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@incollection{scales1996uncertainties,
	Author = {Scales, John A},
	Booktitle = {Inverse methods},
	Date-Added = {2016-11-15 19:35:00 +0000},
	Date-Modified = {2016-11-15 19:35:00 +0000},
	Pages = {79--97},
	Publisher = {Springer},
	Title = {Uncertainties in seismic inverse calculations},
	Year = {1996}}

@unpublished{esser2016tvr,
	Abstract = {We propose an extended full-waveform inversion formulation that includes general convex constraints on the model. Though the full problem is highly nonconvex, the overarching optimization scheme arrives at geologically plausible results by solving a sequence of relaxed and warm-started constrained convex subproblems. The combination of box, total-variation, and successively relaxed asymmetric total-variation constraints allows us to steer free from parasitic local minima while keeping the estimated physical parameters laterally continuous and in a physically realistic range. For accurate starting models, numerical experiments carried out on the challenging 2004 BP velocity benchmark demonstrate that bound and total-variation constraints improve the inversion result significantly by removing inversion artifacts, related to source encoding, and by clearly improved delineation of top, bottom, and flanks of a high-velocity high-contrast salt inclusion. The experiments also show that for poor starting models these two constraints by themselves are insufficient to detect the bottom of high-velocity inclusions such as salt. Inclusion of the one-sided asymmetric total-variation constraint overcomes this issue by discouraging velocity lows to buildup during the early stages of the inversion. To the author{\textquoteright}s knowledge the presented algorithm is the first to successfully remove the imprint of local minima caused by poor starting models and band-width limited finite aperture data.},
	Author = {Ernie Esser and Llu{\'\i}s Guasch and Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann},
	Date-Added = {2016-10-07 00:06:03 +0000},
	Date-Modified = {2016-10-07 00:06:03 +0000},
	Keywords = {constrained optimization, Full-waveform inversion, hinge loss, private, salt, total variation},
	Note = {Submitted on July 13, 2016.},
	Title = {Total-variation regularization strategies in full-waveform inversion},
	Url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2016/esser2016tvr/esser2016tvr.pdf},
	Year = {2016},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2016/esser2016tvr/esser2016tvr.pdf}}

@article{lieberman2010parameter,
	Author = {Lieberman, Chad and Willcox, Karen and Ghattas, Omar},
	Date-Added = {2016-10-04 17:32:14 +0000},
	Date-Modified = {2016-10-04 17:32:14 +0000},
	Journal = {SIAM Journal on Scientific Computing},
	Number = {5},
	Pages = {2523--2542},
	Publisher = {SIAM},
	Title = {Parameter and state model reduction for large-scale statistical inverse problems},
	Volume = {32},
	Year = {2010}}

@article{bruna2013invariant,
	Author = {Bruna, Joan and Mallat, St{\'e}phane},
	Date-Added = {2016-09-08 17:47:31 +0000},
	Date-Modified = {2016-09-08 17:47:31 +0000},
	Journal = {IEEE transactions on pattern analysis and machine intelligence},
	Number = {8},
	Pages = {1872--1886},
	Publisher = {IEEE},
	Title = {Invariant scattering convolution networks},
	Volume = {35},
	Year = {2013}}

@article{brooks1998general,
	Author = {Brooks, Stephen P and Gelman, Andrew},
	Date-Added = {2016-09-07 21:48:12 +0000},
	Date-Modified = {2016-09-07 21:48:12 +0000},
	Journal = {Journal of computational and graphical statistics},
	Number = {4},
	Pages = {434--455},
	Publisher = {Taylor \& Francis},
	Title = {General methods for monitoring convergence of iterative simulations},
	Volume = {7},
	Year = {1998}}

@article{tariyal2016greedy,
	Author = {Tariyal, Snigdha and Majumdar, Angshul and Singh, Richa and Vatsa, Mayank},
	Date-Added = {2016-08-30 18:02:28 +0000},
	Date-Modified = {2016-08-30 18:02:28 +0000},
	Journal = {arXiv preprint arXiv:1602.00203},
	Title = {Greedy Deep Dictionary Learning},
	Year = {2016}}

@article{aravkin2012estimating,
	Author = {Aravkin, Aleksandr Y and Van Leeuwen, Tristan},
	Date-Added = {2016-08-05 23:31:40 +0000},
	Date-Modified = {2016-08-05 23:31:40 +0000},
	Journal = {Inverse Problems},
	Number = {11},
	Pages = {115016},
	Publisher = {IOP Publishing},
	Title = {Estimating nuisance parameters in inverse problems},
	Volume = {28},
	Year = {2012}}

@article{sambridge2006trans,
	Author = {Sambridge, Malcolm and Gallagher, K and Jackson, Andrew and Rickwood, Peter},
	Date-Added = {2016-07-21 22:30:28 +0000},
	Date-Modified = {2016-07-21 22:30:28 +0000},
	Journal = {Geophysical Journal International},
	Number = {2},
	Pages = {528--542},
	Publisher = {Oxford University Press},
	Title = {Trans-dimensional inverse problems, model comparison and the evidence},
	Volume = {167},
	Year = {2006}}

@article{chen2012ensemble,
	Author = {Chen, Yan and Oliver, Dean S},
	Date-Added = {2016-07-21 22:07:06 +0000},
	Date-Modified = {2016-07-21 22:07:06 +0000},
	Journal = {Mathematical Geosciences},
	Number = {1},
	Pages = {1--26},
	Publisher = {Springer},
	Title = {Ensemble randomized maximum likelihood method as an iterative ensemble smoother},
	Volume = {44},
	Year = {2012}}

@article{nocedal1980updating,
	Author = {Nocedal, Jorge},
	Date-Added = {2016-07-18 21:21:48 +0000},
	Date-Modified = {2016-07-18 21:21:48 +0000},
	Journal = {Mathematics of computation},
	Number = {151},
	Pages = {773--782},
	Title = {Updating quasi-Newton matrices with limited storage},
	Volume = {35},
	Year = {1980}}

@book{robert2013monte,
	Author = {Robert, Christian and Casella, George},
	Date-Added = {2016-07-14 22:34:03 +0000},
	Date-Modified = {2016-07-14 22:34:03 +0000},
	Publisher = {Springer Science \& Business Media},
	Title = {Monte Carlo statistical methods},
	Year = {2013}}

