Tracking Vector Magnetograms with the Magnetic Induction Equation
Abstract
The differential affine velocity estimator (DAVE) developed in Schuck (2006) for estimating velocities from lineofsight magnetograms is modified to directly incorporate horizontal magnetic fields to produce a differential affine velocity estimator for vector magnetograms (DAVE4VM). The DAVE4VM’s performance is demonstrated on the synthetic data from the anelastic pseudospectral ANMHD simulations that were used in the recent comparison of velocity inversion techniques by Welsch et al. (2007). The DAVE4VM predicts roughly 95% of the helicity rate and 75% of the power transmitted through the simulation slice. Intercomparison between DAVE4VM and DAVE and further analysis of the DAVE method demonstrates that lineofsight tracking methods capture the shearing motion of magnetic footpoints but are insensitive to flux emergence — the velocities determined from lineofsight methods are more consistent with horizontal plasma velocities than with flux transport velocities. These results suggest that previous studies that rely on velocities determined from lineofsight methods such as the DAVE or local correlation tracking may substantially misrepresent the total helicity rates and power through the photosphere.
1 Introduction
Coronal mass ejections (CMEs) are now recognized as the primary solar driver of geomagnetic storms Gosling (1993). Several theoretical mechanisms have been proposed as drivers of CMEs, including large scale coronal reconnection Sweet (1958); Parker (1957); Antiochos et al. (1999), emerging flux cancellation of the overlying coronal field Linker et al. (2001), flux injection Chen (1989, 1996), the kink instability of filaments Rust & Kumar (1996); Török et al. (2004); Kliem et al. (2004), and photospheric footpoint shearing Amari et al. (2000, 2003a, 2003b); Schrijver et al. (2005). All of these CME mechanisms are driven by magnetic forces. The main differences depend on whether the magnetic helicity and energy are first stored in the corona and later released by reconnection and instability or whether the helicity and Poynting fluxes are roughly concomitant with the eruption. The timing and magnitude of the transport of magnetic helicity and energy through the photosphere provides an important discriminator between the mechanisms. In addition, eruption precursors in the photospheric magnetic field might provide reliable forecasting for space weather events. However, reliable, repeatable photospheric precursors of CMEs have so far eluded detection Leka & Barnes (2003a, b, 2007).
The magnetic helicity and Poynting flux may be estimated from photospheric velocities inferred from a sequence of magnetograms Berger & Ruzmaikin (2000); Démoulin & Berger (2003). However, accurately estimating velocities from a sequence of images is extremely challenging because image motion is ambiguous. The “aperture problem” occurs when different velocities produce image dynamics that are indistinguishable Stumpf (1911); Marr & Ullman (1981); Hildreth (1983, 1984). Optical flow methods solve these underdetermined or illposed problems that have no unique velocity field solution by applying additional assumptions about flow structure or flow properties. Both Schuck (2006) and Welsch et al. (2007) provide an overview of optical flow methods for recovering estimates of photospheric velocities from a sequence of magnetograms Kusano et al. (2002, 2004); Welsch et al. (2004); Longcope (2004); Schuck (2005, 2006); Georgoulis & LaBonte (2006). Currently, most methods for estimating photospheric velocities implement some form of the normal component of the induction equation
(1) 
where the plasma velocity and the magnetic fields are decomposed into a local righthanded Cartesian coordinate system with vertical direction along the axis and the horizontal plane, denoted generically by the subscript “h,” containing the  and axes.
Démoulin & Berger (2003) observed that the geometry of magnetic fields embedded in the photosphere implied that
(2a)  
where denotes the flux transport vector, is the horizontal footpoint velocity or flux transport velocity and is the plasma velocity perpendicular to the magnetic field . The flux transport vectors are composed of two terms and representing shearing due to horizontal motion and flux emergence due to vertical motion respectively. Equation (2a) may be used to transform (1) into a continuity equation for the vertical magnetic field  
(2b) 
where plasma velocity may be written generally in terms of the flux transport velocity as
(3a)  
(3b)  
(3c) 
and the subscripts “” and “” denote plasma velocities parallel and perpendicular to the magnetic field respectively. Equations (1ac) are the algebraic decomposition Welsch et al. (2004) generalized for arbitrary parallel velocity , but the value of does not affect the perpendicular plasma velocity (3b)(3c) or the perpendicular electric field
(4) 
which both depend only on the flux transport velocity .
