Summary Seismic full-waveform inversion (FWI) provides high resolution images of the subsurface by exploiting information in the recorded seismic waveforms. This is achieved by solving a highly nonlinear and nonunique inverse problem. Bayesian inference is therefore used to quantify uncertainties in the solution. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently using optimization. The method has been applied to 2D FWI problems to produce full Bayesian posterior distributions. However, due to higher dimensionality and more expensive computational cost, the performance of the method in 3D FWI problems remains unknown. We apply three variational inference methods to 3D FWI and analyse their performance. Specifically we apply automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), to a 3D FWI problem, and compare their results and computational cost. The results show that ADVI is the most computationally efficient method but systematically underestimates the uncertainty. The method can therefore be used to provide relatively rapid but approximate insights into the subsurface together with a lower bound estimate of the uncertainty. SVGD demands the highest computational cost, and still produces biased results. In contrast, by including a randomized term in the SVGD dynamics, sSVGD becomes a Markov chain Monte Carlo method and provides the most accurate results at intermediate computational cost. We thus conclude that 3D variational full-waveform inversion is practically applicable, at least in small problems, and can be used to image the Earth’s interior and to provide reasonable uncertainty estimates on those images.

Geoscientists often have little information about earth’s subsurface heterogeneities prior to mapping them using seismic or other geophysical data. Marchenko methods are a set of novel, data-driven techniques that help us to project surface seismic data to points in the subsurface, to form seismograms as though they had been created at each point. In so doing, Marchenko methods account for many of the complex, multiply reflected seismic wave interactions that take place in the real earth’s subsurface. The resulting seismograms are the information required to create subsurface images that are more accurate than standard methods. Our aim is to introduce these concepts with the minimum amount of mathematics required to understand how the Marchenko method can iterate to a solution and to provide a well-commented, easily editable MATLAB code package for demonstration and training purposes. Green’s function estimation using the Marchenko method is first illustrated for a constant velocity, variable density, 1D medium, with results that indicate a near-perfect match when compared with true, synthetically modeled solutions. Similar quality results are shown for variable velocity, 2D Green’s function estimation. Finally, we determine how these estimates could be used to create images of the subsurface, which, when compared with standard methods, contain reduced contamination due to multiple-related artifacts. Our code package includes the 2D data set required to reconstruct the relevant figures that we present, and it allows for experimentation with the implementation of the Marchenko method and the application of Marchenko imaging.

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.

Marchenko methods use seismic data acquired at or near the surface of the earth to estimate seismic signals as if the receiver (now a virtual receiver) was at an arbitrary point inside the subsurface of the earth. This process is called redatuming, and it is central to subsurface imaging. Marchenko methods estimate the multiply scattered components of these redatumed signals, which is not the case for most other redatuming techniques that are based on single-scattering assumptions. As a result, images created using Marchenko redatumed signals contain a reduction in the artifacts that usually contaminate migrated seismic images due to improper handling of internal multiples. We exploit recent theoretical advances that enable virtual sources and virtual receivers to be placed at arbitrary points inside the subsurface as a means to incorporate vertical seismic profile (VSP) data into Marchenko methods. The advantage of including this type of data is that the additional acquisition boundary increases subsurface illumination, which in turn enables vertical interfaces and steeply dipping structures to be imaged. We develop this methodology using two synthetic data sets. The first is created using a simple variable density but constant velocity subsurface model. In this example, we find that our newly devised VSP Marchenko imaging methodology enables imaging of horizontal and vertical structures and that optimal results are achieved by combining these images with those created using standard Marchenko imaging. A second example demonstrates that the method can be applied to more realistic subsurface structures, in this case a modified version of the Marmousi 2 model. We determine the applicability of the methods to image fault structures with the final imaging result containing reduced contamination due to internal multiples and an improvement in the imaging of fault structures when compared to other standard imaging methods alone.

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