Propagation Properties of Sunspots Umbral Oscillations in Horizontal and Vertical Directions

We present a study on investigating the propagation characteristics of umbral oscillations in sunspots. In sunspot 1 (located in NOAA AR 12127) with four umbrae, the analysis shows that the oscillations in different umbrae are correlated. The weak correlation (<20%) is attributed to the propagation of umbral oscillations across the umbral boundary to its adjacent umbra in the horizontal direction. We speculate that oscillations in two of the umbrae have a common origin in the sub-photosphere, resulting in a stronger correlation (>30%). Additionally, utilizing the TiO (photosphere), Hα (chromosphere) images provided by BBSO/GST, and the 304 Å (upper chromosphere and lower transition region), 171 Å (upper transition region), 193 Å (corona), and 211 Å (active region corona) images acquired by the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory (SDO), we analyze the vertical propagation of oscillations in the sunspot umbra. Multi-channel observation shows that the umbral oscillations observed in the lower atmosphere of sunspot 1 cannot be detected in the upper atmosphere. However, in sunspot 2 (located in NOAA AR 12132), oscillations in the lower atmosphere can propagate to the upper atmosphere. Using photospheric magnetic field data provided by the Helioseismic and Magnetic Imager on board SDO, potential field extrapolation of the magnetic field for the two sunspots shows that open magnetic field structures allow sunspot oscillations to propagate to higher heights, while closed magnetic field structures do not.


