On: “A Method of Computing Residual Anomalies from Bouguer Gravity Map by Applying Relaxation Technique” by M. K. Paul (GEOPHYSICS, vol. 32, no. 4, p. 708–719, August, 1967)

Geophysics ◽  
1970 ◽  
Vol 35 (2) ◽  
pp. 357-357
Author(s):  
H. A. Meinardus
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Relaxation Technique ◽  
Bouguer Gravity ◽  
Entire Region ◽  
Degree Polynomial ◽  
Entire Area ◽  
Bouguer Map ◽  
Polynomial Formula ◽  
Gravity Map

On page 711 the author, after reference to previous users of the least‐squares method for estimating residuals in a Bouguer gravity map, states, “All of them have used the method for estimating the residual field over the entire area under consideration, while in this case the method will be applied to obtain the same on the boundaries only.” He then proceeds to compute residuals on the boundaries from the second degree polynomial [Formula: see text], (10) representing the regional field over the entire region. However, by this procedure the residuals on the boundaries are influenced by all the gravity observations inside the region, as implied by equation (16) where the vector A is a function of the Bouguer map values over the whole area. In fact, equation (12) could be solved for the vector b, and the condition [Formula: see text] arising from [Formula: see text] (A) could be introduced. The following expression for the regional over the entire area results: [Formula: see text], and there is no need for additional computations by the relaxation technique described.

On: “A Method of Computing Residual Anomalies from Bouguer Gravity Map by Relaxation Technique” by Dr. M. K. Paul (GEOPHYSICS, August 1967, p. 708–719)

Geophysics ◽  
1968 ◽  
Vol 33 (3) ◽  
pp. 527-527
Author(s):  
P. S. Naidu
Keyword(s):  
Least Squares ◽  
Boundary Curve ◽  
Closed Curve ◽  
Relaxation Technique ◽  
Bouguer Gravity ◽  
Residual Field ◽  
Gravity Map ◽  
Anomaly Region

Dr. M. K. Paul has presented a very interesting method of separating the residual and regional fields from a Bouguer gravity map. In essence, his method is the estimation of the residual field along a closed curve enclosing, but away from, the anomaly by the usual least‐squares technique. Next he extrapolates the residual field on the boundary curve into the anomaly region by a relaxation technique.

A METHOD OF COMPUTING RESIDUAL ANOMALIES FROM BOUGUER GRAVITY MAP BY APPLYING RELAXATION TECHNIQUE

Geophysics ◽  
1967 ◽  
Vol 32 (4) ◽  
pp. 708-719 ◽  
Author(s):  
M. K. Paul
Keyword(s):  
Linear Function ◽  
Linear Equations ◽  
Point Of View ◽  
Relaxation Technique ◽  
Bouguer Gravity ◽  
Left Hand ◽  
Analytical Point ◽  
Residual Values ◽  
The Right ◽  
Gravity Map

A new method of computing residual anomalies for gravity prospecting data from a Bouguer gravity map has been evolved. In arriving at the proposed method, we have at first examined the behavior of the regional gravity field from an analytical point of view. With the concepts acquired therefrom in mind, we consider the case of square grids with such separation of stations that in an elementary area, formed by joining the four nearest stations around a central station, the regional field may be represented by a linear function of the Cartesian coordinates in the horizontal surface of observation. Making use of the formal relationship between the residual, regional, and Bouguer gravity values, we have been able to formulate in this case a set of simultaneous linear equations—one for each station of observation—with the residual values at the grid corners as the unknowns in the left hand sides of these equations and some linear function of the Bouger values at the grid corners as the known quantities in the right hand sides. With some plausible estimates of the residual values at the stations on the boundaries at hand, these equations can be solved efficiently with the aid of the relaxation technique as has been exemplified in the cases of theoretical model as well as field data.

On the least‐squares residual anomaly determination

Geophysics ◽  
1985 ◽  
Vol 50 (3) ◽  
pp. 473-480 ◽  
Author(s):  
E. M. Abdelrahman ◽  
S. Riad ◽  
E. Refai ◽  
Y. Amin
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Vertical Cylinder ◽  
Correlation Factor ◽  
Western Desert ◽  
Bouguer Gravity ◽  
Residual Component ◽  
Optimum Order ◽  
Gravity Interpretation ◽  
Residual Maps

This paper discusses an approach to determine the least‐squares optimum order of the regional surface which, when subtracted from the Bouguer gravity anomaly data, minimizes distortion of the residual component of the field. The least‐squares method was applied to theoretical composite gravity fields each consisting of a constant residual component (sphere or vertical cylinder) and a regional component of different order using successively increasing orders of polynomial regionals for residual determination. The overall similarity between each two successive residual maps was determined by computing the correlation factor between the mapped variables. Similarity between residual maps of the lowest orders, verified by good correlation, may generally be considered a criterion for determining the optimum order of the regional surface and consequently the least distorted residual component. The residual map of the lower order in this well‐correlated doublet is considered the most plausible one and may be used for gravity interpretation. This approach was successfully applied to the Bouguer gravity of Abu Roash dome, located west of Cairo in the Western Desert of Egypt.

