Rafael Orozco

Mathias Louboutin, Ziyi Yin, Rafael Orozco et al.

We present the Seismic Laboratory for Imaging and Modeling/Monitoring (SLIM) open-source software framework for computational geophysics and, more generally, inverse problems involving the wave-equation (e.g., seismic and medical ultrasound), regularization with learned priors, and learned neural surrogates for multiphase flow simulations. By integrating multiple layers of abstraction, our software is designed to be both readable and scalable. This allows researchers to easily formulate their problems in an abstract fashion while exploiting the latest developments in high-performance computing. We illustrate and demonstrate our design principles and their benefits by means of building a scalable prototype for permeability inversion from time-lapse crosswell seismic data, which aside from coupling of wave physics and multiphase flow, involves machine learning.

Ali Siahkoohi, Gabrio Rizzuti, Rafael Orozco et al.

Bayesian inference for high-dimensional inverse problems is computationally costly and requires selecting a suitable prior distribution. Amortized variational inference addresses these challenges via a neural network that approximates the posterior distribution not only for one instance of data, but a distribution of data pertaining to a specific inverse problem. During inference, the neural network -- in our case a conditional normalizing flow -- provides posterior samples at virtually no cost. However, the accuracy of amortized variational inference relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems due to the Earth's heterogeneity. In addition, the network is prone to errors if evaluated over out-of-distribution data. As such, we propose to increase the resilience of amortized variational inference in the presence of moderate data distribution shifts. We achieve this via a correction to the latent distribution that improves the posterior distribution approximation for the data at hand. The correction involves relaxing the standard Gaussian assumption on the latent distribution and parameterizing it via a Gaussian distribution with an unknown mean and (diagonal) covariance. These unknowns are then estimated by minimizing the Kullback-Leibler divergence between the corrected and the (physics-based) true posterior distributions. While generic and applicable to other inverse problems, by means of a linearized seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in the number of seismic sources, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty at approximately the same cost as five reverse-time migrations.

Ziyi Yin, Rafael Orozco, Mathias Louboutin et al.

Solving multiphysics-based inverse problems for geological carbon storage monitoring can be challenging when multimodal time-lapse data are expensive to collect and costly to simulate numerically. We overcome these challenges by combining computationally cheap learned surrogates with learned constraints. Not only does this combination lead to vastly improved inversions for the important fluid-flow property, permeability, it also provides a natural platform for inverting multimodal data including well measurements and active-source time-lapse seismic data. By adding a learned constraint, we arrive at a computationally feasible inversion approach that remains accurate. This is accomplished by including a trained deep neural network, known as a normalizing flow, which forces the model iterates to remain in-distribution, thereby safeguarding the accuracy of trained Fourier neural operators that act as surrogates for the computationally expensive multiphase flow simulations involving partial differential equation solves. By means of carefully selected experiments, centered around the problem of geological carbon storage, we demonstrate the efficacy of the proposed constrained optimization method on two different data modalities, namely time-lapse well and time-lapse seismic data. While permeability inversions from both these two modalities have their pluses and minuses, their joint inversion benefits from either, yielding valuable superior permeability inversions and CO2 plume predictions near, and far away, from the monitoring wells.

Rafael Orozco, Mathias Louboutin, Ali Siahkoohi et al.

We present a novel approach to transcranial ultrasound computed tomography that utilizes normalizing flows to improve the speed of imaging and provide Bayesian uncertainty quantification. Our method combines physics-informed methods and data-driven methods to accelerate the reconstruction of the final image. We make use of a physics-informed summary statistic to incorporate the known ultrasound physics with the goal of compressing large incoming observations. This compression enables efficient training of the normalizing flow and standardizes the size of the data regardless of imaging configurations. The combinations of these methods results in fast uncertainty-aware image reconstruction that generalizes to a variety of transducer configurations. We evaluate our approach with in silico experiments and demonstrate that it can significantly improve the imaging speed while quantifying uncertainty. We validate the quality of our image reconstructions by comparing against the traditional physics-only method and also verify that our provided uncertainty is calibrated with the error.

Huseyin Tuna Erdinc, Ipsita Bhar, Rafael Orozco et al.

