This section describes the Magnetic Extrapolation algorithm steps of the IDL Solarsoft 'PFSS' package created by Dr Marc deRosa at LMSAL.

A complete list of the files in the PFSS package can be viewed at: http://orpheus.nascom.nasa.gov/~zarro/xdoc (search on 'pfss').

References:

1. Photospheric and Heliospheric Magnetic Fields - Carolus J Schrijver and Marc L deRosa.

For a mathematical analysis of the algorithm used, refer to Appendix C. 'Source-Surface Model' of Ref 1.

- Read in Magnetogram
- Compute Spherical Harmonic Transform on 2-D input arrays
- Setup integration weights.
- Apply Fast Fourier Transform (FFT) to the array.
- Next, apply Legendre transform. The output corresponds to term F
^{m}_{l}of Ref 1.

- Compute terms A
^{m}_{l}and B^{m}_{l}, corresponding to Eqn (8) of Ref 1.

- Compute Br in r,l,m space.
- Convert to spherical coordinates using inverse spherical transform. See Eqn (9) Ref 1.
- Compute Br in r,l,m space.
- Convert to spherical coordinates using inverse spherical transform. See Eqn (10) Ref 1.
- Compute Br in r,l,m space.
- Convert to spherical coordinates using inverse spherical transform. See Eqn (11) Ref 1.

- Calculate gridpoints from which to start tracing magnetic field lines.
- Compute field start coordinates - source surface inward.
- Setup grid type [simple, random lon, random lat/lon, file, uniform, flux weighted].
- Perform interpolation on input array.

- Compute field start coordinates - photosphere outward.
- Setup grid type simple, random lon, random lat/lon, file, uniform flux weighted.
- Perform interpolation on input array to derive Br of start points.

- Combine both sets of coordinates.

- Compute field start coordinates - source surface inward.
- Begin field tracing
- Compute Brm Bth, Bph at current point using interpolation.
- Determine if field is to be traced 1 way or both ways.
- Determine position of starting point.
- Integrate to form step size, adjusting if too big/too small.
- Loop through previous 4 steps to form collection of field line coordinates.

- Map open fields (coronal hole 'footprint')
- Compute lat/lon of each open field intersection with the photosphere and use to map coronal hole footprint.

To run the extrapolation process and create a .sav file that can be viewed by PFSS (see below) takes approximately 10-12 minutes.

The IDL Solarsoft PFSS package also includes a viewing widget that allows the user to display the extrapolated magnetic lines. The widget displays a flux density map (stored in .sav format) that allows the user to select a region of interest and specify the maximum number of lines to be plotted. Open and closed field lines are differentiated by colour.

-- MikeSmith - 07 Jun 2005

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