Automatic Contouring from Scattered Data Points

It is possible to interpolate in two dimensions from scattered data points using a stochastic process model, giving an interpolating function which is continuous in all derivatives, passes exactly through the points given and does not generate spurious features in regions of no data. Using this interpolating method, contours are produced directly from the data without an intermediate grid. Extensions of the basic model include a two-stage model which allows for a long- range trend. Thus the values y may be calculated once for all. This step involves inverting the n x n correlation matrix for the known points, and if n is large it may be more efficient to partition the points into groups, and calculate the y- values separately for each group, taking into account neighbouring points. The values of y may be considered to be 'uncorrelated' data values, wherein the fact that the data values are spatially correlated has been removed from the data. Thus each interpolating function evaluation requires the calculation of n correlation values, and a multiplica- tion with a constant vector. With little additional work, it is possible to calculate the first and second derivatives of the interpolating function, and these are used in the contour tracing. Details of the theory of this interpolating method are to be found in the other paper. 1 The two parameters, fi and p, may be estimated from the known data points, as described in the earlier paper, or can be chosen to suit the user of the system. The 'correlation distance' p may be considered to be the distance over which spatial correlation between two points is appreciable. The 'grand mean' n is the limiting value to which the interpolating function tends far away from any known data points.