@article{fisher2005equivalence,
	Author = {Fisher, Michael and Leutbecher, Mand and Kelly, GA},
	Date-Added = {2016-07-14 20:15:36 +0000},
	Date-Modified = {2016-07-14 20:15:36 +0000},
	Journal = {Quarterly Journal of the Royal Meteorological Society},
	Number = {613},
	Pages = {3235--3246},
	Publisher = {Wiley Online Library},
	Title = {On the equivalence between Kalman smoothing and weak-constraint four-dimensional variational data assimilation},
	Volume = {131},
	Year = {2005}}

@inproceedings{vigh2013long,
	Author = {Vigh, Denes and Jiao, K and Huang, W and Moldoveanu, N and Kapoor, J},
	Booktitle = {75th EAGE Conference \& Exhibition incorporating SPE EUROPEC 2013},
	Date-Added = {2016-07-13 21:58:13 +0000},
	Date-Modified = {2016-07-13 21:58:13 +0000},
	Title = {Long-offset-aided Full-waveform Inversion},
	Year = {2013}}

@incollection{heinkenschloss2008inexact,
	Author = {Heinkenschloss, M and Ridzal, D},
	Booktitle = {Numerical Mathematics and Advanced Applications},
	Date-Added = {2016-07-13 20:47:45 +0000},
	Date-Modified = {2016-07-13 20:47:45 +0000},
	Pages = {613--620},
	Publisher = {Springer},
	Title = {An inexact trust-region SQP method with applications to PDE-constrained optimization},
	Year = {2008}}

@article{dennis1998trust,
	Author = {Dennis, JE and Heinkenschloss, Matthias and Vicente, Lu{\'\i}s N},
	Date-Added = {2016-07-13 20:46:39 +0000},
	Date-Modified = {2016-07-13 20:46:39 +0000},
	Journal = {SIAM Journal on Control and Optimization},
	Number = {5},
	Pages = {1750--1794},
	Publisher = {SIAM},
	Title = {Trust-region interior-point SQP algorithms for a class of nonlinear programming problems},
	Volume = {36},
	Year = {1998}}

@article{grote2014inexact,
	Author = {Grote, Marcus J and Huber, Johannes and Kourounis, Drosos and Schenk, Olaf},
	Date-Added = {2016-07-13 20:45:32 +0000},
	Date-Modified = {2016-07-13 20:45:32 +0000},
	Journal = {SIAM Journal on Scientific Computing},
	Number = {3},
	Pages = {A1251--A1276},
	Publisher = {SIAM},
	Title = {Inexact interior-point method for PDE-constrained nonlinear optimization},
	Volume = {36},
	Year = {2014}}

@article{haber2000optimization,
	Author = {Haber, Eldad and Ascher, Uri M and Oldenburg, Doug},
	Date-Added = {2016-07-13 20:43:53 +0000},
	Date-Modified = {2016-07-13 20:43:53 +0000},
	Journal = {Inverse problems},
	Number = {5},
	Pages = {1263},
	Publisher = {IOP Publishing},
	Title = {On optimization techniques for solving nonlinear inverse problems},
	Volume = {16},
	Year = {2000}}

@article{mr1763essay,
	Author = {Mr. Bayes and Price, Mr},
	Date-Added = {2016-07-12 21:13:29 +0000},
	Date-Modified = {2016-07-12 21:13:29 +0000},
	Journal = {Philosophical Transactions (1683-1775)},
	Pages = {370--418},
	Publisher = {JSTOR},
	Title = {An essay towards solving a problem in the doctrine of chances. by the late rev. mr. bayes, frs communicated by mr. price, in a letter to john canton, amfrs},
	Year = {1763}}

@conference{fang2016SEGuqw,
	Abstract = {The data of full-waveform inversion often contains noise, which induces uncertainties in the inversion results. Ideally, one would like to run a number of independent inversions with different realizations of the noise and assess model-side uncertainties from the resulting models, however this is not feasible because we collect the data only once. To circumvent this restriction, various sampling schemes have been devised to generate an ensemble of models that fit the data to within the noise level. Such sampling schemes typically involve running multiple inversions or evaluating the Hessian of the cost function, both of which are computationally expensive. In this work, we propose a new method to quantify uncertainties based on a novel formulation of the full-waveform inversion problem {\textendash} wavefield reconstruction inversion. Based on this formulation, we formulate a semidefinite approximation of the corresponding Hessian matrix. By precomputing certain quantities, we are able to apply this Hessian to given input vectors without additional solutions of the underlying partial differential equations. To generate a sample, we solve an auxiliary stochastic optimization problem involving this Hessian. The result is a computationally feasible method that, with little overhead, can generate as many samples as required at small additional cost. We test our method on the synthetic BG Compass model and compare the results to a direct-sampling approach. The results show the feasibility of applying our method to computing statistical quantities such as the mean and standard deviation in the context of wavefield reconstruction inversion.},
	Author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann},
	Date-Added = {2016-07-12 17:28:37 +0000},
	Date-Modified = {2016-07-12 17:28:37 +0000},
	Keywords = {SEG, Uncertainty quantification, WRI},
	Note = {(to be presented at SEG, Dallas)},
	Title = {Uncertainty quantification for {Wavefield} {Reconstruction} {Inversion} using a {PDE} free semidefinite {Hessian} and randomize-then-optimize method},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw.html},
	Year = {2016},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw.html}}

@article{halko2011finding,
	Author = {Halko, Nathan and Martinsson, Per-Gunnar and Tropp, Joel A},
	Date-Added = {2016-07-11 22:33:42 +0000},
	Date-Modified = {2016-07-11 22:33:42 +0000},
	Journal = {SIAM review},
	Number = {2},
	Pages = {217--288},
	Publisher = {SIAM},
	Title = {Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions},
	Volume = {53},
	Year = {2011}}

@article{liberty2007randomized,
	Author = {Liberty, Edo and Woolfe, Franco and Martinsson, Per-Gunnar and Rokhlin, Vladimir and Tygert, Mark},
	Date-Added = {2016-07-11 22:32:18 +0000},
	Date-Modified = {2016-07-11 22:32:18 +0000},
	Journal = {Proceedings of the National Academy of Sciences},
	Number = {51},
	Pages = {20167--20172},
	Publisher = {National Acad Sciences},
	Title = {Randomized algorithms for the low-rank approximation of matrices},
	Volume = {104},
	Year = {2007}}

