@article{Witte2019, author = {Philipp A. Witte and Mathias Louboutin and Fabio Luporini and Gerard J. Gorman and Felix J. Herrmann}, title = {{Compressive least-squares migration with on-the-fly Fourier transforms}}, journal = {GEOPHYSICS}, volume = {84}, number = {5}, pages = {R655-R672}, year = {2019}, doi = {10.1190/geo2018-0490.1}, URL = {https://doi.org/10.1190/geo2018-0490.1}, eprint = {https://doi.org/10.1190/geo2018-0490.1}, abstract = { ABSTRACTLeast-squares reverse time migration is a powerful approach for true-amplitude seismic imaging of complex geologic structures, but the successful application of this method is currently hindered by its enormous computational cost, as well as its high memory requirements for computing the gradient of the objective function. We have tackled these problems by introducing an algorithm for low-cost sparsity-promoting least-squares migration using on-the-fly Fourier transforms. We formulate the least-squares migration objective function in the frequency domain (FD) and compute gradients for randomized subsets of shot records and frequencies, thus significantly reducing data movement and the number of overall wave equations solves. By using on-the-fly Fourier transforms, we can compute an arbitrary number of monochromatic FD wavefields with a time-domain (TD) modeling code, instead of having to solve individual Helmholtz equations for each frequency, which becomes computationally infeasible when moving to high frequencies. Our numerical examples demonstrate that compressive imaging with on-the-fly Fourier transforms provides a fast and memory-efficient alternative to TD imaging with optimal checkpointing, whose memory requirements for a fixed background model and source wavelet are independent of the number of time steps. Instead, the memory and additional computational costs grow with the number of frequencies and determine the amount of subsampling artifacts and crosstalk. In contrast to optimal checkpointing, this offers the possibility to trade the memory and computational costs for image quality or a larger number of iterations and is advantageous in new computing environments such as the cloud, where computing is often cheaper than memory and data movement. } } @INPROCEEDINGS{Lempitsky, author={V. {Lempitsky} and A. {Vedaldi} and D. {Ulyanov}}, booktitle={2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition}, title={{D}eep {I}mage {P}rior}, year={2018}, volume={}, number={}, pages={9446-9454}, keywords={convolution;feedforward neural nets;image restoration;inverse problems;learning (artificial intelligence);neural net architecture;statistical analysis;deep convolutional networks;image generation;randomly-initialized neural network;deep neural representations;deep image prior;image statistics;inverse problems;image restoration;generator network architectures;realistic image priors learning;Image restoration;Image resolution;Noise reduction;Task analysis;Optimization;Generators;Image reconstruction}, doi={10.1109/CVPR.2018.00984}, month={June} } @InProceedings{Cheng_2019_CVPR, author = {Cheng, Zezhou and Gadelha, Matheus and Maji, Subhransu and Sheldon, Daniel}, title = {{A Bayesian Perspective on the Deep Image Prior}}, booktitle = {{The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)}}, month = {June}, year = {2019}, pages = {5443--5451} } @inproceedings{welling2011bayesian, author = {Welling, Max and Teh, Yee Whye}, title = {{Bayesian Learning via Stochastic Gradient Langevin Dynamics}}, year = {2011}, booktitle = {{Proceedings of the 28th International Conference on International Conference on Machine Learning}}, pages = {681--688}, series = {{ICML’11}} } @article{devito-api, author = {Louboutin, M. and Lange, M. and Luporini, F. and Kukreja, N. and Witte, P. A. and Herrmann, F. J. and Velesko, P. and Gorman, G. J.