**PROPOSAL SUMMARY**

**[PROGRAM ELEMENT]{.underline}**: LWS

**[TITLE OF INVESTIGATION]{.underline}**: 3D Reconstruction of White
Light Coronagraph Images

from Two Viewpoints

**[PRINCIPAL INVESTIGATOR/INSTITUTION]{.underline}**: John W. Cook /
Naval Research Laboratory,

Jeffrey S. Newmark (Co-I), Paulett C. Liewer (Co-I)

**[SUMMARY OF PROPOSED INVESTIGATION]{.underline}**:

**Proposal Objectives**: We propose to systematically investigate the 3D
electron density distribution reconstructions from white light
coronagraph images, and their limitations, which are achievable from
only two solar viewpoints in the ecliptic plane. The future STEREO
mission, due for launch in 2005, will carry twin copies of the SECCHI
experiment, each with three white light coronagraphs, onboard two
spacecraft. The coronagraph reconstructions have the special
difficulties of minimally determined solutions (only two viewpoints) and
optically thin lines‑of‑sight. We employ a reconstruction technique
using the PIXON algorithm, a mathematically sophisticated method whose
execution time scales only as the number of voxels N, not as a power of
N, and which should produce reconstructions from 1024x1024 CCD white
light images in less than a day. We will study a progressive series of
scientific problems for reconstruction, using both synthetic CCD images
from model electron density distributions, and also existing SOHO LASCO
coronagraph observations to attempt reconstruction of long‑lived
structures where solar rotation can mimic the effect of two viewpoints
with images taken at different times.

**Research plan**: We will apply our reconstruction technique to
synthetic test data and to existing solar observations for scientific
investigation of polar plumes, equatorial streamers, and CMEs. In
particular, we will develop our 3D reconstruction technique to
incorporate the geometry and physics of solar coronal observations from
arbitrary vantage points; investigate initial reconstructions from
simple geometrical volumes; use reconstructions of existing SOHO LASCO
data to investigate geometry and hydrostatic equilibrium of polar
plumes; use reconstructions of existing SOHO LASCO data to study the
evolution of equatorial streamers, their relationship to active regions,
and their comparison to extrapolated magnetic fields; and investigate
the effects of noise, velocity, viewing angle (elongation), image
contrast, and background models on the reconstructions of synthetic CMEs
produced from MHD models.

**Relevance to NASA programs**: STEREO is one of the first space
missions of the Living With A Star program, and it is imperative that
techniques for analysis of the STEREO observations, including those from
the SECCHI coronagraphs, be available and well-understood before actual
launch of the spacecraft. 3D reconstruction of coronal and heliospheric
structures from coronagraph images supports the development of "new
techniques and models for predicting solar/geospace disturbances which
affect human technology," where it can be used to distinguish and model
Earth-directed CMEs observed by future missions such as STEREO.

**TABLE OF CONTENTS**

Cover Page/Proposal Summary

Table of Contents

Summary of Personnel and Work Efforts

Scientific/Technical /Management Section

Introduction

Description of the Scientific Investigation

Geometrical Considerations

Synthetic Coronagraph Observations

The PIXON Method

Initial Work with our Reconstruction Technique

Scientific Investigations using our Reconstruction Technique

Description of the Proposed Work

Management Approach

Relevance of the Anticipated Results to NASA Programs

References

Facilities and Equipment

###  

**SCIENTIFIC/TECHNICAL/MANAGEMENT SECTION**

**Introduction**

Volumetric reconstruction of structures using imaging from multiple
viewpoints has been a major scientific endeavor for many years now.
Applications arise in medical, industrial, and scientific contexts.
Analysis of solar and heliospheric stereoscopic observations has also
been a topic of current interest, especially with the coming launch in
2005 of the STEREO mission, a pair of spacecraft which will gradually
separate, behind and ahead of Earth's orbital position, over the course
of several years, obtaining pairs of images from the two viewpoints.

Both STEREO spacecraft will carry twin copies of the SECCHI (Sun Earth
Connection Coronal and Heliospheric Investigation) experiment (Howard et
al., 2001), which contains three white light coronagraph instruments and
an EUV disk imager, EUVI. It is imperative that by launch it is well
understood how to best analyze such stereoscopic data.

Stereoscopic analysis can mean one of several different techniques.
Binocular stereoscopic visualization of the solar disk has been achieved
by simultaneous viewing of images of the solar disk which are slightly
separated in angular viewpoint, with each image displayed using a
separate color table or polarization state. Viewing simultaneously both
images using glasses with different color filters or polarizers produces
a very convincing 3D viewing effect (Batchelor, 1994 with Yohkoh SXT
images; Artzner, 2001 with SOHO EIT images).

More scientifically interesting, stereoscopic tie point methods of
actual 3D volumetric reconstruction have been applied to, for example,
LASCO coronagraph images (Liewer et al., 2000). This requires the
identification in two or more 2D images from different viewpoints of
positions of individual identical points (tie points), where the
application of simple trigonometry then provides their 3D locations in
space. This method has also been used by Aschwanden and collaborators
with EUV images of coronal loops from EIT and TRACE (Aschwanden, 1999,
2000a), and is being adapted to 3D reconstruction of images from the
SECCHI EUVI instrument on STEREO.

Finally, it is mathematically possible to invert the individual (CCD)
images to obtain an optimal 3D distribution of matter that would best
reproduce the observed images, usually called tomographic analysis.

