THEORETICAL TRANSFORMS OF THE GRAVITY ANOMALIES OF TWO IDEALIZED BODIES

Geophysics ◽  
1978 ◽  
Vol 43 (3) ◽  
pp. 631-633 ◽  
Author(s):  
Robert D. Regan ◽  
William J. Hinze
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Three Dimensional ◽  
Gravity Anomalies ◽  
Vertical Line ◽  
The Fourier Transform ◽  
Line Elements ◽  
Distribution Of Magnetization

Odegard and Berg (1965) have shown that the interpretational process can be simplified for several idealized bodies by utilizing the Fourier transform of the resultant theoretical gravity anomalies. Additional studies relating similar conclusions for other idealized bodies have been reported by Gladkii (1963), Roy (1967), Sharma et al (1970), Davis (1971), Eby (1972), and Saha (1975), and a summary of the spatial and frequency domain equations is given in Regan and Hinze (1976, Table 1); however, the transforms of the three‐dimensional prism and vertical line elements, often utilized in interpretation, have not been previously examined in this manner. Although Bhattacharyya and Chen (1977) have developed and utilized the transform of the 3-D prism in their method for determining the distribution of magnetization in a localized region, it is still of value to present the interpretive advantages of the transform equation itself.

scholarly journals Fourier transform from momentum space to twistor space

2020 ◽  
pp. 2150032
Author(s):  
Jun-ichi Note
Keyword(s):  
Fourier Transform ◽  
Momentum Space ◽  
Twistor Space ◽  
Three Dimensional ◽  
Yang Mills ◽  
The Fourier Transform ◽  
Complex Integral ◽  
Cech Cohomology ◽  
Complex Projective ◽  
Operator Representations

Several methods use the Fourier transform from momentum space to twistor space to analyze scattering amplitudes in Yang–Mills theory. However, the transform has not been defined as a concrete complex integral when the twistor space is a three-dimensional complex projective space. To the best of our knowledge, this is the first study to define it as well as its inverse in terms of a concrete complex integral. In addition, our study is the first to show that the Fourier transform is an isomorphism from the zeroth Čech cohomology group to the first one. Moreover, the well-known twistor operator representations in twistor theory literature are shown to be valid for the Fourier transform and its inverse transform. Finally, we identify functions over which the application of the operators is closed.

Electron-density maps

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Electron Density ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Unit Cell ◽  
Density Maps ◽  
Wave Cycle ◽  
Density Map ◽  
The Fourier Transform ◽  
Electron Density Map

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

scholarly journals Characterization of Thermal Damage Due to Two-Temperature High-Order Thermal Lagging in a Three-Dimensional Biological Tissue Subjected to a Rectangular Laser Pulse

Polymers ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 922
Author(s):  
Hamdy M. Youssef ◽  
Najat. A. Alghamdi
Keyword(s):  
Fourier Transform ◽  
Biological Tissue ◽  
Thermal Damage ◽  
Three Dimensional ◽  
Bioheat Transfer ◽  
Conduction Model ◽  
Conductive Temperature ◽  
Arrhenius Integral ◽  
The Fourier Transform ◽  
Two Temperature

The use of lasers and thermal transfers on the skin is fundamental in medical and clinical treatments. In this paper, we constructed and applied bioheat transfer equations in the context of a two-temperature heat conduction model in order to discuss the three-dimensional variation in the temperature of laser-irradiated biological tissue. The amount of thermal damage in the tissue was calculated using the Arrhenius integral. Mathematical difficulties were encountered in applying the equations. As a result, the Laplace and Fourier transform technique was employed, and solutions for the conductive temperature and dynamical temperature were obtained in the Fourier transform domain.

