Tutorial: unified 1D inversion of the acoustic reflection response

Acoustic inversion in one-dimension gives impedance as a function of travel time.
Inverting the reflection response is a linear problem. Recursive methods, from top to bottom or vice versa, are known and use a fundamental wave field that is computed from the reflection response. An integral over the solution to the Marchenko equation, on the other hand, retrieves the impedance at any vertical travel time instant. It is a non-recursive method, but requires the zero-frequency value of the reflection response. These methods use the same fundamental wave field in different ways. Combining the two methods leads to a non-recursive scheme that works with finite-frequency bandwidth. This can be used for target-oriented inversion. When a reflection response is available along a line over a horizontally layered medium, the thickness and wave velocity of any layer can be obtained together with the velocity of an adjacent layer and the density ratio of the two layers. Statistical analysis over 1000 noise realizations shows that the forward recursive method and the Marchenko-type method perform well on computed noisy data.