Bayesian Full-waveform Inversion with Weak Priors

In this paper we proposed an accurate and efficient numerical algorithm for solving the acoustic wave equation in three-dimensional heterogeneous media. Numerical solution of the wave equation has been used in various science and engineering applications, such as the seismic full waveform inversion (FWI) problem. FWI is a computationally intensive procedure, in which the acoustic wave equation is numerically solved (forward modelling) repeatedly during the iterative process. Therefore, efficiency and accuracy of the numerical method for solving the acoustic wave equation is critical in the success of seismic full waveform inversion. The new method is obtained by combining the Pade approximation and a novel algebraic manipulation with the Alternative Directional Implicit (ADI) method. Numerical experiments have shown that the new method is accurate, efficient and stable.

<p>The seismic full waveform inversion (FWI), as one of important ways to obtain the seismic wave velocity, has made rapid development in the last decade. In response to problems of cycle-skipping artifacts, dependence on the initial model, and low-frequency information in FWI, researchers have made many improvements, such as multi-scale envelope inversion and low-frequency extension. Recently, deep learning has been also adopted seismic data processing and interpretation, because of its strong nonlinear mapping ability. However, these works depend on labels used for training heavily, especially for the velocity model in the inversion, which prevents them from real application. Referring to these studies, this work combines low-frequency extension commonly as well as multiscale inversion with deep learning, and proposes a multi-scale FWI gradient optimization method based on CNN. CNN we designed is trained to predict the inversion gradient corresponding to the low-frequency band data in FWI, so that multi-scale gradient optimization can be directly used in multi-scale inversion, expanding the low-frequency information in the actual data and reducing the calculation in FWI. With a specially designed dataset, CNN is trained to optimize the gradients computed from the high-frequency band data by predicting the gradients corresponding to the low-frequency band data and the gradients corresponding to the mid-frequency band data, respectively. The predicted gradients are used in different stages of the multi-scale inversion. The low-frequency gradients are used to invert the initial structural construction so as not to rely on a good initial model, and the high-frequency gradients are used to improve the accuracy of the inversion results. In this way, low-frequency expansion and multiscale inversion can be achieved simultaneously. Our method achieves good results on the initial model for a given uniform wave velocity, effectively alleviating the reliance on the initial model in FWI. This study provides a new idea of combining deep learning and full waveform inversion, which will be effectively used in seismic data processing.</p>

Seismic full waveform inversion (FWI) faces cycle-skipping problem when the seismic data lack low-frequency information and the initial velocity model is far from the true model. To address this challenge, we proposed a neural network-based weighted envelope correlation inversion with a learnable envelope power, which is referred to as FWI-NN-WECP. In this method, we constructed the loss function using an envelope-based misfit to enhance low-frequency components during the inversion. We also introduced a learnable envelope-power parameter to control the effective bandwidth of the data envelope. In addition, we applied a regularization term to further constrain the evolution of the power parameter, thereby guiding the inversion to progressively focus from shallow to deep parts of the velocity model. Synthetic data experiments demonstrated that the FWI-NN-WECP method significantly mitigates cycle skipping and successfully reconstructs complex geological structures, even when low-frequency data (below 5 Hz) are absent and the initial model is inaccurate. The field data experiment confirmed the accuracy of the inverted velocity models and demonstrated the potential of the proposed method to mitigate cycle-skipping through by using neural networks and a carefully designed loss function.

Seismic full waveform inversion (FWI) is able to build high-resolution velocity model based on the full information carried by seismic wave. However, FWI requires an accurate enough initial model to ensure convergence. In this paper, we propose a new nonlinear FWI method to mitigate the initial model dependence problem. Specifically, we firstly propose a nonlinear operator within the hybrid model- and data-driven framework based on the frequency controllable envelope operator (FCEO) and a deep learning architecture U-Net. FCEO is used to obtain the envelope of a band-limited data and U-Net realizes the mapping from this envelope to that corresponding to a lower frequency band. The U-Net is trained in a self-supervised manner that avoids the reliance on labeled data and benefits the generalization ability. Based on the nonlinear operator, a nonlinear FWI method is proposed by defining a new misfit function. In addition, the calculation of gradient is derived using the adjoint-state method. Using numerical examples, we investigate the performance of the proposed nonlinear operator and the new nonlinear FWI method. The results clearly demonstrate that the proposed nonlinear operator is effective in obtaining low-frequency envelope data, and the new nonlinear FWI method has advantages over common method in mitigating cycle-skipping and building an initial model for conventional FWI.

