Petroleum Reservoir Simulation

The theory of splines and their applications

The article provides arguments in favor of an alternative approach that uses splines, which is equally justifiable on a theoretical basis, and which offers many practical advantages. To reassure the reader who may be afraid to enter new territory, it is emphasized that one is not losing anything because the traditional theory is retained as a particular case (i.e., a spline of infinite degree). The basic computational tools are also familiar to a signal processing audience (filters and recursive algorithms), even though their use in the present context is less conventional. The article also brings out the connection with the multiresolution theory of the wavelet transform. This article attempts to fulfil three goals. The first is to provide a tutorial on splines that is geared to a signal processing audience. The second is to gather all their important properties and provide an overview of the mathematical and computational tools available; i.e., a road map for the practitioner with references to the appropriate literature. The third goal is to give a review of the primary applications of splines in signal and image processing.

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Splines: a perfect fit for signal and image processing

A gridding method commonly called minimum curvature is widely used in the earth sciences. The method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature. The minimum-curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points. Minimum-curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used. These extraneous inflection points can be eliminated by adding tension to the elastic-plate flexure equation. It is straightforward to generalize minimum-curvature gridding algorithms to include a tension parameter; the same system of equations must be solved in either case and only the relative weights of the coefficients change. Therefore, solutions under tension require no more computational effort than minimum-curvature solutions, and any algorithm which can solve the minimum-curvature equations can solve the more general system. We give common geologic examples where minimum-curvature gridding produces erroneous results but gridding with tension yields a good solution. We also outline how to improve the convergence of an iterative method of solution for the gridding equations.

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Gridding with continuous curvature splines in tension

This paper presents the use of B-splines as a tool in various digital signal processing applications. The theory of B-splines is briefly reviewed, followed by discussions on B-spline interpolation and B-spline filtering. Computer implementation using both an efficient software viewpoint and a hardware method are discussed. Finally, experimental results are presented for illustrative purposes in two-dimensional image format. Applications to image and signal processing include interpolation, smoothing, filtering, enlargement, and reduction.

Cubic splines for image interpolation and digital filtering

Convergence Properties of the Spline Fit

A graphical technique for selecting optimum nozzle contours from a family of truncated perfect nozzles is presented which permits simultaneous consideration of various types of optima. This procedure includes the effects of friction, as well as the thermodynamic properties of the reacting gas. Results are given of hot and cold flow tests, showing the effect of wall contour and wall cooling upon ohlique shock separation.

Truncated Perfect Nozzles in Optimum Nozzle Design

The Approximation of Linear Functionals

Introduction. The past two decades have witnessed an increasing intensity of investigation, [1]-[26], into the properties of spline functions. These functions, which in their simplest form yield the analytic counterpart of the draftsman's tool for drawing a smooth curve through a number of prescribed points, play an important and fundamental role in many parts of numerical analysis. Many of the properties which these functions possess, such as the minimum curvature property [3], [8], [13], [14], are associated with obvious attributes of the elastic beam. Some relate to the best approximation of linear operators [15], [20], and have rather profound meanings in approximation theory. Others, such as orthogonality [24], rate of convergence [12], [17], [23], [25], and completeness [25] of certain bases of splines, are at first rather surprising. A few (cf. [22, p. 241]) are still quite puzzling. The application of spline theory to the approximation of a function analytic interior to a rectifiable Jordan curve and continuous in the corresponding closed region has been considered to a limited extent in an earlier paper, [24], by the authors. There splines were treated which are piecewise polynomial in the arc length s on the curve. The present development is concerned with cubic splines in the complex variable z and provides some insight into the structure of the spline approximation generally. In particular, it serves to establish a connection with the classical theory of approximation to an analytic function. As part of this development, proofs are given for the convergence of the complex cubic spline and its derivatives for the situations in which the approximated function is of class C' (a = 0, 1, 2, 3, or 4) on the boundary. These may be modified in an obvious manner for the standard real cubic spline and for the convergence properties of second and third derivatives. There result noteworthy simplifications over proofs already existing in the literature [12], [17], for a= 2, 3, 4. The convergence properties of cubic splines for cases in which the approximated function is assumed merely to be continuous or to have continuous first derivative constitute significant new developments in spline theory. In addition, a curious spline property is here presented relating to the approximation of the fourth derivative.

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J. H. Ahlberg

Papers

The theory of splines and their applications

Convergence Properties of the Spline Fit

Truncated Perfect Nozzles in Optimum Nozzle Design

The Approximation of Linear Functionals

Complex cubic splines