Solar Tomography

Nancy MezoJocelyn Renner

The Problem:

Ummm. Maybe we should think about non-invasive imaging….

The Corona

The Solar Corona

• The corona is the outer region of the sun’s atmosphere

• Extends from chromosphere to several million km

• Generates and accelerates the solar wind.

Neat facts about the corona:• The amount of light emitted by the solar

corona compares to that of the full moon.

• Nearly a vacuum (very low density)

• VERY hot --- temperatures greater than 1

million degrees Celsius.

Why do we care?

• Solar winds affect satellites, astronauts, and power grids.• Natural laboratory of super heated plasmas in strong fields.• We want to understand the universe.

What do we study?

• Topology

• The magnetic field

• Electron density and temperature

TopologyThe study of the features of a place or object.

Active regions, active region interconnections, coronal holes, bright points, loop structures, and filament

structures.

Magnetic Field(Magnetic fields are at the root of virtually all of the

features we see on and above the Sun.)

Electron Density

Derived from coronography of scattered white light above the limb.

Types of Data

• SXR -- Soft X-Ray• Higher temperatures

• EUV -- Extreme Ultraviolet

• Cooler Plasma, transition regions

• Radio• Free-free emissions• Magnetic field strength

What has been attempted...

• Bracewell• Altschuler & Perry• Hurlburt• Frazin• Komm• Davila

Fourier series Least Squares to find Fourier coefficientsFiltered Back ProjectionRobust Regularized Positive EstimationWaveletsAlgebraic Reconstruction Technique

1956 -- Bracewell used Fourier series to reconstruct sun spots

Altschuler and Perry

Least Squares to find Fourier Coefficients

• First to use solar rotation tomography (SRT) • Reconstruction of electron density distribution• One satellite observation• Assumed static structures on the sun• Two dimensional model--simple elliptic

Fourier Solutions

• Used least squares approach to find best set of coefficients.

• Does not guarantee positive solutions.

• Form of the basis functions imposes unrealistic constraints which are unrelated to the problem.

N.E. Hurlburt

Reconstruction Using FilteredBack Projection

• Reconstruction of Magnetic Field• One satellite observation -- used sun’s rotation• Assumed static structures on the sun• Two dimensional model--simple elliptic• Used SXT data from the Yohkoh Satellite

Filtered Back Projection•Accounts for the sun’s opacity•Row of pixels as a function of time gives

sinusoidal pattern.

Results:

• Reconstructed images on solar poleEquatorial reconstruction is complicated by highly active regions and faster rotation.

• ProblemsTopological change or rapid fluctuations lead to streaking or blurring.

Sudden disappearance of features when they pass behind the x-ray opaque sun.

Hurlburt’s Reconstruction

R.W. Komm

Wavelets

• Analysis of Magnetic Field• Used magnetograms from the Vacuum Telescope at the Kitt Peak Solar Observatory

Why wavelets?

Fourier transforms provide information about frequency, but not location in space. Wavelets provide

both.

Helps eliminate white noise

Shows the smaller magnetic structures are more intermittent, less space filling than larger ones.

Noise fills space at all length scales.

J.M. Davila

Multiple Spacecraft Model

• Simulation• Many spacecraft orbiting at 1 AU• Two dimensional model--simple

elliptic• Created graph of source distribution• Synthetically observed source by

summing the emission along the line of sight (LOS)

Source Distribution

Algebraic Reconstruction Technique (ART)

• Choose any projection and spread the value evenly along LOS

• Chose a second projection angle and spread the reconstructed array along

LOS• Spread difference between actual and

reconstruction along LOS• Repeat for each angle

Advantages:• Easy computationally• Allows for unevenly spaced observation• Number of observations can be varied easily• Unequal pixel size easily accounted for by rebinning

process• Converges to reasonable likeness after one iteration

Disadvantages:• Weighted more strongly toward larger objects• Tends to smear along LOS• No natural way to update pixels that don’t fall along any LOS• Depends on initial guess, unduly sensitive to out-lying points

in data

Results:• Angular placement:

The optimal configuration occurs where the craft are equally spaced.

For four craft, this occurs at 135º.

• Fourier space coverageSufficient samples give full coverage of Fourier Space

While theoretically no polar orbiting craft are needed, limitations on instruments provide a practical reason to employ them.

• Number of spacecraftThe larger number of spacecraft, the better the reconstruction.

An arrangement with four spacecraft is optimal.

• The point source resolution improves with Gaussian Resolution

• Detector NoiseSource of error

Reconstruction is degraded only slightly at attainable signal to noise ratios (O ~ 200 pixels)

Results:

R.A. Frazin

Robust, Regularized, PositiveEstimation

• Reconstruction of emissivity structures • RRPE method is less sensitive to outlying points

in data• Removes problems of dependence on initial

guess, non-positivity, non-robustness, and inappropriateness of Fourier expansion.

Question, comments,cheap shots or

one-liners?