Tutorial on seismic interferometry. Part II: Underlying theory and new advances

In part II of this two-part tutorial we review the underlying theory of seismic interferometry and discuss various new advances. In the 1990’s the method of time-reversed acoustics was developed. This method employs the fact that the acoustic wave equation for a lossless medium is invariant for time-reversal. When ultrasonic responses recorded by piezoelectric transducers are reversed in time and fed simultaneously as source signals to the transducers, they focus at the position of the original source, even when the medium is very complex. In seismic interferometry the time-reversed responses are not physically sent into the earth, but they are convolved with other measured responses. The effect is essentially the same: the time-reversed signals focus and create a virtual source which radiates waves into the medium that are subsequently recorded by receivers. A mathematical derivation, based on reciprocity theory, formalizes this principle: the crosscorrelation of responses at two receivers, integrated over different sources, gives the Green’s function emitted by a virtual source at the position of one of the receivers and observed by the other receiver. The basic Green’s function representations for seismic interferometry assume a lossless non-moving acoustic or elastic medium. We discuss many variants and extensions, including interferometric representations for attenuation and/or moving media, unified representations for waves and diffusion phenomena, bending waves, quantum mechanical scattering, potential fields, elastodynamic, electromagnetic, poroelastic and electroseismic waves. We discuss the relation with the generalized optical theorem, discuss variants for virtual receivers and virtual reflectors and indicate the potential applications of time-lapse interferometry. Finally we discuss the improvements that can be obtained with interferometry by deconvolution. A trace-by-trace deconvolution process compensates for complex source functions and the attenuation of the medium. Interferometry by multidimensional deconvolution compensates in addition for the effects of one-sided and/or irregular illumination.