In this proposal we describe an investigation to study the 3D
tomographic reconstruction of coronal structures seen in white light
coronagraph observations from two viewpoints, such as the STEREO mission
will obtain from the SECCHI coronagraphs COR 1, COR 2, and HI
(Heliospheric Imager) (Howard et al., 2001). We will discuss our
reconstruction technique, using the PIXON algorithm, and show first
results from our preliminary work. There are severe limitations on the
reconstructions achievable from only two viewpoints. The main goal of
our proposal is to systematically examine these limitations by analyzing
a progressive series of scientific problems, starting with modeled
simple geometrical structures of increasing complexity and realism,
going on to actual LASCO coronagraphic observations to reconstruct
long-lived structures where the one LASCO viewpoint and solar rotation
can mimic simultaneous observations from multiple viewpoints (polar
plumes, equatorial streamers), and finally, reconstruction of
theoretical MHD models of coronal mass ejections (CMEs). The study of
Earth-directed CMEs is the primary scientific goal of the STEREO
mission.

Our tomographic reconstruction problem with white light images has
several special features which separate it from most existing
tomographic applications. The most important are an optically thin
integration along the lines-of-sight for each pixel in the brightness
image from any given viewpoint, and the limitation, often, to
reconstruction with images from only two individual viewpoints,
resulting in a minimally determined solution.

The physics of the brightness integration along lines-of sight is well
understood (Billings, 1966): K corona brightness at visible wavelengths
(white light) arises from Thomson scattering by hot electrons of the
incident photospheric spectrum. There is a complicated geometry of solar
surface contributions to the light scattered into the lines-of-sight
toward the detector, and the scattered light is polarized into radial
and tangential components. The angular dependence of the scattering
weights the brightness contributions from each voxel along the
lines-of-sight to emphasize structures nearest the plane of the sky seen
from the given viewpoint. For white light input images, a 3D
reconstruction consists of the electron density distribution within the
spatial volume observed.

Presently, most 3D solar corona reconstruction techniques fall under the
loose term of solar rotational tomography (SRT). Frazin (2000) provides
an excellent review of the history of 3D reconstructions of the white
light corona and the use of SRT. For our concerns, the main limitation
of the various SRT techniques is that they all require many (\>\>3)
viewpoints and assume limited evolution of the reconstructed object
throughout the observation period. Frazin and Janzen (2001) show
reconstructions of the large-scale electron density distribution derived
from LASCO C2 images using SRT. These reconstructions used data taken
over half a solar rotation, and in order to make the problem tractable
by their method used 4x4 binned images of the original 512x512
observations. These authors specifically point out the difficulty that
SRT has in reconstructing dynamic phenomena such as CMEs, and in fact
delete frames containing CME's from their reconstructions. A second
variation of SRT has been used in a series of papers (e.g. Jackson and
Hick, 2000, 1997) in reconstruction of co-rotating structures in the
interplanetary medium using Helios Photometer and Interplanetary
Scintillation (IPS) velocity data. These reconstructions use solar
rotation (time series) and solar wind flow to provide the multiple
viewpoints necessary to constrain the models, and are limited to only
the co-rotating, slowly evolving structures, i.e. not dynamically
changing CMEs. Jackson and Froehling (1995) used a combination of
Solwind coronagraph observations and data from the two Helios spacecraft
to perform a low resolution (40x10x10 elements) reconstruction of a CME.

For our problem, the limitation to only a few viewpoints, often only
two, fundamentally affects the tomographic reconstruction, and this has
not been studied carefully in the context of white light coronagraph
observations. The above examples of reconstructions with SRT techniques
have several limitations for our purposes: they have studied problems
using input images from a number of individual viewpoints, and their
reconstruction calculations take an amount of time, even with severely
reduced 3D volume array sizes, which would be impractical for 1024x1024
pixel CCD array reconstructions (with order 10^9^ volume elements or
voxels), as with SECCHI observations.

We make no claim that the 3D electron density reconstructions we obtain
using the PIXON algorithm are necessarily superior to reconstructions
with other algorithms. The critical point is that the reconstruction
calculation is quick, and scales only as the total number of voxels N,
not as a power of N. Our method offers the promise of achieving
tomographic reconstructions during the STEREO mission at high resolution
in a practical (less than a day of computation) amount of time.

STEREO is one of the first space missions of the Living With A Star
program, and it is imperative that techniques for analysis of the STEREO
observations, including those from the SECCHI coronagraphs, be available
and well-understood before actual launch of the spacecraft. 3D
reconstruction of coronal and heliospheric structures from coronagraph
images supports the LWS goal of development of "new techniques and
models for predicting solar/geospace disturbances which affect human
technology," where it can be used to distinguish and model
Earth-directed CMEs observed by future missions such as STEREO.

We will make publicly available the reconstruction techniques and
results developed in this proposal, and will produce a series of test
problems with starting model coronagraph CCD images and our
reconstructed electron density distribution solutions. These should be
useful for any independent efforts to study the optimal analysis of
white light coronagraph observations from multiple viewpoints.

**Description of the Scientific Investigation**

We propose to develop our existing technique for the 3D reconstruction
of the electron density of coronal structures from white light
coronagraph observations to investigate the reconstructions, and
understand their limitations, which are possible from only two
viewpoints. We will begin with initial work to further develop our
reconstruction program. We will then apply our reconstruction technique
to synthetic test data and to existing solar observations for scientific
investigation of polar plumes, equatorial streamers, and CMEs. In
particular we will:

-   Develop our 3D reconstruction technique to incorporate the geometry
    and physics of solar coronal observations from arbitrary viewpoints;

-   Investigate initial reconstructions of simple geometrical volumes;

-   Use reconstructions of existing SOHO LASCO data to investigate
    geometry and hydrostatic

equilibrium of polar plumes;

-   Use reconstructions of existing SOHO LASCO data to study the
    evolution of equatorial streamers over the solar cycle, their
    relationship to active regions, and their comparison to extrapolated
    magnetic fields;

-   Investigate the effects of noise, velocity, viewing angle
    (elongation), image contrast, and background models on the
    reconstructions of synthetic CMEs produced from MHD models.

