Full-waveform inversion within a deterministic framework commonly uses gradient-based methods to minimize a least-squares error function. Due to the non-linearity of the problem, this function has several local minima. To avoid them, it is necessary to start the optimization procedure from a good initial model. Within a probabilistic framework, Markov Chain Monte Carlo methods are used to sample a probability density function that represents the possible solutions. In high-dimensional problems, traditional MCMC methods become inefficient as random transitions are unlikely to find the regions of high probability. Hamiltonian Monte Carlo appears as a method that can efficiently find these regions without spending computation time on regions of no interest. To evaluate the probability of a sample, the least-squares error is calculated. Synthetic data is obtained through an existing 2D frequency domain full-wave
modelling code and compared to the observed data. Through the Hamiltonian Monte Carlo sampler, the high probability regions are sampled and used to build the probability density function that represents the possible solutions. The algorithm was tested with both synthetic and real data. For the synthetic case a 1D model was inverted. The normalized least-squares error was reduced by 98%. In the deepest section the true model lies outside the uncertainty range of the estimated model. This appears as the result of sampling narrow regions
of the probability density function. Then, the variance of the samples is underestimated and therefore the error. For the real dataset, a 2D model was inverted. This model lacks some of the large-scale features compared to deterministic full-waveform inversion results. However, a good model was found without any a priori information and almost any manual intervention. This suggest that, at least for the studied example, the algorithm can provide a
good starting model for deterministic full-waveform inversion. The results obtained through this approach are almost identical to those obtained with more labour intensive adjustments.

The finite-element method can easily handle complicated geometries because of the application of unstructured meshes. Unlike the Cartesian grid used in the finite-difference method, the unstructured mesh can follow the sharp interfaces that separate two layers of different properties. Therefore, the finite-element method can provide more accurate solutions for the simulation of seismic wave propagation. Meshes of good quality are required for the finite-element simulation. However, it is not trivial to set up an appropriate mesh. First
of all, the mesh should contain elements of good shapes and sizes. In addition, the sharp interfaces should coincide with the edges of the elements instead of intersecting with them. These requirements are formulated as an optimization problem with three terms, measuring the difference between the actual and prescribed scaling field, shape quality, and the area between prescribed curves and the nearest triangle edges. The solution of the optimization problem should provide the desired mesh. The mesh generator MESH2D was applied to obtain an initial mesh. The Matlab function minFunc was used to search for the minimum of the constructed objective function. Three weights balance the three terms in the objective function. When it comes to complicated models, these weights have to be chosen carefully to produce a reasonable mesh.