Recent advances in generative networks have enabled new approaches to subsurface velocity model synthesis, offering a compelling alternative to traditional methods such as Full Waveform Inversion. However, these approaches predominantly rely on the availability of large-scale datasets of high-quality, geologically realistic subsurface velocity models, which are often difficult to obtain in practice. We introduce SAGE, a novel framework for statistically consistent proxy velocity generation from incomplete observations, specifically sparse well logs and migrated seismic images. During training, SAGE learns a proxy posterior over velocity models conditioned on both modalities (wells and seismic); at inference, it produces full-resolution velocity fields conditioned solely on migrated images, with well information implicitly encoded in the learned distribution. This enables the generation of geologically plausible and statistically accurate velocity realizations. We validate SAGE on both synthetic and field datasets, demonstrating its ability to capture complex subsurface variability under limited observational constraints. Furthermore, samples drawn from the learned proxy distribution can be leveraged to train downstream networks, supporting inversion workflows. Overall, SAGE provides a scalable and data-efficient pathway toward learning geological proxy posterior for seismic imaging and inversion. Repo link: https://github.com/slimgroup/SAGE.

Rafael Orozco, Mathias Louboutin, Felix J. Herrmann

Photoacoustic imaging (PAI) can image high-resolution structures of clinical interest such as vascularity in cancerous tumor monitoring. When imaging human subjects, geometric restrictions force limited-view data retrieval causing imaging artifacts. Iterative physical model based approaches reduce artifacts but require prohibitively time consuming PDE solves. Machine learning (ML) has accelerated PAI by combining physical models and learned networks. However, the depth and overall power of ML methods is limited by memory intensive training. We propose using invertible neural networks (INNs) to alleviate memory pressure. We demonstrate INNs can image 3D photoacoustic volumes in the setting of limited-view, noisy, and subsampled data. The frugal constant memory usage of INNs enables us to train an arbitrary depth of learned layers on a consumer GPU with 16GB RAM.

Abhinav Prakash Gahlot, Huseyin Tuna Erdinc, Rafael Orozco et al.

As the global deployment of carbon capture and sequestration (CCS) technology intensifies in the fight against climate change, it becomes increasingly imperative to establish robust monitoring and detection mechanisms for potential underground CO2 leakage, particularly through pre-existing or induced faults in the storage reservoir's seals. While techniques such as history matching and time-lapse seismic monitoring of CO2 storage have been used successfully in tracking the evolution of CO2 plumes in the subsurface, these methods lack principled approaches to characterize uncertainties related to the CO2 plumes' behavior. Inclusion of systematic assessment of uncertainties is essential for risk mitigation for the following reasons: (i) CO2 plume-induced changes are small and seismic data is noisy; (ii) changes between regular and irregular (e.g., caused by leakage) flow patterns are small; and (iii) the reservoir properties that control the flow are strongly heterogeneous and typically only available as distributions. To arrive at a formulation capable of inferring flow patterns for regular and irregular flow from well and seismic data, the performance of conditional normalizing flow will be analyzed on a series of carefully designed numerical experiments. While the inferences presented are preliminary in the context of an early CO2 leakage detection system, the results do indicate that inferences with conditional normalizing flows can produce high-fidelity estimates for CO2 plumes with or without leakage. We are also confident that the inferred uncertainty is reasonable because it correlates well with the observed errors. This uncertainty stems from noise in the seismic data and from the lack of precise knowledge of the reservoir's fluid flow properties.

Ziyi Yin, Rafael Orozco, Mathias Louboutin et al.

We introduce a probabilistic technique for full-waveform inversion, employing variational inference and conditional normalizing flows to quantify uncertainty in migration-velocity models and its impact on imaging. Our approach integrates generative artificial intelligence with physics-informed common-image gathers, reducing reliance on accurate initial velocity models. Considered case studies demonstrate its efficacy producing realizations of migration-velocity models conditioned by the data. These models are used to quantify amplitude and positioning effects during subsequent imaging.

