The amplitude of the cross-covariance function of solar oscillations as a diagnostic tool for wave attenuation and geometrical spreading

摘要

Context. In time-distance helioseismology, wave travel times are measured from the two-point cross-covariance function of solar oscillations and are used to image the solar convection zone in three dimensions. There is, however, also information in the amplitude of the cross-covariance function, for example, about seismic wave attenuation. Aims. We develop a convenient procedure to measure the amplitude of the cross-covariance function of solar oscillations. Methods. In this procedure, the amplitude of the cross-covariance function is linearly related to the cross-covariance function and can be measured even for high levels of noise. Results. As an example application, we measure the amplitude perturbations of the seismic waves that propagate through the sunspot in active region NOAA 9787. We can recover the amplitude variations due to the scattering and attenuation of the waves by the sunspot and associated finite-wavelength effects. Conclusions. The proposed definition of cross-covariance amplitude is robust to noise, can be used to relate measured amplitudes to 3D perturbations in the solar interior under the Born approximation, and provides independent information from the travel times.