We discuss next the further development of our reconstruction technique
in the areas of geometry and white light rendering. We use a
reconstruction algorithm based on "minimum complexity" to reconstruct a
model of the coronal density which is consistent with the white light
observations. We explain this PIXON algorithm, which is the engine of
our reconstruction technique. We discuss our initial calculations for
geometrical cylinders ("polar plumes") and hemispherical shells
("CMEs"), which illustrate several points we wish to make about
reconstructions from only two viewpoints. This powerful tool will allow
us to perform a number of scientific investigations. We discuss our
planned analyses using synthetic models and existing LASCO observations
to study polar plumes, equatorial streamers through the solar cycle, and
CMEs.

To explore the reconstruction of CMEs, more sophisticated models are
needed. We will use CME models computed using the BATS‑R‑US
time‑dependent 3D MHD model developed at the University of Michigan
(Gombosi et al., 1994, and references therein). This code, which runs on
parallel computers, has recently been applied to the study of CME
propagation from the Sun to 1 A.U. (Groth et al., 1999). We will model
CME propagation through a background corona and create a time series of
stereoscopic white light images using the density structure from the MHD
model. The block‑adaptive‑grid BATS‑R‑US code is well suited to CME
modeling because of the many different spatial scales spanned by this
problem. The adaptive grid allows the user to place high‑resolution grid
blocks only in the regions where they are needed.

#### 

#### ***Geometrical Considerations***

The STEREO observations are obtained from two viewpoints (perhaps using
images taken from one moving viewpoint, at different times, with
existing LASCO observations) in approximately circular orbits around the
Sun at a distance of 1 A.U. The fields-of-view of the instruments (real
or synthetic observations) may extend from the Sun to the Earth's orbit;
such large fields-of-view require a full spherical geometry treatment.
Determining the geometrical relations (see Figures 1 and 2) between the
spacecraft observations and the coronal and heliospheric structures is
our first step. We can then define the volume of interest, and determine
the sampling of each voxel in the lines-of-sight.

######### Synthetic Coronagraph Observations

To perform the reconstructions, it is necessary to calculate (render)
from starting electron density distributions the resulting synthetic
coronagraph images. Coronagraphs measure the total brightness or the
polarization brightness pB integrated over the line-of-sight through the
optically thin corona. The polarization brightness is given by:

![](media/image1.wmf)

where B, K, and F are the intensities of the total corona, the K corona,
and the F corona, respectively, and p, p~k~, and p~f~ are the percentage
polarizations of the total corona, the K corona, and the F corona,
respectively. An F corona model has been developed (Lamy et al., 1997)
which is an extension of the F model developed after the Skylab mission
(Saito et al., 1977) and the model of Koutchmy and Lamy (1985).
Neglecting the F corona term results in the familiar formula:

*pB = B~t~ -- B~r~ = C ∫ N~e~(x,y,z) G(x,y,z) dz*

######### ![](media/image2.wmf){width="3.0070866141732284in" height="3.264566929133858in"}![](media/image3.wmf){width="2.985433070866142in" height="4.65748031496063in"}where B~t~ and B~r~ are the tangential and radial components of the polarized intensity, C is a constant involving the Thomson cross-section for electron scattering and various scaling factors, N~e~ is the electron density at a point in the corona, G is a geometric function to the point along the line-of-sight and involves the solar surface limb darkening coefficients, and z is the distance along the line-of-sight (Billings, 1966). 

######### The PIXON Method

We employ the PIXON algorithm in our reconstruction technique. The
tomographic reconstruction of the electron density in each voxel in the
volume of interest is a computationally intensive problem. The number of
voxels N can be 10^9^ for 1024x1024x1024 resolutions. The problem is
difficult because for most methods (i.e. FFT) the number of operations
scales as N^2^. The techniques that can be used for such reconstructions
are in their infancy. In the past, maximum entropy methods have been
used in astronomical problems. For example, the international radio
astronomy community uses maximum entropy methods for the reconstruction
of large images based upon incomplete Fourier-synthesis observations.

The PIXON method utilizes model reconstructions based upon sets of
spatially extended, overlapping, and transparent components called
"pixons" (see Figure 3). Furthermore, by minimizing the number of
degrees of freedom used through constructing the simplest model (minimum
complexity), the PIXON solutions will have less noise and fewer
artifacts than those found with other methods. By applying strict
statistical criteria, the PIXON method minimizes the number of pixons
needed to describe the structure, and is able to achieve better
sensitivity and resolution than other methods.

The PIXON method is an iterative algorithm. With each
iteration![](media/image4.wmf)is calculated from the differences between
the synthetic coronagraph observations and the actual observations. The
PIXON method differs from maximum entropy in the goodness-of-fit metric,
which is defined in terms of the pixon number. The final solution has
the lowest pixon count. At each step strict statistical acceptance
criteria are applied and pixons that do not meet them are rejected. This
is done using a locally adaptive technique. In this manner, the number
of pixons, parameters, and degrees of freedom used to describe the
reconstruction is optimized. The reconstruction itself is derived by
summing the contributions of the various pixons that overlap the voxels
of interest. Finally, the computation time scales as N, not as a power
of N.

