Multifractal interpolation method for spatial data with singularities

This paper introduces the multi fractal interpolation method (MIM)developed for handling singularities in data analysis and for data interpolation. The MIM is a new moving average model for spatial mapping and interpolation. The model decomposes the raw data into two components: singular and nonsingular components. The former can be characterized by a localized singularity index that quantifies the scaling invariance property of measures from a multifractal point of view. The latter is a smooth component that can be estimated using ordinary kriging or other moving average models. The local singularity index characterizes the concave/convex properties of the neighbourhood values. The paper utilizes a binomial multiplicative cascade model todemonstrate the generation of one- and two-dimensional data with multi-scale singularities which can be modelled by asymmetrical multifractal distribution. It then introduces a generalized moving average mathematical model for analysing and interpolating data with singularities. Finally, it is demonstrated by a one-dimensional case study of de Wijs’ data from a profile in a zinc mine, that incorporation of spatial association and singularity can improve the interpolation result, especially for observed values with significant singularities.