Rafael Orozco, Felix J. Herrmann, Peng Chen

Bayesian optimal experimental design (OED) seeks to conduct the most informative experiment under budget constraints to update the prior knowledge of a system to its posterior from the experimental data in a Bayesian framework. Such problems are computationally challenging because of (1) expensive and repeated evaluation of some optimality criterion that typically involves a double integration with respect to both the system parameters and the experimental data, (2) suffering from the curse-of-dimensionality when the system parameters and design variables are high-dimensional, (3) the optimization is combinatorial and highly non-convex if the design variables are binary, often leading to non-robust designs. To make the solution of the Bayesian OED problem efficient, scalable, and robust for practical applications, we propose a novel joint optimization approach. This approach performs simultaneous (1) training of a scalable conditional normalizing flow (CNF) to efficiently maximize the expected information gain (EIG) of a jointly learned experimental design (2) optimization of a probabilistic formulation of the binary experimental design with a Bernoulli distribution. We demonstrate the performance of our proposed method for a practical MRI data acquisition problem, one of the most challenging Bayesian OED problems that has high-dimensional (320 $\times$ 320) parameters at high image resolution, high-dimensional (640 $\times$ 386) observations, and binary mask designs to select the most informative observations.

Rafael Orozco, Philipp Witte, Mathias Louboutin et al.

InvertibleNetworks.jl is a Julia package designed for the scalable implementation of normalizing flows, a method for density estimation and sampling in high-dimensional distributions. This package excels in memory efficiency by leveraging the inherent invertibility of normalizing flows, which significantly reduces memory requirements during backpropagation compared to existing normalizing flow packages that rely on automatic differentiation frameworks. InvertibleNetworks.jl has been adapted for diverse applications, including seismic imaging, medical imaging, and CO2 monitoring, demonstrating its effectiveness in learning high-dimensional distributions.

Rafael Orozco, Ali Siahkoohi, Mathias Louboutin et al.

Due to their uncertainty quantification, Bayesian solutions to inverse problems are the framework of choice in applications that are risk averse. These benefits come at the cost of computations that are in general, intractable. New advances in machine learning and variational inference (VI) have lowered the computational barrier by learning from examples. Two VI paradigms have emerged that represent different tradeoffs: amortized and non-amortized. Amortized VI can produce fast results but due to generalizing to many observed datasets it produces suboptimal inference results. Non-amortized VI is slower at inference but finds better posterior approximations since it is specialized towards a single observed dataset. Current amortized VI techniques run into a sub-optimality wall that can not be improved without more expressive neural networks or extra training data. We present a solution that enables iterative improvement of amortized posteriors that uses the same networks architectures and training data. The benefits of our method requires extra computations but these remain frugal since they are based on physics-hybrid methods and summary statistics. Importantly, these computations remain mostly offline thus our method maintains cheap and reusable online evaluation while bridging the approximation gap these two paradigms. We denote our proposed method ASPIRE - Amortized posteriors with Summaries that are Physics-based and Iteratively REfined. We first validate our method on a stylized problem with a known posterior then demonstrate its practical use on a high-dimensional and nonlinear transcranial medical imaging problem with ultrasound. Compared with the baseline and previous methods from the literature our method stands out as an computationally efficient and high-fidelity method for posterior inference.

Rafael Orozco, Huseyin Tuna Erdinc, Yunlin Zeng et al.

Accurately characterizing migration velocity models is crucial for a wide range of geophysical applications, from hydrocarbon exploration to monitoring of CO2 sequestration projects. Traditional velocity model building methods such as Full-Waveform Inversion (FWI) are powerful but often struggle with the inherent complexities of the inverse problem, including noise, limited bandwidth, receiver aperture and computational constraints. To address these challenges, we propose a scalable methodology that integrates generative modeling, in the form of Diffusion networks, with physics-informed summary statistics, making it suitable for complicated imaging problems including field datasets. By defining these summary statistics in terms of subsurface-offset image volumes for poor initial velocity models, our approach allows for computationally efficient generation of Bayesian posterior samples for migration velocity models that offer a useful assessment of uncertainty. To validate our approach, we introduce a battery of tests that measure the quality of the inferred velocity models, as well as the quality of the inferred uncertainties. With modern synthetic datasets, we reconfirm gains from using subsurface-image gathers as the conditioning observable. For complex velocity model building involving salt, we propose a new iterative workflow that refines amortized posterior approximations with salt flooding and demonstrate how the uncertainty in the velocity model can be propagated to the final product reverse time migrated images. Finally, we present a proof of concept on field datasets to show that our method can scale to industry-sized problems.

