Earth crust vertical movements according to ITRF2000, ITRF2005,ITRF2008 and ITRF2014 coordinate systems and their comparing analysis

The analysis of Earth's crust movement vertical velocities was made both for separate regions, and averaged on regions. As input data coordinates and velocities of earth crust points, obtained in International Coordinate Systems ITRF2000, ITRF2005,ITRF2008 on the base of processing radio interferometric(VLBI), laser(SLR), Doppler(DORIS) and GPS observations was used. For the purpose of global analysis all input velocity values were averaged in the x trapezoids. For filling trapezoids the spherical function expansion to N=36 was made. Expansion harmonic coefficients allowed determine the global characteristics of earth crust movements both for all Earth, and for separate hemispheres, polar and equatorial regions, continents and oceans. It appears, that polar regions were risen, and equatorial ones were lowered, that can indicate the modern Earth oblateness reduction. The constructed maps of vertical velocities were compared with obtained by us map of earth crust strain distribution. It is appeared, that regions of modern earth surface rising(Fennoscandia, Canada, Antarctica) coincide with regions of vertical extension strain, and the lowering regions-with compression regions. Simultaneously with the determination of harmonious coefficients the mean-square error of approximation for expansion of power n=1 N and power dispersion were determined. Whereas the results converge badly and dispersion increased with the n grows, hence we can concludes, that vertical movements do not characterized global earth crust movement and denotes the unrelated character of these movement. Seemingly, the main cause of such movement origin is the local seismic events(earthquake, volcanic eruption and so on). It confirms by our conclusions about correspondence of vertical strain maximal gradient(and therefore vertical velocities) map of the earthquake distribution. In work the detailed analysis of obtained results for separate Earth's regions was made.

An optimized finite difference method based on a polar coordinate system for regional-scale irregular topography

The finite difference method(FDM)is an important numerical approach for simulating the propagation of seismic waves,and some FDMs can be used to study the impact of the Earth’s curvature and topography over large distances.To efficiently model the effects of the Earth’s irregular topography on the propagation of seismic waves,here we optimize a previously proposed grid mesh method and develop a novel two-dimensional boundary-conforming FDM based on a curvilinear polar coordinate system.This method efficiently simulates the propagation of seismic waves in an arc-shaped model with large variations in surface topography.Our method was benchmarked against other reported methods using several global-scale models.The consistency of the results confirms the validity of our proposed optimization strategy.Furthermore,our findings indicate that the proposed optimization strategy improves computational efficiency.

Grid Model for High-accuracy Coordinate Transformation of China Geodetic Coordinate System 2000

After implementing CGCS2000,establishing grid models for high-accuracy coordinate transformation which are mainly used to transform border lines and coordinate grids of topographic maps becomes an important issue in mapping applications.Consequently,a grid model for high-accuracy coordinate transformation of CGCS2000 is proposed.Specifically,we firstly analyze a minimum curvature equation of coordinate transformation,which possesses the characteristics of both the global and local smoothness,achieving better consistency with the consecutive smoothness for the coordinate transformation of map’s linear feature.Then an iterative calculation method of grid nodes and an approach for establishing regional grid models based on collocation by two-step minimization are proposed.Meanwhile,a data structure of grid model is constructed.Finally we give the optimized grid interval and transformation accuracy in China corresponding to the proposed grid model.Using 48 433 points of 2000 National Geodetic Control Network of China,we take the proposed model into practice by constructing grid models for coordinate transformation from BJS54 and XAS80 to CGCS2000,and the external positional accuracies for both models are 0.26 m and 0.03 m respectively.

A Preliminary Study on the Theory of Polar Coordinates Digital Photogrammetry and the Coordinate System of Spatial Information

The development of new aeronautics and astronautics technologies has been constrained by strict mathematical rules for data processing among the diverse methods used to obtain spatial information.The acquisition of spatial information has been affected by various choices including the applied technologies(e.g.,push broom sensors),techniques(e.g.,zoom imaging),and equipment settings(e.g.,swing angle,aerial platform attitude,camera angle)in terms of the convergence,efficiency,and accuracy of the data.Based on the principle of the bionic machine parallax angle and pyramidal projection of the aerial space platform to the surface,this study explored solutions for high-resolution image sparsity,ill-conditioned singularity,and non-convergence by building a set of mathematical models to process the polar coordinates of the parallax angular vector.This study also formed a polar information theory for initial spatial information.This method improved the ranges of accuracy,efficiency,and anti-interference in close-range photogrammetry and the free net bundle adjustment model by several orders of magnitude.The open source code was made globally available more than 3 years ago,and has received positive reactions.The method’s effectiveness was verified using aerophotogrammetry and absolute network adjustment model experiments,and its performance was better than that of the Cartesian coordinate processing method.Finally,the higher-order solution characteristics of various applications and spaceflight platforms were provided,which are expected to provide a foundation for construction of a new polar coordinate system for aerospace multi-scale all-attitude spatial information acquisition,organization,management,storage,processing,and application.