Introduction
Observations of sunspots show that oscillations of different periods commonly exist in the sunspot umbra and penumbra, called umbral oscillations (∼3 minutes) and running penumbral waves (RPWs; ∼5 minutes), respectively.Chromosphere umbral flash is a form of umbral oscillation, for which the first observational evidence came from Beckers & Tallant (1969) in Ca II H and K filtergrams and spectrograms of a sunspot.In the chromosphere observation of sunspots, umbral oscillations are manifested as spiral arms that propagate outwards from inner umbra to the umbral-penumbral boundary.The first observational evidence of RPWs comes from Giovanelli (1972) and Zirin & Stein (1972); it appears as bright concentric rings propagating through the penumbra from the umbral-penumbral boundary to the sunspot boundary.The period of the umbral oscillations increases with distance radially outward, from about 3 minutes at the beginning to about 5 minutes at the umbral-penumbral boundary, which is already close to the period of the RPWs.The relationship between umbral oscillations and RPWs has become an essential issue of concern.Some argue that umbral oscillations and RPWs are two independent phenomena and they have different driving sources (Giovanelli 1972;Moore & Tang 1975).However, others believe that umbral oscillations and RPWs have the same driving sources (Tziotziou et al. 2002).They are the same phenomenon and are different manifestations of the disturbance propagating along the magnetic field lines in different regions of sunspots (Bogdan & Judge 2006;Cho & Chae 2020).Others believe that the umbral oscillations drive the RPWs (Zirin & Stein 1972;Alissandrakis et al. 1992).
The general opinion is that the sunspot oscillations essentially originate from the driving source below the photosphere.Thomas (1984) believed that sunspot chromospheric oscillations and sunspot photosphere oscillations are highly correlated and may be all from the driving source below the photosphere.Gurman (1987) and Georgakilas et al. (2000) believed that the sunspot oscillation is the manifestation of the p-mode wave in the Sun's interior propagating to the sunspot chromosphere.According to Centeno et al. (2006) and Jess et al. (2013), sunspot oscillations are caused by magnetohydrodynamic (MHD) waves propagating along the magnetic field lines of sunspots.Chai et al. (2022) further pointed out that the MHD wave from below the photosphere caused the umbral oscillations.Cho et al. (2021) believed that the umbral oscillation is an observational manifestation of the MHD shock wave generated by the convection cells below the photosphere.When oscillations excited below the photosphere reaches the photosphere, the strong magnetic fields of sunspots can act as waveguides for waves, propagating oscillations from the photosphere to the corona (Reznikova et al. 2012).The research of Yurchyshyn et al. (2020) also confirms this point of view, who proved that the location of the origin of umbral oscillations is highly spatially correlated with the location of the strongest magnetic field in the umbra.As the height increases, the spatial distribution of umbral oscillations gradually expands in the horizontal direction.Sych et al. (2020) speculated that the inclination angle of the inclined magnetic lines of force as a waveguide increases with height and correspondingly will change cutoff frequency (Bel & Leroy 1977).This will affect Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
the appearance of the dependence of period oscillations on the distance to the umbra center in the form of a concentration of ∼3 minute periodicity in the umbra and 5-7 minute periods at the penumbral boundary.The above results indicate that the sunspot oscillations originating from the sub-photosphere propagate upward in the vertical direction and are highly correlated with the magnetic field line distribution of the sunspots.
Recently, high-resolution observations of sunspots revealed a fine structure of umbral oscillations.Sych & Nakariakov (2014) discovered the wavefronts of umbral oscillations exhibit a spiralarm structure.Kang et al. (2019) interpreted umbral oscillation wavefront structure as the superposition of slow magnetoacoustic waves of two different azimuthal modes below the photosphere on the surface of an untwisted and nonrotating magnetic cylinder.Wu et al. (2021) extended this work by adding the condition of twisted magnetic field.Sych et al. (2021) used the spectral preparation of imaging from the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory (SDO/AIA) and showed that broadband ∼3 minute wavefronts consisted of separate narrowband details with different spatial shapes.These details occur when waves pass through the filed-line inhomogeneities of the sunspot.It was assumed that different magnetic waveguides with different inclination angles and, accordingly, different cutoff frequencies, lead to the appearance of these sources.The combination of narrowband spherical and linear sources leads to the appearance of visible spirality.In the horizontal direction, Felipe et al. (2019) noted out that the spiral-arm structure rotates counterclockwise to the umbral-penumbral boundary and evolves into RPWs.Su et al. (2016a) and Priya et al. (2018) have slightly different views; they argued out that the wavefront spiral-arm structure of the umbral oscillations splits when it reaches the umbral boundary; a part returns to the umbra to excite the next round of umbral oscillations, and the other part enters the penumbral to form the RPWs.However, in sunspots with multiple umbrae, the propagation mechanism of oscillations between different umbrae, as well as the possibility of their sharing a common origin in the sub-photosphere, have not been definitively established.We will discuss these problems.
In this work, we selected two sunspots located in two active regions (ARs) for analysis.One of the sunspots has four umbrae, and we can understand the propagation of sunspot oscillations in the horizontal direction by analyzing the correlation of umbral oscillations among different umbrae.We also compare the propagation of umbral oscillations in the vertical direction in the two sunspots to find the factors affecting the propagation of sunspot oscillations.This paper is organized as follows.Section 2 introduces the observation and data reduction methods of the target area.Section 3 explains the main analysis results.Section 4 discusses the findings and their implications.Section 5 summarizes the main conclusions.

Observation and Methods
As shown in Figure 1, sunspot 1 in NOAA AR 12127 observed on 2014 August 1 (located at S09E08, 17:15-17:55 UT) and sunspot 2 in NOAA AR 12132 observed on 2014 August 5 (located at S12E04, 18:20-19:20 UT).The TiO and Hα images are acquired by the Goode Solar Telescope (GST) at BBSO.We used the broadband filter imager on GST to acquire photosphere images every 15 s in TiO (7057 Å), of which the field of view (FOV) is 70″ and each pixel size is 0 029.We used the Visible Imaging Spectrometer of GST to acquire chromosphere images every 23 s by scanning the Hα spectral line from its blue wing −1 Å to red wing +1 Å with a step of 0.2 Å, of which the FOV is 70″ and each pixel size is 0 034.We take the first image at Hα−1.0 Å as a reference to align images of other bandpasses in the Hα band.In order to distinguish oscillations more clearly, we use the one phasespeed filter method (its kernel is the Butterworth filter) to extract the images at a specific velocity range, e.g., the images with v > 14 km s −1 are used for studying umbral waves and those with 4 < v < 14 km s −1 for studying penumbral waves.
We investigated the propagation of sunspot oscillations at higher layers using EUV images acquired by the SDO/AIA with a cadence of 12 s.The 304 Å channel is dominated by the He II lines, corresponding to the chromosphere and transition region.The 171 Å channel is dominated by the Fe IX lines, corresponding to the quiet corona and upper transition region.The 193 Å channel is dominated by the Fe XII, XXIV lines, corresponding to the corona and hot flare plasma.The 211 Å channel is dominated by the Fe XIV lines, corresponding to the AR corona (O'dwyer et al. 2010).We use the potential extrapolation of the photosphere (6173 Å) magnetic field data acquired by the SDO/HMI to analyze the effect of the magnetic field structure of the AR on the propagation of umbral oscillations at different heights.

