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.

Considerations 1.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 4.Noise handling DAVE4VM, DAVE, FLCT handle noise well, but other approaches are possible.

- 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.

Via several methods, we estimated v from N = 7 pairs of magnetograms, with increasing Δt’s. We verified that the ANMHD data were consistent with ∂ t B n =  ∙ (v n B hor - v hor B n ). Here, I show representative results from just a few of the methods tested: 1. Fourier LCT (FLCT, Welsch et al. 2004) 2. Inductive LCT (ILCT, Welsch et al. 2004) 3. Minimum Energy Fit (MEF, Longcope 2004) 4. Differential Affine Velocity Estimator (DAVE, Schuck 2006) … 5. DAVE4VM (Schuck 2008) --- DAVE for vector m’grams

Here are MEF’s estimated v’s plotted over ANMHD’s v. Like MEF, methods have problems at the edges of magnetic flux.

We verified the methods’ inductivity, i.e., that they satisfy ∂ t B n =  ∙ (v n B hor – v hor B n ). FLCT MEF DAVE ILCT

We tested Démoulin & Berger’s relation of u f to v. Estimated u f ’s are highly correlated with ANMHD’s u f. DAVE MEF FLCT ILCT

Defining u f B n = -  φ +  x ψ n means ∂ t B n =  2 φ So, for any given ∂ t B n, you can recover some part of u f B n. From the definition of u f, ∂ t B n = -  ∙ (u f B n ) (5)

Estimated v’s are highly correlated with ANMHD’s v.. DAVE MEF FLCT ILCT

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.

Some methods estimated the direction of v to within ~30º, on average. C VEC and C CS were as defined by Schrijver et al. (2005):

Schuck has developed DAVE4VM, a new version of DAVE meant for vector magnetograms. A manuscript is posted on arXiv.org, at 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 MEF’s.

Considerations 1.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 4.Noise handling DAVE4VM, DAVE, FLCT handle noise well, but other approaches are possible.

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 Δx ≈ 1” Pixels. (I’ll try to use capital P for rebinned Pix.) Typical flows are v typ ~ 1 km/ sec.  Δt ≈ 12 min. (Rotation rate is 2 km/ sec.  Δt ≈ 6 min. But rotation can be systematically removed.) * J. Schou, “Instrument Performance and Requirements,“ HMI Team Mtg. ‘05

This data rate is slow enough that ΔB n (F) from flows exceeds ΔB n (N) from noise. Since ΔB n (F) = Δt min  hor ·(v n B hor - v hor B n )  ΔB n (F) ~ Δt min (B typ v typ ) / Δx > ΔB n (N)  Δt min > ΔB n (N) Δx / (B typ v typ ) If B z meets HMI noise target (Schou 2005), then σ B ~10 G, so ΔB n (N) ~ sqrt(2) σ B ~ 14 G. With B typ ~100 G, Δt min > 100 sec. ~ 1.6 min.  Linear in ΔB z (N)

“Inductivity” might be an objective measure of consistency when flows are not known. “Inductivity” is how well  hor ·(v n B hor - v hor B n ) matches ΔB n /Δt Rieutord et al. (2001) argue that (1) spatial windowing during tracking, and (2) large Δt effectively average smaller-scale velocities. These can undermine the inductivity as a test of consistency.

Inductivity is affected both by averaging B initial and B final to reduce noise, and data’s Δt. Avg. of five 1-min. cadence magnetograms prior to computing ΔB (right) improves inductivity compared to using unaveraged ΔB (left).

Inductivity is affected both by averaging B initial and B final to reduce noise, and data’s Δt. 12 min. 18 min. 36 min. 54 min.

Considerations 1.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 4.Noise handling DAVE4VM, DAVE, FLCT handle noise well, but other approaches are possible.

HMI has N pix ~ 3 x 10 6 Pixels within 60 o of disk center. - 2x rebin of  Pix - w/in ~60 o  (0.866) 2 x Pix ~ 3 MPix Only tracking pixels if |B n | > |B| thresh, for |B| thr = 20 G, ~ 25% of N pix at solar max. for MDI  750 kPix ~ 5% of N pix 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/ |B| thresh = 100G (~1% of N pix ),  t ~ 30 sec. Matching HMI’s 10-minute vector magnetogram cadence is feasible, with DAVE (or FLCT). *Both DAVE and FLCT are trivially parallelizable. ** DAVE4VM will be slower --- probably by a factor of 3.

Considerations 1.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 4.Noise handling DAVE4VM, DAVE, FLCT handle noise well, but other approaches are possible.

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!

Considerations 1.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 4.Noise handling We expect DAVE4VM, DAVE, FLCT to handle noise well (but still need to test this), & other approaches might work.