LEAST SQUARES POLYNOMIAL FITTING TO GRAVITATIONAL DATA AND DENSITY PLOTTING BY DIGITAL COMPUTERS

Geophysics ◽  
1954 ◽  
Vol 19 (2) ◽  
pp. 255-269 ◽  
Author(s):  
Stephen M. Simpson
Keyword(s):  
Least Squares ◽  
Digital Computer ◽  
Gravity Data ◽  
Data Distribution ◽  
Polynomial Fitting ◽  
Normal Equations ◽  
Order Polynomial ◽  
Qualitative Picture ◽  
Residual Maps ◽  
Digital Computers

The fitting of a nth order polynomial in x and y to gravity data by least squares is discussed. A consideration of the normal equations for the general case shows certain simplifications resulting from rectangularity in data distribution. Some sample residual maps are constructed. Density plotting, made possible by the digital computer, is described and illustrated. It is shown that this process can serve as a substitute for contouring when only a qualitative picture is desired.

CORRECTED FIGURE FROM “ LEAST SQUARES POLYNOMIAL FITTING TO GRAVITATIONAL DATA AND DENSITY PLOTTING BY DIGITAL COMPUTERS”

Geophysics ◽  
1954 ◽  
Vol 19 (3) ◽  
pp. 644-644 ◽  
Keyword(s):  
Least Squares ◽  
Polynomial Fitting ◽  
Corrected Figure ◽  
Digital Computers

CORRECTED FIGURE FROM “ LEAST SQUARES POLYNOMIAL FITTING TO GRAVITATIONAL DATA AND DENSITY PLOTTING BY DIGITAL COMPUTERS” by Stephen M. Simpson, Jr. (GEOPHYSICS, Volume X IX, April, 1954, Page 268)

scholarly journals High-Order Polynomial Fitting Assistance for Fast Double-Peak Finding in Brillouin-Distributed Sensing

Sensors ◽  
2020 ◽  
Vol 21 (1) ◽  
pp. 187
Author(s):  
Marcelo A. Soto ◽  
Alin Jderu ◽  
Dorel Dorobantu ◽  
Marius Enachescu ◽  
Dominik Ziegler
Keyword(s):  
Optical Fibers ◽  
High Order ◽  
Polynomial Fitting ◽  
Gaussian Fitting ◽  
Peak Finding ◽  
Fitting Method ◽  
Fold Reduction ◽  
Order Polynomial ◽  
Fitting In ◽  
Distributed Brillouin Sensor

A high-order polynomial fitting method is proposed to accelerate the computation of double-Gaussian fitting in the retrieval of the Brillouin frequency shifts (BFS) in optical fibers showing two local Brillouin peaks. The method is experimentally validated in a distributed Brillouin sensor under different signal-to noise ratios and realistic spectral scenarios. Results verify that a sixth-order polynomial fitting can provide a reliable initial estimation of the dual local BFS values, which can be subsequently used as initial parameters of a nonlinear double-Gaussian fitting. The method demonstrates a 4.9-fold reduction in the number of iterations required by double-Gaussian fitting and a 3.4-fold improvement in processing time.

scholarly journals Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test area

2012 ◽  
Vol 2 (1) ◽  
pp. 53-64 ◽  
Author(s):  
H. Yildiz ◽  
R. Forsberg ◽  
J. Ågren ◽  
C. Tscherning ◽  
L. Sjöberg
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Test Area ◽  
Geoid Determination ◽  
Geopotential Model ◽  
Low Degree ◽  
Residual Terrain Modelling ◽  
The Alps ◽  
Stokes Formula ◽  
Regional Geoid Determination

Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test areaThe remove-compute-restore (RCR) technique for regional geoid determination implies that both topography and low-degree global geopotential model signals are removed before computation and restored after Stokes' integration or Least Squares Collocation (LSC) solution. The Least Squares Modification of Stokes' Formula (LSMS) technique not requiring gravity reductions is implemented here with a Residual Terrain Modelling based interpolation of gravity data. The 2-D Spherical Fast Fourier Transform (FFT) and the LSC methods applying the RCR technique and the LSMS method are tested over the Auvergne test area. All methods showed a reasonable agreement with GPS-levelling data, in the order of a 3-3.5 cm in the central region having relatively smooth topography, which is consistent with the accuracies of GPS and levelling. When a 1-parameter fit is used, the FFT method using kernel modification performs best with 3.0 cm r.m.s difference with GPS-levelling while the LSMS method gives the best agreement with GPS-levelling with 2.4 cm r.m.s after a 4-parameter fit is used. However, the quasi-geoid models derived using two techniques differed from each other up to 33 cm in the high mountains near the Alps. Comparison of quasi-geoid models with EGM2008 showed that the LSMS method agreed best in term of r.m.s.

