The sign of temperature inhomogeneities deduced from time-distance helioseismology

Inhomogeneities in wave propagation conditions near and below the solar surface have been detected by means of time-distance helioseismology. Here we calculate the effect of temperature inhomogeneities on the travel times of sound waves. A temperature increase, e.g., in active regions, not only increases the sound speed but also lengthens the path along which the wave travels because the expansion of the heated layers shifts the upper turning of the waves upward. Using a ray-tracing approximation we find that in many cases the net effect of a temperature enhancement is an increase of the travel times. We argue that the reduced travel times that are observed are caused by a combination of magnetic fields in the active region and reduced subsurface temperatures. Such a reduction may be related to the increased radiative energy loss from small magnetic flux tubes.     

The parabolic wave equation in local helioseismology

In recent years methods of time-distance helioseismology have been used to produce maps of local flows in the surface layers of the Sun. Usually, these studies rely on ray theory to describe the propagation of sound waves. Ray theory, however, is a poor approximation of the acoustic wavefield near the surface of the Sun. In particular, it is inappropriate for the study of scattering and diffraction by inhomogeneities. But an exact solution of the acoustic wave equation in the Sun is not trivial. In this paper I present an approximation to the full wave equation, which transforms it into a parabolic equation. The parabolic equation is commonly used in ocean acoustics and geoseismology because it is much simpler to solve numerically. Here I discuss the parabolic approximation, its limitations and potential applications in helioseismology. Finally, I present some numerical results to demonstrate the capabilities of this method.     

Comparison of Travel-Time and Amplitude Measurements for Deep-Focusing Time–Distance Helioseismology

The purpose of deep-focusing time–distance helioseismology is to construct seismic measurements that have a high sensitivity to the physical conditions at a desired target point in the solar interior. With this technique, pairs of points on the solar surface are chosen such that acoustic ray paths intersect at this target (focus) point. Considering acoustic waves in a homogeneous medium, we compare travel-time and amplitude measurements extracted from the deep-focusing cross-covariance functions. Using a single-scattering approximation, we find that the spatial sensitivity of deep-focusing travel times to sound-speed perturbations is zero at the target location and maximum in a surrounding shell. This is unlike the deep-focusing amplitude measurements, which have maximum sensitivity at the target point. We compare the signal-to-noise ratio for travel-time and amplitude measurements for different types of sound-speed perturbations, under the assumption that noise is solely due to the random excitation of the waves. We find that, for highly localized perturbations in sound speed, the signal-to-noise ratio is higher for amplitude measurements than for travel-time measurements. We conclude that amplitude measurements are a useful complement to travel-time measurements in time–distance helioseismology.     

Travel Time Sensitivity Kernels

We derive, following the standard first Born approximation approach used in the geophysics literature, an expression for the travel time perturbation caused by a perturbation to sound speed. In our simple model we employ a point source at one point and calculate the time taken for a wave packet created at the source to move to a second point. In the first Born approximation the travel time delay caused by a perturbation to the background model can be expressed as the integral over the whole sun of some function, called the travel time sensitivity kernel, multiplied by the perturbation. The sensitivity kernels are zero along the geometrical ray connecting the two points and have maximum weight in a tube around the ray; they are the solar equivalent of `the banana-doughnut' kernels discussed in the geophysics literature. Calculating sensitivity kernels that are more accurate than those derived from ray theory is important for the future of inversions done with time-distance helioseismology data as they will allow greater confidence in the results as well as increased resolution.     

High-Resolution Mapping of Flows in the Solar Interior: Fully Consistent OLA Inversion of Helioseismic Travel Times

To recover the flow information encoded in travel-time data of time?–?distance helioseismology, accurate forward modeling and a robust inversion of the travel times are required. We accomplish this using three-dimensional finite-frequency travel-time sensitivity kernels for flows along with a (2+1)-dimensional (2+1D) optimally localized averaging (OLA) inversion scheme. Travel times are measured by ridge filtering MDI full-disk Doppler data and the corresponding Born sensitivity kernels are computed for these particular travel times. We also utilize the full noise-covariance properties of the travel times, which allow us to accurately estimate the errors for all inversions. The whole procedure is thus fully consistent. Because of ridge filtering, the kernel functions separate in the horizontal and vertical directions, motivating our choice of a 2+1D inversion implementation. The inversion procedure also minimizes cross-talk effects among the three flow components, and the averaging kernels resulting from the inversion show very small amounts of cross-talk. We obtain three-dimensional maps of vector solar flows in the quiet Sun at horizontal spatial resolutions of 7?10 Mm using generally 24 hours of data. For all of the flow maps we provide averaging kernels and the noise estimates. We present examples to test the inferred flows, such as a comparison with Doppler data, in which we find a correlation of 0.9. We also present results for quiet-Sun supergranular flows at different depths in the upper convection zone. Our estimation of the vertical velocity shows good qualitative agreement with the horizontal vector flows. We also show vertical flows measured solely from f-mode travel times. In addition, we demonstrate how to directly invert for the horizontal divergence and flow vorticity. Finally we study inferred flow-map correlations at different depths and find a rapid decrease in this correlation with depth, consistent with other recent local helioseismic analyses.     

