Discrete Data Samples Research Articles

Integral estimation with the ordinary Kriging method using the Gaussian semivariogram function

This paper describes a detailed procedure for integral estimation of an unknown function using the ordinary Kriging method. This method estimates the definite integral of an approximated surface generated using the ordinary Kriging method. The integral of an approximated surface can be computed directly using a set of sampling data for a function value. One of the merits of the proposed method is that the integral can be estimated without iterative summation generally used in a conventional numerical integration technique. In addition to a continuous function, the proposed method enables to estimate the integral using a set of discrete sampling data. In this paper, a basic formulation for an integral estimation using the Kriging method, especially with the Gaussian-type semivariogram model, is introduced. An explicit form of the integral of the Gaussian function is approximated by Williams' method. As test problems, definite integrals of some elementary functions are estimated using the proposed method. In addition, as an engineering example, a flow in a pipe is estimated using the proposed method with less samples of the velocity of fluid. The numerical results illustrate the accuracy and efficiency of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.

Does ak-space span limitation affect MRI resolution?

It is common practice to employ conceptual models for data interpretation that are continuous even for data that are discrete. Conclusions drawn from analytical Fourier transform integrals do not automatically apply for discrete data. The conventional sampling point spread function width of 1.2 pixels in MRI is a prime example of this problem. Direct calculation reveals that the k-space span limitation, expressed in terms of a simple sinc function, makes no impact on discrete data samples in the reciprocal image domain. In this article we show that it is the discretization and discrete Fourier transfor- mation that causes image distortion, rather than the limited k-space coverage. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 29B: 55- 61, 2006

Dynamically knots setting in meshless method for solving time dependent propagations equation

In order to display a function with some finite discrete sampling data efficiently, we require more sampling data where the function is more oscillatory, and less sampling data where the function is more flat. If the function is happen to be a solution of partial differential equation and is solved by finite elements method, then we should construct finer element near the singularity. However, we do not know where the oscillation or even shocks will happen to a function, which is a solution of non-linear partial differential equation. Therefore we cannot preset the finer elements near such points. A trivial method is taking very dense knots or very fine elements everywhere to keep the accuracy. This strategy cost the computation time. Another idea is to move the position of the sampling points according to the varying of the function with the time. We observed that, this approach could be achieved by Lagrangian formulation for finite difference method. However the Lagrangian formulation does not possess shape preserving and variation diminishing properties. It is difficult to achieve the approach for finite elements methods too, because the moving knots will destroy the topology of the mesh. Recently the meshless method becomes to topic to solve partial differential equation numerically. The meshless method does not require any mesh or any structure of the knots (sampling points); therefore we can move the knots freely to simulate the problem. The only restriction is to keep the knots no overlapping. This paper is a test of our approach for the Burger's equation with radial basis quasi-interpolation.

Eocene to Miocene magnetostratigraphy, biostratigraphy, and chemostratigraphy at ODP Site 1090 (sub-Antarctic South Atlantic)

At Ocean Drilling Program (ODP) Site 1090 (lat 42854.89S, long 8854.09E) located in a water depth of 3702 m on the Agulhas Ridge in the sub-Antarctic South Atlantic, ;300 m of middle Eocene to middle Miocene sediments were recovered with the advanced piston corer (APC) and the extended core barrel (XCB). U-channel samples from the 70‐230 meters composite depth (mcd) interval provide a magnetic polarity stratigraphy that is extended to 380 mcd by shipboard whole-core and discrete sample data. The magnetostratigraphy can be interpreted by the fit of the polarity-zone pattern to the geomagnetic polarity time scale (GPTS) augmented by isotope data and bioevents with documented correlation to the GPTS. Three normal-polarity subchrons (C5Dr.1n, C7Ar.1n, and C13r.1n), not included in the standard GPTS, are recorded at Site 1090. The base of the sampled section is correlated to C19n (middle Eocene), although the interpretation is unclear beyond C17r. The top of the sampled section is correlated to C5Cn (late early Miocene), although, in the uppermost 10 m

Numerical Integration of Harmonic Functions with Restricted Sampling Data

Based on the idea of kriging and the radial basis function approximation, we develop in this paper a numerical scheme to integrate harmonic functions with restricted sampling data. To be more precise, the integration is performed by using the function values which are given as discrete sampling data on only part of the boundary. These problems often arise from non-destructive evaluation techniques in the engineering industry. The existence and uniqueness of the solution and the error estimation for the proposed numerical scheme are also discussed. Several numerical experimental results are presented for the verification on the accuracy and convergence of the method.

Identification of cubically nonlinear systems using undersampled data

A practical technique for identification of cubically nonlinear systems using higher order spectra of the discrete data samples of the system input and output is proposed. This technique differs from the conventional one in that it only requires the sampling frequency for the system output to be equal to twice the bandwidth of the system input, instead of six times the bandwidth of the system input. This means the demand for high speed processing and a large amount of data in the conventional approach can be greatly relieved. Two methods are developed: one is suitable for systems with a Gaussian random input, the other is suitable for systems with a non-Gaussian random input. The advantages of the two methods over their conventional counterparts are demonstrated via computer simulation.