On: “The Complex Wiener Filter” by Sven Treitel (GEOPHYSICS, April 1974, p. 169–173)

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 358-359
Author(s):  
A. J. Berkhout
Keyword(s):  
Least Squares ◽  
Autocorrelation Function ◽  
Input Signal ◽  
Solution Method ◽  
Wiener Filter ◽  
Sampling Interval ◽  
Inverse Filtering ◽  
Normal Equations ◽  
Matrix Notation

In this publication Treitel determines the complex normal equations and proposes a solution method (Robinson’s block‐Toeplitz algorithm). With respect to these results, I would like to draw attention to the paper Berkhout (1973). In Appendix B of that paper the complex normal equations were also derived: [Formula: see text]for n = 0, 1, ⋯ , N. In (1a), [Formula: see text] is the autocorrelation function of a complex input signal x(t), [Formula: see text] is the complex least‐squares filter of duration (N + 1)Δt, [Formula: see text] is the complex desired output, and Δt represents the sampling interval. Note that in matrix notation expression (1a) equals expression (3) of Treitel’s paper. In the case of least‐squares inverse filtering, (1a) takes on a different form.

Accuracy of the linear filter coefficients determined by the iteration of the least‐squares method

Geophysics ◽  
1982 ◽  
Vol 47 (2) ◽  
pp. 244-256 ◽  
Author(s):  
Yutaka Murakami ◽  
Toshihiro Uchida
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Wiener Filter ◽  
Sampling Interval ◽  
Filter Method ◽  
Linear Filter ◽  
Rapid Decay ◽  
Convolution Integrals ◽  
Two Factors

The linear filter method is a powerful tool for the estimation of some convolution integrals which are encountered in many aspects of geophysical problems. The accuracy of the method is affected by two factors, apart from the sampling interval. Because of the slowly decaying oscillations in the filter tail, the tail must be truncated at some point. Because of the extensive numerical calculations required for the determination of the filter coefficients, the resultant filter cannot be free from round‐off errors. The Wiener filter analogy, pointed out by Koefoed and Dirks (1979), gives a straightforward and efficient answer to this problem. We show that the Wiener filter technique with the iterative application of the Levinson’s algorithm not only enhances the accuracy of the filter coefficients, but also greatly reduces the oscillations in the filter tail. Because of the very rapid decay of the filter tail, there is no need to truncate it.

LEAST‐SQUARES INVERSE FILTERING

Geophysics ◽  
1966 ◽  
Vol 31 (5) ◽  
pp. 917-926 ◽  
Author(s):  
Wayne T. Ford ◽  
James H. Hearne
Keyword(s):  
Least Squares ◽  
Autocorrelation Function ◽  
Scale Factor ◽  
Inverse Filtering ◽  
Sampled Signal ◽  
The Matrix ◽  
Minimum Delay ◽  
The Given ◽  
Unknown Signal

Suppose we are given the autocorrelation function of a certain unknown sampled signal. Although a number of different signals might produce the given autocorrelation function, only one of these is minimum‐delay. Denoting this minimum‐delay unknown signal by the matrix K, the given autocorrelation may be written in the form K′K where K′ is the transpose of K. It is desired to determine approximately the inverse of this unknown signal K; that is, we wish to determine a vector X so that KX is as close to B′=(1, 0, 0, ⋯, 0) as possible in a least‐squares sense. If K were known, Rice shows that [Formula: see text] At first glance, the above formula appears useless as K′ is unknown. However, although K′ is indeed unknown, K′B has the form K′B=(c, 0, 0, ⋯, 0)′ where the scalar c simply plays the role of a scale factor. Thus, we determine X by simply selecting a convenient multiple of the first column of the inverse of the known matrix K′K. Although we do not present the details of the computer programming involved in the above calculation, we do present some simple examples to illustrate the process.

