THE DISCRETE RADON TRANSFORM WITH APPLICATIONS TO TIME DOMAIN BEAMFORMING
SCHEIBNER, DAVID JAMES
Doctor of Philosophy
Performing a discrete Radon transform (DRT) with an array of space-time samples results in a function of slowness (inverse velocity) and time intercept. Time domain beamforming and semblance, two commonly used methods of obtaining a slowness-time analysis, give outputs that are composed of discrete Radon transforms of the data. Many of the properties satisfied by the continuous Radon transform are also satisfied in the discrete case. Sampling problems give rise to additional considerations not present in the continuous transform. As a function of slowness, beamforming decimates each channel in proportion to its distance from the beamformer reference channel. This viewpoint allows the slowness bandwidth to be expressed as a function of the array spatial locations and the normalized temporal bandwidth of the received waveforms. These relationships determine the temporal sample rate increase at each channel necessary to avoid slowness aliasing. Examples are presented showing the slowness aliasing that results if either the temporal bandwidths or the array spatial aperture are too large. The conclusions regarding the slowness sampling density are applicable as well to frequency domain beamforming, where the beams must also be computed at discrete values of slowness. The slowness sampling rate can be increased using a technique termed interpolation beamforming. A polyphase filter structure is shown to be particularly suited to interpolation beamforming. The polyphase structure results in a pipelined processor, in which computation occurs in parallel filters at the low input sample rate. We describe four methods of inverting the beamformer output to reconstruct the space-time samples. Of the four, one method, a block matrix inversion technique, is new. It is exact when operating on the original data, but fails when the beamformer output is modified. This problem can be alleviated by reducing the dimensionality of the inversion matrix using a singular value decomposition. Time-domain beamforming separates transient plane-waves as a function of their velocity and time-of-arrival. Modifications in the beamformer output domain, such as masking out coherent interfering signals while taking into account both velocity and time information, are more straightforward than with traditional (k-(omega)) velocity filtering methods. Processing in this domain is especially useful if a time-varying velocity filter is required.
Electronics; Electrical engineering