An explicit relationship between time-domain noise correlation and spatial autocorrelation (SPAC) results

The success of recent ambient noise tomographic studies is now understood to arise due to cross-correlation properties documented in the acoustics community since the 1950s. However, despite the fact that Aki's 1957 spatial autocorrelation (SPAC) work yields identical analytical results to certain noise correlation results, the precise relationship between SPAC and time-domain cross-correlation remains not entirely transparent. Here, we present an explicit comparison of the two approaches and clarify that SPAC theory is indeed equivalent to the cross-correlation theory used for recent noise tomography studies. This equivalence allows theoretical work from each field to be applied to the other, and we illustrate a few examples of this.