Equations (1)(1) should be formally distinguished from the inverse problem for determining an estimate of the plasma velocity from vector magnetograms using the normal component of the magnetic induction equation
(5a)  
where  
(5b) 
and the inverse problem for determining flux transport velocity from the evolution of the vertical magnetic field or lineofsight component
(6) 
The notation , denoting an optical flow estimate, emphasizes that determined from (6) is not necessarily immediately identified with the flux transport velocity . Equations (1)(6) are illposed inverse problems because of two ambiguities:

The Helmholtz decomposition of the flux transport vectors Welsch et al. (2004); Longcope (2004)
(7) where is the inductive potential and is the electrostatic potential manifestly demonstrates that only inductive potential may be unambiguously determined from the local evolution of in (1). The electrostatic potential must be constrained by additional assumptions. By analogy, (6) is also illposed for the same reason; is not constrained by the local evolution of .
The first ambiguity may be resolved for (5a) by the induction method (IM) Kusano et al. (2002, 2004), minimum energy fit (MEF) Longcope (2004), or the differential affine velocity estimator for vector magnetograms (DAVE4VM) presented in § 2. These methods produce a unique solution for , but not necessarily the unique solution that corresponds to . The first ambiguity may be resolved for (6) by local optical flow methods such as the differential affine velocity estimator (DAVE) Schuck (2006), its nonlinear generalization Schuck (2005), global methods Wildes et al. (2000), the minimum structure reconstruction (MSR) Georgoulis & LaBonte (2006) which imposes as an assumption, or hybrid localglobal methods such as inductive local correlation tracking (ILCT) Welsch et al. (2004). These methods produce a unique solution for , but not necessarily the unique solution that corresponds to .
Several assumptions have been used either explicitly or implicitly to resolve the second ambiguity. Chae et al. (2001) conjecture that local correlation tracking (LCT) Leese et al. (1970, 1971); November & Simon (1988) provides a direct estimate of the horizontal photospheric plasma velocity: . Démoulin & Berger (2003) conjecture that lineofsight tracking methods, and in particular LCT, estimate the total flux transport velocity . Schuck (2005) formally demonstrated that LCT is consistent with the advection equation
(8) 
not the continuity equation in (6), but that LCT could be modified to be consistent with (6) by direct integration along Lagrangian trajectories in an affine velocity profile. Nonetheless, both conjectures may be considered in the context of (6). Under Chae et al.’s (2001) assumption, the flux transport velocity would be derived from lineofsight optical flow methods via
(9) 
where in principle, might be approximately determined from Doppler velocities near disk center. Under Démoulin & Berger’s (2003) assumption, the total flux transport velocity would be derived from lineofsight optical flow methods via
(10) 
The Ansatz has important implications for solar observations. This conjecture implies that the total helicity and Poynting flux may be estimated by tracking the vertical magnetic field or by tracking the lineofsight component near disk center as a proxy for the vertical magnetic field. Démoulin & Berger’s (2003) Ansatz has largely been accepted by the solar community Welsch et al. (2004, 2007); Kusano et al. (2004); Schuck (2005, 2006); LaBonte et al. (2007); Santos & BÃ¼chner (2007); Tian & Alexander (2008); Zhang et al. (2008); Wang et al. (2008). However, equivalence between and for lineofsight methods has never been practically established. These two different hypotheses (9) and (10) for the interpretation of inferred by DAVE will be considered in § 4.
The second ambiguity usually is not resolved using only information about the magnetic fields. The velocity field inferred by the IM Kusano et al. (2002, 2004) does produce a component of the plasma velocity along the magnetic field, but this was simply subtracted off in Welsch et al. (2007). In the absence of a reference flow, possibly derived from Doppler measurements or LCT, the MEF imposes Longcope (2004). ILCT and the original algebraic decomposition both assume Welsch et al. (2004). Georgoulis & LaBonte (2006) describe a method for inferring from Doppler measurements for MSR. For DAVE4VM the second ambiguity is resolved simultaneously with the first. The DAVE4VM method estimates a field aligned plasma velocity from only magnetic field observations!
Using established computer vision techniques Lucas & Kanade (1981); Lucas (1984); Baker & Matthews (2004), Schuck (2006) developed the DAVE from a short timeexpansion of the modified LCT method discussed in Schuck (2005) for estimating velocities from lineofsight magnetograms. The DAVE locally minimizes the square of the continuity equation (2b) subject to an affine velocity profile. Using “moving paint” experiments, Schuck (2006) demonstrated that this technique was faster and more accurate than existing LCT algorithms for data satisfying (2b). The DAVE method has been used to study the apparent motion of active regions Schuck (2006), flux pile up in the photosphere Litvinenko et al. (2007), and helicity flux in the photosphere Chae (2007). However, nagging questions remain about its performance.
