Hamiltonian Monte Carlo solution of tomographic inverse problems

Abstract

SUMMARY We present the theory for and applications of Hamiltonian Monte Carlo (HMC) solutions of linear and nonlinear tomographic problems. HMC rests on the construction of an artificial Hamiltonian system where a model is treated as a high-dimensional particle moving along a trajectory in an extended model space. Using derivatives of the forward equations, HMC is able to make long-distance moves from the current towards a new independent model, thereby promoting model independence, while maintaining high acceptance rates. Following a brief introduction to HMC using common geophysical terminology, we study linear (tomographic) problems. Though these may not be the main target of Monte Carlo methods, they provide valuable insight into the geometry and the tuning of HMC, including the design of suitable mass matrices and the length of Hamiltonian trajectories. This is complemented by a self-contained proof of the HMC algorithm in Appendix A. A series of tomographic/imaging examples is intended to illustrate (i) different variants of HMC, such as constrained and tempered sampling, (ii) the independence of samples produced by the HMC algorithm and (iii) the effects of tuning on the number of samples required to achieve practically useful convergence. Most importantly, we demonstrate the combination of HMC with adjoint techniques. This allows us to solve a fully nonlinear, probabilistic traveltime tomography with several thousand unknowns on a standard laptop computer, without any need for supercomputing resources.