THE REALIZATION OF THE ITERATIVE METHOD OF THE LEAST SQUARES FOR THE ESTIMATION OF STATIC OBJECT PARAMETERS IN MATLAB ENVIRONMENT

2017 ◽  
pp. 28-36
Author(s):  
Galina Vasil’evna Troshina ◽  
Alexander Aleksandrovich Voevoda
Keyword(s):  
Parameter Estimation ◽  
Iterative Method ◽  
Least Squares ◽  
Input Signal ◽  
Layer Structure ◽  
Estimation Procedure ◽  
Parameter Estimation Procedure ◽  
Noise Simulation ◽  
Static Object ◽  
Object Parameters

It was suggested to use the system model working in real time for an iterative method of the parameter estimation. It gives the chance to select a suitable input signal, and also to carry out the setup of the object parameters. The object modeling for a case when the system isn't affected by the measurement noises, and also for a case when an object is under the gaussian noise was executed in the MatLab environment. The superposition of two meanders with different periods and single amplitude is used as an input signal. The model represents the three-layer structure in the MatLab environment. On the most upper layer there are units corresponding to the simulation of an input signal, directly the object, the unit of the noise simulation and the unit for the parameter estimation. The second and the third layers correspond to the simulation of the iterative method of the least squares. The diagrams of the input and the output signals in the absence of noise and in the presence of noise are shown. The results of parameter estimation of a static object are given. According to the results of modeling, the algorithm works well even in the presence of significant measurement noise. To verify the correctness of the work of an algorithm the auxiliary computations have been performed and the diagrams of the gain behavior amount which is used in the parameter estimation procedure have been constructed. The entry conditions which are necessary for the work of an iterative method of the least squares are specified. The understanding of this algorithm functioning principles is a basis for its subsequent use for the parameter estimation of the multi-channel dynamic objects.

The Estimation of Aquifer Properties From Reservoir Performance in Water-Drive Fields

1977 ◽  
Vol 99 (2) ◽  
pp. 345-352 ◽  
Author(s):  
A. T. Chatas
Keyword(s):  
Least Squares ◽  
Material Balance ◽  
Dimensionless Form ◽  
Simultaneous Solution ◽  
Method Of Least Squares ◽  
Normal Equations ◽  
Aquifer Parameters ◽  
Reservoir Performance ◽  
Analytical Functions ◽  
Aquifer Properties

The purpose of this paper is to indicate a method for estimating values of specified aquifer parameters from an investigation of the reservoir performance of an associated oilfield. To achieve this objective an analysis was made of the simultaneous solution of the material-balance and diffusivity equations, followed by an application of the method of least squares. Three analytical functions evolved, which in dimensionless form were numerically evaluated by computer and tabulated herein. Application of the proffered method requires the simultaneous solution of the three normal equations developed in the paper.

Linear least squares estimation: method of normal equations

2009 ◽  
pp. 99-120
Author(s):  
John M. Lewis ◽  
S. Lakshmivarahan ◽  
Sudarshan Dhall
Keyword(s):  
Least Squares ◽  
Estimation Method ◽  
Least Squares Estimation ◽  
Linear Least Squares ◽  
Normal Equations

II.—On Fitting Polynomials to Data with Weighted and Correlated Errors

1935 ◽  
Vol 54 ◽  
pp. 12-16 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Positive Definite ◽  
Correlated Errors ◽  
Positive Definite Quadratic Form ◽  
Matrix Notation ◽  
Image Position

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

The Solution of Linear Equation Systems

1965 ◽  
Vol 19 (1) ◽  
pp. 78-83
Author(s):  
Peter Wilson
Keyword(s):  
Linear Equation ◽  
Least Squares ◽  
Covariance Matrix ◽  
Error Propagation ◽  
Normal Equations ◽  
The Law ◽  
Linear Equation Systems ◽  
Variance Covariance Matrix

Several methods for solving normal equations in least squares solutions are explained and the variance-covariance matrix is developed from the law of error propagation.

Linear Least-Squares Computations Using Givens Transformations

1983 ◽  
Vol 37 (4) ◽  
pp. 225-233 ◽  
Author(s):  
J. A. R. Blais
Keyword(s):  
Least Squares ◽  
Error Estimates ◽  
Data Storage ◽  
Computational Efficiency ◽  
Numerical Stability ◽  
Direct Method ◽  
Linear Least Squares ◽  
Normal Equations ◽  
Estimation Problems ◽  
A Direct Method

Givens transformations provide a direct method for solving linear least-squares estimation problems without forming the normal equations. This approach has been shown to be particularly advantageous in recursive situations because of characteristics related to data storage requirements, numerical stability and computational efficiency. The following discussion will concentrate on the problem of updating least-squares parameter and error estimates using Givens transformations. Special attention will be given to photogrammetric and geodetic applications.

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