1. HMI, Photospheric Flows and ILCT Brian Welsch, George Fisher, Yan Li, & the UCB/SSL MURI & CISM Teams HMI Team Mtg., 2006M3: Mag Data Products Correlation Tracking Image Deprojection Output Pipeline

2. Velocity inversions generate a 2D map v(x 1,x 2 ) from one 2D image, f 1 (x 1,x 2 ), to another, f 2 (x 1,x 2 ). The map depends upon: 1.the difference  f(x 1,x 2 ) = f 2 (x 1,x 2 ) – f 1 (x 1,x 2 ) 2.assumption(s) relating v(x 1,x 2 ) to  f/  t, e.g.: –continuity equation,  f/  t +  t  (v t f) = 0, or –advection equation,  f/  t + (v t   t )f = 0, etc. Based on the assumption chosen, v(x 1,x 2 ) is not necessarily velocity – e.g., group velocity of interference patterns.

3. 16 Feb 2006HMI+AIA, M3: Mag Data Products3 Local correlation tracking (LCT) finds v(x 1,x 2 ) by correlating subregions; it assumes advection. 1) for ea. (x i, y i ) above |B| threshold … 2) apply Gaussian mask at (x i, y i ) … 3) truncate and cross-correlate… * 4) v(x i, y i ) is inter- polated max. of correlation funct = = =

4. Demoulin & Berger (2003) argued that LCT applied to magnetograms does not necessarily give plasma velocities. u f  v n B h -v h B n is the flux transport velocity u f is the apparent velocity (2 components) v  is the actual plasma velocity (3 comps) The apparent motion of flux on the photosphere, u f, is a combination of horizontal flows and vertical flows acting on non-vertical fields.

5. 16 Feb 2006HMI+AIA, M3: Mag Data Products5 The magnetic induction equation’s normal component relates velocities to dB n /dt.  B n /  t =  h  (v n B h -v h B n ) = -  h  (u f B n ) In fact, -  h  (u LCT B n ) only approximates  B n /  t, so u LCT  u f Inductive LCT (ILCT) finds u f that matches  B n /  t exactly and closely matches u LCT. Writing u f B n = -  h  +  h x(  n), we find –  via  B n /  t =  h 2  –  by assuming u f = u LCT, so  h 2  = -  h x( u LCT B n )

6. Doppler shifts (  v n ) can’t distinguish between flows paral- lel to B, perpendicular to B, or in an intermediate direction. Since  B n /  t =  x (v x B), flows v || along B do not affect  B n /  t, so “inductive flow” methods only determine v . Once v  is known, the measured Doppler shift allows determination of v ||.

7. 16 Feb 2006HMI+AIA, M3: Mag Data Products7 Aside: fundamentally, two components of u f (x 1,x 2 ) cannot determine three components of plasma velocity, v(x 1,x 2 ). Hence, other velocity fields v(x 1,x 2 ) consistent with  B n /  t can be found. Other techniques available include: –Minimum Energy Fit (MEF, Longcope, 2004) –Differential LCT (DLCT) & Differential Affine Velocity Estimator (DAVE) (Schuck, 2006)

8. The FLCT code’s current version combines pro- grams written in IDL & C, and open source code. 1.IDL 2.C 3.Standard C library routines: stdio.h, stdlib.h, math.h 4.Fastest Fourier Transform in the West (FFTW), v. 3.0 The executable has been compiled & tested on several architectures. 1.Linux 2.Solaris 3.Windows 4.Macintosh

9. HMI has N pix ~ 10 7 pixels within 60 o of disk center. - MDI’s 1024 2  HMI’s 4096 2  x 16 - MDI, w/in ~30 o  HMI, w/in ~60 o  x 2.5 We track pixels with |B n | > |B| thresh = 20G ~ 25% of N pix at solar max. ~ 5% of N pix at solar min. FLCT speed is ~linear in N pix correlated. -  t ~ (1 sec/100 pix) x (2.5 x 10 6 pix) ~ 2.5 x 10 4 sec ~ 7 hr! - at solar min., w/ |B| thresh = 100G (~1% of N pix ),  t ~ 20 min. Matching HMI’s 10-minute vector magnetogram cadence will be challenging.

