The development of the Wiener linear least-mean-square-error processing theory for seismic signal enhancement through use of a two-dimensional array of seismometers leads to the theory of three-dimensional filtering. The array processing system for this theory consists of applying individual frequency filters to the outputs of the seismometers in the array before summation.The basic design equations for the optimum frequency filters are derived from the Wiener multichannel theory. However, the development of the three-dimensional frequency and vector-wave-number-filtering theory results in a physical understanding of generalized linear array processing.The three-dimensional filtering theory is illuminated by a theoretical problem of P-wave enhancement in the presence of ambient seismic noise. An analysis of the results shows why optimum three-dimensional filtering gives greater signal-to-noise ratio improvements than achieved by conventional array processing techniques.

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