In a probabilistic formulation, the solution to an inverse problem can be expressed an a posteriori probability density function (pdf) that combines the independent states of information provided by data and a priori information. Here, we define an a posteriori probability density function that defines the solution to a tomographic full waveform inverse problem, which provides a means of obtaining an uncertainty estimate. Unfortunately, no explicit formulation of the solution to this problem can be defined. Therefore, the a posteriori probability density function has to be sampled. The full waveform inverse problem is known to be computationally very hard and is, traditionally, considered out of reach for Monte Carlo sampling strategies. We show that by means of informative a priori information this problem become tractable for a sampling strategy anyways. We outline the theoretical framework for a full waveform inversion strategy that integrates the extended Metropolis algorithm with sequential Gibbs sampling, which allows for arbitrary complex a priori information to be incorporated. At the same time we show how temporally correlated data uncertainties can be taken into account during the inversion. The suggested inversion strategy is tested on synthetic tomographic crosshole ground penetrating radar full waveform data using multiplepoint based a priori information. This is, to our knowledge, the first example of obtaining a posteriori realizations of a full waveform inverse problem.

Seismic full-waveform inversion (FWI) aims to reconstruct a high-resolution velocity model of the subsurface from seismic data. However, it is known to face the problem of ill-posedness and has difficulties in reconstructing deep, low-illumination regions. To overcome these challenges, we use a migration image to regularize the FWI problem. Unlike conventional FWI approaches in which the model regularization is added to the data misfit term, our method uses a U-net to introduce this prior information. We find that the prior information about the structure contained in a migration image can be naturally and effectively incorporated into the generated output; therefore, we use the migration image as input to the U-net. We reparameterize the velocity model as a combination of the U-net output and a background velocity model. The learnable parameters of the U-net are optimized against the FWI data misfit using the gradients calculated by automatic differentiation. Our method does not require pretraining of the network and is time-saving as the automatic differentiation is accelerated by graphic processing units. We implement this method on the synthetic Marmousi and Sigsbee2a models as well as on field seismic data from the Gulf of Mexico. The results indicate a regularization effect that mitigates the problem with local minima, improves convergence speed, and preserves geological structures. Such a regularization effect is well suited to improve velocity reconstruction under poor illumination conditions, such as subsalt structures.

Knowledge of planetary interior structure is essential for understanding the origin and evolution of the Solar System. Seismic full-waveform inversion (FWI) is currently the state-of-the-art method for imaging Earth's interior and has revealed mantle heterogeneities at increasingly finer scales. However, FWI solely images variations in seismic properties and, as a result, the thermal or/and chemical origin of such anomalies is poorly constrained. Mineral physics provides complementary constraints by linking seismic properties to temperature, pressure and composition through laboratory measurements. Over the past two decades, significant work has been done to integrate seismic observations with mineral physics predictions. Nevertheless, direct integration of FWI and thermodynamic calculations has not yet been achieved, limiting our ability to fully exploit the information contained in seismic waveforms to image changes in temperature and composition.In this contribution, we present a novel framework that integrates state-of-the-art FWI techniques with mineral physics to directly invert seismic waveforms for mantle temperature and chemical composition. The forward model is based on pre-calculated tables of mantle properties, which provide the seismic properties to carry out wave propagation. The inverse problem is formulated as an optimisation problem where gradients of the objective function with respect to temperature and composition are required. For the seismic component, we employ the adjoint method. For the thermodynamic component, we developed a formalism accounting for the pressure–density coupling which is combined with the seismic part by exploiting the chain rule. This approach is tested on 2-D synthetic models containing thermal and compositional anomalies where P–SV elastic wave propagation is simulated. The optimisation problem is solved using the L-BFGS algorithm. The proposed framework successfully recovers anomalies in temperature and composition, while revealing a strong trade-off. Such non-uniqueness reflects the importance of taking into account both thermal and compositional variations and a priori information about them. The proposed framework enables the integration of diverse geophysical datasets as well as the incorporation of additional information on the potential origin of certain mantle anomalies based on petrological constraints, which are crucial to tackle the non-uniqueness of the inverse problem.