Welsch et al. (2007) set an important new standard for evaluating scientific optical flow methods used for studying the Sun. For the first time many existing methods for estimating photospheric velocities from magnetograms were tested on a reasonable approximation to synthetic photospheric data from anelastic pseudospectral ANMHD simulations Fan et al. (1999); Abbett et al. (2000, 2004). The methods tested were Lockheed Martin’s Solar and Astrophysical Laboratory’s (LMSAL) LCT code DeRosa (2001), Fourier LCT (FLCT) Welsch et al. (2004), the DAVE Schuck (2006), the IM Kusano et al. (2002, 2004), ILCT Welsch et al. (2004), the MEF Longcope (2004), and MSR Georgoulis & LaBonte (2006). Unfortunately the results were not entirely encouraging. Welsch et al. (2007) treated the velocities estimated from lineofsight methods as the flux transport velocities consistent with the hypothesis of Démoulin & Berger (2003) in (10). Evaluation of the DAVE’s performance on the ANMHD data under this assumption revealed that the DAVE method did not estimate the helicity flux or Poynting flux reliably. In fact none of the pure lineofsight methods: LMSAL’s LCT, FLCT, or the DAVE—estimated these fluxes reliably, reproducing (at best) respectively 11%, 9%, and 23% of the helicity rate, and reproducing respectively 6%, 11%, and 22% of the power injected through the surface.
Of course the ANMHD data have limitations. The simulation models the rise of a buoyant magnetic flux rope in the convection zone and represents the magnetic structure of granulation or supergranulation rather than the dynamics of an active region (See § 2 in Welsch et al., 2007, for a complete discussion). In addition, Welsch et al. (2007) noted that tracking methods performed better on real magnetograms than on the synthetic ANMHD data using “moving paint” experiments where images were simply shifted relative to one another. These results provoked them to comment “that the ANMHD data set either lacks some characteristic present in real solar magnetograms or contains artifacts not present in solar data.” Consequently, the poor performance of tracking methods on ANMHD data might be attributed to the dealiasing method for nonlinear terms in ANMHD (truncating the spatial Fourier spectrum effectively smoothes smallscale structures) or perhaps to the Fourier ringing near strong fields in the ANMHD data set. While these issues are important to resolve, they fail to fully explain the poor performance of the tracking methods to accurately reproduce the quantity they were designed to estimate, namely the helicity flux!
This paper has two primary goals:

Develop a modified DAVE Schuck (2006) that incorporates horizontal magnetic fields, termed the “differential affine velocity estimator for vector magnetograms” (DAVE4VM), and demonstrate its performance on the ANMHD simulation data. DAVE4VM performs much better than the original DAVE technique and roughly on par with the minimum energy fit (MEF) method developed by Longcope (2004) which was deemed to have performed the best overall in Welsch et al.’s (2007) comparison of velocityinversion techniques.

Identify the reasons for the poor performance of DAVE in Welsch et al. (2007).
The paper attempts to follow, as closely as possible, the presentation of the DAVE in Schuck (2006) and the analysis of velocity inversion techniques by Welsch et al. (2007). For the remainder of this paper, lower case variables are used to represent the flux transport vector, flux transport velocity, plasma velocity, electric field, Poynting flux, and helicity flux estimates from the DAVE4VM and DAVE: , , , , and and the corresponding uppercase variables are used to represent the “ground truth” from ANMHD: , , , , , and . The one deviation from this notation involves which denotes an optical flow estimate based on (6). Section (2) describes the DAVE4VM model and § 3 describes its application to the ANMHD data. For the most part, the plots and quantitative analysis presented in Welsch et al. (2007) are produced for the DAVE4VM and DAVE to facilitate intercomparison and comparison to the other methods considered in Welsch et al. (2007). For the DAVE this analysis involves the explicit assumption that . In § 4 the assumption for the DAVE is relaxed and compared with an alternative hypotheses that — that the DAVE produces a biased estimate of the total horizontal plasma velocity.
