Local Sensitivity Analysis of a Numerical Model of Volcanic Plinian Columns through Automatic Differentiation

Numerical simulation of geological phenomena are nowadays widely used to understand them. A good example are the numerous codes developed to model mass motion in different processes. Such models are approximations to the real phenomena because the physical conditions are simplified or because the values of the parameters intervening in the modeling are known only approximately. For this reason, an obligatory step in the application of these models is the assessment of their sensitivity to changes in the parameters. Sensitivity analysis is also required to evaluate the influence of the different input variables in the outcome of the model. In general this analysis is carried out by repeated computation of the model to different sets of input variables and settings of the model. A more fruitful procedure to carry out such analysis can be achieved by automatic differentiation. In this paper we illustrate such approach through its application to the model of an eruptive plinian column. The results of the analysis allow to assess, in a systematic and efficient manner, the relevance of the different boundary values and parameters of the model. The analysis reveals the effects on velocity, temperature, gas fraction, radius and ultimately column height, by changes in their boundary value (i.e. at the vent). The effect of changes in the value of the gas constants, density of pyroclasts and specific heat of the erupted material is also evidenced. Alternatively, the results can be regarded as a mean to estimate the values of the variables for different set of boundary values without recurring to new runs of the model. Automatic differentiation is shown to be a convenient scheme to obtain derivatives of model variables. This type of analysis forms in addition the basis for further work in inversion schemes.