The Hessian matrix plays an important role in correct interpretation of the multiple scattered wave fields inside the FWI frame work. Due to the high computational costs, the computation of the Hessian matrix is not feasible. Consequently, FWI produces overburden related artifacts inside the target zone model, due to the lack of the exact Hessian matrix. We have shown here that Marchenko-based target-oriented Full Waveform Inversion can compensate the need of Hessian matrix inversion by reducing the non-linearity due to overburden effects. This is achieved by exploiting Marchenko-based target replacement to remove the overburden response and its interactions with the target zone from residuals and inserting the response of the updated target zone into the response of the entire medium. We have also shown that this method is more robust with respect to prior information than the standard gradient FWI. Similarly to standard Marchenko imaging, the proposed method only requires knowledge of the direct arrival time from a focusing point to the surface and the reflection response of the medium.
Applications of the Marchenko method have recently been developed for migration, wavefield redatuming, internal multiple subtraction, and primaries estimation. Marchenko methods estimate the subsurface-to-surface point-source Green's functions and the so-called focusing functions. Focusing functions are solutions of the wave equation which focus in time and space at specified subsurface locations. Here, we use these focusing functions as virtual source/receiver surface acquisition wavefields, with the upgoing focusing function being the virtual received wavefield, created when the downgoing focusing function acts as the source. This results in three imaging schemes, one of which allows individual reflectors chosen to be imaged. These methods provide images with certain advantages over current reverse-time migration methods, such as fewer artifacts, and artifacts that occur in different locations. We show that one of these images can be combined with standard images to remove acquisition and multiple-related artifacts. We demonstrate our methods with acoustic and elastic synthetic examples.
Implementations of Markov chain Monte Carlo (MCMC) methods need to confront two fundamental challenges: accurate representation of prior information and efficient evaluation of likelihoods. Principal component analysis (PCA) and related techniques can in some cases facilitate the definition and sampling of the prior distribution, as well as the training of accurate surrogate models, using for instance, polynomial chaos expansion (PCE). However, complex geological priors with sharp contrasts necessitate more complex dimensionality-reduction techniques, such as, deep generative models (DGMs). By sampling a low-dimensional prior probability distribution defined in the low-dimensional latent space of such a model, it becomes possible to efficiently sample the physical domain at the price of a generator that is typically highly non-linear. Training a surrogate that is capable of capturing intricate non-linear relationships between latent parameters and outputs of forward modeling presents a notable challenge. Indeed, while PCE models provide high accuracy when the input-output relationship can be effectively approximated by relatively low-degree multivariate polynomials, this condition is typically not met when employing latent variables derived from DGMs. In this contribution, we present a strategy combining the excellent reconstruction performances of a variational autoencoder (VAE) with the accuracy of PCA-PCE surrogate modeling in the context of Bayesian ground penetrating radar (GPR) traveltime tomography. Within the MCMC process, the parametrization of the VAE is leveraged for prior exploration and sample proposals. Concurrently, surrogate modeling is conducted using PCE, which operates on either globally or locally defined principal components of the VAE samples under examination.
Summary Marchenko redatuming is a novel scheme used to retrieve up- and down-going Green’s functions in an unknown medium. Marchenko equations are based on reciprocity theorems and are derived on the assumption of the existence of so called focusing functions, i.e. functions which exhibit time-space focusing properties once injected in the subsurface. In contrast to interferometry but similarly to standard migration methods, Marchenko redatuming only requires an estimate of the direct wave from the virtual source (or to the virtual receiver), illumination from only one side of the medium, and no physical sources (or receivers) inside the medium. In this contribution we consider a different time-focusing condition within the frame of Marchenko redatuming and show how this can lead to the retrieval of virtual plane-wave-responses, thus allowing multiple-free imaging using only a 1 dimensional sampling of the targeted model. The potential of the new method is demonstrated on a 2D synthetic model.
Estimating image uncertainty is fundamental to guiding the interpretation of geoscientific tomographic maps. We reveal novel uncertainty topologies (loops) which indicate that while the speeds of both low- and high-velocity anomalies may be well constrained, their locations tend to remain uncertain. The effect is widespread: loops dominate around a third of United Kingdom Love wave tomographic uncertainties, changing the nature of interpretation of the observed anomalies. Loops exist due to 2nd and higher order aspects of wave physics; hence, although such structures must exist in many tomographic studies in the physical sciences and medicine, they are unobservable using standard linearized methods. Higher order methods might fruitfully be adopted.
Summary Marchenko redatuming retrieves Green's functions inside an unknown medium, by solving a set of coupled Marchenko equations, which are derived from an under-determined system of equation and two temporal truncations. To constrain the problem, two assumptions are made, which hold reasonably well for acoustic, but not for elastodynamic waves. First, an early part of the inverse transmission field is needed which can be estimated for sufficiently-simple acoustic cases, but remains hard to predict for elastic media without detailed overburden knowledge. Secondly, the scheme assumes temporal separability of up-going focusing and Green's functions, which holds for many acoustic media but easily fails in presence of elastic effects. The impact of the failure to meet these assumptions is a somewhat controllable problem in 1.5D media. Independently, in acoustic media one can use a time-only focusing to retrieve focusing functions which collapse to a single plane wave below the overburden. We apply this approach to elastic data from a very complex almost 1.5D medium. The numerical example shows that the plane-wave approach can also be combined with mitigation of failure to satisfy the aforementioned assumptions and the result could lead to a high-fidelity internal multiple-free image.
Wavefield focusing can be achieved by Time-Reversal Mirrors, which involve in- and output signals that are infinite in time and waves propagating through the entire medium. Here, an alternative solution for wavefield focusing is presented. This solution is based on a new integral representation where in- and output signals are finite in time, and where the energy of the waves propagating in the layer embedding the focal point is reduced. We explore the potential of the proposed method with numerical experiments involving a 1D example and a cranium model consisting of a skull enclosing a brain.
Summary Green's functions in an unknown medium can be retrieved from single-sided reflection data by solving a multidimensional Marchenko equation. This methodology requires knowledge of the direct wavefield throughout the medium, which should include forward-scattered waveforms. In practice, the direct field is often computed in a smooth background model, where such subtleties are not included. As a result, Marchenko-based Green's function retrieval can be inaccurate, especially in severely complex media. In some cases, auxiliary transmission data may be available. In this extended abstract, we show how these data can be used to modify the Marchenko equation so that forward-scattered waveforms can be retrieved without additional knowledge of the medium.
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