Accurate source wavelet estimation using Marchenko focusing functions

Marchenko redatuming is a powerful technique that allows retrieving Green's functions for virtual receivers in the subsurface by calculating so-called focusing functions. One required input for the Green's function retrieval is the surface reflection response that needs to be correctly deconvolved by the source wavelet. Deconvolving with an erroneously estimated or scaled source wavelet leads to artifacts in the redatumed wavefields and images created with redatumed data. Focusing functions can be used to correct an erroneous amplitude and phase spectrum of the source wavelet, enabling an accurate deconvolution, subsequent redatuming, and imaging. Additional events are present in the focusing functions in the case of an incorrect source wavelet amplitude spectrum, whereas an erroneous phase spectrum leads to an overall phase shift of the upgoing focusing function. Therefore, two objective functions are defined, which are minimized separately to invert for the source wavelet amplitude and phase spectrum accurately. Furthermore, optimizing focusing functions for a range of lateral focusing positions in the subsurface extends the approach, enabling inversion of laterally varying scaling errors of the source wavelet. Target-oriented images created with Marchenko redatumed data using the source scaling estimation method based on focusing functions are significantly improved compared to conventional Marchenko images with incorrectly estimated source wavelets.