Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

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

2021 ◽  
Vol 973 (7) ◽  
pp. 2-8
Author(s):  
A.A. Kluykov
Keyword(s):  
Gravitational Field ◽  
Coordinate System ◽  
Gravitational Potential ◽  
Rectangular Coordinate System ◽  
Gradient Tensor ◽  
Gravitational Gradient ◽  
Earth’S Gravitational Field ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Potential Tensor

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.

Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies

Geophysics ◽  
1979 ◽  
Vol 44 (4) ◽  
pp. 730-741 ◽  
Author(s):  
M. Okabe
Keyword(s):  
Gravitational Potential ◽  
Computing Time ◽  
Gravity Anomalies ◽  
Magnetic Anomalies ◽  
Two Dimensions ◽  
First Derivative ◽  
Second Derivatives ◽  
Analytical Expressions ◽  
Derivatives Of ◽  
The Magnetic Field

Complete analytical expressions for the first and second derivatives of the gravitational potential in arbitrary directions due to a homogeneous polyhedral body composed of polygonal facets are developed, by applying the divergence theorem definitively. Not only finite but also infinite rectangular prisms then are treated. The gravity anomalies due to a uniform polygon are similarly described in two dimensions. The magnetic potential due to a uniformly magnetized body is directly derived from the first derivative of the gravitational potential in a given direction. The rule for translating the second derivative of the gravitational potential into the magnetic field component is also described. The necessary procedures for practical computer programming are discussed in detail, in order to avoid singularities and to save computing time.

scholarly journals Advanced analysis in epidemiological modeling: Detection of wave

2021 ◽  
Author(s):  
Abdon Atangana ◽  
Seda IGRET ARAZ
Keyword(s):  
Simple Model ◽  
Reproductive Number ◽  
Second Derivative ◽  
Mathematical Concepts ◽  
Epidemiological Modeling ◽  
Derivative Analysis ◽  
Second Derivatives ◽  
Advanced Analysis ◽  
The Stability ◽  
Derivatives Of

Some mathematical concepts have been used in the last decades to predict the behavior of spread of infectious diseases. Among them, the reproductive number concept has been used in several published papers for study the stability of the spread. Some conditions were suggested to predict there would be either stability or instability. An analysis was also suggested to determine conditions under which infectious classes will increase or die out. Some authors pointed out limitations of the reproductive number, as they presented its inability to fairly help understand the spread patterns. The concept of strength number and analysis of second derivatives of the mathematical models were suggested as additional tools to help detect waves. In this paper, we aim at applying these additional analyses in a simple model to predict the future. Keywords: Strength number, second derivative analysis, waves, piecewise modeling.

Hybrid Position/Force Control of Robot Manipulators Based on Second Derivatives of Position and Force

1996 ◽  
Vol 8 (3) ◽  
pp. 243-251
Author(s):  
Satoshi Komada ◽  
◽  
Muneaki Ishida ◽  
Kouhei Ohnishi ◽  
Takamasa Hori ◽  
...  
Keyword(s):  
Coordinate System ◽  
Force Control ◽  
Disturbance Observer ◽  
Position Control ◽  
Robot Manipulators ◽  
Computational Effort ◽  
Direct Drive ◽  
Second Derivatives ◽  
Parameter Variations ◽  
Derivatives Of

This paper proposes a new robust hybrid position/force control of robot manipulators. The proposed method controls the second derivatives of control variables, such as position and force in a task coordinate system, in order to realize robust and high response control. To this end, the disturbances are estimated by a position-based disturbance observer and a force-based distrubance observer in the task coordinate system, and are compensated by feeding back the estimated distrubances. The proposed method requires less computational effort and is robust against the disturbance and parameter variations. The position-based distrubance observer has been proposed to linearize robot manipulators and has realized robust position control. However, when force control is performed, the force response is influenced by not only the nonlinearity of robot manipulators but also the charactersitics of the environment on which the force is imposed. Therefore, the force-based disturbance observer is developed to realize robust force control. A controller robust against the disturbance and parameter variations is realized by using the position-based disturbance observer and the force-based disturbance observer on performing the position control and the force control respectively. The effectiveness of the proposed method is shown by experiments by using a direct drive robot.

AN INTRODUCTION TO THE SECOND DERIVATIVE CONTOUR METHOD OF INTERPRETING TORSION BALANCE DATA

Geophysics ◽  
1938 ◽  
Vol 3 (3) ◽  
pp. 234-246 ◽  
Author(s):  
H. Klaus
Keyword(s):  
Local Effect ◽  
Torsion Balance ◽  
Resolving Power ◽  
Contour Method ◽  
Second Derivative ◽  
Regional Effects ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Regional Gradient ◽  
Regional Gravity

After auspicious beginnings in the interpretation of torsion balance data, i.e., gradients and curvatures, the balance has been misused as a gravity instrument, the gradients being integrated into gravity, and the curvatures either neglected or not even observed in the field. Gravity was then made the sole basis of interpretation work, the regional effects being determined with more or less luck, subtracted from the total either before or after integration (regional gravity or regional gradient), and the residue held to be “local effect.” This method appears to be now in vogue for most torsion balance and gravity meter work. In contrast to this procedure, the method here described is based on the quantities measured directly by the torsion balance, the gradients and curvatures, or second derivatives, and constitutes a considerable amplification of the original methods of investigating these quantities. Gravity is simply a by‐product of this method, and is not needed at all for its functioning. The essential parts of this method are: 1) the re‐determination of all second derivative components with respect to a new system of rectangular coordinates, one axis of which has been made parallel to the direction of elongation of anomalous features; 2) the contouring of these second derivative components on four separate maps; and 3), the interpretation of the resulting contour patterns. The outstanding advantages of this method over the total gravity methods are the following: 1) full utilization of the two independent aspects of the gravitational field furnished by the gradients and curvatures; 2) virtual independence from regional effects; 3) much greater resolving power when compared to gravity; and 4), complete absence of assumptions, such as are involved in estimating the regional, and in computing gravity from the gradients.

