Marchenko methods can retrieve both Green’s functions and focusing functions from single-sided reflection data and a smooth velocity model, as essential components of a redatuming process. Recent studies also show that a modified Marchenko scheme can reconstruct primary-only reflection responses directly from reflection data without requiring a priori model information. To provide insight into the artifacts that arise when input data are not ideally sampled, we study the effects of subsampling in both types of Marchenko methods in 2D earth and data — by analyzing the behaviour of Marchenko-based results on synthetic data subsampled in sources or receivers. We show with a layered model that for Marchenko redatuming, subsampling effects jointly depend on the choice of integration variable and the subsampling dimension, originated from the integrand gather in the multidimensional convolution process. When reflection data are subsampled in a single dimension, integrating on the other yields spatial gaps together with artifacts while integrating on the subsampled dimension produces aliasing artifacts but without spatial gaps. Our complex subsalt model shows the subsampling may lead to very strong artifacts, which can be further complicated by having limited apertures. For Marchenko-based primary estimation (MPE), subsampling below a certain fraction of the fully-sampled data can cause MPE iterations to diverge, which can to some extent be mitigated by using more robust iterative solvers, such as LSQR. Our results, covering redatuming and primary estimation in a range of subsampling scenarios, provide insights that can inform acquisition sampling choices as well as processing parameterization and quality control, e.g., to set up appropriate data filters and scaling to accommodate the effects of dipole fields, or to help ensuring that the data interpolation achieves the desired levels of reconstruction quality that minimize subsampling artifacts in Marchenko-derived fields and images.@en

The Marchenko method retrieves Green's functions between the acquisition surface and any arbitrary point in the medium. The process generally involves solving an inversion starting with an initial focusing function, e.g., a direct-wave Green's function from the desired subsurface position, typically obtained using an approximate velocity model. We have formulated the Marchenko method in the time-imaging domain. In that domain, we recognize that the traveltime of the direct-wave Green's function is related to the Cheop's traveltime pyramid commonly used in time-domain processing, which in turn can be readily obtained from the local slopes of the common-midpoint gathers. This observation allows us to substitute the velocity-model-based initial focusing operator with that from a data-driven slope estimation process. Moreover, we found that working in the time-imaging domain allows for the specification of the desired subsurface position in terms of vertical time, which is connected to the Cartesian depth position via the time-to-depth conversion. Our results suggest that the prior velocity model is only required when specifying the position in depth, but this requirement can be circumvented by making use of the time-imaging domain within its usual assumptions (e.g., mild lateral heterogeneity). Provided that those assumptions are satisfied, the estimated Green's functions from the proposed method have comparable quality to those obtained with the knowledge of a prior velocity model.

@en

The Marchenko methods can retrieve both Green's functions and focusing functions from single-sided reflection data and an initial focusing function. Recent studies also show that they can be applied to eliminating multiples directly from reflection data without requiring model information as the input. In this work, we study the effect of limited acquisition sampling on both the output focusing functions and the primary estimation results obtained from Marchenko methods. For focusing functions output from the coupled Marchenko system, downsampling in either source or receiver leads to absolute amplitude errors, with the addition of spatial aliasing brought by receiver subsampling. For the projected Marchenko system used in primary estimation, downsampling in these two dimensions can completely contaminate output primaries with multiples, with similar spatial gaps brought by receiver downsampling. Therefore, the acquisition geometry has a significant impact on results of the Marchenko methods, which must be taken into account in realistic acquisition scenarios.@en

The Marchenko method represents a constructive technique toobtain Green�s functions between the acquisition surface andany arbitrary point in the medium. The process generally involvessolving an inversion starting with a direct-wave Green's function from the desired subsurface position, which is typicallyobtained using an approximate velocity model. In thisstudy, we first propose to formulate the Marchenko method inthe time-imaging domain. We recognize that the traveltimeof the direct-wave Green�s function is related to the Cheop�straveltime pyramid commonly used in time-domain processingand can be readily obtained from the local slopes of thecommon-midpoint (CMP) gathers. This observation allowsus to substitute the need for a prior velocity model with thedata-driven slope estimation process. Moreover, we show thatworking in the time-imaging domain allows for the specificationof the desired subsurface position in terms of vertical time,which is connected to the Cartesian depth position via the timeto-depth conversion. Our results suggest that the prior velocitymodel is only required when specifying the position in depthbut this requirement can be circumvented by making use of thetime-imaging domain and its usual assumptions. Provided thatthose assumptions are satisfied, the estimated Green�s functionsfrom the proposed method have comparable quality tothose obtained with the knowledge of a prior velocity@en