A novel trans-dimensional inversion algorithm to model deformation sources with unconstrained shape in finite element domains
Authors: De Paolo Erica , Piana Agostinetti Nicola , Trasatti Elisa 
Published: March 2022

&lt;p&gt;Ground deformation signals, detected by geodetic instruments, can provide valuable insights on subsurface processes. The deformation field patterns, in fact, typically reflect characteristics of the buried source such as the position, depth, shape and volume variation. The increasing accuracy and spatio-temporal density of remote-sensing measurements allow us to map these patterns with unprecedented detail, highlighting the need to quantitatively investigate the processes at the origin. In active volcanic sites, in presence of deep pressurized reservoirs, e.g. magma chambers, the correct interpretation of geodetic signals is essential to define the hazard potential. Inverse modeling techniques are commonly employed for this goal, providing quantitative estimates of parameters describing the volcanic source. However, despite the robustness of the available approaches, a realistic imaging of reservoirs is still challenging. The widely used analytical models return quick but simplistic results, assuming an isotropic and elastic crust and forcing the solution to fit in pre-established geometric shapes. The use of inaccurate assumptions about the source shape can lead to the misinterpretation of other fundamental parameters, affecting the reliability of the solution. A more sophisticated analysis, accounting for the effects of topographic loads, crust inelasticity and the presence of structural discontinuities, requires the employment of numerical models, like those based on finite elements methods (FEM), but also a much higher computational effort. Here, we present a novel approach aimed at overcoming the aforementioned limitations. This method allow us to retrieve deformation sources without a-priori shape constraints, benefiting from the advantages of FEM simulations at a cost-efficient computing effort. We image the deformation source as an assembly of elementary units, each one represented by a cubic element of a regular FE mesh, loaded, in turn, with the six components of the stress tensor. The surface response to each stress component is computed and linearly combined to obtain the total displacement associated to the elementary source. This can be extended to a volume of multiple elements, approximating a deformation source of potentially any shape. Our direct tests prove that the sum of the responses associated to an assembly of solid units, loaded with an appropriate stress tensor, is numerically equivalent to the deformation fields produced by corresponding analytical and FEM cavities with uniform pressures applied at their boundaries. Our ability to simulate pressurized cavities in a continuum domain allow us to pre-compute a library of unitary surface responses, i.e., the Green&amp;#8217;s function matrix, and to avoid complex re-meshing. We develop a Bayesian trans-dimensional inversion algorithm to select, scale and sum the displacements associated to each unit belonging to the assemblies that best fit the observations. In particular, we employ two sets of 3D Voronoi cells to sample the model domain, selecting the elementary units contributing to the source solution and the part belonging to the set representing the crust, which remains inactive. In this contribution, we present the original methodology and preliminary applications.&lt;/p&gt;