Sensitivity kernels for time-distance helioseismology

Abstract

Context. 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 kernels are computed using the ray or the Born approximation. The Born approximation is preferable as it takes finite-wavelength effects into account, although it can be computationally expensive. Aims. We propose a fast algorithm to compute travel-time sensitivity kernels under the assumption that the background solar medium is spherically symmetric. Methods. Kernels are typically expressed as products of Green’s functions that depend upon depth, latitude, and longitude. Here, we compute the spherical harmonic decomposition of the kernels and show that the integrals in latitude and longitude can be performed analytically. In particular, the integrals of the product of three associated Legendre polynomials can be computed. Results. The computations are fast and accurate and only require the knowledge of the Green’s function where the source is at the pole. The computation time is reduced by two orders of magnitude compared to other recent computational frameworks. Conclusions. This new method allows flexible and computationally efficient calculations of a large number of kernels, required in addressing key helioseismic problems. For example, the computation of all the kernels required for meridional flow inversion takes less than two hours on 100 cores.

Numerical evaluation of time-distance helioseismic sensitivity kernels in spherical geometry

“…The sensitivity kernel depends both on the details of the background model such as the thermal structure and the presence of inhomogeneous features such as flows, as well as on specifics of the measurement procedure. Recently, considerable effort has been made to develop frameworks to evaluate sensitivity kernels that describe wave propagation in the first Born approximation while accounting for the spherical geometry of the Sun (Böning et al 2016;Mandal et al 2017;Gizon et al 2017;Fournier et al 2018). The present work builds on the results of Bhattacharya et al (2020) and Bhattacharya (2020), in which the authors described the procedure to evaluate sensitivity kernels by accounting for line-of-sight projections that are inherent in Doppler measurements of wave velocity, as well as the differences in line-formation height between the solar disk center and the limb.…”

Helioseismic diagnostics of solar dynamics in the near-surface layers

“…The active-region flows are converted into north-south helioseismic travel-time shifts by using 3D sensitivity kernels, assuming a radial dependence of the flows. A traveltime sensitivity kernel K connects a perturbation in the flow field v to the shift in helioseismic travel time δτ induced by this flow (see for instance Fournier et al (2018) for a computationally-efficient way to compute kernels in the Born approximation). The traveltime perturbation for waves traveling between points r 1 and r 2 is then the integral over the whole volume of the Sun…”

“….30 -0.15 0.00 0.15 0.30 between the paired arcs, and ∆ the separation distance between the paired points. The travel-time perturbation is thus defined as (e.g., Gizon et al 2017, Fournier et al 2018)…”

“…The computation of the 3D Born sensitivity kernels is based on the approaches from Gizon et al (2017), Fournier et al (2018. The wave field ψ(r, ω) is solution of a scalar wave equation in the frequency domain…”

“…For the sake of simplicity, we now drop the ω in the notation of the cross-covariance and of the Green's function. In a spherically symmetric background, the Green's function depends only on 3.8 Appendix the angular distance between source and receiver and can be obtained from its Legendre coefficients (Fournier et al 2018):…”