The Green's function of waves that propagate between two receivers can be found by cross-correlating multiply scattered waves recorded at these receivers. This technique obviates the need for a source at one of these locations, and is therefore called "passive imaging." This principle has been explained by assuming that the normal modes of the system are uncorrelated and that all carry the same amount of energy (equipartitioning). Here I present an alternative derivation of passive imaging of the ballistic wave that is not based on normal modes. The derivation is valid for scalar waves in three dimensions, and for elastic surface waves. Passive imaging of the ballistic wave is based on the destructive interference of waves radiated from scatterers away from the receiver line, and the constructive interference of waves radiated from secondary sources near the receiver line. The derivation presented here shows that the global requirement of the equipartitioning of normal modes can be relaxed to the local requirement that the scattered waves propagate on average isotropically near the receivers.

S U M M A R YWe present a new method of surface wave tomography based on applying the eikonal equation to observed phase traveltime surfaces computed from seismic ambient noise. The sourcereceiver reciprocity in the ambient noise method implies that each station can be considered to be an effective source and the phase traveltime between that source and all other stations is used to track the phase front and construct the phase traveltime surface. Assuming that the amplitude of the waveform varies smoothly, the eikonal equation states that the gradient of the phase traveltime surface can be used to estimate both the local phase speed and the direction of wave propagation. For each location, we statistically summarize the distribution of azimuthally dependent phase speed measurements based on the phase traveltime surfaces centred on different effective source locations to estimate both the isotropic and azimuthally anisotropic phase speeds and their uncertainties. Examples are presented for the 12 and 24 s Rayleigh waves for the EarthScope/USArray Transportable Array stations in the western USA. We show that (1) the major resulting tomographic features are consistent with traditional inversion methods, (2) reliable uncertainties can be estimated for both the isotropic and anisotropic phase speeds, (3) 'resolution' can be approximated by the coherence length of the phase speed measurements and is about equal to the station spacing, (4) no explicit regularization is required in the inversion process and (5) azimuthally dependent phase speed anisotropy can be observed directly without assuming its functional form.

In coda wave interferometry, one records multiply scattered waves at a limited number of receivers to infer changes in the medium over time. With this technique, we have determined the nonlinear dependence of the seismic velocity in granite on temperature and the associated acoustic emissions. This technique can be used in warning mode, to detect the presence of temporal changes in the medium, or in diagnostic mode, where the temporal change in the medium is quantified.

Traditionally, the Marchenko equation forms a basis for 1D inverse scattering problems. A 3D extension of the Marchenko equation enables the retrieval of the Green’s response to a virtual source in the subsurface from reflection measurements at the earth’s surface. This constitutes an important step beyond seismic interferometry. Whereas seismic interferometry requires a receiver at the position of the virtual source, for the Marchenko scheme it suffices to have sources and receivers at the surface only. The underlying assumptions are that the medium is lossless and that an estimate of the direct arrivals of the Green’s function is available. The Green’s function retrieved with the 3D Marchenko scheme contains accurate internal multiples of the inhomogeneous subsurface. Using source-receiver reciprocity, the retrieved Green’s function can be interpreted as the response to sources at the surface, observed by a virtual receiver in the subsurface. By decomposing the 3D Marchenko equation, the response at the virtual receiver can be decomposed into a downgoing field and an upgoing field. By deconvolving the retrieved upgoing field with the downgoing field, a reflection response is obtained, with virtual sources and virtual receivers in the subsurface. This redatumed reflection response is free of spurious events related to internal multiples in the overburden. The redatumed reflection response forms the basis for obtaining an image of a target zone. An important feature is that spurious reflections in the target zone are suppressed, without the need to resolve first the reflection properties of the overburden.

The motion of a building depends on the excitation, the coupling of the building to the ground, and the mechanical properties of the building. We separate the building response from the excitation and the ground coupling by deconvolving the motion recorded at different levels in the building, and apply this to recordings of the motion in the Robert A. Millikan Library in Pasadena, California. The waveforms obtained from deconvolution with the motion in the top floor show a superposition of one upgoing and one downgoing wave. The waveforms obtained by deconvolution with the motion in the basement can be formulated either as a sum of upgoing and downgoing waves, or as a sum over normal modes. Since these deconvolved waves for late time have a monochromatic character, they are most easily analyzed with normal-mode theory. For this building we estimate a shear velocity c = 322 m/s and a quality factor Q = 20. These values explain both the propagating waves as well as the normal modes. We show for this application of seismic interferometry that deconvolution of waveforms is superior to correlation.