![](media/image5.wmf){width="3.011023622047244in"
height="1.2338582677165355in"}This description of the PIXON method is
necessarily brief. The formalism and methodology are described in detail
by Richard Puetter and his collaborators (Pina and Puetter, 1993;
Puetter, 1995; Puetter, 1996; Puetter, 1997; Puetter and Yahil, 1999). A
previous application of the PIXON method to solar data can be found in
Metcalf et al. (1996), who applied it to Yohkoh HXT data.

#### 

The PIXON algorithm is a general-purpose reconstruction package. For our
application it needs the geometrical, physical, and rendering additions
described above. The package itself must be customized for our problem
in at least four aspects: the definition of the pixon set; acceptance
criteria for retaining pixons; weights for additional constraints
(Lagrange multipliers); and the initial models, which may be uniform,
the result of a prior reconstruction, a physical model, etc.

#### ***Initial Work with our Reconstruction Technique***

PIXON software is being commercially developed by the company PIXON LLC.
We have obtained the PIXON software package and have begun the process
of understanding and utilizing the code, with very useful help from
PIXON LLC. We discuss some of our initial work, which will illustrate
aspects of reconstruction from two viewpoints which we discussed above.

We illustrate 3D 64x64x64 reconstructions of simple geometrical objects.
We can use real physics with two polarization states, but currently only
a point source Sun and geometry for viewing from infinity (all
lines-of-sight from viewpoints in the ecliptic plane are parallel).
Starting model objects have uniform density inside, while outside array
densities are zero. We use some specific PIXON jargon to describe our
work. The CCD white light 2D observation rendered from an electron
density distribution is called the DATA; the 3D electron density
distribution itself is called the IMAGE.

We reconstructed two model tilted cylinders of uniform density (toy
"polar plumes") using the "standard" PIXON code before any customization
was performed. In this early calculation we used only a linear
integration along the line-of-sight with no polarization (no physics).
The input data to PIXON consisted of the 2D rendered views (DATA) as
seen from two viewpoints (x,y axes) at 90 degrees. The PIXON code then
reconstructs a 3D density distribution solution (IMAGE). This
reconstruction process is shown in Figures 4 (rendered DATA views of the
starting model and final solution) and 5 (IMAGE 3D electron density
distribution visualizations of the starting model and final solution).
The IMAGE visualization uses the IDL SHADE_VOLUME procedure with a
chosen view angle which effectively shows the structure. We also show in
Figure 4 a z axis viewpoint DATA rendering of the IMAGE. Because this is
not actually used in the reconstruction, a comparison of the z axis DATA
rendered from the starting model and the DATA rendered from the final
reconstruction shows the difference between the starting and
reconstructed IMAGE along the viewpoint from which the reconstruction
was least constrained.

The initial and reconstructed DATA from the x,y axes viewpoints are
near-identical. However, the z axis DATA, and the reconstructed IMAGE,
show that the solution found from two input viewpoints consists of four
separate structures, not the initial two cylinders. The reconstructed
IMAGE is a perfectly real solution, and renders a near-identical DATA to
the input DATA. For two input viewpoints, it is actually a simpler
solution than the initial two cylinders.

Figures 6 (DATA) and 7 (IMAGE) demonstrate a reconstruction using DATA
rendered from the same two cylinders model, but this time using three
viewpoints (0, 40, 90 degrees). The third viewpoint was added as a test
of our reconstruction technique and to aid in our understanding of the
previous calculation. Now we find a solution IMAGE which is much more
like the initial IMAGE. This illustrates clearly the limitations of
reconstructions from only two viewpoints, where the solution is only
minimally determined. Later in this proposal we will briefly discuss
several ideas we intend to investigate to guide the two viewpoint
reconstruction to a solution which, perhaps, makes more physical sense.
But it is important to realize that the counter-intuitive four structure
solution for two viewpoints is in fact a mathematically justified
solution. In addition, we note that the third viewpoint input DATA might
be provided during the STEREO mission from a still-operating SOHO LASCO
coronagraph.

Finally, Figures 8 (DATA) and 9 (IMAGE) illustrate reconstruction of a
hemispherical shell (toy "CME") at an oblique (30,30 degrees) angle from
the Sun as seen from two viewpoints (x,y axes) at 90 degrees. In this
reconstruction we included the physics of Thomson scattering of photons
from a point source Sun, and tangential and radial polarization states.
The DATA renderings for simplicity show just the total intensity, not
the polarization components. The inputs to this reconstruction then are
four CCD images, two polarizations each from two viewpoints (x,y axes)
at 90 degrees. The superposed circles represent on the scale of this
calculation the solar diameter and a representative occulter at 1.3
R~O~. Again, the difference between the initial model IMAGE and the
final IMAGE solution, seen in the z axis DATA in Figure 8 and the IMAGE
visualizations in Figure 9, this time from two representative view
angles, shows the limitation of reconstructions from only two
viewpoints.

