A method for computing the geoid height contribution of three‐dimensional bodies within a spherical earth

Geophysics ◽  
1983 ◽  
Vol 48 (12) ◽  
pp. 1665-1670 ◽  
Author(s):  
S. J. Hellinger
Keyword(s):  
Gravitational Potential ◽  
Arbitrary Shape ◽  
Three Dimensional ◽  
Geoid Height ◽  
Observation Point ◽  
Simple Shape ◽  
Digital Computation ◽  
Anomalous Body ◽  
Anomalous Density ◽  
Spherical Earth

A method is presented for calculating the geoid height associated with three‐dimensional bodies of arbitrary shape and anomalous density within a spherical earth. Each three‐dimensional body is specified by giving its shape and anomalous density at a number of successive depths. The method calculates the anomalous gravitational potential due to such bodies and then obtains the geoid height using Bruns’ formula. The calculation utilizes the portion of the density model that lies in a hemisphere centered at the observation point. The method is partly analytical and partly numerical, and it is designed for digital computation. Use of the method to compute the anomalous potential in the vicinity of an anomalous body of simple shape shows that the method can give very accurate results.

RAPID COMPUTATION OF GRAVITATIONAL ATTRACTION OF THREE‐DIMENSIONAL BODIES OF ARBITRARY SHAPE

Geophysics ◽  
1960 ◽  
Vol 25 (1) ◽  
pp. 203-225 ◽  
Author(s):  
Manik Talwani ◽  
Maurice Ewing
Keyword(s):  
Gravity Anomaly ◽  
Arbitrary Shape ◽  
High Speed ◽  
Three Dimensional ◽  
Isostatic Compensation ◽  
External Point ◽  
Spherical Earth ◽  
Terrain Corrections ◽  
High Degree ◽  
Special Case

An expression is derived for the gravity anomaly at an external point caused by a horizontal lamina with the boundary of an irregular polygon. This expression is put in a form suitable for computation by a high speed digital computer. By making the number of sides of the polygon sufficiently large, any irregular outline can be closely approximated. Any three dimensional body can be represented by contours. By replacing each contour by a polygonal lamina, the anomaly caused by it can be obtained at any external point. By a system of interpolation between contours combined with a numerical integration the gravity anomaly caused by the three‐dimensional body can be calculated to a high degree of precision. This method may also be used for rapidly computing terrain corrections on a flat earth. By making a small modification it can further be adopted for computing the terrain correction as well as local isostatic compensation on the Airy system up to the external radius of Hayford zone O on a spherical earth. The expression for the anomaly caused by a horizontal polygonal lamina is also obtained for the special case when the sides of the polygon are alternately parallel to the x- and y-axes, that is, the polygonal lamina can be divided into a number of rectangular laminae. A chart is provided for the hand computation of the gravity anomaly in this case.

scholarly journals From three-dimensional weavings to swollen corneocytes

2011 ◽  
Vol 8 (62) ◽  
pp. 1274-1280 ◽  
Author(s):  
Myfanwy E. Evans ◽  
Stephen T. Hyde
Keyword(s):  
Three Dimensional ◽  
Simple Shape ◽  
Outermost Layer ◽  
One Dimensional ◽  
Periodic Arrays ◽  
Novel Technique ◽  
Structural Rigidity ◽  
Mammalian Skin ◽  
Shape Changes ◽  
The Ideal

A novel technique to generate three-dimensional Euclidean weavings, composed of close-packed, periodic arrays of one-dimensional fibres, is described. Some of these weavings are shown to dilate by simple shape changes of the constituent fibres (such as fibre straightening). The free volume within a chiral cubic example of a dilatant weaving, the ideal conformation of the G 129 weaving related to the Σ + rod packing, expands more than fivefold on filament straightening. This remarkable three-dimensional weaving, therefore, allows an unprecedented variation of packing density without loss of structural rigidity and is an attractive design target for materials. We propose that the G 129 weaving (ideal Σ + weaving) is formed by keratin fibres in the outermost layer of mammalian skin, probably templated by a folded membrane.

