BOUGUER GRAVITY CORRECTIONS USING A VARIABLE DENSITY

Geophysics ◽  
1962 ◽  
Vol 27 (5) ◽  
pp. 616-626 ◽  
Author(s):  
F. S. Grant ◽  
A. F. Elsaharty
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Variable Density ◽  
Bouguer Gravity ◽  
Method Of Least Squares ◽  
Surface Conditions ◽  
Worked Example ◽  
Local Gravity

The principle of density profiling as a means of determining Bouguer densities is studied with a view to extending it to include all of the data in a survey. It is regarded as an endeavor to minimize the correlation between local gravity anomalies and topography, and as such it can be handled mathematically by the method of least squares. In the general case this leads to a variable Bouguer density which can be mapped and contoured. In a worked example, the correspondence between this function and the known geology appears to be good, and indicates that Bouguer density variations due to changing surface conditions can be used routinely in the reduction of gravity data.

scholarly journals Evaluation of Gravity Data Derived from Global Gravity Field Models Using Terrestrial Gravity Data in Enugu State, Nigeria

2018 ◽  
Vol 8 (1) ◽  
pp. 145-153 ◽  
Author(s):  
O.I. Apeh ◽  
E.C. Moka ◽  
V.N. Uzodinma
Keyword(s):  
Gravity Field ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Mean Square ◽  
Bouguer Gravity ◽  
Bouguer Anomalies ◽  
The Earth ◽  
Global Gravity ◽  
Enugu State ◽  
Gravity Field Models

Abstract Spherical harmonic expansion is a commonly applied mathematical representation of the earth’s gravity field. This representation is implied by the potential coeffcients determined by using elements/parameters of the field observed on the surface of the earth and/or in space outside the earth in the spherical harmonic expansion of the field. International Centre for Gravity Earth Models (ICGEM) publishes, from time to time, Global Gravity Field Models (GGMs) that have been developed. These GGMs need evaluation with terrestrial data of different locations to ascertain their accuracy for application in those locations. In this study, Bouguer gravity anomalies derived from a total of eleven (11) recent GGMs, using sixty sample points, were evaluated by means of Root-Mean-Square difference and correlation coeficient. The Root-Mean-Square differences of the computed Bouguer anomalies from ICGEMwebsite compared to their positionally corresponding terrestrial Bouguer anomalies range from 9.530mgal to 37.113mgal. Additionally, the correlation coe_cients of the structure of the signal of the terrestrial and GGM-derived Bouguer anomalies range from 0.480 to 0.879. It was observed that GECO derived Bouguer gravity anomalies have the best signal structure relationship with the terrestrial data than the other ten GGMs. We also discovered that EIGEN-6C4 and GECO derived Bouguer anomalies have enormous potential to be used as supplements to the terrestrial Bouguer anomalies for Enugu State, Nigeria.

Shape and depth solutions from gravity data using correlation factors between successive least‐squares residuals

Geophysics ◽  
1993 ◽  
Vol 58 (12) ◽  
pp. 1785-1791 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby
Keyword(s):  
Least Squares ◽  
Shape Factor ◽  
Gravity Data ◽  
Bouguer Anomaly ◽  
Vertical Cylinder ◽  
Gravity Anomalies ◽  
Horizontal Cylinder ◽  
Amplitude Coefficient ◽  
Regional Component ◽  
Correlation Factors

The gravity anomaly expression produced by most geologic structures can be represented by a continuous function in both shape (shape factor) and depth variables with an amplitude coefficient related to the mass. Correlation factors between successive least‐squares residual gravity anomalies from a buried vertical cylinder, horizontal cylinder, and sphere are used to determine the shape and depth of the buried geologic structure. For each shape factor value, the depth is determined automatically from the correlation value. The computed depths are plotted against the shape factor representing a continuous correlation curve. The solution for the shape and depth of the buried structure is read at the common intersection of correlation curves. This method can be applied to a Bouguer anomaly profile consisting of a residual component caused by local structure and a regional component. This is a powerful technique for automatically separating the Bouguer data into residual and regional polynomial components. This method is tested on theoretical examples and a field example. In both cases, the results obtained are in good agreement with drilling results.