}, title = {Devito (v3.1.0): an embedded domain-specific language for finite differences and geophysical exploration}, journal = {{Geoscientific Model Development}}, volume = {12}, year = {2019}, number = {3}, pages = {1165--1187}, url = {https://www.geosci-model-dev.net/12/1165/2019/}, doi = {10.5194/gmd-12-1165-2019} } @article{mosser2018stochastic, title={{Stochastic Seismic Waveform Inversion Using Generative Adversarial Networks as a Geological Prior}}, author={Mosser, Lukas and Dubrule, Olivier and Blunt, M}, doi={10.1007/s11004-019-09832-6}, journal = {{Mathematical Geosciences}}, volume = {84}, number = {1}, pages = {53--79}, year = {2019}, } @conference {herrmann2019NIPSliwcuc, title = {Learned imaging with constraints and uncertainty quantification}, booktitle = {{Neural Information Processing Systems (NeurIPS) 2019 Deep Inverse Workshop}}, year = {2019}, month = {12}, abstract = {We outline new approaches to incorporate ideas from convolutional networks into wave-based least-squares imaging. The aim is to combine hand-crafted constraints with deep convolutional networks allowing us to directly train a network capable of generating samples from the posterior. The main contributions include combination of weak deep priors with hard handcrafted constraints and a possible new way to sample the posterior.}, keywords = {constraint, deep learning, Imaging, Uncertainty quantification}, url = {https://arxiv.org/pdf/1909.06473.pdf}, author = {Felix J. Herrmann and Ali Siahkoohi and Gabrio Rizzuti} } @article{siahkoohi2019transfer, author={Siahkoohi, Ali and Louboutin, Mathias and Herrmann, Felix J.}, title={The importance of transfer learning in seismic modeling and imaging}, month={11}, year={2019}, doi = {10.1190/geo2019-0056.1}, journal = {{GEOPHYSICS}}, number = {6}, pages = {A47--A52}, publisher={{Society of Exploration Geophysicists}}, volume = {84}, } @article{rizzuti2019EAGElis, title = {{Learned iterative solvers for the Helmholtz equation}}, year = {2019}, doi = {10.3997/2214-4609.201901542}, issn = {2214-4609}, journal={{81st EAGE Conference and Exhibition 2019}}, abstract = {We propose a {\textquoteleft}learned{\textquoteright} iterative solver for the Helmholtz equation, by combining traditional Krylov-based solvers with machine learning. The method is, in principle, able to circumvent the shortcomings of classical iterative solvers, and has clear advantages over purely data-driven ap- proaches. We demonstrate the effectiveness of this approach under a 1.5-D assumption, when ade- quate a priori information about the velocity distribution is known.}, keywords = {Helmholtz, Iterative, machine learning, private}, url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2019/rizzuti2019EAGElis/rizzuti2019EAGElis.pdf}, author = {Rizzuti, Gabrio and Siahkoohi, Ali and Herrmann, Felix J.} } @article{wu2019parametric, title={Parametric convolutional neural network-domain full-waveform inversion}, author={Wu, Yulang and McMechan, George A}, journal={{GEOPHYSICS}}, volume={84}, number={6}, pages={R881--R896}, year={2019}, publisher={{Society of Exploration Geophysicists}}, doi={10.1190/geo2018-0224.1} } @article{malinverno2006two, title={Two ways to quantify uncertainty in geophysical inverse problems}, author={Malinverno, Alberto and Parker, Robert L}, journal={{GEOPHYSICS}}, volume={71}, number={3}, pages={W15--W27}, year={2006}, publisher={Society of Exploration Geophysicists}, doi={10.1190/1.2194516} } @article{malinverno2004expanded, title={Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical Bayes}, author={Malinverno, Alberto and Briggs, Victoria A}, journal={{GEOPHYSICS}}, volume={69}, number={4}, pages={1005--1016}, year={2004}, publisher={Society of Exploration Geophysicists}, doi={10.1190/1.1778243} } @article{visser2019bayesian, title={Bayesian transdimensional seismic full-waveform inversion with a dipping layer parameterization}, author={Visser, Gerhard and Guo, Peng and Saygin, Erdinc}, journal={{GEOPHYSICS}}, volume={84}, number={6}, pages={R845--R858}, year={2019}, publisher={Society