The Algorithm

An EUV image is a two dimensional representation of the three dimensional plasmaspheric He⁺ density profile. Under the assumption that that the He⁺ density along L is either constant or can be represented by a known function, the images can be inverted to produce two-dimensional density maps of the He⁺ ion population in the SM equatorial plane. The field is assumed to be dipolar throughout the inversion region. This is not a significant compromise. Generally the plasmasphere extends to large L only during quite times. In active time the plasmapshere is eroded to an L below 4.

An outline of the general steps taken in the inversion of an EUV image is given below. Many of the actions taken are configurable using the EUVSim front-end menu. This allows for a fine tuning of the algorithm.

The reduction of background in the measured image, both instrumental and environmental in the measured image is paramount in the inversion. Both increase the error in the absolute density and are a major source of instability in the inversion process. Because of the importance of background removal the program contains multiple methods by which this can be accomplished. This should be approached with some caution. Background removal is a mixture of science, intuition, and black magic and it is very easy to take a perfectly good image and turn it into perfectly outstanding garbage.

There are several methods to remove background available within the application. These include: instrument background removal, despeckling, and both uniform and value bracketed background subtraction. Were applicable, these are applied in the order: instrument background removal, background subtraction, and despeckling.

The instrument background is removed by computing the average count rate in each column of the measured image within a fixed phase angle interval located above and/or below the area of interest and then subtracting this average from each pixel in the column. The computed averages can be increased or decreased as a percentage of their standard deviations which is computed as the square root of average value. Because this is a far from perfect subtraction it more than not actually increases the noise in the image.

This niose as well as the random environmental noise is often effectively removed or controlled through the use of a despeckling algorithm. Despeckling removes isolate actived pixel clusters in the image. The size of the clusters is user definable.

The inversion proceeds from the cleaned image. It is based on an iterative approach in which an initial guess is made and then modified repeatedly until it converges to a stable solution. Stability is determined by comparing the measured image to a simulated imaged produced from the inversion. The iteration begins with a set of initializations among which is the formation of the zero solution and solution grids. Both grids have identical order and represent either a polar or rectangular griding of the geomagnetic equatorial plane. The coordinate system used is user selectable. A rectangular coordinate system provide equal spatial resolution throughout the equatorial plane while a polar coordinate system provides a high spatial resolution a low radial values which decrease with increasing radius. Both grids are over-dimensioned to give them a higher spatial resolution than the EUV measurements. This is necessary both for stability and smoothness in the final solution.

The solution grid holds the current density He⁺ profile at any step in the inversion. The profile is initialized to a simple power law in R with no azimuthal ($\Lambda$) dependence as:

The zero solution grid holds a status flag for each cell in the solution grid. This flag is either zero or one depending on whether the density of the cell is known to be zero or not. The grid is initialized to all ones. Each EUV line of sight known to have a zero intensity is then mapped onto the zero solution grid and those grids it passes through are set to zero. Since the instrument measures column density any pixel contained in a line of sight which has a zero total intensity associated with it must itself have zero density. How far above an below the equatorial plane the line of sight are followed is selectable. The reason for this is that when inverting images which contain plasma plumes it appears the the plumes may exist in a limited region about the equatorial plane and following the line of sight over a large distance often caused the region of the plume to be marked as having a known zero density.

Also set up in the initialization are the variable cells in the density grid. These are the cells that are modified in each step in the iteration process and from which the new density grid is built in each iteration step. There is one variable cell representing each instrument line of sight. This keeps the problem from being over-constrained in the iteration phase of the inversion. The variable cells form a unique set of unknowns and are the cells through which the lines of sight pass through the equatorial plane.

Each iteration step begins with the construction of a synthetic image from the current solution grid. This is done by first applying the zero solution grid to mask off the zero intensity cells in the solution grid and then computing the column abundance along each of the instrument lines of sight from the modified solution grid. Any line of sight that is found to have only a single non-zero density value along it is set to zero and the zero solution grid is updated. Lines of sight which have only a single non-zero value are generally noise which was not caught in the initial noise and background removal. Lastly, the column densities are modified by the instrument characteristics and the synthetic image is produced.

The mean square deviation between the synthetic and measured image is then computed and compared to mean square deviations computed in previous iteration steps to see if a convergence has been reached. If so, the iterations are terminated and the current density profile is returned.

When convergence has not been obtained the measured image is divided by the synthetic image to produce a correction value for each line of sight. The density grid is rebuilt using only the variable grid values multiplied by their corresponding correction value. All unassigned grids in the density profile grid are refilled using a 2-D linear least square fit algorithm. Last the zero solution mask is applied to give the new density profile and a hot spot check is made. This is a search through the grid for pixels which have intensities larger than their neighbors by some preset value. Such pixels represent solution which are either currently or in the process of becoming unstable. Their density is adjusted to the average value of their nearest neighbors. This concludes the iteration step which restarts with the computation of a new synthetic image.

It should be noted that the inversion makes no special provision for either the dayglow contamination in the near Earth sunlit portion of the EUV images or the dimmer plasmasphere seen within the Earth's shadow. Both are treated in the inversion as real phenomena. This results in an anomalous density enhancement in the sunlit sectors and a density depression directly anti-sunward of the Earth. The features are visible in all inversions.