A total least-squares approach in two stages for semivariogram modeling of aeromagnetic data

Semivariogram analysis and estimation procedures are important for geostatisticians, Geographical Information Systems experts and earth scientists. A mathematical model, selected from a list of admissible functions, is fitted to the vector of empirical semivariogram values. Least-squares techniques (ordinary, weighted, and generalized) are often used in this process to compute the parameters within this mathematical model. However, least-squares fitting techniques consider the lag-distances (on the abscissa) of the empirical semivariogram graph as fixed or error-free; this is an imprecise assumption since these lag-distances are also measured quantities and include random errors. Here, a novel approach is presented that treats the empirical semivariogram lag-distances on the abscissa and the semivariogram values symmetrically under the assumption of random errors in both variables. This approach employs the relatively new Total Least-Squares (TLS) estimation technique, introduced in 1980 by Golub and van Loan, and provides a more accurate fitting with special emphasis on the crucial behavior near the origin. The main formulas and concepts of the TLS approach are described, along with its adaptation to the spatial context. Experiments of fitting semivariogram models to aeromagnetic data collected in Antarctica are presented, and various quality indicators are studied to compare the Weighted Least-Squares (WLS) with the Total Least-Squares methods. These studies demonstrate the general superiority of the TLS approach over the common WLS approach.