Time-distance Helioseismology Research Articles

Recent progress in time–distance helioseismology

Time–distance helioseismology is one of the local helioseismology techniques that are used to derive the interior properties of the Sun. It has been used to study the structures and flow fields beneath sunspots on local scales, as well as used to derive interior rotational rates and meridional flow velocities on global scales. In addition to the efforts in improving time–distance measurements and inversions, theoretical modeling is also carried out to enhance the accuracy of sensitivity kernels. Recently, by use of realistic numerical simulation on solar convection, we have also started to investigate the validity of time–distance studies.

Local Helioseismology and Correlation Tracking Analysis of Surface Structures in Realistic Simulations of Solar Convection

We apply time-distance helioseismology, local correlation tracking and Fourier spatial-temporal filtering methods to realistic supergranule scale simulations of solar convection and compare the results with high-resolution observations from the SOHO Michelson Doppler Imager (MDI). Our objective is to investigate the surface and sub-surface convective structures and test helioseismic measurements. The size and grid of the computational domain are sufficient to resolve various convective scales from granulation to supergranulation. The spatial velocity spectrum is approximately a power law for scales larger than granules, with a continuous decrease in velocity amplitude with increasing size. Aside from granulation no special scales exist, although a small enhancement in power at supergranulation scales can be seen. We calculate the time-distance diagram for f- and p-modes and show that it is consistent with the SOHO/MDI observations. From the simulation data we calculate travel time maps for surface gravity waves (f-mode). We also apply correlation tracking to the simulated vertical velocity in the photosphere to calculate the corresponding horizontal flows. We compare both of these to the actual large-scale (filtered) simulation velocities. All three methods reveal similar large scale convective patterns and provide an initial test of time-distance methods.

Comparison of Large-Scale Flows on the Sun Measured by Time-Distance Helioseismology and Local Correlation Tracking

We present a direct comparison between two different techniques: time-distance helioseismology and a local correlation tracking method for measuring mass flows in the solar photosphere and in a near-surface layer. We applied both methods to the same dataset (MDI high-cadence Dopplergrams covering almost the entire Carrington rotation 1974) and compared the results. We found that, after necessary corrections, the vector flow fields obtained by these techniques are very similar. The median difference between directions of corresponding vectors is 24°, and the correlation coefficients of the results for mean zonal and meridional flows are 0.98 and 0.88, respectively. The largest discrepancies are found in areas of small velocities where the inaccuracies of the computed vectors play a significant role. The good agreement of these two methods increases confidence in the reliability of large-scale synoptic maps obtained by them.

Analytical Models for Cross‐Correlation Signal in Time‐Distance Helioseismology

In time-distance helioseismology, the time signals (Doppler shifts) at two points on the solar surface, separated by a fixed angular distance are cross-correlated, and this leads to a wave packet signal. Accurately measuring the travel times of these wave packets is crucial for inferring the sub-surface properties in the Sun. The observed signal is quite noisy, and to improve the signal-to-noise ratio and make the cross-correlation more robust, the temporal oscillation signal is phase-speed filtered at the two points in order to select waves that travel a fixed horizontal distance. Hence a new formula to estimate the travel times is derived, in the presence of a phase speed filter, and it includes both the radial and horizontal component of the oscillation displacement signal. It generalizes the previously used Gabor wavelet that was derived without a phase speed filter and included only the radial component of the displacement. This is important since it will be consistent with the observed cross-correlation that is computed using a phase speed filter, and also it accounts for both the components of the displacement. The new formula depends on the location of the two points on the solar surface that are being cross correlated and accounts for the travel time shifts at different locations on the solar surface.

