Astronomy and Astrophysics – Astronomy
Scientific paper
May 1998
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998apj...498l..79d&link_type=abstract
Astrophysical Journal Letters v.498, p.L79
Astronomy and Astrophysics
Astronomy
9
Sun: Atmosphere, Sun: Oscillations
Scientific paper
In time-distance helioseismology, the position of the peak of the envelope of a cross-correlation function is identified as the travel time of a wave packet between two spatially separated locations on the Sun. When a cross-correlation function is a sum of two or more cross-correlation functions, the following theorem forbids this identification: THEOREM. The envelope of the analytic signal of a sum of two real functions is not equal to the sum of the envelopes of the analytic signals of the individual functions, unless (1) the ratio of the two functions is equal to the ratio of their Hilbert transforms and the sum of the product of the two functions and the product of their Hilbert transforms is nonnegative or (2) the envelopes of the two functions are disjoint. The individual functions should be isolated before the travel times are identified with their envelope peaks. Typically, in time-distance helioseismology, the envelopes are Gaussians and are never disjoint, nor do the cross-correlation functions and their Hilbert transforms automatically satisfy condition 1. This Letter derives the conditions under which condition 1 is satisfied and the conditions under which the envelope of a sum can be approximated to the sum of the individual envelopes so that the errors committed are minimal. These precautionary measures need to be taken while computing travel times from solar oscillations, so that the subsequent theoretical interpretations that derive the internal structure of the Sun can be reliable.
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