Establishment and application of the anisotropic shale-rock physical model in the observation coordinate system

No shale-rock physical model has been established in the observation coordinate system.To this end,this paper carried out anisotropic wave velocity tests on shale rock and compared the Thomsen,Daley,and Berryman solutions to characterize anisotropic acoustic wave velocity.Finally,the Daley solution was selected.Based on basic rock physical models,such as SCA and DEM methods,and combined with the Daley solution,an anisotropic shale-rock physical model was established in the observation coordinate system and applied in Well B1 in the Luzhou area,Sichuan Basin.Our research conclusions were as follows:1.for the samples from the same core,the P-wave velocities in three directions were in the order VP11>VP45>VP33,shear wave velocity VS11 was the largest,but VS33 and VS45 did not follow the law of Vs33>Vs45 for some samples;2.the Daley solution,which not only considers the accuracy requirements but also has a complete expression of P-,SV-,and SH-waves,is most suitable for characterization of anisotropic wave velocity in this study area;3.the rock physical model constructed in the observation coordinate system has high accuracy,in which the absolute value of the relative error of the P-wave slowness was between 0%and 5.05%(0.55%on average),and that of shear-wave slowness was between 0%and 6.05%(0.59%on average);4.the acoustic waves recorded in Well B1 in the observation coordinate system were very different from those in the constitutive coordinate system.The relative difference of the P-wave was between 6.76%and 30.84%(14.68%on average),and that of the S-wave was between 7.00%and 23.44%(13.99%on average).The acoustic slowness measured in the observation coordinate system,such as in a deviated well or a horizontal well section,must be converted to the constitutive coordinate system before it can be used in subsequent engineering applications;5.the anisotropic shale-rock physical model built in the observation coordinate system proposed in this paper can provide basic data and guidance for subsequent pore pressure prediction,geomechanical modeling,and fracturing stimulation design for deviated and horizontal wells.

Influence of Adoptment of Geocentric 3D Coordinate System on Chinese Existent Maps

If a geocentric 3D coordinate system is adopted in China to replace 2D non-geocentric coordinate system, the coordinates of the surface points will be changed accordingly. The influences on the current maps of China, especially the topographic maps, are discussed due to the replacement of the coordinate systems. Taking the replacement of Xi’an 80 coordinate system by GRS80 for a numerical example, this paper analyzes the changes of latitude, longitude, Gauss plane coordinates of the surface points, as well as the orientation and the length changes between the points on the map, including the changes of map border lines and sheet corner points.

Combined Method of Datum Transformation Between Different Coordinate Systems

The similarity transformation model between different coordinate systems is not accurate enough to describe the discrepancy of them.Therefore,the coordinate transformation from the coordinate frame with poor accuracy to that with high accuracy cannot guarantee a high precision of transformation.In this paper,a combined method of similarity transformation and regressive approximating is presented.The local error accumulation and distortion are taken into consideration and the precision of coordinate system is improved by using the recommended method

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.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

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.

A novel particle tracking algorithm using polar coordinate system similarity

A new algorithm using polar coordinate system similarity （PCSS） for tracking particle in particle tracking velocimetry （PTV） is proposed. The essence of the algorithm is to consider simultaneously the changes of the distance and angle of surrounding particles relative to the object particle. Monte Carlo simulations of a solid body rotational flow and a parallel shearing flow are used to investigate flows measurable by PCSS and the influences of experimental parameters on the implementation of the new algorithm. The results indicate that the PCSS algorithm can be applied to flows subjected to strong rotation and is not sensitive to experimental parameters in comparison with the conventional binary image cross-correlation （BICC） algorithm. Finally, PCSS is applied to images of a real experiment.

Robust Wavelet-Based Facial Image Watermarking Against Geometric Attacks Using Coordinate System Recovery

A coordinate system of the original image is established using a facial feature point localization technique. After the original image transformed into a new image with the standard coordinate system, a redundant watermark is adaptively embedded in the discrete wavelet transform（DWT） domain based on the statistical characteristics of the wavelet coefficient block. The coordinate system of watermarked image is reestablished as a calibration system. Regardless of the host image rotated, scaled, or translated（RST）, all the geometric attacks are eliminated while the watermarked image is transformed into the standard coordinate system. The proposed watermark detection is a blind detection. Experimental results demonstrate the proposed scheme is robust against common and geometric image processing attacks, particularly its robustness against joint geometric attacks.

