Oliver G. Jensen

Joseph geology and seismic deconvolution

Common practice in seismic deconvolution is to assume that the reflection sequence is uncorrelated, that is, that the sequence has a white power spectrum and a delta function autocorrelation. A white spectrum implies that the acoustic impedance function has a power spectrum proportional to 1/f2, which is characteristic of a nonstationary Brownian process (f is frequency). However, the maximum power spectrum permissible for the acoustic impedance function is 1/f; we call a spectrum of this kind a Joseph spectrum. A Joseph spectrum corresponds to a reflection sequence with a power spectrum proportional to f and a negative autocorrelation at small lags. Joseph spectrum behavior for reflection sequences has been seen before and we show it again in a well off Newfoundland and in two wells from Quebec. If the power spectrum is proportional to f, then the first term of the discretized autocorrelation function is −0.405 of the zero‐lag term and higher terms are negligible. We construct a Joseph filter analogous t...

Fractal Velocity Models in Refraction Seismology

The introduction of noise components to a simple crustal velocity model is shown to markedly affect the appearance of synthetic seismograms calculated according to ray theory applied to refraction experiments. Here we simulate noise by a self-similar process with a power spectral density which falls off as inverse wavenumber to a simple power (0–2). The major effect is to destroy the coherency of the arrival branches normally expected from deterministic velocity models; the arrival amplitudes also show large trace-to-trace variations and considerable sensitivity to shot position. Some of these differences can be ascribed to the variety of noise model chosen (i.e., white noise, flicker noise and brown noise). It is argued that there is no clear distinction between coherent noise and geological structure.

The stochastic excitation of reversals in simple dynamos

Abstract Several variations of the αω disc dynamo models are known to exhibit chaotic magnetic field behaviour typical of non-linear, dissipative systems. Although these models demonstrate that geomagnetic reversals can be generated by simplified dynamo equations, the behaviour of the magnetic field itself is generally too simple, showing especially an absence of long polarity epochs in most of the models. We show that the addition of three varieties of stochastic processes (Gaussian, flicker and brown noise) enriches the field evolution and can lead to realistic palaeomagnetic behaviour. We argue that noise processes must be present in the actual fluid core and suggest, from a physical point of view, a flicker noise stimulation of the dynamo. We find two features of the palaeomagnetic record that would favour the presence of noise in the dynamo process, namely the absence of a linear oscillation in field intensity between reversals or, even if present, the absence of an increase in amplitude of this oscillation prior to a reversal. We also consider the addition of a random component to the helicity driving function of the α2 dynamo process and show that various types of reversal can occur. Unfortunately, realistic field behaviour cannot be maintained over long time periods due to the tendency for the magnitudes of the poloidal and toroidal fields to equilibrate during a polarity epoch.

Scaling Geology and Seismic Deconvolution

The reflection seismic signal observed at the surface is the convolution of a wavelet with a reflection sequence representing the geology. Deconvolution of the observations without prior knowledge of the wavelet can be done by making assumptions about the statistics of the reflection sequence. In particular, the widely used prediction error filter is obtained by assuming that the power spectra of reflection sequences are white. However, evidence from well logs suggests that the power spectra are in fact proportional to a power of the frequency f, that is, to f α, with α equal approximately to 1.

Cross-spectral analysis using maximum entropy

A great deal of interest has been shown in the maximum entropy method (MEM) of power spectral analysis originally suggested by Burg (1967, 1968). The application of MEM to problems of geophysical and astronomical interest has met with considerable success (Ulrych, 1972; Ulrych et al., 1973; Smylie et al., 1973; Currie, 1973a, b; and Jensen and Ulrych, 1973). We have recently received a number of enquiries concerning the possibility of computing maximum entropy crosspower spectra. The purpose of this note is to present a method of determining the MEM crosspower spectrum from a knowledge of the MEM autopower spectra of the bivariate time series.

Multichannel Linear Prediction and Maximum-Entropy Spectral Analysis Using Least Squares Modeling

Autoregressive data modeling using the least squares linearprediction method is generalized for multichannel time series. A recursive algorithm is obtained for the formation of the system of multichannel normal equations which determine the least squares solution of the multichannel linear-prediction problem. Solution of these multichannel normal equations is accomplished by the Cholesky factorization method. The corresponding multichannel maximum-entropy spectrum derived from these least squares estimates of the autoregressive-model parameters is compared to that obtained using parameters estimated by a multichannel generalization of Burgs algorithm. Numerical experiments have shown that the multichannel spectrum obtained by the least squares method provides for more accurate frequency determination for truncated sinusoids in the presence of additive white noise.

Excitation of Geophysical Systems with Fractal Flicker Noise

Geophysicists often model their measurements, derived from natural processes, as the linear superposition of a simple rational system function and a purely random excitation process. For many geophysical processes, the assumption of linearity for its deterministic component is sufficient but the assumption of a purely random excitation often and easily leads to a misidentification of the system function. Many geophysical systems are excited by stochastic processes which appear to be stationary even on geological time scales but which possess a preponderance of long period components. Selfsimilar, fractal stochastic processes form a class of possible geophysical excitations having “power spectrum” of the form 1/ |f | k . Of this class, flicker-noise processes, for which k = 1 exist, on the boundary between the stationary and evolutionary subsets. No fractal stationary random excitation can provide for greater weighting of long period components.

Homomorphic deconvolution : Application to gravitational and magnetic field waveforms

The gravitational and magnetic field waveforms which represent potential fields of geologic structures can be expressed as the sum of overlapping, delayed replicas of a certain basic waveform. Such field waveforms, therefore, can be thought of as the outcome of a convolutional process. One way of separating signals which have been combined through convolution or multiplication is by using a set of systems called homomorphic systems. A particular feature of the process called homomorphic deconvolution is that no previous assumption of the character or nature of the basic waveform need be made. It is this feature of homomorphic deconvolution which makes it attractive for applications to gravitational and magnetic field waveforms.

Phase-compensating filtering to reduce distortion caused by in-aircraft reactive integrators

Low‐pass filters based on reactive integrators necessarily introduce temporal phase delays into data. The use of such filters in airborne geophysical surveys, therefore, produces equivalent spatial phase delays which displace the high‐wavenumber Fourier components of anomalies downstream along the flight line, causing distortion of the anomaly shapes. A linear filter is described which eliminates this distortion by introducing a compensating, frequency‐dependent phase advance. The restored data provide anomalies as would be seen from an aircraft moving with zero speed. Exact phase compensation for each stage of reactive low‐pass filtering requires an acausal filter whose coefficients are a weighted sum of zeroth‐ and first‐order modified Hankel functions, the weighting being determined by the integrator’s time constant. A very short and useful approximation to the ideal phase‐compensating filter is also described and subsequently applied in the restoration of data obtained from an airborne electromagnetic...