Optimization of Photospheric Electric Field Estimates for Accurate Retrieval of Total Magnetic Energy Injection

Abstract

Estimates of the photospheric magnetic, electric, and plasma velocity fields are essential for studying the dynamics of the solar atmosphere, for example through the derivative quantities of Poynting and relative helicity flux and using the fields to obtain the lower boundary condition for data-driven coronal simulations. In this paper we study the performance of a data processing and electric field inversion approach that requires only high-resolution and high-cadence line-of-sight or vector magnetograms, which we obtain from the Helioseismic and Magnetic Imager (HMI) onboard Solar Dynamics Observatory (SDO). The approach does not require any photospheric velocity estimates, and the lacking velocity information is compensated for using ad hoc assumptions. We show that the free parameters of these assumptions can be optimized to reproduce the time evolution of the total magnetic energy injection through the photosphere in NOAA AR 11158, when compared to recent state-of-the-art estimates for this active region. However, we find that the relative magnetic helicity injection is reproduced poorly, reaching at best a modest underestimation. We also discuss the effect of some of the data processing details on the results, including the masking of the noise-dominated pixels and the tracking method of the active region, neither of which has received much attention in the literature so far. In most cases the effect of these details is small, but when the optimization of the free parameters of the ad hoc assumptions is considered, a consistent use of the noise mask is required. The results found in this paper imply that the data processing and electric field inversion approach that uses only the photospheric magnetic field information offers a flexible and straightforward way to obtain photospheric magnetic and electric field estimates suitable for practical applications such as coronal modeling studies.

Appendices

Appendix A: Modifications to the Existing Data Processing Methods

1.1 A.1 Tracking Scheme of Active Regions in SHARP Data

As explained by Bobra et al. (2014) and Hoeksema et al. (2014), reprojected vector magnetograms in SHARP time series are created by tracking each HMI Active Region Patch (HARP) using a fixed rotation rate for the center point of the patch. The rotation rate is determined from the latitude of the center point using a differential rotation profile. Instead of the differential rotation profile described in Hoeksema et al. (2014), we employ the “2-day lag” profile from Snodgrass (1983). For the NOAA 11158 vector magnetogram time series used in this paper, our tracking speed is ${\sim}\,2~\mbox{km}\,\mbox{s}^{-1}$ (0.27 deg per day) slower than the SHARP tracking speed. We observe that when using this method, the center of the active region remains slightly better fixed to the center of the patch. As explained by Sun (2013), the final reprojected SHARP magnetogram is created for each frame so that the disambiguated full-disk vector magnetogram is reprojected on the solar surface, i.e. interpolated from image pixels to a new grid that corresponds to a Lambert CEA projection of the solar surface covering the predescribed active region size. The CEA projection is made so that the active region is “viewed directly from above”, i.e. using a heliographic coordinate system where the center of the active region is at the intersection of the central meridian and the equator. The grid spacing is $0.03^{\circ}$ in projected heliographic coordinates, corresponding to the SDO/HMI resolution at the disk center. We have modified this procedure so that instead of a CEA projection, we employ Mercator projection, since conformal mappings are preferred when optical flow methods (such as DAVE4VM) are used (Welsch et al., 2009; Kazachenko et al., 2015). After reprojecting the full-disk magnetogram data, we rotate the interpolated magnetic field vectors from the SDO/HMI image basis $(B_{\xi},B_{\eta},B_{\zeta})$ (uniquely defined by the orthogonal image axes and LOS direction) to the heliographic basis $(B_{r}',B_{\theta }',B_{\phi}')$. The heliographic basis is chosen consistently with the Mercator map projection so that the center point of the active region is at the intersection of the central meridian and the equator. Transformation from the default heliographic basis $(B_{r},B_{\theta},B_{\phi})$ where the true solar equator has latitude $\lambda= 0$ to the patch-centered basis $(B_{r}',B_{\theta}',B_{\phi}')$ can be made using the following transformation (which is not included in the references listed above):

$$\begin{aligned} \left ( \textstyle\begin{array}{c} B_{r}' \\ B_{\theta}' \\ B_{\phi}' \end{array}\displaystyle \right ) =& \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0 \\ 0 & \cos\alpha& -\sin\alpha\\ 0 & \sin\alpha& \cos\alpha \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{c} B_{r} \\ B_{\theta} \\ B_{\phi} \end{array}\displaystyle \right ) \end{aligned}$$

