No abstract

In a long needed paper, R. T. Lacoss (1971) has presented many examples of spectra obtained by the maximum likelihood method and by the maximum entropy method and has shown that these newer techniques are in general superior to the more conventional spectral analysis methods. This short note shows that there exists a simple, exact relationship between maximum entropy spectra and maximum likelihood spectra when the correlation function is known at uniform intervals of lag. The data are of this form in almost all practical cases of time series analysis as well as in the special case of wavenumber spectral analysis of wave propagation as seen by a linear array of equally spaced sensors. The wavenumber case will be explicitly considered in this note since it requires the complex variable form of the theory.

A new technique has been developed which makes it possible to process a seismic record‐section in such a way that all seismic events with dips in a given range are preserved with no alteration over a wide frequency band, while all seismic events with dips outside the specified range are uniformly and severely attenuated. By applying this process to a noisy record‐section, a record‐section may be obtained which has all events within a specified dip range perfectly preserved, and very high‐velocity noise essentially eliminated, a result which is impossible by simple wave‐number filtering or conventional array usage. In structurally complex areas where several steeply dipping events interfere, the technique may be applied to separate the events with different dips. In areas where a normal‐moveout contrast exists between primaries and multiples, the technique may be used for wide‐band multiple attenuation. By application of a “rotating Pie‐Slice” to micro‐spread noise data, seismic noise may be separated on the basis of propagation velocity, and a clearer picture of the seismic noise problem obtained. The “rotating Pie‐Slice” also provides a means of uncovering diffractions and other steeply‐curved events from a record section. The paper discusses the motivation and implementation of the process and its application to both synthetic and actual data.

The development of the W' iener 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.

The spatial correlation characteristics of ambient short‐period (0.5 to 5 cps) noise at Ft. Sill, Oklahoma, and on the Cumberland Plateau in Tennessee were investigated on “permanent” arrays with 3–4 kilometer diameter. Dominant ambient noise at the two locations is spatially organized, and to first order may be treated as a combination of seismic propagating wave trains. At the Tennessee location noise energy above one cps is dominantly propagating with velocities from 3.5 to 4.5 km/sec, and must be carried in deeply trapped, high‐order modes. Generalized multichannel filtering (Burg) can be used to preserve a large class of mantle P‐wave signals, wide‐band, in a single output trace, while at the same time specifically rejecting ambient noise on the basis of its organization. Results of generalized multichannel filtering applied on‐line at the nineteen‐element array in Tennessee and applied off‐line are discussed.