Effects of Horizontal Magnetic Fields on Acoustic Travel Times

Local helioseismology techniques seek to probe the subsurface magnetic fields and flows by observing waves that emerge at the solar surface after passing through these inhomogeneities. Active regions on the surface of the Sun are distinguished by their strong magnetic fields, and techniques such as time-distance helioseismology can provide a useful diagnostic for probing these structures. Above the active regions, the fields fan out to create a horizontal magnetic canopy. We investigate the effect of a uniform horizontal magnetic field on the travel time of acoustic waves by considering vertical velocity in a simple plane-parallel adiabatically stratified polytrope. It is shown that such fields can lower the upper turning point of p-modes and hence influence their travel time. It is found that acoustic waves reflected from magnetically active regions have travel times up to a minute less than for waves similarly reflected in quiet regions. It is also found that sound speeds are increased below the active regions. These findings are consistent with time-distance measurements.

Time-Distance Measurements of Cross-Correlation Asymmetries around NOAA AR 10486

We apply time-distance helioseismology to NOAA AR 10486 in order to examine the effect the presence of magnetic fields has on the resulting cross-correlations. Large-amplitude asymmetries between waves propagating in opposite directions are observed. The asymmetries can be attributed to suppression and absorption of seismic waves by the magnetic field. Unraveling the effects of magnetic field on the waves from the effects of sound-speed perturbations and flows is one of the major challenges facing time-distance helioseismology.

Optimal Gaussian Phase-Speed Filters in Time-Distance Helioseismology

Gaussian phase-speed filters are widely used in time-distance helioseismology to select specific wave packets whose travel times are then measured at the solar surface. This filtering increases the signal-to-noise (S/N) ratio of the temporal cross-covariances that are fitted to derive the travel times. The central phase speeds of these Gaussian filters are prescribed by a solar model; their widths are typically chosen empirically. No systematic study has been published on the effect of this filter width on the S/N ratio of the travel times. Such an analysis requires the ability to generate both noise and signal travel-time perturbations, this is now possible due to the recent introduction of a noise model and Born-approximation sensitivity kernels. These kernels allow for a derivation of travel-time perturbations as functions of a given sound-speed perturbation and are dependent on the phase-speed filters applied to the data, unlike simpler kernels. In this paper, we show that there is indeed an optimum value of the filter width that results in a maximum S/N ratio for the travel-time maps. Narrower filters exclude too much signal to produce useful travel-time perturbation maps, while broader filters are not selective enough.

Scattering of Acoustic Waves by a Magnetic Cylinder: Accuracy of the Born Approximation

With the aim of studying magnetic effects in time-distance helioseismology, we use the first-order Born approximation to compute the scattering of acoustic plane waves by a magnetic cylinder embedded in a uniform medium. We show, by comparison with the exact solution, that the travel-time shifts computed in the Born approximation are everywhere valid to first order in the ratio of the magnetic to the gas pressures. We also show that for arbitrary magnetic field strength, the Born approximation is not valid in the limit where the radius of the magnetic cylinder tends to zero.

Traveling Convective Modes in the Sun's Subsurface Shear Layer

Observations reveal that supergranular convective cells on the Sun rotate faster than the plasma on the solar surface. Recent time-distance helioseismology results from the Solar and Heliospheric Observatory Michelson Doppler Imager suggest that supergranulation has properties of traveling waves. We consider the idea that these properties are related to the steep increase of the angular rotation rate with depth, obtained by helioseismic inversions. The subsurface shear layer causes the convective modes to travel faster than the surface plasma. We calculate the phase speed of unstable modes of convection in the linear approximation for a standard solar model as a function of the velocity gradient in the shear layer. We find that for the helioseismically determined gradient, the calculated phase speed is significantly smaller than the observed speed of the supergranular pattern relative to the surface plasma. This suggests that, while the subsurface shear layer provides a plausible explanation for the wavelike behavior, it is necessary to study nonlinear effects in the dynamics of supergranulation.

