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The interpretation of helioseismic measurements, such as wave travel-time, isbased on the computation of kernels that give the sensitivity of themeasurements to localized changes in the solar interior. These are computedusing the ray or the Born approximation. The Born approximation is preferableas it takes finite-wavelength effects into account, but can be computationallyexpensive. We propose a fast algorithm to compute travel-time sensitivitykernels under the assumption that the background solar medium is sphericallysymmetric. Kernels are typically expressed as products of Green's functionsthat depend upon depth, latitude and longitude. Here, we compute the sphericalharmonic decomposition of the kernels and show that the integrals in latitudeand longitude can be performed analytically. In particular, the integrals ofthe product of three associated Legendre polynomials can be computed thanks tothe algorithm of Dong and Lemus (2002). The computations are fast and accurateand only require the knowledge of the Green's function where the source is atthe pole. The computation time is reduced by two orders of magnitude comparedto other recent computational frameworks. This new method allows for flexibleand computationally efficient calculations of a large number of kernels,required in addressing key helioseismic problems. For example, the computationof all the kernels required for meridional flow inversion takes less than twohours on 100 cores.

Sensitivity kernels for time-distance helioseismology: efficient computation for spherically-symmetric solar models