Interpolation: Multiquadratics

The multiquadric method (Hardy and Gofert, 1975) fits a set of quadric (i.e. hyperbolic or conical) functions to the observations. The fitted function agrees with the observation values at the observation points. The coefficients of the functional fit can be stored for later use.
The generalization of multiquadrics to the sphere has been formulated by Pottmann and Eck (1990). The estimate at point P is:


where R is a user-specified tension parameter and sk is the angular distance between the interpolation point and the kth observation value. The weights Ak are computed so that the estimated function agrees with the observations at the observation points. For n observation points, this requires the solution of n simultaneous equations and the inversion of an nxn matrix. As a result, a multiquadric fit is limited to a reasonable number of points.

In Spherekit, if more than 500 points are entered, the domain is broken up into overlapping regions, and a surface fit is performed on each region. Points lying in more than one region are assigned a value that is a weighted average of the multiple estimates.

An option is available to use reciprocal multiquadric interpolation (RMQ). In this case, the square root in the above equation is replaced by the reciprocal of the square root.


References

Hardy, R. L. and W. M. Gofert (1975), Least squares prediction of gravity anomalies, geoidal undulations, and deflections of the vertical multiquadric harmonic functions, Geophysical Research Letters, 2, 423-426.

Pottmann, H. and M. Eck (1990), Modified multiquadric methods for scattered data interpolation over a sphere, Computer Aided Design, 7, 313-321.