Travel Time Shifts due to Amplitude Modulation in Time-Distance Helioseismology

TL;DR

This paper investigates how amplitude modulation, especially in sunspots, causes systematic shifts in acoustic travel times in helioseismology, affecting the interpretation of solar interior properties.

Contribution

It derives an analytical expression for travel time shifts caused by amplitude modulation combined with phase-speed filtering in time-distance helioseismology.

Findings

Amplitude modulation induces systematic travel time shifts.

Analytical expression explains the observed shifts.

Model aids interpretation of non-uniform oscillation amplitudes.

Abstract

Correct interpretation of acoustic travel times measured by time-distance helioseismology is essential to get an accurate understanding of the solar properties that are inferred from them. It has long been observed that sunspots suppress p-mode amplitude, but its implications on travel times has not been fully investigated so far. It has been found in test measurements using a 'masking' procedure, in which the solar Doppler signal in a localized quiet region of the Sun is artificially suppressed by a spatial function, and using numerical simulations that the amplitude modulations in combination with the phase-speed filtering may cause systematic shifts of acoustic travel times. To understand the properties of this procedure, we derive an analytical expression for the cross-covariance of a signal that has been modulated locally by a spatial function that has azimuthal symmetry, and then filtered by a phase speed filter typically used in time-distance helioseismology. Comparing this expression to the Gabor wavelet fitting formula without this effect, we find that there is a shift in the travel times, that is introduced by the amplitude modulation. The analytical model presented in this paper can be useful also for interpretation of travel time measurements for non-uniform distribution of oscillation amplitude due to observational effects.