Finite-frequency Sensitivity Kernels in Spherical Geometry for Time–Distance Helioseismology

Abstract

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 in parallel ("embarrassingly"), thereby achieving significant speedup in wall-clock time. Kernels are calculated by invoking the first-order Born approximation connecting deviations in the wavefield to perturbations in the operator. Validated flow kernels are shown to produce travel-times within 0.47% of the true value for uniform flows up to 750 m/s. We find that travel-time can be obtained with errors of 1 millisecond or less for flows having magnitudes similar to meridional circulation. Alongside flows, we also compute and validate sensitivity kernel for sound-speed perturbations. These accurate sensitivity kernels might improve the current inferences of sub-surface flows significantly.

Helioseismic Inversion to Infer the Depth Profile of Solar Meridional Flow Using Spherical Born Kernels

“…An efficient approach to compute sensitivity kernels in spherical geometry was proposed by Mandal et al (2017). They showed that Green's function can be expressed as (Equation 12 and 13 from Mandal et al 2017)…”

“…where G rr (r, r s , ω) and G hr (r, r s , ω) = (G θr (r, r s , ω) , G φr (r, r s , ω)) are radial and tangential components of the displacement of a wave with temporal frequency ω, measured at r due to a point source placed at r s , P ℓ the Legendre polynomial of degree ℓ. The terms α ℓω , β ℓω are obtained by solving a coupled differential equation (Equation 10 in Mandal et al 2017) using a finite-difference based scheme for each harmonic degree ℓ and frequency ω. We compute the pair (α ℓω , β ℓω ) once and use them to evaluate Green's function, which is used subsequently in obtaining kernels.…”

“…To compute sensitivity kernels for stream function ψ, we use model S (Christensen-Dalsgaard et al 1996) as the background solar model (Mandal et al 2017). These kernels are highly sensitive to surface layers and very weakly so to the base of the convection zone, making it difficult to determine the profile of meridional circulation at depth.…”

“…The first-Born approximation, an alternative to ray theory, does not suffer from the above limitation. Recently, Mandal et al (2017) computed travel-time sensitivity kernels in the Born limit in spherical geometry (two other alternative approaches were proposed independently by Böning et al 2016;Gizon et al 2017). Using Born kernels, Böning et al (2017) inverted for meridional circulation and found return flow at 0.9 R ⊙ using a SOLA (Pijpers & Thompson 1994) inversion technique.…”

Sensitivity kernels for time-distance helioseismology

“…In this paper, we present a way to reduce the computational time of Born sensitivity kernels in a spherically symmetric background by treating the horizontal variables (the co-latitude θ and the longitude φ) analytically using the properties of the spherical harmonics. Here this approach is demonstrated using the scalar wave equation from Gizon et al (2017), but it could be applied to the normal-mode summation method of Böning et al (2016) or to solving the wave equation using a high-order finitedifference scheme (Mandal et al 2017).…”