@article{aleardi20161d,
	Author = {Aleardi, Mattia and Mazzotti, Alfredo},
	Date-Added = {2016-07-11 21:50:14 +0000},
	Date-Modified = {2016-07-11 21:50:14 +0000},
	Journal = {Geophysical Prospecting},
	Publisher = {Wiley Online Library},
	Title = {1D elastic full-waveform inversion and uncertainty estimation by means of a hybrid genetic algorithm--Gibbs sampler approach},
	Year = {2016}}

@article{adamczyk2014high,
	Author = {Adamczyk, A and Malinowski, M and Malehmir, Alireza},
	Date-Added = {2016-07-11 17:46:35 +0000},
	Date-Modified = {2016-07-11 17:46:35 +0000},
	Journal = {Geophysical Journal International},
	Pages = {ggu070},
	Publisher = {Oxford University Press},
	Title = {High-resolution near-surface velocity model building using full-waveform inversion---A case study from southwest Sweden},
	Year = {2014}}

@article{baeten2013use,
	Author = {Baeten, Guido and de Maag, Jan Willem and Plessix, Ren{\'e}-Edouard and Klaassen, Rini and Qureshi, Tahira and Kleemeyer, Maren and Kroode, Fons ten and Rujie, Zhang},
	Date-Added = {2016-07-11 17:45:08 +0000},
	Date-Modified = {2016-07-11 17:45:08 +0000},
	Journal = {Geophysical Prospecting},
	Number = {4},
	Pages = {701--711},
	Publisher = {Wiley Online Library},
	Title = {The use of low frequencies in a full-waveform inversion and impedance inversion land seismic case study},
	Volume = {61},
	Year = {2013}}

@article{butzer20133d,
	Author = {Butzer, S and Kurzmann, A and Bohlen, T},
	Date-Added = {2016-07-11 17:42:44 +0000},
	Date-Modified = {2016-07-11 17:42:44 +0000},
	Journal = {Geophysical Prospecting},
	Number = {6},
	Pages = {1238--1251},
	Publisher = {Wiley Online Library},
	Title = {3D elastic full-waveform inversion of small-scale heterogeneities in transmission geometry},
	Volume = {61},
	Year = {2013}}

@article{ben2008velocity,
	Author = {Ben-Hadj-Ali, Hafedh and Operto, St{\'e}phane and Virieux, Jean},
	Date-Added = {2016-07-11 17:41:05 +0000},
	Date-Modified = {2016-07-11 17:41:05 +0000},
	Journal = {Geophysics},
	Number = {5},
	Pages = {VE101--VE117},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Velocity model building by 3D frequency-domain, full-waveform inversion of wide-aperture seismic data},
	Volume = {73},
	Year = {2008}}

@article{warner2013anisotropic,
	Author = {Warner, Michael and Ratcliffe, Andrew and Nangoo, Tenice and Morgan, Joanna and Umpleby, Adrian and Shah, Nikhil and Vinje, Vetle and {\v{S}}tekl, Ivan and Guasch, Llu{\'\i}s and Win, Caroline and others},
	Date-Added = {2016-07-11 17:38:57 +0000},
	Date-Modified = {2016-07-11 17:38:57 +0000},
	Journal = {Geophysics},
	Number = {2},
	Pages = {R59--R80},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Anisotropic 3D full-waveform inversion},
	Volume = {78},
	Year = {2013}}

@article{scales2001prior,
	Author = {Scales, John A and Tenorio, Luis},
	Date-Added = {2016-07-11 17:20:28 +0000},
	Date-Modified = {2016-07-11 17:20:28 +0000},
	Journal = {Geophysics},
	Number = {2},
	Pages = {389--397},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Prior information and uncertainty in inverse problems},
	Volume = {66},
	Year = {2001}}

@article{sheng2003finite,
	Author = {Sheng, Jianming and Schuster, Gerard T},
	Date-Added = {2016-07-09 00:14:04 +0000},
	Date-Modified = {2016-07-09 00:14:04 +0000},
	Journal = {Geophysical Journal International},
	Number = {3},
	Pages = {669--676},
	Publisher = {Oxford University Press},
	Title = {Finite-frequency resolution limits of wave path traveltime tomography for smoothly varying velocity models},
	Volume = {152},
	Year = {2003}}

@article{woodward2008decade,
	Author = {Woodward, Marta Jo and Nichols, Dave and Zdraveva, Olga and Whitfield, Phil and Johns, Tony},
	Date-Added = {2016-07-09 00:13:31 +0000},
	Date-Modified = {2016-07-09 00:13:31 +0000},
	Journal = {Geophysics},
	Number = {5},
	Pages = {VE5--VE11},
	Publisher = {Society of Exploration Geophysicists},
	Title = {A decade of tomography},
	Volume = {73},
	Year = {2008}}

@article{dahlen2004resolution,
	Author = {Dahlen, FA},
	Date-Added = {2016-07-09 00:11:22 +0000},
	Date-Modified = {2016-07-09 00:11:22 +0000},
	Journal = {Geophysical Journal International},
	Number = {1},
	Pages = {315--331},
	Publisher = {Oxford University Press},
	Title = {Resolution limit of traveltime tomography},
	Volume = {157},
	Year = {2004}}

@article{neyman1937outline,
	Author = {Neyman, Jerzy},
	Date-Added = {2016-07-09 00:10:51 +0000},
	Date-Modified = {2016-07-09 00:10:51 +0000},
	Journal = {Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences},
	Number = {767},
	Pages = {333--380},
	Publisher = {JSTOR},
	Title = {Outline of a theory of statistical estimation based on the classical theory of probability},
	Volume = {236},
	Year = {1937}}

@article{duijndam1988bayesian,
	Author = {Duijndam, AJW},
	Date-Added = {2016-07-09 00:09:05 +0000},
	Date-Modified = {2016-07-09 00:09:05 +0000},
	Journal = {Geophysical Prospecting},
	Number = {8},
	Pages = {899--918},
	Publisher = {Wiley Online Library},
	Title = {Bayesian Estimation in Seismic Inversion. Part II: Uncertainty ANALYSIS1},
	Volume = {36},
	Year = {1988}}