2 The DAVE4VM Model
The extension of the DAVE for horizontal magnetic fields is straightforward. The plasma velocity is modeled with a threedimensional affine velocity profile:
(11) 
where the hatted variables model the local plasma velocity profile. The coordinate system for the affine velocity profile is not aligned with the magnetic field. Therefore, the velocities are not guaranteed to be orthogonal to . However, the parallel and perpendicular components of the plasma velocity may be determined from
(12a)  
(12b)  
(12c)  
(12d) 
The error metric
(13a)  
(13b)  
characterizes how well the local velocity profile satisfies the magnetic induction equation over a subregion of the magnetogram sequence defined by the window function where is a vector of parameters and . The plasma velocity in (13a) is referenced from the center of the window at so that , , and represent the plasma velocities at the center of the window and the subscripted parameters represent the best fit local shears in the plasma flows, i.e. . The matrix elements of are defined by  
(13c) 
where
(14) 
is a real symmetric positive semidefinite structure tensor where a superscript “*” indicates the matrix transpose. The matrix elements of are provided in Appendix A. The elements correspond to the original DAVE method Schuck (2006) and the remainder represent corrections due to the horizontal components of the magnetic field and flows normal to the surface. The leastsquares solution is
(15) 
when the aperture problem is completely resolved and the velocity field is unambiguous. However, there are important new terms in the structure tensor involving . Situations where because or over the region contained within the window must be considered. In general, the MoorePenrose pseudoinverse provides a numerically stable estimate of the optical flow parameters even when
(16a)  
where  
(16b)  
is defined in terms of the singular value decomposition Golub & Van Loan (1980)  
(16c) 
Here and are orthonormal matrices corresponding to the nine principle directions, is a diagonal matrix containing the nine singular values, and is computed by replacing every nonzero element of by its reciprocal. If , the singular values along the vertical direction are zero and is rank deficient. In this case, the method implemented produces the minimum norm least squares solution resulting in no vertical flows: .
3 Application to ANMHD Simulations
This paper considers the pair of vector magnetograms separated by the shortest time interval s between data dumps of the ANMHD simulation slice archived by Welsch et al. (2007). The “ground truth” data are derived from the timeaveraged velocity and magnetic fields from ANMHD over the shortest time interval. The region of interest in the ANMHD simulations corresponds to a pixel region centered on a convection cell. Welsch et al. (2007) thresholded on the vertical magnetic field and considered only pixels with G for all plots and quantities except for the total helicity where a different masking of results was used Welsch et al. (2008). In a departure from the original presentation of Welsch et al. (2007) this paper considers pixels with G. This corresponds to 7013 pixels or roughly 70% of the region of interest. This difference in thresholding is important because most of the flux emergence and helicity flux in the simulation occurs along the neutral line in regions of weak vertical field that are missed with vertical field thresholding used in the original study. Note that the modified thresholding mask contains weak vertical field regions and contains substantially more points than the roughly 3800 used in Welsch et al. (2007). The difference between the helicity flux in this comparison region and the total simulation is 0.023%.
Since the DAVE4VM is a local optical flow method that determines the plasma velocities within a windowed subregion by constraining the local velocity profile, the choice of window size is a crucial issue for estimating velocities accurately. The window must be large enough to contain enough structure to uniquely determine the coefficients of the flow profile and resolve the aperture problem, but not so large as to violate the affine velocity profile (11) which is only valid locally (See Schuck, 2006, for discussion of the aperture problem in the context of the DAVE). In Welsch et al. (2007), the optimal window size was chosen for the DAVE by examining the Pearson correlation and slope between and from ANMHD. If the method satisfies the induction equation exactly everywhere, the Pearson correlation and slope would both be equal to . However both the DAVE and DAVE4VM were conceived with the recognition that real magnetograms contain noise and should not satisfy the magnetic induction equation exactly; these methods satisfy the induction equation statistically within the window by minimizing the mean squared deviations from the ideal induction equations. Consequently, how well these methods satisfy the magnetic induction equation globally or over a subset of pixels can be used to assess overall performance. In Welsch et al. (2007), the DAVE was “optimized” over a subset of pixels with G that did not include the weak vertical field regions discussed in this study. Using the Pearson correlation and slope, an asymmetric window of performed the best on the ANMHD data. For the present study, the DAVE was “reoptimized” over the new criteria G to provide an “applestoapples” comparison. However, I emphasize that generally this optimization cannot be carried out for the DAVE because this method was proposed for deriving fluxtransport velocities from lineofsight magnetograms. In this situation, the threshold mask can only be applied on the lineofsight component as a proxy for the vertical magnetic field. Nonetheless, as a practical matter, understanding the accuracy limitation of the lineofsight method in comparison to the vector method when synthetic vector magnetograms are available will reveal the relative reliability of helicity flux studies that used optical flow methods that rely exclusively on the lineofsight magnetic field under the “best case scenario” for the tracking methods: the true vertical magnetic field is tracked and regions that contain interesting physics are known apriori. By using the “ground truth” vertical magnetic field, the evaluation is biased to favor the performance of the DAVE method over what would be possible under realistic conditions where only the lineofsight magnetic field is available.
