# Tracking Vector Magnetograms with the Magnetic Induction Equation

P. W. Schuck††affiliation<EMAIL_ADDRESS>Plasma Physics
Division, United States Naval Research Laboratory  4555 Overlook Ave., SW,
Washington, DC 20375-5346

###### Abstract

The differential affine velocity estimator (DAVE) developed in Schuck (2006)
for estimating velocities from line-of-sight 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. Inter-comparison between DAVE4VM and DAVE and further
analysis of the DAVE method demonstrates that line-of-sight tracking methods
capture the shearing motion of magnetic footpoints but are insensitive to flux
emergence — the velocities determined from line-of-sight 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 line-of-sight methods such as the DAVE or local
correlation tracking may substantially misrepresent the total helicity rates
and power through the photosphere.

magnetic fields — Sun: atmospheric motions — methods: data analysis

††slugcomment: Submitted to Ap. J.

##  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 under-determined or ill-posed
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

$\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$V$}}_{h}-V_{z}\,\mbox{\boldmath{$B$}}_{h}\right)=0,$
(1)

where the plasma velocity $V$ and the magnetic fields $B$ are decomposed into
a local right-handed Cartesian coordinate system with vertical direction along
the $z$-axis and the horizontal plane, denoted generically by the subscript
“h,” containing the $x$\- and $y$-axes.

Démoulin & Berger (2003) observed that the geometry of magnetic fields
embedded in the photosphere implied that

$\mbox{\boldmath{$F$}}=\mbox{\boldmath{$U$}}\,B_{z}\equiv{B}_{z}\,\mbox{\boldmath{$V$}}_{h}-V_{z}\,\mbox{\boldmath{$B$}}_{h}=\widehat{\mbox{\boldmath{$z$}}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$V$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)=\widehat{\mbox{\boldmath{$z$}}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$V$}}_{\perp}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right),$
(2a) where $F$ denotes the flux transport vector, $U$ is the horizontal
footpoint velocity or flux transport velocity
$\left(\mbox{\boldmath{$U$}}\cdot\widehat{\mbox{\boldmath{$z$}}}=0\right)$ and
$\mbox{\boldmath{$V$}}_{\perp}$ is the plasma velocity perpendicular to the
magnetic field $\mbox{\boldmath{$V$}}_{\perp}\cdot\mbox{\boldmath{$B$}}=0$.
The flux transport vectors are composed of two terms
${B}_{z}\,\mbox{\boldmath{$V$}}_{h}$ and $V_{z}\,\mbox{\boldmath{$B$}}_{h}$
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
$\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$U$}}\,B_{z}\right)=0,$
(2b)

where plasma velocity may be written generally in terms of the flux transport
velocity as

$V$ $\displaystyle=$
$\displaystyle\mbox{\boldmath{$U$}}-\frac{\left(\mbox{\boldmath{$U$}}\cdot\mbox{\boldmath{$B$}}_{h}\right)\,\mbox{\boldmath{$B$}}}{|\mbox{\boldmath{$B$}}|^{2}}+V_{\parallel}\,\frac{\mbox{\boldmath{$B$}}}{\left|\mbox{\boldmath{$B$}}\right|},$
(3a) $\displaystyle\mbox{\boldmath{$V$}}_{\perp{h}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath{$U$}}-\frac{\left(\mbox{\boldmath{$U$}}\cdot\mbox{\boldmath{$B$}}_{h}\right)\,\mbox{\boldmath{$B$}}_{h}}{|\mbox{\boldmath{$B$}}|^{2}},$
(3b) $\displaystyle\mbox{\boldmath{$V$}}_{\perp{z}}$ $\displaystyle=$
$\displaystyle-\frac{\left(\mbox{\boldmath{$U$}}\cdot\mbox{\boldmath{$B$}}_{h}\right)\,B_{z}}{|\mbox{\boldmath{$B$}}|^{2}},$
(3c)

and the subscripts “$\parallel$” and “$\perp$” denote plasma velocities
parallel and perpendicular to the magnetic field respectively. Equations
(3a-c) are the algebraic decomposition Welsch et al. (2004) generalized for
arbitrary parallel velocity $V_{\parallel}$, but the value of $V_{\parallel}$
does not affect the perpendicular plasma velocity (3b)-(3c) or the
perpendicular electric field

$c\,\mbox{\boldmath{$E$}}_{\perp}=-\mbox{\boldmath{$V$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}=-\overbrace{\mbox{\boldmath{$U$}}\mbox{\boldmath{$\times$}}\widehat{\mbox{\boldmath{$z$}}}\,B_{z}}^{\mbox{$\mbox{\boldmath{$E$}}_{\perp{h}}$}}-\overbrace{\mbox{\boldmath{$U$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}_{h}}^{\mbox{$E_{\perp{z}}$}},$
(4)

which both depend only on the flux transport velocity $U$.

Equations (1)-(3) should be formally distinguished from the inverse problem
for determining an estimate of the plasma velocity $v$ from vector
magnetograms using the normal component of the magnetic induction equation

$\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}\right)=0,$
(5a) where
$\mbox{\boldmath{$f$}}=\mbox{\boldmath{$u$}}\,B_{z}=B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}=\widehat{\mbox{\boldmath{$z$}}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)=\widehat{\mbox{\boldmath{$z$}}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}_{\perp}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right),$
(5b)

and the inverse problem for determining flux transport velocity $u$ from the
evolution of the vertical magnetic field or line-of-sight component

$\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)=0.$
(6)

The notation $\vartheta$, denoting an optical flow estimate, emphasizes that
$\vartheta$ determined from (6) is not necessarily immediately identified with
the flux transport velocity $u$. Equations (5)-(6) are ill-posed inverse
problems because of two ambiguities:

  1. 1.

The Helmholtz decomposition of the flux transport vectors Welsch et al.
(2004); Longcope (2004)

$\mbox{\boldmath{$f$}}=\mbox{\boldmath{$u$}}\,B_{z}=B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}=-\left(\mbox{\boldmath{$\nabla$}}_{h}\phi+\mbox{\boldmath{$\nabla$}}_{h}\psi\mbox{\boldmath{$\times$}}\widehat{\mbox{\boldmath{$z$}}}\right),$
(7)

where $\phi$ is the inductive potential and $\psi$ is the electrostatic
potential manifestly demonstrates that only inductive potential $\psi$ may be
unambiguously determined from the local evolution of $B_{z}$ in (5). The
electrostatic potential $\phi$ must be constrained by additional assumptions.
By analogy, (6) is also ill-posed for the same reason;
$\mbox{\boldmath{$\nabla$}}_{h}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)$
is not constrained by the local evolution of $\partial_{t}B_{z}$.

  2. 2.

For (5a) and (5b) $V_{\parallel}$ is not constrained by the local evolution of
$\partial_{t}B_{z}$. For (6), there is no a priori relationship between
$\vartheta$ and $u$ or $\vartheta$ and $v$ for the inverse problem. However,
if $\vartheta$ is identified with the flux transport velocity $u$ then $u$ and
$v$ will satisfy the same relationships as $U$ and $V$ in (3).

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 $v$, but not
necessarily the unique solution that corresponds to $V$. 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
$\mbox{\boldmath{$v$}}_{\perp{z}}=0$ as an assumption, or hybrid local-global
methods such as inductive local correlation tracking (ILCT) Welsch et al.
(2004). These methods produce a unique solution for $\vartheta$, but not
necessarily the unique solution that corresponds to $u$.

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:
$\mbox{\boldmath{$\vartheta$}}^{\left(\mathrm{LCT}\right)}=\mbox{\boldmath{$V$}}_{h}$.
Démoulin & Berger (2003) conjecture that line-of-sight tracking methods, and
in particular LCT, estimate the total flux transport velocity
$\mbox{\boldmath{$\vartheta$}}^{\left(\mathrm{LCT}\right)}=\mbox{\boldmath{$U$}}$.
Schuck (2005) formally demonstrated that LCT is consistent with the advection
equation

$\partial_{t}B_{z}+\mbox{\boldmath{$\vartheta$}}^{\left(\mathrm{LCT}\right)}\cdot\mbox{\boldmath{$\nabla$}}_{h}{B_{z}}=0,$
(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 line-of-sight optical flow methods via

$\mbox{\boldmath{$u$}}\,B_{z}\equiv\mbox{\boldmath{$\vartheta$}}\,B_{z}-v_{z}\,\mbox{\boldmath{$B$}}_{h},$
(9)

where in principle, $v_{z}$ 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 line-of-sight optical flow
methods via

$\mbox{\boldmath{$u$}}\equiv\mbox{\boldmath{$\vartheta$}}\mbox{ for }B_{z}\neq
0.$ (10)

The Ansatz $\mbox{\boldmath{$u$}}=\mbox{\boldmath{$\vartheta$}}$ 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 line-of-sight 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 & Büchner (2007);
Tian & Alexander (2008); Zhang et al. (2008); Wang et al. (2008). However,
equivalence between $\vartheta$ and $u$ for line-of-sight methods has never
been practically established. These two different hypotheses (9) and (10) for
the interpretation of $\vartheta$ 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 $v_{\parallel}=0$ Longcope (2004). ILCT and the original
algebraic decomposition both assume $v_{\parallel}=0$ Welsch et al. (2004).
Georgoulis & LaBonte (2006) describe a method for inferring $v_{\parallel}$
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
time-expansion of the modified LCT method discussed in Schuck (2005) for
estimating velocities from line-of-sight 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 line-of-sight
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
line-of-sight 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 super-granulation 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 de-aliasing method for nonlinear
terms in ANMHD (truncating the spatial Fourier spectrum effectively smoothes
small-scale 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:

  1. 1.

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 velocity-inversion techniques.

  2. 2.