**Although these results are preliminary, they give us confidence that
our reconstruction technique can accomplish the difficult tasks we
propose.**

**Figure 4. Starting (True) and Reconstructed DATA, from x, y, and z
axis viewpoints, for the two tilted cylinders. Reconstruction uses two
input viewpoints separated by 90 degrees.**

![](media/image8.png){width="2.8006944444444444in"
height="4.127777777777778in"}

![](media/image9.png){width="6.382638888888889in"
height="4.142361111111111in"}

**Figure 6. Starting (True) and Reconstructed DATA, from x, y, and z
axis viewpoints, for the two tilted cylinders. Reconstruction uses three
input viewpoints separated by 40 and 90 degrees.**

![](media/image10.png){width="3.009027777777778in" height="4.18125in"}

![](media/image11.png){width="6.388194444444444in"
height="4.288194444444445in"}

**Figure 8. Starting (True) and Reconstructed DATA, from x, y, and z
axis viewpoints, for the hemispherical shell. Reconstruction uses two
input viewpoints separated by 90 degrees.**

The execution times we found were consistent with the scaling, using
current notebook PCs as the platform, suggested to us by PIXON LLC:
64x64x64 reconstructions in 2 minutes. This scales to 2 min x 16^3^ for
a 1024x1024x1024 reconstruction. We assume that a factor of 4
improvement in chip speeds is reasonable by 2005, which would then give
34 hours execution time. We have so far made no attempt to optimize our
calculations. Remembering that some fraction of the coronagraph field is
occupied by the occulter, and that many white light scenes have only a
limited area of detailed structures, it is reasonable that we can speed
the calculations by a factor of two from applying numerical techniques
for sparse arrays, and perhaps from computational acceleration by use of
a Beowulf network. Our goal is to achieve a practical computational
scheme which performs a 1024x1024x1024 reconstruction in 16 hours or
less.

***Scientific Investigations using our Reconstruction Technique***

We will apply our reconstruction technique to synthetic test models and
to existing solar observations for scientific investigation of polar
plumes, equatorial streamers, and CMEs. We discuss the science goals
which we believe that we can achieve with 3D reconstructions, as well as
some of the associated analysis techniques which we hope to employ.

*SOHO LASCO coronagraph observations*. For sufficiently long-lived
structures, a series of observations from one vantage point can be used
to mimic pairs of observations from two separated viewpoints. In
particular, LASCO coronagraph observations over rotational time periods
can be used to mimic pairs of simultaneous images from two viewpoints in
the ecliptic plane. LASCO images should yield meaningful reconstructions
for polar plume and equatorial streamer observations. Because of their
rapid temporal evolution, LASCO observations of CMEs cannot mimic two
separated vantage points.

*Background components*. We will examine division of the observed
intensities between a slowly evolving background component, effectively
constant on the time scale of interest, and a primary, perhaps evolving,
structure. Background does not refer to emission from structures behind
the structure of primary interest, but instead to a constant specified
component of the total intensity of the heliosphere. In particular cases
the background might be a constant F corona model (Koutchmy and Lamy,
1985); or F corona plus a general ellipsoidal global heliospheric
electron density as a function of radius, polar angle, and activity
level (e.g., Saito, 1950, Saito et al., 1977), to model equatorial
streamers; or F corona, ellipsoidal heliospheric corona, and equatorial
streamers, to model CMEs.

*Extrapolated coronal magnetic fields*. We will use extrapolated coronal
potential field calculations from photospheric magnetograms to better
understand the physical significance of the reconstructed solutions, and
would expect to achieve plausible consistency of the reconstruction and
the extrapolated coronal fields. Specifically, we expect that equatorial
streamers will lie along the coronal current sheet as determined from a
potential field calculation. However, the extrapolated fields are not
accurate enough to use as an actual constraint, since they are
calculated from the longitudinal, not vector, magnetogram, and often
disagree in the low corona with actual observations of the positions of
coronal loops (see illustration in Aschwanden et al., 2000b).

The plasmas in the solar corona and heliosphere are organized into a
variety of morphological structures, which evolve over a range of time
scales. We propose to apply our 3D reconstruction work to the study of
three types of coronal structures: polar plumes, equatorial streamers,
and CMEs. The physical understanding of all three structures can be
greatly furthered by the simple geometrical determination of their
morphologies, which is the most direct product of 3D reconstruction. In
addition, each will aid in the development and testing of our
reconstruction algorithm.

\(1\) *Polar plumes* appear to be linear tubes, anchored in polar
coronal holes, which do not appreciably expand in diameter with
increasing radial distance (Karovska et al., 1999). Plumes appear in 2D
images to converge to a common interior point between disk center and
the pole, and are not truly radial. They often can be observed as an
individual structure over at least a solar rotation.

Science: With a true 3D reconstruction, we will re-examine the above
claims. Does the reconstruction support a hydrostatic equilibrium
solution of density vs. height? If so, for what temperature? How far can
individual streamers be traced outward? Is there any detectable
expansion of the tube? Using the Carrington map technique to aid in
following the histories of individual plumes as far as possible until
disappearance, can we establish statistical properties of lifetimes? The
extrapolated coronal field is usually taken to be radial outward from
the potential source surface, at approximately 2.5 R~O~. Yet the polar
plumes do not appear to be radial, even at these distances. Can we
reconcile these two points, and are there any implications for
extrapolation models?

LASCO provides plentiful observational material; we will select
observations from early in the SOHO mission at solar minimum when the
plumes are most visible and the global structure simplest.

*(2) Equatorial streamers* are usually assumed to be projections of
sheets, extending into the heliosphere, of enhanced density along the
global magnetic field neutral surface (coronal current sheet) (i.e.,
Pneuman and Kopp, 1971, Wang et al., 1997). The coronal current sheet is
more equatorially symmetric at solar minimum, but is deformed at higher
activity levels by the strong active region magnetic fields to form a
more folded, complex structure. The appearance of equatorial streamers
viewed from Earth depends upon the exact line of sight through a complex
structure. In existing LASCO coronagraph observations individual
streamers can be observed, in the long lines of sight above or below the
solar poles, out to at least 60 R~O~. They are approximately constant on
solar rotation time scales.

