A LINE‐INTEGRAL METHOD OF COMPUTING THE GRAVIMETRIC EFFECTS OF TWO‐DIMENSIONAL MASSES

Geophysics ◽  
1948 ◽  
Vol 13 (2) ◽  
pp. 215-225 ◽  
Author(s):  
M. King Hubbert
Keyword(s):  
Opposite Sign ◽  
Gravity Anomalies ◽  
Constant Density ◽  
Two Dimensional ◽  
Horizontal Scale ◽  
Special Equipment ◽  
Line Integral ◽  
Vertical Coordinate ◽  
Plane Normal ◽  
Polar Coordinate

Many computing schemes have been devised for determining the gravity anomalies produced by two‐dimensional masses. Most of these are based upon the evaluation of an areal integral and require specially constructed templates or tables. In the present paper it is shown that the gravity anomaly Δg at the origin of coordinates, produced by a two‐dimensional mass of constant density contrast Δρ, may be obtained quite simply by means of either of the line integrals [Formula: see text] where z is the vertical coordinate, and θ the polar coordinate expressed in radians of a point on the periphery of the mass in a plane normal to its axis and passing through the origin. The line integrals are evaluated around the periphery of the mass and are of opposite sign if taken in the same direction of traverse, or are of the same sign if taken in opposite directions. For use of these integrals no special equipment is required other than a simple template consisting of radial lines, θ=const., and horizontal lines, z=const., which can be constructed in a few minutes with protractor and scale. This can be constructed either for 1:1 or for an exaggerated vertical‐to‐horizontal scale.

Gravity anomalies of two‐dimensional bodies of irregular cross‐section with density contrast varying with depth

Geophysics ◽  
1979 ◽  
Vol 44 (9) ◽  
pp. 1525-1530 ◽  
Author(s):  
I. V. Radhakrishna Murthy ◽  
D. Bhaskara Rao
Keyword(s):  
Gravity Anomaly ◽  
Cross Section ◽  
Cross Sections ◽  
Integral Method ◽  
Gravity Anomalies ◽  
Density Contrast ◽  
Two Dimensional ◽  
Gravity Effect ◽  
Line Integral ◽  
Exponential Increase

The line‐integral method of Hubbert (1948) is extended to obtain the gravity anomalies of two‐dimensional bodies of arbitrary cross‐sections with density contrast varying linearly with depth. The cross‐section is replaced by an N‐sided polygon. The coordinates of two vertices of any given side are used to determine the associated contribution to the gravity anomaly. The gravity contribution of each side is then summed to yield the total gravity effect. The case where density contrast varies exponentially with depth is also considered. This technique is used to obtain the structure of the San Jacinto Graben, California, where sediments filling the graben have an exponential increase in density with depth.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

A Study of Two-Dimensional Supersonic Air Ejector Systems

1973 ◽  
Vol 187 (1) ◽  
pp. 733-743
Author(s):  
R. S. Benson ◽  
V. A. Eustace
Keyword(s):  
Flow Field ◽  
Secondary Flow ◽  
Ambient Pressure ◽  
Performance Characteristics ◽  
Constant Density ◽  
Two Dimensional ◽  
Theoretical Predictions ◽  
Flow Field Characteristics ◽  
Air Ejector

The performance and flow field characteristics for two-dimensional ejector systems are determined theoretically for the condition when operation is independent of ambient pressure. The method considers the detailed inviscid interaction between the primary and secondary streams within the mixing tube and an estimate is made of the secondary flow entrained by the two-stream viscous mixing region. The validity of the theory is tested by comparing the performance characteristics of an experimental ejector facility with theoretical predictions and by comparing the theoretical flow field, in terms of constant density contours, with infinite fringe interferograms.