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: $f$, $u$, $\mbox{\boldmath{$v$}}_{\perp}$,
$\mbox{\boldmath{$e$}}_{\perp}$, $s_{z}$ and $h$ and the corresponding
uppercase variables are used to represent the “ground truth” from ANMHD: $F$,
$U$, $\mbox{\boldmath{$V$}}_{\perp}$, $\mbox{\boldmath{$E$}}_{\perp}$,
$S_{z}$, and $H$. The one deviation from this notation involves $\vartheta$
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 inter-comparison
and comparison to the other methods considered in Welsch et al. (2007). For
the DAVE this analysis involves the explicit assumption that
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$. In § 4 the assumption
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ for the DAVE is relaxed
and compared with an alternative hypotheses that
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$ — 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 straight-forward.
The plasma velocity is modeled with a three-dimensional affine velocity
profile:

$\mbox{\boldmath{$v$}}\left(\mbox{\boldmath{$P$}};\mbox{\boldmath{$x$}}\right)=\left(\begin{array}[]{c}\widehat{u}_{0}\\\
\widehat{v}_{0}\\\
\widehat{w}_{0}\end{array}\right)+\left(\begin{array}[]{cc}\widehat{u}_{x}&\widehat{u}_{y}\\\
\widehat{v}_{x}&\widehat{v}_{y}\\\
\widehat{w}_{x}&\widehat{w}_{y}\end{array}\right)\,\left(\begin{array}[]{c}x\\\
y\end{array}\right),$ (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 $B$. However, the parallel $\left(\parallel\right)$ and perpendicular
$\left(\perp\right)$ components of the plasma velocity may be determined from

$\displaystyle v_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{\left(\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$B$}}\right)\,\mbox{\boldmath{$B$}}}{B^{2}},$
(12a) $\displaystyle\mbox{\boldmath{$v$}}_{\perp}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath{$v$}}-\frac{\left(\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$B$}}\right)\,\mbox{\boldmath{$B$}}}{B^{2}},$
(12b) $\displaystyle\mbox{\boldmath{$v$}}_{\perp{h}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath{$v$}}_{h}-\frac{\left(\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$B$}}\right)\,\mbox{\boldmath{$B$}}_{h}}{B^{2}},$
(12c) $\displaystyle v_{\perp{z}}$ $\displaystyle=$ $\displaystyle
v_{z}-\frac{\left(\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$B$}}\right)\,B_{z}}{B^{2}}.$
(12d)

The error metric

$\displaystyle\mathcal{C}_{\mbox{SSD}}$ $\displaystyle=$
$\displaystyle\int{dt}{dx^{2}}\,w\left(\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}},t-\tau\right)\left\\{\partial_{t}B_{z}\left(\mbox{\boldmath{$x$}},t\right)+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left[B_{z}\left(\mbox{\boldmath{$x$}},t\right)\,\mbox{\boldmath{$v$}}_{h}\left(\mbox{\boldmath{$P$}},\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}}\right)\right.\right.,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\left.-v_{z}\left(\mbox{\boldmath{$P$}},\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}}\right)\,\mbox{\boldmath{$B$}}_{h}\left(\mbox{\boldmath{$x$}},t\right)\right]\right\\}^{2},$
$\displaystyle=$
$\displaystyle\mbox{\boldmath{$\eta$}}\cdot\left\langle\SS\right\rangle\cdot\mbox{\boldmath{$\eta$}}$
(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
$w\left(\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}},t-\tau\right)$ where
$\mbox{\boldmath{$P$}}=\left(\widehat{u}_{0},\widehat{v}_{0},\widehat{u}_{x},\widehat{v}_{y},\widehat{u}_{y},\widehat{v}_{x},\widehat{w}_{0},\widehat{w}_{x},\widehat{w}_{y}\right)$
is a vector of parameters and
$\mbox{\boldmath{$\eta$}}\equiv\left(\mbox{\boldmath{$P$}},1\right)$. The
plasma velocity
$\mbox{\boldmath{$v$}}\left(\mbox{\boldmath{$P$}},\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}}\right)$
in (13) is referenced from the center of the window at
$\mbox{\boldmath{$x$}}=\mbox{\boldmath{$\chi$}}$ so that $\widehat{u}_{0}$,
$\widehat{v}_{0}$, and $\widehat{w}_{0}$ 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.
$\widehat{u}_{x}=\partial_{x}\,\left(\widehat{\mbox{\boldmath{$x$}}}\cdot\mbox{\boldmath{$v$}}\right)$.
The matrix elements of $\left\langle\SS\right\rangle$ are defined by
$\left\langle\SS\right\rangle=\int{dt}\,{dx^{2}}\,w\left(\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}},t-\tau\right)\,\SS\left(\mbox{\boldmath{$\chi$}};x,t\right),$
(13c)

where

$\SS\left(\mbox{\boldmath{$\chi$}};x,t\right)\equiv\left[\begin{array}[]{cc}\mbox{\boldmath{$\mathsf{A}$}}&\mbox{\boldmath{$b$}}\\\
\mbox{\boldmath{$b$}}&\mathcal{G}_{99}\end{array}\right]=\left[\begin{array}[]{cccccccccc}\mathcal{G}_{00}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\mathcal{G}_{10}&\mathcal{G}_{11}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\mathcal{G}_{20}&\mathcal{G}_{21}&\mathcal{G}_{22}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\mathcal{G}_{30}&\mathcal{G}_{31}&\mathcal{G}_{32}&\mathcal{G}_{33}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\mathcal{G}_{40}&\mathcal{G}_{41}&\mathcal{G}_{42}&\mathcal{G}_{43}&\mathcal{G}_{44}&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\mathcal{G}_{50}&\mathcal{G}_{51}&\mathcal{G}_{52}&\mathcal{G}_{53}&\mathcal{G}_{54}&\mathcal{G}_{55}&\cdot&\cdot&\cdot&\cdot\\\
\mathsf{s}_{60}&\mathsf{s}_{61}&\mathsf{s}_{62}&\mathsf{s}_{63}&\mathsf{s}_{64}&\mathsf{s}_{65}&\mathsf{s}_{66}&\cdot&\cdot&\cdot\\\
\mathsf{s}_{70}&\mathsf{s}_{71}&\mathsf{s}_{72}&\mathsf{s}_{73}&\mathsf{s}_{74}&\mathsf{s}_{75}&\mathsf{s}_{76}&\mathsf{s}_{77}&\cdot&\cdot\\\
\mathsf{s}_{80}&\mathsf{s}_{81}&\mathsf{s}_{82}&\mathsf{s}_{83}&\mathsf{s}_{84}&\mathsf{s}_{85}&\mathsf{s}_{86}&\mathsf{s}_{87}&\mathsf{s}_{88}&\cdot\\\
\mathcal{G}_{90}&\mathcal{G}_{91}&\mathcal{G}_{92}&\mathcal{G}_{93}&\mathcal{G}_{94}&\mathcal{G}_{95}&\mathsf{s}_{96}&\mathsf{s}_{97}&\mathsf{s}_{98}&\mathcal{G}_{99}\\\
\end{array}\right],$ (14)

is a real symmetric $\SS=\SS^{*}$ positive semidefinite structure tensor where
a superscript “*” indicates the matrix transpose. The matrix elements of $\SS$
are provided in Appendix A. The elements $\mathcal{G}_{ij}$ correspond to the
original DAVE method Schuck (2006) and the remainder $\mathsf{s}_{ij}$
represent corrections due to the horizontal components of the magnetic field
and flows normal to the surface. The least-squares solution is

$\mbox{\boldmath{$P$}}=-\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle^{-1}\cdot\left\langle\mbox{\boldmath{$b$}}\right\rangle,$
(15)

when the aperture problem is completely resolved
$\det\left(\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle\right)\neq
0$ and the velocity field is unambiguous. However, there are important new
terms in the structure tensor $\left\langle\SS\right\rangle$ involving
$\mbox{\boldmath{$B$}}_{h}$. Situations where
$\det\left(\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle\right)=0$
because $\mbox{\boldmath{$B$}}_{h}=0$ or $B_{z}=0$ over the region contained
within the window must be considered. In general, the Moore-Penrose pseudo-
inverse $\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle^{\dagger}$
provides a numerically stable estimate of the optical flow parameters even
when
$\det\left(\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle\right)=0$

$\mbox{\boldmath{$P$}}=-\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle^{{\dagger}}\cdot\left\langle\mbox{\boldmath{$b$}}\right\rangle,$
(16a) where
$\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle^{\dagger}\equiv\mbox{\boldmath{$\mathsf{V}$}}\,\mbox{\boldmath{$\Sigma$}}^{\dagger}\,\mbox{\boldmath{$\mathsf{U}$}}^{*},$
(16b) is defined in terms of the singular value decomposition Golub & Van Loan
(1980)
$\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle=\mbox{\boldmath{$\mathsf{U}$}}\,\mbox{\boldmath{$\Sigma$}}\,\mbox{\boldmath{$\mathsf{V}$}}^{*}.$
(16c)

Here $\mathsf{U}$ and $\mathsf{V}$ are orthonormal matrices corresponding to
the nine principle directions, $\Sigma$ is a diagonal matrix containing the
nine singular values, and $\mbox{\boldmath{$\Sigma$}}^{\dagger}$ is computed
by replacing every nonzero element of $\Sigma$ by its reciprocal. If
$\mbox{\boldmath{$B$}}_{h}=0$, the singular values along the vertical
direction are zero and
$\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle$ is rank deficient.
In this case, the method implemented produces the minimum norm least squares
solution resulting in no vertical flows: $w_{0}=w_{x}=w_{y}=0$.

##  3 Application to ANMHD Simulations

This paper considers the pair of vector magnetograms $B$ separated by the
shortest time interval $\Delta t\approx 250$ s between data dumps of the ANMHD
simulation slice archived by Welsch et al. (2007). The “ground truth” data are
derived from the time-averaged velocity and magnetic fields from ANMHD over
the shortest time interval. The region of interest in the ANMHD simulations
corresponds to a $101\times 101$ pixel region centered on a convection cell.
Welsch et al. (2007) thresholded on the vertical magnetic field and considered
only pixels with $|B_{z}|>370$ 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 $B=\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}>370$
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
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)$
and $\Delta B_{z}/\Delta t$ from ANMHD. If the method satisfies the induction
equation exactly everywhere, the Pearson correlation and slope would both be
equal to $-1$. 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 $\left|B_{z}\right|>370$ G
that did not include the weak vertical field regions discussed in this study.
Using the Pearson correlation and slope, an asymmetric window of $21\times 39$
performed the best on the ANMHD data. For the present study, the DAVE was “re-
optimized” over the new criteria $\left|\mbox{\boldmath{$B$}}\right|>370$ G to
provide an “apples-to-apples” comparison. However, I emphasize that generally
this optimization cannot be carried out for the DAVE because this method was
proposed for deriving flux-transport velocities from line-of-sight
magnetograms. In this situation, the threshold mask can only be applied on the
line-of-sight component as a proxy for the vertical magnetic field.
Nonetheless, as a practical matter, understanding the accuracy limitation of
the line-of-sight 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 line-of-sight 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 a-priori. 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 line-of-sight magnetic field is available.

