Through use of an objective function that quanitifies the difference between observations and synthetics, waveform inversion becomes an optimization problem.
Conventionally, the optimization procedure is divided into two steps:
Compute search directionDetermine step lengthNonlinear conjugate gradient, quasi-Newton, and truncated-Newton algorithms all provide viable options. In our experience, quasi-Newton algorithms generally provide the best performance.
[example using SeisFlows]() [example using SPECFEM utilities]()
Given a search direction, the line search consists of finding an acceptable step length. With an efficient line search, good step lengths can usually be found with one or two forward simulations.
[example using SeisFlows]() [example using SPECFEM utilities]()
As an optimization problem, waveform inversion is both nonlinear and nonconvex.
Many optimization algorithms were devised for linear problems and only later adapted for nonlinear problems. Numerical errors that accumulate as a result of nonlinearity can be dealt with by discarding the ‘memory’ of the algorithm and starting once again from the steepest descent direction.
Because of nonconvexity, convergence to a point other than global minimum of the objective function is a possibility. Through choice of multiscale and regularization parameters, care must be taken that the updated model remains in the basin of convergence of the global minimum.