A least‐squares minimization approach to depth determination from moving average residual gravity anomalies

Geophysics ◽  
1993 ◽  
Vol 58 (12) ◽  
pp. 1779-1784 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Tarek M. El‐Araby
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Moving Average ◽  
Gravity Anomalies ◽  
The United States ◽  
Random Errors ◽  
Minimization Method ◽  
Residual Gravity ◽  
Average Residual ◽  
Least Squares Minimization

We have developed a least‐squares minimization method to estimate the depth of a buried structure from moving average residual gravity anomalies. The method involves fitting simple models convolved with the same moving average filter as applied to the observed gravity data. As a result, our method can be applied not only to residuals but also to the Bouguer gravity data of a short profile length. The method is applied to synthetic data with and without random errors. The validity of the method is tested in detail on two field examples from the United States and Senegal.

A least‐squares minimization approach to invert gravity data

Geophysics ◽  
1991 ◽  
Vol 56 (1) ◽  
pp. 115-118 ◽  
Author(s):  
E. M. Abdelrahman ◽  
A. I. Bayoumi ◽  
H. M. El‐Araby
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Gravity Data ◽  
Spline Approximation ◽  
Gravity Anomalies ◽  
Approximation Techniques ◽  
Local Gravity ◽  
Fourier Transform Methods ◽  
Minimization Approach ◽  
Least Squares Minimization

The interpretation of gravity data often involves initial steps to eliminate or attenuate unwanted field components in order to isolate the desired anomaly (e.g., residual‐regional separations). These initial filtering operations include, for example, the radial weights methods (Griffin, 1949; Elkins, 1951; Abdelrahman et al., 1990), the fast Fourier transform methods (Bhattacharyya, 1965; Clarke, 1969; Meskó, 1969, 1984, Botezatu, 1970), the rational approximation techniques (Agarwal and Lal, 1971) and recursion filters (Bhattacharyya, 1976), and the bicubic spline approximation techniques (Bhattacharyya, 1969; Inoue, 1986). The derived local gravity anomalies are then geologically interpreted to derive depth estimates, often without properly accounting for the uncertainties introduced by the filtering process. When filters are applied to observed data, the filters often cause serious distortions in the shape of the gravity anomalies (Hammer, 1977). Thus the filtered gravity anomalies generally yield unreliable geologic interpretations (Rao and Radhakrishnamurthy, 1965; Hammer, 1977; Abdelrahman et al., 1985, 1989.

A priori noise and regularization in least squares collocation of gravity anomalies

2013 ◽  
Vol 62 (2) ◽  
pp. 199-216 ◽  
Author(s):  
Wojciech Jarmołowski
Keyword(s):  
Least Squares ◽  
Spatial Resolution ◽  
Gravity Data ◽  
A Priori ◽  
Gravity Anomalies ◽  
Geopotential Model ◽  
Large Set ◽  
Least Squares Collocation ◽  
Gravity Field Model ◽  
Two Parameters

Abstract The paper describes the estimation of covariance parameters in least squares collocation (LSC) by the cross-validation (CV) technique called leave-one-out (LOO). Two parameters of Gauss-Markov third order model (GM3) are estimated together with a priori noise standard deviation, which contributes significantly to the covariance matrix composed of the signal and noise. Numerical tests are performed using large set of Bouguer gravity anomalies located in the central part of the U.S. Around 103 000 gravity stations are available in the selected area. This dataset, together with regular grids generated from EGM2008 geopotential model, give an opportunity to work with various spatial resolutions of the data and heterogeneous variances of the signal and noise. This plays a crucial role in the numerical investigations, because the spatial resolution of the gravity data determines the number of gravity details that we may observe and model. This establishes a relation between the spatial resolution of the data and the resolution of the gravity field model. This relation is inspected in the article and compared to the regularization problem occurring frequently in data modeling.

