Consideration on Interpolation Methods to Characterize a Subsurface Structure from Irregularly Spaced Data

In most cases, geological and geotechnical investigation data are irregularly distributed in horizontal and vertical directions. Therefore proper choice of an automatic contouring method is required to reveal diverse subsurface structures. Many contouring methods have been proposed so far, which can be classified into two categories regarding their principles. The first method is a global fit algorithm termed approximation. The usual global algorithm is a trend surface analysis, which reveals a regional trend in sample (or measured) data through the weighted least squares method. In this method, weighting coefficients are assigned to each data point according to the statistical property of sample data. The second method is a local fit algorithm termed interpolation, which constructs a curved surface passing through or very near to every sample value. The second method can be subdivided into two groups in consideration of the population of sample data. This paper examines several interpolation methods suitable for the sample data which are originated from a single population and therefore satisfy the geological homogeneity. The moving average method, the optimization principle method, kriging, and the optimization principle method combined with kriging were chosen and applied to the reconstruction problem of the defined forthorder polynomial using irregularly spaced sample data. It was revealed that the optimization principle method gives the smallest interpolation error among those methods. The goodness of the interpolation result by the optimization principle method was also judged through two interpolation criterion calculated from the interpolated grid data: the smoothness of curved surface and the error of semivariogram. Furthermore, the interpolation error was found to have a correlation with the total values of three factors, namely, the distance between grid point and data point, the angular distribution pattern of data points, and the deviation of data values.