@incollection{nolet1987seismic,
	Author = {Nolet, Guust},
	Booktitle = {Seismic tomography},
	Date-Added = {2016-07-09 00:08:30 +0000},
	Date-Modified = {2016-07-09 00:08:30 +0000},
	Pages = {1--23},
	Publisher = {Springer},
	Title = {Seismic wave propagation and seismic tomography},
	Year = {1987}}

@book{box2011bayesian,
	Author = {Box, George EP and Tiao, George C},
	Date-Added = {2016-07-09 00:06:33 +0000},
	Date-Modified = {2016-07-09 00:06:33 +0000},
	Publisher = {John Wiley \& Sons},
	Title = {Bayesian inference in statistical analysis},
	Volume = {40},
	Year = {2011}}

@article{osypov2013model,
	Author = {Osypov, Konstantin and Yang, Yi and Fournier, Aim{\'e} and Ivanova, Natalia and Bachrach, Ran and Yarman, Can Evren and You, Yu and Nichols, Dave and Woodward, Marta},
	Date-Added = {2016-07-05 18:11:00 +0000},
	Date-Modified = {2016-07-05 18:11:00 +0000},
	Journal = {Geophysical Prospecting},
	Number = {6},
	Pages = {1114--1134},
	Publisher = {Wiley Online Library},
	Title = {Model-uncertainty quantification in seismic tomography: method and applications},
	Volume = {61},
	Year = {2013}}

@inproceedings{chavanne2008structural,
	Author = {Chavanne, Emmanuel and Brahmantio, Rangga A and Douillard, Anthony and Irving, Alan and others},
	Booktitle = {2008 SEG Annual Meeting},
	Date-Added = {2016-07-04 23:49:49 +0000},
	Date-Modified = {2016-07-04 23:49:49 +0000},
	Organization = {Society of Exploration Geophysicists},
	Title = {Structural uncertainty: a necessary step in geosciences risk analysis},
	Year = {2008}}

@inproceedings{osypov2008uncertainty,
	Author = {Osypov, K and Nichols, D and Woodward, M and Zdraveva, O and Yarman, CE and others},
	Booktitle = {2008 SEG Annual Meeting},
	Date-Added = {2016-07-04 23:47:18 +0000},
	Date-Modified = {2016-07-04 23:47:18 +0000},
	Organization = {Society of Exploration Geophysicists},
	Title = {Uncertainty and resolution analysis for anisotropic tomography using iterative eigendecomposition},
	Year = {2008}}

@inproceedings{osypov2011quantifying,
	Author = {Osypov, K and O'Briain, M and Whitfield, P and Nichols, D and Douillard, A and Sexton, P and Jousselin, P},
	Booktitle = {73rd EAGE Conference and Exhibition incorporating SPE EUROPEC 2011},
	Date-Added = {2016-07-04 23:43:22 +0000},
	Date-Modified = {2016-07-04 23:43:22 +0000},
	Title = {Quantifying structural uncertainty in anisotropic model building and depth imaging: Hild case study},
	Year = {2011}}

@inproceedings{letki2013quantifying,
	Author = {Letki, LP and Ben-Hadj-Ali, H and Desegaulx, P},
	Booktitle = {First EAGE West Africa Workshop 2013-Subsurface Challenges in West Africa},
	Date-Added = {2016-07-04 23:32:55 +0000},
	Date-Modified = {2016-07-04 23:32:55 +0000},
	Title = {Quantifying Uncertainty in Final Seismic Depth Image Using Structural Uncertainty Analysis-Case Study Offshore Nigeria},
	Year = {2013}}

@misc{johnson2011notes,
	Author = {Johnson, Steven G},
	Date-Added = {2016-07-04 23:31:25 +0000},
	Date-Modified = {2016-07-04 23:31:25 +0000},
	Publisher = {MIT Applied Mathematics},
	Title = {Notes on fft-based differentiation},
	Year = {2011}}

@inproceedings{zhu2015uncertainty,
	Author = {Zhu, Hejun and Li, Siwei and Fomel, Sergey and Stadler, Georg and Ghattas, Omar and others},
	Booktitle = {2015 SEG Annual Meeting},
	Date-Added = {2016-07-04 23:28:32 +0000},
	Date-Modified = {2016-07-04 23:28:32 +0000},
	Organization = {Society of Exploration Geophysicists},
	Title = {Uncertainty Estimation for Full Waveform Inversion with a Prior Information from Depth Migration},
	Year = {2015}}

@article{galbally2009nonlinear,
	Author = {Galbally, David and Fidkowski, K and Willcox, K and Ghattas, O},
	Date-Added = {2016-07-04 22:35:28 +0000},
	Date-Modified = {2016-07-04 22:35:28 +0000},
	Publisher = {John Wiley \& Sons, Inc.},
	Title = {Nonlinear model reduction for uncertainty quantification in large-scale inverse problems},
	Year = {2009}}

@article{stark2010primer,
	Author = {Stark, Philip B and Tenorio, Luis},
	Date-Added = {2016-06-28 18:19:24 +0000},
	Date-Modified = {2016-06-28 18:19:24 +0000},
	Journal = {Large Scale Inverse Problems and Quantification of Uncertainty, John Wiley and Sons},
	Publisher = {Citeseer},
	Title = {A primer of frequentist and Bayesian inference in inverse problems},
	Year = {2010}}

@article{stark2015constraints,
	Author = {Stark, Philip B},
	Date-Added = {2016-06-27 20:37:55 +0000},
	Date-Modified = {2016-06-27 20:37:55 +0000},
	Journal = {SIAM/ASA Journal on Uncertainty Quantification},
	Number = {1},
	Pages = {586--598},
	Publisher = {SIAM},
	Title = {Constraints versus priors},
	Volume = {3},
	Year = {2015}}

@article{cotter2013mcmc,
	Author = {Cotter, Simon L and Roberts, Gareth O and Stuart, Andrew M and White, David and others},
	Date-Added = {2016-06-27 20:22:38 +0000},
	Date-Modified = {2016-06-27 20:22:38 +0000},
	Journal = {Statistical Science},
	Number = {3},
	Pages = {424--446},
	Publisher = {Institute of Mathematical Statistics},
	Title = {MCMC methods for functions: modifying old algorithms to make them faster},
	Volume = {28},
	Year = {2013}}