DAVE (Assuming )  DAVE4VM  

Quantities  Spearman  Pearson  Slope  Spearman  Pearson  Slope  
0.34  0.57  0.15  0.88  0.89  0.80  
0.70  0.76  0.71  0.94  0.90  0.89  
0.87  0.85  0.91  0.89  0.88  0.94  
0.93  0.92  1.20  0.94  0.94  1.00  
0.17  0.28  0.07  0.80  0.80  0.79  
0.85  0.95  0.97  0.79  0.94  0.97  
0.70  0.76  0.71  0.94  0.90  0.89  
0.34  0.57  0.15  0.88  0.89  0.80  
0.96  0.96  1.20  0.94  0.97  1.00  
0.20  0.12  0.04  0.88  0.83  0.71 
Five different criteria were used to optimize window selection. Only
symmetric windows were considered and some improvement in the results
can be achieved by implementing asymmetric windows as in
Welsch et al. (2007). Figure 1 shows the optimization curves
for the DAVE (left) and DAVE4VM (right). The top
plots show the Spearman rank order (; dashed black) and
Pearson (; solid black curve) correlations and slope
(; red curve) between
and for DAVE and
and for
DAVE4VM using pixels that satisfy G. The
gradients were computed with 5point leastsquares optimized
derivatives Jähne (2004). Window sizes between 15 and 30 pixels
provide the best balance for achieving performance approaching
. Increasing the window size beyond 30 pixels
continues to improve the Spearman and Pearson correlations but with
diminishing returns while the slopes degrade significantly. The bottom
plots show the power (black curve) and helicity rate
(red curve) as a function of window size for the DAVE and
DAVE4VM. For DAVE these calculations require the assumption that
. The horizontal black and red dashed lines correspond to the
ground truth Poynting and helicity flux from ANMHD. The magnitude of
these fluxes are maximum near 20 pixels with roughly uniform
performance between 15 and 30 pixels. A window size
DAVE (Assuming )  DAVE4VM  

0.830.25  0.470.34  0.61  0.52  0.71  0.380.26  0.090.21  0.91  0.92  0.95  
0.720.30  0.090.27  0.81  0.77  0.87  0.400.25  0.010.14  0.93  0.90  0.92 
3.1 Flux Transport Vectors and Plasma Velocities
Determining the flux transport vectors is equivalent to determining the perpendicular plasma velocities . The accuracy of the flux transport vectors is critical for estimating other MHD quantities: perpendicular plasma velocities, electric field, helicity flux, Poynting flux, etc, since all of these quantities may be derived directly from flux transport vectors.
Flux Transport Velocities and Perpendicular Plasma Velocities
The top left of Figure 2 shows the region of interest from
the ANMHD simulations with grey scale image of vertical magnetic field
overlaid with the horizontal magnetic field vectors in
aqua. The blue contours indicate smoothed neutral lines. The top right
shows the distribution of angles for in the horizontal
plane. The horizontal magnetic field is largely aligned with the
axis as indicated by the aqua vectors in the left panel and
the strong peak in the histogram near
in the right
panel.
Figure 3 shows scatter plots of the from DAVE (left) and from DAVE4VM (right) versus the flux transport vectors from ANMHD. Red and blue are used to distinguish  and components, respectively. The nonparametric Spearman rankorder correlation coefficients (), Pearson correlation coefficients (), and slopes () estimated by the least absolute deviation method are shown for both components of the flux transport vectors. Both visually and quantitatively the DAVE4VM’s correlation with ANMHD is much higher than the DAVE’s. The correlation coefficients even match or exceed the correlation coefficients for the flux transport vectors from the DAVE and MEF reported for the restricted mask G in Welsch et al. (2007). In particular, DAVE does not accurately estimate the flux transport vectors in the direction. The correlation coefficients for this component of the flux transport vectors are and with a slope of . Since that the direction is the predominant direction of the horizontal magnetic for the ANMHD data, the low correlation coefficients suggest that DAVE is insensitive to flux emergence which is proportional to . This will be discussed further in § 4.
