Reflection Coefficient Series Research Articles

INVERSION METHOD IN THE SLANT STACK DOMAIN USING AMPLITUDES OF REFLECTION ARRIVALS*

AbstractAlmost all ray‐tracing methods ignore the analysis of the amplitudes of seismic arrivals and therefore utilize only half of the available information. We propose a method which is a combination of ray‐tracing imaging and transformation of the amplitudes of wide‐aperture data.Seismic data in the conventional X‐T domain are first transformed to the domain of intercept time τ and ray parameter p to recover the plane wave response. The next step is the derivation of a series of plane wave reflection coefficients, which are mapped as a function of τ and p. The reflection coefficients R(τ, p) for two arbitrarily chosen traces can then be used in our inversion method to derive a slowness‐depth and a density‐depth profile. It is shown that the inclusion of amplitudes of seismic arrivals (in this method, we consider the acoustic case) makes the inverse method highly stable and accurate. In a horizontally stratified medium one can recover separate profiles of velocity and density. Since this method utilizes large‐offset data, it can be used for separate recovery of velocity and density to a greater depth.

Statistical pulse compression

The seismic method in petroleum exploration is an echo-location technique to detect interfaces between the subsurface sedimentary layers of the earth. The received seismic reflection record (field trace), in general, may be modeled as a linear time-varying (LTV) system. However, in order to make the problem tractable, we do not deal with the entire field trace as a single unit, but instead subdivide it into time gates. For any time gate on the trace, there is a corresponding vertical section of rock layers within the earth, such that the primary (direct) reflections from these layers all arrive within the gate. Each interface between layers is characterized by a local (or Fresnel) reflection coefficient, which physically must be less than unity in magnitude. Under the hypothesis that the vertical earth section has small reflection coefficients, then within the corresponding time gate the LTV model of the seismic field trace reduces to a linear time-invariant (LTI) system. This LTI system, known as the convolutional model of the seismic trace, says that the field trace is the convolution of a seismic wavelet with the reflection coefficient series. If, in addition, the reflection coefficient series is white, then all the spectral shape of the trace within the gate can be attributed to the seismic wavelet. Thus the inverse wavelet can be computed as the prediction error operator (for unit prediction distance) by the method of least squares. The convolution of this inverse wavelet with the field trace yields the desired reflection coefficients. This statistical pulse compression method, known as predictive deconvolution with unit prediction distance, is also called spike deconvolution. Alternatively, predictive deconvolution with greater prediction distance can be used, and it is known as gapped deconvolution. Other pulse compression methods used in seismic processing are signature deconvolution, wavelet processing, and minimum entropy deconvolution.

Homomorphic deconvolution

The advantages of homomorphic deconvolution are that it does not require the assumptions of minimum‐phase wavelet and of a white random reflection coefficient series. Disadvantages of the method which have been recognized in the public domain are difficulties in unwrapping the phase, in dealing with band‐limited signals, and in handling mixed‐phase reflection coefficient series. These difficulties may be respectively overcome by using an “adaptive numerical integration algorithm” (Tribolet, 1977), frequency transformations (Tribolet, 1979), and exponential weighting of the signal (Tribolet, 1979). There seems to have been some understanding in the literature and among exploration researchers that additive noise would affect the performance of homomorphic deconvolution. However, to the best of our knowledge there have not appeared in the literature any analytical expressions or experiments conclusively showing how additive noise affects homomorphic deconvolution. Analytic and experimental analyses demonstrated that additive noise plays a critical role in homomorphic deconvolution such that homomorphic deconvolution is unreliable whenever the spectral amplitudes of the signal are very small over certain frequency bands and even a small amount of noise is present. This unreliability of the method overshadows its advantages.

