Statistics of the two-point cross-covariance function of solar oscillations

¹ Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
e-mail: nagashima@mps.mpg.de
² National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
³ Department of Astronomical Science, SOKENDAI (the Graduate University for Advanced Studies), Mitaka, Tokyo 181-8588, Japan
⁴ Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

Received: 14 January 2016
Accepted: 17 June 2016

Abstract

Context. The cross-covariance of solar oscillations observed at pairs of points on the solar surface is a fundamental ingredient in time-distance helioseismology. Wave travel times are extracted from the cross-covariance function and are used to infer the physical conditions in the solar interior.

Aims. Understanding the statistics of the two-point cross-covariance function is a necessary step towards optimizing the measurement of travel times.

Methods. By modeling stochastic solar oscillations, we evaluate the variance of the cross-covariance function as function of time-lag and distance between the two points.

Results. We show that the variance of the cross-covariance is independent of both time-lag and distance in the far field, that is, when they are large compared to the coherence scales of the solar oscillations.

Conclusions. The constant noise level for the cross-covariance means that the signal-to-noise ratio for the cross-covariance is proportional to the amplitude of the expectation value of the cross-covariance. This observation is important for planning data analysis efforts.