Unified Coordinate System Model for Performance Calculation of Fix-pad Journal Bearing with Different Pad Preload

Traditional model for calculating performance parameters of a fix-pad journal bearing leads to heavy workload, complicated and changeable formulae as it requires deriving various geometric formulae with different bearing types such as circular journal bearing, dislocated bearing and elliptic bearing. Considering different pad preload ratios for non-standard bearing, traditional model not only becomes more complicated but also reduces scalability and promotion of the calculation programs. For the complexly case of traditional model while dealing with various fix-pad journal bearings, unified coordinate system model for performance calculation of fix-pad journal bearing is presented in the paper. A unified coordinate system with the bearing center at the origin is established, and the eccentricity ratio and attitude angle of axis relative to each pad are calculated through the coordinates of journal center and each pad center. Geometric description of fix-pad journal bearing is unified in this model, which can be used for both various standard bearing and non-standard bearing with different pad preload ratios. Validity of this model is verified with an elliptical bearing. Performance of a non-standard four-leaf bearing with different pad preload ratios is calculated based on this model. The calculation result shows that increasing preload ratio of the pad 1 and keeping that of the left three pads constant improves bearing capacity, stiffness and damping coefficients. This research presents a unified coordinate system model unifies performance calculation of fix-pad journal bearings and studied a non-standard four-leaf bearing with different pad preload ratios, the research conclusions provides new methods for performance calculation of fix-pad journal bearings.

Elastic simulation method in an irregular polar coordinate system

When simulating the propagation of seismic waves in some special structures,such as tunnels and boreholes,finite difference forward modeling in the polar system has higher accuracy than the traditional Cartesian system.In actual situations,the polar space is the most irregular.To solve this problem,a forward modeling method for an irregular polar coordinate system is proposed to improve the simulation accuracy.First,an irregular surface of the polar space was meshed into an irregular polar system.After the transformation,the undulating surface was mapped into a plane one,and the wavefield was then computed in an irregular polar system.The Lebedev staggered grid was used to solve the wave equations in the irregular polar system.In addition,the artificial absorption boundary,cylindrical free boundary,and circumferential boundary conditions were used to absorb the boundary reflection.We selected three polar space models to demonstrate the new method in this study.The results show that the proposed elastic simulation method in an irregular polar coordinate system can produce more accurate and stable simulation results when modeling seismic wave propagation in an irregular polar space.Elastic full waveform inversion further shows that the irregular polar system elastic simulation method can accurately simulate the wavefield in an undulating polar space.

IMPACT STUDY ON THE CALCULATION OF VERTICAL VELOCITY IN DIFFERENT VERTICAL COORDINATE SYSTEMS

With the development of high-resolution and multi-scale unified numerical model, some of techniques about non-hydrostatic meso-scale numerical weather prediction are addressed. The impact of the vertical coordinate system is one of them. In this paper, based on a WRF (Weather Research and Forecast) model, the impact on the calculation of vertical velocity was studied with different vertical coordinates. The simulation results showed that the calculation of vertical velocity is sensitive to vertical coordinates. It is especially more evident when the resolution increased. Due to the close relationships between vertical velocity and precipitation, the difference of vertical velocity inevitably influences model’s description of precipitation. An ideal experiment exhibits that pressure gradient force computations in the pressure terrain- following coordinate are sensitive to surface pressure.

Interaction of Waves, Surface Currents, and Turbulence: the Application of Surface-Following Coordinate Systems

Surface waves comprise an important aspect of the interaction between the atmosphere and the ocean, so a dynamically consistent framework for modelling atmosphere-ocean interaction must take account of surface waves, either implicitly or explicitly. In order to calculate the effect of wind forcing on waves and currents, and vice versa, it is necessary to employ a consistent formula- tion of the energy and momentum balance within the airflow, wave field, and water column. It is very advantageous to apply sur- face-following coordinate systems, whereby the steep gradients in mean flow properties near the air-water interface in the cross-interface direction may be resolved over distances which are much smaller than the height of the waves themselves. We may account for the waves explicitly by employing a numerical spectral wave model, and applying a suitable theory of wave–mean flow interaction. If the mean flow is small compared with the wave phase speed, perturbation expansions of the hydrodynamic equations in a Lagrangian or generalized Lagrangian mean framework are useful: for stronger flows, such as for wind blowing over waves, the presence of critical levels where the mean flow velocity is equal to the wave phase speed necessitates the application of more general types of surface-following coordinate system. The interaction of the flow of air and water and associated differences in temperature and the concentration of various substances (such as gas species) gives rise to a complex boundary-layer structure at a wide range of vertical scales, from the sub-millimetre scales of gaseous diffusion, to several tens of metres for the turbulent Ekman layer. The bal- ance of momentum, heat, and mass is also affected significantly by breaking waves, which act to increase the effective area of the surface for mass transfer, and increase turbulent diffusive fluxes via the conversion of wave energy to turbulent kinetic energy.

