Full waveform inversion is an ill-posed inverse problem whose solution non-uniqueness -- i.e., arising from band-limited, finite-aperture, noisy data -- calls for uncertainty quantification to avoid overconfident geological interpretations. Bayesian inference addresses this need by characterizing the solution as a posterior distribution rather than a single point estimate. Sampling from this distribution, however, remains computationally challenging: Markov chain Monte Carlo and non-amortized variational inference require repeated wave equation solves, while amortized variational inference approaches that avoid repeated solves rely on training data that are inherently scarce in geoscience and face unresolved generalization challenges in high dimensions. To address these limitations, we integrate Stein variational gradient descent with the alternating direction method of multipliers under a dual augmented Lagrangian formulation. By fixing the wave operator at a background model that is updated between frequency batches, it need only be factorized once per particle per frequency, eliminating per-iteration refactorization and reducing the total cost to that of a handful of deterministic inversions while inheriting the favorable conditioning of extended-space formulations. Applied to the Marmousi~II model, the proposed method provides well-calibrated uncertainty estimates and achieves inversion quality comparable to that of the standard augmented Lagrangian SVGD at a fraction of the computational cost.
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 nonnlinear 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 …
It is challenging for full-waveform inversion to determine geologically informative models from field data. An inaccurate wavelet can make it more complicated. We develop a novel misfit function, entitled deconvolutional double-difference misfit measurement to cancel the influence of wavelet inaccuracy on inversion results. Unlike the popular double-difference misfit measurement in which the first difference is evaluated by cross-correlation, the proposed one employs deconvolution to do this step. Numerical examples demonstrate that full-waveform inversion with the new misfit function is resil…
Geophysical models usually contain both sharp interfaces and smooth variations, and it is difficult to accurately account for both of these two types of medium parameter variations using conventional full-waveform inversion methods. In addition, sparse geometry, noisy data and source encoding usually lead to strong inversion artifacts. We develop a novel full-waveform inversion method for acoustic and elastic waves using a total generalized p-variation regularization scheme to address these challenging problems. We decompose the full-waveform inversion into two subproblems and solve these two …
We propose and test the Direct Waveform Inversion (DWI) scheme to simultaneously invert for layered velocity and density profiles, using reflection seismic waveforms recorded on the surface. The recorded data include primary reflections and interbed multiples. DWI is implemented in the time-space domain. By explicitly enforcing the wavefield time-space causality, DWI can recursively determine the subsurface seismic structure in a local fashion for both sharp interfaces as well as the properties of the layers, from shallow to deep depths. DWI is different from the layer stripping methods in the…
Traditional seismic envelope inversion takes use of a nonlinear misfit functional which relates the envelope of seismogram to the observed wavefield records, and then derive the sensitivity kernel of envelope to velocity through the use of waveform Frechet derivative by linearizing the nonlinear functional. We know that the waveform Frechet derivative is based on the Born approximation of wave scattering and can only work for the case of weak scattering. Therefore, the traditional envelope inversion using waveform Frechet derivative has severe limitation in the case of strong-scattering for la…
Seismic full-waveform inversion (FWI) uses full seismic records to estimate subsurface velocity structure. This requires a highly nonlinear and nonunique inverse problem to be solved, and Bayesian methods have been used to quantify uncertainties in the solution. Variational Bayesian inference uses optimization to provide solutions efficiently. The method has been applied to solve a transmission FWI problem using data generated by known earthquake-like sources, with strong prior information imposed on the velocity. Unfortunately such prior information about velocity structure and earthquake sou…
Full waveform inversion (FWI) is a powerful yet computationally expensive technique that can yield subsurface models at high resolution. Randomly selected shots ("mini-batches") can be used to approximate the misfit and the gradient of FWI, thereby reducing its computational cost. Here, we present a methodology to perform mini-batch FWI using the Adam algorithm, an adaptive optimization scheme based on stochastic gradient descent. It provides for stable model updates by smoothing the gradient across iterations and can also account for the curvature of the optimization landscape. We describe em…
Full-waveform inversion (FWI) is an accurate imaging approach for modeling velocity structure by minimizing the misfit between recorded and predicted seismic waveforms. However, the strong non-linearity of FWI resulting from fitting oscillatory waveforms can trap the optimization in local minima. We propose a neural-network-based full waveform inversion method (NNFWI) that integrates deep neural networks with FWI by representing the velocity model with a generative neural network. Neural networks can naturally introduce spatial correlations as regularization to the generated velocity model, wh…
We describe a new method, full waveform inversion by model extension (FWIME) that recovers accurate acoustic subsurface velocity models from seismic data, when conventional methods fail. We leverage the advantageous convergence properties of wave-equation migration velocity analysis (WEMVA) with the accuracy and high-resolution nature of acoustic full waveform inversion (FWI) by combining them into a robust mathematically-consistent workflow with minimal need for user inputs. The novelty of FWIME resides in the design of a new cost function using the variable projection method, and a novel opt…
We have formulated elastic seismic full waveform inversion (FWI) within a deep learning environment. In our formulation, a recurrent neural network is set up with rules enforcing elastic wave propagation, with the wavefield projected onto a measurement surface acting as the synthetic data to be compared with observed seismic data. Gradients for iterative updating of an elastic model, with a variety of parameterizations and misfit functionals, can be efficiently constructed within the network through the automatic differential method. With this method, the inversion based on complex misfits can…
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 …
I train a Generative Adversarial Network to produce realistic seismic wave speed models. I integrate the generator network into seismic Full-Waveform Inversion to reduce the number of model parameters and restrict the inverted models to only those that are plausible. Applying the method to a 2D section of the SEAM model, I demonstrate that it can produce more plausible results than conventional Full-Waveform Inversion.