Data Analysis
Sunspot 1 (located in NOAA AR 12127, see Figure 1(a) and (b)) is a complex sunspot with four umbrae.We analyze the correlation of oscillations in different umbrae to understand the propagation of umbral oscillations in the horizontal direction.Sunspot 2 (located in NOAA AR 12132 (see Figures 1(c) and (d)) is selected as a comparison with sunspot 1 to analyze their difference of oscillation propagation in the vertical direction.

Propagation Process of Umbral Oscillations in the
Horizontal Direction

Correlation of Oscillations in Different Umbrae
The TiO channel is used to determine the umbral boundary of the sunspot, and the Hα channel is used to analyze the oscillation process.The TiO image and Hα−0.4 Å image of sunspot 1 are shown in Figures 1(a) and (b).
In the processed Hα data set, we analyze the mean-intensity oscillations in each green box located in the four umbra centers.The center of the umbra has magnetic field lines with smaller inclination angles, and we tend to search for the localization of the source of 3 minute wavefronts with maximum oscillations here.To avoid the impact of local solar activity, we also analyzed the mean-intensity oscillations in each umbra.The intensity curves of any two green boxes and those of any two umbrae are shown in Figures 2(a) and (b).The mean-intensity curve of the four umbrae has about 12-15 peaks, and each peak corresponds to a period of about 3 minutes of umbral oscillations, which is consistent with the total data length of 40 minutes.The upper three pairs of curves correspond to the mean oscillation intensity curves of two adjacent umbrae, and the lower three pairs of curves correspond to the mean oscillation intensity curves of two nonadjacent umbrae.Table 1 gives the correlation coefficient of all combinations of intensity curves.On the whole, the values are less than 50%, and the correlation of oscillations in each of the two umbrae is weak.The oscillation correlation of adjacent umbrae is larger than that of nonadjacent umbrae.In each two adjacent umbrae, D1 and D2 have a minimum correlation coefficient.We speculate that their lower correlation may be caused by peacock-like jets appearing above the umbra D1 (Su et al. 2016a).In all umbrae, D3 and D4 have the strongest correlation.Further, we use cross-correlation analysis to study the time delay of the meanthe intensity oscillations in any two whole umbrae.As shown in Figures 3 (a)-(f), in ±3 minute range, the correlations between D1 and D2, and D3 and D4 are stronger than the other four pairs.The largest value of correlation is up to 0.4.The correlation between D2 and D3, D2 and D4, and D1 and D4 is slightly weaker than the above two pairs, and that between D1 and D3 is the weakest, less than 0.1.
The Fourier transform is a widely utilized tool for signal analysis in time series.The wavelet transform further extends the temporal domain into the time-frequency domain, enabling more effective extraction of features such as periodic frequencies.For continuous time series, the continuous wavelet transform (CWT) is commonly employed for data analysis.In order to assess the correlation of specific oscillatory intensities in the time series of interest, the cross wavelet transform (XWT; Grinsted et al. 2004) constructed using two CWTs allows us to ascertain their shared power and phase relationship.The mean-intensity oscillations in any two whole umbrae is studied with the XWTs (see Figures 3 (j)-(l) and Table 1).Figures 3 (j)-(l) are the XWTs of the above six couples of curves.The y-axis represents the oscillations period, the x-axis the time since 17:15 UT, and the color bar the relative strength of the occupied components.Generally, the contours with 95% confidence show that the significant common power arises in the 90-250 s (1.6-4.3 minutes) band for the six figures in the whole observation (∼40 minutes).Moreover, the phase difference for D1 and D2, and D2 and D4 in the strongest common power region is about 90°(arrows pointing upward); D1 and D4, and D3 and D4 is mostly about 180°(arrows pointing left); D1 and D3 are ∼0°and 180°; and D2 and D3 have no predominant value.Comparing with Figure 2, D3 and D4, and D1 and D4 display obvious antiphase relationship in the time range of 0-20 minutes.D1 and D3 display in-phase relationship in the time range of 15-20 minutes and antiphase relationship in the time range of 32-40 minutes.