Polynomial Moving Fitting Method for Edge Identification

2011 ◽  
Vol 308-310 ◽  
pp. 2560-2564 ◽  
Author(s):  
Xiang Rong Yuan
Keyword(s):  
Edge Detection ◽  
Curve Fitting ◽  
Polynomial Function ◽  
High Order ◽  
Polynomial Fitting ◽  
Fitting Method ◽  
Order Polynomial ◽  
Better Than ◽  
Low Order

A moving fitting method for edge detection is proposed in this work. Polynomial function is used for the curve fitting of the column of pixels near the edge. Proposed method is compared with polynomial fitting method without sub-segment. The comparison shows that even with low order polynomial, the effects of moving fitting are significantly better than that with high order polynomial fitting without sub-segment.

Highly efficient density inversion of gravity data using nonlinear density polynomial fitting

Geophysics ◽  
2021 ◽  
pp. 1-34
Author(s):  
Guoqing Ma ◽  
Zongrui Li ◽  
Lili Li ◽  
Taihan Wang
Keyword(s):  
Weighting Function ◽  
Gravity Data ◽  
Real Life ◽  
Density Variation ◽  
Density Change ◽  
Inversion Method ◽  
Density Inversion ◽  
Perspective View ◽  
Polynomial Fitting ◽  
Polynomial Coefficients

The density inversion of gravity data is commonly achieved by discretizing the subsurface into prismatic cells and calculating the density of each cell. During this process, a weighting function is introduced to the iterative computation to reduce the skin effect during the inversion. Thus, the computation process requires a significant number of matrix operations, which results in low computational efficiency. We have adopted a density inversion method with nonlinear polynomial fitting (NPF) that uses a polynomial to represent the density variation of prismatic cells in a certain space. The computation of each cell is substituted by the computation of the nonlinear polynomial coefficients. Consequently, the efficiency of the inversion is significantly improved because the number of nonlinear polynomial coefficients is less than the number of cells used. Moreover, because representing the density change of all of the cells poses a significant challenge when the cell number is large, we adopt the use of a polynomial to represent the density change of a subregion with fewer cells and multiple nonlinear polynomials to represent the density changes of all prism cells. Using theoretical model tests, we determine that the NPF method more efficiently recovers the density distribution of gravity data compared with conventional density inversion methods. In addition, the density variation of a subregion with 8 × 8 × 8 prismatic cells can be accurately and efficiently obtained using our cubic NPF method, which can also be used for noisy data. Finally, the NPF method was applied to real gravity data in an iron mining area in Shandong Province, China. Convergent results of a 3D perspective view and the distribution of the iron ore bodies were acquired using this method, demonstrating the real-life applicability of this method.

A Least-Squares Method for Computing Balance Corrections

1964 ◽  
Vol 86 (3) ◽  
pp. 273-277 ◽  
Author(s):  
Thomas P. Goodman
Keyword(s):  
Least Squares ◽  
Digital Computer ◽  
Least Squares Method ◽  
Weighted Least Squares ◽  
Rotating Machinery ◽  
Residual Vibration ◽  
Computing Procedure ◽  
Balance Corrections

To compute final correction masses for multispeed, multiplane balancing of rotating machinery, a least-squares computing procedure has been developed. This procedure uses plain least squares to minimize the rms residual vibration of selected points on the machinery foundation, and then uses weighted least squares to reduce the maximum residual vibration. The computations have been programmed for a digital computer.

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.

THE USE OF LINEAR PROGRAMMING TO FILTER DIGITIZED MAP DATA

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 253-259 ◽  
Author(s):  
E. L. Dougherty ◽  
S. T. Smith
Keyword(s):  
Linear Programming ◽  
Least Squares ◽  
Error Detection ◽  
Linear Programming Problem ◽  
Mineral Resources ◽  
Least Squares Approximation ◽  
Data Errors ◽  
Order Polynomial ◽  
Data Points ◽  
Map Data

The procedure used to discover subsurface formations where mineral resources may exist normally requires the accumulation and processing of large amounts of data concerning the earth’s fields. Data errors may strongly affect the conclusions drawn from the analysis. Thus, a method of checking for errors is essential. Since the field should be relatively smooth locally, a typical approach is to fit the data to a surface described by a low‐order polynomial. Deviations of data points from this surface can then be used to detect errors. Frequently a least‐squares approximation is used to determine the surface, but results could be misleading. Linear programming can be applied to give more satisfactory results. In this approach, the sum of the absolute values of the deviations is minimized rather than the squares of the deviations as in least squares. This paper describes in detail the formulation of the linear programming problem and cites an example of its application to error detection. Through this formulation, once errors are removed, the results are meaningful physically and, hence, can be used for detecting subsurface phenomena directly.

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