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Radial Basis Functions Radial basis interpolation is the name given to a large family of exact interpolators. In many ways the methods applied are similar to those used in geostatistical interpolation (see Section 6.7 ), but without the benefit of prior analysis of variograms. On the other hand they do not make any assumptions regarding the input data points (other than they are not co-linear) and provide excellent interpolators for a wide range of data. See also Section 8.3.2 for an alternative (neural network) approach to modeling with radial basis functions. A great deal of research has been conducted into the quality of these interpolators, across many disciplines. For terrain modeling and Earth Sciences generally the so-called multiquadric function has been found particularly effective, as have thin plate splines. The simplest variant of this method, without smoothing (i.e. as an exact interpolator) can be viewed as a weighted linear function of distance (or inverse distance) from grid point to data point, plus a “bias” factor, m . The model is of the form: or the equivalent model, using the untransformed data values and data weights λ i : In these expressions z p is the estimated value for the surface at grid point p ; φ ( r i ) is the radial basis function selected, with r i being the radial distance from point p to the i th data point; the weights w i and λ i and the bias value m (or Lagrangian multiplier) are estimated from the data points. This requires solving a system of n linear equations. Using the second of the two models above the procedure is then essentially the same as for Ordinary Kriging (see further, Section 6.7.2 ). For clarity we outline this latter procedure below as a series of steps using matrix notation: Compute the n x n matrix, D , of inter-point distances between all ( x,y ) pairs in the source dataset (or a selected subset of these) Apply the chosen radial basis function, φ (), to each distance in D , to produce a new array Φ Augment Φ with a unit column vector and a unit row vector, plus a single entry 0 in position ( n +1),( n +1). Call this augmented matrix A — see below for an illustration Compute the column vector r of distances from the grid point, p , to each of the source data points used to create D Apply the chosen radial basis function to each distance in r , to produce a column vector ϕ and then create the ( n +1) column vector c as ϕ plus a single 1 as the last entry Compute the matrix product b = A -1 c . This provides the set of n weights to be used in the calculation of the estimated value at p , plus the Lagrangian value m using the linear equation cited earlier in this Section. The value m is not used in calculating the estimate at p in this formulation In matrix form the system of linear equations being solved is of the form:

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i.e. Ab = c hence b = A -1 c .
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