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We introduce a generalization of time-domain wavefield reconstruction inversion to anisotropic acoustic modeling. Wavefield reconstruction inversion has been extensively researched in recent years for its ability to mitigate cycle skipping. The original method was formulated in the frequency domain with acoustic isotropic physics. However, frequency-domain modeling requires sophisticated iterative solvers that are difficult to scale to industrial-size problems and more realistic physical assumptions, such as tilted transverse isotropy, object of this study. The work presented here is based on a recently proposed dual formulation of wavefield reconstruction inversion, which allows time-domain propagator that are suitable to both large scales and more accurate physics.
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Most of the seismic inversion techniques currently proposed focus on robustness with respect to the background model choice or inaccurate physical modeling assumptions, but are not apt to large-scale 3D applications. On the other hand, methods that are computationally feasible for industrial problems, such as full waveform inversion, are notoriously bogged down by local minima and require adequate starting models. We propose a novel solution that is both scalable and less sensitive to starting model or inaccurate physics when compared to full waveform inversion. The method is based on a dual (Lagrangian) reformulation of the classical wavefield reconstruction inversion, whose robustness with respect to local minima is well documented in the literature. However, it is not suited to 3D, as it leverages expensive frequency-domain solvers for the wave equation. The proposed reformulation allows the deployment of state-of-the-art time-domain finite-difference methods, and is computationally mature for industrial scale problems.
Solving the wave equation is one of the most (if not the most) fundamental problems we face as we try to illuminate the Earth using recorded seismic data. The Helmholtz equation provides wavefield solutions that are dimensionally reduced, per frequency, compared to the time domain, which is useful for many applications, like full waveform inversion (FWI). However, our ability to attain such wavefield solutions depends often on the size of the model and the complexity of the wave equation. Thus, we use here a recently introduced framework based on neural networks to predict functional solutions through setting the underlying physical equation as a loss function to optimize the neural network parameters. For an input given by a location in the model space, the network learns to predict the wavefield value at that location, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of the Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions of the background wavefield. Providing the neural network (NN) a reasonable number of random points from the model space will ultimately train a fully connected deep NN to predict the scattered wavefield function. The size of the network depends mainly on the complexity of the desired wavefield, with such complexity increasing with increasing frequency and increasing model complexity. However, smaller networks can provide smoother wavefields that might be useful for inversion applications. Preliminary tests on a two-box-shaped scatterer model with a source in the middle, as well as, the Marmousi model with a source on the surface demonstrate the potential of the NN for this application. Additional tests on a 3D model demonstrate the potential versatility of the approach.
Achieving desirable receiver sampling in ocean bottom acquisition is often not possible because of cost considerations. Assuming adequate source sampling is available, which is achievable by virtue of reciprocity and the use of modern randomized (simultaneous-source) marine acquisition technology, we are in a position to train convolutional neural networks (CNNs) to bring the receiver sampling to the same spatial grid as the dense source sampling. To accomplish this task, we form training pairs consisting of densely sampled data and artificially subsampled data using a reciprocity argument and the assumption that the source-site sampling is dense. While this approach has successfully been used on the recovery monochromatic frequency slices, its application in practice calls for wavefield reconstruction of time-domain data. Despite having the option to parallelize, the overall costs of this approach can become prohibitive if we decide to carry out the training and recovery independently for each frequency. Because different frequency slices share information, we propose the use the method of transfer training to make our approach computationally more efficient by warm starting the training with CNN weights obtained from a neighboring frequency slices. If the two neighboring frequency slices share information, we would expect the training to improve and converge faster. Our aim is to prove this principle by carrying a series of carefully selected experiments on a relatively large-scale five-dimensional data synthetic data volume associated with wide-azimuth 3D ocean bottom node acquisition. From these experiments, we observe that by transfer training we are able t significantly speedup in the training, specially at relatively higher frequencies where consecutive frequency slices are more correlated.
The possibility to have results very quickly after, or even during, the collection of electromagnetic data would be important, not only for quality check purposes, but also for adjusting the location of the proposed flight lines during an airborne time-domain acquisition. This kind of readiness could have a large impact in terms of optimization of the Value of Information of the measurements to be acquired. In addition, the importance of having fast tools for retrieving resistivity models from airborne time-domain data is demonstrated by the fact that Conductivity-Depth Imaging methodologies are still the standard in mineral exploration. In fact, they are extremely computationally efficient, and, at the same time, they preserve a very high lateral resolution. For these reasons, they are often preferred to inversion strategies even if the latter approaches are generally more accurate in terms of proper reconstruction of the depth of the targets and of reliable retrieval of true resistivity values of the subsurface. In this research, we discuss a novel approach, based on neural network techniques, capable of retrieving resistivity models with a quality comparable with the inversion strategy, but in a fraction of the time. We demonstrate the advantages of the proposed novel approach on synthetic and field datasets.
Traveltime tomography is a very effective tool to reconstruct acoustic, seismic or electromagnetic wave speed distribution. To infer the velocity image of the medium from the measurements of first arrivals is a typical example of ill-posed problem. In the framework of Tikhonov regularization theory, in order to replace an ill-posed problem by a well-posed one and to get a unique and stable solution, a stabilizing functional (stabilizer) has to be introduced. The stabilizer selects the desired solution from a class of solutions with a specific physical and/or geometrical property; e.g., the existence of sharp boundaries separating media with different petrophysical parameters. Usually stabilizers based on maximum smoothness criteria are used during the inversion process; in these cases the solutions provide smooth images which, in many situations, do not describe the examined objects properly. Recently a new algorithm of direct minimization of the Tikhonov parametric functional with minimum support stabilizer has been introduced; it produces clear and focused images of targets with sharp boundaries. In this research we apply this new technique to real radar tomographic data and we compare the obtained result with the solution generated by the more traditional minimum norm stabilizer.
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