Producing reliable acoustic subsurface velocity models still remains the main bottleneck of the oil and gas industry's traditional imaging sequence. In complex geological settings, the output of conventional ray-based or wave-equation-based tomographic methods may not be accurate enough for full waveform inversion (FWI) to converge to a geologically satisfactory Earth model. We create a new method referred to as full waveform inversion by model extension (FWIME) in which a wave-equation migration velocity analysis (WEMVA) technique is efficiently paired with a modified version of FWI. We show …
The augmented Lagrangian (AL) method provides a flexible and efficient framework for solving extended-space full-waveform inversion (FWI), a constrained nonlinear optimization problem whereby we seek model parameters and wavefields that minimize the data residuals and satisfy the wave equation constraint. The AL-based wavefield reconstruction inversion, also known as iteratively refined wavefield reconstruction inversion, extends the search space of FWI in the source dimension and decreases sensitivity of the inversion to the initial model accuracy. Furthermore, it benefits from the advantages…
We describe a novel framework for estimating subsurface properties, such as rock permeability and porosity, from time-lapse observed seismic data by coupling full-waveform inversion, subsurface flow processes, and rock physics models. For the inverse modeling, we handle the back-propagation of gradients by an intrusive automatic differentiation strategy that offers three levels of user control: (1) at the wave physics level, we adopted the discrete adjoint method in order to use our existing high-performance FWI code; (2) at the rock physics level, we used built-in operators from the $\texttt{…
Data acquisition is changing to incorporate more components of the recorded wavefield. An example of this is so-called vector data in which both the pressure and particle velocity are recorded in marine data. We present a vector acoustic full waveform inversion (VAFWI) method. We demonstrate the connection of the VAFWI adjoint operator with inverse wavefield extrapolation and show that under the assumption of a plane wave propagating towards an infinite flat recording surface at normal incidence, the VAFWI adjoint is equivalent to inverse extrapolation of the normal derivative of the recorded …
Derivatives of the displacement tensor with respect to the independent model parameters of the subsurface, also called Fréchet derivatives (or sensitivity kernels), are a key ingredient for seismic full-waveform inversion with a local-search optimization algorithm. They provide a quantitative measure of the expected changes in the seismograms due to perturbations of the subsurface model parameters for a given survey geometry. Since 2.5-D wavefield modeling involves a real point source in a 2-D geological model with 3D (spherical) wave properties, it yields synthetic data much closer to the act…
Bayesian full waveform inversion with learned prior using deep convolutional autoencoder
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Halliburton Landmark（哈里伯顿陆盾）；Idaho National Laboratory（爱达荷国家实验室）；School of Earth Sciences, Yunnan University（云南大学地球科学学院）；Peking University（北京大学）
In the last ten years, full-waveform inversion has emerged as a robust and efficient high-resolution velocity model-building tool for seismic imaging, with the unique ability to recover complex subsurface structures. Originally based on a data fitting process using a least-squares cost function, it suffered from high sensitivity to cycle-skipping and was therefore of poor efficiency in handling large time shifts between observed and modelled seismic events. To tackle this problem, a common practice is to start the inversion using the low temporal frequencies of the data and selecting diving wa…
Elastic full-waveform inversion (FWI) when successfully applied can provide accurate and high-resolution subsurface parameters. However, its high computational cost prevents the application of this method to large-scale field-data scenarios. To mitigate this limitation, we propose a target-oriented elastic FWI methodology based on a redatuming step that relies upon an extended least-squared migration process. In our approach, the surface-reflection data can be attributed to a given subsurface portion when mapped into the image space. This process allows us to reconstruct reflection data genera…
Full--waveform inversion (FWI) is a method used to determine properties of the Earth from information on the surface. We use the squared Wasserstein distance (squared $W_2$ distance) as an objective function to invert for the velocity of seismic waves as a function of position in the Earth, and we discuss its convexity with respect to the velocity parameter. In one dimension, we consider constant, piecewise increasing, and linearly increasing velocity models as a function of position, and we show the convexity of the squared $W_2$ distance with respect to the velocity parameter on the interval…
Full waveform inversion (FWI) aims at estimating subsurface medium properties from measured seismic data. It is usually cast as a non-linear least-squares problem that incorporates uncertainties in the measurements. In exploration seismology, extended formulations of FWI that allow for uncertaties in the physics have been proposed. Even when the physics is modelled accurately, these extensions have been shown to be beneficial because they reduce the non-lineary of the resulting data-fitting problem. In this note, I derive an alternative (but equivalent) formulation of extended full waveform in…
Seismic waves bring information from the physical properties of the earth to the surface. Full waveform inversion (FWI) is a local optimization technique which tries to invert the recorded wave fields to the physical properties. An efficient forward-modelling engine along with local differential algorithm, to compute the gradient and Hessian operators, are two key ingredients of FWI approach. FWI can be done in time or frequency domain. Each method has its own pros and cons. Here, we only discuss frequency domain method with Krylov subspace solvers for time-harmonic wave equation. Nonlinearity…
Source footprint elimination in full-waveform inversion by model extension: Application to elastic guided waves recorded by distributed acoustic sensing in unconventional reservoir
Full waveform inversion (FWI) strongly depends on an accurate starting model to succeed. This is particularly true in the elastic regime: The cycle-skipping phenomenon is more severe in elastic FWI compared to acoustic FWI, due to the short S-wave wavelength. In this paper, we extend our work on extrapolated FWI (EFWI) by proposing to synthesize the low frequencies of multi-component elastic seismic records, and use those "artificial" low frequencies to seed the frequency sweep of elastic FWI. Our solution involves deep learning: we separately train the same convolutional neural network (CNN) …