@article{pereyra2015proximal,
	Author = {Pereyra, Marcelo},
	Date-Added = {2016-06-27 20:19:56 +0000},
	Date-Modified = {2016-06-27 20:19:56 +0000},
	Journal = {Statistics and Computing},
	Pages = {1--16},
	Publisher = {Springer},
	Title = {Proximal markov chain monte carlo algorithms},
	Year = {2015}}

@article{vigh2014elastic,
	Author = {Vigh, Denes and Jiao, Kun and Watts, Dave and Sun, Dong},
	Date-Added = {2016-06-24 20:03:23 +0000},
	Date-Modified = {2016-06-24 20:03:23 +0000},
	Journal = {Geophysics},
	Number = {2},
	Pages = {R63--R77},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Elastic full-waveform inversion application using multicomponent measurements of seismic data collection},
	Volume = {79},
	Year = {2014}}

@article{operto2013guided,
	Author = {Operto, St{\'e}phane and Gholami, Yaser and Prieux, V and Ribodetti, Alessandra and Brossier, R and Metivier, L and Virieux, Jean},
	Date-Added = {2016-06-24 20:00:57 +0000},
	Date-Modified = {2016-06-24 20:00:57 +0000},
	Journal = {The Leading Edge},
	Number = {9},
	Pages = {1040--1054},
	Publisher = {Society of Exploration Geophysicists},
	Title = {A guided tour of multiparameter full-waveform inversion with multicomponent data: From theory to practice},
	Volume = {32},
	Year = {2013}}

@article{esser2016constrained,
	Author = {Esser, Ernie and Guasch, Llu{\'\i}s and Herrmann, Felix J and Warner, Mike},
	Date-Added = {2016-06-24 19:56:42 +0000},
	Date-Modified = {2016-06-24 19:56:42 +0000},
	Journal = {The Leading Edge},
	Number = {3},
	Pages = {235--239},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Constrained waveform inversion for automatic salt flooding},
	Volume = {35},
	Year = {2016}}

@article{li2012fast,
	Author = {Li, Xiang and Aravkin, Aleksandr Y and van Leeuwen, Tristan and Herrmann, Felix J},
	Date-Added = {2016-06-24 18:56:42 +0000},
	Date-Modified = {2016-06-24 18:56:42 +0000},
	Journal = {Geophysics},
	Number = {3},
	Pages = {A13--A17},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Fast randomized full-waveform inversion with compressive sensing},
	Volume = {77},
	Year = {2012}}

@article{bui2013computational,
	Author = {Bui-Thanh, Tan and Ghattas, Omar and Martin, James and Stadler, Georg},
	Date-Added = {2016-06-24 17:56:41 +0000},
	Date-Modified = {2016-06-24 17:56:41 +0000},
	Journal = {SIAM Journal on Scientific Computing},
	Number = {6},
	Pages = {A2494--A2523},
	Publisher = {SIAM},
	Title = {A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion},
	Volume = {35},
	Year = {2013}}

@article{gouveia1998bayesian,
	Author = {Gouveia, Wences P and Scales, John A},
	Date-Added = {2016-06-24 17:52:33 +0000},
	Date-Modified = {2016-06-24 17:52:33 +0000},
	Journal = {Journal of Geophysical Research: Solid Earth},
	Number = {B2},
	Pages = {2759--2779},
	Publisher = {Wiley Online Library},
	Title = {Bayesian seismic waveform inversion: Parameter estimation and uncertainty analysis},
	Volume = {103},
	Year = {1998}}

@article{kaufl2013probabilistic,
	Author = {K{\"a}ufl, Paul and Fichtner, Andreas and Igel, Heiner},
	Date-Added = {2016-06-21 21:48:19 +0000},
	Date-Modified = {2016-06-21 21:48:19 +0000},
	Journal = {Geophysical Journal International},
	Pages = {ggs131},
	Publisher = {Oxford University Press},
	Title = {Probabilistic full waveform inversion based on tectonic regionalization---development and application to the Australian upper mantle},
	Year = {2013}}

@article{pereyra2016survey,
	Author = {Pereyra, Marcelo and Schniter, Philip and Chouzenoux, Emilie and Pesquet, Jean-Christophe and Tourneret, Jean-Yves and Hero, Alfred O and McLaughlin, Steve},
	Date-Added = {2016-06-21 21:44:37 +0000},
	Date-Modified = {2016-06-21 21:44:37 +0000},
	Journal = {IEEE Journal of Selected Topics in Signal Processing},
	Number = {2},
	Pages = {224--241},
	Publisher = {IEEE},
	Title = {A survey of stochastic simulation and optimization methods in signal processing},
	Volume = {10},
	Year = {2016},
	Bdsk-File-1 = {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}}

@article{golub2003separable,
	Author = {Golub, Gene and Pereyra, Victor},
	Date-Added = {2016-06-20 21:50:07 +0000},
	Date-Modified = {2016-06-20 21:50:07 +0000},
	Journal = {Inverse problems},
	Number = {2},
	Pages = {R1},
	Publisher = {IOP Publishing},
	Title = {Separable nonlinear least squares: the variable projection method and its applications},
	Volume = {19},
	Year = {2003}}

@inproceedings{warner2013full,
	Author = {Warner, Mike and Nangoo, Tenice and Shah, Nikhil and Umpleby, Adrian and Morgan, Joanna and others},
	Booktitle = {2013 SEG Annual Meeting},
	Date-Added = {2016-06-15 17:31:39 +0000},
	Date-Modified = {2016-06-24 20:10:20 +0000},
	Organization = {Society of Exploration Geophysicists},
	Title = {Full-waveform inversion of cycle-skipped seismic data by frequency down-shifting},
	Year = {2013}}

@article{cordua2012monte,
	Author = {Cordua, Knud Skou and Hansen, Thomas Mejer and Mosegaard, Klaus},
	Date-Added = {2016-06-09 23:01:27 +0000},
	Date-Modified = {2016-06-09 23:01:27 +0000},
	Journal = {Geophysics},
	Number = {2},
	Pages = {H19--H31},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Monte Carlo full-waveform inversion of crosshole GPR data using multiple-point geostatistical a priori information},
	Volume = {77},
	Year = {2012}}

@article{malinverno2004expanded,
	Author = {Malinverno, Alberto and Briggs, Victoria A},
	Date-Added = {2016-06-05 22:54:25 +0000},
	Date-Modified = {2016-06-05 22:54:25 +0000},
	Journal = {Geophysics},
	Number = {4},
	Pages = {1005--1016},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical Bayes},
	Volume = {69},
	Year = {2004}}

