Statistical analysis of bubble and crystal size distributions: Formulations and procedures
Bubble and crystal size distributions have previously been described only by either exponential or power law functions. Within this limited framework, it has not been possible to characterize size distributions in a fully quantitative manner. We have developed an analytical and computational formulation with which to characterize and study crystal and bubble size distributions (BSD). This formulation demonstrates that all distributions known to date belong to the logarithmic family of statistical distributions. Four functions within the logarithmic family are best suited to natural bubbles and crystals (log normal, logistic, Weibull, and exponential). This characterization is supported by the fact that the power law function widely used for crystal and bubble size analysis is not a statistical distribution function, but rather represents an approximation of the upper regions (larger bubbles/crystals) of the logistic distribution, whose sizes are much larger than the mode.
The coefficients for each of the four logarithmic functions can be derived by 1) best fit exceedance function of the logarithmic distribution, and 2) best fit of the linear transformation of the distribution probability density. A close match of the coefficients derived by the above two methods can be used as an indicator of correct function fitting (choice of initial values). Function fitting by exceedance curves leads to the most accurate statistical results, but has certain strict limitations, including 1) a requirement to rescale the base distribution function; 2) a higher failure rate for function fitting than that for distribution density; 3) uncertainty in observational data error estimates; and 4) unsuitability for visual interpretation. The most productive approach to visualization and interpretation of size distributions is through linear transformation of logarithmic distributions on the basis of probability densities. This also makes it possible to 1) clearly discern bimodal distributions; 2) assess the range of observed objects relative to the full range of the indicated distribution; 3) determine number densities for each mode directly; and 4) integrate to obtain total volume fraction for comparison with available observations. The latter could, in some cases, provide more accurate results than many measurement methods.
Unambiguous definition of Bubble Number Density (BND) must be based on the number of bubbles per melt volume (not number of bubbles per bulk volume), so that like is done with crystals, it can be directly used as an indicator of basic vesiculation processes such that: a) nucleation leads to increase of BND, b) diffusive or decompressive bubble growth keeps BND constant, and c) coalescence decreases BND.
Earth Systems Research Center
Journal of Volcanology and Geothermal Research
Digital Object Identifier (DOI)
Proussevitch, A. A., Sahagian, D. L., & Tsentalovich, E. P. (2007). Statistical analysis of bubble and crystal size distributions: Formulations and procedures. Journal of Volcanology and Geothermal Research, 164(3), 95-111. doi:10.1016/j.jvolgeores.2007.04.007
© 2007 Elsevier B.V. All rights reserved.