(21)

$$\begin{aligned} \alpha(\lambda,\phi,\lambda_{c},\phi_{c}) =& \operatorname{sgn}(\lambda_{c})\operatorname{sgn}(\phi- \phi_{c}) \\ &{}\times\cos^{-1} \bigl[ \bigl(\cos\lambda_{c} \cos\lambda+ \sin \lambda_{c} \sin\lambda\cos(\phi- \phi_{c}) \bigr)/ \cos\lambda' \bigr] \end{aligned}$$

(22)

$$\begin{aligned} \lambda' =& \sin^{-1} \bigl[\sin\lambda\cos\lambda_{c} - \sin\lambda_{c} \cos\lambda\cos (\phi- \phi_{c} ) \bigr], \end{aligned}$$

(23)

where $\lambda'$ is the latitude in heliographic coordinates where the patch center $(\lambda_{c}',\phi_{c}') = (0,0)$.

This is different from SHARP data in which the heliographic basis corresponds to Carrington coordinates in which the equator is the true solar equator (Sun, 2013). Finally, we further transform into local Cartesian basis where we treat the projected active region patch as a flat surface:

$$\begin{aligned} \left ( \textstyle\begin{array}{c} B_{x} \\ B_{y} \\ B_{z} \end{array}\displaystyle \right ) =& \left ( \textstyle\begin{array}{c} B_{\phi}' \\ -B_{\theta}' \\ B_{r}', \end{array}\displaystyle \right ) \end{aligned}$$

(24)

where we drop the apostrophes to simplify notation.

1.2 A.2 Algorithm for the Removal of Spurious Flips in the Azimuth of the Magnetic Field

Welsch, Fisher, and Sun (2013) defined spurious flips of the azimuth of the magnetic field $\phi$ as sudden jumps where $\phi$ jumps approximately from one disambiguation to the other $\phi\rightarrow\phi+ 180^{\circ}$ and then quickly back. In a time series, this flipping can be seen as rapid blinking in the transverse magnetic field components. The authors proposed a temporal smoothing algorithm to recognize and remove such spurious flips. The algorithm detects these flips using two conditions. First, the change in azimuth between successive frames must be significant enough (over $120^{\circ}$):

$$ \bigl\| \phi(x_{m},y_{n},t_{k}) -\phi(x_{m},y_{n},t_{k-1})\bigr\| > 120^{\circ}, $$

(25)

where $\|x\|$ refers to the absolute acute angle of $x$:

$$ \|x\| = \left\{ \textstyle\begin{array}{l} |x|, |x| \leq180^{\circ} \\ 360^{\circ} - |x|, |x| > 180^{\circ}. \end{array}\displaystyle \right . $$

(26)

Second, if a spurious azimuth flip candidate at $t_{k}$ is removed by flipping the azimuth back to the “correct” value ($\phi\rightarrow\phi+ 180^{\circ}$), the procedure must decrease the sum of unsigned azimuth differences over the nearby frames:

$$ S = \sum_{l = -R}^{R} \bigl\| \phi(x_{m},y_{n},t_{k+l}) - \phi (x_{m},y_{n},t_{k})\bigr\| , $$

(27)

i.e. the procedure must smooth the azimuth in the time domain. Here $R$ is the expected length of that spurious flip in time, and it gives the upper limit for the number of frames the spurious flip may last in order to be recognized by the algorithm. $R$ is the only free parameter in the algorithm, and Welsch, Fisher, and Sun (2013) suggested two values for it: $R=2$ or $R=4$, of which they used the former. We chose to use $R=4$ instead, since it produces a more stable evolution of the azimuth in visual inspection. Another major qualitative departure from the work of Welsch, Fisher, and Sun (2013) is that we applied the flip-removal procedure after interpolating the SDO/HMI data to a local Cartesian system in Mercator projection (Appendix A.1), whereas they applied the procedure in native HMI pixels. Thus, in our approach the erroneous values of pixels where the azimuth had a spurious flip propagated in the interpolation. We therefore employed an additional step after recognizing and fixing the spurious flips: we smoothed each fixed pixel and all of its eight neighboring pixels using a Gaussian smoother with $\sigma= 1$ pixel truncated at $1\sigma$ as a way to mitigate the effect of propagating the erroneous azimuth to the neighboring pixels in the interpolation. An additional motivation for this smoothing is that it removes some of the artifacts of the flip-removal procedure itself: the procedure leaves some ragged structures and singular pixels of inconsistent azimuth values to the data, which we consider spurious, particularly when taking into account that the minimum energy disambiguation tries to reduce currents, i.e. to smooth the azimuth spatially.

