Helioseismic inferences of large-scale flows in the solar interior necessitate accounting for the curvature of the Sun, both in interpreting systematic trends introduced in measurements as well as the sensitivity kernel that relates photospheric measurements to subsurface flow velocities. Additionally the inverse problem that relates measurements to model parameters needs to be well-posed to obtain accurate inferences, which necessitates a sparse set of parameters. Further, the sensitivity functions need to be computationally easy to evaluate. In this work we address these issues by demonstrating that the sensitivity kernels for flow velocities may be computed efficiently in a basis of vector spherical harmonics. We are also able to account for line-of-sight projections in Doppler measurements, as well as center-to-limb differences in line-formation heights. We show that given the assumed spherical symmetry of the background model, it is often cheap to simultaneously compute the kernels for pairs of observation points that are related by a rotation. Such an approach is therefore particularly well-suited to inverse problems for large-scale flows in the Sun, such as meridional circulation.
As helioseismology matures and turns into a precision science, modeling finite-frequency, geometric and systematical effects is becoming increasingly important. Here we introduce a general formulation for treating perturbations of arbitrary tensor rank in spherical geometry using fundamental ideas of quantum mechanics and their extensions in geophysics. We include line-of-sight projections and center-to-limb differences in line-formation heights in our analysis. We demonstrate the technique by computing a travel-time sensitivity kernel for sound-speed perturbations. The analysis produces the s…
Helioseismic waves observable at the solar surface can be used to probe the properties of the Sun's interior. By measuring helioseismic travel times between different location on the surface, flows and other interior properties can be inferred using so-called sensitivity kernels which relate the amount of travel-time shift with variations in interior proporties. In particular, sensitivity kernels for flows have been developed in the past, using either ray or Born approximation, and have been used to infer solar interior flows such as the meridional circulation which is of particular interest f…
Context: Helioseismic analysis of large-scale flows and structural inhomogeneities in the Sun requires the computation of sensitivity kernels that account for the spherical geometry of the Sun, as well as systematic effects such as line-of-sight projection. Aim: I aim to develop a code to evaluate helioseismic sensitivity kernels for flows using line-of-sight projected measurements. Methods: I decomposed the velocity field in a basis of vector spherical harmonics and computed the kernel components corresponding to the coefficients of velocity in this basis. The kernels thus computed are radial…
Motivated by the solar composition problem and by using the recently developed Linear Solar Model approach, we analyze the role of opacity and metals in the sun. After a brief discussion of the relation between the effects produced by a variation of composition and those produced by a modification of the radiative opacity, we calculate numerically the opacity kernels that, in a linear approximation, relate an arbitrary opacity variation to the corresponding modification of the solar observable properties. We use these opacity kernels to discuss the present constraints on opacity (and compositi…
Seismic traveltime measurements are a crucial tool in the investigation of the solar interior, particularly in the examination of fine-scale structure. Traditional analysis of traveltimes relies on a geometrical ray picture of acoustic wave propagation, which assumes high frequencies. However, it is well-known that traveltimes obtained from finite-frequency waves are sensitive to variations of medium parameters in a wide Fresnel zone around the ray path. To address this problem, Fréchet traveltime sensitivity kernels have previously been developed. These kernels use a more realistic approximat…
Accurate inference of solar meridional flow is of crucial importance for the understanding of solar dynamo process. Wave travel times, as measured on the surface, will change if the waves encounter perturbations e.g. in the sound speed or flows, as they propagate through the solar interior. Using functions called sensitivity kernels, we may image the underlying anomalies that cause measured shifts in travel times. The inference of large-scale structures e.g meridional circulation requires computing sensitivity kernels in spherical geometry. Mandal et al. (2017) have computed such spherical ker…
