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Deliverable eSDO-1121: Magnetic Field Extrapolation

M. Smith, E.Auden, L. van Driel-Gesztelyi

10 August 2005

The Magnetic field extrapolation process will calculate magnetic field lines between Solar active regions using vector and line-of-sight magnetograms from the SDO HMI instrument. The extrapolated fields will be shown as an overlay on the magnetogram and their footpoints displayed in tabular form.

The primary investigation will concentrate on Wiegelmann's [1] improved 'Optimization' method of extrapolation, which uses the nonlinear magnetic model. The nonlinear model provides the most accurate representation of magnetic fields, particularly around active regions. The Optimization technique, originally developed by Wheatland [2], computes the magnetic field over a predefined volume, or box, using full-disk vector magnetogram data for the extrapolation. The magnetogram is also used to determine the bottom boundary conditions of the box, while the lateral and top boundaries are computed from a Potential Field model, which provides a valid approximation of magnetic field activity high in the the Sun's corona.

Wiegelmann's Optimization method improves on other nonlinear methods by introducing the concepts of a boundary layer and weighting function. These diminish the influence of the lateral and top boundary conditions on the calculation of the magnetic field over the region of interest.

The vector magnetogram data will be preprocessed in two stages before it is submitted to the Optimization method for extrapolation. These are (in order):

- Azimuth Disambiguation
- Wiegelmann and Sakurai's preprocessing procedure

'Azimuth Disambiguation' (Georgoulis [3]) is a new technique designed to resolve the Pi ambiguity, one of several problems inherent in modern vector magnetograms. The Pi ambiguity describes the case where the azimuth angle of the transverse component of the magnetic field has two equally likely values 180 degrees apart. Azimuth Disambiguation will typically add a 1-2 minute processing time penalty to the overall extrapolation computation on an average desktop computer.

Another drawback of currently available vector magnetograms is the level of noise that is sometimes apparent in the transverse components. Such noisy data can violate the force-free assumptions used in the Optimization extrapolation and invalidate the results. Wiegelmann, Inhester and Sakurai [4] have recently developed a numerical technique that detects any data lying outside force-free constraints and 'smooths' them into a force-free condition. The amount of additional processing time incurred by this technique is unknown at present.

Should the Optimization method prove computationally expensive, then the 6 nonlinear force-free algorithms discussed at the Lockheed Martin NLFFF modelling meeting in May 2005 will be reviewed; the algorithm which provides the best balance between speed and accuracy will be deployed.

Again, if none of these is acceptably fast, then the simpler (but less accurate) Potential Field model will be used.

- HMI full-disk vector magnetogram.
- HMI line-of-sight magnetogram.

- Extrapolated fields overlayed on the original magnetogram.
- Tabular extension - footprints for each extrapolated field. A footprint will be described in terms of solar coordinates and radius.

The Magnetic Field Extrapolation tool will be included as 'C' code in the HMI pipeline at Stanford University. Calling syntax:

The Magnetic Field Extrapolation 'C' code will also be wrapped as an IDL application and distributed through the SolarSoft gateway at MSSL. Users will be able to access the extrapolation tool through SolarSoft installations at their institutions; access may potentially be available through the CoSEC VO project as well.

The tool's 'C' code can also be wrapped with the AstroGrid CEA software and made accessible as a web service should space be available on a suitably enabled machine within the UK.

The user wants to view extrapolated magnetic fields superimposed on a vector magnetogram image, and also a description of each field's footpoint that includes solar coordinates and radius.

- The user identifies the image taken during the specified time period.
- The user input the full-disk magnetogram HMI image to the Magnetic Field Extrapolation algorithm.
*(Currently there are no user-specified variables to describe for the loop recognition algorithm to be distributed through SolarSoft or deployed as an AstroGrid CEA service.)* - The algorithm is run and returns a FITS file to the user.
- The user can view an image within the FITS file displaying extrapolated magnetic fields superimposed in colour over the original HMI magnetogram image.
- The user can also view a table of footprints for each of the fields identified. The footprints are described in terms of solar coordinates and radius.

- The Magnetic Field Extrapolation algorithm receives a full-disk vector magnetogram as input.
- Georgoulis's Azimuth Disambiguation technique is applied and used to remove the Pi Ambiguity of the transverse components of the vector magnetogram data.
- Wiegelmann, Inhester and Sakurai's preprocessing procedure is applied to check for and 'smooth' magnetogram data that lie outside force-free constraints.
- A computational box is defined, i.e. a bounded volume which represents the region in which the magnetic field will be calculated.
- A physical domain is defined inside the box which represents the volume in which the nonlinear magnetic field will be calculated using vector magnetogram data.
- A boundary layer is defined which stretches from the edge of the physical domain to the computational box boundary.
- The measured normal component of the magnetic field, Bz, is used to calculate a potential magnetic field over the whole box using Seehafer's method [5] and is used to define the lateral and top boundary conditions.
- The vector magnetogram data is used to determine the bottom boundary (photosphere) conditions of the box.
- The nonlinear magnetic field is computed from vector magnetogram data using the Landweber iteration method [6]. A weighting cosine function is applied which is set to unity within the physical domain, decreases within the boundary layer and falls to zero at the lateral and top boundaries of the computational box.

- Thumbnail image of HMI magnetogram with superimposed magnetic fields (JPEG)
- Full sized low resolution image with superimposed magnetic fields (JPEG)

[1] T.Wiegelmann, 'Optimization code with weighting function for the reconstruction of coronal magnetic fields', Solar Physics, 219, 87-108 (2004)

[2] M.S.Wheatland, P.A. Sturrock and G.Roumeliotis, AstroPhysical Journal 540, 1150-1155.

[3] M. Georgoulis, A New Technique for a Routine Azimuth Disambiguation of Solar Vector Magnetograms, AstroPhysical Journal 629:L69-72.

[4] T. Wiegelmann, B.Inhester, T.Sakurai, 'Preprocessing of vector magnetograph data for a nonlinear force-free magnetic field reconstruction', Solar Physics, In Press (2005).

[5] N.Seehafer, Solar Physics, 58, 215 (1978).

[6] A.K.Louis, 'Inverse und schlecht gestellte Probleme', Teubner Studienbuecher, ISBN 3-519-02085-X (1989) (discusses Landweber's iteration method).

-- MikeSmith - 19 Aug 2005

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