A Penalization-Regularization-Operator Splitting Method for Eikonal Based Traveltime Tomography

We propose a new methodology for carrying out eikonal based traveltime tomography arising from important applications such as seismic imaging and medical imaging. The new method formulates the traveltime tomography problem as a variational problem for a certain cost functional explicitly with respect to both traveltime and sound speed. Furthermore, the cost functional is penalized to enforce the nonlinear equality constraint associated with the underlying eikonal equation, biharmonically regularized with respect to traveltime, and harmonically regularized with respect to sound speed. To overcome the difficulty associated with the inherent nonlinearity of the eikonal equation, the Euler-Lagrange equation of the penalized-regularized variational problem is reformulated into an equivalent, mixed optimality system. This mixed system is associated with an initial value problem which is solved by an operator-splitting based solution method, and the splitting approach effectively reduces the optimality system into three nonlinear subproblems and three linear subproblems. Moreover, the nonlinear subproblems can be solved pointwise, while the linear subproblems can be reduced to linear second-order elliptic problems. Numerical experiments show that the new method can carry out traveltime tomography successfully and recover sound speeds efficiently.