Figure 1: Optimization curves for the DAVE (left) and DAVE4VM (right). (top)
The Spearman rank order (dashed black curve) and Pearson (solid black curve)
correlations and slope (red curve) between
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)$
and $\Delta B_{z}/\Delta t$ for DAVE and
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$u$}}\,B_{z}\right)$
and $\Delta B_{z}/\Delta t$ for DAVE4VM as a function of window size. (bottom)
The power (black) and helicity (red) as a function of window size; this
assumes $\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ for DAVE. The
horizontal black and red dashed lines correspond to the ground truth power and
helicity rate through the simulation slice from ANMHD. The vertical dot-dashed
lines indicate the “optimal” window size of 23 pixels chosen for both DAVE and
DAVE4VM. Table 1: Comparison between the DAVE and DAVE4VM over the 7013 pixels
that satisfy $\left|\mbox{\boldmath{$B$}}\right|>370$ G in Figure 2. This mask
contain regions of weak vertical field not considered in Welsch et al. (2007).

|  | DAVE (Assuming $\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$) | DAVE4VM
---|---|---|---
Quantities | Spearman | Pearson | Slope | Spearman | Pearson | Slope
$u_{x}\,B_{z}$ | $U_{x}\,B_{z}$ | 0.34 | 0.57 | 0.15 | 0.88 | 0.89 | 0.80
$u_{y}\,B_{z}$ | $U_{y}\,B_{z}$ | 0.70 | 0.76 | 0.71 | 0.94 | 0.90 | 0.89
$v_{\perp{x}}$ | $V_{\perp{x}}$ | 0.87 | 0.85 | 0.91 | 0.89 | 0.88 | 0.94
$v_{\perp{y}}$ | $V_{\perp{y}}$ | 0.93 | 0.92 | 1.20 | 0.94 | 0.94 | 1.00
$v_{\perp{z}}$ | $V_{\perp{z}}$ | 0.17 | 0.28 | 0.07 | 0.80 | 0.80 | 0.79
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$u$}}\,B_{z}\right)$ | $\Delta B_{z}/\Delta t$ | -0.85 | -0.95 | -0.97 | -0.79 | -0.94 | -0.97
$e_{\perp{x}}$ | $E_{\perp{x}}$ | 0.70 | 0.76 | 0.71 | 0.94 | 0.90 | 0.89
$e_{\perp{y}}$ | $E_{\perp{y}}$ | 0.34 | 0.57 | 0.15 | 0.88 | 0.89 | 0.80
$e_{\perp{z}}$ | $E_{\perp{z}}$ | 0.96 | 0.96 | 1.20 | 0.94 | 0.97 | 1.00
$s_{z}$ | $S_{z}$ | 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 ($\rho$; dashed black) and Pearson ($C$; solid
black curve) correlations and slope ($S$; red curve) between
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)$
and $\Delta B_{z}/\Delta t$ for DAVE and
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$u$}}\,B_{z}\right)$
and $\Delta B_{z}/\Delta t$ for DAVE4VM using pixels that satisfy
$\left|\mbox{\boldmath{$B$}}\right|>370$ G. The gradients were computed with
5-point least-squares optimized derivatives Jähne (2004). Window sizes between
15 and 30 pixels provide the best balance for achieving performance
approaching $C=\rho=S=-1$. 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 $\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$. 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
size111Only windows with odd numbers of pixels on each side of the window are
possible in these implementations of the DAVE and DAVE4VM. of 23 pixels was
chosen for both the DAVE and DAVE4VM as indicated by the vertical dot-dashed
lines in Figure 1. These objective metrics for evaluation of global
performance can be implemented without knowledge of ground truth. Only future
tests with more realizations of synthetic data can reveal whether they are
robust metrics for optimizing window choice. Table 3 presents a summary of the
correlation coefficients on the mask $\left|\mbox{\boldmath{$B$}}\right|>370$
G characterizing the accuracy of the DAVE and DAVE4VM for the quantities
discussed in this section.

Table 2: Comparison of accuracy of the velocity estimates between the DAVE and
DAVE4VM over the 7013 pixels that satisfy
$\left|\mbox{\boldmath{$B$}}\right|>370$ G in Figure 2. This mask contains
regions of weak vertical field not considered in Welsch et al. (2007). Here
$\left\langle\left|\delta\smash{\widetilde{\mbox{\boldmath{$f$}}}}\right|\right\rangle$
is the average fractional error,
$\left\langle\delta\left|\smash{\widetilde{\mbox{\boldmath{$f$}}}}\right|\right\rangle$
is the average error in magnitude, $C_{\mathrm{vec}}$ is the vector
correlation, $C_{\mathrm{CS}}$ is the direction correlation, and
$\left\langle\cos\theta\right\rangle_{W}$ is weighted direction cosine.

###  3.1 Flux Transport Vectors and Plasma Velocities

Figure 2: Top left: Grey scale image of the vertical magnetic field $B_{z}$
overlaid with the horizontal magnetic field vectors
$\mbox{\boldmath{$B$}}_{h}$ in aqua. Blue contours indicate smoothed neutral
lines. Top right: Distribution of angles for $\mbox{\boldmath{$B$}}_{h}$ in
the horizontal plane. Bottom: (red arrows)
$\mbox{\boldmath{$\vartheta$}}\,B_{z}$ for the DAVE (left) and
$\mbox{\boldmath{$u$}}\,B_{z}$ for the DAVE4VM (right) plotted over ANMHD’s
flux transport vectors $\mbox{\boldmath{$U$}}\,B_{z}$ (green arrows). Vectors
are shown only for pixels in which $|B|>370$ G, and for clarity, only every
third vector is displayed.

Determining the flux transport vectors
$\mbox{\boldmath{$f$}}=\mbox{\boldmath{$u$}}\,B_{z}$ is equivalent to
determining the perpendicular plasma velocities
$\mbox{\boldmath{$v$}}_{\perp}$. 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.

####  3.1.1 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 $\mbox{\boldmath{$B$}}_{h}$ in aqua. The
blue contours indicate smoothed neutral lines. The top right shows the
distribution of angles for $\mbox{\boldmath{$B$}}_{h}$ in the horizontal
plane. The horizontal magnetic field is largely aligned with the
$\widehat{\mbox{\boldmath{$x$}}}$-axis as indicated by the aqua vectors in the
left panel and the strong peak in the histogram near
$\arctan\left(B_{y},B_{x}\right)\approx 0^{\circ}$ in the right
panel.222$\arctan\left(y,x\right)\equiv\arctan\left(y/x\right)$. There is also
significant alignment of the magnetic field with $\pm 60^{\circ}$ and
alignment of weak fields with $-140^{\circ}$. The bottom panels show
$\mbox{\boldmath{$\vartheta$}}\,B_{z}$ from DAVE (left) and
$\mbox{\boldmath{$u$}}\,B_{z}$ from DAVE4VM in red (right) and the flux
transport vectors $\mbox{\boldmath{$F$}}=\mbox{\boldmath{$U$}}\,B_{z}$ from
ANMHD in green. The improvement between the DAVE and DAVE4VM is manifest —
finding a region where the DAVE4VM performs qualitatively worse than the DAVE
is difficult. The DAVE4VM performs the worst in the region $140\\--160\times
150\\--170$ where the flux transport vectors run roughly anti-parallel to the
horizontal magnetic field and there is little structure in the vertical
component.

Figure 3: Scatter plots of $\mbox{\boldmath{$\vartheta$}}\,B_{z}$ for the DAVE
(left) and $\mbox{\boldmath{$u$}}\,B_{z}$ for the DAVE4VM (right) versus
ANMHD’s flux transport vectors,
$\mbox{\boldmath{$F$}}=\mbox{\boldmath{$U$}}\,B_{z}$. Red and blue are used to
distinguish $x$\- and $y$-components, respectively. The nonparametric Spearman
rank-order correlation coefficients ($\rho$), Pearson correlation coefficients
($C$), and slopes ($S$) estimated by the least absolute deviation method are
shown for both components of the flux transport vectors.

Figure 3 shows scatter plots of the $\mbox{\boldmath{$\vartheta$}}\,B_{z}$
from DAVE (left) and $\mbox{\boldmath{$u$}}\,B_{z}$ from DAVE4VM (right)
versus the flux transport vectors $\mbox{\boldmath{$U$}}\,B_{z}$ from ANMHD.
Red and blue are used to distinguish $x$\- and $y$-components, respectively.
The nonparametric Spearman rank-order correlation coefficients ($\rho$),
Pearson correlation coefficients ($C$), and slopes ($S$) 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
$\left|B_{z}\right|>370$ G in Welsch et al. (2007). In particular, DAVE does
not accurately estimate the flux transport vectors in the
$\widehat{\mbox{\boldmath{$x$}}}$-direction. The correlation coefficients for
this $\widehat{\mbox{\boldmath{$x$}}}$-component of the flux transport vectors
are $\rho_{x}=0.34$ and $C_{x}=0.57$ with a slope of $S_{x}=0.15$. Since that
the $\widehat{\mbox{\boldmath{$x$}}}$-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 $v_{z}\,\mbox{\boldmath{$B$}}_{h}$. This will be discussed
further in § 4.