Time-Distance Inversion Methods and Results – (Invited Review)

The current interpretations of the travel-time measurements in quiet and active regions on the Sun are discussed. These interpretations are based on various approximations to the 3-D wave equation such as the Fermat principle for acoustic rays and the Born approximation. The ray approximation and its modifications have provided the first view of the 3-D structures and flows in the solar interior. However, more accurate and computationally efficient approximations describing the relation between the wave travel times and the internal properties are required to study the structures and flows in detail. Inversion of the large three-dimensional datasets is efficiently carried out by regularized iterative methods. Some results of time-distance inversions for emerging active regions, sunspots, meridional flows and supergranulation are presented. An active region which emerged on the solar disk in January 1998, was studied from SOHO/MDI for eight days, both before and after its emergence at the surface. The results show a complicated structure of the emerging region in the interior, and suggest that the emerging flux ropes travel very quickly through the depth range of our observations. The estimated speed of emergence is about 1.3 km s^–1. Tomographic images of a large sunspot reveal sunspot `fingers' - long narrow structures at a depth of about 4 Mm, which connect the sunspot with surrounding pores of the same polarity.     

Organized Subsurface Flows near Active Regions

Local helioseismic techniques, such as ring analysis and time-distance helioseismology, have already shown that large-scale flows near the surface converge towards major active regions. Ring analysis has further demonstrated that at greater depths some active regions exhibit strong outflows. A critique leveled at the ring-analysis results is that the Regularized Least Squares (RLS) inversion kernels on which they are based have negative sidelobes near the surface. Such sidelobes could result in a surface inflow being misidentified as a diverging outflow at depth. In this paper we show that the Optimally Located Averages (OLA) inversion technique, which produces kernels without significant sidelobes, generates flows markedly similar to the RLS results. Active regions are universally zones of convergence near the surface, while large complexes evince strong outflows deeper down.     

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.     

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.     

Time – Distance Modelling in a Simulated Sunspot Atmosphere

In time?–?distance helioseismology, wave travel times are measured from the cross-correlation between Doppler velocities recorded at any two locations on the solar surface. However, one of the main uncertainties associated with such measurements is how to interpret observations made in regions of strong magnetic field. Isolating the effects of the magnetic field from thermal or sound-speed perturbations has proved to be quite complex and has yet to yield reliable results when extracting travel times from the cross-correlation function. One possible way to decouple these effects is by using a 3D sunspot model based on observed surface magnetic-field profiles, with a surrounding stratified, quiet-Sun atmosphere to model the magneto-acoustic ray propagation, and analyse the resulting ray travel-time perturbations that will directly account for wave-speed variations produced by the magnetic field. These artificial travel-time perturbation profiles provide us with several related but distinct observations: i) that strong surface magnetic fields have a dual effect on helioseismic rays?–?increasing their skip distance while at the same time speeding them up considerably compared to their quiet-Sun counterparts, ii) there is a clear and significant frequency dependence of both skip-distance and travel-time perturbations across the simulated sunspot radius, iii) the negative sign and magnitude of these perturbations appears to be directly related to the sunspot magnetic-field strength and inclination, iv) by “switching off” the magnetic field inside the sunspot, we are able to completely isolate the thermal component of the travel-time perturbations observed, which is seen to be both opposite in sign and much smaller in magnitude than those measured when the magnetic field is present. These results tend to suggest that purely thermal perturbations are unlikely to be the main effect seen in travel times through sunspots, and that strong, near-surface magnetic fields may be directly and significantly altering the magnitude and lateral extent of sound-speed inversions of sunspots made by time?–?distance helioseismology.     

The Inversion Problem of Phase Travel Time Perturbations in Acoustic Imaging

The three-dimensional distribution of change in phase travel time of an active region below the solar surface can be constructed with the technique of acoustic imaging. The interpretation of the distribution of measured phase travel time perturbation suffers from the finite spatial resolution of the acoustic lenses. In the ray approximation, the phase travel time perturbation measured in acoustic imaging can be expressed as an integral of the product of the relative sound-speed perturbation and a kernel. Forward computations show that the vertical resolution of phase travel time perturbation is poor even in the ray approximation. In this study, we discuss the inversion of phase travel time perturbations to estimate the relative sound-speed perturbation with a regularized least-squares inversion method. The tests with model perturbations of sound speed show that the inversion reasonably recovers the distribution of the model perturbation. We also apply the inversion method to the measured phase travel time perturbation of active region NOAA 7981.     

