A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described. The quadric surfaces are located at significant points throughout the region to be mapped. Procedures are given for solving multiquadric equations of topography that are based on coordinate data. Contoured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived.Topography can be represented by various analytical, numerical, and digital methods, in addition to the classical contour map. The extremes in generalization or detail that result from use of these methods are perhaps demonstrated best by Lee and Kaula [1967] and by Gilbert [1968]. Lee and Kaula described the topography of the whole earth in the form of thirty-sixth-degree spherical harmonics. Gilbert reported the magnetic tape storage of more than six million increments of height information in digital form, measured or interpolated from one ordinary map sheet.In Lee and Kaula's work we have an extreme generalization of existing topographic information over a wide area by highly analytical methods, whereas Gilbert's work is extremely detailed but scarcely analytical. As valuable as these techniques are in certain cases, they are related more to map utilization than to map making. Basically, the problem they solve is: given continuous topographic information in a certain region, reduce it to an equivalent set of discrete data, e.g., spherical harmonic coefficients or digital terrain increments.Other investigators, including myself, are concerned with a procedural inverse of the above problem, namely: given a set of discrete data on a topographic surface, reduce it to a satisfactory continuous function representing the topographic surface. Practical solutions to this problem will tend to eliminate the classical Copyright @ 1971 by the American Geophysical Union. contour map as the first step in representing terrain information.An equation of topography can be evaluated digitally or analytically without its having been reduced to graphical form. The same equation can be treated analytically for the automatic production of contoured maps. Automatic contouring can become a computer-plotter problem in analytical geometry, i.e., to determine and plot the intercept equations of horizontal planes passed through a three-dimensional equation of topography. This approach could also lead to reconsideration of the need for digitized map data. Problems involving map use, such as determining unobstructed lines of sight, areas of deftlade, volumes of earth, and minimum length of surface curves, may involve the more direct application of analytical geometry and calculus to the interrelationship of these parameters with a mathematical surface of topography. For these reasons, the question of representing a topographic surface in detail by unique equations deserves increased consideration.
•NUMERICAL SURFACE TEC•NmUESFourier and polynomial ser...
Least squares prediction with MQ (multiquadric) functions is conceptually different from least squares prediction using covariance functions. MQ kernels are based on geometric or physical considerations rather than stochastic processes, and were found to be superior to covariance functions in topographic applications. This may be true also for gravity anomalies or other phenomena which result from marginally stationary, or non‐stationary random processes. The MQ harmonic kernel is used to develop a formula for estimating the best depth of point mass anomalies as a function of their number and areal extent on a sphere. Functional relationships between geoidal surface parameters are developed which provide linear equation analogs for the solution of Stokes and Vening Meinesz Integral Formulas as well as for the inversion of these classic problems. These relationships are extended to solutions at exterior equipotential surfaces.
Properties of the potential are considered for applications in gravity, geomagnetic, and thermal field studies. It is found that classical theory is somewhat limited because of the common difficulty in evaluating potential within a material body containing the sources. The theory of a new type of representation of disturbing potential, a biharmonic form, is given which eliminates this difficulty. It is shown that multiquadric equations provide us with a physically valid numerical approximation of the formal integral representation. Error bounds are derived. The results of tests with real gravity anomalies are given, which compare classical methods with the new biharmonic form. In summary, the new approach eliminates the classical singularities associated with collocation of points of measurement (or prediction) and the sources of disturbing potential. It also improves computational efficiency.
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