Marchenko methods in a 3-D world

Abstract

SUMMARYMarchenko methods are a suite of geophysical techniques that convert seismograms of energy created by surface sources and measured by surface receivers into seismograms that would have been recorded by a virtual receiver at an arbitrary point inside the subsurface—an operation called redatuming. In principle these redatumed seismograms contain all contributions from direct, primary (singly-reflected) and multiply-reflected waves that would have been recorded by a real subsurface receiver, without requiring prior information about interfaces that generated the reflections. The potential of these methods for seismic imaging and redatuming has been demonstrated extensively in previous literature, but only using 1-D and 2-D Marchenko methods. There remain aspects of the methods that are poorly understood when applied in a 3-D world, so we investigate the application of Marchenko methods to 3-D data, subsurface structures and wavefields. We first show that for waves propagating in three dimensions, Marchenko methods can be applied to seismic data collected using both linear (so-called 2-D seismic) and areal (3-D seismic) acquisition arrays. However, for 2-D acquisition arrays the Marchenko workflow requires additional dimensionality correction factors to obtain accurate solutions, even in a subsurface that only varies with depth. Without these correction factors phase errors occur in redatumed Marchenko estimates; these errors propagate through the Marchenko algorithm and create depth errors in the Marchenko images. Furthermore, applying Marchenko methods to fully 3-D seismic wavefields recorded by linear (2-D seismic) arrays that contain out-of-plane reflections deteriorates surface-to-subsurface Green’s function estimates with spurious energy and resulting images are less accurate than those created using ‘conventional’ imaging methods. The application of fully 3-D Marchenko methods using data recorded on areal arrays solves both of the above problems, creating accurately redatumed wavefields and images with reduced artefact contamination. However, it appears that source–receiver spacing at most of $\lambda _A /4$ is required for accurate results using existing Marchenko methods, where λA is the dominant wavelength and in many real 3-D seismic acquisition scenarios this is impractical.

3D virtual seismology

“…The second row shows the pressure wavefield that is obtained using the open boundary retrieval scheme from Equation (14). When the source is located below the virtual receiver, as is the case in Figure 4(d), the result shows a good match to the reference solution in both amplitude and arrival time.…”

“…However, using it in Equation ( 14) is still valid, as no time-reversal is applied. We rewrite Equation (14) for this purpose as…”

“…While many applications of the Marchenko method have been performed on 2D data, recently more applications on 3D data have been achieved [16,22]. Especially in areas where there are strong out-of-plane effects, the 2D approximation on 3D data can cause errors in the result [14].…”

On the retrieval of forward-scattered waveforms from acoustic reflection and transmission data with the Marchenko equation

“…Further, we have chosen to demonstrate the validity of our workflow for 2D wave propagation only. Various publications have emerged recently on the implementation of the Marchenko equation for 3D wave propagation problems [50], [51], [52]. Building on these developments, we prospect that a 3D implementation of the auxiliary equation should also be feasible.…”

Unified elimination of 1D acoustic multiple reflection

“…Most field test results are with two‐dimensional (2D) implementations on line data (Jia et al ., 2018; Staring et al ., 2018; Brackenhoff et al ., 2019). The effect of line data on 2D methods and out‐of‐plane effects as well as sampling conditions for three‐dimensional (3D) applications have been analysed (Jia et al ., 2019; Lomas and Curtis, 2020). A 3D example of approximate double focusing can be found in Staring and Wapenaar (2020).…”