We first illustrate an impressive example of work already done that
incorporates the tie point methods we have earlier described. Figure 10
shows results by Liewer et al. (2000), using LASCO coronagraph images
and a coronal potential field calculation at the time of Carrington
rotation 1935. The heliospheric coordinates of 8 individual radial
equatorial streamers were determined from synoptic coronagraph
observations. At top left is a Carrington format map of the coronal
field at 2.5 R~O~, taken as the potential source surface, where the dark
band is the coronal current sheet. The positions at 2.5 R~O~ of the 8
equatorial streamers are shown. At bottom left the coronal field at 2.5
R~O~ is shown on a sphere together with the projected radial paths of
the 8 streamers. At top right the synoptic photospheric magnetogram is
shown, while at bottom right the synoptic magnetogram is re-mapped onto
the 1 R~O~ solar sphere. Again, the same 8 streamers are shown on both.
The paths of the flux lines located on the source surface at 2.5 R~O~ at
the 8 points are followed along the extrapolated field back to the
photospheric surface. (Note that the x-y planes of the two bottom
figures are rotated!) This is a beautiful 3D geometric illustration that
equatorial streamers at 2.5 R~O~ are connected by field lines tracing
back to flux concentrations at the photosphere.

![](media/image14.png){width="3.2125984251968505in"
height="4.248031496062992in"}![](media/image15.png){width="3.160236220472441in"
height="4.2507874015748035in"}

In our study of limitations on tomographic reconstructions from only two
viewpoints, we intend to investigate whether it may be possible to
incorporate tie point identifications into the reconstruction as
additional constraints on the solution.

Science: We will study the evolution over the solar cycle of our
reconstructions of the equatorial streamer, and in particular, the
relationship with the coronal global magnetic field from extrapolated
photospheric magnetograms. Are the reconstructed equatorial streamers
along the global coronal current sheet throughout the solar cycle? How
do active regions complicate the streamer geometry? What is the relative
importance in streamer visibility between local density enhancement in
the streamer, perhaps associated with active regions (Liewer et al.,
2000), vs. lines of sight through multiple streamer folds (Wang et al.,
1997)? The synoptic LASCO observations throughout the mission cover
solar minimum in 1996 to the current peak activity levels of this cycle.
Using observations over a solar rotation to mimic two vantage points, we
can produce a series of streamer reconstructions and extrapolated
coronal magnetic fields for every Carrington rotation.

\(3\) *Coronal mass ejections* (CMEs) are disturbances originating from
the lower solar atmosphere which travel outwards through the
heliosphere. They are often viewed by the outer C3 coronagraph on LASCO
at up to 30 R~O~, but clearly continue to travel to much greater
distances. They evolve at very short time scales.

Observation and understanding of CMEs is really the principal goal of
the upcoming STEREO mission. The primary difficulty in the 3D
reconstruction of CMEs is their rapid evolution and high speeds.
Although LASCO has obtained extensive observations of CMEs over a range
of activity levels, here observations over a rotational cycle cannot
substitute for simultaneous observations from two viewpoints. Instead,
we will generate synthetic CME images, with and without added noise
various backgrounds, and image constrasts, for use in testing our
reconstruction technique. The advantage is that we know in this case the
exact solution. We described earlier in our **Introduction** the
calculational effort for the MHD CME models.

CMEs which are directed toward Earth can have major terrestrial effects.
The Heliospheric Imagers of the SECCHI experiment on STEREO are intended
to view at two orbital positions the Sun-Earth line from 12 -- 318 R~O~
(well past Earth at 215 R~O~), specifically to study the propagation of
CMEs which are Earth-directed. 3D reconstructions at these great
elongations push the geometrical and other constraints to extremes, and
we will investigate the accuracy of reconstructions from such high
elongation observations.

Science: We will examine with synthetic data the effect of rapid
temporal evolution and high outward velocities (1000 km s^-1^) on the
reconstruction. We will examine the degree of agreement of a modeled CME
structure, such as an evolving Chen model CME flux rope (Chen et al.,
1999, Wood et al., 1999), with existing LASCO CME observations. We will
investigate model accelerations and trajectories into the outer corona
and heliosphere using synthetic CME observations, and the reliability of
reconstructions in matching the known model solution. What are the
constraints on the viewpoint separation needed to best reconstruct the
model from the synthetic observations?

**DESCRIPTION OF THE PROPOSED WORK**

This proposal will fund a three year effort. The goals for each year
are:

1\. In the first year we will complete development of our reconstruction
program. We have already included the Thomson scattering physics for
both polarization states, but are using a point source Sun and geometry
for viewing from infinity (all lines-of-sight are parallel). We will:

-   Include a real solar and heliospheric geometry, and incorporate the
    solar photospheric spectral distribution observed through a model
    instrumental passband;

-   Incorporate the true heliospheric lines-of-sight geometry for
    arbitrary viewpoints;

-   Investigate reconstructions using input model DATA generated from
    simple geometrical model structures such as cylinders and
    hemispherical shells;

-   Organize the effort for the MHD model CME calculations.

2\. The focus of the second year will be on the application of our
reconstruction technique to LASCO observations of polar plumes and
equatorial streamers. We will:

-   Investigate questions of geometry and hydrostatic equilibrium within
    polar plumes;

-   Investigate the reconstruction of equatorial streamers over the
    solar cycle, and their responses to increasing levels of solar
    activity. We will compare the streamer results with extrapolated
    coronal field calculations;

-   Produce the CME models.

