GNGTS 2019 - Atti del 38° Convegno Nazionale

GNGTS 2019 S essione 3.2 705 space. In this work we apply an HMC algorithm for inverting Rayleigh waves dispersion curves on synthetic and experimental tests. We inverted for Vs , Vp/Vs ratio and layer thicknesses, whereas the density is kept fixed during the inversion at a constant value. The implemented HMC algorithm requires the number of layers be defined as input to the inversion. However, the limited computational cost of the HMC inversion allows us to perform different inversions with different model space parameterizations. Then, standard statistical tools (such as χ 2 probability or the Bayesian information criterion “BIC”) can be used to define the most appropriate model parameterization to use. Method. The HMC relies on the Bayesian inversion framework. In this context the solution of an inverse problem is the posterior probability density (PPD) function that is defined as follows: (1) where d is N -dimensional observed data vector, and q is the Q -dimensional model parameter vector. The left-hand side term of equation (1) represents the target PPD that could be numerically estimated from the ensemble of models sampled during the Monte Carlo sampling. HMC algorithm treats a model as the mechanical analogue of a particle that moves from its current position (current model) to a new position (proposed model) along a trajectory. This trajectory is determined by the potential energy ( U ), the kinetic energy ( K ) and the mass matrix ( M ). The potential energy is the negative natural logarithm of the posterior (see equation 1) or in other terms is the misfit function associated to the inverse problem. In this context more plausible models with large values of the posterior are associated to low potential energies. HMC determines the kinetic energy by introducing an auxiliary variable (momentum variable) p that is defined over a Q-dimensional space: (2) where M is the Q × Q mass matrix that must be accurately set to ensure the convergence of the HMC algorithm (see Fictner et al. 2019). The vectors p and q define the so-called phase space. After defining the kinetic and potential energies, the model q moves through the 2×Q-phase space according to Hamilton’s equations: (3) where t indicates the artificially introduced time variable. For each current model q , and for each iteration, HMC executes the following steps: 1. Randomly draw the Q momenta p i from the normal distribution ; 2. Derive the proposed model q ( t ) and the new momenta p ( t ) by solving Hamilton’s equations (3) for a given propagation time t . In this work we use the leap-frog method to solve this equation (Betancourt, 2017). The propagation time t , together with the mass matrix plays a crucial influence on the convergence of the sampling; 3. Accept the proposed model with probability α: (4) where the total energy or Hamiltonian of the model is the sum of kinetic and potential energies ( H = U + K ). If accepted, the proposed q ( t ) point constitutes the starting models for the next trajectory ( q = q ( t )). Otherwise, the current model q is again used as the