A novel traveltime tomography approach has been developed to invert both velocity and anisotropy restricted to 2D geometry. The fundamental equation whose solution gives us the first-arrival traveltimes between a source and a receiver is the Eikonal equation, which becomes more complex when anisotropy is considered. In order to solve the Eikonal, an Eulerian formulation based on ... [Show full abstract] element-discretization discontinuous Galerkin method is adopted. The use of a direct solver allows us to obtain the total solution of the Eikonal, this includes diffraction events that may occur in the presence of large-velocity contrasts, while the widely used ray solution does not include these events. For the inverse part, an iterative local gradient-based optimization is chosen, where a least-square misfit function between picked and synthetic traveltimes need to be minimized. Contrary to other tomography approaches that usually compute the expensive sensitivity matrix, we avoid this computation by using the adjoint-state method. The adjoint formulation allows us to obtain the gradient efficiently by solving a transport equation that propagates the residuals from receivers to each source location, thus describing the sensitivity of the data to the model. We have developed a workflow that includes model regularization and data-weight matrix. Anisotropy is obtained under the elliptical assumption, thus two parameters are inverted simultaneously with an optimal parametrization that includes vertical and horizontal velocities, this choice being driven by a sensitivity analysis and synthetic examples. The code was used for active seismic and electromagnetic data acquired in carbonates both at the field and laboratory scale with different acquisition configurations. A first example concerns crosshole GPR acquisitions performed at the field scale within the Laboratoire Souterrain à Bas Bruit (LSBB) facilities, where the presence of a deep gallery makes the inversion challenging. In this weak anisotropy environment, the results are confronted to full wave inversion results and to geological data. At the laboratory scale, a multi-physics acquisition including seismic and GPR data was tested on a cubic rock sample. It underlines some issues related to the size of the sources/receivers compared to the dimension of the sample, which must be tackled before considering any inversion. Then, the datasets are inverted and velocity/anisotropy images are obtained and discussed in terms of heterogeneity and potential localized fractures.