Title: Characterization of Multi-scale Invariant Random Fields,
Abstract: Applying certain flexible geometric sampling of a multi-scale invariant (MSI)
field we provide a multi-dimensional multi-selfsimilar field which has a one to
one correspondence with such sampled MSI field. This sampling enables us to
characterize harmonic-like representation and spectral density function of the
sampled MSI field. Imposing Markov property for the MSI field, we find that the
covariance function and spectral density matrix of such sampled Markov MSI
field are characterized by the covariance functions of samples of the first
scale rectangle. We present an example of MSI field as two-dimensional simple
fractional Brownian motion. We consider a real data example of the
precipitation in some area of Brisbane in Australia for some special period. We
show that precipitation on this area has MSI property and estimate time
dependent scale and Hurst parameters of this MSI field in three dimension as
latitude, longitude and time. Our method enables one to predict precipitation
in time and place.