Figure 4: Scatter plots of the estimated perpendicular plasma velocities
$\mbox{\boldmath{$v$}}_{\perp}$ from DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and DAVE4VM
(right) versus the perpendicular plasma velocities
$\mbox{\boldmath{$V$}}_{\perp}$ from ANMHD. Red, blue, and black correspond to
the $x$-, $y$\- and $z$-components respectively. The nonparametric Spearman
rank-order correlation coefficients ($\rho$), Pearson correlation coefficients
($C$), and slopes ($S$) are shown for each component of the perpendicular
plasma velocities.

Figure 4 shows scatter plots of the estimated perpendicular plasma velocities
$\mbox{\boldmath{$v$}}_{\perp}$ from the DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and DAVE4VM
(right) versus ANMHD’s perpendicular plasma velocities
$\mbox{\boldmath{$V$}}_{\perp}$. Red, blue, and black correspond to the $x$-,
$y$-, and $z$-components respectively. The nonparametric Spearman rank-order
correlation coefficients ($\rho$), Pearson correlation coefficients ($C$), and
slopes ($S$) 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 $\mbox{\boldmath{$v$}}_{\perp{z}}$ which exceeds the correlation for
the DAVE by roughly $0.5\\--0.6$. 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 $f$ and the true vector $F$ at the $i$th pixel is

$\left|\delta\smash{\widetilde{\mbox{\boldmath{$f$}}}}_{i}\right|\equiv\frac{\left|\mbox{\boldmath{$f$}}_{i}-\mbox{\boldmath{$F$}}_{i}\right|}{\left|\mbox{\boldmath{$F$}}_{i}\right|},$
(17a) whereas the fractional error in magnitude is
$\delta\left|\smash{\widetilde{\mbox{\boldmath{$f$}}}}_{i}\right|\equiv\frac{\left|\mbox{\boldmath{$f$}}_{i}\right|-\left|\mbox{\boldmath{$F$}}_{i}\right|}{\left|\mbox{\boldmath{$F$}}_{i}\right|}.$
(17b)

The moments of these error metrics or any quantity $q$ may be accumulated over
the $N$ pixels within the masks (either
$\left|\mbox{\boldmath{$B$}}\right|>370$ G or $\left|B_{z}\right|>370$ G)
producing the average

$\left\langle{q}\right\rangle\equiv\frac{1}{N}\sum_{i=1}^{N}q_{i},$ (18a) and
the variance
$\sigma^{2}_{q}\equiv\frac{1}{N-1}\sum_{i=1}^{N}\left(q_{i}-\left\langle
q\right\rangle\right)^{2}.$ (18b)

For perfect agreement between the estimates and the “ground truth” from ANMHD,
$\left\langle\left|\delta\smash{\widetilde{\mbox{\boldmath{$f$}}}}\right|\right\rangle$,
$\left\langle\delta\left|\smash{\widetilde{\mbox{\boldmath{$f$}}}}\right|\right\rangle$,
and their associated variances would be zero. Two measures of directional
error are considered, the vector correlation

$C_{\mathrm{vec}}=\frac{\left\langle\mbox{\boldmath{$f$}}\cdot\mbox{\boldmath{$F$}}\right\rangle}{\sqrt{\left\langle\mbox{\boldmath{$f$}}^{2}\right\rangle\,\left\langle\mbox{\boldmath{$F$}}^{2}\right\rangle}},$
(19a) and the direction correlation
$C_{\mathrm{CS}}=\left\langle\frac{\mbox{\boldmath{$f$}}\cdot\mbox{\boldmath{$F$}}}{\sqrt{\mbox{\boldmath{$f$}}^{2}\,\mbox{\boldmath{$F$}}^{2}}}\right\rangle\equiv\left\langle\cos\theta\right\rangle.$
(19b)

Both metrics range from $-1$ for antiparallel vector fields, to $0$ for
orthogonal vector fields, and to $1$ for parallel vector fields (perfect
agreement). Table 3 shows these metrics for the DAVE and DAVE4VM over the mask
$\left|\mbox{\boldmath{$B$}}\right|>370$ G. The DAVE4VM has fractional errors
less than or equal to $0.4$ whereas the fractional errors for the DAVE exceed
$0.7$ 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 $-0.09$ and $-0.47$ for DAVE4VM and DAVE respectively which
corresponds to a factor of $5$ 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 $9$ improvement. The vector correlation is larger
for DAVE4VM than for DAVE. For DAVE4VM $C_{\mathrm{vec}}\gtrsim 0.9$ 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 $C_{\mathrm{vec}}=0.61$ and the perpendicular
plasma velocities with $C_{\mathrm{vec}}=0.81$. Finally the directional errors
are smaller for the DAVE4VM than for the DAVE. For the DAVE4VM
$C_{\mathrm{CS}}\gtrsim 0.9$ 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
$C_{\mathrm{CS}}=0.52$ and the perpendicular plasma velocities with
$C_{\mathrm{CS}}=0.77$.

Figure 5: Histograms of the angles between
$\mbox{\boldmath{$\vartheta$}}\,B_{z}$ and $\mbox{\boldmath{$U$}}\,B_{z}$ for
the DAVE (left) and between $\mbox{\boldmath{$u$}}\,B_{z}$ and
$\mbox{\boldmath{$U$}}\,B_{z}$ for the DAVE4VM (right). The mean (bias) and
standard deviation are reported in the upper left-hand corners.

The direction correlation $C_{\mathrm{CS}}$ is difficult to translate into
average angular error because it is a nonlinear function of $\theta$ 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

$\theta=\arctan\left[\left(\mbox{\boldmath{$u$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$U$}}\right)_{z},\mbox{\boldmath{$u$}}\cdot\mbox{\boldmath{$U$}}\right],$
(20)

can be more informative. Figure 5 shows histograms of the angles between
$\mbox{\boldmath{$u$}}\,B_{z}$ and $\mbox{\boldmath{$U$}}\,B_{z}$ 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 $0^{\circ}$, whereas the DAVE produces a multi-peaked distribution
with the largest peak at $80^{\circ}$ and a variance that is more than twice
as large as DAVE4VM.

Metrics such as (17) and (19) 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

$\left\langle\cos\theta\right\rangle_{W}\equiv\frac{\left\langle{W}\,\cos\theta\right\rangle}{\left\langle{W}\right\rangle}$
(21)

where $W$ represents weights. For the flux transport velocity, the errors in
the orientation of $u$ are more important where
$\left|\mbox{\boldmath{$U$}}\,B_{z}\right|$ is large and less important where
$\left|\mbox{\boldmath{$U$}}\,B_{z}\right|$ is small which suggests a
weighting factor
$W_{i}=\left|\left(\mbox{\boldmath{$U$}}\,B_{z}\right)\right|$. For perfect
agreement $\left\langle\cos\theta\right\rangle_{W}=1$. The weighted direction
cosines for the flux transport vectors and the plasma velocities are reported
in Table 3. Comparing the values of $C_{\mathrm{CS}}$ and
$\left\langle\cos\theta\right\rangle_{W}$ 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.

####  3.1.2 Parallel Velocity

Under ideal conditions, the magnetic field is only affected by
$\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}\right)$.
Consequently, only the inductive potential $\phi$ in (7) may be uniquely
determined from the evolution of $B$ alone. The electrostatic potential $\psi$
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 $\psi$ from the nonlocal structure of the inductive potential
$\phi$. 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.

Figure 6: Schematic diagram of a uniform plasma flow across a diverging
magnetic field above the photosphere at $h=0$. The black arrows indicate the
strength and direction of the magnetic field, the red arrows indicate the
direction of the spatially uniform total plasma velocity $v$, and the blue
arrows indicate the magnitude and direction of the perpendicular plasma
velocity $\mbox{\boldmath{$v$}}_{\perp}$. The aperture in the photosphere is
indicated by the gray box.

While researchers have widely recognized that estimating the electrostatic
potential $\psi$ from (5a) requires additional assumptions, they have not
generally recognized that the parallel velocity $v_{\parallel}$ 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:
$v_{\parallel}=0$. 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 field-aligned
component of the plasma velocity at the center of the aperture.

Consider the simplified two-dimensional situation illustrated by the schematic
diagram in Figure 6 of spatially uniform plasma flow across a diverging
magnetic field above the photosphere at $h=0$. 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 $v$, and the blue
arrows indicate the magnitude and direction of the perpendicular plasma
velocity $\mbox{\boldmath{$v$}}_{\perp}$. 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
$\widehat{\mbox{\boldmath{$z$}}}$-direction at $x=0$ may be determined from
the components of the perpendicular plasma velocity in the
$\widehat{\mbox{\boldmath{$z$}}}$ direction at other locations within the
aperture. Using the $N$ pixels in the window aperture results in an
overdetermined system for the total plasma velocity:

$\underbrace{\left[\begin{array}[]{cc}\frac{B_{z}^{2}\left(\mbox{\boldmath{$x$}}_{1}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{1}\right)}&-\frac{B_{x}\left(\mbox{\boldmath{$x$}}_{1}\right)\,B_{z}\left(\mbox{\boldmath{$x$}}_{1}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{1}\right)}\\\
-\frac{B_{x}\left(\mbox{\boldmath{$x$}}_{1}\right)\,B_{z}\left(\mbox{\boldmath{$x$}}_{1}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{1}\right)}&\frac{B_{x}^{2}\left(\mbox{\boldmath{$x$}}_{1}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{1}\right)}\\\
\vdots&\vdots\\\
\frac{B_{z}^{2}\left(\mbox{\boldmath{$x$}}_{N}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{N}\right)}&-\frac{B_{x}\left(\mbox{\boldmath{$x$}}_{N}\right)\,B_{z}\left(\mbox{\boldmath{$x$}}_{N}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{N}\right)}\\\
-\frac{B_{x}\left(\mbox{\boldmath{$x$}}_{N}\right)\,B_{z}\left(\mbox{\boldmath{$x$}}_{N}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{N}\right)}&\frac{B_{x}^{2}\left(\mbox{\boldmath{$x$}}_{N}\right)}{B^{2}\left(\mbox{\boldmath{$x$}}_{N}\right)}\end{array}\right]}_{\mbox{\boldmath{$\mathsf{D}$}}}\,\left(\begin{array}[]{cc}v_{x}\\\
v_{z}\end{array}\right)=\underbrace{\left[\begin{array}[]{cc}v_{\perp{x}}\left(\mbox{\boldmath{$x$}}_{1}\right)\\\
v_{\perp{z}}\left(\mbox{\boldmath{$x$}}_{1}\right)\\\ \vdots\\\
v_{\perp{x}}\left(\mbox{\boldmath{$x$}}_{N}\right)\\\
v_{\perp{z}}\left(\mbox{\boldmath{$x$}}_{N}\right)\end{array}\right]}_{{\mbox{\boldmath{$\mathsf{d}$}}}},$
(22a) which has the solution Golub & Van Loan (1980)
$\left(\begin{array}[]{cc}\widehat{v}_{x}\\\
\widehat{v}_{z}\end{array}\right)=\left({\mbox{\boldmath{$\mathsf{D}$}}}^{*}\,{\mbox{\boldmath{$\mathsf{D}$}}}\right)^{-1}\,{\mbox{\boldmath{$\mathsf{D}$}}}^{*}{\mbox{\boldmath{$\mathsf{d}$}}}.$
(22b)