3\. In the third year we will investigate the reconstructions of our
model CMEs using the input synthetic CME observations and compare with
the initial model CME, including the quality of the reconstructions at
large elongations, out to 1 A.U. (i.e., Earth). Other aspects of the
importance of stereographic angle on these more complicated structures,
including the improvement to be gained by adding a third vantage point
(e.g. STEREO plus LASCO, SDO, ground-based coronagraphs and eclipse
observations, solar sentinels), will be examined.

**MANAGEMENT APPROACH**

All members of the proposal team are very experienced with coronal
physics, coronagraph observations, and scientific analysis of coronal
observations, and have participated in the LASCO and EIT experiments on
SOHO. The proposal team members also are closely associated, as Co-I's,
with the SECCHI investigation on the STEREO mission.

John Cook is a member of the Solar Physics Branch of the Space Science
Division at NRL and head of the Solar Radiation Section. He has worked
on both LASCO and EIT scientific investigations over the last several
years, and especially on emission measure analyses of coronal
temperature structure and instrumental calibration, using EIT
observations, in collaboration with Jeff Newmark. He will work on
development of the 3D reconstruction program and of model input
observations. His main focus during the second and third years will be
the scientific interpretation of the polar plume, streamer, and CME
reconstructions. He will be responsible for the execution of the effort
outlined for this proposal, but will work closely with the proposal
co-investigators in all phases to make best use of the extensive
knowledge of the individual team members. This proposal would fund three
months for this work for three years.

Jeff Newmark is a member of the Solar Physics Branch of the Space
Science Division at NRL. He is a Co-Investigator on the SOHO EIT
experiment, and a Co-Investigator on the SECCHI instrument for STEREO.
He developed the in-flight calibration of SOHO EIT, and has worked on
emission measure analyses of coronal observations, both in collaboration
with John Cook. He will work on development of the 3D reconstruction
program, supervise most of the numerical calculations, and participate
in the interpretation of the reconstruction results at all stages of the
effort. This proposal would fund three months for this work for three
years.

Paulett Liewer is in the Space Physics Element, and Deputy Program
Manager for Space Physics, at JPL. Her principal task is to provide the
MHD theoretical model of a CME to use in our CME reconstruction work.
She will organize her calculational effort in the first year, produce
the CME models the second year, and participate the third year in the
analysis of the reconstructed CME structures and comparison with the
initial model CMEs. This proposal would fund two weeks for this work for
three years.

###### RELEVANCE OF THE ANTICIPATED RESULTS FOR NASA PROGRAMS

This proposal will systematically investigate the 3D reconstructions
possible with white light coronagraph observations from two viewpoints,
which will be available from the SECCHI instrument on the STEREO
mission. STEREO is one of the first space missions of the Living With A
Star program, and it is imperative that techniques for analysis of the
STEREO observations, including those from the SECCHI coronagraphs, be
available and well-understood before actual launch of the spacecraft. 3D
reconstruction of coronal and heliospheric structures from coronagraph
images supports the development of "new techniques and models for
predicting solar/geospace disturbances which affect human technology,"
where it can be used to distinguish and model Earth-directed CMEs
observed by future missions such as STEREO.

This proposal is not intended to develop the actual code that will
perform the SECCHI-specific reconstruction analysis. That will be done
using the experiment development funding. We will investigate our
approach to reconstruction techniques which must be implemented by the
SECCHI software before launch, and develop the understanding necessary
to make scientifically optimized observing plans for the SECCHI
coronagraphs over all phases, and associated viewpoint angles, of the
STEREO mission.

This LWS Targeted Research and Techniques opportunity calls for the
exploitation of data from present and past missions to improve
knowledge. Our study of equatorial streamers over a solar cycle
addresses the LWS science objectives "How and why does the Sun vary?"
and "To improve knowledge of space environmental conditions and
variations over solar cycle".

**PROPOSAL REFERENCES**

Artzner, G. 2001, LASCO/EIT Consortium Meeting.

Aschwanden, M.J., Newmark, J.S., et al.: 1999, \"3-D-Stereoscopic
Analysis of Solar Active Region Loops: I. SOHO/EIT 171 A Observations at
Temperatures of 1.0-1.5 MK,\" Ap. J. 515, 842.

Aschwanden, M.J., Nightingale, R.W., and Alexander, D., 2000a, "Evidence
for Nonuniform Heating of Coronal Loops Inferred from Multithread
Modeling of TRACE Data," Ap. J. 541, 1059.

Aschwanden, M.J., Hurlburt, N., Alexander, D., Newmark, J.S., Neupert,
W.M., Klimchuck,

> J.A.,& Gary, G.A., 2000b, \"3D-Stereoscopic Analysis of Solar Active
> Region
>
> Loops: II. SOHO/EIT Observations at Temperatures of 1.5-2.5 MK,\" Ap.
> J., 531, 1129.

Batchelor, D.: 1994, "Quasi-Stereoscopic Imaging of the Solar X-Ray
Corona", Solar Phys. 155,

57-61.

Billings, D. E.: 1966, A Guide to the Solar Corona, New York: Academic.

Chen, J., Howard, R.A., Brueckner, G.E. Santoro, R., Krall, J.,
Passwaters, S.E., St. Cyr, O.C., Schween, R., Lamy, P., and Simnett,
G.M.: 1999, "Evidence of a Erupting Magnetic Flux Rope: LASCO Coronal
Mass Ejection of 1997 April 13", Ap. J. 490, L191-L194.

Frazin, R.A. 2000, Tomography of the Solar Corona I. A Robust,
Regularized, Positive

Estimation Method, Ap. J. 530, 1026.

Frazin, R.A. and Janzen, P., 2001, Tomography of the Solar Corona II.
RRPE Reconstruction

of the 3D Electron Density Distribution from LASCO-C2 Polarized
White-Light Images, Ap. J. submitted.