On: “Method of Computing Residual Anomalies from Bouguer Gravity Map by Applying Relaxation Technique” by M. K. Paul (GEOPHYSICS, August 1967, p. 708–719)

Geophysics ◽  
1970 ◽  
Vol 35 (1) ◽  
pp. 160-161
Author(s):  
N. F. Uren
Keyword(s):  
Gravity Field ◽  
Theoretical Basis ◽  
Two Dimensions ◽  
Relaxation Technique ◽  
Bouguer Gravity ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Regional Gravity ◽  
Gravity Map

Dr. M. K. Paul suggests that further investigation of this method may be necessary. The theoretical basis for this method is that [Formula: see text] i.e., that the regional gravity field obeys Laplace’s equation in two dimensions over the plane of observation.

On: “A Method of Computing Residual Anomalies from Bouguer Gravity Map by Applying Relaxation Technique,” by M. K. Paul (GEOPHYSICS, August 1967, p. 708–719)

Geophysics ◽  
1969 ◽  
Vol 34 (3) ◽  
pp. 480-483
Author(s):  
K. Biesheuvel
Keyword(s):  
Digital Computer ◽  
High Speed ◽  
New Method ◽  
Relaxation Technique ◽  
Bouguer Gravity ◽  
Gravity Map

The following comments on this new method indicate some limitations of the technique as well as a more practical means of solving the equations involved by high‐speed digital computer.

Fitting of Reference Surfaces for Engineering Surfaces by Nonlinear Least-Squares Technique

2006 ◽  
Vol 6 (4) ◽  
pp. 349-354 ◽  
Author(s):  
Jagadeesha Kumar ◽  
M. S. Shunmugam
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Basis Functions ◽  
Reference Surface ◽  
Degree Polynomial ◽  
Simultaneous Separation ◽  
Sinusoidal Functions ◽  
Nonlinear Least Squares Method ◽  
First Time

Engineering surfaces comprise form and waviness errors, which are separated from the measured surface by establishing a reference surface that represents these errors. An attempt is made for the first time to fit a reference surface for simultaneous separation of form and waviness errors. A second-degree polynomial and a set of sinusoidal functions are taken as basis functions to represent form and waviness, respectively, and fitting is done using a nonlinear least-squares method. Different examples of surfaces are considered and a comparison is also made with 3D Gaussian filter to bring out the nature of reference surfaces obtained by the present fitting approach.

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method

An Augmented Iteratively Re-weighted and Refined Least Squares Method for lp Minimization Problem in Inverse Modeling of Potential Field Magnetic Data

2020 ◽  
Vol 1 (3) ◽  
Author(s):  
Maysam Abedi
Keyword(s):  
Magnetic Field ◽  
Magnetic Susceptibility ◽  
Least Squares ◽  
Minimization Problem ◽  
Potential Field ◽  
Field Data ◽  
Least Squares Method ◽  
Porphyry Copper ◽  
Magnetic Data ◽  
Magnetic Field Data

The presented work examines application of an Augmented Iteratively Re-weighted and Refined Least Squares method (AIRRLS) to construct a 3D magnetic susceptibility property from potential field magnetic anomalies. This algorithm replaces an lp minimization problem by a sequence of weighted linear systems in which the retrieved magnetic susceptibility model is successively converged to an optimum solution, while the regularization parameter is the stopping iteration numbers. To avoid the natural tendency of causative magnetic sources to concentrate at shallow depth, a prior depth weighting function is incorporated in the original formulation of the objective function. The speed of lp minimization problem is increased by inserting a pre-conditioner conjugate gradient method (PCCG) to solve the central system of equation in cases of large scale magnetic field data. It is assumed that there is no remanent magnetization since this study focuses on inversion of a geological structure with low magnetic susceptibility property. The method is applied on a multi-source noise-corrupted synthetic magnetic field data to demonstrate its suitability for 3D inversion, and then is applied to a real data pertaining to a geologically plausible porphyry copper unit.  The real case study located in  Semnan province of  Iran  consists  of  an arc-shaped  porphyry  andesite  covered  by  sedimentary  units  which  may  have  potential  of  mineral  occurrences, especially  porphyry copper. It is demonstrated that such structure extends down at depth, and consequently exploratory drilling is highly recommended for acquiring more pieces of information about its potential for ore-bearing mineralization.

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