Unambiguous signal recovery above the Nyquist
Authors: Wisecup R. Daniel 
Published: October 2008

Abstract
A method is presented for processing multi-channel seismic data that recovers, simultaneously and unambiguously, signal frequencies above and below the temporal Nyquist of the input data, By exploiting signal stationarity and redundancy, I show that the sample interval of a single input channel does not uniquely determine the maximum recoverable frequency. The method is tested using normal moveout (NMO) and stack on synthetic model data. Frequencies of 10 to 225 Hz are recovered from input data sampled at 4 ms. I then apply the method to real data. The results demonstrate that antialias strategies based on the one-dimensional Shannon-Whittaker sampling theorem (l) impose an unnecessary limit on the ability to recover high frequencies and maximize signal resolution. The method is applicable wherever the trajectory of the signal to be imaged irregularly intersects the sampling grid. It is appropriate for resolving both temporal and spatial aliasing concerns.

Introduction
Since the inception of digital signal processing, concerns about signal aliasing have played a major role in determining useable signal bandwidth and the cost of obtaining that bandwidth. For example, the authors of a 1991 article in The Leading Edge observed that "..the major use of Nyquist's work in geophysics is the elimination of alias frequencies on digitally recorded seismic data. The Nyquist frequency is the highest frequency that can be obtained for a given sampling interval." (2). Such concerns have led the seismic industry to systematically restrict potential signal frequencies to below the Nyquist frequency by the application of antialiasing filters based on the Whittaker-Shannon sampling theorem. This paper describes a simple methodology for the recovery of signal components in excess of the Nyquist limit predicted by the one-dimensional Whittaker-Shannon sampling theorem. With this method, an optimum balance of image bandwidth and signal-to-noise ratio may be achieved by simple processing parameter choices. The methodology is applicable whenever the signal trajectory is irregularly intersected by a sampling grid of two or more dimensions.

Aliasing
It is generally accepted that the digital sampling interval and the upper limit of the recoverable signal spectrum are inexorably linked. Authors usually cite the one-dimensional Whittaker-Shannon sampling theorem which states that the maximum recoverable frequency (? Nyq) in an evenly-sampled function is given by:(mathematical equation)(available in full paper) where ?t is the sampling interval (in seconds). Frequencies in the input function prior to sampling, which are in excess of this Nyquist frequency, are, if not removed prior to sampling, said to be aliased (or folded) in the sampled output. Aliasing is generally understood to mean that those frequencies above ? Nyq are irretrievably lost due to being "mixed" with those below ? Nyq.
Commonly, aliasing has been dealt with by one or both of two methods:

Alias reduction method #1
Antialias filtering is almost always used to significantly reduce contamination by aliased frequencies whenever analog data are sampled (e.g., during field recording) or whenever the sampling interval is increased. While usually successful in preventing anticipated aliasing problems, this filtering can result in a loss of valuable signal both above and below ? Nyq. Also, additional equipment and/or processing costs are usually incurred.