The magnetic field about a three-dimensional block neodymium magnet

2021 ◽  
Vol 62 ◽  
pp. 386-405
Author(s):  
Graham John Weir ◽  
George Chisholm ◽  
Jerome Leveneur
Keyword(s):  
Magnetic Field ◽  
Permanent Magnets ◽  
Hard Disk Drives ◽  
Three Dimensional ◽  
Transverse Magnetic ◽  
Observation Point ◽  
Magnetization Direction ◽  
Potential Approach ◽  
Neodymium Magnet ◽  
The Magnetic Field

Neodymium magnets were independently discovered in 1984 by General Motors and Sumitomo. Today, they are the strongest type of permanent magnets commercially available. They are the most widely used industrial magnets with many applications, including in hard disk drives, cordless tools and magnetic fasteners. We use a vector potential approach, rather than the more usual magnetic potential approach, to derive the three-dimensional (3D) magnetic field for a neodymium magnet, assuming an idealized block geometry and uniform magnetization. For each field or observation point, the 3D solution involves 24 nondimensional quantities, arising from the eight vertex positions of the magnet and the three components of the magnetic field. The only unknown in the model is the value of magnetization, with all other model quantities defined in terms of field position and magnet location. The longitudinal magnetic field component in the direction of magnetization is bounded everywhere, but discontinuous across the magnet faces parallel to the magnetization direction. The transverse magnetic fields are logarithmically unbounded on approaching a vertex of the magnet.   doi:10.1017/S1446181120000097

scholarly journals Technical Note: Three-dimensional transient groundwater flow due to localized recharge with an arbitrary transient rate in unconfined aquifers

2016 ◽  
Vol 20 (3) ◽  
pp. 1225-1239 ◽  
Author(s):  
Chia-Hao Chang ◽  
Ching-Sheng Huang ◽  
Hund-Der Yeh
Keyword(s):  
Groundwater Flow ◽  
Integral Transform ◽  
Three Dimensional ◽  
Observation Point ◽  
Technical Note ◽  
Specific Yield ◽  
Vertical Flow ◽  
Dimensional Flow ◽  
Special Cases ◽  
Present Solution

Abstract. Most previous solutions for groundwater flow induced by localized recharge assumed either aquifer incompressibility or two-dimensional flow in the absence of the vertical flow. This paper develops a new three-dimensional flow model for hydraulic head variation due to localized recharge in a rectangular unconfined aquifer with four boundaries under the Robin condition. A governing equation describing spatiotemporal head distributions is employed. The first-order free-surface equation with a source term defining a constant recharge rate over a rectangular area is used to depict water table movement. The solution to the model for the head is developed with the methods of Laplace transform and double-integral transform. Based on Duhamel's theorem, the present solution is applicable to flow problems accounting for arbitrary time-dependent recharge rates. The solution to depth-average head can then be obtained by integrating the head solution to elevation and dividing the result by the aquifer thickness. The use of a rectangular aquifer domain has two merits. One is that the integration for estimating the depth-average head can be analytically achieved. The other is that existing solutions based on aquifers of infinite extent can be considered as special cases of the present solution before the time when the aquifer boundary had an effect on head predictions. With the help of the present solution, the assumption of neglecting the vertical flow effect on the temporal head distribution at an observation point outside a recharge region can be assessed by a dimensionless parameter related to the aquifer horizontal and vertical hydraulic conductivities, initial aquifer thickness, and the shortest distance between the observation point and the edge of the recharge region. The validity of assuming aquifer incompressibility is dominated by the ratio of the aquifer specific yield to its storage coefficient. In addition, a sensitivity analysis is performed to investigate the head response to the change in each of the aquifer parameters.

Three-dimensional magnetotelluric modelling in spherical Earth

2019 ◽  
Vol 217 (1) ◽  
pp. 532-557 ◽  
Author(s):  
Alexander V Grayver ◽  
Martin van Driel ◽  
Alexey V Kuvshinov
Keyword(s):  
Three Dimensional ◽  
Spherical Earth
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