Figure 4 shows scatter plots of the estimated perpendicular plasma velocities from the DAVE assuming (left) and DAVE4VM (right) versus ANMHD’s perpendicular plasma velocities . Red, blue, and black correspond to the , , and components respectively. The nonparametric Spearman rankorder correlation coefficients (), Pearson correlation coefficients (), and slopes () estimated by the least absolute deviation method are shown for each component of the perpendicular plasma velocities. The DAVE4VM’s correlation coefficients match or exceed the correlation coefficients for the perpendicular plasma velocities from the DAVE. Particularly striking is the DAVE4VM’s relatively higher correlation for the perpendicular vertical plasma velocity which exceeds the correlation for the DAVE by roughly . The improvement in the DAVE4VM’s estimate is due to the explicit inclusion of horizontal magnetic fields and vertical flows. The flux transport and perpendicular plasma velocity estimates are further quantified by considering the metrics used by Schrijver et al. (2006), Welsch et al. (2007), and Metcalf et al. (2008). The fractional error between the estimated vector and the true vector at the th pixel is
(17a)  
whereas the fractional error in magnitude is  
(17b) 
The moments of these error metrics or any quantity may be accumulated over the pixels within the masks (either G or G) producing the average
(18a)  
and the variance  
(18b) 
For perfect agreement between the estimates and the “ground truth” from ANMHD, , , and their associated variances would be zero. Two measures of directional error are considered, the vector correlation
(19a)  
and the direction correlation  
(19b) 
Both metrics range from for antiparallel vector fields, to for orthogonal vector fields, and to for parallel vector fields (perfect agreement). Table 2 shows these metrics for the DAVE and DAVE4VM over the mask G. The DAVE4VM has fractional errors less than or equal to whereas the fractional errors for the DAVE exceed for both the flux transport vectors and the perpendicular plasma velocities. The average bias error in the magnitude is improved for the DAVE4VM over the DAVE. For the flux transport vectors the bias error in magnitude is and for DAVE4VM and DAVE respectively which corresponds to a factor of improvement. For the plasma velocity, the bias error in magnitude is 0.01 and 0.09 for DAVE4VM and DAVE respectively which corresponds to a factor of improvement. The vector correlation is larger for DAVE4VM than for DAVE. For DAVE4VM for both the flux transport velocity and the perpendicular plasma velocities. In contrast, for the DAVE there is a substantial difference in the accuracy of the flux transport vectors with and the perpendicular plasma velocities with . Finally the directional errors are smaller for the DAVE4VM than for the DAVE. For the DAVE4VM for both the flux transport vectors and the perpendicular plasma velocities. Again, for the DAVE there is a substantial difference in the accuracy of the flux transport vectors with and the perpendicular plasma velocities with .
The direction correlation is difficult to translate into average angular error because it is a nonlinear function of and does not indicate whether the estimated vectors “lead” or “lag” the “ground truth” on average. For the 2D flux transport vectors the moments of the distribution of angular errors
(20) 
can be more informative. Figure 5 shows histograms of the angles between and for the DAVE (left) and DAVE4VM (right). This is a quantitative estimate of the errors in directions of the flux transport vectors. The DAVE4VM represents a dramatic improvement over the DAVE. The DAVE4VM produces a nearly unimodal distribution peaked near , whereas the DAVE produces a multipeaked distribution with the largest peak at and a variance that is more than twice as large as DAVE4VM.
Metrics such as (3.1.1) and (3.1.1) weight all estimates equally. To address this Metcalf et al. (2008) suggested weighting the errors. For example, the weighted direction cosine between an inferred vector and the ground truth vector may be defined as
(21) 
where represents weights. For the flux transport velocity, the errors in the orientation of are more important where is large and less important where is small which suggests a weighting factor . For perfect agreement . The weighted direction cosines for the flux transport vectors and the plasma velocities are reported in Table 2. Comparing the values of and demonstrates that weighting the direction cosine improves the apparent performance of DAVE but the results for DAVE4VM are essentially unchanged. This suggests that DAVE4VM estimates velocities better than DAVE in regions of weak flux transport.