Oscillations Propagating from One Umbra to Another
In 40 minutes, we have observed many events of oscillations propagating in between two adjacent umbrae, and the results are shown in Table 2. Most of the events occurred in between D1 and D2, D2 and D3, and only a few events occurred in between D3 and D4.We select two typical events to illustrate in Figure 4.In order to distinguish the propagation process, we

Propagation of Umbral Oscillations in the Vertical Direction
In Figures 1(c)-(d), we present TiO and Hα images of sunspot 2. Sunspot 2 is analyzed together for comparison with sunspot 1.

Umbral Oscillations in the Chromosphere and the Transition Region
As shown in the Figure 5, the top eight maps correspond to the NOAA AR 12127, and the bottom eight maps correspond to the NOAA AR 12132.We analyzed the velocity-filtered images of Hα (Chromosphere) and 304 Å (transition region) in the two ARs to trace the umbral oscillations, with a velocity range of v > 14 km s −1 .Panels (a)-(d) shows that in the Hα images, four moments with clear umbral oscillations are selected.If the propagation of umbral oscillations from the chromosphere to transition region is not hindered, a similar wavefront structure should appear in panels (a)-(d) and panels (e)-(h).Considering the time delay of wave propagation in the vertical direction, the corresponding times in the two channel images differ.However, the analysis shows that no wavefront structure similar to that in the Hα image is found in the 304 Å image.This may indicate that in the NOAA AR 12127, umbral oscillations cannot propagate vertically from the chromosphere to the transition region.
Similar analysis can be made for the bottom eight maps.Panels (i)-(l) are the Hα images at four moments in NOAA AR 12132, and the wavefront structure of the umbral oscillations is clear.The wavefront structure of panels (m)-(p) is very similar to those of panels (i)-(l), and the time delay between them is about 80 s, which may indicate the time required for the propagation of umbral oscillations from the chromosphere to transition region.In order to explain the difference between the oscillation phenomena in the two ARs, we analyzed the oscillation power at different heights and frequencies.

Oscillations Power of Different Frequencies from Photosphere to Corona
We first analyzed the umbral oscillations power from photosphere to transition region.Figure 6 shows the 3 minute oscillation power map of two sunspots in the channels of TiO, Hα−1.0,−0.8,−0.6,−0.4,−0.2,−0.0 Å, and 304 Å.The power increases with height, and the strongest power appears at the Hα−0.4Å channel.After this channel, the power strength begins to decay.Although oscillation power of 3 minutes in the 304 Å channel is detected in both ARs, their distributions are different.Panel (h) and panel (p) show that the power of sunspot 1 distributes loosely in and around its umbrae, while that of sunspot 2 distributes mainly in its umbra.This shows that the energy of the 3 minute oscillations in sunspot 1 has leaked out of the umbra in the horizontal direction, while in sunspot 2 it is still confined in the vertical direction.By closely inspecting panel (p), we find there is some leakage located in the lower right of the panel, which is along the direction of the magnetic field (see Figure 7).
We study the spatial distribution of the Fourier power in the four narrow bands of 304 Å, 171 Å,193 Å, and 211 Å.The center frequency is 5, 5.5, 6, and 7 m Hz, and the corresponding period is about 3.33, 3.03, 2.78, and 2.38 minutes, respectively.As shown in the Figure 8, the first column on the left is the original image of 304 Å, 171 Å, 193 Å, and 211 Å, and the other four columns on the right are the spatial distribution of the power of the four frequencies of the corresponding channel.Figure 7 shows a similar image of spatial distribution of the Fourier power of sunspot 2. Compared with the filtered images of Figures 8 and 7, for 304 Å, 193 Å, and 211 Å channels, the most obvious difference is that the power distribution of sunspot 1 is scattered, which is roughly concentered in D2 and the areas around it, but the distribution of power of sunspot 2 is almost located in the umbra and the corona loop extending from the umbra.In sunspot 1, with the increase of height, the distribution of oscillation power gradually scatters in the horizontal direction and is not confined in the positions of the umbra and its vicinity.This trend is particularly evident in the 193 Å to 211 Å images.In sunspot 2, for 211 Å, although we can still discern this trend extending outward from the center of the umbra, the power distribution also began to scatter.Therefore, compared with Figure 8, the leakage of oscillation power outside the umbra occurs at a higher height.Such power distribution has very clear regularity; we try to check the magnetic field structure of sunspots to explain this regularity and the difference in the distribution of oscillation, power in the two sunspots.