Development of the geodetic coordinate system in Antarctica

Defining a universal geodetic coordinate system is one of the fundamental challenges of geodesy. We present a review of the basic general coordinate systems -- the space rectangular coordinate system, the geodetic coordinate system, the topocentric coordinate system, and the plane coordinate system. We then look ： t the World Geodetic System WGS72 and WGS84 and the In- ternational Terrestrial Reference Frames ITRF2000 and 1TRF2005, which were introduced when space technology became avail- able. The history of international geodetic coordinate systems in the Antarctic region is briefly reviewed and the development of the geodetic coordinate systems in the Chinese Great Wall Station and Zhongshan Station in Antarctica is outlined. Finally, the issue of coordinate system transformation is discussed.

ON THE OBJECTIVE STRESS RATE IN CO-MOVING COORDINATE SYSTEM

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

Simulation of Surface Wave with Large Eddy Simulation in σ-Coordinate System

A three dimensional numerical model in the σ coordinate system is developed to study the problem of waves. Turbulence effects are modeled by a subgrid scale (SGS) model with the concept of large eddy simulation (LES). The σ coordinate transformation is introduced to map the irregular physical domain of the wavy free surface and uneven bottom onto the regular computational domain of the shape of rectangular prism. The operator splitting method, which splits the solution procedure into the advection, diffusion, and propagation steps, is used to solve the modified Navier Stokes Equation. The model is used to simulate the propagation of solitary wave and wave passing over a submerged breakwater. Numerical results are compared with available analytical solutions and experimental data in terms of velocity profiles, free surface displacement, and energy conservation. Good agreement is obtained. The method is proved to be of high accuracy and efficiency in simulating surface wave propagation and wave structure interaction. It is suitable for the large and irregular physical domain, and requiring the non uniform grid system. The present work provides a foundation for further studies of random waves, wave structure interaction, wave discharge interaction, etc.

Parabolic Wave Propagation Modelling in Orthogonal Coordinate Systems

This paper presents a numerical model study of the propagation of water waves using the parabolic approximation of the mild slope equation in the orthogonal coordinate system. Two types of coordinate systems are studied: (a) a general form of orthogonal coordinate system and (b) the conformal system, a special form of orthogonal coordinate system. Two typical examples, namely, expanded breakwaters and a circular channel, are studied to validate the model. First, the examples are studied by use of the general orthogonal coordinates. Then the same examples are computed by use of the conformal system. The computational results show that the conformal coordinate system generally gives better predictions than the general orthogonal system. A numerical technique for generating the conformal grid is combined with the numerical model to improve the practicability of the model. The comparison between the result from the numerical grid system and that from the analytical grid system shows that reliable computational results can be obtained by use of the numerical conformal grid system.

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.

Adaptive preprocessing algorithms of corneal topography in polar coordinate system

New adaptive preprocessing algorithms based on the polar coordinate system were put forward to get high-precision corneal topography calculation results. Adaptive locating algorithms of concentric circle center were created to accurately capture the circle center of original Placido-based image, expand the image into matrix centered around the circle center, and convert the matrix into the polar coordinate system with the circle center as pole. Adaptive image smoothing treatment was followed and the characteristics of useful circles were extracted via horizontal edge detection, based on useful circles presenting approximate horizontal lines while noise signals presenting vertical lines or different angles. Effective combination of different operators of morphology were designed to remedy data loss caused by noise disturbances, get complete image about circle edge detection to satisfy the requests of precise calculation on follow-up parameters. The experimental data show that the algorithms meet the requirements of practical detection with characteristics of less data loss, higher data accuracy and easier availability.

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.

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.