Fulton Koehler

Complex seismic trace analysis

The conventional seismic trace can be viewed as the real component of a complex trace which can be uniquely calculated under usual conditions. The complex trace permits the unique separation of envelope amplitude and phase information and the calculation of instantaneous frequency. These and other quantities can be displayed in a color-encoded manner which helps an interpreter see their interrelationship and spatial changes. The significance of color patterns and their geological interpretation is illustrated by examples of seismic data from three areas.

Velocity spectra-digital computer derivation and applications of velocity functions

Multifold ground coverage by seismic techniques such as the common reflection point method provides a multiplicity of wave travel path information which allows direct determination of root‐mean‐square velocities associated with such paths. Hyperbolic searches for semblance among appropriately gathered arrays of traces form the basis upon which velocities are estimated. Measured semblances are presented as a velocity spectral display. Interpretation of this information can give velocities with meaningful accuracy for primary as well as multiple events. In addition, the velocity data can help correctly label events. This paper outlines the fundamental principles for calculating velocity spectra displays. Examples are included which demonstrate the depth and detail of geological information which may be obtained from the interpretation of such displays.

ESTIMATION AND CORRECTION OF NEAR‐SURFACE TIME ANOMALIES

The problem of computing static corrections for CDP seismic reflection data is discussed. A new approach is presented and it is related to various existing approaches. The approach consists of using crosscorrelation computations to find time shifts which appear to align the traces of each common‐depth‐point. These shifts are expressed in terms of surface corrections, one for each source and receiver position; a residual NMO correction for each common‐depth‐point; and a fixed correction for each common‐depth‐point. These simultaneous equations form an overdetermined set which can be solved for the unknown static and NMO corrections. The least‐square‐error solution to these equations has an important indeterminancy which is discussed. Methods for its resolution are proposed. Application of the technique to real data is illustrated by several examples. Validity of the corrections is demonstrated by velocity analyses before and after correction of the traces.

Surface consistent corrections

Amplitudes of seismic reflections have been of interest since the first days of exploration seismology. Any change of amplitude or anomalous behavior may be significant, so it is important that the zones of interest be free from outside disturbances, such as those caused by the near‐surface layers. Surface consistent factors may be divided into source, receiver, offset, and subsurface components, and these may be divided further into amplitude and phase (or time shift) factors. Correction of trace amplitudes using multiplication by a scale factor is similar to correction of phase distortions by a static shift, and both corrections enhance seismic data. Displays of surface consistent components for time and amplitude corrections provide an additional diagnostic for the geophysicist.

Multi‐dimensional seismic imaging

Seismic traces synthesizing the response of subsurface formations to a cylindrical or plane wave are obtained for a succession of shotpoint locations along a seismic line of profile. The traces obtained are then wavefront steered and the steered traces and original trace for each shotpoint are summed. Groups of these traces for a line of profile are assembled to form a steered section. A number of these sections are then individually imaged or migrated, and the migrated sections are summed to form a migrated two-dimensional stack of data from cylindrical or plane wave exploration. Reflectors may then be located by finding common tangents. The traces for those shotpoints of the several lines which lie in planes perpendicular to the lines are then assembled and processed in the foregoing manner to obtain three dimension migrated seismic data.

The use of the conjugate-gradient algorithm in the computation of predictive deconvolution operators

A number of excellent papers have been published since the introduction of deconvolution by Robinson in the middle 1950s. The application of the Wiener‐Levinson algorithm makes deconvolution a practical and vital part of today’s digital seismic data processing. We review the original formulation of deconvolution, develop the solution from another perspective, and demonstrate a general and rigorous solution that could be implemented. By “general” we mean a deterministic time‐varying and multichannel operator design, and by “rigorous” we mean the straightforward least‐squares error solution without simplifying to a Toeplitz matrix. Also we show that the conjugate‐gradient algorithm used in conjunction with the least‐squares problem leads to a satisfactory simplification; that in the computation of the operators, the square matrix involved in the normal equations need not be computed. Furthermore, the product of this matrix with a column matrix can be obtained directly from the data as a result of two cascad...

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 trans...

Estimation of unbiased delays

A large number of exploration processing procedures need the solution to the problem stated as, “Given a set of seismic traces, determine the common element.” This element could be the seismic wavelet, as in a set of common‐shot traces or as in a set of stack traces that need to be matched with synthetic seismograms. It could also be a pilot trace representing a noise‐free estimate of traces in a common‐depth‐point (CDP) gather to be used during an automatic time and phase statics computation. Most present computations use iteratively improved estimates of one sort or another. They assume that initial estimates are close enough to ensure that subsequent steps will produce more accurate results. The problem, of course, is the accuracy of the first unbiased estimation: when it fails, the rest becomes uncertain. In this paper we introduce a method that provides a robust solution to the problem of finding the common element. We also show that the method may be used to estimate both common wavelets and pilot t...

Reply by author to J. G. Saha, N. D. J. Rao, and M. C. Agrawal.

In reply to the discussion on the “Complex Seismic Trace” coauthored by Taner, Koehler, and Sheriff, we have no substantial disagreement with the remarks of Saha, Rao, and Agrawal. Sharp termination of the Hilbert operator does lead to inaccuracies.

Complex seismic trace analysis; discussion and reply

In Appendix B, the authors derive an analytical expression for the conjugate component of a Ricker wavelet and state that the different attributes of a wavelet with a peak frequency of 25 Hz are listed in Table 1 of their paper. The quadrature component of a Ricker wavelet should be read as:f*(t)=(2/π)1/2∫0∞ω2e-ω2/2Sinωtdω=2(2/π)1/2e-t2/2t-(2m-3)(2m+1)!t2m+1,where (2m-3)=1 for m=1 and 1.3…, (2m-3) for m⩾.