Conjugate complex variables method for the computation of gravity anomalies

Geophysics ◽  
1989 ◽  
Vol 54 (12) ◽  
pp. 1629-1637 ◽  
Author(s):  
Yue‐Kuen Kwok
Keyword(s):  
Gravity Anomalies ◽  
Higher Order ◽  
Complex Variables ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
Horizontal Distance ◽  
Contour Integrals ◽  
Vertical Gravity ◽  
Analytical Expressions ◽  
Derivatives Of

Using conjugate complex variables, a generalized method is presented to derive formulas to calculate first‐ and higher‐order derivatives of the gravity potential due to selected mass models. Double integrals in the computation of gravity‐gradient anomalies are transformed into complex contour integrals. Analytical expressions for higher‐order derivatives of the gravitational potential in arbitrary directions due to two‐dimensional (2‐D) polygonal mass models are derived. The method is extended to 2‐D polygonal bodies whose density contrasts vary with depth and horizontal distance and can be generalized to deal with 2‐D bodies of any shape. The vertical gravity field and its first derivatives due to a homogeneous radially symmetric body are also computed using conjugate complex variables. Derivation of gravity and gravity gradient formulas generally is greatly simplified by the use of complex variables.

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.

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.

Reduction to the pole of the North American magnetic anomalies

Geophysics ◽  
1990 ◽  
Vol 55 (2) ◽  
pp. 218-225 ◽  
Author(s):  
J. Arkani‐Hamed ◽  
W. E. S. Urquhart
Keyword(s):  
North America ◽  
Gravity Anomalies ◽  
Magnetic Anomalies ◽  
Gravity Gradient ◽  
Plate Boundaries ◽  
Gravity Gradients ◽  
The North ◽  
Vertical Gravity Gradient ◽  
Regional Tectonic ◽  
Vertical Gravity

Magnetic anomalies of North America are reduced to the pole using a generalized technique which takes into account the variations in the directions of the core field and the magnetization of the crust over North America. The reduced‐to‐the‐pole magnetic anomalies show good correlations with a number of regional tectonic features, such as the Mid‐Continental rift and the collision zones along plate boundaries, which are also apparent in the vertical gravity gradient map of North America. The magnetic anomalies do not, however, show consistent correlation with the vertical gravity gradients, suggesting that magnetic and gravity anomalies do not necessarily arise from common sources.

scholarly journals IDENTIFICATION OF BURIED ARCHEOLOGICAL OBJECTS WITH RADIAL DERIVATIVES OF MICRO GRAVITY DATA

2019 ◽  
Vol 2 (2) ◽  
pp. 84
Author(s):  
Muhammad Zuhdi ◽  
Bakti Sukrisna ◽  
Syamsuddin Syamsuddin
Keyword(s):  
Gravity Data ◽  
Synthetic Data ◽  
Gravity Anomalies ◽  
Density Contrast ◽  
Horizontal Distance ◽  
Radial Derivative ◽  
Horizontal Derivative ◽  
Almost All ◽  
Derivatives Of ◽  
Micro Gravity

The development of recent gravimetric technology allows us to measure gravity anomalies with accuracy of micro Gal. Micro gravity is able to detect very small gravity anomalies such as anomaly due to buried archeological objects below the earth surface. Radial Derivatives of gravity data is used to sharpen anomaly due to lateral changes of density contrast. Horizontal derivatives carried out by previous researchers have some weaknesses, i.e. the loss of derivative values in certain directions and inconsistence values at the source boundary of the same anomaly edge. To solve the horizontal derivative problem, a radial derivative is made. Radial derivative is derivative of gravity anomaly over horizontal distance in the radial direction from a certain point which is considered as the center of anomaly. There are two kind of radial derivative i.e. First Radial Derivative (FRD) and Second Radial Derivative (SRD). Blade Pattern is another way to enrich the ability of SRD to detect boundary of anomaly source. Synthetic gravity data of buried archeological object was made by counting the response of forward modelling. All of programs and calculation of the models in this research is performed based on Matlab® program. The results of the tests on the synthetic data of the model show that the radial derivative is able to detect the boundaries in buried temples due to density contrast. The advantage of radial derivatives which is a horizontal derivative in the direction of radial compared to ordinary horizontal derivatives is the ability to detect vertical boundaries of various anomaly due to horizontal layers and capable of showing density contrast in almost all directions.

