Determination of the geodetic latitude and height by spatial geocentric coordinates

The authors propose to derive the formulas given in [1, 2] for determining the height and latitude based on the Cartesian rectangular coordinates X, Y, Z, giving an accuracy for the geodetic height H of 1 mm for heights up to 50 km and for geodetic latitude B of 0,0001 arc seconds for H < 10 km. The formulas proposed in [1, 2] apply to all values of latitude and longitude (B and L). In [3], we propose two new formulas for H and B. In this paper, it is shown that the formulas proposed in [3] apply to points of ellipsoid surface and points with geodetic latitude of 0° and 90°. For the same formulas proposed in [3], the corrections are derived to ensure an accuracy of H of 1 mm at H ≤ 10 km, which apply to all values of B and L. Basing on the presented geometric conclusions, calculations and analyzes, a new solution for H and B respectively is proposed for given X, Y, Z, which provides an accuracy for H less than 1 mm for H ≤ 100 km and for B of 0,0001 arc seconds for H ≤ 50 km.

Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines

We propose a numerical solution to the heat equation in polar cylindrical coordinates by using the meshless method of lines approach. The space variables are discretized by multiquadric radial basis function, and time integration is performed by using the Runge-Kutta method of order 4. In radial basis functions (RBFs), much of the research are devoted to the partial differential equations in rectangular coordinates. This work is an attempt to explore the versatility of RBFs in nonrectangular coordinates as well. The results show that application of RBFs is equally good in polar cylindrical coordinates. Comparison with other cited works confirms that the present approach is accurate as well as easy to implement to problems in higher dimensions.

Technology of determining the Earth’s gravitational field parameters using gradiometric measurements Part 6. Calculation the components of the gravitational potential tensor in the earth’s spatial rectangular coordinate system

This is the sixth one in a series of articles describing the technology of determining the Earth’s gravitational field parameters through gradiometric measurements performed with an onboard satellite electrostatic gradiometer. It provides formulas for calculating the components of the gravitational potential tensor in a geocentric spatial rectangular earth coordinate system in order to convert them into a gradiometric one and obtain a free term for the equations of correcting gradiometric measurements when determining the parameters of the Earth’s gravitational field. The components of the gravitational gradient tensor are functions of test masses accelerations measured by accelerometers and relate to the gradiometer coordinate system, while the desired parameters of the Earth’s gravitational field model relate to the Earth’s coordinate one. The components of the gravitational gradient tensor are the second derivatives of the gravitational potential in rectangular coordinates. The calculated values of the gravitational potential tensor components in the earth’s spatial rectangular coordinate system are obtained through double differentiation of the gravitational potential formula. Basing on the obtained formulas, an algorithm and a program in the Fortran algorithmic language were developed. Using this program, experimental calculations were performed, the results of which were compared with the data of the EGG_TRF_2 product.

3D brain tumor segmentation using a two-stage optimal mass transport algorithm

Optimal mass transport (OMT) theory, the goal of which is to move any irregular 3D object (i.e., the brain) without causing significant distortion, is used to preprocess brain tumor datasets for the first time in this paper. The first stage of a two-stage OMT (TSOMT) procedure transforms the brain into a unit solid ball. The second stage transforms the unit ball into a cube, as it is easier to apply a 3D convolutional neural network to rectangular coordinates. Small variations in the local mass-measure stretch ratio among all the brain tumor datasets confirm the robustness of the transform. Additionally, the distortion is kept at a minimum with a reasonable transport cost. The original $$240 \times 240 \times 155 \times 4$$ 240 × 240 × 155 × 4 dataset is thus reduced to a cube of $$128 \times 128 \times 128 \times 4$$ 128 × 128 × 128 × 4 , which is a 76.6% reduction in the total number of voxels, without losing much detail. Three typical U-Nets are trained separately to predict the whole tumor (WT), tumor core (TC), and enhanced tumor (ET) from the cube. An impressive training accuracy of 0.9822 in the WT cube is achieved at 400 epochs. An inverse TSOMT method is applied to the predicted cube to obtain the brain results. The conversion loss from the TSOMT method to the inverse TSOMT method is found to be less than one percent. For training, good Dice scores (0.9781 for the WT, 0.9637 for the TC, and 0.9305 for the ET) can be obtained. Significant improvements in brain tumor detection and the segmentation accuracy are achieved. For testing, postprocessing (rotation) is added to the TSOMT, U-Net prediction, and inverse TSOMT methods for an accuracy improvement of one to two percent. It takes 200 seconds to complete the whole segmentation process on each new brain tumor dataset.

