The Marchenko method offers a new perspective on eliminating internal multiples. Instead of predicting internal multiples based on events, the Marchenko method formulates an inverse problem that is solved for an inverse transmission response. This approach is particularly advantageous when internal multiples generate complicated interference patterns, such that individual events cannot be identified. Moreover, the retrieved inverse transmissions can be used for a wide range of applications. For instance, we present a numerical example of the single-sided homogeneous Green's function representation in elastic media. These applications require a generalization of the Marchenko method beyond the acoustic case. Formally these extensions are nearly straightforward, as can be seen in the chapter on plane-wave Marchenko redatuming in elastic media. Despite the formal ease of these generalizations, solving the aforementioned inverse problem becomes significantly more difficult in the elastodynamic case. We analyze fundamental challenges of the elastodynamic Marchenko method. Elastic media support coupled wave-modes with different propagation velocities. These velocity differences lead to fundamental limitations, which are due to differences between the temporal ordering of reflection events and the ordering of reflectors in depth. Other multiple-elimination methods such as the inverse scattering series encounter similar limitations, due to violating a so-called monotonicity assumption. Nevertheless, we show that the Marchenko method imposes a slightly weaker form of the monotonicity assumption because it does not rely on event-based multiple prediction. Another challenge arises from the initial estimate that is required by the Marchenko method. In the acoustic case, this initial estimate can be as simple as a direct transmission from the recording surface to the redatuming level. In the presence of several wave-modes, an acoustic direct transmission generalizes to a so-called forward-scattered transmission, which is not a single event but a wavefield with a finite temporal duration. Former formulations of the elastodynamic Marchenko method require this forward-scattered transmission as an initial estimate. However, in practice, this initial estimate is often unknown. We present an alternative formulation of the elastodynamic Marchenko method that simplifies the initial estimate to a trivial one. This approach replaces the inverse transmission, which is often referred to as a focusing function, by a so-called backpropagated focusing function. This strategy allows us to remove internal multiples, however, unwanted forward-scattered waves persist in the data. This insight suggests that forward-scattered waves cannot be predicted by the Marchenko method: either they are provided as prior knowledge, or they remain unaddressed. The remaining forward-scattered waves may be eliminated by exploiting minimum-phase behavior as additional constraint. This approach is inspired by recent developments of the acoustic Marchenko method that use a minimum-phase constraint to handle short-period multiples. Generalizing this strategy to the elastodynamic case is challenging because wavefields are no longer described by scalars but by matrices. Hence, we start by analyzing the meaning of minimum-phase in a multi-dimensional sense. This investigation illustrates that the aforementioned backpropagation turns the focusing function into a minimum-phase object. This insight suggests that, from a mathematical view point, the backpropagated focusing function can be seen as a more fundamental version of the focusing function. Moreover, we present attempts of using this property as additional constraint to remove unwanted forward-scattered waves. Given the remaining theoretical challenges of the elastodynamic Marchenko method, we analyze the performance of an acoustic approximation. We evaluate the effect of applying the acoustic Marchenko method to elastodynamic reflection data. For this analysis, we look for geological settings where an acoustic approximation could be impactful. The Middle East is a promising candidate because, due to its nearly horizontally-layered geology, elastic scattering effects are weaker for short-offsets, which are the main contributors to structural images. Therefore, we construct a synthetic Middle East model based on regional well-log data as well as knowledge about the regional geology. In contrast to field data examples, the synthetic study allows us to include or exclude elastic effects. Hence, we can inspect the artifacts caused by an acoustic approximation. The results indicate that the acoustic Marchenko method can be sufficient for multiple-free structural imaging in geological settings akin to the Middle East.