M. Sambridge et al., GEOPHYSICAL PARAMETRIZATION AND INTERPOLATION OF IRREGULAR DATA USINGNATURAL NEIGHBORS, Geophysical journal international, 122(3), 1995, pp. 837-857
An approach is presented for interpolating a property of the Earth (fo
r example temperature or seismic velocity) specified at a series of 'r
eference' points with arbitrary distribution in two or three dimension
s. The method makes use of some powerful algorithms from the field of
computational geometry to efficiently partition the medium into 'Delau
nay' triangles (in 2-D) or tetrahedra (in 3-D) constructed around the
irregularly spaced reference points. The field can then be smoothly in
terpolated anywhere in the medium using a method known as natural-neig
hbour interpolation. This method has the following useful properties:
(1) the original function values are recovered exactly at the referenc
e points; (2) the interpolation is entirely local (every point is only
influenced by its natural-neighbour nodes); and (3) the derivatives o
f the interpolated function are continuous everywhere except at the re
ference points. In addition, the ability to handle highly irregular di
stributions of nodes means that large variations in the scale-lengths
of the interpolated function can be represented easily. These properti
es make the procedure ideally suited for 'gridding' of irregularly spa
ced geophysical data, or as the basis of parametrization in inverse pr
oblems such as seismic tomography. We have extended the theory to prod
uce expressions for the derivatives of the interpolated function. Thes
e may be calculated efficiently by modifying an existing algorithm whi
ch calculates the interpolated function using only local information.
Full details of the theory and numerical algorithms are given. The new
theory for function and derivative interpolation has applications to
a range of geophysical interpolation and parametrization problems. In
addition, it shows much promise when used as the basis of a finite-ele
ment procedure for numerical solution of partial differential equation
s.