Tad J. Ulrych

Time series modelling and maximum entropy

This paper briefly reviews the principles of maximum entropy spectral analysis and the closely related problem of autoregressive time series modelling. The important aspect of model identification is discussed with particular emphasis on the representation of harmonic processes with noise in terms of autoregressive moving-average models. It is shown that this representation leads to a spectral estimator proposed by Pisarenko in 1973.

Phase estimation using the bispectrum

The information which is of importance in the reflection seismic method resides in the reflectivity series. In order to extract this information about the subsurface, the blurring effect of the seismic wavelet must first be removed. Since this signature is generally unknown, various wavelet estimation schemes have been developed. The one most currently used in the seismic industry is based on the assumption that the seismic wavelet has the minimum-phase property. This restrictive assumption is often incorrect. The purpose of this paper is to explore the use of the bispectrum in order to obviate the minimum-phase requirement. Specifically, using synthetic examples, we develop and compare three different algorithms which determine the wavelet phase from the bispectrum of the reflection seismogram. An important aspect of the problem not treated before is the application of the bispectral technique to band-limited data.

Application of singular value decomposition to vertical seismic profiling

An essential part of the interpretation of vertical seismic profiles (VSP) is the separation of the upgoing and downgoing waves. This paper presents a new approach which is based on the decomposition of time-shifted VSP sections into eigenimages, using singular value decomposition (SVD). The first few eigenimages of the time-shifted VSP section contain the contributions of the horizontally aligned downgoing waves. The last few eigenimages contain the contribution of uncorrelated noise components. The separated upgoing waves are recovered as a partial sum of the eigenimages. !mportant aspects of this approach are that regular sampling of the recording levels is not required, that the first-break times need not be measured with extreme accuracy, that noise rejection may be automatically included in the processing, and that eigenimages or sums of eigenimages which may be computed as part of the approach can provide important additional information.

Minimum relative entropy: Forward probabilistic modeling

The pioneering work of Jaynes in Bayesian/maximum entropy methods has been successfully explored in many disciplines. The principle of maximum entropy (PME) is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution. Minimum relative entropy (MRE) is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included. In this paper we use MRE to determine the prior probability density function (pdf) of a set of model parameters based on limited information. The resulting pdf is used in Monte Carlo simulations to provide expected values in field variables such as concentration, and confidence limits. We compare the probabilistic results from a traditional advection-dispersion (ADE) model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process. The results suggest that although Naffs (1990) model satisfies the observed data to a better degree than ADE model, the upper and lower confidence bands about the mean value are larger than the ADE results. This result we attribute to the fact that Naffs (1990) model simply contains more parameters, each of which is unknown and has to be estimated. There is no statistical difference between the expected values of second-spatial moments for the two models. The examples presented in this paper illustrate problems associated with assigning Gaussian pdfs as priors in a probabilistic model. First, such an assumption for the input parameters actually injects more information into the problem than may actually exist, whether consciously or unconsciously. This fact is borne out by comparison with minimum relative entropy theory. Second, the output parameters as suggested from the Monte Carlo analysis cannot be assumed to be Gaussian distributed even when the prior pdf is Gaussian in form. In a practical setting, the significance of this result and the approximation of Gaussian form would depend on the toxicity and environmental standards that apply to the problem.

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.

HIGH RESOLUTION 2-DIMENSIONAL POWER SPECTRAL ESTIMATION

This paper presents an investigation of a variety of high resolution spectral estimators. We explore different prediction geometries as possible approximations to the 2-D maximum entropy spectrum and we develop a 2-D estimator which is determined from sets of 1-D maximum entropy estimators. Of particular interest is the development of a least-squares estimate of the 2-D autocorrelation matrix which we show has advantageous resolution properties for short realizations of noisy harmonic processes.

GPR studies of pyroclastic deposits: Subsurface information where there are no outcrops

We have collected ground penetrating radar (GPR) data to evaluate its applications and limitations for obtaining subsurface information about pyroclastic deposits where there are no exposures. The surveys were conducted mainly in the Ubehebe hydro-volcanic field (Death Valley National Park, California, U.S.) that is composed of well preserved deposits left by base surges—turbulent flows generated by explosive interaction between magma and water. Base surges have high (i.e., magmatic) temperatures and move horizontally on the surface of the ground with hurricane velocity, causing extensive damage and loss of life. The main vent in this area, Ubehebe Crater, is a spectacular feature about 700–800 m in diameter and 235 m deep. Other, much smaller, tuff rings are south and west of the main crater.

Discrete convolution by means of forward and backward modeling

An approach to discrete convolution is presented which obviates the zero assumption. The method is structurally similar to the method of J.P. Burg (Stanford Univ., May 1975), which estimates the autocorrelation coefficients of a series in a manner which does not require a predefinition of the behavior of the signal outside of the known interval. The basic principle of the present approach is that each term of the convolution is recursively determined from previous terms in a manner consistent with the optimal modeling of one signal in terms of the other. The recursion uses forward and backward modeling together with the algorithm of M. Morf et al. (ibid., vol.ASSP-25, p.429-43, Oct. 1977) for computation of the prediction error filter. The method is illustrated by application to the computation of the analytic signal and its derivative. >

Multiple dynamic matching: a new approach to well log correlation

Abstract A new method of nonlinear correlation or matching is presented which overcomes some of the difficulties of other techniques. ‘Multiple dynamic matching’ is based on a constrained optimization approach but contains some important innovations. Rather than finding only one solution, the present method generates a ‘function network’ of probable interpretations. These are then ranked by an objective function which quantifies an interpreters qualitative matching criteria through a similarity and simplicity measure. The top ranking matching function thus determined is used to correlate the zones which are structurally equivalent in sonic well logs, and an example of an automatic structural interpretation is given.

Compound median filtering applied to sonic logs

LOW pass filtering of sonic logs is often desired to remove high frequency measurement errors, to allow a valid comparison with a particular inversion scheme, and simply to make interpretation easier. However, because abrupt changes in velocity require high frequencies for their represenation, conventional low pass filtering tends to smear the important features of the signal. This has lead to non linear techniques such as median filtering, which however, can also degrade edges producing ‘edge artifacts’, especially for longer filter lengths. Recently the signal conditions necessary for the production of edge artifacts have been established (Leaney and Clrych 1987) and an extension to median filtering, compound median filtering, has been shown to preserve signal edges. In addition, compound median filtering has lead to a signal decomposition which allows greater flexibiltiy in the filtering operation. In this paper compound median filtering and decomposition are discussed and two applications of the decomposition, significant event reconstruction and time varying compound median filtering, are demonstrated as applied to sonic well log data.