Magnetic Field Structure of Two Sunspots
Based on the above analysis, we reconstructed the coronal magnetic fields of the two sunspots with the potential field model based on SDO/HMI photospheric field measurements as a boundary condition (see Figure 9).The grid size selected for the two regions is 500 × 400 × 400, and each grid size is 360 km.As the central region of interest is the sunspot umbra, both images mainly contain magnetic lines originating within it.
As shown in Figure 9, there is a significant difference in the height of the magnetic lines of the umbra of the two sunspots, i.e., in the same height, sunspot 1 has closed magnetic field structure, while sunspot 2 has open magnetic structure.Bloomfield et al. (2007) pointed out that the sunspot oscillations results from slow magnetoacoustic waves propagating along the sunspot magnetic lines.Compared with the magnetic lines of sunspot 1, the magnetic lines with a smaller inclination angle of sunspot 2 allow oscillation propagation to a higher height.Panels (c) and (d) are 171 Å images containing the FOV (white box) of the magnetic field images.In the region near the umbra, the coronal loops with two sunspots as footpoints have significant differences in height.Consistent with the magnetic field structure, the matter and oscillations in sunspot 2 are guided by magnetic field lines, reaching higher heights in the vertical direction.
In Figure 5, the oscillations of sunspot 1 obviously visible in Hα images is indistinguishable in 304 Å images; sunspot 2 has highly similar oscillations waveforms in Hα images and 304 Å images.This shows that when the magnetic field line of sunspot 1 reaches the transition region (304 Å), the closed magnetic field structure can no longer provide a waveguide for the oscillations; in the same height, the magnetic lines of sunspot 2 still have an open structure, which makes the oscillations propagate to a higher height in the vertical direction than sunspot 1.This can also be used to explain the difference in power distribution in Figures 8 and 7.For sunspot 1, with the increase of height, the distribution of oscillation power is no longer consistent with the magnetic lines; Figures 8(q)-(t) show the leakage of oscillation power in the horizontal direction.For sunspot 2, with the increase of height, the oscillation power always keeps at the position extending from the center of the figure to the lower right, which is more consistent with the trend of the magnetic lines in Figure 9(b).The structure of the magnetic lines leads to the difference of the vertical propagation of the oscillations in the two sunspots.