Parallel Velocity
Under ideal conditions, the magnetic field is only affected by . Consequently, only the inductive potential in (7) may be uniquely determined from the evolution of alone. The electrostatic potential must and can be estimated with additional judicious assumptions. These additional assumptions correspond to the minimum photospheric velocity consistent with (7) for the global method MEF and to the prescribed affine form of the local plasma velocity for the local method DAVE4VM. The constraint of the affine velocity profile permits DAVE4VM to determine the electrostatic potential from the nonlocal structure of the inductive potential . DAVE4VM uses unambiguous “pieces” of the plasma velocity within the window aperture to reconstruct the total plasma velocity at the center of the aperture. Within the notation of the Helmholtz decomposition, DAVE4VM estimates the local electrostatic field from the structure of the nonlocal inductive field within the aperture window by imposing a smoothness constraint on the velocity (the affine velocity profile). The accuracy of this estimate for the electrostatic field depends on the validity of the local affine velocity profile and the amount of structure in the local magnetic field; local methods cannot detect motion in regions of uniform magnetic field.
While researchers have widely recognized that estimating the electrostatic potential from (5a) requires additional assumptions, they have not generally recognized that the parallel velocity may be estimated by the analogous arguments. In the absences of a reference flow, the MEF constrains the velocity to be perpendicular to the magnetic field: . In contrast to the velocity estimated by DAVE4VM is not constrained to be perpendicular to the local magnetic field. Instead, DAVE4VM fits an affine velocity model to the magnetic induction equation in an aperture window. This affine velocity model couples the dynamics across pixels within the window aperture. If there is sufficient structuring in the direction of the magnetic field within the aperture, i.e., the perpendicular plasma velocity points in different directions at different pixels within the aperture, then DAVE4VM can resolve the ambiguity in the fieldaligned component of the plasma velocity at the center of the aperture.
Consider the simplified twodimensional situation illustrated by the schematic diagram in Figure 6 of spatially uniform plasma flow across a diverging magnetic field above the photosphere at . The black arrows indicate the magnitude and direction of the magnetic field, the red arrows indicate the direction of the spatially uniform total plasma velocity , and the blue arrows indicate the magnitude and direction of the perpendicular plasma velocity . The aperture in the photosphere is indicated by the gray box. Within the aperture, the perpendicular plasma velocity captures a different component of the total plasma velocity at different locations; this is a consequence of the structuring of the magnetic field. Under the smoothness assumption of a uniform velocity profile, the velocity along the magnetic field in the direction at may be determined from the components of the perpendicular plasma velocity in the direction at other locations within the aperture. Using the pixels in the window aperture results in an overdetermined system for the total plasma velocity:
(22a)  
which has the solution Golub & Van Loan (1980)  
(22b) 
Note that is analogous to and is analogous to in (15).
This pedagogical example illustrates how DAVE4VM may analogously estimate the fieldaligned plasma velocity for the more general case of a spatially variable plasma flow in an inhomogeneous magnetic field for (15). The accuracy of the estimate of the parallel velocity will be limited by the structuring in direction of the magnetic field within the aperture; if the magnetic field has a uniform orientation in the aperture window, no useful estimate of the fieldaligned plasma velocity can be made from the magnetic measurements alone. The quality of the estimate may be assessed with the conditioning of in (22b) for the pedagogical example or in (15) for the full system.
Figure (7) shows scatter plots of (left) the estimated parallel plasma velocities from DAVE4VM versus the parallel plasma velocities from ANMHD and (right) the estimated total plasma velocity from DAVE4VM versus the total plasma velocities from ANMHD. The nonparametric Spearman rankorder correlation coefficients (), Pearson correlation coefficients (), and slopes () estimated by the least absolute deviation method are shown. The commaseparated pairs of numbers in the right plot, corresponding to correlations between and and and respectively, represent the relative improvement in total velocity estimate over the simple null hypothesis that the total plasma velocity is the perpendicular plasma velocity. The correlation of the component of total velocity is significantly improved over the null hypothesis . This improvement id interesting since the horizontal magnetic field is predominantly aligned with the axis (See Figure 2). The correlation of the component of total velocity is slightly worse than the null hypothesis. Finally, the correlation of the component of total velocity is mixed with the Spearman correlation slightly worse than the null hypothesis and the Pearson correlation slightly better than the null hypothesis. However, the slopes of all three components are improved over the null hypothesis.