Three‐dimensional Inversion of Sound Speed below a Sunspot in the Born Approximation

We revise the inversion of acoustic travel times for the three-dimensional sound-speed structure below the solar NOAA Active Region 8243 of 1998 June. We benefit from recent progress in time-distance helioseismology that provides us with more reliable tools to infer subsurface solar properties. Among the improvements we implement here are the use of Born approximation-based travel-time sensitivity kernels that take into account finite-wavelength effects and thus are more accurate than the previously employed ray-path kernels, the inclusion of solar noise statistical properties in the inversion procedure through the noise covariance matrix, and the use of the actual variance of the noise in the temporal cross-covariances in the travel-time fitting procedure. Of these three improvements, the most significant is the application of the Born approximation to time-distance helioseismology. This puts the results of this discipline at the same level of confidence as those of global helioseismology based on inversion of normal-mode frequencies. Also, we compare inversion results based on ray-path and Born approximation kernels. We show that both approximations return a similar two-region structure for sunspots. However, the depth of inverted structures may be offset by 1 or 2 Mm, and the spatial resolution of the results is more accurately estimated with the more realistic Born sensitivity kernels. Finally, using artificial realizations of Doppler velocities of the quiet Sun, we are now able to estimate the statistical uncertainties of these inversion results.

Helioseismic observations of magnetic flux emergence and flare effects

Time-distance helioseismology and data from SOHO/MDI are used for obtaining 3D images of subsurface sound-speed perturbations and maps of plasma flows, associated with emerging magnetic flux and flaring activity of large active regions in October 2003. The results reveal extremely complicated dynamical processes in the upper convection zone and indicate that subsurface shear flows may play an important role in magnetic energy release in solar flares. Strong X-class flares generated impulsive seismic waves (“sunquakes”), traveling through surrounding sunspots, thus providing new insight into the interaction of seismic waves with magnetic fields.

Comparison of subsurface sound-speed structures of three active regions

We analyze three solar active regions observed with the MDI instrument onboard SoHO (Scherrer et al. 1995). We apply the time-distance helioseismology formalism to derive the travel times of acoustic waves propagating through these active regions. The inversion of these acoustic travel times gives us access to the 3D sound-speed structure below the sunspots. We compare the main characteristics of these inversion results as a function of the active region size and magnetic field strength.

Surface Magnetism Effects in Time‐Distance Helioseismology

Recent observations of helioseismic holography revealed that magnetic fields that are inclined relative to the line-of-sight direction could cause systematic variations in measured acoustic phase shifts (hereafter, inclined magnetic field effect) and that the presence of surface magnetic field may shift the phases and impair the coherence of acoustic waves (known as the showerglass effect), thus complicating the interpretation of acoustic wave propagation time through the solar interior. In this paper we examine how these two observational effects affect time-distance helioseismology measurements in magnetic regions. It is confirmed that the inclined magnetic field could cause variations in time-distance measured acoustic travel times inside sunspot penumbra as well; however, inversions of the measured times for the wave propagation show that this effect only slightly shifts the location of negative sound-speed variations near the solar surface but basically does not change the inverted deeper interior structures. Further measurements using continuum intensitygrams and line-depth data from the MDI on board SOHO illustrate that the inclined magnetic field does not cause any obvious systematic travel-time variations in these observations. Regarding to the showerglass effect, we find that outgoing and ingoing travel-time perturbations through sunspots from our typical time-distance measurements are significantly smaller than those reported from helioseismic holography and also strongly depend on the propagation depth indicating deep changes. In addition, our second-skip cross-correlation experiments demonstrate that inside sunspots, the half of the double-skip travel times are very similar to the mean single-skip travel times, indicating that acoustic signals observed inside sunspots do not introduce detectable phase shifts after applying proper phase-speed filtering.

Evolution of Solar Subsurface Meridional Flows in the Declining Phase of Cycle 23

We study the evolution of meridional flows in the solar convection zone extending to a depth of 0.793 R☉ in the period 2000-2003 with helioseismic data taken with the Taiwan Oscillation Network (TON) using the technique of time-distance helioseismology. The meridional flows of each hemisphere formed a single-cell pattern in the convection zone at the solar minimum. An additional divergent flow was created at active latitudes in both hemispheres as the activity developed. The amplitude of this divergent flow correlates with the sunspot number: it increased from solar minimum to maximum (from 1996 to 2000), and then decreased from 2000 to 2003 with the sunspot number. The amplitude of the divergent flow increases with depth from 0.987 R☉ to a depth of about 0.9 R☉, and then decreases with depth at least down to 0.793 R☉.

