lowest range gate where the minimum of diagonal elements of matrix (22) is greater than 8 times the maximum of the off-diagonal elements. Thus in m = Mrs to 58 range we ignore the small off-diagonal matrix elements to obtain the last term of (27). Mp denoting the highest range with acceptable differential phase data is calculated as in Feng et al. [2003], but in the situation of MpMrs we set Mp = Mrs − 1. Mi to 58 describes the range where the correlation between δPθm and δPϕm is negligible as established by the condition min(Pequation image, Pequation image) > 10(Requation image + Sequation image). In that range Pθm and Pϕm data are used separately while below that range they are used in ∣g2Pθ + Pϕ combination as in Feng et al. [2003].

[13] Overall, then, the global misfit (27) is defined in such a way that we make most convenient use of the data available at different range gates with varying SNR and correlation values. In the next section we will present examples of the covariance matrix (22) computed with lower and higher F region data Pθm, Pϕm, Rm and Sm and demonstrate the observed values of diagonal and off-diagonal elements of the matrix justify the misfit definition (27) introduced above.

[14] So far our misfit definition represents spectral data measured at a single frequency bin. But since the data errors at individual frequency bins are nearly independent it is possible to use in our inversions spectral data averaged over N such bins. In that case all of the above equations are still valid provided that factors I are all replaced by IN. Thus in subsequent interpretation of (27) data values Pθm, Pϕm, Rm and Sm should be viewed as spectral averages over N = 127 frequency bins rather than spectral samples. This interpretation also applies for differential phase data ψm = tan−1equation image which are calculated with frequency averaged values of Rm and Sm. Finally, 127I is used in place of I.

[15] Finally, in view of available prior information we choose to augment (27) as

equation image

where MT corresponds to 460 km range gate above which we assume μm ≃ 1. Most of the details of how the misfit (28) is evaluated in our data inversions (e.g., evaluation of Γ in (18)–(19), the way μm unknowns are handled, specification of 〈δψequation image〉 etc.) are identical to what has been described in Feng et al. [2003]. The main exception concerns the channel noise parameters Aθ,ϕ which contribute to Pθ,ϕ but not R and S.

4. Experimental Results and Discussion

[16] Figure 1 plots the elements of the covariance matrix for the signals received by the north antenna between 09:05 and 09:10 LT on May 31, 2002. In the range of altitudes where differential-phase data and power data are used, the ratio of 〈δPθδPϕ〉 over 〈δPθδPθ〉 or 〈δPϕδPϕ〉 decreases with increasing altitude. After 485 km, where 〈δPθδPϕ〉 is negligible compared to 〈δPθδPθ〉 or 〈δPϕδPϕ〉, we start to fit the Pθm and Pϕm data to separate models. Similarly, after 620 km, where all of the off-diagonal elements of the data covariance matrix are negligible compared to the diagonal elements, the measured Rm and Sm data are used for fitting purposes instead of the differential-phase data.

Figure 1.

Covariance matrix elements of north antenna between 09:10 and 09:15 LT on May 31, 2002. In the graph, dPx2: 〈δPθδPθ〉; dPy2: 〈δPϕδPϕ〉; dPxdPy: 〈δPθδPϕ〉; dR2: 〈δRδR〉; dS2: 〈δSδS〉; dPxdR: 〈δPθδR〉; dPxdS: 〈δPθδS〉; dPydR: 〈δPϕδR〉; dPydS: 〈δPϕδS〉; dRdS: 〈δRδS〉.

[17] The inversion results shown in this section were obtained by minimizing the misfit defined in (28). For the sake of convenience, we call the inversion procedure developed above method-1, compared to method-0 described in Feng et al. [2003]. In addition to fitting the complex cross correlation data at high altitudes, method-1 relaxes the constraints about noise levels used in method-0. The search for the minimum of χ2 is carried out using two nonlinear least squares algorithms. One is the Levenberg-Marquardt algorithm [e.g., Press et al., 1988], and the other is the trust region algorithm [e.g., Branch et al., 1999]. Both algorithms give nearly identical results, but the convergence rate is faster with the trust region algorithm. In this section, only the results from the trust region algorithm are presented. Figure 2 shows some of the fitting details for the same time interval as shown in Figure 1. Furthermore, the density inversions for the two-day long data are shown in Figures 3 and 4. Method-1 gives better results for Ne(z) at very high altitudes than method 0 since it determines the noise level more accurately. The plots of Nmax versus time in Figure 5a and the heights of Nmax versus time in Figure 5b show comparisons of inversion results with ionosonde data.

Figure 2.

Fitting details for the same data taken between 09:05 and 09:10 LT, on May 31, 2002. χ2/ν = 0.87. αreg = 150. (a) Total power ∣g2Pθm + Pϕm (stars, ∣g∣ is the fitted parameter, Pθm, and Pϕm are the measured data), 2Pθm(crosses), 2Pϕm(triangles), differential phase data ψm and their corresponding fitted models. (b) RmRe{Ac}, SmIm{Ac} and their fitted models. (c) Measured data of phase difference between neighboring heights, its fitted model and the fitted μms. (d) Inversion result of the profile of electron density Ne(z) and its error estimates.

Figure 3.

Logarithmic map plot of Ne(z, t) estimates obtained with method-1 at Jicamarca on May 31 through June 1, 2002.

Figure 4.

Surface plot of the density estimates shown in Figure 3. (a) Day 151 (May 31, 2002). (b) Day 152 (June 1, 2002).

Figure 5.

(a) Nmax and (b) heights of Nmax versus time obtained with the transverse-beam incoherent scatter technique method-1 and the Jicamarca ionosonde on May 31 through June 1, 2002.

[18] The resulting Ne(z)'s of this inversion method – method-1 – agree with the ionosonde data reasonably well in the daytime. During the nighttime, the Nmax obtained by the Jicamarca ionosonde is generally larger than our inversion results. One possible reason for the discrepancy is that ionosonde measures the all-sky Nmax, while we only look at the direction nearly transverse to the geomagnetic field. Another possibility is that Nmax and the heights of Nmax measured by the ionosonde may be biased. The reason that we think part of the blame is in ionosonde data is as follows: The observed differential-phase derivative peaks above the height of Nmax obtained by the ionosonde. However, by running simulations, we have determined Ne(z) should peak slightly higher (by 15 km to 30 km) than the peak of the differential phase height derivative, which agrees with our inversion results. In summary, small discrepancies between inversion results and ionosonde data may be due to a mixture of biases present in both techniques. Further refinements of the differential-phase method using a frequency domain approach will hopefully reduce the inversion biases. Frequency domain inversion results will be reported in a future publication.

Acknowledgments

[19] We appreciate many helpful suggestions from S. J. Franke. We also thank the staff and engineers of the Jicamarca Radio Observatory for their assistance with the observations. The Jicamarca Radio Observatory is operated by the Instituto Geofísico del Perú, with support from the U.S. NSF cooperative agreement ATM 99-11209 through Cornell University. This work was supported by the Aeronomy Program, Division of Atmospheric Sciences of the NSF, through grant ATM 02-15426 to the University of Illinois.

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