David Korda^{1} & Michal Švanda^{1, 2}

1. Astronomical Institute of Charles University, Faculty of Mathematics and Physics, V Holešovičkách 2, Praha 8 180 00, Czech Republic

2. Astronomical Institute of Czech Academy of Sciences, Fričova 298, Ondřejov 25165, Czech Republic

The Sun’s internal structure and the dynamics of plasma are key factors in understanding the solar activity, which in turn influences the interplanetary space (including effects on human infrastructures). Helioseismology is a method that allows us to study subsurface layers of the Sun; these layers, at least to some extent, reflect mostly the properties of the convection zone. Knowledge of the structure and dynamics of plasma in these regions allows us to put important constraints on theories of the solar dynamo and on the formation and evolution of magnetic fields, including sunspot models, and others.

Helioseismology is used to study the solar interior via interpretation of the dispersion relations of acoustic and surface gravity waves. These waves are generated randomly via the vigorous convection in the convection zone. The time-distance method^{[1]} is a tool to study the perturbations of travel times of waves caused by inhomogeneities of plasma parameters relative to the reference model.

The usual task of time-distance helioseismology is to invert for the horizontal components and/or spatial variations of the vertical component of the vector of plasma flows, or the sound-speed perturbations separately, using different sets of travel-time measurements. The separate inversions are justified under the assumption of independent actions of various perturbations to the wave travel times. In the inverse modeling, however, it is not certain that this assumption holds, because the inversions never provide us with an exact and unique answer. Also, in the realistic models of solar plasma, the perturbations are correlated. For instance, in the mass-conserving convective flows, the upflows (in the vertical direction) in the cell centers are accompanied by outflows (in the horizontal direction). Thus, these two quantities are naturally correlated. The large upflow velocities are usually located in the interior of the convective cells, where the temperature is greater compared to the surroundings. The greater temperature implies an increase in the sound speed, which naturally follows from the ideal-gas equation of state. Thus we should expect a correlation between the vertical velocity and the sound speed.

Figure 1|Top row: Vertical flows inverted from f-modes only with our improved approach. Middle row: Inverted flows with the difference geometries only (corresponding the traditional inversion schemes). Bottom row: diference of two aforementioned inversions. Left column: Inversion results. Right column: Ideal answers of the inversions. Note that the colour bars are not the same.

In time-distance helioseismology, the travel times are usually averaged over the annulus around the cell center, which, together with the correlation described above, leads to a leakage — the cross-talk — of one perturbation into the inversion for another one. Separate inversions do not allow one to quantify the possible cross-talk contributions. Information about the cross-talk is extremely important for proper interpretation of results. In the past, it turned out that only the cross-talk minimization allowed to successfully invert for subtle variations of the vertical velocity near the surface^{[2]}, which is otherwise buried in the cross-talk from the horizontal flow components with much larger magnitudes.

Figure 2a|4-D averaging kernel for the inversion for the sound-speed perturbations. The panels in the top row always represent a cut in the target depth (about 0.5 Mm in this case), whereas the panels in the bottom show the cuts in the vertical plane. The columns display the contributions (the leakage) from the horizontal flow (v_{x}= west-east, v_{y}=south-north), vertical flow (v_{z}) and sound-speed perturbation. Red contours correspond to the half-maximum of the target function, yellow corresponds to the half-maximum of the averaging kernel and the blue solid and dotted lines correspond to the +5% and -5% of the maximum of the averaging kernel, respectively. In the right part one sees the values of the trade-off parameters and the standard deviation of the inverted quantity when assuming travel times averaged over 24 hours. This plot is for the inversion contaminated by the cross-talk.

Building on the previous studies we improved the time-distance inversion pipeline to be able to invert for 3-D vector flows and the speed of sound at a same time^{[3]}. That allowed us not only to evaluate (and minimize) the cross-talks, but also to invert for the full vertical velocity component, unlike just its horizontal variations as before.

We validated our inversions by using a snapshot from the simulation of the fully developed solar convection provided by Matthias Rempel^{[4,5]}. From this snapshot, we forward-modeled the synthetic travel times, added a realistic estimate of the travel-time noise and applied our improved inversion. Since the inputs to the inversions were known, we could easily compare and study all the components of the inversion one-by-one. For the purpose of this study, we utilized the surface gravity waves (f-mode) travel times only.

Figure 2b|Same as Figure 2a, but for the inversions with cross-talk minimized.

We found that our new methodology does not constitute an improvement to what has been used recently by other authors in the case of the inversion for the horizontal flow. On the other hand, it does not worsen the results either. For the vertical flow, it is essential to minimize the cross-talk, otherwise, the results are nullified by the leakage mainly from the horizontal components. The leakage from the sound-speed perturbations is acceptably small. An obvious advantage of our new methodology is that one can also quantify the large-scale (with scales larger than the typical horizontal extent of the averaging kernel) component of the vertical flow, in addition to its horizontal variations as before (Figure 1). Thus, by using the improved methodology, we achieved significant improvements in inversions for the vertical flow. In the case of the inversion for the speed of sound, we found that there is a positively correlated cross-talk from the flow, which leads to an overestimation of the magnitude of the sound-speed variations by almost a factor of two in the near-surface layers (Figure 2a and 2b). By using our method we obtain the correct magnitude of the sound-speed perturbations. The method is ready to be applied to HMI observations.

### References

[1] Duvall, Jr., T. L., Jefferies, S. M., Harvey, J. W., & Pomerantz, M. A. 1993, *Nature*, **362**, 430

[2] Švanda, M., Gizon, L., Hanasoge, S. M., & Ustyugov, S. D. 2011, *A&A*, **530**, A148

[3] Korda, D., Švanda, M. 2019, *A&A*, **622**, A163

[4] Rempel, M. 2014, *ApJ*, **789**, 132

[5 http://download.hao.ucar.edu/pub/rempel/sunspot_models/Helioseismology/quiet_sun_98x98x18Mm_ 64x64x32km/