ON THE OBJECTIVE STRESS RATE IN CO-MOVING COORDINATE SYSTEM

The objective stress rate is a rather important problem in mechanics of finitedeformation.In this paper,the objective stress rate in co-moving coordinate is derived byapplying nonlinear geometric field theory of deformation.Problems.such ax largeextension coupled with rotation,and large shear deformation,are exemplified by using thenew formula.Comparing with Jaumann’s stress rate and other formulae presented incurrent literature,the new result appears to be the reasonable one in co-moving coordinatesystem.

Curve integral with path independent in orthogonal curvilinear coordinate system

It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.

Plücker Coordinate System of Screws and the Study of Generating Gears

After an introductiou of the Plucker coordinate geometry, a discussion is made of the expression of screws in Plucker coordiuates and the addition of screws’ As a result, the geometry or geuerating gears is re-studied and a formula is derived for calculating the axode leugth or hypoid gears’

Uzbekistan's coordinate system transformation from CS42 to WGS84 using distortion grid model

Currently,a non-geocentric geodetic coordinate system introduced in 1942(CS42),based on the Krasovsky ellipsoid and the Baltic system of normal heights introduced in 1977,is used for geodetic works in the Republic of Uzbekistan.The development of the GNSS network proposes a task of transition to a new geocentric system based on the World Geodetic System 1984(WGS84)ellipsoid.Many software products adopt 3-or 7-parameter conversions,which can cause errors up to several meters and are not suitable for many applications.In this case,local transformations using a grid of differences between observation points with known coordinates in both datums would give the best accuracy.In this paper,we discuss various interpolation methods(Kriging,Minimum Curvature,Inverse Distance to a Power and Radial Basis Function)to solve the distortion modeling between CS42 and WGS84 systems for national datum improvement.The results show that the distortion models share a common tendency for all interpolation methods:the maximum horizontal displacements are concentrated along the West Tien Shan lineament,which is the boundary of the relief lowering.The discrepancies between the grid-based(calculated)and GPS-measured coordinates are evaluated.Statistical and spatial analysis has confirmed that for the co-ordinate transformation from CS42 to WGS84 and vice versa,grid-based transformation with Radial Basis Function interpolation has a high accuracy transformation.Analysis of the available data across the eastern part of the country shows that some positional distortions existed between the CS42 and WGS84 datums.For the best RBF method,the magnitude of these distortions is about 0.019-0.755 m with a standard deviation of 0.015 m.

Engineering coordinate system for vertical mine region based on GPS technique

For the problems existing in control network of vertical mine region established by conventional way, this paper advances a new scheme of establishing engineering coordinate system for vertical mine region. Based on the GPS positioning technique, through the relationship of coordinate transformation about a group of connecting points, the transforming parameters between the WGS-84 system and the Beijing54 system or Xi’an80 system about the region were calculated. The plane coordinate systems about the region can be maintained with a few of GPS control points using the method of GPS level, and fits the quasi-geoid about the region based on the Ellipsoid WGS-84. Huanghai Vertical Datum 1985 about the region can be maintained with the Ellipsoid WGS-84 as datum plane. Practice proves that there are advantages using the scheme to establish engineering coordinate system: the coordinate system as engineering foundation for vertical mine region will be in preservation for a long time; it is convenient and easy to carry out daily surveying work based on this coordinate system.

GENERALIZATION TO CYLINDRICAL COORDINATE SYSTEM FOR THE HIGHER ORDER KDV AND KP EQUATION

Two new model equations which we call a higher order cylindrical KdV equation and a higher order cylindrical KP equation are proposed when we consider the 2-dimensional modulation of water waves with surface tension in cylindrical coordinate system.Some exact (2+1)-dimensional solitary wave solutions have been obtained by relating them to the higher order KdV equation through an appropriate variable transformation.

Restricted Three-Body Problem in Different Coordinate Systems

In this paper of the series, the equations of motion for the spatial circular restricted three-body problem in sidereal spherical coordinates system were established. Initial value procedure that can be used to compute both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedure was illustrated by numerical example and graphical representations of the variations of the two sidereal coordinate systems.

Subdomain Chebyshev Spectral Method for 2D and 3D Numerical Differentiations in a Curved Coordinate System

A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.

Spatial Distribution of Seismicity: Relationships with Geomagnetic Z-Component in Geocentric Solar Magnetospheric Coordinate System

For 173477 epicenters of earthquakes with М ≥ 4.5, which occurred at the globe in 1973-2010, the geomagnetic Z-component in Geocentric Solar Magnetospheric (GSM) coordinate system were evaluated for the moment of earthquake occurrence on the base of the International Geomagnetic Reference Field model (IGRF-10). It is found that in the regions, where the ZGSM reaches large positive value (low and middle latitudes), earthquake occurrence is higher than in the regions where ZGSM is mainly negative (high latitudes). In the area of strongest seismicity at the globe, which is located in the longitudinal ranges of about 1200E - 1700W, the values of ZGSM are the most high at the globe. It is found that statistically significant dependence, with correlation coefficient R = 0.91, exists between the maximal possible magnitude of earthquake (Mmax) and the logarithm of absolute value of ZGSM . We suggest that earthquake occurrence is triggered by the perturbations, which in first occur at the magnetopause due to reconnection of the magnetic field of the solar wind with the Earth’s magnetic field, and then propagate into the solid earth via the GEC, which is considered at present as a main applicant for a physical mechanism of solar-terrestrial relationships. It is clear that much work remains to further verify this speculative assertion and to find the physical processes linking seismicity with the main geomagnetic field structure.