@inproceedings{warner2014adaptiveSEG,
	Author = {Warner, Mike and Guasch, Llu{\'\i}s},
	Booktitle = {2014 SEG Annual Meeting},
	Date-Added = {2016-06-04 23:48:38 +0000},
	Date-Modified = {2016-06-29 18:55:33 +0000},
	Organization = {Society of Exploration Geophysicists},
	Title = {Adaptive waveform inversion: Theory},
	Year = {2014}}

@inproceedings{warner2014adaptive,
	Author = {Warner, Mike and Guasch, Llu{\'\i}s},
	Booktitle = {76th EAGE Conference and Exhibition 2014},
	Date-Added = {2016-06-04 23:48:24 +0000},
	Date-Modified = {2016-06-05 21:41:22 +0000},
	Title = {Adaptive Waveform Inversion-FWI Without Cycle Skipping-Theory},
	Year = {2014}}

@article{symes2008migration,
	Author = {Symes, William W},
	Date-Added = {2016-06-04 23:47:54 +0000},
	Date-Modified = {2016-06-04 23:47:54 +0000},
	Journal = {Geophysical prospecting},
	Number = {6},
	Pages = {765--790},
	Publisher = {Wiley Online Library},
	Title = {Migration velocity analysis and waveform inversion},
	Volume = {56},
	Year = {2008}}

@article{sava2004wave,
	Author = {Sava, Paul and Biondi, Biondo},
	Date-Added = {2016-06-04 23:47:28 +0000},
	Date-Modified = {2016-06-04 23:47:28 +0000},
	Journal = {Geophysical Prospecting},
	Number = {6},
	Pages = {593--606},
	Publisher = {Wiley Online Library},
	Title = {Wave-equation migration velocity analysis. I. Theory},
	Volume = {52},
	Year = {2004}}

@article{pratt1999seismic,
	Author = {Pratt, R Gerhard},
	Date-Added = {2016-06-04 23:46:59 +0000},
	Date-Modified = {2016-06-04 23:46:59 +0000},
	Journal = {Geophysics},
	Number = {3},
	Pages = {888--901},
	Publisher = {Society of Exploration Geophysicists},
	Title = {Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model},
	Volume = {64},
	Year = {1999}}

@article{plessix2006review,
	Author = {Plessix, R-E},
	Date-Added = {2016-06-04 23:43:27 +0000},
	Date-Modified = {2016-06-04 23:43:27 +0000},
	Journal = {Geophysical Journal International},
	Number = {2},
	Pages = {495--503},
	Publisher = {Oxford University Press},
	Title = {A review of the adjoint-state method for computing the gradient of a functional with geophysical applications},
	Volume = {167},
	Year = {2006}}

@article{vanleeuwen2015IPpmp,
	Abstract = {Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the nonlinearity of the problem and is less sensitive to the initial iterate.},
	Author = {Tristan van Leeuwen and Felix J. Herrmann},
	Date-Added = {2016-05-12 18:06:35 +0000},
	Date-Modified = {2016-05-12 18:06:35 +0000},
	Journal = {Inverse Problems},
	Keywords = {Inverse problems, Optimization, PDE, penalty method},
	Month = {12},
	Note = {(Inverse Problems)},
	Number = {1},
	Pages = {015007},
	Title = {A penalty method for {PDE}-constrained optimization in inverse problems},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InverseProblems/2015/vanleeuwen2015IPpmp/vanleeuwen2015IPpmp.pdf},
	Url2 = {http://stacks.iop.org/0266-5611/32/i=1/a=015007},
	Volume = {32},
	Year = {2015},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InverseProblems/2015/vanleeuwen2015IPpmp/vanleeuwen2015IPpmp.pdf}}

@article{rue2001fast,
	Author = {Rue, H{\aa}vard},
	Date-Added = {2016-05-06 21:13:07 +0000},
	Date-Modified = {2016-05-06 21:13:07 +0000},
	Journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
	Number = {2},
	Pages = {325--338},
	Publisher = {Wiley Online Library},
	Title = {Fast sampling of Gaussian Markov random fields},
	Volume = {63},
	Year = {2001}}

@article{orieux2012sampling,
	Author = {Orieux, Fran{\c{c}}ois and F{\'e}ron, Olivier and Giovannelli, J-F},
	Date-Added = {2016-05-06 21:09:16 +0000},
	Date-Modified = {2016-05-06 21:09:16 +0000},
	Journal = {Signal Processing Letters, IEEE},
	Number = {5},
	Pages = {251--254},
	Publisher = {IEEE},
	Title = {Sampling high-dimensional Gaussian distributions for general linear inverse problems},
	Volume = {19},
	Year = {2012}}

@article{papandreou2010gaussian,
	Author = {Papandreou, George and Yuille, Alan L},
	Booktitle = {Advances in Neural Information Processing Systems},
	Date-Added = {2016-05-06 21:07:19 +0000},
	Date-Modified = {2016-05-06 21:08:28 +0000},
	Pages = {1858--1866},
	Title = {Gaussian sampling by local perturbations},
	Year = {2010}}

@article{bardsley2015randomize,
	Author = {Bardsley, Johnathan M and Sepp{\"a}nen, Aku and Solonen, Antti and Haario, Heikki and Kaipio, Jari},
	Date-Added = {2016-04-28 18:21:37 +0000},
	Date-Modified = {2016-04-28 18:24:08 +0000},
	Journal = {SIAM/ASA Journal on Uncertainty Quantification},
	Keywords = {Nonlinear sample},
	Number = {1},
	Pages = {1136--1158},
	Publisher = {SIAM},
	Title = {Randomize-Then-Optimize for Sampling and Uncertainty Quantification in Electrical Impedance Tomography},
	Volume = {3},
	Year = {2015}}

@article{solonen2014optimization,
	Author = {Solonen, Antti and Bibov, Alexander and Bardsley, Johnathan M and Haario, Heikki},
	Journal = {International Journal for Uncertainty Quantification},
	Number = {4},
	Publisher = {Begel House Inc.},
	Title = {Optimization-based sampling in ensemble Kalman filtering},
	Volume = {4},
	Year = {2014}}