Helioseismology of Sunspots: Confronting Observations with Three-Dimensional MHD Simulations of Wave Propagation

The propagation of solar waves through the sunspot of AR?9787 is observed by using temporal cross-correlations of SOHO/MDI Dopplergrams. We then use three-dimensional MHD numerical simulations to compute the propagation of wave packets through self-similar magnetohydrostatic sunspot models. The simulations are set up in such a way as to allow a comparison with observed cross-covariances (except in the immediate vicinity of the sunspot). We find that the simulation and the f-mode observations are in good agreement when the model sunspot has a peak field strength of 3 kG at the photosphere and less so for lower field strengths. Constraining the sunspot model with helioseismology is only possible because the direct effect of the magnetic field on the waves has been fully taken into account. Our work shows that the full-waveform modeling of sunspots is feasible.     

Recent Developments in Local Helioseismology

Local helioseismology is providing new views of subphotospheric flows from supergranulation to global-scale meridional circulation and for studying structures and dynamics in the quiet Sun and active regions. In this short review we focus on recent developments, and in particular on a number of current issues, including the sensitivity of different measures of travel time and testing the forward modelling used in local helioseismology. We discuss observational and theoretical concerns regarding the adequacy of current analyses of waves in sunspots and active regions, and we report on recent progress in the use of numerical simulations to test local helioseismic methods.     

Subsurface Supergranular Vertical Flows as Measured Using Large Distance Separations in Time–Distance Helioseismology

As large-distance rays (say, 10?–?24°) approach the solar surface approximately vertically, travel times measured from surface pairs for these large separations are mostly sensitive to vertical flows, at least for shallow flows within a few Mm of the solar surface. All previous analyses of supergranulation have used smaller separations and have been hampered by the difficulty of separating the horizontal and vertical flow components. We find that the large-separation travel times associated with supergranulation cannot be studied using the standard phase-speed filters of time–distance helioseismology. These filters, whose use is based upon a refractive model of the perturbations, reduce the resultant travel-time signal by at least an order of magnitude at some distances. More effective filters are derived. Modeling suggests that the center–annulus travel-time difference [δt _oi] in the separation range Δ=10?–?24^° is insensitive to the horizontally diverging flow from the centers of the supergranules and should lead to a constant signal from the vertical flow. Our measurement of this quantity, 5.1±0.1 seconds, is constant over the distance range. This magnitude of the signal cannot be caused by the level of upflow at cell centers seen at the photosphere of 10 m?s^?1 extended in depth. It requires the vertical flow to increase with depth. A simple Gaussian model of the increase with depth implies a peak upward flow of 240 m?s^?1 at a depth of 2.3 Mm and a peak horizontal flow of 700 m?s^?1 at a depth of 1.6 Mm.     

Large-Scale Zonal Flows Near the Solar Surface

Migrating bands of weak, zonal flow, associated with the activity bands in the solar cycle, have been observed at the solar surface for some time. More recently, these flows have been probed deep within the convection zone using global helioseismology and examined in more detail close to the surface with the techniques of local helioseismology. We compare the near-surface results from global and local helioseismology using data from the Michelson Doppler Imager and the Global Oscillation Network Group with surface Doppler velocity measurements from the Mount Wilson 150-foot tower and find that the results are in reasonable agreement, with some explicable differences in detail. All of the data sets show zones of faster rotation approaching the equator from mid-latitudes during the solar cycle, with a variation at any given location that can be approximately, but not completely, described by a single sinusoid and an amplitude that does not drop off steeply below the surface.     

Helioseismic Travel-Time Definitions and Sensitivity to Horizontal Flows Obtained from Simulations of Solar Convection

We study the sensitivity of wave travel times to steady and spatially homogeneous horizontal flows added to a realistic simulation of the solar convection performed by Robert F. Stein, Ake Nordlund, Dali Georgobiani, and David Benson. Three commonly used definitions of travel times are compared. We show that the relationship between travel-time difference and flow amplitude exhibits a non-linearity depending on the travel distance, the travel-time definition considered, and the details of the time – distance analysis (in particular, the impact of the phase-speed filter width). For times measured using a Gabor wavelet fit, the travel-time differences become nonlinear in the flow strength for flows of about 300 m s⁻¹, and this non-linearity reaches almost 60% at 1200 m s⁻¹ (relative difference between actual travel time and expected time for a linear behavior). We show that for travel distances greater than about 17 Mm, the ray approximation predicts the sensitivity of travel-time shifts to uniform flows. For smaller distances, the ray approximation can be inaccurate by more than a factor of three.     

