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This paper is a further extension of the author’s wavelet theory of seismogram structure. The wavelet functions used in the previous paper are extended, a modified notation is adopted, and twenty four of the polynomials are given, together with their roots. Expressions are given for the displacement, velocity, and acceleration type wavelets.

In recent years, improvement has been achieved in vibration signal processing, using wavelet analysis for condition monitoring and fault diagnosis. The use of wavelet analysis has proven to be efficient to detect faults in vibration signals with nonstationary, transient characteristics/components. An experimental data set is used to compare the diagnostic capability of the fast Fourier transform power spectrum to the wavelet envelope power spectrum as respectively computed using Laplace and Morlet wavelet functions. The gear testing apparatus was used for experimental studies to obtain the vibration signal from a healthy gear and a faulty gear.

The general problem of seismic resolution is discussed. A seismogram, free from distortions, is an elaborate wavelet complex, and the analysis of a seismogram consists in breaking it down into its wavelet components. Distortions introduced by conventional seismographs are discussed and the conditions to be imposed upon seismic apparatus for distortionless wavelet transmission are set down. Three distortionless systems are possible: flat response; the wavelet contractor; and the wavelet expander. Wavelet contraction is discussed, whereby the individual wavelets which go to make up the seismogram may be contracted to a lesser breadth without altering the relative arrival times of the wavelet centers. Laboratory studies of the wavelet contractor are described and examples of wavelet contraction and the resolution of wavelet complexes are given. A description of the wavelet contracting seismograph used in the evaluation of the device in the field is given and results of the evaluation program are discussed. The wavelet contractor delineates subsurface formations with greater precision than does the conventional seismograph and it is able to carry pinchouts and truncations farther than is possible with conventional seismographs.