Reference Coordinate System of Nonlinear Beam Element Based on the Geometrically Exact Formulation under Large Spatial Rotations and Deformations

Analysis of slender beam structures in a three-dimensional space is widely applicable in mechanical and civil engineering. This paper presents a new procedure to determine the reference coordinate system of a beam element under large rotation and elastic deformation based on a newly introduced physical concept: the zero twist sectional condition, which means that a non-twisted section between two nodes always exists and this section can reasonably be regarded as a reference coordinate system to calculate the internal element forces. This method can avoid the disagreement of the reference coordinates which might occur under large spatial rotations and deformations. Numerical examples given in the paper prove that this procedure guarantees the numerical exactness of the inherent formulation and improves the numerical efficiency, especially under large spatial rotations.

A Polar Coordinate System Based Grid Algorithm for Star Identification

In Cartesian coordinate systems, the angular separation-based star identification algorithms involve much trigon- ometric function computing. That delays the algorithm process. As in a polar coordinate system, the coordinates are denoted by angular values, it is potential to speed up the star identification process by adopting a polar coordinate sys-tem. An angular polar coordinate system is introduced and a grid algorithm based on the coordinate system is proposed to enhance the performances of the star identification process. The simulations demonstrate that the algorithm in the angular polar coordinate system is superior to the grid algorithm in the rectangle Cartesian coordinate system in com-puting cost and identification rate. It can be used in the star sensors for high precision and high reliability in spacecraft navigation.

Derivations of Vector Area and Volume Elements in Curved Coordinate Systems for Flux Vector Fields Helping Eye Disease

Our aim in this paper is to interest retinal eye specialists in preventing dry macula degeneration by a special flurry vector field through open or closed curved surfaces. The flux of vector fields through surfaces is based on vector element area and volume element. Therefore, we explain a few geometrical derivations of area and volume elements in curved orthogonal coordinate systems. We hope that by derivation of a spatial vector field flurry against drusen through open or closed surfaces due to the Gauss theorem might select drusen under eye retina cells without destroying the cells and prevent macula degeneration. A changed flurry of a magnetic or electric vector field through a closed line causes an electric or magnetic vector field on the surface closed by the line. We also hope that derivation by Stokes’ and Greens’ theorems, with the help of iron, might help eye cells to get in life.

Riemannian Acceleration in Oblate Spheroidal Coordinate System

The planetary bodies are more of a spheroid than they are a sphere thereby making it necessary to describe motions in a spheroidal coordinate system. Using the oblate spheroidal coordinate system, a more approximate and realistic description of motion in these bodies can be realized. In this paper, we derive the Riemannian acceleration for motion in oblate spheroidal coordinate system using the golden metric tensor in oblate spheroidal coordinates. The Riemannian acceleration in the oblate spheroidal coordinate system reduces to the pure Newtonian acceleration in the limit of c0 and contains post-Newtonian correction terms of all orders of c-2. The result obtained thereby opens the way for further studies and applications of the motion of particles in oblate spheroidal coordinate system.

The Establishment of the Projections of Four to Seven-Dimensional Rectangular Coordinate Systems into the Three-Dimensional Space

Starting from the finite rotation group,the author makes a penetrating study of 4 to 7-dimensional hypercube so that we have acquired the projection models of 4 to 7-dimeneional spatial rectangular coordinate systems into the three-dimensional space to have the 4 to 7-dimensional geometric figures demonstrated correctly.

Light-Cone Coordinate System in General Relativity

If there exists a null gradient field in 3 + 1 dimensional space-time, we can set up a kind of light-cone coordinate system in the space-time. In such coordinate system, the metric takes a simple form, which is helpful for simplifying and solving the Einstein’s field equation. This light-cone coordinate system has wonderful properties and has been used widely in astrophysics to calculate parameters. We discuss the structure of space-time with light-cone coordinate system in detail. We show how to construct the light-cone coordinate system and obtain the conditions of its existence, and then explain their geometrical and physical meanings.

Stochastic programming based coordinated expansion planning of generation,transmission,demand side resources,and energy storage considering the DC transmission system

With the increasing penetration of wind and solar energies,the accompanying uncertainty that propagates in the system places higher requirements on the expansion planning of power systems.A source-grid-load-storage coordinated expansion planning model based on stochastic programming was proposed to suppress the impact of wind and solar energy fluctuations.Multiple types of system components,including demand response service entities,converter stations,DC transmission systems,cascade hydropower stations,and other traditional components,have been extensively modeled.Moreover,energy storage systems are considered to improve the accommodation level of renewable energy and alleviate the influence of intermittence.Demand-response service entities from the load side are used to reduce and move the demand during peak load periods.The uncertainties in wind,solar energy,and loads were simulated using stochastic programming.Finally,the effectiveness of the proposed model is verified through numerical simulations.