<p>Imaging of hyperextended zone and exhumed continental mantle rocks can improve our understanding of the tectonics of the final stages of rifting. In the Deep Galicia margin, the upper and lower crust are coupled allowing the normal faults to cut through the brittle crust and penetrate to the mantle leading to serpentinization of the mantle. Localized extensional forces caused extreme thinning and elongation of crystalline continental crust causing the continental blocks to slip over a lithospheric-scale detachment fault called the S-reflector.  </p><p>A high-resolution velocity model obtained using seismic full waveform inversion gives us deeper insights into the rifting process. In this study, we present results from three dimensional acoustic full waveform inversion performed using wide-angle seismic data acquired in the deep water environments of the Deep Galicia margin using ocean bottom seismometers. We performed full waveform inversion in the time domain, starting with a velocity model obtained using travel-time tomography, of dimensions 78.5 km x 22.1 km and depth 12 km. The high-resolution modelling shows short-wavelength variations in the velocity, adding details to the travel-time model. We superimposed our final model, converted to two-way time, on pre-stack time-migrated three-dimensional reflection data from the same survey. Compared to the starting model, our model shows improved alignment of the velocity variations along the steeply dipping normal faults and a sharp velocity contrast across the S-reflector. We validated our result using checkerboard tests, by tracking changes in phases of the first arrivals during the inversion and by comparing the observed and the synthetic waveforms. We observe a clear evidence for preferential serpentinization (45 %) of the mantle with lower velocities in the mantle correlating with the fault intersections with the S-reflector.</p>

Summary Full-waveform inversion can be severely compromised by problems of cycle skipping; this occurs when predicted and observed data differ by more than half a cycle, and it leads the inversion to recover a local rather than the global minimum model. Overcoming cycle skipping normally requires both a good starting model and low-frequency content in the field data. Here we present a scheme that uses a non-linear extrapolation to add missing low- frequencies into the field data. We demonstrate the scheme using a 3D OBC field dataset, and show that it can invert to recover the global minimum model even when the original un-extrapolated field dataset is significantly cycle skipped. Introduction Full-waveform inversion (FWI) is an optimization scheme that seeks to find the global minimum of the misfit between field seismic data and synthetic data generated by a model. It proceeds towards the global minimum by a series of local linearized iterations from a starting model. The true relationship between model and data however is non-linear, and several of these non-linearities can lead the inversion scheme to become trapped within local minima rather than reaching the desired global solution. Amongst these non-linearities, the most important is that produced by the oscillatory nature of the seismic data themselves. A cycle-skipped starting model, that is one that produces data that differ by more than half a cycle with respect to the field data, will normally lead the inversion in entirely the wrong direction towards a local minimum. This problem is well known; its practical solution typically requires a highly accurate starting model combined with low-frequency field data. In these circumstances, the predicted starting data are not cycle skipped at the lowest frequencies, and the global solution can be found by starting FWI using the lowest frequencies in the field data. In this paper, we present a FWI scheme that relaxes the requirement for low-frequency field data and a good starting model, and that is able to reach the global minimum solution even when the starting data are cycle skipped. We do this by using the observed field data to synthesize low-frequency data that were not present in the original field data. We then invert these invented low- frequencies, which has the effect of dragging the starting model towards the global minimum. We then switch to invert the true field data which are now no longer cycle skipped, and so reach the global minimum.

