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: “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.

Gravity interpretation using correlation factors between successive least‐squares residual anomalies

Geophysics ◽  
1989 ◽  
Vol 54 (12) ◽  
pp. 1614-1621 ◽  
Author(s):  
E. M. Abdelrahman ◽  
A. I. Bayoumi ◽  
Y. E. Abdelhady ◽  
M. M. Gobashy ◽  
H. M. El‐Araby
Keyword(s):  
Least Squares ◽  
Bouguer Anomaly ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
The United States ◽  
Horizontal Cylinder ◽  
Regional Component ◽  
Gravity Interpretation ◽  
Regional Gravity ◽  
Correlation Factors

The correlation factors between successive least‐squares residual (or regional) gravity anomalies from a buried sphere, a two‐dimensional (2‐D) horizontal cylinder, and a vertical cylinder and the first horizontal derivative of the gravity from a 2‐D thin faulted layer are computed. Correlation values are used to determine the depth to the center of the buried structure, and the radius of the sphere or the cylinder and the thickness of the fault are estimated. The method can be applied not only to residuals but also to the Bouguer‐anomaly profile consisting of the combined effect of a residual component due to a purely local structure and a regional component represented by a polynomial of any order. The method is easy to apply and may be automated if desired. It can also be applied to the derivative anomalies of the gravity field. The validity of the method is tested on two field examples from the United States and Denmark.

New methods for shape and depth determinations from SP data

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1202-1210 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby ◽  
Abdel‐Rady G. Hassaneen ◽  
Mahfooz A. Hafez
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Least Squares Method ◽  
Random Noise ◽  
Theoretical Data ◽  
Derivative Analysis ◽  
Higher Derivatives ◽  
Optimum Order ◽  
Simple Linear Equation ◽  
Sp Anomaly

We have extended our earlier derivative analysis method to higher derivatives to estimate the depth and shape (shape factor) of a buried structure from self‐potential (SP) data. We show that numerical second, third, and fourth horizontal‐derivative anomalies obtained from SP data using filters of successive window lengths can be used to simultaneously determine the depth and the shape of a buried structure. The depths and shapes obtained from the higher derivatives anomaly values can be used to determine simultaneously the actual depth and shape of the buried structure and the optimum order of the regional SP anomaly along the profile. The method is semi‐automatic and it can be applied to residuals as well as to observed SP data. We have also developed a method (based on a least‐squares minimization approach) to determine, successively, the depth and the shape of a buried structure from the residual SP anomaly profile. By defining the zero anomaly distance and the anomaly value at the origin, the problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of form f(z) = 0. Knowing the depth and applying the least‐squares method, the shape factor is determined using a simple linear equation. Finally, we apply these methods to theoretical data with and without random noise and on a known field example from Germany. In all cases, the depth and shape solutions obtained are in good agreement with the actual ones.

‘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.

Theoretical accuracy of the last squares method in the isotope dilution analysis

1984 ◽  
Vol 49 (4) ◽  
pp. 805-820
Author(s):  
Ján Klas
Keyword(s):  
Least Squares ◽  
Isotope Dilution ◽  
Least Squares Method ◽  
Isotope Dilution Analysis ◽  
Straight Line ◽  
The One ◽  
Two Parameter ◽  
Theoretical Accuracy

The accuracy of the least squares method in the isotope dilution analysis is studied using two models, viz a model of a two-parameter straight line and a model of a one-parameter straight line.The equations for the direct and the inverse isotope dilution methods are transformed into linear coordinates, and the intercept and slope of the two-parameter straight line and the slope of the one-parameter straight line are evaluated and treated.

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