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\begin{document}

\title{Asymptotic analysis for late coda correlations under different geometric
distributions of earthquakes}



\author[1]{Yingjie Xia}%
\author[1]{Xuping Feng}%
\author[1]{Xiaofei Chen}%
\affil[1]{Southern University of Science and Technology}%


\vspace{-1em}



  
  \date{May 24, 2024}


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\begin{abstract}
Some features in late coda correlations have now been commonly treated
as ``the inter- station body waves''. In general, however, large
earthquakes releasing coda waves mostly situate at the continental
boundaries. It remains unclear as to how such a discrete and non-
uniform distribution of earthquakes influences these features. To
understand the impacts, here we introduce geometric ray theory to
explore the body wave cross-correlation. In the stationary phase
integral, we show that the distribution geometry of earthquakes and the
dimension of the stationary phase zone significantly influence the
correlation phases. The dimension of the stationary phase zone is
inversely proportional to the k-\selectlanguage{greek}κ \selectlanguage{english}coefficient which, as a newly-proposed
terminology, is composed of the seismic wave-number and the coda
propagation distance. In late coda correlations, most of the large
earthquakes situate in the stationary phase zone for constructing the
inter-station wave due to the small k- \selectlanguage{greek}κ \selectlanguage{english}coefficient. However, because
earthquakes are not always at the stationary points, the correlation
signals may appear a little earlier than their counterparts in Green's
function. We have verified the theoretical analyses with the synthetic
and realistic coda correlations.This theory is also applicable in other
physics fields allowing for geometric ray theory. It demonstrates that
the event-receiver geometry can result in the travel time variation up
to 1/6 of the body wave correlation period. Thus, researchers should
carefully investigate the impacts when utilizing the correlation signals
as inter-station body waves for the future work of illuminating the
Earth's discontinuities.%
\end{abstract}%



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