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.

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.

A versatile solution for the gravity anomaly of 3D prism-meshed bodies with depth-dependent density contrast

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. G77-G86 ◽  
Author(s):  
Li Jiang ◽  
Jianzhong Zhang ◽  
Zhibing Feng
Keyword(s):  
Los Angeles ◽  
Half Space ◽  
Numerical Stability ◽  
Analytic Solution ◽  
Gravity Anomalies ◽  
Variable Density ◽  
Density Contrast ◽  
Contrast Variation ◽  
Stability Range ◽  
Limit Values

We have developed a generalized solution for computing the gravity anomalies of 3D irregular-mass bodies with complicated density-contrast variation. The 3D irregular-shaped bodies can be approximated flexibly by a collection of finite-juxtaposed right-rectangular prisms. The complicated density-contrast variation of each prism can be well-represented by a depth-dependent polynomial function. A novel analytic solution of gravity anomalies due to a right-rectangular prism with an arbitrary order of polynomial density-contrast function of depth is then derived. The solution is singularity free in the upper half-space over the prism, and its singularity in the lower half-space containing the prism is resolved by assigning their limit values to the singular terms. The numerical stability of the solution is also evaluated through numerical tests. Hence, the solution can be used to compute the gravity anomalies of 3D irregular bodies with variable density contrasts without singularities when computation points are within the numerical stability range. Based on synthetic models with variable density contrast, our solution is validated by using other solutions in the literature. We also simulated the gravity anomalies of the Los Angeles basin and compared them with the observed anomalies and with those computed using the analytic solutions of other workers. These tests confirm the accuracy and efficiency of our analytic solution.

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.

scholarly journals Solutions of Conformable Fractional-Order SIR Epidemic Model

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Atimad Harir ◽  
Said Malliani ◽  
Lalla Saadia Chandli
Keyword(s):  
Fractional Order ◽  
Epidemic Model ◽  
Analytic Solution ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Transformation Method ◽  
Conformable Fractional Derivative ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Fractional Differential

In this paper, the conformable fractional-order SIR epidemic model are solved by means of an analytic technique for nonlinear problems, namely, the conformable fractional differential transformation method (CFDTM) and variational iteration method (VIM). These models are nonlinear system of conformable fractional differential equation (CFDE) that has no analytic solution. The VIM is based on conformable fractional derivative and proved. The result revealed that both methods are in agreement and are accurate and efficient for solving systems of OFDE.

A Study of Tracking System Using Multiple Emitter Towers

2013 ◽  
Vol 694-697 ◽  
pp. 927-935 ◽  
Author(s):  
Yi Sun ◽  
Tao Ma ◽  
Chia Yung Han ◽  
Joseph Ross ◽  
William Wee
Keyword(s):  
Coordinate Transformation ◽  
Motion Tracking ◽  
Tracking System ◽  
Transformation Method ◽  
Tracking Accuracy ◽  
Working Space ◽  
Frontal Part ◽  
Motion Tracking System ◽  
Coordinate Transformation Method

This paper presents a simple and accurate coordinate transformation method for extending the tracking space of the Intersense IS-900 spatial and motion tracking system using multiple pre-configured emitter towers to form the emitter constellation, but without resorting to the use of a surveyor machine. The proposed approach uses the differences of positional coordinate readings from each emitter tower among a set of commonly viewed spatial points to calculate the parameters needed to define the coordinate transformation. By applying this method, the tracking accuracy using the entire emitter constellation can be achieved by less than 0.5 inches error in most of the working space, and as low as 0.2 inches error in the frontal part of the working space.

Study on NURBS Surface Tool Trajectory Planning of 6R Engraving Robot

2013 ◽  
Vol 711 ◽  
pp. 422-425 ◽  
Author(s):  
Yu Hu Zuo
Keyword(s):  
Trajectory Planning ◽  
Coordinate Transformation ◽  
Inverse Kinematics ◽  
End Milling ◽  
Transformation Method ◽  
Nurbs Surface ◽  
Calculation Algorithm ◽  
Tool Trajectory ◽  
Coordinate Transformation Method ◽  
Cutting Point

A NURBS surface tool trajectory planning method of engraving robot is proposed. The calculation algorithm including NURBS surface tool trajectory, cutting point and effective cutting radius of end milling cutter and inverse kinematics transform is discussed in detail using Taylor and coordinate transformation method. It is the foundation to further applied to the engraving robot tool trajectory planning or off-line programming.

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