Comparison of the least‐squares criterion and the Cauchy criterion in frequency‐wavenumber inversion

Geophysics ◽  
1991 ◽  
Vol 56 (12) ◽  
pp. 2027-2035 ◽  
Author(s):  
Lasse Amundsen
Keyword(s):  
Lower Bound ◽  
Least Squares ◽  
Scale Parameter ◽  
Least Squares Method ◽  
Random Noise ◽  
Synthetic Data ◽  
Error Criterion ◽  
Numerical Examples ◽  
Wavenumber Domain ◽  
Weather Noise

One alternative to the least‐squares inversion technique is the use of a Cauchy error criterion. We show how inversion algorithms of the Gauss‐Newton type based on the least‐squares method can be modified to handle the Cauchy norm. A criterion for the lower bound of the scale parameter in the Cauchy norm is given. We compare the least‐squares and Cauchy error criteria by inverting synthetic data corrupted by random noise and weather noise. The data are transformed to the frequency‐wavenumber domain before the inversion starts. The numerical examples show that the algorithm based on the Cauchy criterion is more robust in the presence of the noise tested here. Per iteration, the computer costs of the two algorithms are approximately the same.

Magnetic interpretation using a least-squares, depth-shape curves method

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. L23-L30 ◽  
Author(s):  
El-Sayed M. Abdelrahman ◽  
Khalid S. Essa
Keyword(s):  
Least Squares ◽  
Magnetic Anomaly ◽  
Least Squares Method ◽  
Random Noise ◽  
Synthetic Data ◽  
Theoretical Data ◽  
Magnetic Anomalies ◽  
Shape Factors ◽  
Characteristic Points ◽  
Horizontal Cylinders

We have developed a least-squares approach to depth determination from residual magnetic anomalies caused by simple geologic structures. By normalizing the residual magnetic anomaly using three characteristic points and their corresponding distances on the anomaly profile, the problem of determining depth from residual magnetic anomalies has been transformed into finding a solution to a nonlinear equation of the form z = f(z). Formulas have been derived for spheres, horizontal cylinders, thin dikes, and contacts. The method is applied to synthetic data with and without random noise. We have also developed a method using depth-shape curves to simultaneously define the shape and depth of a buried structure from a residual magnetic anomaly profile. The method is based on determining the depth from the normalized residual anomaly for each shape factor using the least-squares method mentioned above. The computed depths are plotted against the shape factors on a graph. The solution for the shape and depth of the buried structure is read at the common intersection of the depth-shape curves. The depth-shape curves method was successfully tested on theoretical data with and without random noise and applied to a known field example from Ontario.

scholarly journals Effective fault detection in structural health monitoring systems

2019 ◽  
Vol 11 (9) ◽  
pp. 168781401987323 ◽  
Author(s):  
Marwa Chaabane ◽  
Majdi Mansouri ◽  
Kamaleldin Abodayeh ◽  
Ahmed Ben Hamida ◽  
Hazem Nounou ◽  
...  
Keyword(s):  
Fault Detection ◽  
Least Squares ◽  
Partial Least Squares ◽  
Least Squares Method ◽  
Random Noise ◽  
Ratio Test ◽  
Multiscale Representation ◽  
Kernel Partial Least Squares ◽  
Partial Least Squares Model ◽  
Partial Least Squares Method

A new fault detection technique is considered in this article. It is based on kernel partial least squares, exponentially weighted moving average, and generalized likelihood ratio test. The developed approach aims to improve monitoring the structural systems. It consists of computing an optimal statistic that merges the current information and the previous one and gives more weight to the most recent information. To improve the performances of the developed kernel partial least squares model even further, multiscale representation of data will be used to develop a multiscale extension of this method. Multiscale representation is a powerful data analysis way that presents efficient separation of deterministic characteristics from random noise. Thus, multiscale kernel partial least squares method that combines the advantages of the kernel partial least squares method with those of multiscale representation will be developed to enhance the structural modeling performance. The effectiveness of the proposed approach is assessed using two examples: synthetic data and benchmark structure. The simulation study proves the efficiency of the developed technique over the classical detection approaches in terms of false alarm rate, missed detection rate, and detection speed.

