We present results of Capon's method for estimation of in-beam images of
ionospheric scattering structures observed by a small, low-power coherent
backscatter interferometer. The radar interferometer operated in the
equatorial site of São Luís, Brazil (2.59

Equatorial
spread F (ESF) refers to a broad spectrum of electron density irregularities
occurring in the equatorial and low-latitude F region ionosphere. ESF is
associated with interchange plasma instabilities, which can create large-scale (tens
of kilometers) sizes plasma density perturbations. Secondary plasma
instabilities then create smaller-scale (down to centimeter) irregularities

Ground-based ionospheric radars have contributed significantly to our understanding of ionospheric plasma and irregularities. Ionospheric radars are used to measure the scatter from electron density irregularities matching the Bragg scatter condition. Because these radars usually operate in the VHF and UHF bands, the scatter comes from irregularities in scale sizes ranging from a few meters down to a few tens of centimeters.

Incoherent scatter radars are high-power, large-aperture systems that can measure extremely weak echoes produced by thermal electron density fluctuations. Coherent scatter radars, on the other hand, are smaller systems capable of measuring echoes produced by non-thermal electron density fluctuations, that is, fluctuations caused by plasma instabilities. Conventional coherent radar measurements are capable of providing information about the altitudinal distribution of irregularities, as a function of time, as well as the mean radial phase velocity of the irregularities producing the echoes. Additionally, conventional coherent radars provide information about the strength of plasma turbulence at the wavelength matching the Bragg condition.

Radar measurements made using spaced antennas can provide additional
information about the irregularities causing echoes

Coherent backscatter radar imaging has been used for investigations of
equatorial spread F. These investigations have been performed, predominantly,
in the Peruvian longitude sector using measurements made by the radar and
antenna modules of the Jicamarca Radio Observatory

In the present study, we revisit the Capon method and investigate its performance when applied to measurements made by a small, low-power coherent backscatter radar interferometer located at the equatorial site of São Luís, Brazil. We show that the method can produce high-resolution images, which reproduce features predicted by numerical simulations and previous observations of interchange plasma instabilities. This report is presented as follows: Sect. 2 describes the setup used by the São Luís radar for interferometric measurements. Section 3 provides information about the Fourier and Capon methods for interferometric imaging. In Sect. 4, we present and discuss the results of analyses. Numerical simulations are used to evaluate the performance of the Capon method. We also present and discuss the results of applying the Capon algorithm to actual measurements made by the São Luís radar during a typical pre-midnight ESF event. The features in the scattering structures resolved by the Capon method are discussed in light of our current understanding of ESF. Finally, Sect. 5 summarizes the main results of our study.

The measurements available for this study were made by a 30 MHz coherent
backscatter radar interferometer. The radar was installed at the equatorial
site of São Luís, Brazil (2.59

Diagram describing the distribution of the antenna sets used for the
interferometric observations. Each antenna set is formed by a four by four array of Yagi antennas. A, B, C and D represent the antenna sets.
Transmissions are made with sets A and B. Reception is made with all of the
antenna sets. The magnetic declination is approximately
21

For F region observations, we used 28 bit coded pulses, with a 9.33 ms inter-pulse period (IPP). The baud length and sampling were 2.5 km. A total of 250 samples were collected per IPP. This observation setup allowed us to make measurements of the F region from 200 to 825 km altitude with a range resolution of 2.5 km.

Coherent scatter radar imaging techniques have been used to determine the
distribution of scatterers as a function of height and zenith angle (

The brightness distribution is closely related to the cross correlation of
scattered signals received by antennas spaced by a distance

In imaging, the spatial cross-correlations

The visibility is the correlation of the scattered signals received by
antennas spaced by a distance

One can recognize that Eq. (2) represents a Fourier-type integral, and,
therefore, the brightness can be estimated from the visibility function using
the inverse Fourier transform:

In the practical case of a finite number of receivers (

Ionospheric radar images are obtained by computing the brightness function for each range gate independently. The images, therefore, represent the distribution of ionospheric irregularities causing echoes, as a function of range and zenith angle. The brightness functions can also be obtained for different Doppler shifts adding a new dimension to the images. A sequence of images can be used to track the appearance, development and decay of scattering regions and to understand the dynamics of the irregularity structures.

The angular resolution of the
images obtained with the Fourier method is limited by the length of the
longest baseline.

Before obtaining imaging results from actual measurements, we investigated the ability of the Capon method to produce accurate estimates of the distribution of the irregularities. We considered the specific case of the São Luís baseline setup and created synthetic visibility distributions for different scattering geometries (brightness distributions) under different levels of measurement errors.

Numerical simulations of the Fourier and Capon methods under
different SNR conditions. The simulated brightness
function is described by a Gaussian function centered at 1.5

Measurement errors for normalized correlations were taken into account
considering typical SNR values
observed during ESF events and the number of incoherent integrations (

The expected value for the error variances of the normalized correlation
functions (

In order to evaluate the potential of the Capon method, two scenarios were considered in our simulation analysis. These two scenarios illustrate the weaknesses and strengths of the Capon method.

