A multiquadric‐biharmonic representation and approximation of disturbing potential

Abstract

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.

Biharmonic spline interpolation of GEOS‐3 and SEASAT altimeter data

“…In three dimensions the Green function has discontinuous slope at the origin. To avoid this discontinuity in the multiquadric method, the Green functions are placed on a plane below all of the data points [Hardy and Nelson, 1986]. in four or more dimensions the Green functions are unbounded at the origin so the values of their coefficients are undetermined.…”

Biharmonic spline interpolation of GEOS-3 and Seasat altimeter data

“…In three dimensions, it corresponds to multiquadric interpolation [Hardy, 1971;Hardy and Nelson, 1986]. This method has been used quite successfully by a number of Copyright 1987 by the American Geophysical Union.…”

PHLST5: A Practical and Improved Version of Polyharmonic Local Sine Transform

“…The values of the first order normal derivative are computed using the method described in Section 2.3 again. In addition, we also compare the results of LLST and PHLST5 with the component u computed from the radial basis function transform (RDT) [9,10,11,17,6,20]. In RDT, we choose the most often applied radial basis function-a multiquadric φ(x) = ||x|| 2 2 + 1.…”

“…In particular we compare PHLST5, LLST with an approximation method using radial basis functions [9,10,11,17,6,20]. Radial basis functions are extremely useful when interpolating scattered data, especially of high dimensions.…”