of Exploration Geophysicists}, doi={10.1190/geo2018-0785.1} } @article{zhu2018seismic, title={Seismic inversion and uncertainty quantification using transdimensional Markov chain Monte Carlo method}, author={Zhu, Dehan and Gibson, Richard}, journal={{GEOPHYSICS}}, volume={83}, number={4}, pages={R321--R334}, year={2018}, publisher={Society of Exploration Geophysicists}, doi={10.1190/geo2016-0594.1} } @incollection{NEURIPS2019_9015, title = {{PyTorch: An Imperative Style, High-Performance Deep Learning Library}}, author = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Kopf, Andreas and Yang, Edward and DeVito, Zachary and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith}, booktitle = {Advances in Neural Information Processing Systems 32}, pages = {8024--8035}, year = {2019}, url = {http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf} } @article {fang2018uqfip, title = {Uncertainty quantification for inverse problems with weak partial-differential-equation constraints}, journal={{GEOPHYSICS}}, volume = {83}, number = {6}, year = {2018}, pages = {R629--R647}, doi = {10.1190/geo2017-0824.1}, author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann} } @article{doi:10.1046/j.1365-246X.2002.01847.x, author = {Malinverno, Alberto}, title = {Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem}, journal = {Geophysical Journal International}, volume = {151}, number = {3}, pages = {675-688}, keywords = {Bayesian inversion, geophysical inversion, Markov chain Monte Carlo, MCMC, resistivity}, doi = {10.1046/j.1365-246X.2002.01847.x}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1046/j.1365-246X.2002.01847.x}, eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1046/j.1365-246X.2002.01847.x}, abstract = {Summary A key element in the solution of a geophysical inverse problem is the quantification of non-uniqueness, that is, how much parameters of an inferred earth model can vary while fitting a set of measurements. A widely used approach is that of Bayesian inference, where Bayes' rule is used to determine the uncertainty of the earth model parameters a posteriori given the data. I describe here, a natural extension of Bayesian parameter estimation that accounts for the posterior probability of how complex an earth model is (specifically, how many layers it contains). This approach has a built-in parsimony criterion: among all earth models that fit the data, those with fewer parameters (fewer layers) have higher posterior probabilities. To implement this approach in practice, I use a Markov chain Monte Carlo (MCMC) algorithm applied to the nonlinear problem of inverting DC resistivity sounding data to infer characteristics of a 1-D earth model. The earth model is parametrized as a layered medium, where the number of layers and their resistivities and thicknesses are poorly known a priori. The algorithm obtains a sample of layered media from the posterior distribution; this sample measures non-uniqueness in terms of how many layers are effectively resolved by the data and of the range of layer thicknesses and resistivities consistent with the data. Because the complexity of the model is effectively determined by the data, the solution does not need to be regularized. This is a desirable feature, because requiring the solution to be smooth beyond what is implied by prior information can lead to underestimating posterior uncertainty. Letting the number of layers be a free parameter, as done here, broadens the space of earth models possible a priori and makes the determination of posterior uncertainty less dependent on the parametrization.