VERTICAL SEISMIC PROFILES: THEIR APPLICATIONS IN EXPLORATION GEOPHYSICS*

AbstractDuring the past four or five years, Vertical Seismic Profiles have been run in a large number of wells and in a variety of geological provinces with the object of assisting the seismic data processor and improving the geological interpretation.The special properties of the Vertical Seismic Profile, which allows the separation of the upward and downward travelling components of the recorded waveform, provides a means of studying the acoustic response of the earth in detail. Deconvolution of the data in particular gives a clearer understanding of the mechanics of multiple reflections and the way in which they may obscure primary reflections in the vicinity of the well.The primary reflection response of the earth can be observed with greater resolution than conventional seismic data permits, making correlation with lithology more precise, and allowing a good estimate of the reflection coefficient series to be determined. Inversion of the seismic traces to produce an acoustic impedance log leads to better definition of the sedimentary sequence and is of particular interest in predicting lithologic variations ahead of the drilling bit.In addition, Vertical Seismic Profile data can be used to assess parameters in areas such as deconvolution and signal band width for the evaluation of conventional seismic data.

Reservoir Studies and Inversion of Seismic Traces: ABSTRACT

Inverting seismic traces is the means of finding a velocity or acoustic-impedance model from the recorded traces. Data are processed in preserved-amplitude mode. A wavelet inverse filtering results in a series of reflection coefficients. Reflection coefficients are simple functions of velocities and densities. A transformation of the series of reflection coefficients into relative velocities provides a display where the inverted seismic traces at a well location can be directly correlated to the sonic log. Prediction of lateral changes affecting the formations intersected by the well is the most direct application. Provided the original surface seismic data have a sufficiently broad frequency spectrum, the inversion improves the detection of thin beds and lateral facies changes. Two examples of reservoir studies illustrate the application of the method: the first is in a deltaic sandstone and shale sequence; the second is a shelf-carbonate province. End_of_Article - Last_Page 513------------

Synthetic sonic logs—a process for stratigraphic interpretation

Modern seismic reflection data may be processed to approximate closely the reflection coefficient series of a sedimentary section. Inversion of the series will produce a low‐cut filtered impedance log. Extension of the technique to include density correction and replacement of missing low‐frequency components leads to generation of a synthetic sonic log having dimensions and characteristics similar to a conventional borehole sonic log. A synthetic sonic log section provides several advantages over conventional seismic sections, particularly for stratigraphic exploration. Direct depth display largely eliminates the vertical scale distortion inherent in time sections and simplifies integration with other subsurface data. Resolution is generally improved. The process is well adapted to automated generation of stratigraphic sections. Individual rock units are mapped by contours of constant transit‐time, to show facies changes and depositional sequences in detail. The detailed measurements of velocity may be used to interpret rock type. Comparison of synthetic sonic log sections to the conventional seismic data from which they were derived, clearly demonstrates improved understanding of subsurface geology.

The Fine Structure of the Normal Incidence Synthetic Seismogram

Summary. The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient co, where |co|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, co=±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well. If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.

DIRECT AND INVERSE PROBLEMS RELATING REFLECTION COEFFICIENTS AND REFLECTION RESPONSE FOR HORIZONTALLY LAYERED MEDIA

If a plane wave is normally incident on the boundary of a horizontally layered medium, the reflection and transmission responses can be found in terms of the reflection coefficients at the interfaces of the medium. Recursion formulas for finding these responses have been given by various authors: Goupillaud (1961), Sherwood (1962), Kunetz and d’Erceville (1962), Sherwood and Trorey (1965), and Claerbout (1968). This paper gives explicit solutions of the recursion formulas for the numerator and denominator of the reflection and transmission operators. The solutions are expressed as polynomials in the variable z of the z-transform, with coefficients given as explicit sums of products of reflection coefficients. These formulas are equivalent to those given by Goupillaud. We also give the operator which removes all effects of any given set of layers at the top, the operator being equivalent to one given by Goupillaud. The inverse problem of finding the reflection coefficients from a reflection record or transmission record has been solved in two different, equivalent forms by Kunetz and d’Erceville and Claerbout. This paper gives further results derived along the lines of Claerbout’s paper. Since the solutions are developed for the noise‐free case, the effects of timing errors, scaling errors, and random noise on the reflection coefficient series computation are investigated. Examples are given which show the zones of stabilities. The Appendix gives the general form of the energy conservation equation in elastic wave propagation in strict analogy with the Poynting formulation for electromagnetic wave propagation. The main body of the paper includes a proof that Claerbout’s polynomial identity is an energy conservation equation for the case when no free surface is present.