Preprocessing vector magnetograms for NLFFF

NLFFF extrapolation wants a lower boundary that is close to force-free and torque-free.

The measured boundary is a photospheric vector magnetogram. That is the awkward part: the photosphere has $\beta \sim 1$, so pressure, gravity, and flows still matter. It is not perfectly force-free.

Preprocessing is the step where the magnetogram is adjusted so it becomes a more consistent boundary for a coronal force-free model.

The key honesty point is simple: preprocessing can improve the model input, but it edits the data. That has to be stated.

Why the raw boundary is inconsistent

A force-free field above the plane $z=0$ cannot sit on arbitrary boundary values. Aly showed that the boundary must satisfy integral constraints expressing no net force and no net torque.

For a magnetogram area where the field decays at infinity, the basic constraints are below.

Flux balance:

$$\int B_{z}dxdy=0.$$

This says the patch should not contain a net magnetic monopole moment.

Net force vanishes:

$$\int B_x{B}_{z}dxdy=0,\int B_y{B}_{z}dxdy=0,\int (B_x^2+B_y^2-B_z^2)dxdy=0.$$

Net torque vanishes:

$$\int x(B_x^2+B_y^2-B_z^2)dxdy=0,$$ $$\int y(B_x^2+B_y^2-B_z^2)dxdy=0,$$ $$\int (y{B}_x{B}_z-x{B}_y{B}_z)dxdy=0.$$

Real magnetograms fail these constraints because of noise, projection effects, the 180-degree azimuth ambiguity, and genuine non-force-free photospheric physics.

Feeding the raw magnetogram directly into an NLFFF method means asking the solver to satisfy inconsistent demands.

Wiegelmann preprocessing

Wiegelmann, Inhester, and Sakurai recast preprocessing as a minimisation problem.

Find a boundary field that is:

• close to the observed magnetogram;
• closer to force-free and torque-free consistency;
• smoother than the raw data.

The functional is:

$$L={\mu }_1{L}_{\text{force}}+{\mu }_2{L}_{\text{torque}}+{\mu }_3{L}_{\text{data}}+{\mu }_4{L}_{\text{smooth}}.$$

The force term is:

$$\begin{array}{rl}{L}_{\text{force}}& =(\sum B_x{B}_z{)}^2+(\sum B_y{B}_z{)}^2+(\sum (B_z^2-B_x^2-B_y^2){)}^2.\end{array}$$

The torque term is:

$$\begin{array}{rl}{L}_{\text{torque}}& =(\sum x(B_z^2-B_x^2-B_y^2){)}^2+(\sum y(B_z^2-B_x^2-B_y^2){)}^2\\ & +(\sum (y{B}_x{B}_z-x{B}_y{B}_z){)}^2.\end{array}$$

The data and smoothness terms are:

$$L_{\text{data}}=\sum (\mathbf{B}-{\mathbf{B}}_{\text{obs}}{)}^2,L_{\text{smooth}}=\sum ({\nabla }^2\mathbf{B}{)}^2.$$

Read the terms as a trade-off:

• $L_{\text{force}}$ and $L_{\text{torque}}$ make the boundary more compatible with a force-free volume.
• $L_{\text{data}}$ keeps the boundary near the observation.
• $L_{\text{smooth}}$ removes small-scale noise the coronal model should not chase.

The ${\mu }_i$ weights decide how much the data are allowed to move. They are modelling choices, not harmless defaults.

The cost

Preprocessing is standard and defensible, but it changes the boundary.

That has three consequences:

1. After preprocessing, the model is fitting a modified magnetogram, not the raw observation.
2. Boundary-agreement metrics should say whether they are measured against the raw or preprocessed field.
3. The preprocessing weights and smoothing length become extra hyperparameters in the experiment.

It cannot manufacture missing coronal information. It trades measurement fidelity for consistency with the force-free model.

How NF2 relates to preprocessing

NF2 mostly moves this trade-off into the training loss.

The boundary enters as a soft penalty:

$${\lambda }_b⟨|\mathbf{B}-{\mathbf{B}}_{\text{obs}}{|}^2⟩.$$

So the network is not forced to satisfy the boundary exactly. It can settle on a nearby boundary that better matches a force-free interior. In effect, some of the preprocessing role is absorbed into the optimisation.

NF2 can also use measurement-error maps to down-weight noisy pixels. That is a local, soft version of saying “trust this part of the boundary less”.

The important conclusion is not that NF2 escapes the photosphere-corona mismatch. It does not. It simply handles the mismatch through the loss function rather than only through a separate preprocessing stage.

A clean experiment would compare:

raw boundary -> NF2
preprocessed boundary -> NF2

using the same joint physical metrics.

Related

• Photosphere versus corona force-free assumption
• From MHD to the force-free equations
• Classical NLFFF extrapolation methods
• Classical optimisation and the NF2 loss
• NLFFF quality metrics
• HMI SHARP CEA vector components
• HMI SHARP field inclination azimuth disambig