The significance of the correlations in the left plot may be tested against the null hypothesis by the Fisher permutation test. Fieller et al. (1957) have demonstrated with analysis backed MonteCarlo simulation that Fisher’s transform of the correlation coefficient
(23) 
produces approximately normally distributed values. For example, permuting the values of and 10,000 times generates the null hypothesis distribution with . The Spearman correlation coefficient has a transform of which is roughly 50 standard deviations from the mean of the null distribution indicating that the parallel velocity correlation is statistically significant and not due to sampling error. However, the correlation is small and the parallel velocity estimates may not be scientifically significant for accurately predicting the parallel velocity. The plasma velocities may be further constrained by introducing Doppler velocities, but this is beyond the scope of the present discussion.
Are and Redundant?
DAVE4VM has incorporated an additional component of the velocity over DAVE by introducing three additional variables , and . Consequently, one may reasonably wonder “are the terms and redundant for DAVE4VM?” The answer is a clear “No” for the ANMHD data. Equation (2a) is composed of two terms and describing shearing motion and emergence respectively. Figure 8 shows scatterplots of the two terms for the (left) and (right) components of (2a). The scatterplots indicate the lack of correlation between the terms describing shearing motions and emergence . Red points indicate the results for DAVE4VM and blue points indicate the results for ANMHD. The Spearman rank order () and Pearson () correlations between the two terms, summarized in Table 3, are very low for both components of the flux transport velocities from DAVE4VM or ANMHD. These terms describe different physics, that are uncorrelated, and which require independent variables to describe.
Correlates  DAVE4VM  ANMHD  

Spearman  Pearson  Spearman  Pearson  
0.23  0.08  0.12  0.01  
0.07  0.11  0.09  0.02  
0.09  0.10  0.15  0.15  
0.34  0.33  0.30  0.20 
3.2 Induction Equation and Electric Fields
Figure 9 shows from the DAVE (left) and from DAVE4VM (right) versus from ANMHD. The derivatives for these plots were estimated from 5point optimized least squares. These plots indicate how well the two methods satisfy the MHD induction equation globally. The DAVE has higher correlations than the DAVE4VM but the slopes are equivalent. For the DAVE4VM, the most significant deviations from the MHD induction equation occur near . Neither the DAVE nor DAVE4VM satisfy the induction equation exactly. This is by design, because real magnetogram data are likely to contain significant noise which will contaminate velocity estimates if the induction equation is satisfied exactly. Furthermore, how well a method satisfies the induction equation will generally depend on the differencing template. Consequently, if the velocity estimates are to be used as boundary values for ideal MHD coronal field models, then the velocities of any method will have to be adjusted to satisfy the induction equation on the differencing template implemented by the simulation. Using the Helmholtz decomposition (7), the inductive potential may be computed for the simulation directly from the magnetogram sequence (on the simulation differencing template)
(24a)  
and the electrostatic potential may be derived from the flux transport vectors determined by the optical flow method Welsch et al. (2004)  
(24b) 
Incorporating photospheric velocity estimate into boundary conditions for a coronal MHD simulation, in a minimally consistent way with the normal component of the magnetic induction equation, requires solving two Poisson equations on the photospheric boundary using the differencing template of the MHD code.
Figure 10 shows scatter plots of the estimated perpendicular electric fields from DAVE assuming (left) and from DAVE4VM (right) versus the electric fields from ANMHD. Red, blue, and black correspond to the , , and components, respectively. The nonparametric Spearman rankorder correlation coefficients () and Pearson correlation coefficients () are shown for each component of the electric field. On the present mask the DAVE4VM estimates improve or essentially match the correlation and slopes of the DAVE’s estimates for all three components of the electric field. Particularly dramatic is the improvement in the component of the electric field which the DAVE does not estimate accurately either on the present mask G or the restricted mask G Welsch et al. (2007).
3.3 Poynting and Helicity Fluxes
Démoulin & Berger (2003) show that the Poynting flux can be expressed concisely in terms of the flux transport vectors
(25) 
Figure 11 shows scatterplots of the estimated Poynting flux from the DAVE assuming (left) and DAVE4VM (right) versus ANMHD’s Poynting flux . The correspondence for DAVE4VM, or lack there of for DAVE, indicates the accuracy of the velocity estimates in the direction of the horizontal magnetic field . The nonparametric Spearman rankorder correlation coefficients (), Pearson correlation coefficients (), and slopes () estimated by the least absolute deviation method are shown, as is the ratio of the integrated estimated Poynting flux to the integrated ANMHD Poynting flux