GEOSTATISTICAL MEANING OF THE INDEFINITE LAGRANGE MULTIPLIER IN THE ORDINARY AND DOUBLE KRIGING METHODS

Abstract

Relevance. When developing new methods of volumetric modeling of the material composition of geological environment according to borehole and surface observations, the authors previously proposed the possibility of an alternative estimate of the error using the Lagrange multiplier. In practice, due to small number of borehole measurements and large distance between boreholes, it is important not only to use effective model of the medium, but also to estimate its accuracy in the interwell space. The use of simple and understandable estimates of model accuracy is the key to reducing the risks of making incorrect decisions on subsequent drilling. The method of indefinite Lagrange multipliers is used in solving extreme problems with constraints of equality type. In such sciences as economics, statistical and classical mechanics, Lagrange multipliers have economic or physical meaning and become of practical importance. However, in geostatistics, when solving the problem of selection of optimal methods of interpolation of spatial data, Lagrange multipliers are avoided at the first stages of solution of systems of linear equations. In this relation there are no both analytical and applied researches of its value. Objective: to clarify the meaning of the Lagrange multiplier when solving systems of ordinal and double kriging; to develop new ways of obtaining normalized estimates of geostatistical modeling error; to carry out investigations of the obtained expressions on the real materials, namely, in solving the tasks of geological media material composition prediction. Methods: ordinary kriging, double kriging, Lagrange multiplier method, Cramer's method. Results. In order to conduct research it was propose to add an unknown component to the model of the set of known values of the predicted parameter. Based on this model, analytical and numerical studies were conducted, which led to new expressions linking the weight of the unknown component and the covariance properties of the measurements with the Lagrange multiplier. Estimates of the weight function of the unknown component were proposed. The practical importance of the Lagrange multiplier in the analysis of modeling errors is shown on the example of real data for the solution of the problem of predicting the material composition of geological environments.

Keywords

• modeling geological environment parameters
• modeling errors
• geostatistics
• kriging
• indefinite lagrange multipliers
• statistical estimates