Seismic interferometry with scattered waves: Stationary-phase analysis of correlation formulations

Seismic interferometry retrieves inter-receiver Green's functions by correlating wave fields from distributed sources, but practical implementations differ in how amplitudes are handled. This study analyzes three correlation formulations-crosscorrelation, deconvolution, and crosscoherence-using a stationary-phase approximation to clarify how normalization choices affect direct and scattered waves. I show that crosscorrelation reproduces the causal and acausal Green's functions, with spurious terms canceling under the generalized optical theorem. In contrast, deconvolution and crosscoherence generate additional pseudo arrivals that do not correspond to physical propagation between receivers, and crosscoherence further modulates direct-wave amplitudes through scattering-dependent factors. A finite-difference numerical experiment with uniformly distributed, uncorrelated sources confirms these predictions: crosscorrelation reconstructs both direct and scattered arrivals, whereas deconvolution and crosscoherence produce late, unphysical phases while diminishing scattered-wave energy. These results clarify the kinematic content and limitations of amplitude normalization in interferometric processing and provide guidance for interpreting coda and scattered-wave signals and for designing filters that suppress pseudo arrivals.