@article {siahkoohi2021dbif,
	title = {Deep Bayesian inference for seismic imaging with tasks},
	journal = {Geophysics},
	volume = {87},
	number = {5},
	year = {2022},
	note = {(Geophysics)},
	month = {07},
	pages = {281-302},
	abstract = {We propose to use techniques from Bayesian inference and deep neural networks to translate uncertainty in seismic imaging to uncertainty in tasks performed on the image, such as horizon tracking. Seismic imaging is an ill-posed inverse problem because of unavoidable bandwidth and aperture limitations, which that is hampered by the presence of noise and linearization errors. Many regularization methods, such as transform-domain sparsity promotion, have been designed to deal with the adverse effects of these errors, however, these methods run the risk of biasing the solution and do not provide information on uncertainty in the image space and how this uncertainty impacts certain tasks on the image. A systematic approach is proposed to translate uncertainty due to noise in the data to confidence intervals of automatically tracked horizons in the image. The uncertainty is characterized by a convolutional neural network (CNN) and to assess these uncertainties, samples are drawn from the posterior distribution of the CNN weights, used to parameterize the image. Compared to traditional priors, in the literature it is argued that these CNNs introduce a flexible inductive bias that is a surprisingly good fit for many diverse domains in imaging. The method of stochastic gradient Langevin dynamics is employed to sample from the posterior distribution. This method is designed to handle large scale Bayesian inference problems with computationally expensive forward operators as in seismic imaging. Aside from offering a robust alternative to maximum a posteriori estimate that is prone to overfitting, access to these samples allow us to translate uncertainty in the image, due to noise in the data, to uncertainty on the tracked horizons. For instance, it admits estimates for the pointwise standard deviation on the image and for confidence intervals on its automatically tracked horizons.},
	keywords = {deep priors, horizon tracking, seismic imaging, Uncertainty quantification},
	doi = {10.1190/geo2021-0666.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2022/siahkoohi2021dbif/paper.html},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Felix J. Herrmann}
}
@conference {herrmann2021KAUSTdbi,
	title = {Deep Bayesian Inference for Task-based Seismic Imaging},
	booktitle = {KAUST},
	year = {2021},
	note = {(KAUST, virtual)},
	month = {03},
	abstract = {High dimensionality, complex physics, and lack of access to the ground truth make seismic imaging one of the most challenging inversion problems in the computational imaging sciences. If these challenges were not bad enough, modern applications of computational imaging increasingly call for the assessment of uncertainty on the image itself and on subsequent tasks conducted on these images. By making use of a Markov-chain Monte Carlo (McMC) sampling technique, we demonstrate how uncertainty in the data can be propagated to the task of seismic horizon tracking. While this example shows that uncertainty can in principle be handled in a systematic manner by using neural nets as deep priors, it unfortunately also reveals that the McMC method fundamentally lacks ability to scale to relevant problem sizes. To overcome this shortcoming, we introduce a variational formulation based on normalizing flows. In this approach, invertible neural networks are trained to generate samples from the posterior. There are strong indications that this approach has the capability to scale better to problems where the forward operator is expensive to evaluate.},
	keywords = {Bayesian inference, Inverse problems, seismic imaging, Uncertainty quantification},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/KAUST/2021/herrmann2021KAUSTdbi/Kaust.pdf},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Mathias Louboutin and Philipp A. Witte and Felix J. Herrmann}
}
@article {rizzuti2020dfw,
	title = {A dual formulation of wavefield reconstruction inversion for large-scale seismic inversion},
	journal = {Geophysics},
	volume = {86},
	number = {6},
	year = {2021},
	note = {(Geophysics)},
	month = {10},
	pages = {1ND-Z3},
