What is known about small scale structure of photospheric magnetic fields?
The magnetic field in the solar photosphere is very inhomogeneous, and has fine structure with spatial scales of about 10km (see [1,2] and references therein for a review). Therefore, magnetograms produced by instruments, such as SDO/HMI or Hinode/SOT, show not the actual magnetic field, but the field averaged over a volume of around 100x100km (spatial resolution of an instrument) x500km (thickness of the photosphere) (Figure 1). To complicate things further, the observed magnetic field is weighted by a function of depth, or the contribution function. Since different spectral lines used for magnetic field measurements have different magnetic field sensitivities and different contribution functions, the maps produced by different magnetographs can differ substantially, particularly in active regions, where the magnetic field is expected to be complex.
To a first approximation, the small-scale magnetic field structure can be described using the so-called two-component model, where photospheric magnetic flux is carried by thin fluxtubes with field strength Breal and the filling factor α (α represents the fraction of volume penetrated by Breal). Hence, the observed magnetic field is
Bobs = α Breal.
Although the two-component approximation sounds like an oversimplification, in fact, it is not very far from the reality. Simulations show that the magneto-convective collapse results in two populations of magnetic field : strong magnetic elements with kG strength (usually concentrated at the photospheric network boundaries) and weak ambient field (Figure 2).
Why is this important?
The difference between the observed and real distributions of magnetic field in the photosphere can affect all sorts of measurements. For instance, in terms of the two-component model, ignoring the filling factor α while estimating the magnetic energy density or the Poynting flux in the photosphere would result in them being underestimated by factor of 1/α. Similarly, the current density would be underestimated by factor of about α-1.5.
Can we do anything about it?
There are a few different ways to evaluate the “real” magnetic field (or the filling factor). Firstly, the classical Magnetic Line Ratio (MLR) method based on the comparison of the Zeeman Effect in spectral lines with different magnetic field sensitivities (see  for an in-depth review).
Recently, Gordovskyy et al. [5,6] have developed an alternative method for diagnostics of unresolved field: the Stokes V Width (SVW) method. It links the filling factor with the width of the Stokes V component in some classical magnetometric spectral lines, such as Fe I 5247 Å and 6301 Å. An important advantage of this new method is that, unlike MLR, it requires only one spectral line. The SVW method has been tested using the magnetoconvection models of the photosphere and appears to be, generally, as reliable as the classical MLR method (Figure 3). Comparison of these methods applied to different spectral lines show that MLR is usually more reliable for lower values of Bobs (typically, up to about 500G), while SVW is more reliable for higher Bobs values.
Can we do better?
The methods discussed above cannot properly account for the temperature and velocity variations, which affect spectral line profiles and, hence, the reliability of both the MLR and SVW methods. Stokes inversion yields much better quality [e.g. 7]. The idea of the Stokes inversion approach is to find the line-of-sight distribution of thermodynamic parameters, LOS velocity, magnetic field components, and filling factors (for B and VLOS) providing the best fit to the observed Stokes components of selected spectral lines. However, this approach is computationally expensive, and usually applied only to relatively small patches of the solar surface (order of 100×100 arcsec or so). Therefore, the Stokes inversion can be used for more accurate analysis of smaller areas, while MLR and SVW can be used for fast “on-the-fly” analysis of large areas (a big active region or the whole solar disk) or for calibration of large-area magnetograms.
Can we do even better?
The methods discussed above can only evaluate Bobs or α. To find the sizes and shapes of small-scale magnetic elements, we need direct high-resolution observations.
During the last decade the spatial resolution of solar optical observations have improved greatly, currently reaching ~0.1arcsec (70km) [e.g. 8]. The Daniel K. Inouye Solar Telescope (DKIST), which is entering operation this year, and the planned European Solar Telescope (EST) will push this boundary even further: DKIST spatial resolution will be 35km, while EST is expected to resolve scales as small as 20-25km. These two instruments are likely to be game-changers, finally revealing the fine structure of solar magnetic fields.
-  Frazier, E.N. & Stenflo, J.O., 1972, Solar Phys., 27, 330.
-  de Wijn, A.G., Stenflo, J.O., Solanki, S.K. & Tsuneta, S., 2009, SSRv, 144, 275.
-  Vogler, A., Shelyag, S., Schussler, M., Cattaneo, F. et al., 2005, A&A, 429, 335..
-  Smitha H.N. & Solanki S.K., 2017, A&A, 608, A111.
-  Gordovskyy M., Shelyag, S., Browning P.K. & Lozitsky V.G., 2018, A&A, 619, A164.
-  Gordovskyy M., Shelyag, S., Browning P.K. & Lozitsky V.G., 2020, A&A, 633, A136.
-  Kobel, P., Solanki, S.K. & Borrero, J.M., 2011, A&A, 531, A112.
-  Keys, P.H., Reid, A., Mathioudakis, M., Shelyag, S., Henriques, V.M.J. et al., 2020, A&A, 633, A60.