A General Approach for Solving Volcano Deformation Inverse Problems by Applying Finite Element Method and Metamodel Techniques

The inverse problem also known as parameters estimation or parameters identification is a common but very important task in the world of science and engineering. Its applications can be found in many different fields such as geoscience, nondestructive material testing, aerodynamics, etc. Solving an inverse problem means to find a proper set of parameters of a model, to be well fitted, to a given data set, a task very difficult and time-consuming for complex models, especially when the relationships among the parameters are highly nonlinear.

Our work is dedicated to an efficient solver for inverse problems by applying finite element method together with metamodel techniques.

In recent decades, thanks to the rapid development of geodetic techniques such as GPS and InSAR, the number of observations of ground deformation in volcanic areas increased drastically. The improvement of these datasets, both in term of spatial and temporal distribution, higher resolution and better accuracy, provides invaluable observations of the surface deformation that can be used to better understand volcanic processes and possibly improve our ability of forecasting the behavior and the hazards associated with a given volcano.

Usually, volcano deformation is modeled using simple analytical solutions. In reality, the complexity of volcanic processes is highly oversimplified by these models, and the estimated sources of deformation could be significantly biased. The use of more complex models, as finite element method, allows a more realistic representation of the complex geophysical system and a more reliable simulation of a volcanic system more compatible with the improved observations. However, finite element models usually take long time to run and are not directly suitable to traditional inversion schemes. During the parameter identification phase, solving the inverse problems requires running the underlying model a significant number of times. The long time needed to run finite element models makes this approach inefficient. To overcome this issue, we suggest the introduction of the idea of metamodel. A metamodel is a mathematical approximation of the underlying system, which is very efficient in computation and keeps relatively good accuracy compared to the original model. With the self-updating procedure, metamodels can greatly reduce the number of model runs needed for the parameter estimation. In this way, the efficiency of the optimization process is significantly improved.

Here we present some example of the use of methamodels with synthetics data.