# Inverse Problems in Local Helioseismology

by Majid Pourabdian

Date of Examination:2020-02-17

Date of issue:2020-08-04

Advisor:Prof. Dr. Laurent Gizon

Referee:Prof. Dr. Laurent Gizon

Referee:Prof. Dr. Thorsten Hohage

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## Abstract

### English

Helioseismology is the main tool to infer the physical properties in the solar interior. In time-distance helioseismology, measurements of wave travel times are extracted from the cross-correlation of the oscillation signal (e.g. the Doppler velocity) between pairs of points on the solar surface. These measurements must then be inverted (the inverse problem) to infer the solar subsurface properties. Helioseismic inferences are based on a relationship between the perturbations in solar properties with respect to a reference solar model and the corresponding changes in the helioseismic measurements (the forward problem). Measurements of wave travel times are very noisy and suffer from systematic errors. These have led to conflicting results, in particular in the deeper layers of the Sun and many open questions about the solar internal structure. A particularly challenging problem is the inference of the solar meridional flow, which is a crucial ingredient in models of the solar dynamo. There is no consensus about the radial profile of the solar meridional flow. This dissertation mainly focuses on a better understanding of the solar meridional flow deep inside the convection zone by performing helioseismic inversions of wave travel times. In doing so, improved methods of inversion are developed. In a first study, we consider acoustic waves propagating in a homogeneous medium to investigate the deep-focusing time-distance technique in terms of signal and noise. The aim of the deep-focusing time-distance helioseismology is to construct seismic measurements that inform us about the physical conditions at a well-defined target point in the solar interior. In this technique, pairs of points on the solar surface are chosen in a way that their acoustic ray paths intersect at the target point. We compare two measurement quantities extracted from the deep-focusing cross-covariance functions: travel times and amplitudes. Using the first Born approximation which is a single-scattering approximation, we find the deep-focusing travel-time measurements have zero sensitivity at the target location and maximum sensitivity in a surrounding shell around the target location. On the other hand, the sensitivity of deep-focusing amplitude measurements peaks at the target location. The measurements have noise due to the stochastic excitation of the waves. In the case of a highly localized sound-speed perturbation, we find that the signal-to-noise ratio of deep-focusing amplitude measurements is higher than for deep-focusing travel-time measurements. These results obtained for a homogeneous medium, suggest that amplitude measurements may be used in local helioseismology in addition to the travel times. In the main part of this thesis, we perform inversions of helioseismic travel times to infer the profile of the solar meridional flow. The observations cover two solar cycles from 1996 until 2019. Employing the constraint of mass conservation, we find that the solar meridional flow has a single-cell structure in each hemisphere: poleward at the surface and equatorward at the base of the convection zone with an amplitude of approximately 4 m/s at latitude 45 deg. At the base of the convection zone, the velocity is equatorward with a functional form approximately given by $U_\theta = U_b \sin 2 \theta$, with $U_b=4.8\pm1.0$ m/s for cycle 23 and $U_b=3.6\pm1.0$ m/s for cycle 24. The flow switches sign at a depth of about $0.79$ solar radius. Confidence in the results is provided by the agreement between GONG and SOHO/MDI data during the period $2001-2011$. According to a flux-transport dynamo model, the inferred meridional flow is able to explain the migration of sunspots towards the equator in each hemisphere. The details of the inversion procedure and additional tests with synthetic data are presented in a complementary chapter. The inversions are tuned and validated using different test cases. We find that mass conservation is a necessary constraint to reconstruct the radial component of the meridional flow. A regularization term must also be introduced to avoid fast variations in latitude. Finally, we discuss future developments in helioseismic inversions. Particularly promising are full-waveform inversions, which ought to provide improvements in both localization and noise levels.**Keywords:**Sun: helioseismology; Sun: interior; Sun: oscillations; Inverse problems; Meridional flow