Interpolated Distance Weighted

In this chapter, the application of multifractal inverse distance weighted (MIDW) interpolation method and a fractal filtering technique, named spatial and spectral analysis or simply (S-A) method (Cheng, 2003), will be illustrated to evaluate geochemical background at the regional and local scale. For this purpose, two case studies will be discussed for two different Italian areas ... 

Gradient methods discussed above use a quadratic function (energy, gradient and approximate Hessian) to model the energy surface near the transition state. Distance-weighted interpolants provide a more flexible functional form that can interpolate arbitrarily spaced points with a smooth differentiable function. For a gradient-based optimization, the Shepard interpolation functions seem appropriate... 

In the bi-hnear interpolation, a simplistic rationale is followed, i.e., for calculating the pixel value of a particular position X,Y), four adjacent pixel values are used [21]. The closer the pixel is to the position (X,Y), the more influence (weight) it will carry. The method is not merely a falhng function of distance from the pixel. Rather it considers a weighted approach based on its spatial locations in a two-dimensional space [47]. The derivation of bi-linear interpolation weights can be expressed as follows [48, 49] ... 

Data are interpolated to fixed pressure intervals, e.g., to 2 dbar for WOCE resolution of the final data. One method is to average all values around the centre value within one interpolation interval (basketing). Values are weighted by distance firom the centre value. 

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