}, year = {2002} } @inproceedings{li2016preconditioned, title={{Preconditioned stochastic gradient Langevin dynamics for deep neural networks}}, author={Li, Chunyuan and Chen, Changyou and Carlson, David and Carin, Lawrence}, booktitle={{Thirtieth AAAI Conference on Artificial Intelligence}}, year={2016} } @article {siahkoohi2020EAGEdlb, title = {A deep-learning based Bayesian approach to seismic imaging and uncertainty quantification}, year = {2020}, month = {6}, journal={{82nd EAGE Conference and Exhibition 2020}}, 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/Submitted/2020/siahkoohi2020EAGEdlb/siahkoohi2020EAGEdlb.html}, author = {Ali Siahkoohi and Gabrio Rizzuti and Felix J. Herrmann} } @conference{sun2019extrapolated, title={Extrapolated full waveform inversion with convolutional neural networks}, doi={10.1190/segam2019-3197987.1}, booktitle = {{SEG Technical Program Expanded Abstracts 2019}}, author={Sun, Hongyu and Demanet, Laurent}, year={2019} } @article{zhang2019regularized, title={Regularized elastic full waveform inversion using deep learning}, author={Zhang, {Zhen-Dong} and Alkhalifah, Tariq}, journal = {{GEOPHYSICS}}, volume={84}, number={5}, pages={1SO--Z28}, doi={10.1190/geo2018-0685.1}, year={2019}, month={9}, publisher={{Society of Exploration Geophysicists}} } @article{ovcharenko2019deep, title={Deep learning for low-frequency extrapolation from multi-offset seismic data}, author={Ovcharenko, Oleg and Kazei, Vladimir and Kalita, Mahesh and Peter, Daniel and Alkhalifah, Tariq Ali}, year={2019}, month={11}, doi = {10.1190/geo2018-0884.1}, journal = {{GEOPHYSICS}}, number = {6}, pages = {R989--R1001}, volume = {84}, publisher={{Society of Exploration Geophysicists}} } @conference{siahkoohi2019dlwr, author = {Ali Siahkoohi and Rajiv Kumar and Felix J. Herrmann}, title = {{Deep-learning based ocean bottom seismic wavefield recovery}}, booktitle = {{SEG Technical Program Expanded Abstracts 2019}}, pages = {2232--2237}, year = {2019}, month = {8}, doi = {10.1190/segam2019-3216632.1} } @conference{siahkoohi2019srmedl, author = {Ali Siahkoohi and Dirk J. Verschuur and Felix J. Herrmann}, title = {{Surface-related multiple elimination with deep learning}}, booktitle = {{SEG Technical Program Expanded Abstracts 2019}}, pages = {4629--4634}, year = {2019}, month = {8}, doi = {10.1190/segam2019-3216723.1} } @inproceedings{gadelha2019shape, title={Shape Reconstruction Using Differentiable Projections and Deep Priors}, author={Gadelha, Matheus and Wang, Rui and Maji, Subhransu}, booktitle={Proceedings of the IEEE International Conference on Computer Vision}, pages={22--30}, year={2019} } @article{devito-compiler, author = { {Luporini}, F. and {Lange}, M. and {Louboutin}, M. and {Kukreja}, N. and {H{\"u}ckelheim}, J. and {Yount}, C. and {Witte}, P. and {Kelly}, P.~H.~J. and {Herrmann}, F.~J. and {Gorman}, G.~J.}, title = { Architecture and performance of Devito, a system for automated stencil computation }, journal = { CoRR }, volume = { abs/1807.03032 }, month = { jul }, year = { 2018 }, url = { http://arxiv.org/abs/1807.03032 }, archivePrefix = { arXiv }, eprint = { 1807.03032 } } @article{adler2018task, title={Task adapted reconstruction for inverse problems}, author={Adler, Jonas and Lunz, Sebastian and Verdier, Olivier and Sch{\"o}nlieb, Carola-Bibiane and {\"O}ktem, Ozan}, journal={arXiv preprint arXiv:1809.00948}, year={2018} } @article{wu2018least, author = {Xinming Wu and Sergey Fomel}, title = {Least-squares horizons with local slopes and multi-grid correlations}, journal = {GEOPHYSICS}, volume = {83}, issue = {4}, pages = {IM29-IM40}, year = {2018}, doi = {10.1190/geo2017-0830.1}, URL = {https://doi.org/10.1190/geo2017-0830.1}, } @techreport{Veritas2005, title = {{Parihaka 3D Marine Seismic Survey - Acquisition and Processing Report}}, number = {New Zealand Petroleum Report 3460}, year = {2005}, institution = {New Zealand Petroleum \& Minerals, Wellington}, author={Veritas} } @techreport{WesternGeco2012, title = {{Parihaka 3D PSTM Final Processing Report}}, number = {New Zealand Petroleum Report 4582}, year = {2012}, institution = {New Zealand Petroleum \& Minerals, Wellington}, author={WesternGeco.