2D vector gravity potential and line integrals for the gravity anomaly caused by a 2D mass of depth-dependent density contrast

Geophysics ◽  
2008 ◽  
Vol 73 (6) ◽  
pp. I43-I50 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Density Function ◽  
Case Studies ◽  
Two Dimensions ◽  
Density Contrast ◽  
Gravity Potential ◽  
Gravity Modeling ◽  
Desktop Computer ◽  
Contrast Function ◽  
Dependent Density

Using line integrals (LIs) used to calculate the gravity anomaly caused by a 2D mass of complicated geometry and spatially variable density contrast is a computationally efficient algorithm, that reduces the calculation from two dimensions to one dimension. This work has developed a mechanism for defining LIs systematically for different types of density functions. Two-dimensional vector gravity potential is defined as a vector, the net circulation of which, along the closed contour bounding a 2D mass, equals the gravity anomaly caused by the 2D mass. Two representative types of LIs are defined: an LI with an arctangent kernel for any depth-dependent density-contrast function, which has been studied historically; and an LI with a simple algebraic kernel for any integrable density-contrast function. The present work offers (1) a vectorial-based derivation of formulas that do not suffer from the arbitrary sign conventions found in some historical approaches; and (2) a simple algebraic kernel in line integrals as an alternative to the historical arctangent kernel, with the possibility of extension to more general cases. The concept of 2D vector gravity potential provides a useful tool for defining LIs systematically for any mass density function, helping us understand how dimensions can be reduced in a calculating gravity anomaly, especially when the density contrast varies with space. LIs have been tested in case studies. The maximum differences in calculated gravity anomalies by different LIs for the case studies were between [Formula: see text] and [Formula: see text]. Processing time required per station per segment of the 2D polygon of a 2D mass using LIs is [Formula: see text] on a Dell Optiplex GX 620 desktop computer, almost independent of the density function. The results indicate that the two types of LIs provide very fast, efficient, and reliable algorithms in gravity modeling or inversion for various types of density-contrast functions.

General line integrals for gravity anomalies of irregular 2D masses with horizontally and vertically dependent density contrast

Geophysics ◽  
2009 ◽  
Vol 74 (2) ◽  
pp. I1-I7 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Gravity Anomalies ◽  
Density Contrast ◽  
Vertical Position ◽  
Horizontal Position ◽  
Surface Integral ◽  
Cross Sectional ◽  
Contrast Model ◽  
Contrast Function ◽  
Dependent Density

Line integrals (LIs) are an efficient tool in calculating the gravity anomaly caused by an irregular 2D mass body because the 2D surface integral is reduced to a 1D LI. Historically, LIs have been derived for 2D mass bodies of depth-dependent density contrast. I derive LIs for 2D mass bodies with density contrast dependent on (1) horizontal and (2) horizontal and vertical directions. Assuming the density contrast depends only on horizontal position, two types of representative LIs are derived: LIs with logarithmic kernel and density-integrated LIs for any integrable density-contrast function. A general density-contrast model that depends on horizontal and vertical directions is developed to include three components: a function of horizontal position, a function of vertical position, and a sum of crossterms of horizontal and vertical positions. Based on the general density-contrast model defined and proper selection of 2D vector gravity potentials, general LIs are derived to calculate the gravity anomaly. The newly developed LI method is then compared with two cases from the literature in calculating gravity anomaly, and agreement is obtained. However, the new LI method allows for more general 2D density-contrast variations and can be used to calculate the gravity anomaly of a 2D mass body. Such a mass body can have any cross-sectional profile that can be approximated by a polygonal cross section with any density-contrast function that can be approximated by a rich set of basis functions.

Analytic solution of the gravity anomaly of irregular 2D masses with density contrast varying as a 2D polynomial function

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. I11-I19 ◽  
Author(s):  
Xiaobing Zhou
Keyword(s):  
Gravity Anomaly ◽  
Density Function ◽  
Coordinate Transformation ◽  
Numerical Stability ◽  
Analytic Solution ◽  
Polynomial Function ◽  
Transformation Method ◽  
Density Contrast ◽  
Vertical Position ◽  
Analytic Methods

The analytic solution of the gravity anomaly caused by a 2D irregular mass body with the density contrast varying as a polynomial function in the horizontal and vertical directions is extrapolated from a historical version in which the analytic solution for the gravity anomaly was given only at the origin of the coordinate system to any point for the density function in terms of variables relative to that origin. To calculate the gravity anomaly at stations that are not at origins, a coordinate transformation is performed, in which case the polynomial density contrast function must also be expressed in the transformed coordinates, or a transformed solution must be obtained. These analytic solutions can be obtained at any station using (1) a solution transformation method, in which the density function and boundary of a mass body are kept intact, or (2) a coordinate transformation method, in which polynomial coefficient and boundary of a mass body are transformed accordingly. The issue of singularity and instability of the analytic methods has been related to case studies. Caution should be exercised in modeling or interpreting the gravity survey data using the analytic methods for large target-distance-to-target-size ratios outside the range of numerical stability. Compared with other published methods, the analytic solution results agree very well with other numerical or seminumerical methods, indicating the solution is correct and can be applied for any gravity anomaly calculation caused by an irregular 2D mass body with the density-contrast approximated as a polynomial function of horizontal position and/or vertical position when the observation is within the range of numerical stability.

