Seismic interferometry is a technique for estimating the Green's function that accounts for wave propagation between receivers by correlating the waves recorded at these receivers. We present a derivation of this principle based on the method of stationary phase. Although this derivation is intended to be educational, applicable to simple media only, it provides insight into the physical principle of seismic interferometry. In a homogeneous medium with one horizontal reflector and without a free surface, the correlation of the waves recorded at two receivers correctly gives both the direct wave and the singly reflected waves. When more reflectors are present, a product of the singly reflected waves occurs in the crosscorrelation that leads to spurious multiples when the waves are excited at the surface only. We give a heuristic argument that these spurious multiples disappear when sources below the reflectors are included. We also extend the derivation to a smoothly varying heterogeneous background medium.

A practical method is devised to calculate the elastic wave field in a layer‐over‐half‐space medium with an irregular interface, when plane waves are incident from below. This method may be used for studying the interface shape of the M discontinuity, for example, using the observed spectral amplitude and phase‐delay anomalies due to teleseismic body waves. The method is also useful for the engineering‐seismological study of earthquake motions of soft superficial layers of various cross sections. The scattered field is described as a superposition of plane waves, and application of the continuity conditions at the interface yields coupled integral equations in the spectral coefficients. The equations are satisfied in the wave‐number domain when the interface shape is made periodic and the equations are Fourier transformed and truncated. Frequency smoothing by using complex frequencies reduces lateral interferences associated with the periodic interface shape and permits comparison of computed results with those obtained from finite bandwidth observations. Analyses of the residuals in the interface stress and displacement, performed for each computed solution, provided estimates of the errors. The relative root‐mean‐square residual errors were generally less than 5% and often less than 1% for problems in which the amplitude of the interface irregularity and the shortest wavelength were comparable. The method is applied to several models of ‘soft basins’ ‘dented M discontinuity’ and ‘stepped M discontinuity’ The results are compared with those derived from the flat‐layer theory and from the ray theory. In addition to vertical interference effects familiar in the flat‐layer theory, we observe the effects of lateral interference as well as those of ray geometry on the motion at the surface.

We apply the cross-coherence method to the seismic interferometry of traffic noise, which originates from roads and railways, to retrieve both body waves and surface-waves. Our preferred algorithm in the presence of highly variable and strong additive random noise uses cross-coherence, which uses normalization by the spectral amplitude of each of the traces, rather than crosscorrelation or deconvolution. This normalization suppresses the influence of additive noise and overcomes problems resulting from amplitude variations among input traces. By using only the phase information and ignoring amplitude information, the method effectively removes the source signature from the extracted response and yields a stable structural reconstruction even in the presence of strong noise. This algorithm is particularly effective where the relative amplitude among the original traces is highly variable from trace to trace. We use the extracted, reflected shear waves from the traffic noise data to construct a stacked and migrated image, and we use the extracted surfacewaves (Love waves) to estimate the shear velocity as a function of depth. This profile agrees well with the interval velocity obtained from the normal moveout of the reflected shear waves constructed by seismic interferometry. These results are useful in a wide range of situations applicable to both geophysics and civil engineering.