Note that
${\mbox{\boldmath{$\mathsf{D}$}}}^{*}\,{\mbox{\boldmath{$\mathsf{D}$}}}$ is
analogous to $\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle$ and
${\mbox{\boldmath{$\mathsf{D}$}}}^{*}{\mbox{\boldmath{$\mathsf{d}$}}}$ is
analogous to $\left\langle\mbox{\boldmath{$b$}}\right\rangle$ in (15).

This pedagogical example illustrates how DAVE4VM may analogously estimate the
field-aligned 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 field-
aligned plasma velocity can be made from the magnetic measurements alone. The
quality of the estimate may be assessed with the conditioning of
${\mbox{\boldmath{$\mathsf{D}$}}}^{*}\,{\mbox{\boldmath{$\mathsf{D}$}}}$ in
(22b) for the pedagogical example or
$\left\langle\mbox{\boldmath{$\mathsf{A}$}}\right\rangle$ in (15) for the full
system.

Figure 7: Scatter plots of (left) the estimated parallel plasma velocities
$v_{\parallel}$ from DAVE4VM versus the parallel plasma velocities
$V_{\parallel}$ from ANMHD and (right) the estimated total plasma velocity $v$
from DAVE4VM versus the total plasma velocities $V$ from ANMHD. The
nonparametric Spearman rank-order correlation coefficients ($\rho$), Pearson
correlation coefficients ($C$), and slopes ($S$) estimated by the least
absolute deviation method are shown. The pairs of numbers represent
correlations between $v$ and $V$ and $\mbox{\boldmath{$v$}}_{\perp}$ and $V$.

Figure (7) shows scatter plots of (left) the estimated parallel plasma
velocities $v_{\parallel}$ from DAVE4VM versus the parallel plasma velocities
$V_{\parallel}$ from ANMHD and (right) the estimated total plasma velocity $v$
from DAVE4VM versus the total plasma velocities $V$ from ANMHD. The
nonparametric Spearman rank-order correlation coefficients ($\rho$), Pearson
correlation coefficients ($C$), and slopes ($S$) estimated by the least
absolute deviation method are shown. The comma-separated pairs of numbers in
the right plot, corresponding to correlations between $v$ and $V$ and
$\mbox{\boldmath{$v$}}_{\perp}$ and $V$ respectively, represent the relative
improvement in total velocity estimate over the simple null hypothesis
$H_{0}:\mbox{\boldmath{$v$}}=\mbox{\boldmath{$v$}}_{\perp}$ that the total
plasma velocity is the perpendicular plasma velocity. The correlation of the
$\widehat{\mbox{\boldmath{$x$}}}$-component of total velocity is significantly
improved over the null hypothesis $H_{0}$. This improvement id interesting
since the horizontal magnetic field is predominantly aligned with the $x$-axis
(See Figure 2). The correlation of the
$\widehat{\mbox{\boldmath{$y$}}}$-component of total velocity is slightly
worse than the null hypothesis. Finally, the correlation of the
$\widehat{\mbox{\boldmath{$z$}}}$-component of total velocity is mixed with
the Spearman correlation $\rho$ slightly worse than the null hypothesis and
the Pearson correlation $C$ 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 $H_{0}:\rho=0$ by the Fisher permutation test. Fieller et
al. (1957) have demonstrated with analysis backed Monte-Carlo simulation that
Fisher’s $z$-transform of the correlation coefficient

$z_{\mathrm{S}}\left(\rho\right)=\frac{1}{2}\,\log\left|\frac{1+\rho}{1-\rho}\right|,$
(23)

produces approximately normally distributed values. For example, permuting the
values of $v_{\parallel}$ and $V_{\parallel}$ 10,000 times generates the null
hypothesis distribution with
$\left\langle{z_{\mathrm{S}}}\right\rangle=0.00\pm 0.01$. The Spearman
correlation coefficient $\rho=0.53$ has a $z$-transform of
$z_{\mathrm{S}}\left(0.53\right)=0.60$ 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 $\rho=0.53$ 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.

####  3.1.3 Are $\mbox{\boldmath{$v$}}_{h}$ and $v_{z}$ Redundant?

Figure 8:  Scatterplots for the $\widehat{\mbox{\boldmath{$x$}}}$ (left) and
$\widehat{\mbox{\boldmath{$y$}}}$ (right) components of (2a). The scatterplots
indicate the lack of correlation between the terms describing shearing motion
${B}_{z}\,\mbox{\boldmath{$V$}}_{h}$ and emergence
$V_{z}\,\mbox{\boldmath{$B$}}_{h}$. Red points indicate the results for
DAVE4VM and blue points indicate the results for ANMHD. The Spearman rank
order ($\rho$) and Pearson ($C$) correlations between the two terms are very
low for both components of the flux transport velocities.

DAVE4VM has incorporated an additional component of the velocity over DAVE by
introducing three additional variables $\widehat{w}_{0}$, $\widehat{w}_{x}$
and $\widehat{w}_{y}$. Consequently, one may reasonably wonder “are the terms
$\mbox{\boldmath{$v$}}_{h}\,B_{z}$ and $v_{z}\,\mbox{\boldmath{$B$}}_{h}$
redundant for DAVE4VM?” The answer is a clear “No” for the ANMHD data.
Equation (2a) is composed of two terms ${B}_{z}\,\mbox{\boldmath{$V$}}_{h}$
and $V_{z}\,\mbox{\boldmath{$B$}}_{h}$ describing shearing motion and
emergence respectively. Figure 8 shows scatterplots of the two terms for the
$\widehat{\mbox{\boldmath{$x$}}}$ (left) and $\widehat{\mbox{\boldmath{$y$}}}$
(right) components of (2a). The scatterplots indicate the lack of correlation
between the terms describing shearing motions
${B}_{z}\,\mbox{\boldmath{$V$}}_{h}$ and emergence
$V_{z}\,\mbox{\boldmath{$B$}}_{h}$. Red points indicate the results for
DAVE4VM and blue points indicate the results for ANMHD. The Spearman rank
order ($\rho$) and Pearson ($C$) correlations between the two terms,
summarized in Table 3.1.3, 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.

Table 3: The Spearman rank order ($\rho$) and Pearson ($C$) correlations
between the terms in the flux transport velocity describing shearing motion
and flux emergence.

###  3.2 Induction Equation and Electric Fields

Figure 9: Scatter plots of
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)$
from the DAVE (left) and
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$u$}}\,B_{z}\right)$
from the DAVE4VM (right) versus $\Delta B_{z}/\Delta t$ from ANMHD. The
nonparametric Spearman rank-order correlation coefficients ($\rho$), Pearson
correlation coefficients ($C$), and slopes ($S$) estimated by the least
absolute deviation method are shown.

Figure 9 shows
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$\vartheta$}}\,B_{z}\right)$
from the DAVE (left) and
$\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(\mbox{\boldmath{$u$}}\,B_{z}\right)$
from DAVE4VM (right) versus $\Delta B_{z}/\Delta t$ from ANMHD. The
derivatives for these plots were estimated from 5-point 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 $\Delta B_{z}/\Delta
t\approx 0$. 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)

$\partial_{t}{B_{z}}=\nabla_{h}^{2}\phi,$ (24a) and the electrostatic
potential may be derived from the flux transport vectors determined by the
optical flow method Welsch et al. (2004)
$\nabla_{h}^{2}\psi=\widehat{\mbox{\boldmath{$z$}}}\cdot\left[\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\left(\mbox{\boldmath{$u$}}\,B_{z}\right)\right].$
(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: Scatter plots of the estimated perpendicular electric field
$\mbox{\boldmath{$e$}}_{\perp}$ from DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and from DAVE4VM
(right) versus the electric field $\mbox{\boldmath{$E$}}_{\perp}$ from ANMHD.
Red, blue, and black correspond to the $x$-, $y$-, and $z$-components
respectively. The nonparametric Spearman rank-order correlation coefficients
($\rho$), Pearson correlation coefficients ($C$), and slopes ($S$) estimated
by the least absolute deviation method are shown for each component of the
electric field.

Figure 10 shows scatter plots of the estimated perpendicular electric fields
$\mbox{\boldmath{$e$}}_{\perp}$ from DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and from DAVE4VM
(right) versus the electric fields $\mbox{\boldmath{$E$}}_{\perp}$ from ANMHD.
Red, blue, and black correspond to the $x$-, $y$-, and $z$-components,
respectively. The nonparametric Spearman rank-order correlation coefficients
($\rho$) and Pearson correlation coefficients ($C$) 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 $\widehat{\mbox{\boldmath{$y$}}}$ component
of the electric field which the DAVE does not estimate accurately either on
the present mask $\left|\mbox{\boldmath{$B$}}\right|>370$ G or the restricted
mask $\left|B_{z}\right|>370$ G Welsch et al. (2007).

