Compute wave propagation through the subsurface from measurements at the surface.

What is it about?

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. Here, we describe 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 up- and downgoing focusing functions are updated with these terms. After convergence of the scheme, the resulting up- and downgoing focusing functions are used to compute the up- and downgoing Green's functions with virtual-source position in the subsurface and receivers at the surface. Here, we describe this algorithm in detail and provide an implementation that reproduces the presented examples. The software fits into the Seismic Unix (SU) software suite of Colorado School of Mines.

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Why is it important?

In this article we explain and illustrate the method with simple examples. The included source code can be used to reproduce all examples in the paper and will be a good first step to get familiar with the Marchenko method.

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