A COMPREHENSIVE SYSTEM OF AUTOMATIC COMPUTATION IN MAGNETIC AND GRAVITY INTERPRETATION

Geophysics ◽  
1960 ◽  
Vol 25 (3) ◽  
pp. 569-585 ◽  
Author(s):  
Roland G. Henderson
Keyword(s):  
Effective Means ◽  
Gravity Anomalies ◽  
Comprehensive System ◽  
Diagnostic Features ◽  
Special Cases ◽  
Desk Calculator ◽  
Concentric Circles ◽  
Gravity Interpretation ◽  
Magnetic And Gravity Anomalies ◽  
Derivatives Of

In the interpretation of magnetic and gravity anomalies, downward continuation of fields and calculation of first and second vertical derivatives of fields have been recognized as effective means for bringing into focus the latent diagnostic features of the data. A comprehensive system has been devised for the calculation of any or all of these derived fields on modern electronic digital computing equipment. The integral for analytic continuation above the plane is used with a Lagrange extrapolation polynomial to derive a general determinantal expression from which the field at depth and the various derivatives on the surface and at depth can be obtained. It is shown that the general formula includes as special cases some of the formulas appearing in the literature. The process involves a “once for all depths” summing of grid values on a system of concentric circles about each point followed by application of the appropriate one or more of the 19 sets of coefficients derived for the purpose. Theoretical and observed multilevel data are used to illustrate the processes and to discuss the errors. The coefficients can be used for less extensive computations on a desk calculator.

A LEAST‐SQUARES PROCEDURE FOR GRAVITY INTERPRETATION

Geophysics ◽  
1965 ◽  
Vol 30 (2) ◽  
pp. 228-233 ◽  
Author(s):  
Charles E. Corbató
Keyword(s):  
Least Squares ◽  
High Speed ◽  
Gravity Anomalies ◽  
The Body ◽  
Two Dimensional ◽  
Partial Derivatives ◽  
Dimensional Gravity ◽  
Gravity Interpretation ◽  
Relationship Of ◽  
Derivatives Of

A procedure suitable for use on high‐speed digital computers is presented for interpreting two‐dimensional gravity anomalies. In order to determine the shape of a disturbing mass with known density contrast, an initial model is assumed and gravity anomalies are calculated and compared with observed values at n points, where n is greater than the number of unknown variables (e.g. depths) of the model. Adjustments are then made to the model by a least‐squares approximation which uses the partial derivatives of the anomalies so that the residuals are reduced to a minimum. In comparison with other iterative techniques, convergence is very rapid. A convenient method to use for both the calculation of the anomalies and the adjustments is the two‐dimensional method of Talwani, Worzel, and Landisman, (1959) in which the outline of the body is polygonized and the anomalies and the partial derivatives of the anomaly with respect to the depth of a vertex on the body can be expressed as functions of the coordinates of the vertex. Not only depths but under certain circumstances regional gravity values may be evaluated; however, the relationship of the disturbing body to the gravity information may impose certain limitations on the application of the procedure.

scholarly journals Rate of Convergence for Ibragimov-Gadjiev-Durrmeyer Operators

2017 ◽  
Vol 50 (1) ◽  
pp. 119-129 ◽  
Author(s):  
Tuncer Acar
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
General Class ◽  
Durrmeyer Operators ◽  
Special Cases ◽  
Derivatives Of

Abstract The present paper deals with the rate of convergence of the general class of Durrmeyer operators, which are generalization of Ibragimov-Gadjiev operators. The special cases of the operators include somewell known operators as particular cases viz. Szász-Mirakyan-Durrmeyer operators, Baskakov-Durrmeyer operators. Herewe estimate the rate of convergence of Ibragimov-Gadjiev-Durrmeyer operators for functions having derivatives of bounded variation.

scholarly journals Review of Environmental and Geological Microgravity Applications and Feasibility of Its Employment at Archaeological Sites in Israel

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Lev V. Eppelbaum
Keyword(s):  
Gravity Anomalies ◽  
Gravity Potential ◽  
Archaeological Sites ◽  
Useful Signal ◽  
Geometric Size ◽  
Archaeological Models ◽  
Vertical Gravity ◽  
Transformation Methods ◽  
Derivatives Of ◽  
Archaeological Objects

Microgravity investigations are widely applied at present for solving various environmental and geological problems. Unfortunately, microgravity survey is comparatively rarely used for searching for hidden ancient targets. It is caused mainly by small geometric size of the desired archaeological objects and various types of noise complicating the observed useful signal. At the same time, development of modern generation of field gravimetric equipment allows to register promptly and digitally microGal (10-8 m/s2) anomalies that offer a new challenge in this direction. An advanced methodology of gravity anomalies analysis and modern 3D modeling, intended for ancient targets delineation, is briefly presented. It is supposed to apply in archaeological microgravity the developed original methods for the surrounding terrain relief computing. Calculating second and third derivatives of gravity potential are useful for revealing some closed peculiarities of the different Physical-Archaeological Models (PAMs). It is underlined that physical measurement of vertical gravity derivatives in archaeological studying has a significant importance and cannot be replaced by any transformation methods. Archaeological targets in Israel have been ranged by their density/geometrical characteristics in several groups. The performed model computations indicate that microgravity investigations might be successfully applied at least in 20–25% of archaeological sites in Israel.