Consolidation analysis for unsaturated soils

2004 ◽  
Vol 41 (4) ◽  
pp. 599-612 ◽  
Author(s):  
Enrico Conte
Keyword(s):  
Fourier Transform ◽  
Unsaturated Soils ◽  
Time Integration ◽  
Three Dimensional ◽  
Finite Element Approximation ◽  
Spatial Dimension ◽  
Element Approximation ◽  
Field Equations ◽  
Actual Solution ◽  
The Fourier Transform

This paper deals with the multidimensional consolidation of unsaturated soils when both the air phase and water phase are continuous. Following the approach proposed by D.G. Fredlund and his coworkers, the differential equations governing the coupled and uncoupled consolidation are first derived and then solved numerically. The solution is achieved using a procedure that depends on the transformation of the field equations by using the Fourier transform. This transformation has the effect of reducing a two- or three-dimensional problem to a problem involving only a single spatial dimension. The transformed equations are solved using a finite element approximation that makes use of simple one-dimensional elements. Once the solution in the transformed domain is obtained, the actual solution is achieved by inversion of the Fourier transform. The time integration process is formulated in a stepwise form. Results are presented to point out some aspects of the consolidation in unsaturated soils. Moreover, it is shown that the results obtained using the simple uncoupled theory are of sufficient accuracy for practical purposes.Key words: coupled consolidation, uncoupled consolidation, unsaturated soils, Fourier transform.

FOURIER TRANSFORMS OF FINITE LENGTH THEORETICAL GRAVITY ANOMALIES

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1450-1457 ◽  
Author(s):  
Robert D. Regan ◽  
William J. Hinze
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Finite Length ◽  
Fourier Transforms ◽  
Gravity Anomalies ◽  
Mathematical Structure ◽  
Practical Application ◽  
Convolution Integrals ◽  
The Fourier Transform ◽  
Interpretation Method

The mathematical structure of the Fourier transformations of theoretical gravity anomalies of several geometrically simple bodies appears to have distinct advantages in the interpretation of these anomalies. However, the practical application of this technique is dependent upon the transformation of an observed gravity anomaly of finite length. Ideally, interpretation methods similar to those for the transformations of the theoretical gravity anomalies should be developed for anomalies of a finite length. However, the mathematical complexity of the convolution integrals in the transform calculations of theoretical anomaly segments indicate that no general closed analytical solution useful for interpretation is available. Thus, in order to utilize the Fourier transform interpretation method, the data must be of sufficient length for the finite transform to closely approximate the theoretical transforms.

A GENERAL EXPRESSION FOR THE FOURIER TRANSFORM OF THE GRAVITY ANOMALY DUE TO A FAULT

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1458-1461 ◽  
Author(s):  
Bijon Sharma ◽  
T. K. Bose
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
General Expression ◽  
Fault Plane ◽  
Gravity Anomalies ◽  
Arbitrary Angle ◽  
Vertical Fault ◽  
The Fourier Transform

The application of the method of the Fourier transform in interpreting gravity anomalies of faults has so far been based upon the Fourier transform of the gravity anomaly due to a single semi‐infinite block cut by a vertical fault. A general expression for the Fourier transform of the fault anomaly is here derived which is valid for an arbitrary angle of inclination of the fault plane. For deriving the general expression, the gravity anomaly of the fault is first separated into a constant and a variable term. The transforms of the two terms are calculated separately and then added to give the general expression for the Fourier transform of the fault anomaly.

A PUBLIC MESH WATERMARKING ALGORITHM BASED ON ADDITION PROPERTY OF FOURIER TRANSFORM

2006 ◽  
Vol 06 (01) ◽  
pp. 35-43 ◽  
Author(s):  
LI LI ◽  
ZHIGENG PAN ◽  
DAVID ZHANG
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Spatial Domain ◽  
Experimental Results ◽  
Original Image ◽  
The Fourier Transform

This paper presents a public mesh watermarking algorithm whereby the resultant watermarked image minus the original image is the watermark information. According to the addition property of the Fourier transform, a change of spatial domain will cause a change in the frequency domain. The watermark information is then scaled down and embedded in one part of the x-coordinate of the original mesh. Finally, the x-coordinate of the test mesh is amplified before extraction. Experimental results prove that our algorithm is resistant to a variety of attacks without the need for any preprocessing.

LP -bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with D∞ Type Singularities

2020 ◽  
pp. 350-359
Author(s):  
Nigina A. Soleeva
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
The Fourier Transform

Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erd¨osh and Salmhofer

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