Multifractal Distribution of Eigenvalues and Eigenvectors from 2D Multiplicative Cascade Multifractal Fields

Two-dimensional fields (maps) generated by isotropic and anisotropic multiplicative cascade multifractal processes are common in many fields including oceans, atmosphere, the climate and solid earth geophysics. Modeling the anisotropic scaling property and heterogeneity of these types of fields are essential for understanding the underlying processes. The paper explicitly derives the eigenvalues and eigenvectors from these types of fields and proves that the eigenvalues and eigenvectors are described by non-conservative multifractal distributions. This results in a new multifractal model implemented in eigen domain to characterize 2D fields not only with respect to overall heterogeneity and singularity as characterized by the ordinary multifractal model applied to the field itself, but also with respect to orientational heterogeneity of the field. It may also result in a new way to characterize the distribution of extreme large eigenvalues involved in other studies such as principal component analysis. A newly defined operator and its properties as derived in this paper may be useful for studying other types of multifractal cascade processes.