Modeling thickness variability in tephra deposition

Abstract

The attenuation of tephra fall thickness is most commonly estimated after contouring isolated and often irregular field measurements into smooth isopachs, with varying degrees of subjectivity introduced in the process. Here, we present an explicit description of the variability introduced into a semiempirical tephra attenuation relation. This opens the way to fitting models to actual tephra observations through maximum likelihood estimation, rather than using weighted least-squares estimation on the isopachs. The method is illustrated for small-scale basaltic explosive eruptions using a simple, but typical, data set of the actual tephra thickness data published from the 1973 Heimaey eruption. Of the distributions considered to describe variability in these measurements, the lognormal performed poorly, due to its tendency to predict a small number of greatly over-thickened deposits. The Weibull and gamma distributions fitted the data to a very similar degree and produced very similar estimates for the “effective volume,” mean wind direction, and mass/thickness attenuation rate. The latter can be inverted to obtain an estimate of the mean column height. The estimated wind direction, and the column height derived from the estimated thickness attenuation parameter, agreed very well with the direct observations made during the eruption. Augmented by a mixture framework allowing for the incorporation of multiple lobes and/or vents, the model was able to identify the source and direction of tephra deposition for the 1977 Ukinrek Maars eruptions from only the tephra thickness data.