Theoretical estimation of roll acceleration in beam seas using PDF line integral method

2020 ◽  
Author(s):  
Atsuo Maki ◽  
Leo Dostal ◽  
Yuuki Maruyama ◽  
Masahiro Sakai ◽  
Kei Sugimoto ◽  
...  
Keyword(s):  
Integral Method ◽  
Theoretical Estimation ◽  
Line Integral ◽  
Beam Seas

Gravitational attraction of a rectangular prism with depth‐dependent density

Geophysics ◽  
1992 ◽  
Vol 57 (3) ◽  
pp. 470-473 ◽  
Author(s):  
J. García‐Abdeslem
Keyword(s):  
Cross Sections ◽  
Integral Method ◽  
Three Dimensional ◽  
Density Contrast ◽  
Gravitational Attraction ◽  
Two Dimensional ◽  
Gravity Effect ◽  
Line Integral ◽  
Closed Expression ◽  
Dependent Density

The gravity effect produced by two and three‐dimensional bodies with nonuniform density contrast has been treated by several authors. One of the first attempts in this direction made by Cordell (1973), who developed a method to compute the gravity effect due to a two‐dimensional prism whose density decreases exponentially with depth. A different approach was proposed by Murthy and Rao (1979). They extended the line‐integral method to obtain the gravity effect for bodies of arbitrary cross‐sections, with density contrast varying linearly with depth. Chai and Hinze (1988) have derived a wavenumber‐domain approach to compute the gravity effect due to a vertical prism whose density contrast varies exponentially with depth. Recently, Rao (1990) has developed a closed expression of the gravity field produced by an asymmetrical trapezoidal body whose density varies with depth following a quadratic polynomial.

3D vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3D variable density contrast

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. I43-I53 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Integral Method ◽  
Numerical Solutions ◽  
Vertical Component ◽  
Building Blocks ◽  
Variable Density ◽  
Rectangular Prism ◽  
Density Contrast ◽  
Gravity Potential ◽  
Line Integral

Three-dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3D mass bodies with spatially variable density contrasts. A 3D vector gravity potential is defined for a 3D rectangular prism with density contrast varying in depth and horizontally. The vertical component of the gravity anomaly equals the flux of the 3D vector gravity potential through the enclosed surface of the prism. Thus, the 3D integral for the gravity anomaly is reduced to a 2D surface integral. In turn, a 2D vector gravity potential is defined. The vertical component of the gravity anomaly equals the net circulation of the 2D vector gravity potential along the enclosed contour bounding the surfaces of the prism. The 3D integral for the gravity anomaly is reduced to 1D line integrals. Further analytical or numerical solutions can then be obtained from the line integrals, depending on the forms of the density contrast functions. If an analytical solution cannot be obtained, the line-integral method is semianalytical, requiring numerical quadratures to be carried out at the final stages. Singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them. Then the vector-potential line-integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast, varying simply in one direction and sophisticatedly in three directions. The advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast.

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.

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