@conference{fang2015EAGEuqw,
	Author = {Fang, Zhilong and Lee, C and Silva, C and Herrmann, F and Kuske, Rachel},
	Booktitle = {77th EAGE Conference and Exhibition 2015},
	Title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	Year = {2015}}

@article{bardsley2014randomize,
	Author = {Bardsley, Johnathan M and Solonen, Antti and Haario, Heikki and Laine, Marko},
	Journal = {SIAM Journal on Scientific Computing},
	Number = {4},
	Pages = {A1895--A1910},
	Publisher = {SIAM},
	Title = {Randomize-then-optimize: A method for sampling from posterior distributions in nonlinear inverse problems},
	Volume = {36},
	Year = {2014}}

@article{Tarantola1982Bayesian,
	Author = {Tarantola, Albert and Valette, Bernard},
	Journal = {Journal of Geophysics},
	Pages = {159-170},
	Title = {Inverse Problems = Quest for Information},
	Volume = {50},
	Year = {1982}}

@article{VirieuxOverview2009,
	Author = {Virieux, Jean and Operto, St{\'e}phane},
	Date-Modified = {2016-06-24 20:25:02 +0000},
	Doi = {10.1190/1.3238367},
	Eprint = {http://dx.doi.org/10.1190/1.3238367},
	Journal = {GEOPHYSICS},
	Number = {6},
	Pages = {WCC1-WCC26},
	Title = {An overview of full-waveform inversion in exploration geophysics},
	Url = {http://dx.doi.org/10.1190/1.3238367},
	Volume = {74},
	Year = {2009},
	Bdsk-Url-1 = {http://dx.doi.org/10.1190/1.3238367}}

@article{Tarantola1982FWI,
	Author = {Tarantola, Albert and Valette, Bernard},
	Doi = {10.1029/RG020i002p00219},
	Issn = {1944-9208},
	Journal = {Reviews of Geophysics},
	Number = {2},
	Pages = {219--232},
	Title = {Generalized nonlinear inverse problems solved using the least squares criterion},
	Url = {http://dx.doi.org/10.1029/RG020i002p00219},
	Volume = {20},
	Year = {1982},
	Bdsk-Url-1 = {http://dx.doi.org/10.1029/RG020i002p00219}}

@conference{aravkin2012EAGErobust,
	Abstract = {Source estimation is an essential component of full waveform inversion. In the standard frequency domain formulation, there is closed form solution for the the optimal source weights, which can thus be cheaply estimated on the fly. A growing body of work underscores the importance of robust modeling for data with large outliers or artifacts that are not captured by the forward model. Effectively, the least-squares penalty on the residual is replaced by a robust penalty, such as Huber, Hybrid {\textquoteleft}1-{\textquoteleft}2 or Student{\textquoteright}s t. As we will demonstrate, it is essential to use the same robust penalty for source estimation. In this abstract, we present a general approach to robust waveform inversion with robust source estimation. In this general formulation, there is no closed form solution for the optimal source weights so we need to solve a scalar optimization problem to obtain these weights. We can efficiently solve this optimization problem with a Newton-like method in a few iterations. The computational cost involved is of the same order as the usual least-squares source estimation procedure. We show numerical examples illustrating robust source estimation and robust waveform inversion on synthetic data with outliers.},
	Author = {Aleksandr Y. Aravkin and Tristan van Leeuwen and Henri Calandra and Felix J. Herrmann},
	Booktitle = {EAGE},
	Keywords = {EAGE},
	Month = {06},
	Presentation = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/aravkin2012EAGErobust/aravkin2012EAGErobust_pres.pdf},
	Title = {Source estimation for frequency-domain {FWI} with robust penalties},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/aravkin2012EAGErobust/aravkin2012EAGErobust.pdf},
	Url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=59196},
	Year = {2012},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/aravkin2012EAGErobust/aravkin2012EAGErobust.pdf}}

@article{vanLeeuwen2014GEOPcav,
	Author = {Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann},
	Doi = {10.1190/geo2013-0466.1},
	Journal = {Geophysics},
	Keywords = {variable projection, waveform inversion},
	Month = {05},
	Note = {(discussion by Tristan van Leeuwen, Aleksandr Y. Aravkin, and Felix J. Herrmann)},
	Number = {3},
	Pages = {X11-X17},
	Title = {Comment on: {\textquotedblleft}Application of the variable projection scheme for frequency-domain full-waveform inversion{\textquotedblright} (M. Li, J. Rickett, and A. Abubakar, Geophysics, 78, no. 6, R249{\textendash}R257)},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2014/vanLeeuwen2014GEOPcav/vanLeeuwen2014GEOPcav.pdf},
	Volume = {79},
	Year = {2014},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2014/vanLeeuwen2014GEOPcav/vanLeeuwen2014GEOPcav.pdf},
	Bdsk-Url-2 = {http://dx.doi.org/10.1190/geo2013-0466.1}}

@article{MaokunLi2013srcest,
	Author = {Li, M. and Rickett, J. and Abubakar, A.},
	Doi = {10.1190/geo2012-0351.1},
	Eprint = {http://dx.doi.org/10.1190/geo2012-0351.1},
	Journal = {GEOPHYSICS},
	Number = {6},
	Pages = {R249-R257},
	Title = {Application of the variable projection scheme for frequency-domain full-waveform inversion},
	Url = {http://dx.doi.org/10.1190/geo2012-0351.1},
	Volume = {78},
	Year = {2013},
	Bdsk-Url-1 = {http://dx.doi.org/10.1190/geo2012-0351.1}}

