Testing the Resolution of Rotation Inversion Methods at the Tachocline

Resumen

We present a study of the spatial resolution that rotation inversion methodologies can achieve when using a realistic data set of rotational splittings. To this effect, we computed artificial rotational splittings, using various profiles of the solar internal rotation and a forward computation using eigenfrequencies derived from a standard solar model and an adiabatic pulsation code. The rotation profiles we used are similar to the measured solar rotation, namely a solid body rotation below the convection zone and a differential rotation in the convection zone. We also included a small near surface shear and varied the thickness of the transition region between solid body and differential rotation, aka the tachocline. Artificial rotational splittings were computed for the same range in l and n as observed when fitting a long time series. We then applied two types of rotation inversion methodologies to infer the rotation profile from artificial rotational splittings. One a regularized least squares (RLS) with a weighted second derivation smoothing to lift the singular nature of the inverse problem, while the other is an iterative method based on the Simultaneous Algebraic Reconstruction Technique (SART), a methodology originally implemented to solve linear systems in image reconstruction. In order to test the resolution potential of our rotation inversion methodologies, we inverted first noiseless artificial splittings, and then injected some random noise. Since the inversion methodologies weight the input rotational splittings by their uncertainties, and since the observed uncertainty is a strong function of the splittings indices (n, l, m), we used the observed uncertainties measured from fitting the same long time series used to define the splittings set, and injected random noise for each splitting proportional to these observed uncertainties.