scholarly journals Usporedba različitih pristupa Bouguer-ove redukcije na temelju satelitskih gravimetrijskih podataka

Geofizika ◽  
2020 ◽  
Vol 37 (2) ◽  
pp. 237-261
Author(s):  
Fan Luo ◽  
Xin Tao ◽  
Guangming Fu ◽  
Chong Zhang ◽  
Kun Zhang ◽  
...  
Keyword(s):  
Gravity Anomaly ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Gravitational Effect ◽  
Seismic Profile ◽  
Bouguer Gravity ◽  
Seismic Profiles ◽  
Satellite Gravity ◽  
Free Air ◽  
Free Air Gravity

Satellite gravity data are widely used in the field of geophysics to study deep structures at the regional and global scales. These data comprise free-air gravity anomaly data, which usually need to be corrected to a Bouguer gravity anomaly for practical application. Bouguer reduction approaches can be divided into two methods based on the coordinate system: the spherical coordinates method (SBG) and the Cartesian coordinates method; the latter is further divided into the CEBG and CBG methods, which do and do not include the Earth’s curvature correction. In this paper, free-air gravity anomaly data from the eastern Tibetan Plateau and its adjacent areas were used as the basic data to compare the CBG, CEBG, and SBG Bouguer gravity correction methods. The comparison of these three Bouguer gravity correction methods shows that the effect of the Earth’s curvature on the gravitational effect increases with increasing elevation in the study area. We want to understand the inversion accuracy for the data obtained by different Bouguer gravity reduction approaches. The depth distributions of the Moho were obtained by the interface inversion of the Bouguer gravity anomalies obtained by the CBG, CEBG, and SBG, and active seismic profiles were used as references for comparison and evaluation. The results show that the depths of the Moho obtained by the SBG inversion are more consistent with the measured seismic profile depths. Therefore, the SBG method is recommended as the most realistic approach in the process of global or regional research employing gravity data.

scholarly journals Modelling Local Gravity Anomalies from Processed Observed Gravity Measurements for Geodetic Applications

2019 ◽  
pp. 144-162
Author(s):  
Eteje S. O. ◽  
Oduyebo O. F. ◽  
Oluyori P. D.
Keyword(s):  
Applied Sciences ◽  
Gravity Anomalies ◽  
Absolute Gravity ◽  
Bouguer Gravity ◽  
Benin City ◽  
Bouguer Gravity Anomalies ◽  
Free Air ◽  
Local Gravity ◽  
Gravity Measurements ◽  
Mean Square Errors

As the application of gravity data in applied sciences such as geodesy, geodynamics, astronomy, physics and geophysics for earth shape determination, geoid model determination, computation of terrestrial mass displacement, orbit computation of natural and artificial celestial bodies, realization of force standards and derived quantities and density distribution in the different layers in the upper crust and having considered the cost of direct gravity survey, the study presents modelling local gravity anomalies from processed observed gravity measurements for geodetic application in Benin City. A total of 22 points were used. The points were respectively observed with CHC900 dual frequency GNSS receivers and SCINTREX CG-5 Autograv to obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid to obtain their local gravity anomalies. The free air and the Bouguer corrections were applied to the computed gravity anomalies to obtain the free air and the Bouguer gravity anomalies of the points. Least squares adjustment technique was applied to obtain the model variables coefficient/parameters, as well as to fit the fifth-degree polynomial interpolation surface to the computed free air and the Bouguer gravity anomalies. Kriging method was applied using Surfer 12 software to plot the computed and the models' free air and Bouguer gravity anomalies. Microsoft Excel programs were developed for the application of the models in the study area. The Root Mean Square Errors (RMSEs) and the standard errors of the two models were computed to obtain the dependability, as well as reliability of the models. It is recommended that whenever either free air or Bouguer gravity anomalies of points within Benin City are to be obtained for application in applied sciences, the determined models should be applied.