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies

Geophysics ◽  
1979 ◽  
Vol 44 (4) ◽  
pp. 730-741 ◽  
Author(s):  
M. Okabe
Keyword(s):  
Gravitational Potential ◽  
Computing Time ◽  
Gravity Anomalies ◽  
Magnetic Anomalies ◽  
Two Dimensions ◽  
First Derivative ◽  
Second Derivatives ◽  
Analytical Expressions ◽  
Derivatives Of ◽  
The Magnetic Field

Complete analytical expressions for the first and second derivatives of the gravitational potential in arbitrary directions due to a homogeneous polyhedral body composed of polygonal facets are developed, by applying the divergence theorem definitively. Not only finite but also infinite rectangular prisms then are treated. The gravity anomalies due to a uniform polygon are similarly described in two dimensions. The magnetic potential due to a uniformly magnetized body is directly derived from the first derivative of the gravitational potential in a given direction. The rule for translating the second derivative of the gravitational potential into the magnetic field component is also described. The necessary procedures for practical computer programming are discussed in detail, in order to avoid singularities and to save computing time.

scholarly journals Long Periodic Variation of Orbital Elements of A Satellite Perturbed by Discrete Gravity Anomalies

1978 ◽  
Vol 41 ◽  
pp. 240-240
Author(s):  
M.P. Ananda
Keyword(s):  
Gravity Field ◽  
Gravity Anomalies ◽  
Large Data ◽  
Periodic Variation ◽  
Static Case ◽  
Orbital Elements ◽  
Disturbing Potential ◽  
Data Set ◽  
Simple Matrix ◽  
Derivatives Of

AbstractA method for generating long periodic variations in satellite orbital elements when perturbed by discrete gravity anomalies is presented. The method consists of developing a disturbing potential as a function of orbital and gravity anomaly parameters, and generating partial derivatives of the potential with respect to the orbital elements. The partials are averaged over the period of the satellite to eliminate the short periodic variations. The averaged partials are substituted into the variation of parameter equations to give the mean orbital rates. Classically orbital elements are used in generating gravity field and thus the method is dynamic in nature. The problem is extremely cumbersome and complex when multi-state parameters have to be estimated from a considerably large data set. However, when mean orbital rates are used, the problem reduces to a simple linear static case, where only the gravity parameters have to be estimated, and it is a simple matrix inversion problem. Thus the method developed here was utilized in reducing Appolo 15 and 16 subsatellite radio tracking data to produce a lunar gravity field represented by point masses.

Improved inversion of geopotential field anomalies for lithospheric investigations

Geophysics ◽  
1988 ◽  
Vol 53 (3) ◽  
pp. 375-385 ◽  
Author(s):  
R. R. B. von Frese ◽  
D. N. Ravat ◽  
W. J. Hinze ◽  
C. A. McGue
Keyword(s):  
Large Scale ◽  
Superposition Principle ◽  
Gravity Anomalies ◽  
Damping Factor ◽  
Stable Model ◽  
Memory Integration ◽  
Unstable Solutions ◽  
Ill Posed ◽  
Anomaly Analysis ◽  
Magnetic And Gravity Anomalies

Instabilities and the large matrices which are common to inversions of regional magnetic and gravity anomalies often complicate the use of efficient least‐squares matrix procedures. Inversion stability profoundly affects anomaly analysis, and hence it must be considered in any application. Wildly varying or unstable solutions are the products of errors in the anomaly observations and the integrated effects of observation spacing, source spacing, elevation differences between sources and observations, geographic coordinate attributes, geomagnetic field attitudes, and other factors which influence the conditioning of inversion. Solution instabilities caused by ill‐posed parameters can be efficiently minimized by ridge regression with a damping factor large enough to stabilize the inversion, but small enough to produce an analytically useful solution. An effective choice for the damping factor is facilitated by plotting damping factors against residuals between observed and modeled anomalies and by then comparing this curve to curves of damping factors plotted against solution variance or the residuals between predicted anomaly maps representing the processing objective (e.g., downward continuation, differential reduction to the radial pole, etc.). To obtain accurate and efficient large‐scale inversions of anomaly data, a procedure based on the superposition principle of potential fields may be used. This method involves successive inversions of residuals between the observations and various stable model fields which can be readily accommodated by available computer memory. Integration of the model fields yields a well‐resolved representation of the observed anomalies corresponding to an integrated model which normally could not be obtained by direct inversion because the memory requirements would be excessive. MAGSAT magnetic anomaly inversions over India demonstrate the utility of these procedures for improving the geologic analysis of potential field anomalies.

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