Abhinav Prakash Gahlot, Rafael Orozco, Felix J. Herrmann

Geological Carbon Storage (GCS) is a key technology for achieving global climate goals by capturing and storing CO2 in deep geological formations. Its effectiveness and safety rely on accurate monitoring of subsurface CO2 migration using advanced time-lapse seismic imaging. A Digital Shadow framework integrates field data, including seismic and borehole measurements, to track CO2 saturation over time. Machine learning-assisted data assimilation techniques, such as generative AI and nonlinear ensemble Bayesian filtering, update a digital model of the CO2 plume while incorporating uncertainties in reservoir properties. Compared to 2D approaches, 3D monitoring enhances the spatial accuracy of GCS assessments, capturing the full extent of CO2 migration. This study extends the uncertainty-aware 2D Digital Shadow framework by incorporating 3D seismic imaging and reservoir modeling, improving decision-making and risk mitigation in CO2 storage projects.

Zijun Deng, Rafael Orozco, Abhinav Prakash Gahlot et al.

Reducing CO$_2$ emissions is crucial to mitigating climate change. Carbon Capture and Storage (CCS) is one of the few technologies capable of achieving net-negative CO$_2$ emissions. However, predicting fluid flow patterns in CCS remains challenging due to uncertainties in CO$_2$ plume dynamics and reservoir properties. Building on existing seismic imaging methods like the Joint Recovery Method (JRM), which lacks uncertainty quantification, we propose the Probabilistic Joint Recovery Method (pJRM). By estimating posterior distributions across surveys using a shared generative model, pJRM provides uncertainty information to improve risk assessment in CCS projects.

Huseyin Tuna Erdinc, Rafael Orozco, Felix J. Herrmann

In this study, we introduce a novel approach to synthesizing subsurface velocity models using diffusion generative models. Conventional methods rely on extensive, high-quality datasets, which are often inaccessible in subsurface applications. Our method leverages incomplete well and seismic observations to produce high-fidelity velocity samples without requiring fully sampled training datasets. The results demonstrate that our generative model accurately captures long-range structures, aligns with ground-truth velocity models, achieves high Structural Similarity Index (SSIM) scores, and provides meaningful uncertainty estimations. This approach facilitates realistic subsurface velocity synthesis, offering valuable inputs for full-waveform inversion and enhancing seismic-based subsurface modeling.

Rafael Orozco, Abhinav Gahlot, Felix J. Herrmann

CO$_2$ sequestration is a crucial engineering solution for mitigating climate change. However, the uncertain nature of reservoir properties, necessitates rigorous monitoring of CO$_2$ plumes to prevent risks such as leakage, induced seismicity, or breaching licensed boundaries. To address this, project managers use borehole wells for direct CO$_2$ and pressure monitoring at specific locations. Given the high costs associated with drilling, it is crucial to strategically place a limited number of wells to ensure maximally effective monitoring within budgetary constraints. Our approach for selecting well locations integrates fluid-flow solvers for forecasting plume trajectories with generative neural networks for plume inference uncertainty. Our methodology is extensible to three-dimensional domains and is developed within a Bayesian framework for optimal experimental design, ensuring scalability and mathematical optimality. We use a realistic case study to verify these claims by demonstrating our method's application in a large scale domain and optimal performance as compared to baseline well placement.

Rafael Orozco, Ali Siahkoohi, Mathias Louboutin et al.

We present an iterative framework to improve the amortized approximations of posterior distributions in the context of Bayesian inverse problems, which is inspired by loop-unrolled gradient descent methods and is theoretically grounded in maximally informative summary statistics. Amortized variational inference is restricted by the expressive power of the chosen variational distribution and the availability of training data in the form of joint data and parameter samples, which often lead to approximation errors such as the amortization gap. To address this issue, we propose an iterative framework that refines the current amortized posterior approximation at each step. Our approach involves alternating between two steps: (1) constructing a training dataset consisting of pairs of summarized data residuals and parameters, where the summarized data residual is generated using a gradient-based summary statistic, and (2) training a conditional generative model -- a normalizing flow in our examples -- on this dataset to obtain a probabilistic update of the unknown parameter. This procedure leads to iterative refinement of the amortized posterior approximations without the need for extra training data. We validate our method in a controlled setting by applying it to a stylized problem, and observe improved posterior approximations with each iteration. Additionally, we showcase the capability of our method in tackling realistically sized problems by applying it to transcranial ultrasound, a high-dimensional, nonlinear inverse problem governed by wave physics, and observe enhanced posterior quality through better image reconstruction with the posterior mean.