Recently the Global Ultraviolet Imager (GUVI) on-board the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite has detected far ultraviolet (FUV) images of plasma depletions in the low-latitude and equatorial ionosphere. A model of GUVI observation geometry was developed to simulate radiance observations of a model ionosphere. We report on results in reconstructing multi-dimensional electron density profiles from GUVI brightness measurements through the use of statistical inversion techniques. These results enable the global observation and characterization of the structure of plasma bubbles and provide a means to quantify the level of depletion in the structures. Results are compared with corresponding JULIA observations for validation. The ability to globally image and characterize equatorial plasma bubbles provides a powerful tool for understanding this elusive space weather phenomenon.
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 The plasma instabilities known as “Equatorial Spread F” have been observed and studied systematically since measurements were compiled at the Jicamarca Radio Observatory in Peru in 1970 [Farley et al., 1970]. After several decades of observation, modeling, and theory, ESF is still elusive. The ability to understand, characterize, and predict these plasma depletions is still highly desirable.
Woodman and LaHoz  observed spread F in incoherent scatter radar measurements at the Jicamarca. Using range-time intensity plots, they were able to create some of the first images of these plasma irregularities. Since that time, equatorial plasma bubbles have been observed with ground-based instruments including radars and airglow cameras [e.g., Kelley, 1989, and references therein].
 It is generally believed that ESF evolution is driven by the gravitational Rayleigh-Taylor (R-T) instability [e.g. Sultan, 1996], where plasma density gradients in the post-sunset ionosphere are gravitationally unstable. The growth rate of ESF bubbles is dependent on the height of the F region layer and the steepness of the vertical density gradient. A shear in plasma flow causes the bubbles to develop an increasing westward tilt with time [Mendillo and Tyler, 1983; Kelley, 1989].
 Some types of ESF events are termed Equatorial Plasma Bubbles (EPBs) which describes regions of depleted plasma density that typically originate in the bottomside post-sunset ionosphere and, while longitudinally thin, extend latitudinally along magnetic field lines. EPBs can extend vertically to altitudes above 1000 km. When viewed from above, as with GUVI, a ‘C’-shaped figure can develop as spread F plumes tilt to the West with increasing altitude. We study images of EPBs with the intent of elucidating, on a global basis, and to the extent possible, the conditions that characterize the occurrence, growth, decay, and morphology of EPBs.
 Airglow cameras have provided a means of effectively imaging the structure and development of EPBs from the ground. By observing an optical emission as a proxy for electron density, these cameras can provide high-resolution movies of plasma bubble formation and drift over a small area of the globe. An altitude profile of the bubbles can be obtained by mapping the camera's field of view along field lines to the magnetic equator [Kelley et al., 2002]. Although these observations can be coordinated with space-based observations [Kelley et al., 2003], they are limited in their ability to quantitatively measure the depleted electron density and are restricted to the field of view of the instrument.
 In situ plasma density measurements have been used to study equatorial plasma bubbles for over a decade [Huang et al., 2002]. These observations have provided a global climatology of plasma bubble occurrence and are able to provide quantitative measures of depleted electron density. However, satellite in situ measurements are restricted to plasma measurements along the orbit path of the satellite and provide no information about the three-dimensional structure of the bubbles.
 While EPBs have been globally detected and locally imaged, GUVI now enables global imaging of plasma bubbles. Reconstructed images from GUVI provide an altitude vs. longitude image of electron density that can serve as a unique and valuable data set for evaluating these theories and identifying the stages of development of ESF bubbles. Both the height of the F region layer and the vertical density gradient can be observed in these reconstructed images, albeit with limited resolution. The tilt and height profile of bubbles can also be determined, thereby providing insight into the stage of development of an observed depletion. The global coverage of GUVI allows these images to be produced at all longitudes, providing data over regions without ground based imagers or ISR data or regions that have different orientations of the magnetic equator with respect to the geographic equator. Here we report on the technique for producing multi-dimensional images of plasma bubbles with GUVI and discuss the resulting images.