Discussion
In Section 3.1, we found that for sunspot 1 with four umbrae, the oscillations in different umbrae are correlated.The correlation analysis of the mean oscillation intensity curves (whole and green box) of any two umbrae show that the overall correlation is weak (less than 50%; see Table 1).The correlation coefficient can be divided into three categories: less than 10% (D1 and D3; D2 and D4; D1 and D4), 10%-20% (D1 and D2; D2 and D3), and 30%-40% (D3 and D4).It is obvious that the correlations of oscillations between any two adjacent umbrae are significantly higher than those between any two nonadjacent umbrae.
In addition, numerous oscillations are observed wavefront propagating in the horizontal direction from one umbra across the boundary of the umbra into another in the velocity-filtered images of Hα (see Figure 4), while a time resolution of 23 s might not be sufficient to conclusively establish whether these wavefronts belong to the same wave train.There even exists a coincidence where the oscillation begins in one umbra just as the oscillation in another umbra arrives.However, this does not prevent us from arriving at an intuitive conclusion that oscillations in different umbrae can propagate horizontally, which also leads to their correlation.However, there may be damping effects between the umbra that we are not yet clear about, which leads to a weak correlation between oscillations of different umbrae.It cannot be ruled out that the oscillation of one umbra can further propagate to the next umbra after propagating to adjacent umbra, which explains the lower correlation between nonadjacent umbra.It should be noted that the horizontal propagation of oscillations in different umbrae is an observational result.We will discuss whether this is a real physical process or a visual effect of oscillation propagation in sunspots.
Table 2 shows that with the increase of the distance between the four umbrae (the width of light bridge for D1 and D2 ∼1200 km; for D2 and D3 ∼1500 km; and for D3 and D4 ∼2700 km), the propagation events decrease, while the correlation coefficient of oscillations in the adjacent umbra increases.Especially for D3 and D4 with the maximum distance, there are the fewest propagation events of oscillations through the boundary of the umbra, but they exhibit the highest correlation.Similar to the oscillations in the umbra and penumbra caused by the common source (Bogdan & Judge 2006), we speculate that the oscillations of D3 and D4 also have a common origin in sub-photosphere (see Figure 10), and they propagate upward from the sub-photosphere along different paths to D3 and D4 of the sunspot.Considering the vertical propagation speed (10 km s −1 ) of slow magnetoacoustic waves from the photosphere to the chromosphere and the time delay (3 minutes) of oscillations in D3 and D4, the distance difference between the vertical propagation of the oscillations to D3 and D4 is about 1800 km.
We have another reason to consider this inference.SDO/ HMI data indicate that the sunspot 1 used for analyzing horizontal oscillations is located within NOAA 12127, which had only one umbra when it was closer to the edge of the Sun on 2014 July 29 and four umbrae on 2014 August 1.Considering the Parker model, the magnetic field structure of this sunspot may still converge in the sub-photosphere, and the magnetic field lines gradually begin to separate with increasing height.There is a possibility that the oscillations in the four umbrae have a common source.Due to the convergence of the magnetic field lines of D3 and D4 at relatively higher positions, they have a higher correlation.
At the height from the chromosphere to the corona, we attribute the difference of oscillations in the propagation of the two sunspots to the open and closed structure of the sunspot magnetic field.Compared with sunspot 2, at the same height, sunspot 1 does not have a greatly vertical magnetic field line as a waveguide to propagate the oscillations to a higher height.The influence of magnetic field structure on oscillation is also shown in the change of cutoff frequency.In layered media, only waves with frequencies higher than the cutoff frequency can propagate (Horace 1909).The factors affecting the cutoff frequency of MHD waves in the solar atmosphere include temperature and magnetic field inclination.Reznikova et al. (2012) further pointed out that in the sunspot umbra, the influence of magnetic field inclination on the distribution of oscillation frequency is dominant.In addition, the increase of height also affects the cutoff frequency of MHD wave in the sunspot umbra (Felipe et al. 2018).
In Figure 2, the mean-intensity curves of umbral oscillations in each group exhibit consistent correlations during certain time intervals, while at other times, they may display completely opposite phases.This may indicate a difference in the frequency of oscillations in different umbrae, which may comes from the cutoff frequency.The magnetic field lines of sunspot 1 converge in the sub-photosphere layer, and as the height increases, the magnetic field lines gradually separate into multiple groups with different inclination angle.Changes in the magnetic field inclination angle can lead to changes in the cutoff frequency of the allowed waves, which leads to differences in the oscillation frequency in different umbrae.Chae et al. (2017) pointed out that the power of the umbral oscillation is significantly enhanced near the umbral dots and the light bridge.This may imply the contribution of oscillations on the umbral dots and light bridges to the umbral oscillation of sunspots.As shown in Figure 1, the area we initially analyzed is roughly located at the center of each umbra (marked by a green box), which is the region with the strongest magnetic field in the umbra.In order to avoid the impact of local activities on the umbra on the analysis, we ultimately selected the range as the mean oscillation intensity of each umbra as a whole.This actually reduces the impact of oscillations near the umbra dots on the entire analysis.Su et al. (2016b) pointed out that oscillations within a certain period range in the umbra quickly damped near the light bridges (2.2-2.6 minutes), while oscillations within certain period ranges can propagate outside the umbra (>2.6 minutes).In addition, cross wavelet analysis of the oscillations on both sides of the light bridges showed a common significant power region with a phase difference of approximately 90°.We have expanded the selection scope for this.Taking the light bridges of umbrae D3 and D4 as examples, the mean oscillation intensity of a portion of the regions of umbrae D3 and light bridges, as well as a portion of the regions of umbra D4 and light bridges, were selected for analysis.The results are similar to the oscillation intensity of the analysis of umbrae D3 and D4, with correlation and frequency differences.Furthermore, considering that we did not find any significant instances of oscillations propagating from the light bridges to the two side umbrae in the velocityfiltered image of Hα, we speculate that oscillations on the light bridges do not have a significant impact on the umbral oscillation, at least within the analyzed time and region.
In all channels shown in Figure 8, with the increase of frequency, the distribution of oscillation decreases gradually.In the first three channels in Figure 7, this trend is not obvious; the distribution of oscillation power starts to decrease when it reaches 211 Å.We infer that in sunspot 2, the inclination of the magnetic field line starts to increase significantly at the height corresponding to 211 Å.But for sunspot 1, similar magnetic field structure changes occur at a lower height.The cutoff frequency of sunspot 1 is lower compared to that in sunspot 2, implying less obstruction to the propagation of oscillations in the former.This observation indicates that the cutoff frequency does not play a decisive role in dictating the distribution of oscillation power across different heights in the atmosphere of sunspots.In addition, the plasma temperature/density and the magnetic field intensity/intensity gradient may also affect the vertical propagation of oscillation, which may provide ideas for future research.