###  3.3 Poynting and Helicity Fluxes

Figure 11: Scatter plots of the estimated Poynting flux from DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and DAVE4VM
(right) versus the Poynting flux from ANMHD. The nonparametric Spearman rank-
order correlation coefficients ($\rho$), Pearson correlation coefficients
($C$), and slopes ($S$) 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.

Démoulin & Berger (2003) show that the Poynting flux can be expressed
concisely in terms of the flux transport vectors
$\mbox{\boldmath{$u$}}\,B_{z}$

$s_{z}\left(\mbox{\boldmath{$x$}}\right)=-\frac{1}{4\,\pi}\,\mbox{\boldmath{$B$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}\right)=-\frac{\mbox{\boldmath{$B$}}_{h}\cdot\left(\mbox{\boldmath{$u$}}\,\,B_{z}\right)}{4\,\pi}.$
(25)

Figure 11 shows scatterplots of the estimated Poynting flux $s_{z}$ from the
DAVE assuming $\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and
DAVE4VM (right) versus ANMHD’s Poynting flux $S_{z}$. 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
$\mbox{\boldmath{$B$}}_{h}$. The nonparametric Spearman rank-order correlation
coefficients ($\rho$), Pearson correlation coefficients ($C$), and slopes
($S$) 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 $\mathcal{R}_{{s}_{z}}=\sum{s}_{z}/\sum{S}_{z}$. The DAVE4VM’s
estimate of Poynting flux is a significant improvement over the DAVE’s. The
correlations have improved by roughly a factor of $4\\--6$, the slope has
improved by nearly a factor of 18, and the ratio of the totals has improved by
nearly a factor of 5. Again, DAVE does not reliably estimate the flux
transport velocity in the direction of the horizontal magnetic field
$\mbox{\boldmath{$B$}}_{h}$ suggesting that DAVE is insensitive to flux
emergence which is proportional to $v_{z}\,\mbox{\boldmath{$B$}}_{h}$. The
“ground truth” total power through the mask is $dP/dt=\sum{S_{z}}=7.7\times
10^{28}\,\mathrm{ergs/s}$.

Figure 12: Scatter plots of the estimated helicity flux DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and DAVE4VM
(right) versus ANMHD’s helicity flux. The nonparametric Spearman rank-order
correlation coefficients ($\rho$), Pearson correlation coefficients ($C$), and
slopes ($S$) estimated by the least absolute deviation method are shown, as is
the ratio of the integrated estimated helicity flux to the integrated ANMHD
helicity flux.

Démoulin & Berger (2003) show that the gauge-invariant helicity flux Berger &
Field (1984) can be expressed concisely in terms of the flux transport vectors
$\mbox{\boldmath{$u$}}\,B_{z}$

$g_{A}\left(\mbox{\boldmath{$x$}}\right)=-2\,\mbox{\boldmath{$A$}}_{p}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}\right)=-2\,\mbox{\boldmath{$A$}}_{p}\cdot\left(\mbox{\boldmath{$u$}}\,B_{z}\right)$
(26)

where
$\mbox{\boldmath{$A$}}_{p}=\widehat{\mbox{\boldmath{$z$}}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$\nabla$}}\Phi_{p}$
is the potential reference field (with zero helicity) which satisfies

$\widehat{\mbox{\boldmath{$z$}}}\cdot\left(\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}_{p}\right)=\nabla_{h}^{2}\Phi_{p}=B_{z},$
(27)

and
$\mbox{\boldmath{$\nabla$}}\cdot\mbox{\boldmath{$A$}}_{p}=\widehat{\mbox{\boldmath{$z$}}}\cdot\mbox{\boldmath{$A$}}_{p}=0$.
To estimate the helicity flux density, $\Phi_{p}$ was computed on a $257\times
257$ square centered on the region of interest with Dirichlet boundary
conditions using MUDPACK Adams (1993). While interpretation of maps of
helicity flux $g_{A}\left(\mbox{\boldmath{$x$}}\right)$ through the
photosphere is problematic Pariat et al. (2005, 2007), a comparison of
$g_{A}\left(\mbox{\boldmath{$x$}}\right)$ estimated from the DAVE or DAVE4VM
verses $G_{A}\left(\mbox{\boldmath{$x$}}\right)$ calculated from ANMHD
indicates the accuracy of the estimated flux transport vectors in the
direction of the vector potential. Figure 12 shows scatter plots of the
estimated helicity flux from DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (left) and DAVE4VM
(right) versus ANMHD’s helicity flux. The nonparametric Spearman rank-order
correlation coefficients ($\rho$) and Pearson correlation coefficients ($C$)
are shown, as is the ratio of the integrated estimated helicity flux to the
integrated ANMHD helicity flux $\mathcal{R}_{g_{A}}=\sum{g_{A}}/\sum{G_{A}}$.
The DAVE4VM’s estimates represent a significant improvement over the DAVE’s,
improving the correlation coefficients by roughly $0.1$ and the slope by
$0.2$. Furthermore, the ratio of totals has improved by roughly a factor of 4
from $0.22$ for the DAVE to $0.94$ for the DAVE4VM. The “ground truth”333This
estimate differs by about 10% from the helicity estimate in Welsch et al.
(2007). The discrepancy is caused by the different methodologies and
boundaries used for computing the vector potential
$\mbox{\boldmath{$A$}}_{p}$. MUDPACK was used in this study with (27) whereas
Welsch et al. (2007) used a Green’s function scheme to compute
$\mbox{\boldmath{$A$}}_{p}$. helicity injected through the surface is
$dH_{A}/dt=\sum{G_{A}}=-2.8\times 10^{37}\,\mathrm{Mx}^{2}/\mathrm{s}$.

##  4 Discussion and Conclusions

Table 4: Comparison between the DAVE and DAVE4VM over the 3815 pixels that
satisfy $\left|B_{z}\right|>370$ G in Figure 2. This corresponds roughly to
the mask used in Welsch et al. (2007). Table 5: Comparison of accuracy of the
velocity estimates between the DAVE and DAVE4VM over the 3815 pixels that
satisfy $\left|B_{z}\right|>370$ G in Figure 2. This corresponds roughly to
the mask used in Welsch et al. (2007).

For completeness, Tables 4 and 4 provide a summary of metrics and correlation
coefficients for the DAVE and DAVE4VM on the original mask
$\left|B_{z}\right|>370$ G used by Welsch et al. (2007) in the same format as
in Tables 3 and 3. MEF performed the best overall in the original study by
Welsch et al. (2007) although there were some metrics where the DAVE
outperformed MEF such as in the accuracy of the plasma velocities listed in
Table 4 (compare with Figure 8 in Welsch et al. (2007)). The DAVE4VM’s
estimates are a substantial improvement over the results of the DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ on this mask. Comparing
the rank-order Spearman correlation coefficients for the flux transport
vectors, perpendicular plasma velocity, and electric field in Table 4, the
DAVE4VM equals or out-performs MEF. Particularly, the DAVE4VM’s estimate of
the vertical perpendicular plasma velocity is substantially better than MEF
with a rank-order of 0.76 in the former and 0.61 in the latter case. Accurate
vertical flows are necessary to diagnose flux emergence and accurately
estimate the helicity flux. The one area where MEF exhibits superiority is in
the estimate of the Poynting flux where the DAVE4VM captures 76% and MEF
captures 100% Welsch et al. (2007). The fractional errors
$\left\langle\left|\delta\smash{\widetilde{\mbox{\boldmath{$f$}}}}\right|\right\rangle$
and
$\left\langle\delta\left|\smash{\widetilde{\mbox{\boldmath{$f$}}}}\right|\right\rangle$
are substantially lower than the DAVE for both the flux transport vectors and
plasma velocities. The DAVE had the largest vector correlation
$C_{\mathrm{vec}}$ and the direction correlation $C_{\mathrm{CS}}$ in the
original study and the DAVE4VM improves over this performance exhibiting
correlation coefficients of roughly $0.9$. The improvement for the flux
transport vectors is particularly dramatic. The plasma velocities are more
accurate and exhibit considerably less bias than those reported for MEF in
Welsch et al. (2007).

The DAVE4VM offers some minor advantages over MEF. The DAVE4VM is somewhat
faster than MEF; the DAVE4VM(DAVE) requires 30(10) seconds to process444The
routines were all coded in Interactive Data Language IDL (2002) and the
computations were performed on a dual processor AMD Opteron 240 running at 1.4
GHz with a one megabyte memory cache and ten gigabytes of Random Access
Memory. the full $288\times 288$ pixel frame from ANMHD whereas MEF requires
roughly 10 minutes to converge on a reduced mask of the ANMHD data (private
communication with Belur Ravindra). The DAVEVM is local and directly estimates
velocities across neutral lines and across broader weak field regions whereas
MEF is a iterative global method that requires judicious choice of boundaries
to ensure convergence. In concert, the DAVE4VM’s velocity estimate might be
used with MEF either as an initial guess for the electrostatic potential
$\psi$ via (24b) or as ancillary inaccurate velocity measurements in the MEF
variational term that constrains the photospheric plasma velocities Longcope
(2004). Since the DAVE4VM is fast and does not require supervision beyond
choosing a window size (and even this could be automated according to the
criteria discussed in § 3), this approach is appropriate for real-time
monitoring of helicity and energy fluxes through the photosphere from
observatories such as Solar Dynamics Observatory.

What is responsible for the DAVE4VM’s improved performance? The only
differences between the DAVE and DAVE4VM are the terms $\mathsf{s}_{ij}$ in
the structure tensor (14) that describe the local structure of the horizontal
magnetic fields necessary for the description of vertical flows. There are two
circumstances when line-of-sight methods such as the DAVE, LMSAL’s LCT, and
FLCT will produce accurate estimates of the flux transport velocity:

  1. 1.

$\mbox{\boldmath{$B$}}_{h}=0$: The magnetic field is purely vertical and
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}=\mbox{\boldmath{$v$}}_{\perp{h}}$.
If the horizontal magnetic fields and their associated derivatives are zeroed,
the DAVE and DAVE4VM produce identical flux transport and perpendicular plasma
velocity estimates. The DAVE is consistent with the assumption that the
magnetic field is purely vertical

$\lim_{\mbox{\boldmath{$B$}}_{h}\rightarrow
0}\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}\right)=\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}\right).$
(28a)

* 2.

