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

TL;DR

This paper models the statistics of the two-point cross-covariance function of solar oscillations, revealing that its variance is constant in the far field, which impacts how signal-to-noise ratios are understood in helioseismology.

Contribution

It provides a theoretical analysis of the variance of the cross-covariance function, showing its independence from time-lag and distance in the far field, aiding in data analysis optimization.

Findings

Variance of cross-covariance is constant in the far field.

Signal-to-noise ratio is proportional to the expectation value amplitude.

Constant noise level impacts helioseismic data interpretation.

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, i.e., 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.