Coronal Tomography with STEREO (+TRACE)

Charles Kankelborg

November 6, 2007


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!)


  1. The STEREO mission
  2. Extreme UltraViolet Solar Imager (EUVI) instrument
  3. EUVI observations; anaglyph movie
  4. Prior attempts (very little published):
  5. Tomographic reconstruction (why is it hard?)
  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:

Dynamic equilibrium between smoothing and ideal fit to the data!

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


Image data


The following movies are square root scaled. They compare the original volume model (left) to the SMART reconstruction (right).

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.


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!