} } @inproceedings{shi2020deep, title={Deep learning parameterization for geophysical inverse problems}, author={Shi, Yunzhi and Wu, Xinming and Fomel, Sergey}, booktitle={{SEG 2019 Workshop: Mathematical Geophysics: Traditional vs Learning, Beijing, China, 5-7 November 2019}}, pages={36--40}, year={2020}, doi = {10.1190/iwmg2019_09.1}, organization={Society of Exploration Geophysicists} } @article{liu2019deep, title={Deep-seismic-prior-based reconstruction of seismic data using convolutional neural networks}, author={Liu, Qun and Fu, Lihua and Zhang, Meng}, journal={arXiv preprint arXiv:1911.08784}, year={2019} } @inproceedings{brosse2018promises, title={{The promises and pitfalls of stochastic gradient Langevin dynamics}}, author={Brosse, Nicolas and Durmus, Alain and Moulines, Eric}, booktitle={{Advances in Neural Information Processing Systems}}, pages={8268--8278}, year={2018} } @book{tarantola2005inverse, title={Inverse problem theory and methods for model parameter estimation}, author={Tarantola, Albert}, year={2005}, publisher={SIAM}, doi={10.1137/1.9780898717921}, ISBN={978-0-89871-572-9} } @article{fomel2014structural, title={Structural uncertainty of time-migrated seismic images}, author={Fomel, Sergey and Landa, Evgeny}, journal={Journal of Applied Geophysics}, volume={101}, pages={27--30}, year={2014}, publisher={Elsevier} } @article{schaaf2019quantification, title={Quantification of uncertainty in {3-D} seismic interpretation: implications for deterministic and stochastic geomodeling and machine learning}, author={Schaaf, Alexander and Bond, Clare E}, journal={Solid earth}, year={2019} } @article{poliannikov2016effect, title={The effect of velocity uncertainty on migrated reflectors: Improvements from relative-depth imaging}, author={Poliannikov, Oleg V and Malcolm, Alison E}, journal={Geophysics}, volume={81}, number={1}, pages={S21--S29}, year={2016}, publisher={Society of Exploration Geophysicists} } @article{thore2002structural, title={Structural uncertainties: Determination, management, and applications}, author={Thore, Pierre and Shtuka, Arben and Lecour, Magali and Ait-Ettajer, Taoufik and Cognot, Richard}, journal={Geophysics}, volume={67}, number={3}, pages={840--852}, year={2002}, publisher={Society of Exploration Geophysicists} } @article{louseismic, title={Seismic horizon picking by integrating reflector dip and instantaneous phase attributes}, author={Lou, Yihuai and Zhang, Bo and Lin, Tengfei and Cao, Danping}, journal = {GEOPHYSICS}, volume = {85}, issue = {2}, pages = {O37-O45}, year = {2020}, doi = {10.1190/geo2018-0303.1}, URL = {https://doi.org/10.1190/geo2018-0303.1}, } @conference{zhao2019gradient, title={A gradient based {MCMC} method for {FWI} and uncertainty analysis}, author={Zhao, Zeyu and Sen, Mrinal K}, booktitle = {{SEG Technical Program Expanded Abstracts 2019}}, pages={1465--1469}, year = {2019}, month = {8}, doi={10.1190/segam2019-3216560.1} } @conference{izzatullah2019bayesian, title={Bayesian uncertainty estimation for full waveform inversion: {A} numerical study}, author={Izzatullah, Muhammad and van Leeuwen, Tristan and Peter, Daniel}, booktitle = {{SEG Technical Program Expanded Abstracts 2019}}, pages={1685--1689}, year = {2019}, month = {8}, doi={10.1190/segam2019-3216008.1} } @article{MartinMcMC2012, Author = {Martin, James and Wilcox, Lucas C. and Burstedde, Carsten and Ghattas, OMAR}, Date-Modified = {2016-06-24 20:16:07 +0000}, Eprint = {http://epubs.siam.org/doi/pdf/10.1137/110845598}, Journal = {SIAM Journal on Scientific Computing}, Number = {3}, Pages = {A1460-A1487}, Title = {A {S}tochastic {N}ewton {MCMC} {M}ethod for {L}arge-Scale {S}tatistical {I}nverse {P}roblems with {A}pplication to {S}eismic {I}nversion}, Url = {http://epubs.siam.org/doi/abs/10.1137/110845598}, Volume = {34}, Year = {2012}, Bdsk-Url-1 = {http://epubs.siam.org/doi/abs/10.1137/110845598} } @article{Ely2018, author = {Gregory Ely and Alison Malcolm and Oleg V. Poliannikov}, title = {Assessing uncertainties in velocity models and images with a