Gombosi, T.L., K.G. Powell, and D.L. De Zeeuw, \"Axisymmetric modeling
of cometary mass

loading on an adaptively refined grid: MHD results\", J. Geophys. Res.
99, 21525, 1994.

Groth, C.P.T., D.L. De Zeeuw, K.G. Powell, T.L. Gombosi, and Q.F. Stout,
\"A parallel

> solution‑adapter scheme for ideal magnetohydrodynamics\", in Proc.
> AIAA 14th Computational Fluid Dynamics Conference, AIAA Paper No.
> 99‑3273, 1999.

Howard, R.A., Moses, J.D., Socker, D.G., Dere, K.P., Cook, J.W., 2001,
Sun Earth Coronal and

Heliospheric Investigation (SECCHI), Adv. Space Res., in press.

Jackson, B.V. and Hick, P.L., 2000, Three Dimensional Tomography of
Heliospheric Features

Using Global Thomson Scattering Data, Adv. Space Res. 25, No. 9, 1875.

Jackson, B.V. and Hick, P.L., 1997, Heliospheric Tomography Using
Interplanetary Scintillation

Observations, Physics and Chemistry of the Earth 22 (5), 425-434.

Jackson, B.V. and Froehling, H.R., 1995, Three Dimensional
Reconstruction of a Coronal Mass

Ejection, A&A 299, p. 885.

Karovska, M., Wood, B., Cook, J.W., Brueckner, G.E.: 1999, "Study of
Dynamical Properties of

Coronal Structures in the Polar Regions", Space Sci. Rev. 87, 219.

Koutchmy, S., and Lamy, P.L.: 1985, "The F-Corona and the Circum-Solar
Dust Evidences and Properties", in Properties and Interactions of
Interplanetary Dust, ed. Giese, R.H. and Lamy, P., New York: Reidel, pp.
63-74.

Lamy, P., Quemerais, E., Llebaria, A., Bout, M., Howard, R, Schwenn, R.,
and Simnett, G.: 1997

> \"Electron densities in coronal holes from LASCO-C2 images\", in
> Proceedings of the 5^th^ SOHO Workshop, ESA SP-404, p 491.

Liewer, P.C., Hall, J.R., De Jong, E.M., Socker, D.G., Howard, R.A.,
Vourlidas, A., Crane, P., Reiser, P., and Rich, N.,: 2000,
"Determination of three-dimensional structure of coronal streamers and
relationship to the solar magnetic field", JGR, in press.

#### Metcalf, T.R. et al.: 1996, "PIXON-BASED Multiresolution Image Reconstruction for

YOHKOH'S Hard X-Ray Telescope," Ap.J. 466, 585.

Pina, R.K., and Puetter, R.C.: 1993, "Bayesian Image Reconstruction: The
Pixon and Optimal Image Modeling," Pub. Astron. Soc. Pac. 105, 630-637.

Pneuman, G.W., and Kopp, R.A.: 1971, "Gas-Magnetic Field Interactions in
the Solar Corona", Solar Phys. 18, 258-270.

Puetter, R.C.: 1995, " Pixon-Based Multiresolution Image Reconstruction
and the Quantification of Picture Information Content," Int. J. Image
Sys. & Tech. 6, 314-331.

Puetter, R.C.: 1996, "Information, Language, and Pixon-Based Bayesian
Image Reconstruction," in Digital Image Recovery and Synthesis III, P.S.
Idell and T.J. Schulz, eds., Proc. S.P.I.E. 2827, 12-31.

Puetter, R.C.: 1997, "Modern methods of image reconstruction," in
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Wang, Y.-M., Sheely, Jr., N.R., Howard, R.A., Kraemer, J.R., Rich, N.B.,
Andrews, M.D., Brueckner, G.E., Dere, K.P., Kooman, M.J., Korendyke,
C.M., Michels, D.J., Moses, J.D., Passwaters, S.E., Socker, D.G., Wang,
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G.M.: 1997, "Origin and Evolution of Coronal Streamer Structure During
the 1996 Minimum Activity Phase", Ap.J. 485, 875-889.

Wood, B.E., Karovska, M., Chen, J., Brueckner, G.E., Cook, J.W., and
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512, 484.

###### FACILITIES AND EQUIPMENT

## Naval Research Laboratory

At NRL we have access to existing computer resources, which include a
network of Sun workstations with a large amount of disk space, CDROM
writing and storage (jukebox carousel) capabilities, and networked
printers. A high end (e.g. Pentium 4, 2 GHz, 2 GB RAM, networked)
dedicated PC is being purchased from another funded NRL project. The
initial software development and testing will use this machine. It is
possible as the project matures beyond the initial software development
that we will take advantage of additional NRL facilities. A resource
co-located in the NRL Space Science Division and available to us will be
a Large Memory Computer which is an 8 Alpha processor Compaq computer
with approximately 128 GB of memory. This workstation is intended for
large data sets, including assembling large CCD images into movies or
data cubes and 3D modeling and reconstruction.

## Jet Propulsion Laboratory

All of the computational resources necessary for this project, including
workstations and parallel supercomputers, are available to Liewer
through JPL without additional investments. Most of the computations
will be done on the JPL 128 processor SGI Origin 2000 or the JPL Cray
SV1 16 processor vector computer. The JPL Solar System Visualization
Project 40-node Beowulf cluster (2 cpu/node, 1 GHz/cpu, 1 GB/node) is
also available to us for this project. JPL also has several SGI
workstations available for analysis.