Scattered data interpolation

Each horizon is picked in this way. The resulting scattered data sets are then passed to a mapping package, an embodiment of one or more of the many scattered data interpolation algorithms reviewed in Section 4. 

M. S. Eloater and A. iske (1996) Multistep scattered data interpolation using compactly supported radial basis functions. Journal of Computational and Applied Mathematics 73, 65-78. 

R. Eranke (1982) Scattered data interpolation tests of some methods. Math. Comp. 38, 181-199. 

T. Werther (2003) Characterization of semi-Hilbert spaces with application in scattered data interpolation. Curve and Surface Fitting Saint-Malo 2002, A. Cohen, J.-L. Merrien, and L.L. Schumaker (eds.), Nashboro Press, Brentwood, 374-383. 

Figure 3. Computer plot obtained by isochronous interpolation of the experimental light-scattering data (O) of HEC during endocellulase attack. The Langrangian interpolation functions are given by the horizontal curves ana the isochronous interpolated Kc/Re values by A. The quadratic least squares extrapolations to zero angle ( ) are given by the...

Figure 4. Changes in Mw as a function of the enzymic hydrolysis times, calculated from the light-scattering data by isochronous interpolation and by subsequent extrapolation to zero concentration using Equation 17.

Nealen, A. (2004.). An as-short-as-possible introduction to the least squares, weighted least squares and moving least squares methods for scattered data approximation and interpolation. Technical Report, TU Darmstadt. 

Note that the problem of scattered data set interpolation is discussed in many other contexts for example in weather forecasting [39] and in oceanography [160]. 

The classical multiquadric method of [79] has often been used in the geosciences. See [63] for a review of classical methods of scattered data set interpolation. 

Thus for large sets of scattered data and with a need to evaluate the interpolant at a large number of points on a grid one might be better served, when D is small, by the numerical solution of a partial differential equation as in the Briggs technique. However, the recent research into radial basis functions with compact support [61] and application of the fast multipole method [29] do provide efficient methods. 

A detailed discussion of how to interpolate scattered data has already been given. This interpolation problem can be considered an example of an inverse problem. Indeed the situation is very often that measurements of diagnostic functions are available and a few direct measurements of the properties p are also available at a small number of points. Then it makes sense to combine interpolation techniques with the general inverse problem. 

The link between kriging, radial basis functions and maximum probability interpolants could be investigated in a much deeper way than in Section 4. The huge effort to analyse and develop radial basis function methods would be made more valuable if the participants in the growing radial basis function literature were more aware of the need for statistical considerations in the scattered data problem. Deterministic approaches to problems with sparse data are not applicable in most of the problems encountered in the geosciences. 

The reconstruction of high order polynomials on unstructured triangulations is much more difficult than the reconstruction on one-dimensional intervals or multi-dimensional Cartesian grids. In fact, polynomial reconstruction from scattered data requires the solution of multi-dimensional interpolation problems, which typically tend to be ill-conditioned. This problem becomes even more critical with increasing order of the reconstruction. 

C.A. MiccheUi (1986) Interpolation of scattered data distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11-22. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

In Figure 4 the results from the three different groups are in excellent agreement for butanol concentrations of 90 wt% and greater, although the data from the Russian group scatter somewhat more around our results than do the values interpolated from Westmeier s data.(14.16). At lower amphiphile concentrations the isoperibolic calorimeter measurements are in noticeably better agreement with the data of ref. 16 than with the Russian work (14-16). However, almost all results fall within the 95% confidence interval (dashed lines) for our results. 

Geometrical and flexibility data pertaining to the same polymers are also given in Table 1, namely the persistence length and the average chain-to-chain interaxial distance D. The first five polymers in Table 1 have D values smaller than 6 A, unlike all the following polymers (i.e., no. 6 to 19 in Table 1, Class II). This is a consequence of the relatively bulky substituents carried by Class II polymer chains. For some of the polymers in Table 1 the C0o and P literature values are widely scattered or unavailable. In those cases lower-limit values of P from experimentally determined geometrical parameters, are predicted from our model by suitable interpolation and reported within parentheses. 

The light scatter assay may be used to determine absolute numbers of viable cells if flow cytometry data from cell suspensions of known concentration are used to construct a standard curve. For that purpose, cell concentrations should be determined in a series of graded, standard cell suspensions with the use of a Coulter counter. A plot of those standard concentrations versus the number of events (Hght scatter signals) acquired during a specified acquisition interval in the flow cytometer may then be used to interpolate cell concentrations for test samples that have been assayed by the light scatter procedure. 

Kinetic Light-Scattering Method. Isochronous Interpolation. When high-activity samples of endocellulase are used, the reaction proceeds so quickly that, since the measurements of scattered intensity at different angles are not performed at the same extent of reaction, the extrapolations to zero angle and the subsequent calculations are erroneous. For this reason Kratochvil et al. (27) have proposed an isochronous interpolation method, whereby the Kc/Re values are plotted against sin21 (0/2) + kft. As in the double-extrapolation method of Zimm, the value of k may be chosen arbitrarily in order to space the experimental data. 

Experimental data of the coagulation coefficient for NaCI aerosols of different sizes obtained by Shon et al. (14) are compared with the upper bound, with the lower bound (for a Hamaker constant of 10 12 erg) and with the Fuchs interpolation formula in Fig. 9. The experimental data exhibit considerable scatter, the upper bound for the coagulation coefficient agrees somewhat better with the experimental data than the Fuchs interpolation formula. 

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