Deep-learning techniques appear to be poised to play very important roles in our processing flows for inversion and interpretation of seismic data. The most successful seismic applications of these complex pattern-identifying networks will, presumably, be those that also leverage the deterministic physical models on which we normally base our seismic interpretations. If this is true, algorithms belonging to theory-guided data science, whose aim is roughly this, will have particular applicability in our field. We have developed a theory-designed recurrent neural network (RNN) that allows single- and multidimensional scalar acoustic seismic forward-modeling problems to be set up in terms of its forward propagation. We find that training such a network and updating its weights using measured seismic data then amounts to a solution of the seismic inverse problem and is equivalent to gradient-based seismic full-waveform inversion (FWI). By refining these RNNs in terms of optimization method and learning rate, comparisons are made between standard deep-learning optimization and nonlinear conjugate gradient and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimized algorithms. Our numerical analysis indicates that adaptive moment (or Adam) optimization with a learning rate set to match the magnitudes of standard FWI updates appears to produce the most stable and well-behaved waveform inversion results, which is reconfirmed by a multidimensional 2D Marmousi experiment. Future waveform RNNs, with additional degrees of freedom, may allow optimal wave propagation rules to be solved for at the same time as medium properties, reducing modeling errors.
The determination of subsurface elastic property models is crucial in quantitative seismic data processing and interpretation. This problem is commonly solved by deterministic physical methods, such as tomography or full-waveform inversion. However, these methods are entirely local and require accurate initial models. Deep learning represents a plausible class of methods for seismic inversion, which may avoid some of the issues of purely descent-based approaches. However, any generic deep learning network capable of relating each elastic property cell value to each sample in a seismic data set would require a very large number of degrees of freedom. Two approaches might be taken to train such a network: first, by invoking a massive and exhaustive training data set and, second, by working to reduce the degrees of freedom by enforcing physical constraints on the model-data relationship. The second approach is referred to as “physics-guiding.” Based on recent progress in wave theory-designed (i.e., physics-based) networks, we have developed a hybrid network design, involving deterministic, physics-based modeling and data-driven deep learning components. From an optimization standpoint, a data-driven model misfit (i.e., standard deep learning) and now a physics-guided data residual (i.e., a wave propagation network) are simultaneously minimized during the training of the network. An experiment is carried out to analyze the trade-off between two types of losses. Synthetic velocity building is used to examine the potential of hybrid training. Comparisons demonstrate that, given the same training data set, the hybrid-trained network outperforms the traditional fully data-driven network. In addition, we performed a comprehensive error analysis to quantitatively compare the fully data-driven and hybrid physics-guided approaches. The network is applied to the SEG salt model data, and the uncertainty is analyzed, to further examine the benefits of hybrid training.
Full-waveform inversion (FWI) has emerged as a powerful strategy for estimating subsurface model parameters by iteratively minimizing the difference between synthetic data and observed data. The Hessian-free (HF) optimization method represents an attractive alternative to Newton-type and gradient-based optimization methods. At each iteration, the HF approach obtains the search direction by approximately solving the Newton linear system using a matrix-free conjugate-gradient (CG) algorithm. The main drawback with HF optimization is that the CG algorithm requires many iterations. In our research, we develop and compare different preconditioning schemes for the CG algorithm to accelerate the HF Gauss-Newton (GN) method. Traditionally, preconditioners are designed as diagonal Hessian approximations. We additionally use a new pseudo diagonal GN Hessian as a preconditioner, making use of the reciprocal property of Green’s function. Furthermore, we have developed an [Formula: see text]-BFGS inverse Hessian preconditioning strategy with the diagonal Hessian approximations as an initial guess. Several numerical examples are carried out. We determine that the quasi-Newton [Formula: see text]-BFGS preconditioning scheme with the pseudo diagonal GN Hessian as the initial guess is most effective in speeding up the HF GN FWI. We examine the sensitivity of this preconditioning strategy to random noise with numerical examples. Finally, in the case of multiparameter acoustic FWI, we find that the [Formula: see text]-BFGS preconditioned HF GN method can reconstruct velocity and density models better and more efficiently compared with the nonpreconditioned method.
Frequency-dependent seismic field data anomalies, appearing in association with low-[Formula: see text] targets, have, on occasion, been attributed to the presence of a strong absorptive reflection coefficient. This “absorptive reflectivity” represents a potent, and largely untapped, source of information for determining subsurface target properties. It would most likely be encountered where a predominantly elastic/nonattenuating overburden suddenly is interrupted by a highly attenuative target. Series expansions of absorptive reflection coefficients about small parameter contrasts and incidence angles can expose these anomalies to analysis, either frequency-by-frequency (amplitude variation with frequency [AVF]) or angle-by-angle (amplitude variation with angle of incidence [AVA]). Within this framework, variations in P-wave velocity and [Formula: see text] can be estimated separately through a range of direct formulas, both linear and with nonlinear corrections. The latter come to the fore when a contrast from an incidence medium [Formula: see text] (i.e., acoustic/elastic) to a target medium [Formula: see text] is encountered, in which case the linearized estimate can be in error by as much as 50%. Algorithmically, it is a differencing of the reflection coefficient across frequencies that separates [Formula: see text] variations from variations in other parameters. This holds for both two-parameter (P-wave velocity and [Formula: see text]) problems and five-parameter anelastic problems, and would appear to be a general feature of direct absorptive inversion.
Simultaneous use of data within relatively broad frequency bands is essential to discriminating between velocity and [Formula: see text] errors in the construction of viscoacoustic full-waveform inversion (QFWI) updates. Individual frequencies or narrow bands in isolation cannot provide sufficient information to resolve parameter crosstalk issues in a surface seismic acquisition geometry. At the same time, too broad a frequency band introduces significant problems in the presence of modeling errors. The risk of modeling errors arising in QFWI is high because of the range of very different geologic contributors to attenuation and dispersion and the variety of available mathematical descriptions. We perform numerical tests that suggest that by relaxing the typical requirement that the frequency dependence of the assumed intrinsic attenuation model be self-consistent across the full spectrum, significant improvement in the fidelity of the inversion results can be obtained in cases where the attenuation model assumed in the inversion differs substantially from the true subsurface behavior. We find that the size of the frequency bands used in this inversion approach is a useful hyperparameter controlling trade-off between crosstalk reduction and flexibility in coping with uncertain physics.
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