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.

COMPUTATION WITH THE HELP OF A DIGITAL COMPUTER OF MAGNETIC ANOMALIES CAUSED BY BODIES OF ARBITRARY SHAPE

Geophysics ◽  
1965 ◽  
Vol 30 (5) ◽  
pp. 797-817 ◽  
Author(s):  
Manik Talwani
Keyword(s):  
Computer Program ◽  
Gravitational Potential ◽  
Digital Computer ◽  
Arbitrary Shape ◽  
Complex Shape ◽  
Magnetic Anomalies ◽  
Simple Geometry ◽  
Second Derivatives ◽  
First Vertical Derivative ◽  
Derivatives Of

Formulas are derived for the magnetic anomalies caused by irregular polygonal laminas. These are used to obtain the three components of the magnetic anomalies caused by a finite homogeneously magnetized body of arbitrary shape. There is no restriction to the direction of magnetization; in general, it may not be the same as that of the earth’s field. Total‐intensity anomalies are also obtained. Use of these formulas in a computer program is discussed and illustrated by computing the anomaly caused by Caryn Seamount. Simplified, formulas are presented for the anomalies caused by finite rectangular laminas. In addition to bodies of complex shape, the computer program can also be profitably used for computing the magnetic anomalies caused by bodies of relatively simple geometry. The second derivatives of the gravitational potential of a massive body, that is, quantities familiarly known as gradient and curvature in torsion‐balance work and the first vertical derivative in gravity work are also obtained by this method.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

MAGNETIC FIELD DEPENDENCE OF THE POINT-CONTACT SPECTRA OF CeNi5

1993 ◽  
Vol 07 (01n03) ◽  
pp. 222-225 ◽  
Author(s):  
Y. NAIDYUK ◽  
M. REIFFERS ◽  
A. G. M. JANSEN ◽  
I. K. YANSON ◽  
P. WYDER ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Intermetallic Compound ◽  
Field Dependence ◽  
Magnetic Field Dependence ◽  
Point Contact ◽  
Spin Fluctuations ◽  
Point Contacts ◽  
Second Derivatives ◽  
Derivatives Of ◽  
The Magnetic Field

The intermetallic compound CeNi 5, where Ce is not magnetic and in the maximum valence state, is an enhanced Pauli paramagnet in which spin fluctuations are present. For point contacts between a CeNi 5 single crystal and Cu, the second derivatives of the I-V characteristics reveal a maximum at low voltages (around 3 mV) which disappears in applied magnetic fields above 10 T. This behaviour in the point-contact spectra is ascribed to the quenching of spin fluctuations by the magnetic field.

Overview on the spectral combination of integral transformations

2020 ◽  
Author(s):  
Martin Pitoňák ◽  
Michal Šprlák ◽  
Pavel Novák ◽  
Robert Tenzer
Keyword(s):  
Gravitational Potential ◽  
Gravity Anomalies ◽  
Kernel Functions ◽  
Combination Method ◽  
Harmonic Series ◽  
Directional Derivatives ◽  
Theoretical Point ◽  
The Earth ◽  
Derivatives Of ◽  
Surface Integrals

<p>Geodetic boundary-value problems (BVPs) and their solutions are important tools for describing and modelling the Earth’s gravitational field. Many geodetic BVPs have been formulated based on gravitational observables measured by different sensors on the ground or moving platforms (i.e. aeroplanes, satellites). Solutions to spherical geodetic BVPs lead to spherical harmonic series or surface integrals with Green’s kernel functions. When solving this problem for higher-order derivatives of the gravitational potential as boundary conditions, more than one solution is obtained. Solutions to gravimetric, gradiometric and gravitational curvature BVPs (Martinec 2003; Šprlák and Novák 2016), respectively, lead to two, three and four formulas. From a theoretical point of view, all formulas should provide the same solution, but practically, when discrete noisy observations are exploited, they do not.</p><p>In this contribution we present combinations of solutions to the above mentioned geodetic BVPs in terms of surface integrals with Green’s kernel functions by a spectral combination method. We investigate an optimal combination of different orders and directional derivatives of potential. The spectral combination method is used to combine terrestrial data with global geopotential models in order to calculate geoid/quasigeoid surface. We consider that the first-, second- and third-order directional derivatives are measured at the satellite altitude and we continue them downward to the Earth’s surface and convert them to the disturbing gravitational potential, gravity disturbances and gravity anomalies. The spectral combination method thus serves in our numerical procedures as the downward continuation technique. This requires to derive the corresponding spectral weights for the n-component estimator (n = 1, 2, … 9) and to provide a generalized formula for evaluation of spectral weights for an arbitrary N-component estimator. Properties of the corresponding combinations are investigated in both, spatial and spectral domains.</p><p> </p>

Investigation of the Relationship of Muscle Mechanical Characteristics with Biosignal Energy

2006 ◽  
Vol 113 ◽  
pp. 151-156
Author(s):  
Mečislovas Mariūnas ◽  
Kristina Kojelyte
Keyword(s):  
Mechanical Characteristics ◽  
Extreme Points ◽  
Shift Function ◽  
Energy Equality ◽  
Second Derivatives ◽  
Digital Methods ◽  
Relationship Of ◽  
The Relationship ◽  
Analytical Expressions ◽  
Derivatives Of

Based on biosignal energy equality to the work performed by a muscle, in the estimation of energy dissipation within a biolotronic system, analytical expressions have been derived that have helped to calculate major mechanical characteristics of the muscle including elongation of the muscle, speed and acceleration of such elongation. The paper presents the analysis of relevant digital methods alongside with the formulae for the calculation of major mechanical characteristics when the relationship of a muscle biosignal that is presented in a graphical way. As shown, fatigue of the muscle is characterized by one of extreme points of the phase shift function. The values of the first and the second derivatives of the elongation function may be used for the evaluation of functional capacity of the muscle.

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 On Differentiating Eigenvalues and Eigenvectors

1985 ◽  
Vol 1 (2) ◽  
pp. 179-191 ◽  
Author(s):  
Jan R. Magnus
Keyword(s):  
Eigenvalues And Eigenvectors ◽  
First Derivative ◽  
Differentiable Functions ◽  
Second Derivatives ◽  
Derivatives Of ◽  
Square Matrix

Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . We obtain the first and second derivatives of λ(X) and the first derivative of u(X). Two alternative expressions for the first derivative of λ(X) are also presented.

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.

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