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Introduction

When analyzing seismic traces, it is often useful to know what frequencies are present in the data. Filtering and smoothing of data should be done with knowledge of the frequency content. In the standard approach to spectral analysis, the Fourier transform of the trace (amplitude spectrum) is computed. This approach is quite reliable for long data sequences (1000 or more data points) and is satisfactory for somewhat shorter sequences. Unfortunately, this technique becomes unreliable for very short time samples due to the increased importance of end effects: (a) the resolution of true peaks in the spectrum becomes poor and (b) spurious peaks may be introduced because of the implicit (and incorrect) assumption often made that the known data sequence is repeated periodically in time.

A different approach to spectral anaysis was introduced into the geophysical literature by (). His idea was to obtain an estimate of the power spectrum (square of the amplitude spectrum) by maximizing the spectral entropy with the known autocorrelation values as constraints. In principle, this approach should give a power spectrum that is consistent with the available information, but maximally noncommittal with regard to the unavailable information. It turns out that the resulting mathematical problem can be solved exactly using linear matrix theory. In fact, the method requires computation of the minimum phase deconvolution operator [also known as the ``prediction error filter'' (, )], which has received much attention in the geophysical literature. The power spectrum is then given by the square inverse of the operator's Fourier transform. Burg's method is known as maximum entropy spectral analysis (MESA) and is closely related both to deconvolution and to autoregressive analysis of stationary random time series.

The method of computing the spectrum in MESA can be easily understood in terms of filter theory. If we apply a prediction error filter to an input time series, the output will be a white spectrum. It is well-known that the spectrum of the output is the spectrum of the input times the spectrum of the filter. Since a white spectrum is constant, an estimate of the input spectrum is given by the inverse of the spectrum of the prediction error filter.

MESA has one principal advantage over the standard Fourier transform method of spectral analysis: resolution of peaks in the power spectrum is enhanced for short data sequences. MESA has two principal disadvantages: (a) computation time is increased (substantially for long data sequences) and (b) the best choice of for the operator length is not known (poor choices can give misleading results for short data samples). A possible solution to this second problem is discussed in the section on Choosing the Operator Length.

At least two other approaches to spectral analysis are possible. (a) The maximum likelihood method or MLM () has been shown by () to be the inverse of the arithmetic average of inverse maximum entropy spectra of increasing operator length. Thus, MLM weights the strongest peaks of MESA the least and cannot give very good resolution. (b) Using the terms of stochastic theory (, ), the ordinary power spectrum assumes that the underlying process is a moving average (MA) process. Using MESA can be viewed as being equivalent to assuming the process is autoregressive (AR). In fact, a discretely sampled geophysical time series is most likely to be a combination of the two, namely an autoregressive-moving-average (ARMA) process. It is possible to estimate the spectrum under the ARMA assumption; however, a substantial increase in computation time is required (over MESA), while the resolution of peaks should remain nearly the same.

A brief discussion of the theory and practice of MESA has appeared previously in (). An expanded version of this account is given in the following pages. The work presented here leads to the conclusion that for short time series MESA may well be a useful tool, and that MESA is probably the best available alternative to standard methods for such short data processing problems.


next up previous [pdf]

Next: The Variational Principle Up: Berryman: MESA Previous: Berryman: MESA

2009-04-13