Seismic full waveform inversion (FWI) has been applied to simple elastic problems with certain symmetries, such as isotropic, transverse isotropic or vertical transversely isotropic media. In this study, the FWI concept is extended to the most general anisotropic case with 21 independent elastic material parameters and no symmetry plane (triclinic). Beside a short description of the 3-D finite-difference scheme to solve the forward problem and the FWI optimization algorithm, we present a sensitivity study for a simple anisotropic medium. This test problem consists of a homogenous triclinic anisotropic full space, which contains 21 spatially separated spheres. In each sphere one component of the elastic tensor deviates by 5 per cent from the background medium. The resolution of the different spheres, ambiguities between the different elastic parameters, as well as the effect of the acquisition geometry can be systematically investigated. Due to the high computational costs of the triclinic forward problem a few compromises have to be made regarding the acquisition geometries. Point sources are replaced by plane wave sources which lead to a limitation of incidence angles and therefore a strong decrease in resolution of the nondiagonal elastic tensor components. It is shown that, despite these limitations, a tomographic acquisition geometry would be able to resolve to some extent a monoclinic symmetry via FWI. Restricting the acquisition geometries (e.g. VSP combined with reflection seismic or reflection seismic only) significantly reduces the number of resolvable tensor elements in strict dependence of the covered incidence angles.

Seismic full waveform inversion (FWI) is a powerful tool for monitoring subsurface changes during carbon capture and storage (CCS) operations, but its ill-posed nature makes uncertainty quantification (UQ) essential for reliable interpretation. Sampling-based Bayesian methods such as Markov chain Monte Carlo (McMC) provide rigorous UQ but are computationally demanding. In conventional FWI, elastic properties are assigned to densely discretized space-filling cells, resulting in a high-dimensional parameterization that makes large-scale elastic FWI computationally infeasible. To address this challenge, we propose a rock-physics-guided parameter-reduction strategy that compactly represents the CO2 plume geometry using cubic splines controlled by a limited number of nodes. This parsimonious parameterization not only significantly reduces the number of model parameters and the forward simulations required for an effective UQ using sampling methods but also has the potential to improve the practicality and efficiency of other types of UQ methods. Numerical experiments on a cross-well synthetic scenario and a field-scale case based on the Aquistore storage site in Saskatchewan, Canada, demonstrate that the method efficiently reconstructs the plume shape and its extent and that it converges to consistent posterior distributions across multiple Markov chains.

Grain shoals are one of the principal reservoir facies of carbonate rocks in oil and gas exploration domains. With the discovery of oil and gas in the grain shoal reservoirs of the Longwangmiao Formation in the central Sichuan paleo-uplift, the characteristics and distribution of grain shoal reservoirs are still unclear in the northern slope of the central Sichuan paleo-uplift. Through integrated analysis of wireline logs and high-quality 3D seismic data, combined with seismic forward modeling and waveform classification, reservoir seismic facies, reservoir classification, and the reservoir distribution of different grades are investigated. The results show that (1) under the thin thickness (i.e., the thickness is around 80 m) of the Longwangmiao Formation (i) when reservoirs are not developed, the seismic waveform is manifested as a single peak, and the top interface of the Longwangmiao Formation corresponds to 1/8 λ above the single peak. (ii) When a single set of reservoirs is merely developed in the second member of the Longwangmiao Formation (SMLF), the waveform displays a single peak, and the top peak moves downward. In addition, when multiple sets of reservoirs are developed in the SMLF, the seismic reflections of multiple reservoirs are consistent with those of a single reservoir with the same cumulative thickness. (iii) When the reservoirs are developed in the first member of the Longwangmiao Formation (FMLF) and SMLF, the single peak moves downward, and either the reservoir thickness of the SMLF is greater than that of the FMLF or the reservoir velocity of the SMLF is lower than that of the FMLF. (2) Under the thick thickness (i.e., the thickness is approximately 110 m) of the Longwangmiao Formation, (i) when reservoirs are not developed, the waveform of the Longwangmiao Formation exhibits a double peak, and the top interface of the Longwangmiao Formation corresponds to the extreme value of the upper peak. (ii) When only a single set of reservoirs or multiple sets of reservoirs are developed in the SMLF, the Longwangmiao Formation exhibits a single peak, and the top interface of the Longwangmiao Formation corresponds to the trough. (iii) With the reservoir development in the first and second members of the Longwangmiao Formation, the waveform of the Longwangmiao Formation shows an asymmetric low-frequency peak, and either the reservoir thickness of the SMLF is greater than that of the FMLF or the reservoir velocity of the SMLF is lower than that of the FMLF. (3) Using the waveform classification method, seismic waveforms corresponding to the Longwangmiao Formation could be summarized into four types (i.e., two types of waveforms associated with reservoir development and two types of waveforms associated with reservoir undevelopment). Two reservoir waveforms are distributed as two strips oriented approximately east–west and spread in the central study area, according to the planar study results that combine seismic attributes and waveform classification. The reservoir stripes near well PS1 represent the better-developed reservoir zones in the study area.

