7.6 Gridding and Contouring Background

The previous data sets were all stored in equally-spaced two-dimensional arrays. Most data in earth sciences, however, are obtained on an irregular sampling pattern. Therefore, irregular-spaced data have to be interpolated, i.e., we compute a smooth and continuous surface from our measurements in the field. Surface estimation is typicall carried out in two major steps. Firstly, the number of control points needs to be selected. Secondly, the grid points have to be estimated. Control points are irregularly-space field measurements, such as the thicknesses of sandstone units at different outcrops or the concentrations of a chemical tracer in water wells. The data are generally represented as xyz triplets, where x and y are spatial coordinates and z is the variable of interest. In such cases, most gridding methods require continuous and unique data. However, the spatial variables in earth sciences are often discontinuous and spatially nonunique. As an example, the sandstone unit may be faulted or folded. Furthermore, gridding requires spatial autocorrelation. In other words, the neighboring data points should be correlated with each other by a certain relationship. It is not sensible to use random z variable for the surface estimation if the data are not autocorrelated. Having selected the control points, the calculation of the z values at the equallyspaced grid points varies from method to method.

Various techniques exist for selecting the control points (Fig. 7.5a). Most methods make arbitrary assumptions on the autocorrelation...