An analytical and computational method is described which
has broad applicability to studies of multiscale phenomena, such as
turbulence, with regard to fractal dimensions as well as their
scale-dependent extensions known as generalized fractal dimensions as
functions of scale. The mathematical basis of the method is the
analytical relation between the shortest-distance probability density
function and the generalized fractal dimension function. The shortest
distance refers to the distance between any randomly chosen point
location, within a reference boundary, and the nearest part of the
multiscale object of interest. These shortest distances, in addition
to being analytically related to the dimension, provide a means to
characterize the scales of level crossing sets of fluctuating fields
or of the fields themselves. We demonstrate aspects of the method
using exact analytical examples and computational tests.