Estimating Surface Flows from HMI Magnetograms

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Estimating Surface Flows from HMI Magnetograms

• Brian Welsch, SSL UC-Berkeley

GOAL Consider techniques available to estimate
flows from HMI vector magnetograms, to recommend
which to employ in the HMI pipeline.
2
Considerations

• Accuracy of estimated flows
• MEF DAVE4VM DAVE/FLCT inductive correction
• 2. Data rate
• 20 min. --- if inductivity matters, and if MDI
  is any guide
• 3. Processing rate
• DAVE4VM, DAVE, FLCT are fast, and parallelizable
• Noise handling
• DAVE4VM, DAVE, FLCT handle noise well,
• but other approaches are possible.

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- We created synthetic magnetograms from
ANMHD simulations of an emerging flux rope. -
In these data, both v B are known exactly.
Recently, we conducted quantitative tests of
accuracy using several available methods.
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Via several methods, we estimated v from N 7
pairs of magnetograms, with increasing ?ts.

• We verified that the ANMHD data were consistent
  with ?t Bn ? (vnBhor - vhorBn).
• Here, I show representative results from just a
  few of the methods tested
• Fourier LCT (FLCT, Welsch et al. 2004)
• Inductive LCT (ILCT, Welsch et al. 2004)
• Minimum Energy Fit (MEF, Longcope 2004)
• Differential Affine Velocity Estimator (DAVE,
• Schuck 2006)
• 5. DAVE4VM (Schuck 2008) --- DAVE for vector
  mgrams

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Here are MEFs estimated vs plotted over ANMHDs
v.
Like MEF, methods have problems at the edges of
magnetic flux.
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We verified the methods inductivity, i.e., that
they satisfy ?tBn ? (vnBhor vhorBn).
ILCT
FLCT
DAVE
MEF
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We tested Démoulin Bergers relation of uf to v.
FLCT
ILCT
MEF
DAVE

• Estimated ufs are highly correlated with ANMHDs
  uf.

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From the definition of uf, ?t Bn - ? (ufBn)
(5)

• Defining ufBn - ?f ? x ? n means ?t Bn
  ?2f
• So, for any given ?t Bn, you can recover some
  part of ufBn.

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Estimated vs are highly correlated with ANMHDs
v..
FLCT
ILCT
MEF
DAVE
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Not surprisingly, the methods performance
worsened as the time between magnetograms
increased.
vector errors (direction magnitude) were at
least 50 (!!!). speed errors (magnitude) were
smaller, but biases were seen.
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Some methods estimated the direction of v to
within 30º, on average.
CVEC and CCS were as defined by Schrijver et al.
(2005)
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Schuck has developed DAVE4VM, a new version of
DAVE meant for vector magnetograms.
Flows from DAVE4VM are as accurate as the best of
the methods tested by Welsch et al. (2007),
though its Poynting flux estimates are slightly
worse than MEFs.

• A manuscript is posted on arXiv.org, at
  http//arxiv.org/abs/0803.3472

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Considerations

• Accuracy of estimated flows
• MEF DAVE4VM DAVE/FLCT inductive correction
• 2. Data rate
• 20 min. --- if inductivity matters, and if MDI
  is any guide
• 3. Processing rate
• DAVE4VM, DAVE, FLCT are fast, and parallelizable
• Noise handling
• DAVE4VM, DAVE, FLCT handle noise well,
• but other approaches are possible.

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Pixel size and timescales of rotation magnetic
evolution affect optimal data rate.

• Target HMI resolution is 1 (Schou 2005),
• or 725 km at the Sun (cf., pixel size
  0.5)
• Here, I assume rebinning, so ?x 1 Pixels.
• (Ill try to use capital P for rebinned Pix.)
• Typical flows are vtyp 1 km/ sec. ? ?t 12
  min.
• (Rotation rate is 2 km/ sec. ? ?t 6 min.
• But rotation can be systematically removed.)

J. Schou, Instrument Performance and
Requirements, HMI Team Mtg. 05
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This data rate is slow enough that ?Bn(F) from
flows exceeds ?Bn(N) from noise.

