Prediction Error Filter Research Articles

White-noise estimators for seismic data processing in oil exploration

This paper is motivated by a problem from seismic data processing in oil exploration. We develop a Kalman filtering approach to obtaining optimal smoothed estimates of the so-called reflection coefficient sequence. This sequence contains important information about subsurface geometry. Our problem is shown to be equivalent to that of estimating white-plant noise for a linear dynamic system. By means of the equations which are derived herein, it is possible to compute fixed-interval, fixed-point, or fixed-lag optimal smoothed estimates of the reflection coefficient sequence, as well as respective error covariance-matrix information. Our optimal estimators are compared with an ad hoc error filter, (PEF) which has recently been reported on in the geophysics literature. We show that one of our estimators performs at least as well as, and in most cases, better than the prediction error filter. Simulation results are given which support and illustrate the theoretical developments.

Numerical estimation of the complex Wiener prediction filter

A complete algorithm is presented for numerical estimation of the complex prediction error filter from a finite number of data samples. The method is a generalization of Andersen's results for real samples [3]. It is also demonstrated that, using Anderson's method, a nonincreasing mean error as a function of filter size cannot be arbitrarily guaranteed.

Maximum entropy spectral analysis and autoregressive decomposition

The duality between the maximum entropy method (MEM) of spectral analysis and the autoregressive (AR) representation of the data allows the application of recent advances in AR analysis to MEM in an attempt to obviate some shortcomings in this method of spectral decomposition. Specifically, this paper investigates the work of Akaike (1969a, b) on a criterion for choosing the length of the required prediction error filter and compares two methods of determining the filter coefficients. Recent work by Kromer (1970) on asymptotic properties of the AR spectral estimator is also of importance. Some preliminary results of the splitting of the normal modes of the earth are presented as an illustration of the application of MEM to geophysics.

A comparison of power spectral estimates and applications of the maximum entropy method

A new spectral estimate, called the maximum entropy method, is described. This estimate was originated by John Parker Burg for use in seismic wave analysis. In the maximum entropy method the entropy, or information, of a signal is maximized under the constraint that the estimated autocorrelation function of the signal is the Fourier transform of the spectral power density. The spectral estimates are calculated in two ways: (1) by minimization of the error power to obtain the coefficients of the prediction error filter, as suggested by Burg, and (2) by a direct solution of the matrix equation using an algorithm due to Norman Levinson. For comparison a Blackman-Tukey technique, calculated with a Hamming window, is used also. We illustrate these three methods by applying them to a composite signal consisting of four sinusoids of unit amplitude: one each at high and low frequencies and two at moderate frequencies with respect to the Nyquist frequency, to which is added white noise of 0.5 amplitude. Results are shown to indicate that the best correspondence with the input spectrum is provided by the Burg technique. Applications of the maximum entropy method to geomagnetic micropulsations reveal complex multiplet structure in the Pc 4, 5 range. Such structure, not previously resolved by conventional techniques, has been predicted by a recent theory of magnetospheric resonances. In a period range 7 orders of magnitude longer than micropulsation periods, analysis of annual sunspot means shows that the 11-yr band is composed of at least three distinct lines. With each of these lines is associated a harmonic sequence. Long periods of the order of 100 yr also are revealed.

Experiments with maximum entropy power spectra of sinusoids

Burg's maximum entropy technique is applied to the estimation of power spectra of short sinusoids. The estimated spectral maximum was found to shift in relation to the true maximum. The frequency shifts depend upon the initial phase and the length of the sinusoid. The influence of the number of terms in the prediction error filter, the number of the data samples, and the noise intensity are also considered.

PREDICTION ERROR FILTERS, WHITE NOISE AND ORTHOGONAL COORDINATES*

AbstractOptimum filters can be computed using orthogonal coordinates obtained from the eigenvalues and eigenvectors of the autocorrelation matrix. The method is used to obtain unit distance prediction error filters. The output of a unit distance prediction error filter when applied to the input wavelet is an impulse at zero time. The effect on the output of added white noise is easily obtained using the approach through the orthogonal coordinates. The added white noise results in output wavelets which are no longer impulses at zero time. The decrease in time resolution gives a filter that does not increase undesirable high frequency noise as much as filters computed without white noise.Orthogonal coordinates with little signal energy can be omitted from the filter computation resulting in output wavelets resembling those computed using added white noise.

PREDICTION ERROR AS A CRITERION FOR OPERATOR LENGTH

Prediction error filtering has been widely used for deconvolution. The mean squared error in prediction is a monotonically nonincreasing function of operator length, and the value of the error is readily available from the Wiener‐Levinson algorithm. In general, the value of this error for the infinitely long operator is not known a priori. It is shown that the final value of the error can be obtained by considering the Kolmogorov spectrum factorization. Simple criteria can then be established for operator effectiveness and length.

Speech Enhancement Based on Linear Prediction and Correlation-Inputting Bias Free Equation Error ADF

In this paper, a speech enhancement technique to reduce background noise in noisy speech is proposed. We investigated the noise reconstruction system (NRS) based on linear prediction and system identification as a speech enhancement. Assuming that the background noise is generated from white noise by exciting a linear filter, the system identification estimates the background noise from estimated white noise. However, the white noise estimated by a linear prediction error filter (LPEF) includes residual speech, then the estimation accuracy of background noise is degraded at the system identification and the quality of enhanced speech is deteriorated. In order to reduce the influence of the residual speech, a lattice filter and a bias free equation error adaptive digital filter (ADF) are respectively introduced to the LPEF and system identification. The residual speech is reduced by the lattice filter which approximates a vocal-tract filter well. On the other hand, the bias free equation error ADF uses the cross-correlation between the whitened noise and a desired signal as a tap input. Since the speech does not have the correlation from the desired signal, the tap coefficients converge without the influence of speech.