Satellitenbild Siziliens mit dem rauchenden Ätna am 7. November 2013.
Sicily: The volcano Etna, 7th of November 2013. (Photo: ESA/VITO)

The quality of the estimate depends on how complete and how chronologically accurate and precise past eruptions are documented.

To qualify for analysis, the eruption record needs to fulfill a number of conditions:

1.    The eruption records needs to be complete. This may sound unnecessary to mention at first glance, but in many cases it is not guaranteed that really all eruptions that took place at a volcano are listed in the eruption record. For the recent past, there are usually quite reliable and complete records. Often, direct observations are available, provided either by monitoring devices, satellites or eyewitnesses' reports.
2.    Volcanic eruptions reaching further back in time are reconstructed from their deposits. There is a risk of in particular small eruptions to be overlooked. Very often, the deposits may have been partially or completely removed in the course of years to thousands of years by erosion through rain, snow and wind; or destroyed to a degree that they can not be identified and assigned to a particular eruption any more. Moreover, as for the historical eruptions, accurate dates are needed to ensure that the statistical analysis is based on a correct data set. Ages of eruptions of the geological past are obtained using numerical (usually radiometric) dating methods. In addition, relative ages can be derived from stratigraphic relationships, i.e., from the succession of deposits in the field and sedimentation rates of the deposits between the volcanic rocks.
3.    The eruptions need to have happened independently of each other, implying that the probability of an eruption does not depend on the point of time of the previous eruption. Technically, this would be described as memoriless system.
4.    Furthermore, it has to be possible to express the eruption record along its entire length with the same statistical parameters, meaning that there is no time trend. Each randomly selected section must exhibit the same statistical properties. Should this not be the case, a piecewise analysis is performed.

If these conditions are fulfilled, the eruption time series is transferred into a numerical model. In the processing of the temporal eruption sequence as a statistical time series, the following can be taken into consideration:

1.    Is the activity of a volcano generally increasing or declining?
2.    Are any competing processes possibly delaying or preventing the eruptive activity?

As in medical sciences, volcanologists use the expression "survival time". In the medical context, it refers literally to the time patients survive subsequent to a particular type of therapy, for example. For a volcano, it corresponds to the time interval elapsing between two successive eruptions. In other words, this is the period of time that the "repose time" "survives". Statistical tools such as the so-called Kaplan-Meier estimator and several different distribution functions are used to numerically describe the repose time distribution. After confirming that the models represent the data adequately well, the probability of a future eruption is determined from the mathematical survival function.

Abbildung 1: Kumulative Anzahl der Eruptionen über die letzten 300 Jahre am Vulkan San Miguel in El Salvador.
Abbildung 2: Verteilung der Zeitintervalle zwischen zwei Eruptionen, abstrahiert durch einen Kaplan-Meier Schätzer der Überlebenszeitfunktion. Die farbigen Kurven stellen die optimierten Fitfunktionen dar: grün: Exponentialfunktion; blau: Weibullfunktion; orange: Log-logistische Funktion.
Abbildung 3: Wahrscheinlichkeiten einer zukünftigen Eruption über Zeit, abgeleitet von der Exponentialfunktion (grün) und der Weibullfunktion (blau).
Figure 1: Cumulative number of eruptions over the past 300 years at the San Miguel volcano in El Salvador. (Illustration: GEOMAR)
Figure 2: Distribution of time intervals between two eruptions estimated by a Kaplan-Meier estimator of the survival time function. The coloured curves represent the optimised fit func-tions: green: exponential function; blue: Weibull function; orange: log-logistic function. (Illustration: GEOMAR)
Figure 3: Probabilities of a future eruption over time, derived from the exponential function (green) and the Weibull function (blue). (Illustration: GEOMAR)

Text: Dr. Heidi Wehrmann,GEOMAR Helmholtz-Zentrum für Ozeanforschung Kiel

text, eskp.de CC BY;photographs and graphics, if there are no conflicts with other licences as well: eskp.de CC BY