First derivatives of fuzzy surfaces

Abstract

The presented research shows how the first derivatives (slope and aspect) can be calculated from a fuzzy surface by the means of fuzzy arithmetic within the geographic information system. The proposed method works with fuzzy numbers of arbitrary shape which helps with more precise specification of input values as well as more exact calculation of results. Three most important methods of partial derivatives calculation based on finite elements approximation of a surface are presented and discussed. The presented approach provides an alternative for uncertainty propagation that is commonly performed by the utilization of statistics and the Monte Carlo method in geographic applications. The example calculation shows the differences between the obtained results calculated with the utilization of fuzzy arithmetic and the Monte Carlo method.