Multifractal objective analysis: conditioning and interpolation

We investigate various ways of statistically estimating multifractal fields from sparse data. First, the problem is set in the general framework of conditional expectations, and the effect of (multi) fractal sampling on the statistics of the measured process is investigated, showing how analytical expressions describing the statistical properties of the phenomenon should be modified by the sampling. Then, several techniques are introduced, our goal being to estimate the intensity of a field at resolution lambda, given samples of the process collected by networks at higher resolutions Lambda > lambda. The general strategy underlying all the estimating techniques presented is to approximate the unknown field values at resolution lambda by means of most likely estimates conditional to the available information at resolution Lambda > lambda. Finally, the procedures are tested on simulated lognormal multifractal fields sampled by means of fractal networks, and the propagation of the errors in a scaling framework is also discussed. These techniques are necessary for estimating geophysical processes in regions where no monitoring stations are present, a scenario often encountered in practice, and may also be of great help in studying natural hazards and risk assessment.