3.2.4 Radial Basis Function
RBF is a deterministic interpolation technique; the interpolation
surface is formed in such a way that it passes through all the
observation points. Wong et al (2002) and Fornberg (2006) has defined
the function as:
P(r) = \(\sum_{j=1}^{N}{\lambda_{j}dz}(||r-r_{j}||)\) (4)
Where s(.) definite positive RBF;\(\mathbf{||}\). \(\mathbf{||}\) = Euclidian norm;
λj = set of unknown weights
λjis calculated using P(rj) = f
(\(r_{j}\)) j = 1,2……., N ( 5)
(4) and (5) together form a new system of equation which is the form
Ѕᴧ = ϴ (6)
Where Ѕ is a NxN matrix of RBF values it is also known
as interpolation matrix and ᴧ = [\(\lambda_{j}\)] whose weights are
unknown [fj] = column matrix of observed
values.
The interpolation widely depends upon the basis function chosen. The
available choices of basic functions are thin plate spline, multi-log,
inverse multi-quadric, natural cubic spline. The basis function further
depends upon Euclidean distance between r and rj, and a
smoothing parameter “R”. Hardy (1971) has discussed how to evaluate R.
Later, Folly (1987) and Franke (1982) also discussed the range of values
that can be taken for R. In this paper R is taken as
R2= d2/(25N). Where d is the
diagonal distance of the grid in which the interpolation is taking
place.