scholarly journals Calibration for Parameter Estimation of Signals with Complex Noise via Nonstationarity Measure

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Zhiming Zhou ◽  
Zhengyun Zhou ◽  
Liang Wu
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Least Squares Method ◽  
Random Noise ◽  
Calibration Method ◽  
Numerical Studies ◽  
Chaotic Signals ◽  
Inaccurate Estimation ◽  
Non Gaussian ◽  
First Time

The signals in numerous complex systems of engineering can be regarded as nonlinear parameter trend with noise which is identically distributed random signals or deterministic stationary chaotic signals. The commonly used methods for parameter estimation of nonlinear trend in signals are mainly based on least squares. It can cause inaccurate estimation results when the noise is complex (such as non-Gaussian noise, strong noise, and chaotic noise). This paper proposes a calibration method for this issue in the case of single parameter via nonstationarity measure from the perspective of the stationarity of residual sequence. Some numerical studies are conducted for validation. Results of numerical studies show that the proposed calibration method performs well for various models with different noise strengths and types (including random noise and chaotic noise) and can significantly improve the accuracy of initial estimates obtained by least squares method. This is the first time that the nonstationarity measure is applied to the parameter calibration. All these results will be a guide for future studies of other parameter calibrations.

scholarly journals Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ting Ke ◽  
Guo Jiang ◽  
Mengting Deng
Keyword(s):  
Integral Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Least Squares ◽  
Algebraic Equation ◽  
Volterra Integral Equation ◽  
Linear Algebraic Equation ◽  
Least Squares Method ◽  
Numerical Examples ◽  
Block Pulse Function

In this paper, a method based on the least squares method and block pulse function is proposed to solve the multidimensional stochastic Itô-Volterra integral equation. The Itô-Volterra integral equation is transformed into a linear algebraic equation. Furthermore, the error analysis is given by the isometry property and Doob’s inequality. Numerical examples verify the effectiveness and precision of this method.

scholarly journals Approximate Solutions for a Class of Nonlinear Fredholm and Volterra Integro-Differential Equations Using the Polynomial Least Squares Method

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2692
Author(s):  
Bogdan Căruntu ◽  
Mădălina Sofia Paşca
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Analytical Solutions ◽  
General Class ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Test Problems ◽  
Numerical Examples ◽  
Approximate Analytical Solutions ◽  
Very High

We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its precision for this type of equations is very high, a fact that is illustrated by the numerical examples presented. The comparison with previous approximations computed for the included test problems emphasizes the method’s simplicity and accuracy.

scholarly journals Polynomial Least Squares Method for the Solution of Nonlinear Volterra-Fredholm Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Bogdan Căruntu ◽  
Constantin Bota
Keyword(s):  
Least Squares ◽  
Integral Equations ◽  
Least Squares Method ◽  
Fredholm Integral Equations ◽  
Nonlinear Integral Equations ◽  
Straightforward Manner ◽  
Fredholm Type ◽  
Numerical Examples ◽  
Polynomial Solutions ◽  
Approximate Polynomial

The present paper presents the application of the polynomial least squares method to nonlinear integral equations of the mixed Volterra-Fredholm type. For this type of equations, accurate approximate polynomial solutions are obtained in a straightforward manner and numerical examples are given to illustrate the validity and the applicability of the method. A comparison with previous results is also presented and it emphasizes the accuracy of the method.

Simultaneous multivintage time-shift estimation

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. V109-V121 ◽  
Author(s):  
Ehsan Zabihi Naeini ◽  
Henning Hoeber ◽  
Gordon Poole ◽  
Hamid R. Siahkoohi
Keyword(s):  
Continental Shelf ◽  
Least Squares ◽  
Least Squares Method ◽  
Synthetic Data ◽  
Time Lapse ◽  
Time Shift ◽  
Data Sets ◽  
Data Set ◽  
Time Alignment ◽  
Multiple Repeat