Conclusion
This work investigates the propagation characteristics of umbral oscillations in the two sunspots (named sunspot 1 in NOAA AR 12127 and sunspot 2 in NOAA AR 12132).The main conclusions are as follows.
(1) In the observation, sunspot 1 with multiple umbrae and umbral oscillations can propagate into the adjacent umbra through the umbral boundary and cause a new round of umbral oscillations.Analysis indicates that the possible cause is that the magnetic field lines gathered in the sub-photosphere gradually separate with increasing height, and the difference in the time that the oscillations propagate to different umbrae along different magnetic lines in the vertical direction leads to the visual effect.
(2) The magnetic field lines in different umbrae may share one common origin in the sub-photosphere; oscillations from the same source propagate vertically along different paths, resulting in the correlation of oscillations in different umbra.
D3 and D4 had the highest correlation in the control group, but hardly horizontally propagating wavefronts were observed in the chromosphere.We speculate that the oscillations in the umbra are damped in the horizontal direction, making it almost impossible for them to propagate horizontally.
(3) Propagation of umbral waves with height was studied earlier in detail using even longer GST times series (Yurchyshyn et al. 2015).In our work, we did not analyze the specific heights at which the oscillations of two sunspots propagate in the vertical direction; we only compared the relative heights at which they can propagate.By comparing the differences in the magnetic field structures of the sunspots in the two ARs, we show that open magnetic field structures allow sunspot oscillations to propagate to higher heights, while closed magnetic field structures do not.

