Sampling Theory and Wave Propagation

Abstract

In the seismic reflection method, the seismic signal does vary in amplitude, shape, frequency and phase versus the propagation time.

Besides various artefacts due to the recording and processing methods, these variations reflect the frequency dependence of wave propagation, due to the following phenomena:

• absorption in heterogeneous media,

• scattering in multilayered and heterogeneous media.

From theoretical studies, it appears that, in high resolution seismic, the propagation of high frequency waves may be very sensitive to 1D scattering phenomena due to intrabed multiples. Such phenomena induce velocity dispersion, resulting, after standard processing, in attenuation of high frequencies.

On the other hand, backscattered waves may carry useful informations related to thicknesses of thin layers and layering of thin layered series. Extracting these informations from the seismic records gives us a tool to increase the seismic resolution.

In this aim, we need to develop multifrequency-band processing methods, involving multichannel interferometric methods in adjacent narrow frequency bands.

Such methods involve a time-frequency representation of the seismic trace, in other words “Instantaneous Frequency Spectra”.

In this paper, we describe a practical 2D time-frequency sampling method for the seismic traces, suitable for optimization of the information/bit ratio. This sampling method is based on a new 2D transform, developed as an extension of the time-frequency Gabor expansion for complex signals.

In this new expansion, the 2D sampling grid is regular when using as the two coordinates the cycles (or unwrapped phases) and octaves rather than the times and frequencies. We may then call it the “Cycle-Octave” expansion.

Using this expansion, we may quantify the complex amplitude as a function of time and frequency (i.e. instantaneous energy and phase for any frequency).

Furthermore, using the “Uncertainty Principle”, we may quantify the resolution of the method.

Finally, after theoretical considerations based on Quantum Mechanics, it appears that the practical value for the dimensions of each elementary cell in the sampling grid must be:

• 2 sampling points per cycle,

• 4 sampling points per octave,

an in each cell of the sampling grid, the sampled value represents a complex amplitude.

Practical methods to compute both direct and inverse transforms are given.

To avoid problems of dynamic range in field recording, it will be useful to develop new recorders for direct 2D sampling of the seismic traces, using the “Cycle-Octave” expansion.