First-arrival delayed tomography using 1st and 2nd order adjoint-state method

International audienceWe compare the standard procedure of delayed first-arrival time tomography based on two nested loops workflow and ray tracing with two alternative ray-free approaches based on eikonal solver and adjoint formulations. The standard approach requires the explicit computation and storage of Fréchet derivatives. Then the workflow is composed of an outer loop on velocity model updating and an inner loop for the iterative estimation of the model perturbation (usually LSQR). The adjoint approaches avoid computation and storage of Fréchet matrix, and use instead an efficient estimation of the gradient of the data misfit function, which is enough for steepest-descent or l-BFGS quasi-Newton methods. In the l-BFGS approach, successive gradients are stored and used for an approximate estimation of the influence of the inverse Hessian. Such a workflow is only composed of a single loop optimization. Using a second-order adjoint formulation, we can also obtain adjoint formulas to compute efficiently Hessian-vector product. Those formulas yields a new manner to perform the two-nested-loops workflow (either in the Gauss-Newton approximation or with the full Hessian matrix). The composition of the memory-demanding Fréchet derivative matrix is not required and the computational cost is independent of the number of receivers. This leads to the so-called matrix-free Truncated Newton optimization method. Through a synthetic 2D example and a real example of travel-time tomography in the Nankai Trough, we show that the results provided by the l-BFGS quasi-Newton method and the truncated Newton method are similar. This promotes the use of the l-BFGS method for the mono-parameter velocity model reconstruction as it is based on a single loop formalism and is therefore less computationally expensive