On the assumption of translational invariance

The justification for spectral factorization rests upon an assumption of translational invariance. This assumption runs counter to many applications of time-distance helioseismology, where the interest comes in three-dimensional structure. The assumption of translation invariance may be partly overcome by working with patches of data with small spatial extent.

Time-distance measurements by crosscorrelation may not seem to have this perceived disadvantage; however, studies Kosovichev and Duvall (1997) have shown that significant amounts of averaging are required to produce signal-to-noise levels high enough to make reliable measurements. There is an implied assumption of invariance in this averaging procedure.