Wave propagation and sampling theory—Part I: Complex signal and scattering in multilayered media

Abstract

From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time. To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail. We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media. As wavelet shapes, we use zero‐phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two‐dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as “instantaneous frequency spectra.” Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio. For layered media, as found in sedimentary basins, we can distinguish two main types of series: (1) progressive series, and (2) cyclic or quasi‐cyclic series. The second type is of high interest in hydrocarbon exploration. Progressive series do not involve noticeable distortions of the seismic signal. We studied, therefore, the wave propagation in cyclic series and, first, simple models made up of two components (binary media). Such periodic structures have a spatial period. We present synthetic traces computed in the time domain using the Goupillaud‐Kunetz model of propagation for one‐dimensional (1-D) synthetic seismograms. Three different cases appear for signal scattering, depending upon the value of the ratio wavelength of the signal/spatial period of the medium. (1) Large wavelengths The composite medium is fully transparent, but phase delaying. It acts like an homogeneous medium, with an “effective velocity” and an “effective impedance.” (2) Short wavelengths For wavelengths close to twice the spatial period of the medium, the composite medium strongly attenuates the transmission, and superreflectivity occurs as counterpart. (3) Intermediate wavelengths For intermediate values of the frequency, velocity dispersion versus frequency appears. All these phenomena are studied in the frequency domain, by analytic formulation of the transfer functions of the composite media for transmission and reflection. Such phenomena are similar to Bloch waves in crystal lattices as studied in solid state physics, with only a difference in scale, and we checked their conformity with laboratory measurements. Such models give us an easy way to introduce the use of effective velocities and impedances which are frequency dependent, i.e., complex. They will be helpful for further developments of “complex deconvolution.” The above results can be extended to quasi‐cyclic media made up of a random distribution of double layers. For signal transmission, quasi‐cyclic series act as a high cut filter with possible time delay, velocity dispersion, and “constant Q” type of law for attenuation. For signal reflection they act as a low cut filter, with possible superreflections. These studies could be extended to three‐dimensional (3-D) binary models (grains and pores in a porous reservoir), in agreement with well‐known acoustic properties of gas reservoirs (theory of bright spots). We present some applications to real well data.