The spatial displacement gradient of a seismic wave is related to displacement and velocity through two spatial coefficients for any one dimension. One coefficient gives the relative change of wave geometrical spreading with distance and the other gives the horizontal slowness and its change with distance. The essential feature of spatial gradient analysis is a time-domain relation between three seismograms that yields information on the amplitude and phase behavior of a seismic wave. Filter theory is used to find these coefficients for data from 2D areal arrays of seismometers, termed gradiometers. A finite-difference star is used to compute the displacement gradient for irregularly shaped gradiometers, and a relation for the frequency-dependent error in the displacement gradient is obtained and applied to ensure accurate estimates. This kind of array analysis is useful for gradiometers at any distance from a source and yields a variety of time-domain and frequency-domain views of wave-amplitude changes and horizontal phase velocity estimates across the gradiometer. For example, time-dependent horizontal slowness and wave-azimuth plots are natural results of the analysis. These time-domain maps may be used in conjunction with time–distance and horizontal slowness–distance models to locate seismic sources or may be used directly to study earth structure. These methods are demonstrated by using data from a small-aperture (∼40 m) seismic gradiometer.

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