Pinsker estimators for local helioseismology: inversion of travel times for mass-conserving flows
Damien Fournier (Institute for numerical and applied mathematics Göttingen), Laurent Gizon (Max-Planck-Institut for solar system Göttingen), Martin Holzke (Institut for numerical and applied mathematics Göttingen), Thorsten Hohage (Institut for numerical and applied mathematics Göttingen)

The reconstruction of the 3D velocity field of supergranules below the solar surface is a difficult task in time-distance helioseismology. In particular, the vertical component of velocity is generally poorly reconstructed. We investigate the difficulties of the classical inversion methods (RLS and SOLA) and propose a new method based on the Pinsker estimator (Pinsker 1980), which minimizes the risk (mean square error) of the estimator. To compare the methods, we use synthetic travel times generated by convolving Born kernels with a model of the averaged supergranule. A realistic noise level corresponding to 4 days of observations is added. We focus on the inversion procedure and do not discuss possible bias in the forward modeling. We show that all inversion methods can recover the horizontal part of the velocity but RLS and SOLA fail to reconstruct the vertical velocity vz. The Pinsker method is better but does not recover properly the amplitude of vz. Implementing a mass conservation constraint in the Pinsker inversion method improves the reconstruction, giving the correct amplitude of the vertical velocity down to a depth of 5.5 Mm.