In order to predict future performance of subsurface fluid reservoirs under possible operating scenarios, a dynamic, porous‐medium flow simulation model must be tuned to include representative properties of the reservoir. Estimating subsurface reservoir properties given remotely sensed or borehole‐based observations typically involves finding the solution to a challenging inverse problem. We compare Monte Carlo random sampling to variational inference methods which use optimization to constrain parametrized uncertainties in nonlinear Bayesian inversions. We use them to estimate the posterior probability distribution of reservoir permeability given fluid pressure and seismic measurements. The methods include automatic differentiation variational inference (ADVI), Stein variational gradient descent (SVGD), and a Monte Carlo method called stochastic SVGD (sSVGD), all of which we benchmark against results from Metropolis‐Hastings McMC. We also test an ADVI variant called physically structured variational inference (PSVI): in our implementation this method estimates only spatially‐local correlations between model parameters based on the intuition that such correlations are strong in remote sensing problems in which data only inform about spatial‐averages of local dynamics. We apply the methods to two‐ and three‐dimensional inverse problems of carbon dioxide storage, inspired by the Endurance field, located in the UK North Sea. Results show that PSVI achieves a good balance between mean‐field ADVI and full‐rank ADVI in terms of accuracy of the posterior approximation and computational efficiency. SVGD and sSVGD offer more accurate approximations of the target posterior distribution, but at far higher computational cost. Between them, sSVGD outperforms SVGD, exhibiting better computational efficiency and mitigating the problems of mode collapse and spurious correlations.

Shape reconstruction in seismic full waveform inversion using a level set approach and time reversal

Full waveform inversion (FWI) is a technique used to estimate subsurface model parameters by minimizing the difference between observed and calculated seismic data. Sparsity-promoting regularization are useful tools for traditional FWI methods to tackle complex subsurface structures. Since the traditional regularization techniques can only impose some fixed priors, it is necessary to develop a regularization strategy to obtain more flexible priors. In this way, we develop a structural sparse representation method that exploits the non-local self-similarity prior of the model, which is achieved by grouping similar patches using graph matching operators and a dynamic group selection strategy. A group-based dictionary is trained with the aim of providing the best sparse representation of complex features and variations in the entire model perturbation. The dynamic selection strategy of the training method can balance computational efficiency and inversion accuracy by constantly updating and retaining groups during the processing. In addition, two loop algorithm framework is utilized to enhance the robustness and the efficiency of the proposed method. Numerical experiments are presented to demonstrate that the proposed method outperforms the total variation regularization method and the adaptive dictionary learning with non-local self-similarity in terms of robustness and resolution. This structural sparsity-promoting regularization is incorporated into the FWI problem through a two-loop algorithm framework, enhancing the robustness and efficiency of FWI results.