Implementation of the marchenko method

The Marchenko method makes it possible to compute subsurface-to-surface Green's functions from reflection measurements at the surface. Applications of the Marchenko method have already been discussed in many papers, but its implementation aspects have not yet been discussed in detail. Solving the Marchenko equation is an inverse problem. The Marchenko method computes a solution of the Marchenko equation by an (adaptive) iterative scheme or by a direct inversion. We have evaluated the iterative implementation based on a Neumann series, which is considered to be the conventional scheme. At each iteration of this scheme, a convolution in time and an integration in space are performed between a so-called focusing (update) function and the reflection response. In addition, by applying a time window, one obtains an update, which becomes the input for the next iteration. In each iteration, upgoing and downgoing focusing functions are updated with these terms. After convergence of the scheme, the resulting upgoing and downgoing focusing functions are used to compute the upgoing and downgoing Green's functions with a virtual-source position in the subsurface and receivers at the surface. We have evaluated this algorithm in detail and developed an implementation that reproduces our examples. The software fits into the Seismic Unix software suite of the Colorado School of Mines.