Time-distance helioseismology uses cross-covariances of wave motions on the solar surface to determine the travel times of wave packets moving from one surface location to another. We review the methodology to interpret travel-time measurements in terms of small, localized perturbations to a horizontally homogeneous reference solar model. Using the first Born approximation, we derive and compute 3D travel-time sensitivity (Fréchet) kernels for perturbations in sound-speed, density, pressure, and vector flows. While kernels for sound speed and flows had been computed previously, here we extend …
Here we use synthetic data to explore the performance of forward models and inverse methods for helioseismic holography. Specifically, this work presents the first comprehensive test of inverse modeling for flows using lateral-vantage (deep-focus) holography. We derive sensitivity functions in the Born approximation. We then use these sensitivity functions in a series of forward models and inversions of flows from a publicly available magnetohydrodynamic quiet-Sun simulation. The forward travel times computed using the kernels generally compare favorably with measurements obtained by applying …
Normal modes are coupled by the presence of perturbations in the Sun, providing a novel and under-appreciated helioseismic technique with which to image the solar interior. The process of measuring coupling between normal modes is straightforward, much more so when compared with other prevalent helioseismic techniques. The theoretical framework to interpret these measurements is well developed with the caveat that it applies only in the case where the entire surface of the Sun is observed. In practice however, the limited visibility of the Sun and line-of-sight related effects diminish the res…
In this article, we derive and compute the sensitivity of measurements of coupling between normal modes of oscillation in the Sun to underlying flows. The theory is based on first-Born perturbation theory, and the analysis is carried out using the formalism described by \citet{lavely92}. Albeit tedious, we detail the derivation and compute the sensitivity of specific pairs of coupled normal modes to anomalies in the interior. Indeed, these kernels are critical for the accurate inference of convective flow amplitudes and large-scale circulations in the solar interior. We resolve some inconsiste…
The interpretation of helioseismic measurements, such as wave travel-time, is based on the computation of kernels that give the sensitivity of the measurements to localized changes in the solar interior. These are computed using the ray or the Born approximation. The Born approximation is preferable as it takes finite-wavelength effects into account, but can be computationally expensive. We propose a fast algorithm to compute travel-time sensitivity kernels under the assumption that the background solar medium is spherically symmetric. Kernels are typically expressed as products of Green's fun…
We present a 3-dimensional (3D) numerical solver of the linearized compressible Euler equations (GALE -- Global Acoustic Linearized Euler), used to model acoustic oscillations throughout the solar interior. The governing equations are solved in conservation form on a fully global spherical mesh ($0 \le ϕ\le 2π$, $0 \le θ\le π$, $0 \le r \le R_{\odot}$) over a background state generated by the standard Solar Model S. We implement an efficient pseudo-spectral computational method to calculate the contribution of the compressible material derivative dyad to internal velocity perturbations, comput…
Context. In local helioseismology, the travel times of acoustic waves propagating in opposite directions along the same meridian inform us about horizontal flows in the north-south direction. The longitudinal averages of the north-south helioseismic travel-time shifts vary with the sunspot cycle. Aims. We aim to study the contribution of inflows into solar active regions to this solar-cycle variation. Methods. To do so, we identify the local flows around active regions in the horizontal flow maps obtained from correlation tracking of granulation in SDO/HMI continuum images. We compute the forw…
The subsurface structure of an "average" supergranule is derived from existing HMI pipeline time-distance data products and compared to the best helioseismic flow model detailed in Duvall and Hanasoge (2013). We find that significant differences exist between them. Unlike the shallow structure predicted by the model, the average HMI supergranule is very extended in depth, exhibiting horizontal outflow down to $7$--$10$~Mm, followed by a weak inflow reaching a depth of $\sim20$~Mm below the photosphere. The maximal velocities in the horizontal direction for the average supergranule are much sma…
The inference of internal properties of the Sun from surface measurements of wave travel times is the goal of time-distance helioseismology. A critical step toward the accurate interpretation of travel-time shifts is the computation of sensitivity functions linking seismic measurements to internal structure. Here we calculate finite-frequency sensitivity kernels in spherical geometry for two-point travel-time measurements. We numerically build Green's function by solving for it at each frequency and spherical-harmonic degree and summing over all these pieces. These computations are performed i…