fast nonlinear uncertainty quantification method}, journal = {GEOPHYSICS}, volume = {83}, number = {2}, pages = {R63-R75}, year = {2018}, doi = {10.1190/geo2017-0321.1}, URL = {https://doi.org/10.1190/geo2017-0321.1}, eprint = {https://doi.org/10.1190/geo2017-0321.1}, abstract = {Seismic imaging is conventionally performed using noisy data and a presumably inexact velocity model. Uncertainties in the input parameters propagate directly into the final image and therefore into any quantity of interest, or qualitative interpretation, obtained from the image. We considered the problem of uncertainty quantification in velocity building and seismic imaging using Bayesian inference. Using a reduced velocity model, a fast field expansion method for simulating recorded wavefields, and the adaptive Metropolis-Hastings algorithm, we efficiently quantify velocity model uncertainty by generating multiple models consistent with low-frequency full-waveform data. A second application of Bayesian inversion to any seismic reflections present in the recorded data reconstructs the corresponding structures’ position along with its associated uncertainty. Our analysis complements rather than replaces traditional imaging because it allows us to assess the reliability of visible image features and to take that into account in subsequent interpretations. } } @article{peters2019, author = {Bas Peters and Justin Granek and Eldad Haber}, title = {Multiresolution neural networks for tracking seismic horizons from few training images}, journal = {Interpretation}, volume = {7}, number = {3}, pages = {SE201-SE213}, year = {2019}, doi = {10.1190/INT-2018-0225.1}, URL = {https://doi.org/10.1190/INT-2018-0225.1}, eprint = {https://doi.org/10.1190/INT-2018-0225.1}, abstract = {Detecting a specific horizon in seismic images is a valuable tool for geologic interpretation. Because hand picking the locations of the horizon is a time-consuming process, automated computational methods were developed starting three decades ago. Until now, most networks have been trained on data that were created by cutting larger seismic images into many small patches. This limits the networks ability to learn from large-scale geologic structures. Moreover, currently available networks and training strategies require label patches that have full and continuous horizon picks (annotations), which are also time-consuming to generate. We have developed a projected loss function that enables training on labels with just a few annotated pixels and has no issue with the other unknown label pixels. We use this loss function for training convolutional networks with a multiresolution structure, including variants of the U-net. Our networks learn from a small number of large seismic images without creating patches. Training uses all seismic data without reserving some for validation. Only the labels are split into training/testing. We validate the accuracy of the trained network using the horizon picks that were never shown to the network. Contrary to other work on horizon tracking, we train the network to perform nonlinear regression, not classification. As such, we generate labels as the convolution of a Gaussian kernel and the known horizon locations that communicate uncertainty in the labels. The network output is the probability of the horizon location. We examine the new method on two different data sets, one for horizon extrapolation and another data set for interpolation. We found that the predictions of our methodology are accurate even in areas far from known horizon locations because our learning strategy exploits all data in large seismic images. } } @article{birgin2000nonmonotone, title={Nonmonotone spectral projected gradient methods on convex sets}, author={Birgin, Ernesto G and Mart{\'\i}nez, Jos{\'e} Mario and Raydan, Marcos}, journal={SIAM Journal on Optimization}, volume={10}, number={4}, pages={1196--1211}, year={2000}, publisher={SIAM} }