COMPUTATION WITH THE HELP OF A DIGITAL COMPUTER OF MAGNETIC ANOMALIES CAUSED BY BODIES OF ARBITRARY SHAPE

Geophysics ◽  
1965 ◽  
Vol 30 (5) ◽  
pp. 797-817 ◽  
Author(s):  
Manik Talwani
Keyword(s):  
Computer Program ◽  
Gravitational Potential ◽  
Digital Computer ◽  
Arbitrary Shape ◽  
Complex Shape ◽  
Magnetic Anomalies ◽  
Simple Geometry ◽  
Second Derivatives ◽  
First Vertical Derivative ◽  
Derivatives Of

Formulas are derived for the magnetic anomalies caused by irregular polygonal laminas. These are used to obtain the three components of the magnetic anomalies caused by a finite homogeneously magnetized body of arbitrary shape. There is no restriction to the direction of magnetization; in general, it may not be the same as that of the earth’s field. Total‐intensity anomalies are also obtained. Use of these formulas in a computer program is discussed and illustrated by computing the anomaly caused by Caryn Seamount. Simplified, formulas are presented for the anomalies caused by finite rectangular laminas. In addition to bodies of complex shape, the computer program can also be profitably used for computing the magnetic anomalies caused by bodies of relatively simple geometry. The second derivatives of the gravitational potential of a massive body, that is, quantities familiarly known as gradient and curvature in torsion‐balance work and the first vertical derivative in gravity work are also obtained by this method.

Reanalysis Method for Second Derivatives of Static Displacement

2019 ◽  
Vol 17 (08) ◽  
pp. 1950056 ◽  
Author(s):  
Wenjie Zuo ◽  
Jiaxin Fang ◽  
Zengming Feng
Keyword(s):  
Numerical Results ◽  
Second Derivative ◽  
Static Displacement ◽  
Normalized Error ◽  
Second Derivatives ◽  
Algebraic Operations ◽  
Combined Approximations Method ◽  
Derivatives Of ◽  
Combined Approximations

The reanalysis method to obtain the second derivatives of static displacement is innovatively proposed in this paper. This method is based on the combined approximations method. The reanalysis formulations of the second derivative of static displacement are derived to provide a programmatic procedure of formulations construction. Besides, the normalized error and the number of algebraic operations are considered to evaluate the accuracy and efficiency, respectively. Finally, three typical numerical results verify the accuracy and robustness of the proposed method.

Traveltime and wavefront curvature calculations in three‐dimensional inhomogeneous layered media with curved interfaces

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1466-1494 ◽  
Author(s):  
H. Gjøystdal ◽  
J. E. Reinhardsen ◽  
B. Ursin
Keyword(s):  
Taylor Series ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Reference Source ◽  
Second Derivative ◽  
Second Derivatives ◽  
Derivative Matrix ◽  
Three Dimensional Models ◽  
Derivatives Of

The seismic rays and wavefront curvatures are determined by solving a system of nonlinear ordinary differential equations. For media with constant velocity and for media with constant velocity gradient, simplified solutions exist. In a general inhomogeneous medium these equations must be solved by numerical approximations. The integration of the ray‐tracing and wavefront curvature equations is then performed by a modified divided difference form of the Adams PECE (Predict‐Evaluate‐Correct‐Evaluate) formulas and local extrapolation. The interfaces between the layers are represented by bicubic splines. The changes in ray direction and wavefront curvature at the interfaces are computed using standard formulas. For three‐dimensional media, two quadratic traveltime approximations have been proposed. Both are based on a Taylor series expansion with reference to a ray from a reference source point to a reference receiver point. The first approximation corresponds to expanding the square of the traveltime in a Taylor series and taking the square root of the result. The second approximation corresponds to expanding the traveltime in a Taylor series. The two traveltime approximations may be expressed in source‐receiver coordinates or in midpoint‐half‐offset coordinates. Simplified expressions are obtained when the reference source and receiver coincide, giving zero‐offset approximations, for which the reference ray is a normal‐incidence ray. A new method is proposed for computing the second derivatives of the normal‐incidence traveltime with respect to the source‐receiver midpoint coordinates. By considering a beam of normal‐incidence rays it is shown that the second‐derivative matrix may be found by computing the wavefront curvature along a reference normal‐incidence ray starting at the reflection point with the wavefront curvature equal to the curvature of the reflecting interface. From this second‐derivative matrix the normal moveout velocity can be computed for any seismic line through the reference source‐receiver midpoint. It is also shown how a reverse wavefront curvature calculation may be used, in a time‐to‐depth migration scheme, to compute the curvature of the reflecting interface from the estimated second derivatives of the normal‐incidence traveltime. Numerical results for different three‐dimensional models indicate that the first traveltime approximation, based on an expansion of the square of the traveltime, is the most accurate for shallow reflectors and for simple models. For deeper reflectors the two approximations give comparable results, and for models with complicated velocity variations the second approximation may be slightly better than the first one, depending on the particular model chosen. A simplified traveltime approximation may be used in a three‐dimensional seismic velocity analysis. Instead of estimating the stacking velocity one must estimate three elements in a [Formula: see text] symmetric matrix. The accuracy and range of validity of the simplified traveltime approximation are investigated for different three‐dimensional models.

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