Time-shift estimation is a key step in seismic time-lapse processing as well as in many other signal-processing applications. We consider the time-shift problem in the setting of multiple repeat surveys that must be aligned consistently. We introduce an optimized least-squares method based on the Taylor expansion for estimating two-vintage time shifts and compare it to crosscorrelation. The superiority of the proposed algorithm is demonstrated with synthetic data and residual time-lapse matching on a U. K. continental shelf data set. We then discuss the shortcomings of cascaded time alignment in multiple repeat monitor surveys and propose an approach to estimate simultaneous multivintage time shifts that uses a constrained least-squares technique combined with elements of network theory. The resulting time shifts are consistent across all vintages in a least-squares sense, improving overall alignment when compared to the classical flow of alignment in a cascaded manner. The method surpasses the cascaded approach, as noted with sample synthetic and three-vintage U. K. continental shelf time-lapse data sets.

scholarly journals Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomial

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
D. Sh. Mohammed
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Least Squares ◽  
Chebyshev Polynomial ◽  
Least Squares Method ◽  
Numerical Examples ◽  
Shifted Chebyshev Polynomial ◽  
Theoretical Results

We investigate the numerical solution of linear fractional integro-differential equations by least squares method with aid of shifted Chebyshev polynomial. Some numerical examples are presented to illustrate the theoretical results.

Theoretical study on inversing the anisotropic permeability of VTI two-phase media by flexural waves in a fluid-filled borehole

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. D161-D173
Author(s):  
Fuli Xie ◽  
Weiguo Lv ◽  
Shouguo Yan ◽  
Kexie Wang ◽  
Bixing Zhang
Keyword(s):  
Least Squares Method ◽  
Random Noise ◽  
Synthetic Data ◽  
Simultaneous Estimation ◽  
Flexural Waves ◽  
Two Phase ◽  
Symmetry Principle ◽  
Waveform Data ◽  
Vertical Permeability ◽  
Horizontal Permeability

We developed a theoretical method of estimating the vertical and horizontal permeability of a transversely isotropic two-phase formation with a vertical symmetry principle axis (VTI) by using the attenuation characteristic of the dipole-flexural waves in a fluid-filled borehole. The attenuation value of flexural waves was extracted from synthetic data. The inversion was operated by the least-squares method. Two main conclusions were obtained based on the analysis of the results of inversion implemented on multiple sets of synthetic waveform data. First, the simultaneous estimation of vertical and horizontal permeability was feasible for the slow VTI formation, which means its shear-wave velocity is slower than the acoustic velocity of fluid in a borehole that is different from the previous understanding in some studies. Second, the fast formation corresponded to the opposite case from the slow formation, the precision of the estimated horizontal permeability relates to the accuracy of the vertical permeability, but the vertical permeability is difficult to estimate by the attenuation of the dipole-flexural waves. We added random noise with different intensities to synthetic data in order to simulate the errors that might exist in actual logging data. We adopted Prony’s method to extract attenuation to eliminate the effect of noise to a certain degree.

Three least‐squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data

Geophysics ◽  
2001 ◽  
Vol 66 (4) ◽  
pp. 1105-1109 ◽  
Author(s):  
E. M. Abdelrahman ◽  
H. M. El‐Araby ◽  
T. M. El‐Araby ◽  
E. R. Abo‐Ezz
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Least Squares Method ◽  
Gravity Data ◽  
Synthetic Data ◽  
Gravity Anomalies ◽  
Maximum Error ◽  
The United States ◽  
Random Errors ◽  
Amplitude Coefficient

Three different least‐squares approaches are developed to determine, successively, the depth, shape (shape factor), and amplitude coefficient related to the radius and density contrast of a buried structure from the residual gravity anomaly. By defining the anomaly value g(max) at the origin on the profile, the problem of depth determination is transformed into the problem of solving a nonlinear equation, [Formula: see text]. Formulas are derived for spheres and cylinders. Knowing the depth and applying the least‐squares method, the shape factor and the amplitude coefficient are determined using two simple linear equations. In this way, the depth, shape, and amplitude coefficient are determined individually from all observed gravity data. A procedure is developed for automated interpretation of gravity anomalies attributable to simple geometrical causative sources. The method is applied to synthetic data with and without random errors. In all the cases examined, the maximum error in depth, shape, and amplitude coefficient is 3%, 1.5%, and 7%, respectively. Finally, the method is tested on a field example from the United States, and the depth and shape obtained by the present method are compared with those obtained from drilling and seismic information and with those published in the literature.

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