	abstract = {Many of the seismic inversion techniques currently proposed that focus on robustness with respect to the background model choice are not apt to large-scale 3D applications, and the methods that are computationally feasible for industrial problems, such as full waveform inversion, are notoriously limited by convergence stagnation and require adequate starting models. We propose a novel solution that is both scalable and less sensitive to starting models or inaccurate parameters (such as anisotropy) that are typically kept fixed during inversion. It is based on a dual reformulation of the classical wavefield reconstruction inversion, whose empirical robustness with respect to these issues is well documented in the literature. While the classical version is not suited to 3D, as it leverages expensive frequency-domain solvers for the wave equation, our proposal allows the deployment of state-of-the-art time-domain finite-difference methods, and is potentially mature for industrial-scale problems.},
	keywords = {3D, finite-difference, Full-waveform inversion, wave equation},
	doi = {10.1190/geo2020-0743.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2021/rizzuti2020dfw/rizzuti2020dfw.html},
	author = {Gabrio Rizzuti and Mathias Louboutin and Rongrong Wang and Felix J. Herrmann}
}
@presentation {rizzuti2021ML4SEISMICdfw,
	title = {A dual formulation of wavefield reconstruction inversion for large-scale seismic inversion},
	year = {2021},
	month = {11},
	abstract = {Many of the seismic inversion techniques currently proposed that focus on robustness with respect to the background model choice are not apt to large-scale 3D applications, and the methods that are computationally feasible for industrial problems, such as full waveform inversion, are notoriously limited by convergence stagnation and require adequate starting models. We propose a novel solution that is both scalable and less sensitive to starting models or inaccurate parameters (such as anisotropy) that are typically kept fixed during inversion. It is based on a dual reformulation of the classical wavefield reconstruction inversion, whose empirical robustness with respect to these issues is well documented in the literature. While the classical version is not suited to 3D, as it leverages expensive frequency-domain solvers for the wave equation, our proposal allows the deployment of state-of-the-art time-domain finite-difference methods, and is potentially mature for industrial-scale problems.},
	keywords = {FWI, ML4SEISMIC, SLIM, wave-equation, WRI},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/ML4SEISMIC/2021/rizzuti2021ML4SEISMICdfw/Tue-10-55-Rizzuti.pdf},
	author = {Gabrio Rizzuti and Mathias Louboutin and Rongrong Wang and Felix J. Herrmann}
}
@presentation {siahkoohi2020SEGCHAPTERGTdlbuq,
	title = {A deep-learning based Bayesian approach to seismic imaging and uncertainty quantification},
	year = {2020},
	note = {(SEG Student Chapter)},
	abstract = {Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is via formulating the posterior distribution. Unfortunately, it is often not possible to formulate a prior distribution that precisely encodes our prior knowledge about the unknown. Furthermore, adherence to handcrafted priors may greatly bias the outcome of the Bayesian analysis. To address this issue, we propose to use the functional form of a randomly initialized convolutional neural network as an implicit structured prior, which is shown to promote natural images and excludes images with unnatural noise. In order to incorporate the model uncertainty into the final estimate, we sample the posterior distribution using stochastic gradient Langevin dynamics and perform Bayesian model averaging on the obtained samples. Our synthetic numerical experiment verifies that deep priors combined with Bayesian model averaging are able to partially circumvent imaging artifacts and reduce the risk of overfitting in the presence of extreme noise. Finally, we present pointwise variance of the estimates as a measure of uncertainty, which coincides with regions that are more difficult to image.},
	keywords = {deep learning, seismic imaging, stochastic gradient Langevin dynamics, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Lectures/SEGCHAPTERGT/2020/siahkoohi2020SEGCHAPTERGTdlbuq/siahkoohi2020SEGCHAPTERGTdlbuq.pdf},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Felix J. Herrmann}
}
@conference {siahkoohi2020EAGEdlb,
	title = {A deep-learning based Bayesian approach to seismic imaging and uncertainty quantification},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2020},
	note = {Accepted in EAGE},
	month = {1},