scholarly journals Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150272 ◽  
Author(s):  
Longtao Xie ◽  
Chuanzeng Zhang ◽  
Jan Sladek ◽  
Vladimir Sladek
Keyword(s):  
Three Dimensional ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Higher Order ◽  
The Novel ◽  
Elastic Materials ◽  
Linear Elastic ◽  
Second Derivatives ◽  
Analytical Expressions ◽  
Derivatives Of

Novel unified analytical displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions for three-dimensional, generally anisotropic and linear elastic materials are presented in this paper. Adequate integral expressions for the displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions are evaluated analytically by using the Cauchy residue theorem. The resulting explicit displacement fundamental solutions and their first and second derivatives are recast into convenient analytical forms which are valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. The correctness and the accuracy of the novel unified and closed-form three-dimensional anisotropic fundamental solutions are verified by using some available analytical expressions for both transversely isotropic (non-degenerate or partially degenerate) and isotropic (fully degenerate) linear elastic materials.

scholarly journals The surface layer integral method for the modelling of the radial gravity gradient component over sea surface

2019 ◽  
Vol 9 (1) ◽  
pp. 127-132
Author(s):  
D. Zhao ◽  
Z. Gong ◽  
J. Feng
Keyword(s):  
Surface Layer ◽  
Integral Method ◽  
Integral Formula ◽  
Radial Component ◽  
Second Order ◽  
Sea Surface ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
Disturbing Potential ◽  
Gradient Component

Abstract For the modelling and determination of the Earth’s external gravity potential as well as its second-order radial derivatives in the space near sea surface, the surface layer integral method was discussed in the paper. The reasons for the applicability of the method over sea surface were discussed. From the original integral formula of disturbing potential based on the surface layer method, the expression of the radial component of the gravity gradient tensor was derived. Furthermore, an identity relation was introduced to modify the formula in order to reduce the singularity problem. Numerical experiments carried out over the marine area of China show that, the modi-fied surface layer integral method effectively improves the accuracy and reliability of the calculation of the second-order radial gradient component of the disturbing potential near sea surface.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

2017 ◽  
Vol 21 (6) ◽  
pp. 1820-1842
Author(s):  
Wu Zhen ◽  
Ma Rui ◽  
Chen Wanji
Keyword(s):  
Transverse Shear ◽  
Equilibrium Equation ◽  
Three Dimensional ◽  
Higher Order ◽  
Triangular Element ◽  
Shear Stresses ◽  
Sandwich Plates ◽  
Transverse Shear Stress ◽  
Transverse Shear Stresses ◽  
Derivatives Of

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

Expansions of the mildly relativistic dispersion function for nearly perpendicular electron cyclotron ray tracing

1999 ◽  
Vol 61 (1) ◽  
pp. 121-128 ◽  
Author(s):  
I. P. SHKAROFSKY
Keyword(s):  
Ray Tracing ◽  
Computer Programs ◽  
Higher Order ◽  
Electron Cyclotron ◽  
Dispersion Function ◽  
Relativistic Dispersion ◽  
Plasma Dispersion ◽  
Derivatives Of ◽  
Cyclotron Harmonic ◽  
Experienced Difficulty

To trace rays very close to the nth electron cyclotron harmonic, we need the mildly relativistic plasma dispersion function and its higher-order derivatives. Expressions for these functions have been obtained as an expansion for nearly perpendicular propagation in a region where computer programs have previously experienced difficulty in accuracy, namely when the magnitude of (c/vt)2 (ω−nωc)/ω is between 1 and 10. In this region, the large-argument expansions are not yet valid, but partial cancellations of terms occur. The expansion is expressed as a sum over derivatives of the ordinary dispersion function Z. New expressions are derived to relate higher-order derivatives of Z to Z itself in this region of concern in terms of a finite series.

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