Appendix B: Numerical Implementation of the Electric Field Inversion Methods in ELECTRICIT

In the PDFI method, Faraday’s law (Equation 4) is uncurled using the poloidal–toroidal decomposition of the magnetic field (Fisher et al., 2010):

$$\begin{aligned} \boldsymbol {A} =& {\nabla} \times{P\hat{\mathbf {z}} } + T\hat{\mathbf {z}} \end{aligned}$$

(28)

$$\begin{aligned} \boldsymbol {E}_{\mathrm{I}} =& -{\nabla} \times{\dot{P}\hat{\mathbf {z}} } - \dot {T}\hat{\mathbf {z}} , \end{aligned}$$

(29)

where $\boldsymbol {A}$ is the magnetic vector potential ${\nabla} \times {\boldsymbol {A}} = \boldsymbol {B}$, and $\dot{P}$ and $\dot{T}$ are the partial time derivatives of two-dimensional poloidal and toroidal potentials, respectively. They can be solved from the Poisson equations of which source terms are determined by the $\partial \boldsymbol {B}/\partial t$ field. The time derivative of the poloidal potential can be solved from

$$ \nabla_{h}^{2} \dot{P} = - \frac{\partial B_{z}}{\partial t} , $$

(30)

where $\nabla_{h}^{2}$ is the horizontal Laplacian $\partial_{x}^{2} + \partial_{y}^{2}$. We here employed only the horizontal components of the inductive electric field $\boldsymbol {E}_{h}^{\mathrm{I}} = (E_{x}^{\mathrm{I}},E_{y}^{\mathrm{I}})$, which are determined solely from $\dot{P}$. We solved Equation 30 using the same homogeneous Neumann boundary conditions and the numerical solver (FISHPACK, Swarztrauber and Sweet, 1975) as Kazachenko, Fisher, and Welsch (2014). As noted by Kazachenko, Fisher, and Welsch (2014), FISHPACK modifies the source term of a Poisson equation by adding a constant value to it, if it is found to be inconsistent with the homogeneous Neumann boundary conditions. As a result of this correction, the output electric field is slightly inconsistent with Faraday’s law. To remove this inconsistency, we added a post facto correction to the horizontal electric field components, as done by Fisher et al. (2010). To further ensure the consistency with Faraday’s law, we also decided to use different finite-difference approximations for Equations 28 and 29 than Kazachenko, Fisher, and Welsch (2014). The FISHPACK solver employs the five-point stencil for the Laplacian, which implicitly assumes that the first-order spatial derivatives of the solution of the Poisson equation are determined in half-grid points. Thus, if the first-order derivatives in Equation 29 and Faraday’s law (Equation 4) for the horizontal components of $\boldsymbol {E}_{\mathrm{I}}$ are calculated using a typical central difference scheme, $\partial B_{z}/\partial t$ is not reproduced exactly. Kazachenko, Fisher, and Welsch (2014) argued that the error introduced by this is small and employed a central difference scheme also for the first-order derivatives. However, we wish to reproduce $\partial B_{z}/\partial t$ from Faraday’s law exactly, and therefore employed the following consistent finite-difference formulas for the spatial derivatives (also used in a similar fashion by Yeates, 2017):

$$\begin{aligned} E_{x}^{\mathrm{I}}(x_{m},y_{n+1/2}) =& - \frac{\partial\dot{P}}{\partial y}(x_{m},y_{n+1/2}) = -\frac{\dot {P}(x_{m},y_{n+1}) - \dot{P}(x_{m},y_{n})}{\Delta y} \end{aligned}$$

(31)

$$\begin{aligned} E_{y}^{\mathrm{I}}(x_{m+1/2},y_{n}) =& \frac{\partial\dot{P}}{\partial x}(x_{m+1/2},y_{n}) = \frac{\dot {P}(x_{m+1},y_{n}) - \dot{P}(x_{m},y_{n})}{\Delta x} \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial B_{z}(x_{m},y_{n})}{\partial t} =& -(\nabla\times\mathbf {E}_{\mathrm{I}})_{z}(x_{m},y_{n}) = \frac{\partial E_{x}^{\mathrm{I}}}{\partial y}(x_{m},y_{n}) - \frac{\partial E_{y}^{\mathrm{I}}}{\partial x}(x_{m},y_{n}) \\ =& \frac{E_{x}^{\mathrm{I}}(x_{m},y_{n+1/2}) - E_{x}^{\mathrm{I}}(x_{m},y_{n-1/2})}{\Delta y} \\ &{} - \frac{E_{y}^{\mathrm{I}}(x_{m+1/2},y_{n}) - E_{y}^{\mathrm{I}}(x_{m-1/2},y_{n})}{\Delta x}. \end{aligned}$$

