Jack Capon

High-resolution frequency-wavenumber spectrum analysis

The output of an array of sansors is considered to be a homogeneous random field. In this case there is a spectral representation for this field, similar to that for stationary random processes, which consists of a superposition of traveling waves. The frequency-wavenumber power spectral density provides the mean-square value for the amplitudes of these waves and is of considerable importance in the analysis of propagating waves by means of an array of sensors. The conventional method of frequency-wavenumber power spectral density estimation uses a fixed-wavenumber window and its resolution is determined essentially by the beam pattern of the array of sensors. A high-resolution method of estimation is introduced which employs a wavenumber window whose shape changes and is a function of the wavenumber at which an estimate is obtained. It is shown that the wavenumber resolution of this method is considerably better than that of the conventional method. Application of these results is given to seismic data obtained from the large aperture seismic array located in eastern Montana. In addition, the application of the high-resolution method to other areas, such as radar, sonar, and radio astronomy, is indicated.

Multidimensional maximum-likelihood processing of a large aperture seismic array

The experimental Large Aperture Seismic Array (LASA) represents an attempt to improve the capability to monitor underground nuclear weapons tests and small earthquakes by making a large extrapolation in the existing art of building arrays of spaced and interconnected seismic transducers. The LASA is roughly equivalent to 21 separate subarrays, each consisting of 25 sensors, spread over an aperture of 200 km. The present work considers the problem of designing a linear filter which combines the outputs of the 25 sensors in a subarray so as to suppress the noise without distorting the signal, or event. This filter provides a minimum-variance unbiased estimate of the signal which is the same as the maximum-likelihood estimate of the signal if the noise is a multidimensional Gaussian process. An extensive discussion of the theory of multidimensional maximum-likelihood processing is given. A computer program implementation of the maximum-likelihood filter is presented which employs the cross-correlation matrix of noise measured just prior to the arrival of the event. This time-domain synthesis procedure requires relatively large amounts of computer time to synthesize the filter, and is quite sensitive to the assumption of noise stationarity. The asymptotic theory of maximum-likelihood filtering is also given. An asymptotically optimum frequency-domain synthesis technique is given for two-sided multidimensional filters. This procedure is well suited to machine computation and has the advantage with respect to the time-domain procedure of requiring about 10 times less computation time. A description of a computer program implementation of the frequency-domain synthesis method is given which employs the spectral matrix of the noise estimated just before the arrival of the event. The experimental results obtained by processing several events recorded at the LASA are presented, as well as a comparison of the performance of the frequency-domain method relative to the time-domain synthesis technique. It is found that the signal-to-noise improvement given by the frequency-domain procedure is within 2 dB of the gain obtained with the time-domain procedure, and that the frequency-domain method is relatively insensitive to the assumption of noise stationarity.

Probability distributions for estimators of the frequency-wavenumber spectrum

An exact derivation, subject to certain assumptions, is given for the probability distribution of the conventional and high-resolution estimators for the frequency-wavenumber power spectrum. These results are compared with those derived previously using approximations recommended by Blackman and Tukey.