2. Experimental Configuration
2.1. GUVI Instrument
 TIMED orbits at 625 km with a 74° inclination. The GUVI instrument [Paxton et al., 1999; Christensen et al., 2003; Paxton et al., 2004] has a 140° cross-track scan perpendicular to the orbit plane, as illustrated in Figure 1. The TIMED orbit allows global coverage of the ionosphere with each orbit occurring with a local time about one minute earlier than the previous one, covering all local times every 60 days. The sensitivity of the GUVI instrument at 1356 Å is approximately 0.5 counts/sec/Rayleigh/pixel [Humm et al., 1998]. The detector is quantized into 14 spatial elements and 160 spectral elements, with selected spectral elements co-added as “colors”. The instrument has an 11.8° field of view over the 14 spatial pixels. There is a 127.2° cross-track scan for disk images over 159 pixels. Each of these pixels has an integration time of 0.064 s. With the improved GUVI sensitivity and appropriate binning of the spatial pixels, GUVI yields brightness measurements at 1356 Å in the equatorial anomaly with a SNR of approximately 10 dB (J. M. Comberiate et al., A Tomographic model For ionospheric imaging with the Global Ultraviolet Imager, submitted to Radio Science, 2005) (hereinafter referred to as Comberiate et al., submitted manuscript, 2005). Allowing for a roughly 1:1 ratio of measurements to reconstructed electron density values, a 10° segment of orbit can support a reconstructed grid with 20 km altitude resolution and 40 km longitude resolution. The 140° cross-track scan supports a reconstruction spanning from 90 to 630 km in altitude and a 10° span in longitude centered around the satellite position.
2.2. Forward Model
 Reconstruction of electron densities with GUVI requires the construction of a forward model of GUVI observation geometry that projects the discrete-resolution grid of electron density values onto the series of GUVI brightness measurements. A three-dimensional region of the ionosphere containing a plasma bubble is selected for a 10° segment along the orbit path. Since equatorial plasma bubbles extend along magnetic field lines, the model of the ionosphere uses magnetic coordinates. An offset, tilted dipole was assumed, therefore defining a point in space in terms of three magnetic coordinates corresponding to altitude, latitude, and longitude. This magnetic coordinate system is used in the SAMI2 model [Huba et al., 2000] and is a common approach to defining magnetic coordinates. The forward model can then be simplified by assuming that the ionosphere is constant along magnetic field lines over the region of interest; magnetic field lines are often used to reduce the ionosphere to two dimensions when studying ESF [e.g., Eccles, 1999]. 10° is roughly the span of the equatorial anomaly in a single hemisphere. If chosen properly, a 10° section is a short enough section of an orbit to ignore variations along magnetic field lines while still providing multiple overlapping scans.
 A two-dimensional slice of the ionosphere can then be divided into discrete sections, with each section having constant ne. If the sections of constant squared electron density are arranged into a single vector x, a series of observations y can then be modeled through the following matrix equation.
where A is proportional to the length of the line of sight for the observation in each section of x and q is an additive noise term. Figure 2 shows a 3-D perspective of the discrete observation model.
 Since the ionosphere is assumed to be constant for the 11 scans used to create the y vector, calculation of the A matrix can be reduced to a two-dimensional problem. The longitudinal variation of the position of the TIMED satellite is retained, effectively yielding overlapping measurements from a moving sensor. The combined effect of these approximations casts the problem as a two-dimensional limited-angle tomography problem, with geometry illustrated in Figure 3. A two-dimensional electron density profile cross-section image of a plasma bubble can be extended to three dimensions by projecting the electron densities along magnetic field lines.
 Extracting two-dimensional electron densities in the observation plane from GUVI brightness measurements requires a tomographic reconstruction through the inversion of a projection matrix A that is ill-conditioned (nearly singular). The existence of A−1 is required for a perfect reconstruction. However, multiple scans of the GUVI instrument produce some rows in A which are approximately linear combinations of others, resulting in a rank-deficient, ill-conditioned projection matrix A.
 The standard algebraic approach for reconstructing x in the case of rank-deficiency is through a pseudoinverse calculated, for example, via the Singular Value Decomposition (SVD). The ill-conditioning of the matrix leads to the pseudoinverse being susceptible to noise. In that case, there are a variety of approaches to cope with the problem of ill-conditioning of the system matrix including truncating the sum in the SVD reconstruction.
 In general, any effective approach will involve the use of additional constraints on the solution based on prior knowledge [Kamalabadi et al., 2002]. In this case, the tomographic inversion is constrained through use of regularization techniques and projection on convex sets. A systematic approach to regularization leads to the minimization of an appropriately formulated cost function. Regularization functionals are included in the cost function to enforce smoothness while preserving edges. In order to effectively image plasma bubbles, the regularization must penalize small gradients in electron density between neighboring pixels while allowing steep gradients. This approach produces a smooth background ionosphere but still allows the sharp gradients typical of bubbles to persist in the image.
 The following regularization functional is used
where [Dx]n is the value of the gradient for an element of x and ϕ is a weighting function. The optimal ϕ is a nonconvex gradient weighting function that will preserve edges
where t is the value of the gradient. The parameter T controls the shape of the weighting function and its optimal value depends on the peak electron density value and does not vary greatly with each reconstruction.