THE USE OF CRACOVIAN COMPUTATION IN ESTIMATING THE REGIONAL GRAVITY

Geophysics ◽  
1959 ◽  
Vol 24 (3) ◽  
pp. 465-478 ◽  
Author(s):  
Zbigniew Fajklewicz
Keyword(s):  
Least Squares ◽  
Gravity Field ◽  
Gravity Anomalies ◽  
Second Order ◽  
Method Of Least Squares ◽  
Electronic Computers ◽  
Regional Gravity ◽  
Work Done ◽  
Very High ◽  
Digital Computers

The author uses the method of least squares in cracovian form and second order polynomials for estimating the regional gravity field. Expressions yielding the regional field are obtained very rapidly by using the inverse cracovians of the coefficients as given in the present paper, and there is no need of electronic digital computers for the computation. The equivalent of the entire work done by a computer of this kind in constructing the formula of the regional field, when effected by this method, takes no more than 20 minutes. The method is exemplified by the treatment of two gravity anomalies from the territory of Poland. The author stresses the fact that electronic computers adapted to the use of cracovians and characterized by a very high versatility may be applied in the method.

scholarly journals A least-squares minimization approach to interpret gravity anomalies caused by a 2D thick, vertically faulted slab

2019 ◽  
Vol 49 (3) ◽  
pp. 229-247
Author(s):  
El-Sayed Abdelrahman ◽  
Mohamed Gobashy
Keyword(s):  
Least Squares ◽  
Gravity Data ◽  
Gravity Anomalies ◽  
Central Valley ◽  
Real Data ◽  
Horizontal Gradient ◽  
Vertical Fault ◽  
Independent Observations ◽  
Minimization Approach ◽  
Least Squares Minimization

Abstract We present a least-squares minimization approach to estimate simultaneously the depth to and thickness of a buried 2D thick, vertically faulted slab from gravity data using the sample spacing – curves method or simply s-curves method. The method also provides an estimate for the horizontal location of the fault and a least-squares estimate for the density contrast of the slab relative to the host. The method involves using a 2D thick vertical fault model convolved with the same finite difference second horizontal gradient filter as applied to the gravity data. The synthetic examples (noise-free and noise affected) are presented to illustrate our method. The test on the real data (Central Valley of Chile) and the obtained results were consistent with the available independent observations and the broader geological aspects of this region.

A least‐squares derivatives analysis of gravity anomalies due to faulted thin slabs

Geophysics ◽  
2003 ◽  
Vol 68 (2) ◽  
pp. 535-543 ◽  
Author(s):  
El‐Sayed M. Abdelrahman ◽  
Hesham M. El‐Araby ◽  
Tarek M. El‐Araby ◽  
Eid Ragab Abo‐Ezz
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Gravity Data ◽  
Bouguer Anomaly ◽  
Theoretical Data ◽  
Gravity Anomalies ◽  
Random Errors ◽  
Fault Structure ◽  
Amplitude Coefficient ◽  
Buried Fault

This paper presents two different least‐squares approaches for determining the depth and amplitude coefficient (related to the density contrast and the thickness of a buried faulted thin slab from numerical first‐, second‐, third‐, and fourth‐horizontal derivative anomalies obtained from 2D gravity data using filters of successive graticule spacings. The problem of depth determination has been transformed into the problem of finding a solution to a nonlinear equation of the form f(z) = 0. Knowing the depth and applying the least‐squares method, the amplitude coefficient is determined using a simple linear equation. In this way, the depth and amplitude coefficient are determined individually from all observed gravity data. The depths and the amplitude coefficients obtained from the first‐, second‐, third‐, and fourth‐ derivative anomaly values can be used to determine simultaneously the actual depth and amplitude coefficient of the buried fault structure and the optimum order of the regional gravity field along the profile. 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 fault structure (shallow or deep) and a regional component represented by a polynomial of any order. The method is applied to theoretical data with and without random errors and is tested on a field example from Egypt.

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