Junwei Zhao1, Andrey Stejko2, Ruizhu Chen1,3
1. W.W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, USA
2. Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102, USA
3. Department of Physics, Stanford University, Stanford, CA 94305-4060, USA
Center-to-limb variation in the time-distance measured travel times 1,2 is a systematic effect that demands a better understanding, as this effect is directly related to a reliable and robust inference of the solar meridional-flow profile in the deep interior. Time–distance analysis typically considers two types of wave travel-times: the mean travel-time [τmn], which is the average of travel times along two oppositely traveling directions and is believed to be related to interior structures; and the travel-time difference [τdiff], which is the difference between travel times of two opposite directions and is typically related to interior flows. Both τmn and τdiff exhibit systematic center-to-limb variations. Different mechanisms were proposed to explain this effect. In this work, we examine whether foreshortening is able to cause the center-to-limb effect in either τmn or τdiff.
Figure 1 | Sample maps showing the process of making data “C”. (a) Acoustic power map for a region located at the disk center and mapped to Postel’s coordinates; (b) acoustic power map for the same region, but placed at 60°N latitude and observed from the angle of HMI; (c) acoustic power map for the same region, but remapped from 60°N latitude to Postel’s coordinates.
Foreshortening is the decrease of observational spatial resolution with the increase of distance to the solar disk center due to the geometric curvature and projection. To examine its effect on our measurements, we prepare three types of data: without foreshortening, with observed foreshortening, and with mimicked foreshortening. SDO/HMI Doppler velocity data were used, and the period of 1 – 30 June 2015 was chosen, for which the solar B-angle is negligible. For each period of eight hours, data “A”, an area of 30°×30° at the disk center, is tracked, and data “B” is tracked at 60°N latitude with the same area. Data “C” is prepared using data “A” to mimic the foreshortening by following the following procedure: every image in data “A” is taken away from the disk center and pasted in a high-latitude area, i.e. with the center of the map located at 60°N and central meridian. Viewed again from the angle of HMI, this region becomes smaller, losing spatial resolution through the entire image, although the loss rate is disproportionate along the latitudinal direction. Furthermore, assuming the oscillations in data “A” are mostly radial, we project the velocity into the line-of-sight direction. Figure 1 illustrates the procedure of this coordinate transform.
The l-ν power-spectrum diagrams and time-distance travel times are calculated from the three types of the data. Figure 2 shows the comparison of all l-ν power-spectrum diagrams. Data “A” have the strongest acoustic power and longest visible power ridges; data “B” have the strongest convective power and noise below 1.0 mHz and the ridges are visibly longer than in data “C”; data “C” have substantially lower high-l and high-ν power than data “A”. The power spectrum of data “C”, which supposedly mimics the geometry near 60° latitude, differs significantly from that of data “B”, which were observed at 60° latitude. It is remarkable that the line-amplitude for each power ridge in data “B” is apparently lower than that in data “C”. That is, although the total power is mostly stronger in data “B” than in “C”, the acoustic power, seen in the line-amplitudes, is weaker. Substantial background power that is most likely due to observational noise and complicated optical spectrum-line formation in near-limb areas boosts the total power for data “B”.
Figure 3 | Comparisons of measured acoustic travel times. The left panels show comparisons of δτmn for selected measurement distances: (a) 1.08° –1.32°, (b) 2.40° –2.88°, and (c) 4.08° –4.80°. The right panels show comparisons of δτns for the same selected distances. The δτns for data “B” is not comparable with others, thus not plotted. All error bars are plotted as 3σ . Note the different vertical scales for different panels.
For time-distance measured travel times, Figure 3 compares δτmn and δτns as a function of relative latitude for three selected measurement distances. Apparently, the δτmn from different types of data are significantly different. The δτmn measured from data “C” are generally shorter than those from data “A” by as much as five seconds under some circumstances. The differences between measurements from these two data types also display some degree of latitude dependence, increasing with the latitude. The δτmn from data “B” show very strong latitude dependence, a clear sign of the center-to-limb effect in τmn. Data “C” do not fully reproduce all of the travel-time variations in data “B”, demonstrating that foreshortening can reduce mean travel-times, but does not account for all of the travel-time changes observed in the near-limb areas. Sharply different from δτmn, the δτns from different data types remains nearly identical across most latitudes, and it only exhibits tiny differences at the higher end of the latitude.
Overall, our exercise shows that foreshortening can reduce measured acoustic travel-times through redistributing acoustic power along the harmonic degree l, but it does not seem to play a role in causing the center-to-limb effect observed in the oppositely traveling waves. For details of this work, please refer to our recent publication Ref. 3.
 Duvall, T. L., Jr. 2003, in Proc. SOHO12/GONG+ 2002 Local and Global Helioseismology: the Present and Future , SP-517, 259
 Zhao, J., Nagashima, K., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., Jr. 2012, ApJ Lett, 749, L5
 Zhao, J., Stejko, A., Chen, R. 2016, Sol Phys, 291, 731