  After 100 years of theoretical treatment of speckle patterns from coherent
illumination, there remain some open questions about the nature of ultrasound
speckle from soft vascularized tissues. A recent hypothesis is that the fractal
branching vasculature is responsible for the dominant echo pattern from organs
such as the liver. In that case an analysis of cylindrical scattering
structures arranged across a power law distribution of sizes is warranted.
Using a simple model of echo strength and basic transformation rules from
probability, we derive the first order statistics of speckle considering the
amplitude, the intensity, and the natural log of amplitude. The results are
given by long tailed distributions that have been studied in the statistics
literature for other fields. Examples are given from simulations and animal
studies, and the theoretical fit to these preliminary data support the overall
framework as a plausible model for characterizing ultrasound speckle
statistics.
