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.