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.

The Politics of E-Learning

2018 ◽  
pp. 109-127
Author(s):  
Mahesh S. Raisinghani ◽  
Celia Romm Livermore ◽  
Pierluigi Rippa
Keyword(s):  
Case Studies ◽  
Learning Strategies ◽  
Two Dimensions ◽  
The Political ◽  
Political Strategy ◽  
Political Strategies ◽  
Organizational Contexts ◽  
Different Types ◽  
E Learning

The goal of this chapter was to study the political strategies utilized in the context of e-learning. The research is based on the e-learning political strategies (ELPoS) model. The model is based on two dimensions: (1) the direction of the political strategy (upward or downward) and (2) the scope of the political strategy (individual or group based). The model assumes that the interaction between these dimensions will define four different types of e-learning political strategies, which, in turn, will lead to different outcomes. The model is presented in the context of the literature on e-learning and is accompanied with four short case studies that demonstrate its political strategies. The discussion and conclusions section integrates the findings from the case studies and outlines the rules that govern the utilization of political e-learning strategies in different organizational contexts.

Syntactic Argumentation

2019 ◽  
pp. 20-735
Author(s):  
Bas Aarts
Keyword(s):  
Case Studies ◽  
Final Section ◽  
Two Dimensions ◽  
English Grammar

The aim of this chapter is to discuss the role of syntactic argumentation in the description of English grammar. The chapter first discusses some of the general principles that are used in syntactic argumentation, namely economy and elegance, which are regarded as two dimensions of simplicity. These principles are then exemplified in a number of case studies. The final section discusses how argumentation can be used to establish constituency in clauses.

Efficient gravity inversion of basement relief using a versatile modeling algorithm

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. G23-G34 ◽  
Author(s):  
João B. C. Silva ◽  
Darcicléa F. Santos
Keyword(s):  
Gravity Anomaly ◽  
Region Of Interest ◽  
Sedimentary Basins ◽  
Density Contrast ◽  
Gravity Inversion ◽  
Deep Basin ◽  
Novel Approach ◽  
Green’S Operator ◽  
Translation Symmetry ◽  
Observation Position

We have developed a novel approach to compute, in an efficient and versatile way, the gravity anomaly produced by an arbitrary, discrete 3D distribution of density contrast. The method allows adjustable precision and is particularly suited for the interpretation of sedimentary basins. Because the gravity field decays with the square of the distance, we use a discrete Green’s operator that may be made much smaller than the whole study area. For irregularly positioned observations, this discrete Green’s operator must be computed just at the first iteration, and because each of its horizontal layers presents a center of symmetry, only one-eighth of its total elements need to be calculated. Furthermore, for gridded data on a plane, this operator develops translation symmetry for the whole region of interest and has to be computed just once for a single arbitrary observation position. The gravity anomaly is obtained as the product of this small operator by any arbitrary density contrast distribution in a convolution-like operation. We use the proposed modeling to estimate the basement relief of a [Formula: see text] basin with density contrast varying along [Formula: see text] only using a smaller Green’s operator at all iterations. The median of the absolute differences between relief estimates, using the small and a large operator (the latter covering the whole basin) has been approximately 9 m for a 3.6 km deep basin. We also successfully inverted the anomaly of a simulated basin with density contrast varying laterally and vertically, and a real anomaly produced by a steeply dipping basement. The proposed modeling is very fast. For 10,000 observations gridded on a plane, the inversion using the proposed approach for irregularly spaced data is two orders of magnitude faster than using an analytically derived fitting, and this ratio increases enormously with the number of observations.

scholarly journals Level Sets of Random Fields and Applications: Specular Points and Wave Crests

2010 ◽  
Vol 2010 ◽  
pp. 1-22
Author(s):  
Esteban Flores ◽  
José R. León R
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Random Fields ◽  
Level Sets ◽  
Two Dimensions ◽  
Level Curve ◽  
Curvilinear Integral ◽  
The Waves ◽  
Rice Formula

We apply Rice's multidimensional formulas, in a mathematically rigorous way, to several problems which appear in random sea modeling. As a first example, the probability density function of the velocity of the specular points is obtained in one or two dimensions as well as the expectation of the number of specular points in two dimensions. We also consider, based on a multidimensional Rice formula, a curvilinear integral with respect to the level curve. It follows that its expected value allows defining the Palm distribution of the angle of the normal of the curve that defines the waves crest. Finally, we give a new proof of a general multidimensional Rice formula, valid for all levels, for a stationary and smooth enough random fields X:ℝd→ℝj(d>j).

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