Subsurface characteristics of sunspots

Time–distance helioseismology provides 3-D subphotospheric maps of sunspots, emerging magnetic structures and associated convective flows. This gives us new insights into the basic physical processes inside the Sun, formation of magnetic structures in the solar plasma and mechanisms of solar activity. Recent results from the MDI instrument on SOHO reveal complicated stratification and flow patterns of solar plasma beneath sunspots, with a characteristic temperature increase in deeper layers and multiple vortex systems, providing evidence for a cluster model of sunspots. The helioseismic observations provide important clues for understanding the mechanisms of the formation of sunspots and active regions, and the subphotospheric dynamics of convective flows shearing and twisting magnetic fields in the corona.

Time‐Distance Helioseismology: Inversion of Noisy Correlated Data

In time-distance helioseismology most inversion procedures ignore the correlations in the data errors. Here we simulate the travel-time perturbations of wavepackets that result from known distributions of sound speed inhomogeneities. The forward and inverse problems are carried out using recently developed Born approximation sensitivity kernels. A realistic solar noise component, with the correct statistics, is added to the data. We then apply a three-dimensional inversion procedure based on an improved multichannel deconvolution algorithm that includes the full covariance matrix of the simulated data and constrains the solution both in the vertical and horizontal directions. The validation of the inversion is achieved through comparison of the inferred sound speed distributions with the exact solutions. We show that including the covariance matrix matters for sound speed inhomogeneities varying on a length scale smaller than the correlation length. We also find that the inversion procedure is improved by adding horizontal regularization.

Helioseismic Determination of the Solar Rotation Axis

Time-distance helioseismology analysis of Dopplergrams from the Michelson Doppler Imager instrument on board the Solar and Heliospheric Observatory spacecraft provides a very accurate measurement of meridional flows. This shows a sinusoidal variation with a period of 1 yr and was identified by P. Giles to be due to an error in the determination of the solar rotation axis. The phase and amplitude of the annual variation of meridional flows are used to determine the error in the Carrington elements. After comparing with other recent estimates of the Carrington elements, we suggest the following values: i = 7155 ± 0002 and Ω = -735 ± 01.

Observations and Interpretation of Subsurface Magnetic Structures

AbstractImmediate subsurface local solar structures – flow, magnetic, and thermal – especially those associated with active regions, are of considerable interest, and are potentially accessible using high ℓ helioseismology. In recent years, classical modal seismology has been supplemented with new approaches, such as time-distance helioseismology, Hankel analysis, and acoustic holography. Results from time-distance helioseismology on shallow subsurface wave speed are at odds with those from the other methods. A simple model is presented which yields good agreement with the observed Hankel phase shifts of p modes passing through sunspots.

Local Helioseismology

We review the current status of local helioseismology, covering both theoretical and observational results. After a brief introduction to solar oscillations and wave propagation through in-homogeneous media, we describe the main techniques of local helioseismology: Fourier-Hankel decomposition, ring-diagram analysis, time-distance helioseismology, helioseismic holography, and direct modeling. We discuss local helioseismology of large-scale flows, the solar-cycle dependence of these flows, perturbations associated with regions of magnetic activity, and solar supergranulation.

Accuracy of the Born and Ray Approximations for Time‐Distance Helioseismology of Flows

Time-distance helioseismology has been used extensively to study mass flows (for example, flows associated with supergranulation), active regions, and sunspots. In nearly all of these studies the sensitivity of travel times to subsurface flows has been modeled using the first-order ray approximation. The Born approximation has been proposed as a more accurate alternative. In this paper we study the accuracy of the ray and Born approximations by comparison with numerical simulations of acoustic wave propagation through simple flows in two dimensions. The numerical experiments show that the Born approximation captures the basic behavior of the travel times except when the flow is strong enough that a significant reflected wave is seen. As expected, the first-order ray approximation fails to capture scattering effects.