$v_{z}=0$: There are no net upflow/downflows and
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}\neq\mbox{\boldmath{$v$}}_{\perp{h}}$.
In this situation, there must be projected vertical flows along the magnetic
field to cancel any projected vertical flow perpendicular to the magnetic
field with
$v_{\perp{z}}=-v_{\parallel{z}}=-B_{z}\,\left(\mbox{\boldmath{$v$}}_{h}\cdot\mbox{\boldmath{$B$}}_{h}\right)/B^{2}$.
Consequently, $\mbox{\boldmath{$v$}}_{\parallel}\neq 0$. The DAVE is
consistent with the assumption that there are no vertical flows $v_{z}=0$:

$\lim_{v_{z}\rightarrow
0}\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}-v_{z}\,\mbox{\boldmath{$B$}}_{h}\right)=\partial_{t}B_{z}+\mbox{\boldmath{$\nabla$}}_{h}\cdot\left(B_{z}\,\mbox{\boldmath{$v$}}_{h}\right).$
(28b)

Both limits (28a) and (28b) are isomorphic with (6). By induction, (6) is
consistent with the assumptions leading to (28a) and (28b). Since DAVE does
not consider corrections $\mathsf{s}_{ij}$ due to the horizontal magnetic
field, $\vartheta$ should generally be considered a biased estimate of the
horizontal plasma velocity
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$ and not the flux
transport velocity! Formally the alternative hypothesis
$H_{1}:\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$ may be tested
against the null hypothesis
$H_{0}:\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ of Démoulin &
Berger (2003).

Figure 13: Scatter plots of the estimated velocities
$\mbox{\boldmath{$\vartheta$}}\,B_{z}$ from DAVE versus the horizontal plasma
velocities $\mbox{\boldmath{$V$}}_{h}\,B_{z}$ from ANMHD. The nonparametric
Spearman rank-order correlation coefficients ($\rho$), Pearson correlation
coefficients ($C$), and slopes ($S$) estimated by the least absolute deviation
method are shown.

The null hypothesis
$H_{0}:\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ is represented by
the left panel in Figure 3. The alternative hypothesis, represented by
$H_{1}:\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$, is
characterized by the scatter plot of the estimated velocities
$\mbox{\boldmath{$\vartheta$}}\,B_{z}$ for DAVE versus the horizontal plasma
velocities $\mbox{\boldmath{$V$}}_{h}\,B_{z}$ from ANMHD in Figure 13. The
nonparametric Spearman rank-order correlation coefficients ($\rho$), Pearson
correlation coefficients ($C$), and slopes ($S$) estimated by the least
absolute deviation method are all significantly better for the alternative
hypothesis than for the null hypothesis. The null hypothesis that the
velocities inferred by DAVE represent the flux transport velocities may be
rejected in favor555This does not imply that the alternative hypothesis is
correct. of the alternative hypothesis
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$.

Figure 14: (left) Scatter plots of the estimated Poynting flux (red) and
helicity flux (blue) from DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ versus the Poynting and
helicity flux from ANMHD. (right) Scatter plots of the estimated Poynting flux
(red) and helicity flux (blue) combining DAVE assuming
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$ with the emergence
term from ANMHD versus the Poynting and helicity flux from ANMHD. The
nonparametric Spearman rank-order correlation coefficients ($\rho$), Pearson
correlation coefficients ($C$), and slopes ($S$) estimated by the least
absolute deviation method are shown.

These results explain why $V_{\perp{z}}$ and $v_{\perp{z}}$ are poorly
correlated for the DAVE in Figure 4 and the slope between them is nearly zero
— the DAVE is consistent with the assumption $v_{z}=0$ when
$\mbox{\boldmath{$B$}}_{h}\neq 0$. Generally, in regions of flux emergence,
the accuracy $v_{\perp{z}}$ is critical for estimating the flux transport
vectors which in turn is critical for estimating the helicity and Poynting
fluxes. When horizontal magnetic fields and vertical flows are present, the
flux transport vectors estimated from methods that rely exclusively on the
line-of-sight or vertical component (DAVE, LMSAL’s LCT, FLCT) cannot be
trusted to provide the total fluxes. This is particularly true along neutral
lines where flux is emerging or submerging! Under the best case scenarios,
only the shearing or “horizontal fluxes”666Démoulin & Berger (2003) terms
these fluxes the “tangential fluxes” but “horizontal” is more appropriate in
the context of Welsch’s terminology used in this paper. across the
photosphere.

$\left.\frac{dp}{dt}\right|_{h}=-\frac{1}{4\,\pi}\,\int_{S}{dx^{2}}\,\mbox{\boldmath{$B$}}_{h}\cdot\left(\mbox{\boldmath{$v$}}_{h}\,B_{z}\right),$
(29a) and
$\left.\frac{dh_{A}}{dt}\right|_{h}=-2\,\int_{S}{dx^{2}}\,\mbox{\boldmath{$A$}}_{p}\cdot\left(\mbox{\boldmath{$v$}}_{h}\,B_{z}\right),$
(29b)

may be estimated from the line-of-sight tracking methods. In the best case
scenarios only the shearing fluxes are captured by line-of-sight tracking
methods in partial agreement with the Ansatz of Chae (2001) and in
disagreement with the geometrical arguments of Démoulin & Berger (2003) who
argue that line-of-sight tracking methods capture both the shearing and
emergence. Again, for the energy and helicity, the alternative hypothesis
$H_{1}:\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$ (shearing) may
be tested against the null hypothesis
$H_{0}:\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$u$}}$ (shearing and
emergence). The left panel of Figure 14, representing the null hypothesis is a
combination of the left-hand panels of Figure 11 and Figure 12. The right
panel of Figure 14, representing the alternative hypothesis, combines the
shearing term estimated from DAVE with the emergence term from ANMHD.
Generally, in the presence of vertical flows and horizontal magnetic fields,
line-of-sight tracking methods do not accurately capture the complete
footpoint dynamics and the null hypothesis that the velocities inferred by
DAVE represent the flux transport velocities may be rejected in favor of the
alternative hypothesis
$\mbox{\boldmath{$\vartheta$}}=\mbox{\boldmath{$v$}}_{h}$.

The implementation of vector magnetograms in optical flow methods presents
practical challenges. First, the transverse magnetic field components are
known to be noisier than the line-of-sight component and the noise variance
will likely change from pixel to pixel due to variable photon statistics
(heteroscedastic errors). Second the line-of-sight component and transverse
components are determined from different polarizations and require inter-
calibration. Third, the orientation of the transverse component is ambiguous
by $180^{\circ}$. The first issue may be addressed within the total least
squares framework discussed by Schuck (2006), Branham Jr. (1999) and others
(See references in Schuck, 2006). The main obstacle to resolving the first
issue is estimating a covariance matrix for the structure tensor
$\left\langle\SS\right\rangle$.

The second and third issues both may be interpreted as inter-calibration bias
where the estimated horizontal magnetic field
$\widehat{\mbox{\boldmath{$B$}}}_{h}=\alpha\,\mbox{\boldmath{$B$}}_{h}$ is
proportional to the true horizontal magnetic field
$\mbox{\boldmath{$B$}}_{h}$; these errors are not random. The flux transport
velocities estimated from DAVE4VM are robust to overall inter-calibration
errors which include the $180^{\circ}$ ambiguity resolution errors. Changing
the overall magnitude or sign of $\mbox{\boldmath{$B$}}_{h}$ has no effect on
the flux transport velocities because (5a) is invariant with respect to the
transformation $v_{z}\rightarrow{v_{z}}/\alpha$ and
$\mbox{\boldmath{$B$}}_{h}\rightarrow\alpha\mbox{\boldmath{$B$}}_{h}$ (private
communication with Pascal Demoulin). However, the estimated vertical
perpendicular plasma velocity and vertical perpendicular electric field will
be anti-correlated with the ground truth when $\alpha<0$. Nonetheless, an
overall rescaling of the horizontal magnetic field will have no effect on the
helicity flux. However, the Poynting flux will be incorrect by a factor of
$\alpha$ including perhaps a sign error because of the rescaling horizontal
magnetic field which is inherent in the energy estimate (25). More troublesome
are the effects of spatially varying bias errors in inter-calibration or
ambiguity resolution. The consequences of these errors, particularly along the
boundaries between proper and improper ambiguity resolution, are presently
unknown and should be investigated with future end-to-end analysis of
synthetic magnetograms. However, local methods such as DAVE4VM are probably
more robust than global methods to spatially dependent errors in inter-
calibration or ambiguity resolution because local methods inherently localize
the effect of bias errors by isolating subregions with the window aperture
whereas global methods couple the entire solution region together permitting
bias errors in one subregion to influence the solution in other subregions.

In light of the DAVE4VM’s dramatic improvement in performance by simply
including horizontal magnetic fields, speculation that the ANMHD simulation
data are not appropriate for testing tracking methods cannot be correct.
Rather, aside from issues of image structure, the ANMHD simulation data
represent an ideal case for the line-of-sight methods because the vertical
magnetic field is known (not simply the line-of-sight component). The results
of this study suggest that horizontal magnetic fields and vertical flows will
render velocity estimates from “pure” tracking methods inaccurate if they are
treated as the flux transport velocities $u$. This conclusion holds equally
true for velocity estimates near disk center as the ANMHD simulations
represents disk-center data! The good agreement between the performance of MEF
and the DAVE4VM on the ANMHD data implies that incorporating the right physics
is more important for producing accurate velocity estimates than is the
particular method used the solve the equations.

Presently, the only way to explore the “image” physics is by testing the
“optical flow” methods on synthetic data from well-designed MHD simulations
that attempt to reproduce the physics of the Sun. Naive “moving paint”
experiments Schuck (2006) cannot critically test optical flow methods for
magnetograms because the test data are consistent with the two circumstances
when pure tracking methods will certainly perform well:
$\mbox{\boldmath{$B$}}_{h}=0$ and $v_{z}=0$. Consequently, good performance of
an optical flow method in naive “moving paint” experiments should not be
considered evidence that a method will produce accurate estimates of plasma
physics quantities.