Sensitivity Kernels for Time-Distance Inversion

Inversion of local-area helioseismic time-distance data has so far only been done in the ray approximation (Kosovichev, 1996). Since this is a high-frequency approximation its applicability can be questioned for the solar case. Bogdan (1997) showed that for a simple solar model the localized wave packets do follow, but are not confined to, the ray path. We use an approximation based on the first Fresnel zone that has been developed in geophysics by Snieder and Lomax (1996) to go beyond the ray approximation in the inversions. We have calculated sensitivity kernels using both approximations. To test them we use a finite-difference forward modeling of the whole wave field in an acoustic medium reminiscent of the Sun. We use the finite-difference modeling to calculate sensitivity kernels for the full wave field and compare this with the other kernels. The results show that the Fresnel-zone-based kernels are in good agreement with the sensitivity obtained from the modeling. Thus these new kernels represent a significant step forward in the inversion of time-distance data.     

Near-Surface Flow Fields Deduced Using Correlation Tracking and Time-Distance Analyses

Near-photospheric flow fields on the Sun are deduced using two independent methods applied to the same time series of velocity images observed by SOI-MDI on SOHO. Differences in travel times between f modes entering and leaving each pixel measured using time-distance helioseismology are used to determine sites of supergranular outflows. Alternatively, correlation tracking analysis of mesogranular scales of motion applied to the same time series is used to deduce the near-surface flow field. These two approaches provide the means to assess the patterns and evolution of horizontal flows on supergranular scales even near disk center, which is not feasible with direct line-of-sight Doppler measurements. We find that the locations of the supergranular outflows seen in flow fields generated from correlation tracking coincide well with the locations of the outflows determined from the time-distance analysis, with a mean correlation coefficient after smoothing of _s=0.890. Near-surface velocity field measurements can be used to study the evolution of the supergranular network, as merging and splitting events are observed to occur in these images. The data consist of one 2048-min time series of high-resolution (0.6 pixels) line-of-sight velocity images taken by MDI on 1997 January 16–18 at a cadence of one minute.     

Polarization of Microwave Radio Emission of Flare-Producing Solar Active Regions

On the basis of our multiwavelength observations made with the one-dimensional RATAN-600 radio telescope, we study the inversion of the circular polarization in the solar microwave emission at different frequencies. The inversion is detected in the emission of flare-producing active regions (FPARs) at various stages of their development, starting from the pre-flare stage. During the latest 23rd solar cycle maximum, numerous FPARs revealed spectral inhomogeneities in their polarized microwave radiation (Bogod and Tokhchukova, 2003, Astron. Lett. 29, 263). Here, we discuss a particular case of such inhomogeneities, the frequency-dependent double inversion of the sign of circular polarization, which probably reflects some essential processes in FPARs. We consider several mechanisms for the double inversion: linear interaction of waves in the region of a quasitransverse magnetic field, the propagation of waves through a region of zero magnetic field, the scattering of radio waves on waves of high-frequency plasma turbulence, the influence of the current fibrils on the propagation of the radio emission, and the magnetic “dips,” in which the direction of magnetic field lines changes the sign relative to the observer. All of them have shortcomings, but the last mechanism explains the observations the best.     

Structure and Evolution of Supergranulation from Local Helioseismology

Supergranulation is visible at the solar surface as a cellular pattern of horizontal outflows. Although it does not show a distinct intensity pattern, it manifests itself indirectly in, for example, the chromospheric network. Previous studies have reported significant differences in the inferred basic parameters of the supergranulation phenomenon. Here we study the structure and temporal evolution of a large sample of supergranules, measured by using local helioseismology and SOHO/MDI data from the year 2000 at solar activity minimum. Local helioseismology with f modes provides maps of the horizontal divergence of the flow velocity at a depth of about 1 Mm. From these divergence maps supergranular cells were identified by using Fourier segmentation procedures in two dimensions and in three dimensions (two spatial dimensions plus time). The maps that we analyzed contain more than 10⁵ supergranular cells and more than 10³ lifetime histories, which makes possible a detailed analysis with high statistical significance. We find that the supergranular cells have a mean diameter of 27.1 Mm. The mean lifetime is estimated to be 1.6 days from the measured distribution of lifetimes (three-dimensional segmentation), with a clear tendency for larger cells to live longer than smaller ones. The pair and mark correlation functions do not show pronounced features on scales larger than the typical cell size, which suggests purely random cell positions. The temporal histories of supergranular cells indicate a smooth evolution from their emergence and growth in the first half of their lives to their decay in the second half of their lives (unlike exploding granules, which reach their maximum size just before they fragment).