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. 2019 Feb 5;9:1449. doi: 10.1038/s41598-019-38595-7

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© The Author(s) 2019

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Q_s prediction model using model selection analysis. (a) Statistical relationship between Q_s derived from fallout deposits, Q_a derived from satellite-based measurements, and H (above the vent) derived from observations, in a three-dimensional natural logarithm space. The best Q_s prediction model is shown as the coloured plane and the related equation is given in natural scale following a power-law at the top of the plot. It was selected by the AICc (Corrected Akaike Information Criterion) which gives a robust evaluation of the goodness-of-the-fit for small datasets. The error factor and related RMSE are provided at a 95% prediction interval. See Supplementary Information Table 2 for all the goodness-of-the-fit evaluation criteria. (b) Error factor contour levels related to the MER estimation plotted on the two-dimensional plane H vs. Q_a in natural logarithm, and showing the anisotropy of the error distribution. Red triangles represent eruptions (12 over 22) for which the error factor value is ∼2 or less. Practically, this means that estimations of MER for future eruptions falling in this range of Q_a (∼1 × 10³ to 1 × 10⁵ kg/s) and H (∼7 to 21 km) have a 95% probability to fall within an error factor of 2 only.