In the interest of reproducibility Joyner & Stein (2007), all of the software
used to perform the calculations, create the figures, and draw the conclusions
for this paper are archived with the Astrophysical Journal as a tgz file.
Updates to the DAVE/DAVE4VM software are also
available.777http://wwwppd.nrl.navy.mil/whatsnew/.

I thank the referee for constructive criticism, Graham Barnes for encouraging
the publication of this work, Bill Abbett for providing the ANMHD data that
formed the core of this research, and Pascal Demoulin and Brian Welsch for
encouraging the clarification of several issues discussed in this manuscript.
I also gratefully acknowledge useful conversations with George Fisher, Bill
Amatucci, and Etienne Pariat. I thank Julie Schuck for editing the manuscript.
This work was supported by NASA LWS TR&T grant NNH06AD87I, LWS TR&T Strategic
Capability grant NNH07AG26I, and ONR.

##  Appendix A Matrix Elements of $\SS$

$\displaystyle\mathcal{G}_{00}$ $\displaystyle=$
$\displaystyle{\left(\partial_{x}{B_{z}}\right)}^{2}$
$\displaystyle\mathcal{G}_{10}$ $\displaystyle=$
$\displaystyle\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right)$
$\displaystyle\mathcal{G}_{11}$ $\displaystyle=$
$\displaystyle{\left(\partial_{y}{B_{z}}\right)}^{2}$
$\displaystyle\mathcal{G}_{20}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{x}{B_{z}}\right)+{\left(\partial_{x}{B_{z}}\right)}^{2}{\,}x^{\prime}$
$\displaystyle\mathcal{G}_{21}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{y}{B_{z}}\right)+\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathcal{G}_{22}$ $\displaystyle=$ $\displaystyle
B_{z}^{2}+2{\,}{B_{z}}{\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}+{\left(\partial_{x}{B_{z}}\right)}^{2}{\,}x^{\prime
2}$ $\displaystyle\mathcal{G}_{30}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{x}{B_{z}}\right)+\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathcal{G}_{31}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{y}{B_{z}}\right)+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}y^{\prime}$
$\displaystyle\mathcal{G}_{32}$ $\displaystyle=$ $\displaystyle
B_{z}^{2}+{B_{z}}{\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}+{B_{z}}{\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}+\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathcal{G}_{33}$ $\displaystyle=$ $\displaystyle
B_{z}^{2}+2{\,}{B_{z}}{\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}y^{\prime
2}$ $\displaystyle\mathcal{G}_{40}$ $\displaystyle=$
$\displaystyle{\left(\partial_{x}{B_{z}}\right)}^{2}{\,}y^{\prime}$
$\displaystyle\mathcal{G}_{41}$ $\displaystyle=$
$\displaystyle\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathcal{G}_{42}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}+{\left(\partial_{x}{B_{z}}\right)}^{2}{\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathcal{G}_{43}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}+\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime
2}$ $\displaystyle\mathcal{G}_{44}$ $\displaystyle=$
$\displaystyle{\left(\partial_{x}{B_{z}}\right)}^{2}{\,}y^{\prime 2}$
$\displaystyle\mathcal{G}_{50}$ $\displaystyle=$
$\displaystyle\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathcal{G}_{51}$ $\displaystyle=$
$\displaystyle{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}x^{\prime}$
$\displaystyle\mathcal{G}_{52}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}+\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime
2}$ $\displaystyle\mathcal{G}_{53}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathcal{G}_{54}$ $\displaystyle=$
$\displaystyle\left(\partial_{x}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathcal{G}_{55}$ $\displaystyle=$
$\displaystyle{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}x^{\prime 2}$
$\displaystyle\mathsf{s}_{60}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right)-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right)$
$\displaystyle\mathsf{s}_{61}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right)-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right)$
$\displaystyle\mathsf{s}_{62}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}{B_{z}}-\left(\partial_{y}{B_{z}}\right){\,}{B_{z}}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathsf{s}_{63}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}{B_{z}}-\left(\partial_{y}{B_{z}}\right){\,}{B_{z}}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathsf{s}_{64}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathsf{s}_{65}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathsf{s}_{66}$ $\displaystyle=$
$\displaystyle{\left(\partial_{x}{B_{x}}\right)}^{2}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right)+{\left(\partial_{y}{B_{z}}\right)}^{2}$
$\displaystyle\mathsf{s}_{70}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}\left(\partial_{x}{B_{z}}\right)-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathsf{s}_{71}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}\left(\partial_{y}{B_{z}}\right)-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathsf{s}_{72}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}{B_{z}}-\left(\partial_{x}{B_{x}}\right){\,}{B_{z}}{\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}{B_{z}}{\,}x^{\prime}-{B_{x}}{\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime
2}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime
2}$ $\displaystyle\mathsf{s}_{73}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}{B_{z}}-\left(\partial_{x}{B_{x}}\right){\,}{B_{z}}{\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}{B_{z}}{\,}x^{\prime}-{B_{x}}{\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathsf{s}_{74}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathsf{s}_{75}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime
2}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime
2}$ $\displaystyle\mathsf{s}_{76}$ $\displaystyle=$
$\displaystyle{B_{x}}{\,}\left(\partial_{x}{B_{x}}\right)+{B_{x}}{\,}\left(\partial_{y}{B_{z}}\right)+{\left(\partial_{x}{B_{x}}\right)}^{2}{\,}x^{\prime}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}x^{\prime}$
$\displaystyle\mathsf{s}_{77}$ $\displaystyle=$ $\displaystyle
B_{x}^{2}+2{\,}{B_{x}}{\,}\left(\partial_{x}{B_{x}}\right){\,}x^{\prime}+2{\,}{B_{x}}{\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}+{\left(\partial_{x}{B_{x}}\right)}^{2}{\,}x^{\prime
2}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime
2}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}x^{\prime 2}$
$\displaystyle\mathsf{s}_{80}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}\left(\partial_{x}{B_{z}}\right)-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathsf{s}_{81}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}\left(\partial_{y}{B_{z}}\right)-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathsf{s}_{82}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}{B_{z}}-{B_{y}}{\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}{B_{z}}{\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}{B_{z}}{\,}y^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathsf{s}_{83}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}{B_{z}}-\left(\partial_{x}{B_{x}}\right){\,}{B_{z}}{\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}{B_{z}}{\,}y^{\prime}-{B_{y}}{\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime
2}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime
2}$ $\displaystyle\mathsf{s}_{84}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime
2}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime
2}$ $\displaystyle\mathsf{s}_{85}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathsf{s}_{86}$ $\displaystyle=$
$\displaystyle\left(\partial_{x}{B_{x}}\right){\,}{B_{y}}+{B_{y}}{\,}\left(\partial_{y}{B_{z}}\right)+{\left(\partial_{x}{B_{x}}\right)}^{2}{\,}y^{\prime}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}y^{\prime}$
$\displaystyle\mathsf{s}_{87}$ $\displaystyle=$
$\displaystyle{B_{x}}{\,}{B_{y}}+\left(\partial_{x}{B_{x}}\right){\,}{B_{y}}{\,}x^{\prime}+{B_{y}}{\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}+{B_{x}}{\,}\left(\partial_{x}{B_{x}}\right){\,}y^{\prime}+{B_{x}}{\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle+{\left(\partial_{x}{B_{x}}\right)}^{2}{\,}x^{\prime}{\,}y^{\prime}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}{\,}y^{\prime}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}x^{\prime}{\,}y^{\prime}$
$\displaystyle\mathsf{s}_{88}$ $\displaystyle=$ $\displaystyle
B_{y}^{2}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}{B_{y}}{\,}y^{\prime}+2{\,}{B_{y}}{\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}+{\left(\partial_{x}{B_{x}}\right)}^{2}{\,}y^{\prime
2}+2{\,}\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime
2}+{\left(\partial_{y}{B_{z}}\right)}^{2}{\,}y^{\prime 2}$
$\displaystyle\mathcal{G}_{90}$ $\displaystyle=$
$\displaystyle\left(\partial_{t}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right)$
$\displaystyle\mathcal{G}_{91}$ $\displaystyle=$
$\displaystyle\left(\partial_{t}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right)$
$\displaystyle\mathcal{G}_{92}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{t}{B_{z}}\right)+\left(\partial_{t}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathcal{G}_{93}$ $\displaystyle=$
$\displaystyle{B_{z}}{\,}\left(\partial_{t}{B_{z}}\right)+\left(\partial_{t}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathcal{G}_{94}$ $\displaystyle=$
$\displaystyle\left(\partial_{t}{B_{z}}\right){\,}\left(\partial_{x}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathcal{G}_{95}$ $\displaystyle=$
$\displaystyle\left(\partial_{t}{B_{z}}\right){\,}\left(\partial_{y}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathsf{s}_{96}$ $\displaystyle=$
$\displaystyle-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{t}{B_{z}}\right)-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{t}{B_{z}}\right)$
$\displaystyle\mathsf{s}_{97}$ $\displaystyle=$
$\displaystyle-{B_{x}}{\,}\left(\partial_{t}{B_{z}}\right)-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{t}{B_{z}}\right){\,}x^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{t}{B_{z}}\right){\,}x^{\prime}$
$\displaystyle\mathsf{s}_{98}$ $\displaystyle=$
$\displaystyle-{B_{y}}{\,}\left(\partial_{t}{B_{z}}\right)-\left(\partial_{x}{B_{x}}\right){\,}\left(\partial_{t}{B_{z}}\right){\,}y^{\prime}-\left(\partial_{y}{B_{z}}\right){\,}\left(\partial_{t}{B_{z}}\right){\,}y^{\prime}$
$\displaystyle\mathcal{G}_{99}$ $\displaystyle=$
$\displaystyle{\left(\partial_{t}{B_{z}}\right)}^{2}$ The primed coordinates
are defined relative to the center of the aperture
$\mbox{\boldmath{$x$}}^{\prime}=\mbox{\boldmath{$x$}}-\mbox{\boldmath{$\chi$}}$.

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