Eike Rietsch

Euclid and the art of wavelet estimation, Part I: Basic algorithm for noise-free data

An algorithm borrowed from polynomial algebra for finding the common factors of two or more polynomials can be used to find the wavelet that several seismic traces have in common. In the implementation described in this first part of a two‐part work, a matrix is constructed from the autocorrelations and crosscorrelations of these seismic traces. The number of zero eigenvalues of this matrix is equal to the number of samples of the wavelet, and the eigenvectors associated with these eigenvalues are related to the reflection coefficients. The method, which works well if the noise is not too high, is illustrated by means of a synthetic example. Part II of this two‐part work shows how this method is affected by noise and gives field‐data examples.

Euclid and the art of wavelet estimation, Part II: Robust algorithm and field-data examples

In this second part of a two-part work, a more robust algorithm is derived and used for the estimation of the seismic wavelet as the common signal of two or more seismic traces. It is based on the properties of the eigenvectors with zero eigenvalue of a matrix derived in the first part, whose elements are the samples of the autocorrelation functions and crosscorrelation functions of these seismic traces for a number of lags. The noise resistance of this algorithm is illustrated by means of a synthetic-data example and then demonstrated on field data. In one field-data example, the so-called Euclid wavelet is compared with one derived deterministically by means of an impedance log. The other example relates three quite different Euclid wavelets determined in three different time zones on a seismic line to one another by showing that their differences can be explained by absorption.

Seismic system and method

The apparatus of the present invention includes a group of at least three seismic detectors. Each seismic detector detects vibrations and provides a seismic signal in accordance with the sensed vibration. A multiplexer multiplexes the seismic signals to provide a multiplexed signal. The multiplexed signal is amplified and converted to digital signals. A memory stores the digital signals according to the detector of origin so that the stored digital signals for a detector are in effect a sample of the seismic signals from the detector. A microprocessor connected to the memory means derives a statistical reference from the stored samples. Samples that are outlying with respect to the statistical reference are discarded. The remaining samples are then combined in a predetermined manner to provide an enhanced seismic signal from the group of detectors.

Geophone sensitivities for Chebyshev optimized arrays

In 1963 Holzman published a theoretical analysis on so‐called Chebyshev optimized geophone arrays, i.e., geophone patterns whose reject band response has lobes of equal amplitude. For equidistant geophones in a linear array, this reject band response can be realized only if the geophone sensitivities are suitably weighted. One of the results of Holzman’s paper is a method for computing weights (relative sensitivities) for the geophones to achieve this desired reject band behavior (here, for simplicity, the term “geophone” is used in a generic sense for any seismic source and receiver).

A Fourier series kernel based on Chebyshev polynomials

In most cases evaluation of Fourier series requires that special summation methods be applied or that the coefficients of the series be suitably modified to suppress strong oscillations at discontinuties of the approximated function. All these methods may be described as a substitution of the Dirichlet kernel by other kernels. In this paper eight of these kernels are briefly reviewed and compared with a ninth kernel which is based on Chebyshev polynominals. A closed-form representation has been derived for the Fourier coefficients of this kernel as well as a recursive relation for their practical computation. Furthermore, an error criterion is given which allows the determination of an upper bound of the difference between Fourier series and approximated function provided upper limits on both the variation and the second derivative of the latter are known.ZusammenfassungFür die praktische Auswertung von Fourierreihen ist es in den meisten Fällen erforderlich, spezielle Summationsverfahren zu verwenden oder die Koeffizienten der Reihe in geeigneter Weise zu modifizieren, um starke Oszillationen der Fourierreihen an Diskontinuitäten der approximierten Funktion zu unterdrücken. Alle diese Verfahren lassen sich als ein Ersetzen des Dirichletkerns durch andere Kerne beschreiben. In dieser Arbeit werden acht dieser Kerne kurz besprochen und mit einem neunten, auf den Tschebyscheffschen Polynomen beruhenden Kern verglichen. Für die Fourierkoeffizienten dieses Kerns wurde sowohl eine Darstellung in geschlossener Form als auch eine für ihre praktische Berechnung geeignetere rekursive Beziehung abgeleitet. Weiters wird ein Fehlerkriterium angegeben, das es gestattet, eine obere Grenze für die Differenz zwischen Fourierreihe und approximierter Funktion festzulegen, vorausgesetzt, daß obere Grenzen sowohl für die Schwankung als auch für die zweite Ableitung der letzteren bekannt sind.