Parabolic coordinates

An important first step in understanding or solving a problem can be the selection of coordinates. Insight can be gained from finding invariants within a class of coordinate systems. For example, an important feature of rectangular coordinates is that the Euclidean distance between two points is an invariant of a change to another rectangular system by a rigid motion, which consists of translations, rotations and reflections. Indeed, the form of the distance function is an invariant. In calculus courses, students learn about polar coordinates, so that useful curves can be simply expressed and more easily studied.

Distributed Multistatic Sky-Wave Over-The-Horizon Radar Based on the Doppler Frequency for Marine Target Positioning

Maritime safety issues have aroused great attention, and it has become a difficult problem to use the sky-wave over-the-horizon radar system to locate foreign targets or perform emergency rescue quickly and timely. In this paper, a distributed multi-point sky-wave over-the-horizon radar system is used to locate marine targets. A positioning algorithm based on the Doppler frequency is proposed, namely, the two-step weighted least squares (2WLS) method. This algorithm first converts the WGS-48 geodetic coordinates of the transceiver station to spatial rectangular coordinates; then, introduces intermediate variables to convert the nonlinear optimization problem into a linear problem. In the 2WLS method, four mobile transmitters and four mobile receivers are set up, and the Doppler frequency is calculated by transmitting and receiving signals at regular intervals; it is proven that the 2WLS algorithm has always maintained a better positioning accuracy than the WLS algorithm as the error continues to increase with a certain ionospheric height measurement error and the Doppler frequency measurement error. This paper provides an effective method for the sky-wave over-the-horizon radar to locate maritime targets.

ALGORITHMS FOR CALCULATING GEODETIC HEIGHTS AND LATITUDES BY RECTANGULAR COORDINATES IN THE MERIDIAN ELLIPSE PLANE

Owing to the widespread use of GNSS technologies in geodetic practice, the problem arises of transition from rectangular spatial coordinates of points to spatial geodetic coordinates, which are necessary for the transition to flat rectangular coordinates in the Gauss-Kruger projection. The authors proposed five algorithms for converting rectangular coordinates of points in the plane of the meridian ellipse into geodetic heights and latitudes. The first two algorithms are geometrically related to the intersection point of the ellipse with the normal passing through the point at which the rectangular spatial coordinates were obtained. The formulas of the other three algorithms are based on the geometric relationships of the point of intersection of the meridian ellipse with the straight line connecting the point with the center of curvature of the meridian. As a result of the experiments, deviations of the calculated latitudes and heights from the reference values of the given grid of geodetic coordinates were obtained. The formulas were tested not only for points under and on the earth's surface, but also outside the earth at different heights up to an altitude of 20,000 km.

Asymptotic approximation of the solution of a perturbed differential equation system

The article is devoted to the determination of second-order perturbations in rectangular coordinates and components of the body motion to be under study. The main difficulty in solving this problem was the choice of a system of differential equations of perturbed motion, the coefficients of the projections of the perturbing acceleration are entire functions with respect to the independent regularizing variable. This circumstance allows constructing a unified algorithm for determining perturbations of the second and higher order in the form of finite polynomials with respect to some regularizing variables that are selected at each stage of approximation. Special points are used to reduce the degree of approximating polynomials, as well as to choose regularizing variables. The problem of generation of an asymptotic approximation of the solution of a perturbed differential equation system is considered in the case where a bifurcation occurs in the “fast motions” equation when the parameter changes: two equilibrium positions merge, followed by a change in stability.