Numerical simulations of the Fourier and Capon methods under
different SNR conditions. The simulated brightness
function is described by two Gaussian functions centered at

The first simulation scenario considered a broad distribution of scatterers
(brightness) that is described by a Gaussian function, centered at
1.5

The second simulation scenario presented here considered a scattering region
that is described by two Gaussian distributions centered at

Figure

Simulation results describing the performance of the Fourier and
Capon methods as a function of SNR conditions for two scattering
distribution scenarios:

Figure

Figure

Panel

The results shown in Fig.

We have also analyzed the results of the Capon method applied to actual interferometric measurements made by the São Luís radar. We applied our Fourier and Capon algorithms to ESF measurements made by the São Luís radar. For this study, we present results of our analysis of a well-developed ESF event observed by the São Luís radar on the night of 24 November 2005. We present and discuss some of the features of the scattering structures resolved by our Capon algorithm.

Figure

The São Luís data consisted of cross-spectral measurements for four
spectral bins, which allowed us to infer information about the line-of-sight
Doppler velocity of the scatterers, in addition to their location.
Two-dimensional in-beam radar images are constructed by stacking
one-dimensional brightness distributions

Figure

The estimated images are scaled by the SNR of the echoes. The Doppler information is encoded in the pixel colors. The green components represent small mean Doppler velocities. Red and blue components represent irregularities moving away and towards the radar, respectively.

Both images show a topside ESF structure that, as will be shown later, drifts from west (left-hand side) to east. The images also show that the structure is tilted to the west, and this is, presumably, caused by height variations in the zonal plasma drifts. More importantly, one can see that the Capon method produces a better-resolved image than the Fourier method. In some range gates, the Fourier images show artifacts in the brightness that are intrinsic to the method. For instance, the simulations in Fig. 3 show an increased brightness outside the scattering region produced by the sidelobes of the Fourier estimates even under high SNR conditions.

Panel

The Capon image shows the scattering region is narrower (in the zonal direction) than what is presented by the Fourier image. The Fourier image shows scattering features that are broad and diffuse; a result of the short baselines used for visibility functions. This is particularly true for the strong scattering channel seen above 350 km altitude. The colors also indicate that irregularities are ascending with large velocities within the scattering structure.

Finally, we must point out that the images were produced with the same number
of incoherent integrations used in our simulations (

Figure

Panel

At 20:59:59 LT one can start to identify two vertically developed ESF
structures, spaced in the zonal direction, within the radar field of view.
The two structures are more clearly seen at 21:01:45 LT. The first
(easternmost) structure reaches about 400 km altitude, while the second
structure reaches over 500 km altitude. The measurements were made during
low solar flux conditions when the altitudinal reach of ESF structures is
limited

We point our that the multiple structures seen in the in-beam images could not be inferred from the RTI map. The RTI map alone would suggest that a single radar plume passed over the radar site around 21:00 LT. Also, the detection of multiple plumes within the radar field of view is possible because of the large east–west beamwidths of the antennas used by the São Luís interferometer.

Figure

In addition to bifurcation, the images produced by the Capon method are
capable of resolving the C-shape structure of the ionospheric perturbation,
which has been predicted by numerical models of ESF

We investigated the application of the Capon method

Numerical simulations were used to evaluate the performance of the Capon method. The numerical simulations show that, for broad scattering structures and the São Luís antenna configuration, the Capon method does not outperform the simple Fourier method. The simulations also show, however, that the strength of the Capon method lies in its ability to identify localized (narrow) scattering structures. We found that images with an angular resolution of a fraction of a degree can be obtained for typical equatorial spread F (ESF) measurement conditions found with the São Luís radar setup using the Capon algorithm. In practice, the resolution is also controlled by the magnitude of zonal irregularity drift during the integration time.

Following the simulation analyses, we applied the Capon method to actual measurements made by the São Luís interferometer during a typical ESF event detected on 24 November 2005. The Capon technique produces sharper images than those created by the Fourier method and better resolves the actual widths of the scattering structures.

Sequences of images show consistency from one inversion to the next indicating the robustness of the method. The sequence of images also shows the occurrence of ionospheric phenomena with better spatial and temporal resolution than would be possible with other types of instruments (e.g., scanning radars or airglow imagers). The images show scattering channels with zonal widths of a few kilometers. As expected from previous observations, the scattering structures move to the east as they evolve in time. Scattering channels spaced by only a few tens of kilometers in the zonal direction can be resolved, and their occurrence should be taken into consideration in theories and the description of ESF. The Capon images are also able to reveal the occurrence of bifurcation in ESF structures as well as variations in the zonal tilt of the scattering channels.

It is believed that the MaxEnt technique can produce images with more
accuracy than the Capon method, particularly for conditions of low SNR

The interferometric radar data used in this study are available upon request from Fabiano Rodrigues (email: fabiano@utdallas.edu).

The authors declare that they have no conflict of interest.

This work was supported by NSF (AGS-1261107) and AFOSR (FA9550-13-1-0095). Eurico R. de Paula acknowledges the support from CNPq under process number 310802/2015-6. The authors would like to thank A. Cunha for their efforts in the operation and maintenance of the São Luís radar. The topical editor, P. J. Erickson, thanks two anonymous referees for help in evaluating this paper.