• Since ?Bn(F) ?tmin ?hor(vn Bhor - vhor Bn) ?
• ?Bn(F) ?tmin (Btyp vtyp) / ?x gt ?Bn(N) ?
• ?tmin gt ?Bn(N) ?x / (Btyp vtyp)
• If Bz meets HMI noise target (Schou 2005), then
  sB 10 G, so ?Bn(N) sqrt(2) sB 14 G.
• With Btyp100 G, ?tmin gt 100 sec. 1.6 min.

? Linear in ?Bz(N)
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Inductivity might be an objective measure of
consistency when flows are not known.

• Inductivity is how well ?hor(vn Bhor - vhor
  Bn) matches ?Bn/?t
• Rieutord et al. (2001) argue that
• (1) spatial windowing during tracking, and
• (2) large ?t
• effectively average smaller-scale velocities.
• These can undermine the inductivity as a test of
  consistency.

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Inductivity is affected both by averaging
Binitial and Bfinal to reduce noise, and datas
?t.

• Avg. of five 1-min. cadence magnetograms prior to
  computing ?B (right) improves inductivity
  compared to using unaveraged ?B (left).

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Inductivity is affected both by averaging
Binitial and Bfinal to reduce noise, and datas
?t.

• 12 min. 18 min.

36 min. 54 min.
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Considerations

• Accuracy of estimated flows
• MEF DAVE4VM DAVE/FLCT inductive correction
• 2. Data rate
• 20 min. --- if inductivity matters, and if MDI
  is any guide
• 3. Processing rate
• DAVE4VM, DAVE, FLCT are fast, and parallelizable
• Noise handling
• DAVE4VM, DAVE, FLCT handle noise well,
• but other approaches are possible.

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Matching HMIs 10-minute vector magnetogram
cadence is feasible, with DAVE (or FLCT).
Both DAVE and FLCT are trivially parallelizable.

• HMI has Npix 3 x 106 Pixels within 60o of disk
  center.
• - 2x rebin of 40962 ? 20482 Pix
• - w/in 60o ? (0.866)2 x 20482 Pix 3 MPix
• Only tracking pixels if Bn gt Bthresh , for
  Bthr 20 G,
• 25 of Npix at solar max. for MDI ? 750 kPix
• 5 of Npix at solar min. for MDI ? 150 kPix
• DAVE tracks 4 kPix/sec in IDL (!), with one
  CPU
• - FLCT tracks 1kPix/sec in C, with one CPU
• - ?t (1 sec/1 kPix) x (750 kPix) 750 sec
  12 min!
• - at solar min., w/ Bthresh 100G (1 of
  Npix), ?t 30 sec.

DAVE4VM will be slower --- probably by a
factor of 3.
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Considerations

• Accuracy of estimated flows
• MEF DAVE4VM DAVE/FLCT inductive correction
• 2. Data rate
• 20 min. --- if inductivity matters, and if MDI
  is any guide
• 3. Processing rate
• DAVE4VM, DAVE, FLCT are fast, and parallelizable
• Noise handling
• DAVE4VM, DAVE, FLCT handle noise well,
• but other approaches are possible.

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The induction equation can be solved to roundoff
error, but data noise can make this undesirable.

• DAVE, DAVE4VM, and LCT codes solve for
  window-averaged flows, so average over noise ---
  a good thing!
• DAVE DAVE4VM use least-squares fitting to
  determine flows --- also good for dealing with
  noise.
• ILCT and MEF solve the induction equation
  exactly, which tends to produce spiky flows.
• Regularization to enforce smoothness, or the
  Kalman filter might enable combining local
  tracking results with global exact methods.
  This is ongoing research!

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Considerations

• Accuracy of estimated flows
• MEF DAVE4VM DAVE/FLCT inductive correction
• 2. Data rate
• 20 min. --- if inductivity matters, and if MDI
  is any guide
• 3. Processing rate
• DAVE4VM, DAVE, FLCT are fast, and parallelizable
• Noise handling
• We expect DAVE4VM, DAVE, FLCT to handle noise
  well (but still need to test this), other
  approaches might work.