 The resulting cost function is:
 The first term controls data fidelity (i.e., how faithful the reconstruction is to the data), whereas the second term (the regularization term) controls how well the reconstruction matches our prior knowledge of the solution [Kamalabadi et al., 2002]. The regularization parameter λ is selected in accordance with the discrepancy principle [Engl, 1987] to maximize the amount of regularization allowed by the uncertainty from the noise in the GUVI measurement. This parameter is the most sensitive, hence the emphasis on edge-preservation to ensure that the plasma bubble imaging ability is robust even when the inversion requires substantial regularization. Details of the inversion cost function and its solution are discussed by Comberiate et al. (submitted manuscript, 2005). Estimated mean absolute error in reconstructed electron density values is 1.2 × 105 cm−3.
 Additional constraints are enforced using the technique of projection on convex sets [Sharif and Kamalabadi, 2005]. To ensure a realistic ionospheric altitude profile, the regularized reconstruction is then projected to within a distance ε of a reference ionosphere generated with PIM [Daniell et al., 1995]. This POCS constraint enhances the edges of the reconstructed profile where GUVI measurements are sparse but a large value of ɛ is chosen to ensure that the structure of the reconstructed bubble is not significantly altered. A positivity constraint is also enforced.
3. Results and Comparison With JULIA Observations
 A two-dimensional electron density profile containing a plasma bubble is shown in Figure 4. The F-region peak is clearly visible with a peak density of nearly 1.5 × 106 cm−3. The plasma bubble has a width of approximately 120 km with areas of electron density depleted below 2 × 105 cm−3. The lack of a westward tilt with increasing altitude suggests that this plasma bubble has recently formed. Mapping this two-dimensional structure along magnetic field lines could provide an image of the plasma bubble in three dimensions.
 The JULIA(Jicamarca Unattended Long-term Investigations of the Ionosphere and Atmosphere) radar is located at Jicamarca near the magnetic equator [Hysell and Burcham 1998]. JULIA images detect coherent echoes from plasma disturbances often correlated with equatorial plasma bubbles. Figures 5a, 5c, 5e, 5g are images of equatorial plasma bubbles as seen by JULIA. Signal-to-noise ratios are shown in conventional range-time-intensity (RTI) grayscale format. Electron density profiles were reconstructed from the corresponding GUVI orbits, as shown in Figures 5b, 5d, 5f, 5h.
 Boxes are added to the JULIA radar images to indicate the portion that corresponds to the GUVI reconstruction. The width of the box assumes that each hour corresponds to 15 degrees in longitude. Figures 5b, 5d, 5f, 5h are taken at 2044 LT, 2110 LT, 2139 LT, and 1949 LT, respectively. There are significant similarities that can be observed in the corresponding images. Figures 5a and 5b both show a single bubble structure tilting westward with increasing altitude. Figures 5c and 5d both show a bifurcated bubble structure that is wider to the East. Figures 5e and 5f both show a single bubble structure that tilts westward but then bends back to the East about 500 km. Figures 5g and 5h both show a large bubble that tilts westward with some additional smaller features to the West of the main bubble. In general, the tilt, width, and shape of structures in GUVI reconstructions agree with JULIA observations.
 However, there are significant differences in the finer structure of the plasma bubbles. Hysell and Burcham  note that JULIA RTI plots should only be used as two-dimensional images for the identification of broad features. JULIA images tend to record most ESF activity before 22 LT, when plasma bubbles are actively developing and small-scale perturbations are still present. Electron density images from GUVI often show plasma bubbles several hours after sunset when plasma bubbles are most often passively drifting through the ionosphere and do not register in JULIA images. Any comparison is limited to broad features by the indirect relationship between coherent scatter power and depleted electron density and the time-variation of the plasma bubble structure as it drifts over Jicamarca. GUVI is well-suited to providing electron density information when ESF is present.
4. Summary and Conclusions
 TIMED/GUVI data can be used to reconstruct multi-dimensional profiles of equatorial plasma bubbles. These reconstructions indicate the width, tilt, and depth of depletion of the plasma bubble. Coincident observations with the JULIA radar demonstrate that the structures seen in the GUVI reconstructions are valid. These GUVI reconstructions provide a powerful tool for global imaging and characterization of equatorial plasma bubbles.
 This work was supported in part by the National Science Foundation under Grant ATM 01-35073 to the University of Illinois. We thank the Jicamarca staff for their time and efforts operating the JULIA radar. The Jicamarca Radio Observatory is a facility of the Instituto Geofisico del Peru and is operated with support from the NSF Cooperative Agreement ATM-0432565 through Cornell University.