GRAVITY ANOMALIES OF TWO‐DIMENSIONAL FAULTS

Geophysics ◽  
1966 ◽  
Vol 31 (2) ◽  
pp. 372-397 ◽  
Author(s):  
L. P. Geldart ◽  
Denis E. Gill ◽  
Bijon Sharma
Keyword(s):  
Fault Plane ◽  
Marked Effect ◽  
Gravity Anomalies ◽  
Two Dimensional ◽  
Gravity Effect ◽  
Fault Density ◽  
Wide Range ◽  
Bed Thickness ◽  
Two Sides ◽  
Simplified Formula

A simplified formula is given for the gravity effect of a horizontal semi‐infinite block truncated by a dipping plane. This formula is used to obtain curves illustrating the gravity anomalies for blocks having different thicknesses and depths truncated by planes dipping at various angles. By combining two blocks, results are obtained for faulted horizontal beds for a wide range of bed thicknesses and depths, fault displacements and dips. These should be useful as guides in interpreting fault anomalies, and in planning gravity programs intended to map faults. The most striking feature of the curves is the marked effect of the dip of the fault plane on the curves for faulted beds. The asymmetry of the fault curves is related mainly to the dip and can be used to determine dips between 30 and 90 degrees. If the dip of the fault, density contrast, and bed thickness are known, the depths to the bed on the two sides of the fault are given by the sizes and positions of the gravity maximum and minimum.

A Two-Dimensional Cellular Automaton Crystal with Irrational Density

2003 ◽  
Author(s):  
David Griffeath ◽  
Dean Hickerson
Keyword(s):  
Cellular Automaton ◽  
Asymptotic Density ◽  
Coarse Grain ◽  
Constant Density ◽  
Mathematical Framework ◽  
Two Dimensional ◽  
Empty Cell ◽  
Asymptotic Shape ◽  
Conway’S Game Of Life ◽  
Asymptotic Densities

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

scholarly journals Theoretical Gravity Anomalies of Glaciers Having Parabolic Cross-Sections

1964 ◽  
Vol 5 (38) ◽  
pp. 255-257 ◽  
Author(s):  
Charles E. Corbató
Keyword(s):  
Cross Sections ◽  
Lower Boundary ◽  
Gravity Anomalies ◽  
Vertical Axis ◽  
Two Dimensional ◽  
Axis Of Symmetry ◽  
Upper Boundary

AbstractEquations and a graph are presented for calculating gravity anomalies on a two-dimensional glacier model having a horizontal upper boundary and a lower boundary which is a parabola with a vertical axis of symmetry.

TWO-DIMENSIONAL VISUALIZATION OF THE PROPAGATION SPEED OF CORTICAL SPREADING DEPRESSION IN RAT CORTEX

2010 ◽  
Vol 03 (01) ◽  
pp. 75-80
Author(s):  
TINGTING XU ◽  
PENGCHENG LI ◽  
SHANGBIN CHEN ◽  
WEIHUA LUO
Keyword(s):  
Cortical Spreading Depression ◽  
Spreading Depression ◽  
Time Lag ◽  
Polar Angle ◽  
Propagation Speed ◽  
Least Square ◽  
Data Matrix ◽  
Radial Coordinate ◽  
Two Dimensional ◽  
Polar Coordinate

Cortical spreading depression (CSD), which is a significant pathological phenomenon that correlates with migraines and cerebral ischemia, has been characterized by a wave of depolarization among neuronal cells and propagates across the cortex at a rate of 2–5 mm/min. Although the propagation pattern of CSD was well-investigated using high-resolution optical imaging technique, the variation of propagation speed of CSD across different regions of cortex was not well-concerned, partially because of the lack of ideal approach to visualize two-dimensional distribution of propagation speed of CSD over the whole imaged cortex. Here, we have presented a method to compute automatically the propagation speed of CSD throughout every spots in the imaged cortex. In this method, temporal clustering analysis (TCA) and least square estimation (LSE) were first used to detect origin site where CSD was induced. Taking the origin site of CSD as the origin of coordinates, the data matrix of each image was transformed into the corresponding points based on the polar-coordinate representation. Then, two fixed-distance regions of interest (ROIs) are sliding along with the radial coordinate at each polar angle within the image for calculating the time lag with correlating algorithm. Finally, we could draw a two-dimensional image, in which the value of each pixel represented the velocity of CSD when it spread through the corresponding area of the imaged cortex. The results demonstrated that the method can reveal the heterogeneity of propagation speed of CSD in the imaged cortex with high fidelity and intuition.

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