On precisely modelling surface deformation due to interacting magma chambers and dykes

Combined data sets of InSAR and GPS allow us to observe surface deformation in volcanic settings. However, at the vast majority of volcanoes, a detailed 3-D structure that could guide the modelling of deformation sources is not available, due to the lack of tomography studies, for example. Therefore, volcano ground deformation due to magma movement in the subsurface is commonly modelled using simple point (Mogi) or dislocation (Okada) sources, embedded in a homogeneous, isotropic and elastic half-space. When data sets are too complex to be explained by a single deformation source, the magmatic system is often represented by a combination of these sources and their displacements fields are simply summed. By doing so, the assumption of homogeneity in the half-space is violated and the resulting interaction between sources is neglected. We have quantified the errors of such a simplification and investigated the limits in which the combination of analytical sources is justified. We have calculated the vertical and horizontal displacements for analytical models with adjacent deformation sources and have tested them against the solutions of corresponding 3-D finite element models, which account for the interaction between sources. We have tested various double-source configurations with either two spherical sources representing magma chambers, or a magma chamber and an adjacent dyke, modelled by a rectangular tensile dislocation or pressurized crack. For a tensile Okada source (representing an opening dyke) aligned or superposed to a Mogi source (magma chamber), we find the discrepancies with the numerical models to be insignificant (<5 per cent) independently of the source separation. However, if a Mogi source is placed side by side to an Okada source (in the strike-perpendicular direction), we find the discrepancies to become significant for a source separation less than four times the radius of the magma chamber. For horizontally or vertically aligned pressurized sources, the discrepancies are up to 20 per cent, which translates into surprisingly large errors when inverting deformation data for source parameters such as depth and volume change. Beyond 8 radii however, we demonstrate that the summation of analytical sources represents adjacent magma chambers correctly.