Surface deformation measured by geodetic data is the sum of single-strain sources deforming at depth. A combination of volume changes from several analytical models (e.g. a point source or dislocation along a plane) can be used to model the different sources. However, solving for the best fit of volume variations, dislocations, position and orientation parameters of all sources is a nonlinear problem, and its solution is generally non-unique. This problem can be converted into a linear one by assimilating the sum of sources to a simplified model formed by three orthogonal planes of dislocations at fixed position and orientation. This strain source model is equivalent to having all neighbouring deformation sources contained in a small size volume. The determination of the strain tensor components can be performed by inverting geodetic data. Because of their high resolution, tiltmeters are well adapted to survey shallow deformation of volcanoes and geological reservoirs. However, they are known to display unknown long-term drift. We propose an approach to jointly estimate the temporal evolution of the strain source and time-dependent instrumental parameters. We verify the approach using synthetic data, giving confidence intervals for each component of the strain tensor. Finally, we link geological information to the internal deformation by interpreting the strain tensor as principal directions of deformation. This approach seems promising for the identification of fracture onset and fault reactivation in geothermal, hydrocarbon exploitations or volcanic systems.