The coefficient of coherence between two stationary time series was introduced by Wiener in 1930. It is related to the signal‐to‐noise ratio, to the minimum prediction error, and has important invariance properties. As an estimate of this parameter, most geophysicists have used the so‐called “sample coherence.” An approximate distribution of the sample coherence for Gaussian data has been derived by N. R. Goodman. We have tested this distribution by means of Monte Carlo experiments for validity and robustness (insensitivity to the Gaussian assumption). It has passed the tests. The Goodman distribution provides a means of constructing estimates of the true coherence which are better than the widely used sample coherence. It can also be used to calculate confidence intervals. Finally, it forms a basis for choosing the lag window and data window necessary for best estimation of the true coherence. For good estimates of the true coherence, two precautions must be observed: 1. The cross‐spectrum and power spectra of the two time series must be smoothly varying over the width of the spectral window. 2. The ratio of the length of the data window to the lag window must be large. For most seismic work the second requirement severely limits the spectral resolution. Examples show that large errors can result if this resolution is not sufficient to satisfy the first requirement. In many geophysical studies the parameter of interest is the signal‐to‐noise ratio. Because of its relation to the coherence, the Goodman distribution provides a basis for its estimation as well.
A long‐spacing velocity log contains almost the same information as an ideal short‐spacing log, but in a distorted form with added noise. The distortion can be thought of as a moving average or smoothing filter. Its inverse, called a “sharpening” filter by astronomers, amplifies noise. If the inverse is to be useful, it must be designed with a balance between errors due to noise amplification and those due to incomplete sharpening. The Wiener optimum filter theory gives a prescription for achieving this balance. The result is called an optimum inverse filter. We have calculated finite‐memory optimum inverse filters using the IBM 704. We have applied them to actual data, digitized in the field, to produce synthetic short‐spacing velocity logs. These we have compared with their field counterparts. The synthetic logs have less calibration error and are free from noise spikes. The general agreement is good.
Optimum systems have been developed to correspond to the sub‐optimum moveout discrimination systems presented previously by several authors. The seismic data on the lth trace is assumed to be additive signal S with moveout [Formula: see text], coherent noise N with moveout [Formula: see text], and incoherent noise [Formula: see text], expressed [Formula: see text] where S, N, and [Formula: see text] are independent, second order stationary random processes and [Formula: see text] and [Formula: see text] are random variables with prescribed probability density functions. The signal estimate S⁁ is produced by filtering each trace with its corresponding filter [Formula: see text] and summing the outputs [Formula: see text] We choose the system of filters [Formula: see text] to make the signal estimate optimum in the Wiener sense (minimum mean‐square error of the signal ensemble). For the special cases discussed, the moveouts are linear functions of the trace number l determined by the moveout/trace τ for signal and [Formula: see text] for noise. Thus, the optimum system is determined by the probability densities of τ and [Formula: see text] together with the noise/signal power spectrum ratios [Formula: see text] and [Formula: see text]. In comparison, suboptimum systems are controlled completely by the cut‐off moveout/trace [Formula: see text]. Events whose moveout/trace falls within [Formula: see text] of the expected dip moveout/trace are accepted, and those falling outside this range are suppressed. Suboptimum systems can be derived from optimum systems by choosing probability densities for τ and [Formula: see text] that are uniform within the above ranges and letting [Formula: see text] be very large. Optimum systems have increased flexibility over suboptimum systems due to control over the probability density functions and the power spectrum ratios and allow increased noise suppression in selected regions of f‐k space.
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