Time-distance helioseismology is the method of the study of the propagation of waves through the solar interior via the travel times of those waves. The travel times of wave packets contain information about the conditions in the interior integrated along the propagation path of the wave. We introduce an improved methodology of the time-distance helioseismology which allows us to invert for a full 3D vector of flows and the sound-speed perturbations at once. Using this methodology one can also derive the mean value of the vertical component of flows and the cross-talk between the flows and the…
According to time-distance helioseismology, information about internal fluid motions is encoded in the travel times of solar waves. The inverse problem consists of inferring 3-D vector flows from a set of travel-time measurements. Here we investigate the potential of time-distance helioseismology to infer 3-D convective velocities in the near-surface layers of the Sun. We developed a new Subtractive Optimally Localised Averaging (SOLA) code suitable for pipeline pseudo-automatic processing. Compared to its predecessor, the code was improved by accounting for additional constraints in order to …
We extend an existing Born approximation method for calculating the linear sensitivity of helioseismic travel times to flows from Cartesian to spherical geometry. This development is necessary for using the Born approximation for inferring large-scale flows in the deep solar interior. In a first sanity check, we compare two $f-$mode kernels from our spherical method and from an existing Cartesian method. The horizontal and total integrals agree to within 0.3 %. As a second consistency test, we consider a uniformly rotating Sun and a travel distance of 42 degrees. The analytical travel-time dif…
The solar coronal magnetic field produces solar activity, including extremely energetic solar flares and coronal mass ejections (CMEs). Knowledge of the structure and evolution of the magnetic field of the solar corona is important for investigating and understanding the origins of space weather. Although the coronal field remains difficult to measure directly, there is considerable interest in accurate modeling of magnetic fields in and around sunspot regions on the Sun using photospheric vector magnetograms as boundary data. In this work, we investigate effects of the size of the domain chos…
We test a nonlinear force-free field (NLFFF) optimization code in spherical geometry using an analytical solution from Low and Lou. Several tests are run, ranging from idealized cases where exact vector field data are provided on all boundaries, to cases where noisy vector data are provided on only the lower boundary (approximating the solar problem). Analytical tests also show that the NLFFF code in the spherical geometry performs better than that in the Cartesian one when the field of view of the bottom boundary is large, say, $20^\circ \times 20^\circ$. Additionally, We apply the NLFFF mode…
The solar meridional flow is a crucial ingredient in modern dynamo theory. Seismic estimates of this flow have, however, been contradictory in deeper layers below about $0.9\,R_\odot$. Results from time-distance helioseismology have so far been obtained using the ray approximation. Here, we perform inversions using the Born approximation. The initial result is similar to the result previously obtained by Jackiewicz et al. (2015) using ray kernels while using the same set of GONG data and the SOLA inversion technique. However, we show that the assumption of uncorrelated measurements used in ear…
Accurate measurements of deep solar meridional flow are of vital interest for understanding the solar dynamo. In this paper, we validate a recently developed method for obtaining sensitivity functions (kernels) for travel-time measurements to solar interior flows using the Born approximation in spherical geometry, which is expected to be more accurate than the classical ray approximation. Furthermore, we develop a numerical approach to efficiently compute a large number of kernels based on the separability of the eigenfunctions into their horizontal and radial dependence. The validation is per…
Direct seismic imaging of sub-surface flow, sound-speed and magnetic field is crucial for predicting flux tube emergence on the solar surface, an important ingredient for space weather. The sensitivity of helioseismic mode-amplitude cross-correlation to $p$- and $f$-mode oscillations enable formal inversion of such sub-photospheric perturbations. It is well-known that such problems are written in the form of an integral equation that connects the perturbations to the observations via ``sensitivity kernels". While the sensitivity kernels for flow and sound-speed have been known for decades and …
