Splines in geophysics

Modeling and contouring of geophysical data often require distributions of regularly spaced values. Splines have been shown to be the most accurate methods to obtain such distributions. We emphasize the general problem of interpolating random distributions of data on a given surface. Splines are classified into unidimensional, quasi‐bidimensional, and strictly bidimensional; based on this classification, a systematic derivation of the corresponding interpolating techniques is conducted. Two approaches are presented to obtain unidimensional splines: one based on the continuity of the first and second derivatives of the polynomials involved, and the other based on a variational approach. Quasi‐bidimensional splines are constructed based on the unidimensional approach, while strictly bidimensional splines are generated by minimizing the bidimensional curvature. Quasi‐bidimensional splines can be used for processing data distributions along nearly parallel lines; linear projections and parameterization are the techniques used in interpolating this type of distribution. Strictly bidimensional splines minimize curvature through the analytic solution of the Euler‐Lagrange equation or by a finite‐difference algorithm. The maximum error, mean error, and standard deviation between interpolated data and exact field values produced by various prisms show that quasi‐bidimensional splines are 2.7 percent more accurate in the maximum error than strictly bidimensional splines when both techniques are applied to regularly spaced data. However, for irregularly spaced data, three examples containing 300, 600, and 900 random data points show the superiority of the thin‐plate approach over the quasi‐bidimensional splines. A comparison between various interpolation densities on regular grids, starting from a set of 327 randomly distributed magnetic stations, illustrates some differences between geophysically meaningful interpolations and interpolations carried out only for contouring purposes.

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

Geodesy and Cartography ◽

10.22389/0016-7126-2017-919-1-7-12 ◽

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.

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Some new generalizations for GA-convex functions

Filomat ◽

10.2298/fil1704009a ◽

2017 ◽

Vol 31 (4) ◽

pp. 1009-1016 ◽

Cited By ~ 2

Author(s):

Ahmet Akdemir ◽

Özdemir Emin ◽

Ardıç Avcı ◽

Abdullatif Yalçın

Keyword(s):

Convex Functions ◽

Integral Identity ◽

Integral Inequalities ◽

General Integral ◽

Second Derivatives ◽

Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

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System of thermodynamic quantities in the McMillan-Mayer solution theory

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19850791 ◽

1985 ◽

Vol 50 (4) ◽

pp. 791-798 ◽

Cited By ~ 1

Author(s):

Vilém Kodýtek

Keyword(s):

Free Energy ◽

Generating Function ◽

Simple Form ◽

Unit Volume ◽

Thermodynamic Quantities ◽

Solution Theory ◽

Pure Solvent ◽

Second Derivatives ◽

Definition Of ◽

Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

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Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

Advances in Difference Equations ◽

10.1186/s13662-019-2385-9 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 5

Author(s):

A. Khalid ◽

M. N. Naeem ◽

P. Agarwal ◽

A. Ghaffar ◽

Z. Ullah ◽

...

Keyword(s):

Numerical Approximation ◽

Analytic Solution ◽

Linear Equations ◽

Computational Cost ◽

Spline Method ◽

System Of Linear Equations ◽

Approximation Solution ◽

B Spline ◽

Novel Technique ◽

Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

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A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2894143 ◽

1990 ◽

Vol 112 (1) ◽

pp. 83-87 ◽

Cited By ~ 25

Author(s):

R. H. Fries ◽

B. M. Coffey

Keyword(s):

Numerical Simulation ◽

White Noise ◽

State Space ◽

Spectral Characteristics ◽

The State ◽

Second Derivatives ◽

Time Histories ◽

Sample Interval ◽

Rail Irregularities ◽

Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

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Investigation of a Technique for the Differentiation of Empirical Data

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3140656 ◽

1983 ◽

Vol 105 (3) ◽

pp. 200-202 ◽

Cited By ~ 6

Author(s):

D. M. Trujillo ◽

H. R. Busby

Keyword(s):

Dynamic Programming ◽

Differentiable Function ◽

Empirical Data ◽

Numerical Experiments ◽

Random Noise ◽

Second Derivatives ◽

Derivatives Of

A dynamic programming filter is derived to estimate the first and second derivatives of empirical data. A series of numerical experiments are conducted using a known differentiable function with various amounts of added random noise.

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Fast automatic differentiation as applied to the computation of second derivatives of composite functions

Computational Mathematics and Mathematical Physics ◽

10.1134/s0965542506110029 ◽

2006 ◽

Vol 46 (11) ◽

pp. 1835-1859

Author(s):

E. S. Zasukhina

Keyword(s):

Automatic Differentiation ◽

Fast Automatic Differentiation ◽

Composite Functions ◽

Second Derivatives ◽

Derivatives Of

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Automatic 3-D interpretation of potential field data using analytic signal derivatives

Geophysics ◽

10.1190/1.1444149 ◽

1997 ◽

Vol 62 (1) ◽

pp. 87-96 ◽

Cited By ~ 56

Author(s):

Nicole Debeglia ◽

Jacques Corpel

Keyword(s):

Signal Amplitude ◽

Vertical Gradient ◽

Analytic Signal ◽

Magnetic Data ◽

Practical Implementation ◽

Potential Field Data ◽

Gravity And Magnetic ◽

Second Derivatives ◽

Gravity And Magnetic Data ◽

Derivatives Of

A new method has been developed for the automatic and general interpretation of gravity and magnetic data. This technique, based on the analysis of 3-D analytic signal derivatives, involves as few assumptions as possible on the magnetization or density properties and on the geometry of the structures. It is therefore particularly well suited to preliminary interpretation and model initialization. Processing the derivatives of the analytic signal amplitude, instead of the original analytic signal amplitude, gives a more efficient separation of anomalies caused by close structures. Moreover, gravity and magnetic data can be taken into account by the same procedure merely through using the gravity vertical gradient. The main advantage of derivatives, however, is that any source geometry can be considered as the sum of only two types of model: contact and thin‐dike models. In a first step, depths are estimated using a double interpretation of the analytic signal amplitude function for these two basic models. Second, the most suitable solution is defined at each estimation location through analysis of the vertical and horizontal gradients. Practical implementation of the method involves accurate frequency‐domain algorithms for computing derivatives with an automatic control of noise effects by appropriate filtering and upward continuation operations. Tests on theoretical magnetic fields give good depth evaluations for derivative orders ranging from 0 to 3. For actual magnetic data with borehole controls, the first and second derivatives seem to provide the most satisfactory depth estimations.

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