Geoscience surface modelling: a summary of some mathematical methods

The purpose of surface modelling is to estimate from scattered data points the value of a variable at a point where the variable has not been measured. In the discussion that follows we take the variable to be the height of the surface but the conclusions can be applied to other variables such as the density of a substance in the rock. The surface model provides a smooth surface that goes through all the data points. It represents an averaged elevation and, as such, is smooth even though the actual surface is rough. The fitted surface should not introduce features for which there is no evidence and should not vary wildly between data points.
A good model should make some estimate of the likely uncertainty in the estimation of a surface. This uncertainty comes from several sources.
1. The errors inherent in the data.
2. The uncertainty in the estimation due to the distribution of data points.
3. The uncertainty caused by the roughness of the surface.
4. Discrepancies between the mathematical model and the real geology.
This document reviews some techniques currently in use as well as some that may be used in the future, giving a short description of each with a brief account of the background mathematics in smaller print.