Figure 1 .
Figure 1.Maps of NOAA ARs 12127 and 12132 and the white contours display their umbral regions.Panels (a) and (b) show, respectively, the TiO and Hα−0.4 Å images of BBSO/GST for the NOAA AR 12127 on 2014 August 1, in which there are four umbrae, from left to right being D1, D2, D3, and D4.The green box analyzes the oscillations in it to compare with the mean oscillation intensity in the whole umbra.Symmetrical yellow lines are used to estimate the width of the light bridge.Panels (c) and (d) show, respectively, the TiO and Hα−0.4 Å images of BBSO/GST for the NOAA AR 12132 on 2014 August 5.

Figure 2 .
Figure 2. In NOAA AR 12127, oscillation intensity in umbra pairs of D1-D2, D2-D3, D3-D4, D1-D3, D2-D4, and D1-D4.Umbra corresponding to each pair of curves is marked from top to bottom on the left side, and the horizontal axis is the time series of the entire data.Panel (a) shows the oscillation intensity in the partial umbra (green box; see Figure 1(a)); panel (b) shows the oscillation intensity in the whole umbra.

Figure 3 .
Figure 3. Panels (a)-(f) show the time delay of the above paired umbral intensities.Panels (j)-(m) show the cross wavelet transform of the paired umbral intensities, where the 95% confidence level against noise is shown with thick contours.The phase difference is shown as arrows (with in-phase pointing right and antiphase pointing left).
performed velocity filtering to the Hα−0.4Å images and kept the image components within the velocity range of v > 14 km s −1 .The first event propagating from D1 to D2 occurred in 17:33:35-17:36:14 UT, as shown in Figures 4(a)-(h).The wavefront first appeared on the left side of D1 and has a clockwise rotation (see Figures 4(a)-(b)).The wavefront in Figure 4 (c) reaches the lower boundary of D1 and then starts to propagate into the boundary of D2 in Figure 4(d).In Figures 4(e)-(h), the wavefront has completely entered D2.The second event propagating from D4 to D3 occurred in 17:20:21-17:23:00 UT, as shown in Figures 4(i)-(p).The wavefront first appeared in the middle of D4, then rotated counterclockwise toward its lower boundary (see Figure 4(i)).When the wavefront reaches the left boundary of D4 in Figures 4(j)-(l), it continues to propagate into the lower-left boundary of D3 in Figure 4(m).In Figures 4(n)-(p), the wavefront has completely entered D3.The above two events are selected because they exhibit a very clear propagation process and can be taken as typical events of wavefronts crossing umbral boundaries.

Figure 4 .
Figure 4. Waves propagate in between two umbrae.The top eight panels show the propagation of an umbral wavefront from umbra D1 to D2 in NOAA AR 12127, where the red circle marks the wavefront spiral-arm structure.The bottom eight panels show the propagation of an umbral wavefront from umbra D4 to D3, where the blue circle marks the wavefront spiral-arm structure.

Figure 5 .
Figure 5. Umbral oscillations propagating from chromosphere to transition region.The top eight panels show the phase-speed filtered images of v > 14 km s −1 in −0.4 Å and 304 Å at four moments in AR NOAA AR 12127.The bottom eight panels show the images in NOAA AR 12132.The position the yellow circular mark represents a similar wavefront.

Figure 7 .
Figure 7. Similar to Figure 5 but for NOAA AR 12132.

Figure 8 .
Figure 8.The first column shows the maps of NOAA AR 12127 in the 304 Å, 171 Å, 193 Å, and 211 Å channels of SDO/AIA.The other four columns are their corresponding oscillation powers of 5, 5.5, 6, and 7 mHz, respectively.The umbral region is marked by a white contour in the figure.

Figure 9 .
Figure 9. Panels (a) and (b) are magnetic field structures derived with potential field extrapolation for the region containing NOAA AR 12127 and NOAA AR 12132.Their FOV is 180 Mm×144 Mm, and the maximum height is 180 Mm.Panels (c) and (d) are 171 Å image of two sunspots, with white boxes indicating the FOV of the magnetic field structures and white outlines indicating the umbra.

Figure 10 .
Figure10.The oscillations propagate in the vertical direction of the magnetic line.Specifically, the magnetic lines of D3 and D4 converge at higher heights relative to D1 and D2.

Table 2
Time of Oscillations Propagating in Different Umbrae