Adjoint state method

Abstract: SUMMARY Estimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Frechet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Frechet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjoint-state variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved. Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established.

Abstract: In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of O(N) for N grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that 2 n Gauss-Seidel iterations is enough for the distance function in n dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.

Abstract: Preface. 1. Forward Problems in Groundwater Modeling. 2. An Introduction to Inverse Problems. 3. Classical Definition of Inverse Problems. 4. Indirect Methods for the Solution on Inverse Problems. 5. Direct Methods for the Solution of Inverse Problems. 6. The Adjoint State Method. 7. The Stochastic Method for Solving Inverse Problems. 8. Experimental Design, Extended Identifiabilities and Model Structure Identification. Appendix A: Mapping, Space and Norm. Appendix B: Probabilities and Random Fields. Appendix C: A Fortran Program. References. Index.

Abstract: Classical algorithms used for traveltime tomography are not necessarily well suited for handling very large seismic data sets or for taking advantage of current supercomputers. The classical approach of first-arrival traveltime tomography was revisited with the proposal of a simple gradient-based approach that avoids ray tracing and estimation of the Frechet derivative matrix. The key point becomes the derivation of the gradient of the misfit function obtained by the adjoint-state technique. The adjoint-state method is very attractive from a numerical point of view because the associated cost is equivalent to the solution of the forward-modeling problem, whatever the size of the input data and the number of unknown velocity parameters. An application on a 2D synthetic data set demonstrated the ability of the algorithm to image near-surface velocities with strong vertical and lateral variations and revealed the potential of the method.