(33)

As can be implicitly read from the equations above, the spatial derivatives consistent with the five-point stencil give the electric field components in a staggered grid (Yee mesh; Yee, 1966) with respect to $B_{z}$ and $\partial B_{z}/\partial t$ at $(x_{m},y_{n})$. Since many of the TMF simulations employ such a staggered grid (e.g. van Ballegooijen, Priest, and Mackay, 2000; Cheung and DeRosa, 2012), this makes the electric field inversions of ELECTRICIT instantly suitable to be used as the boundary condition data of such simulations, while being simultaneously consistent with the observed time evolution of $B_{z}$.

The numerical approach described above was also used to solve the $P$ potential from $B_{z}$ and the resulting horizontal components of the vector potential $\boldsymbol {A}_{p}^{h}$ (see Fisher et al., 2010 for details). Thus, our $\boldsymbol {A}_{p}^{h}$ is determined in a staggered grid that is also exactly consistent with $B_{z}$.

The non-inductive component $\psi$ is solved similarly as the $\dot {P}$ potential in Equation 30 (including the post facto corrections) for each Assumption 1 and 2 (Section 2.3). When calculating the current density $j_{z}$ for Assumption 2, the spatial derivatives of $B_{x}$ and $B_{y}$ were calculated using a central difference scheme, since we wish to keep $j_{z}$ and thereby also $\psi$ cospatial with the magnetogram data. However, since $\boldsymbol {E}_{\mathrm{I}}$ is defined on a staggered grid, the gradients ${\nabla} _{x}{\psi}$ and ${\nabla} _{y}{\psi}$ are not cospatial with $E_{x}^{\mathrm{I}}$ and $E_{y}^{\mathrm{I}}$. This can be fixed by interpolating $\psi(x_{m},y_{n})$ to $\psi(x_{m+1/2},y_{n+1/2})$, after which the gradients

$$\begin{aligned} {\nabla} _{x}{\psi}(x_{m},y_{n+1/2}) =& \frac{\psi(x_{m+1/2},y_{n+1/2}) - \psi(x_{m-1/2},y_{n+1/2})}{\Delta x} \end{aligned}$$

(34)

$$\begin{aligned} {\nabla} _{y}{\psi}(x_{m+1/2},y_{n}) =& \frac{\psi(x_{m+1/2},y_{n+1/2}) - \psi(x_{m+1/2},y_{n-1/2})}{\Delta y} \end{aligned}$$

(35)

are cospatial with $E_{x}^{\mathrm{I}}$ and $E_{y}^{\mathrm{I}}$ in Equations 31 and 32. Moreover, this approach ensures that $({\nabla} \times{ \boldsymbol {E}})_{z}$ in Faraday’s law is unaffected by $-\nabla_{h} \psi$.

Finally, it should be noted that the use of a staggered grid in the ELECTRICIT solutions of $\boldsymbol {E}$ and $\boldsymbol {A}_{p}$ also requires (linear) interpolation back to a centered grid when Poynting or relative helicity fluxes are calculated.

Appendix C: Detection of Bad Pixels in our NOAA 11158 Series

When creating the NOAA 11158 magnetogram time series (Section 2.5), our algorithm detected at most 133 and on average 16 bad pixels in a single magnetogram cutout, which had an average shape of $745 \times 630$ pixels covering 4% of the solar disk. Ninetynine percent of these pixels were detected in weak-field pixels where $B < 250~\mbox{Mx}\,\mbox{cm}^{-2}$ (using the noise threshold chosen in Section 2.3). Maximum and average numbers of bad pixels in strong-field pixels were 6 and 0.2, bad pixels covering at most 0.03% and on average $7 \times 10^{-6}$ of the strong-field pixels. The number of bad pixels has a clear periodic behavior that follows the orbital motion of the SDO spacecraft and resulting variations in the formal error $\sigma_{\mathrm{B}}$. The number of bad pixels detected in the weak-field region spikes consecutively with $\sigma_{\mathrm{B}}$ formal error, which is a direct consequence of using a fixed threshold for $\sigma_{\mathrm{B}}$ to detect the bad pixels. On the other hand, the number of bad pixels in the strong-field pixels spiked at the negative extremum of the spacecraft velocity when the $\sigma_{\mathrm{B}}$ was close to its minimum.

We have published the bitmaps of bad pixels and the corresponding $B_{\mathrm{LOS}}$ full-disk cutouts with the other relevant datasets of this paper at https://zenodo.org/record/1034404 .