	abstract = {Uncertainty quantification is essential when dealing with ill-conditioned inverse problems due to the inherent nonuniqueness of the solution. Bayesian approaches allow us to determine how likely an estimation of the unknown parameters is via formulating the posterior distribution. Unfortunately, it is often not possible to formulate a prior distribution that precisely encodes our prior knowledge about the unknown. Furthermore, adherence to handcrafted priors may greatly bias the outcome of the Bayesian analysis. To address this issue, we propose to use the functional form of a randomly initialized convolutional neural network as an implicit structured prior, which is shown to promote natural images and excludes images with unnatural noise. In order to incorporate the model uncertainty into the final estimate, we sample the posterior distribution using stochastic gradient Langevin dynamics and perform Bayesian model averaging on the obtained samples. Our synthetic numerical experiment verifies that deep priors combined with Bayesian model averaging are able to partially circumvent imaging artifacts and reduce the risk of overfitting in the presence of extreme noise. Finally, we present pointwise variance of the estimates as a measure of uncertainty, which coincides with regions that are difficult to image.},
	keywords = {deep learning, EAGE, seismic imaging, stochastic gradient Langevin dynamics, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2020/siahkoohi2020EAGEdlb/siahkoohi2020EAGEdlb.html},
	author = {Ali Siahkoohi and Gabrio Rizzuti and Felix J. Herrmann}
}
@conference {rizzuti2019SIAMGStwri,
	title = {A dual formulation for time-domain wavefield reconstruction inversion},
	booktitle = {SIAM Conference on Mathematical and Computational Issues in the Geosciences},
	year = {2019},
	note = {(SIAM GS)},
	month = {03},
	abstract = {Wavefield reconstruction inversion (WRI) is based on the combination of data misfit and PDE-residual penalty terms, and it aims at the reconstruction of the optimal property model and field pair. A full-space optimization would require the storage of field variables for every frequency or time sample (and every source). Remedies via variable projection are well studied, but they entail the solution of a modified wave equation, which is costly both in frequency and time domain. Here, we explore an efficient alternative to WRI, where properties are complemented, instead, by elements of the dual data space. The starting point of our proposal is the denoising reformulation of WRI: the PDE compliance is optimized under the constraint of a given data fit. Since the field variable can be eliminated from the associated saddle-point problem, we obtain a new objective functional of only properties and Lagrange multipliers. Its evaluation, and gradient computation, involves the solution of the original wave equation, which is amenable to time-domain discretization. The optimization scheme of choice can now leverage on affordable storage of the iterates.},
	keywords = {WRI},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SIAMGS/2019/rizzuti2019SIAMGStwri/rizzuti2019SIAMGStwri.pdf},
	author = {Gabrio Rizzuti and Mathias Louboutin and Rongrong Wang and Emmanouil Daskalakis and Felix J. Herrmann}
}
@conference {rizzuti2019SEGadf,
	title = {A dual formulation for time-domain wavefield reconstruction inversion},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2019},
	note = {(SEG, San Antonio)},
	month = {09},
	pages = {1480-1485},
	abstract = {We illustrate a dual formulation for full-waveform inversion potentially apt to large 3-D problems. It is based on the optimization of the wave equation compliance, under the constraint of data misfit not exceeding a prescribed noise level. In the Lagrangian formulation, model and wavefield state variables are complemented with multipliers having the same dimension of data ("dual data" variables). Analogously to classical wavefield reconstruction inversion, the wavefield unknowns can be projected out in closed form, by solving a version of the augmented wave equation. This leads to a saddle-point problem whose variables are only model and dual data. As such, this formulation represents a model extension, and is potentially robust against local minima. The classical downsides of model extension methods and wavefield reconstruction inversion are here effectively mitigated: storage of the dual variables is affordable, the augmented wave equation is amenable to time-marching finite-difference schemes, and no continuation strategy for penalty parameters is needed, with the prospect of 3-D applications.},
	keywords = {3D, Full-waveform inversion, SEG, Time-domain},
	doi = {10.1190/segam2019-3216760.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2019/rizzuti2019SEGadf/rizzuti2019SEGadf.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2019/rizzuti2019SEGadf/rizzuti2019SEGadf_pres.pdf},
	software = {https://github.com/slimgroup/Software.rizzuti2019SEGadf},
	author = {Gabrio Rizzuti and Mathias Louboutin and Rongrong Wang and Emmanouil Daskalakis and Felix J. Herrmann}
}
