Coronal Tomography with STEREO (+TRACE)

I will present some preliminary results on 3D coronal reconstructions from STEREO+TRACE data, using an algorithm that is surprisingly simple. The data are all simulated at this stage, because I need a known answer to test the algorithm (and I'm not yet ready to deal with instrumental issues in my software!)

Introduction

1. The STEREO mission
• Launched Oct 25, 2006; nominally 2-year mission.
• By orbital gymnastics, A stereoscopic view of the Sun and heliosphere.
• Where is STEREO now?
2. Extreme UltraViolet Solar Imager (EUVI) instrument
• FOV: full Sun, to 1.6 solar radii
• 1.6 arc second pixels
• Wavelengths: 171, 195, 284, 304 Angstroms (just like SOHO/EIT)
3. EUVI observations; anaglyph movie
4. Prior attempts (very little published):
• Reference list on the EUVI site
• Loop-by-loop approach: Aschwanden review paper
• Magnetic stereoscopy: Wiegelman and Inhester (2006 Sol Phys 236:25-40), SJC 9/5/06
5. Tomographic reconstruction (why is it hard?)
• m*N^2 vs. N^3
• Analytic solution using 2 spacecraft
• Perhaps a third view would help? TRACE observes in 171, 195, 284 from LEO.
6. Enter SMART, an algorithm I developed for MOSES spectral inversions.

SMART Algorithm

SMART = Smooth Multiplicative Algebraic Reconstruction Technique. After inventing SMART (I even had the acronym, believe it or not), I learned that it is a close relative of an existing algorithm with almost the same name, MART (e.g. Takayuki 1991, Verhoeven 1993, and references therein).

Here is how the algorithm works:

1. We are given projections at 3 angles theta: Id(theta, x, z), integrated through an unknown object O(x, y, z), on some finite domain D.
2. Initial guess G(x, y, z) = 1 on D.
3. Initialize undercorrection parameters, gamma(theta) = 1/3.

   ---

4. Rotate G and project to form Ip(theta, x, z).
5. Calculate reduced chi-squared, Xr(theta), for each theta.
6. gamma(theta) = gamma(theta) Xr(theta), but ensure gamma < 1/3.
7. Apply smoothing: G = G * K.
8. Correction factor Crot(theta, x, y, z) = (Ip/Id), constant in y!
9. Rotate Crot by -theta to form C(theta, x, y, z).
10. G = G prod_theta [ C(theta, x, y, z) ^ gamma(theta) ]
11. return to step 4. Iterate until Xr = 1 for all theta.

If the above does not make sense, it may help to ignore the z-dimension. z is perpendicular to the ecliptic plane. Each value of z corresponds to an essentially independent tomography problem, because the rotations (theta) are performed about z.

Some important points:

• Correction scheme is inherently positive.
• Smoothing also provides numerical stability.
• Feedback in step 6 ensures convergence to Xr = 1.

Fake Data

Why not real data? Well, real data is harder. But more importantly, when you are done analyzing real data, you don't know whether the answer is right!

Magnetic field

• Four sub-photosphere magnetic charges.
• Coronal null.

Loops

• Heated strands identified by photospheric mask.
• Emissivity goes as the number of heated field lines per voxel.
• Light smoothing.

Image data

• Randomly chosen clocking angle and heliographic longitude.
• 3 projections, maximum 3000 counts.
• Poisson noise.

Results

• Number of iterations is set to 15.
• Converges to Xr=1 (usually).
• The result is not sensitive to smoothing (the only adjustable parameter).
• The solution is stable for any number of iterations after convergence.
• Computation time: a few minutes on my MacBook Pro for a 169^3 volume.
• Most or all features recovered. Minimal artifacts.

• movie 4
• movie 3
• movie 2
• movie 1

The first three of these movies use the same volume model, with different randomly chosen clocking angles and heliographic longitudes. The last movie started with a slightly different magnetic configuration, with one loop that encounters the coronal null and spreads out in a fan to the edges of the domain. This demonstrates very clearly that the algorithm can handle arbitrary geometries, unlike previously developed algorithms for STEREO data.

Discussion

What do the results mean?

1. Inversion basically works. Better than I expected!
2. Even oddly-shaped structures are recovered, because there are no prior assumptions about geometry.
3. There is some blurring, which is to be expected.
4. Some ghosting is visible in the reconstruction. Can be improved if the volume is restricted to y > 0.

There is lots still to do. Next step is an ApJ letter. Perhaps a SECCHI Guest Investigator proposal. Applying to observations is a big task!