The solution of the coupled Marchenko equations for up- and downgoing focusing wavefields allows for the creation of virtual sources inside a mostly unknown medium, which holds great potential for seismic imaging below complex overburdens. However, to solve the coupled Marchenko equations, accurate knowledge and deconvolution of the source wavelet from the reflection response is required. Here, we present an approach that circumvents such requirements and accommodates for band-limited reflection data by solving the coupled Marchenko equations using sparse inversion. Provided with an initial estimate of the source wavelet, the proposed scheme simultaneously inverts for the Marchenko focusing functions and the source wavelet contained in the reflection response. A 1D synthetic example demonstrates that the inversion successfully converges to the true focusing functions and correct source wavelet after a few iterations.


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