10. 16 Feb 2006HMI+AIA, M3: Mag Data Products10 IVM difference images of B LOS in AR 9026, with a ~4 min. cadence, show large-scale, alternating field fluctuations that inhibit accurate tracking. Velocity estimates work from difference images, so temporal artifacts must be removed.

11. Accurate velocity estimates also require deprojection of full-disk magnetograms. Away from disk center, flows with a component along LOS are foreshortened by curvature of the solar surface. Conformal deprojections, e.g., Mercator, locally preserve angles; scales are distorted, but easily fixed. This is optimal for tracking, since neither flow component is biased by the deprojection. (Apparent changes in lengths perpendicular to the LOS from center-to-limb are negligible.)

12. FLCT was initially tested using a known image. We found FLCT could accurately reconstruct the imposed flow.

13. 16 Feb 2006HMI+AIA, M3: Mag Data Products13 FLCT was also tested on magnetograms with imposed differential rotation – again, recovering the input flow. White dots are imposed differential rotation profile; red dots are raw velocities from Mercator projection; green are properly rescaled; white diamonds are latitudinally binned averages of green dots.

14. 16 Feb 2006HMI+AIA, M3: Mag Data Products14 We have implemented a preliminary, automated “Magnetic Evolution Pipeline” (MEP). http://solarmuri.ssl.berkeley.edu/~welsch/public/data/Pipeline/ cron checks for new magnetograms with wget New magnetograms are downloaded, deprojected, and tracked using FLCT. The output stream includes deprojected m-grams, FLCT flows (.png graphics files & ASCII data files), and tracking parameters. Full documentation & all codes are on line.

15. 16 Feb 2006HMI+AIA, M3: Mag Data Products15 Sub-pixel interpolation was made more efficient. Correlation is now accomplished by spawning a C subroutine that employs FFTW. FLCT is readily parallelizable; we envision this “soon.” Computing velocities in neighborhoods, as opposed to each pixel, is another way to increase speed. Several performance-enhancing modifications to FLCT were implemented and more are planned.

16. 16 Feb 2006HMI+AIA, M3: Mag Data Products16 Conclusions Accurate flow estimates will require –deprojection of full-disk magnetograms, and –careful temporal filtering. Matching planned data cadences will be challenging. Solutions: –parallelization –find v(x 1,x 2 ) on tiles, not every pixel –more restricitve |B n | thresholding Essential tools for an LCT pipeline are in place.

17. 16 Feb 2006HMI+AIA, M3: Mag Data Products17 References Démoulin & Berger, 2003: Magnetic Energy and Helicity Fluxes at the Photospheric Level, Démoulin, P., and Berger, M. A. Sol. Phys., v. 215, # 2, p. 203-215. Longcope, 2004: Inferring a Photospheric Velocity Field from a Sequence of Vector Magnetograms: The Minimum Energy Fit, ApJ, v. 612, # 2, p. 1181-1192. Schuck, 2005: Tracking Magnetic Footpoints with the Magnetic Induction Equation, ApJ (submitted, 2006) Welsch et al., 2004: ILCT: Recovering Photospheric Velocities from Magnetograms by Combining the Induction Equation with Local Correlation Tracking, Welsch, B. T., Fisher, G. H., Abbett, W.P., and Regnier, S., ApJ, v. 610, #2, p. 1148-1156.

18. Yang’s e-mail. “It would be great if you can talk about your ILCT method/ code during the session. Because this session is ‘data products’ session, … briefly summarize your algorithm first, and then focus on addressing following issues: 1.Nature of the codes (Language, etc); 2.Additional supporting software (IDL, MATHLIB,...); 3.Computational requirements (run time estimate, system requirements, etc); 4.Requirements for the input data & format of the output products; 5.Potential challenges, test procedures, target date for completion of codes, etc... Time is 15 minutes, but … leave 5 minutes for further discussion.”