@article{vanLeeuwen2013GJImlm,
	Abstract = {Wave equation based inversions, such as full-waveform
                  inversion and reverse-time migration, are
                  challenging because of their computational costs,
                  memory requirements and reliance on accurate initial
                  models. To confront these issues, we propose a novel
                  formulation of wave equation based inversion based
                  on a penalty method. In this formulation, the
                  objective function consists of a data-misfit term
                  and a penalty term, which measures how accurately
                  the wavefields satisfy the wave equation. This new
                  approach is a major departure from current
                  formulations where forward and adjoint wavefields,
                  which both satisfy the wave equation, are correlated
                  to compute updates for the unknown model
                  parameters. Instead, we carry out the inversions
                  over two alternating steps during which we first
                  estimate the wavefield everywhere, given the current
                  model parameters, source and observed data, followed
                  by a second step during which we update the model
                  parameters, given the estimate for the wavefield
                  everywhere and the source. Because the inversion
                  involves both the synthetic wavefields and the
                  medium parameters, its search space is enlarged so
                  that it suffers less from local minima. Compared to
                  other formulations that extend the search space of
                  wave equation based inversion, our method differs in
                  several aspects, namely (i) it avoids storage and
                  updates of the synthetic wavefields because we
                  calculate these explicitly by finding solutions that
                  obey the wave equation and fit the observed data and
                  (ii) no adjoint wavefields are required to update
                  the model, instead our updates are calculated from
                  these solutions directly, which leads to significant
                  computational savings. We demonstrate the validity
                  of our approach by carefully selected examples and
                  discuss possible extensions and future research.},
	Author = {Tristan van Leeuwen and Felix J. Herrmann},
	Doi = {10.1093/gji/ggt258},
	Eprint = {http://gji.oxfordjournals.org/content/early/2013/07/30/gji.ggt258.full.pdf+html},
	Journal = {Geophysical Journal International},
	Month = {10},
	Pages = {661-667},
	Title = {Mitigating local minima in full-waveform inversion by expanding the search space},
	Url1 = {http://gji.oxfordjournals.org/content/195/1/661},
	Url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/GeophysicalJournalInternational/2013/vanLeeuwen2013GJImlm/vanLeeuwen2013mlm.pdf},
	Volume = {195},
	Year = {2013},
	Bdsk-Url-1 = {http://dx.doi.org/10.1093/gji/ggt258}}

@unpublished{vanLeeuwen2013Penalty2,
	Abstract = {We present a method for solving PDE constrained
                  optimization problems based on a penalty
                  formulation. This method aims to combine advantages
                  of both full-space and reduced methods by exploiting
                  a large search-space (consisting of both control and
                  state variables) while allowing for an efficient
                  implementation that avoids storing and updating the
                  state-variables. This leads to a method that has
                  roughly the same per-iteration complexity as
                  conventional reduced approaches while dening an
                  objective that is less non-linear in the control
                  variable by implicitly relaxing the constraint. We
                  apply the method to a seismic inverse problem where
                  it leads to a particularly ecient implementation
                  when compared to a conventional reduced approach as
                  it avoids the use of adjoint
                  state-variables. Numerical examples illustrate the
                  approach and suggest that the proposed formulation
                  can indeed mitigate some of the well-known problems
                  with local minima in the seismic inverse problem.},
	Author = {Tristan van Leeuwen and Felix J. Herrmann},
	Institution = {UBC},
	Keywords = {waveform inversion, optimization, private},
	Month = {Apr},
	Title = {A penalty method for PDE-constrained optimization},
	Url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2013/vanLeeuwen2013Penalty2/vanLeeuwen2013Penalty2.pdf},
	Year = {2013},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2013/vanLeeuwen2013Penalty2/vanLeeuwen2013Penalty2.pdf}}

@conference{leeuwen2014EAGEntf,
	Abstract = {We discuss a recently proposed novel method for waveform inversion: Wavefield Reconstruction Inversion (WRI). As opposed to conventional FWI {\textendash} which attempts to minimize the error between observed and predicted data obtained by solving a wave equation {\textendash} WRI reconstructs a wave-field from the data and extracts a model-update from this wavefield by minimizing the wave-equation residual. The method does not require explicit computation of an adjoint wavefield as all the necessary information is contained in the reconstructed wavefield. We show how the corresponding model updates can be interpreted physically analogously to the conventional imaging-condition-based approach.},
	Author = {Tristan van Leeuwen and Felix J. Herrmann and Bas Peters},
	Booktitle = {EAGE},
	Keywords = {EAGE, Full-waveform inversion, Optimization, penalty method, Wavefield Reconstruction Inversion},
	Month = {01},
	Organization = {UBC},
	Publisher = {UBC},
	Title = {A new take on FWI: Wavefield Reconstruction Inversion},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/leeuwen2014EAGEntf.pdf},
	Year = {2014},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/leeuwen2014EAGEntf.pdf}}

@conference{peters2014EAGEweb,
	Abstract = {In this paper we make a comparison between wave-equation based inversions based on the adjoint-state and penalty methods. While the adjoint-state method involves the minimization of a data-misfit and exact solutions of the wave-equation for the current velocity model, the penalty-method aims to first find a wavefield that jointly fits the data and honours the physics, in a least-squares sense. Given this reconstructed wavefield, which is a proxy for the true wavefield in the true model, we calculate updates for the velocity model. Aside from being less nonlinear{\textendash}the acoustic wave equation is linear in the wavefield and model parameters but not in both{\textendash}the inversion is carried out over a solution space that includes both the model and the wavefield. This larger search space allows the algortihm to circumnavigate local minima, very much in the same way as recently proposed model extentions try to acomplish. We include examples for low frequencies, where we compare full-waveform inversion results for both methods, for good and bad starting models, and for high frequencies where we compare reverse-time migration with linearized imaging based on wavefield-reconstruction inversion. The examples confirm the expected benefits of the proposed method.},
	Author = {Bas Peters and Felix J. Herrmann and Tristan van Leeuwen},
	Booktitle = {EAGE},
	Doi = {10.3997/2214-4609.20140704},
	Keywords = {EAGE, Full-waveform inversion, Imaging, Optimization, penalty method},
	Month = {06},
	Presentation = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb_pres.pdf},
	Title = {Wave-equation based inversion with the penalty method: adjoint-state versus wavefield-reconstruction inversion},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb.pdf},
	Year = {2014},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb.pdf},
	Bdsk-Url-2 = {http://dx.doi.org/10.3997/2214-4609.20140704}}

@article{TristanFWI2013,
	Author = {van Leeuwen, Tristan and Herrmann, Felix J.},
	Issn = {1365-2478},
	Journal = {Geophysical Prospecting},
	Keywords = {Full-waveform inversion, Source-encoding, Stochastic optimization},
	Pages = {10--19},
	Publisher = {Blackwell Publishing Ltd},
	Title = {Fast {W}aveform {I}nversion without {S}ource-{E}ncoding},
	Url = {http://dx.doi.org/10.1111/j.1365-2478.2012.01096.x},
	Volume = {61},
	Year = {2013},
	Bdsk-Url-1 = {http://dx.doi.org/10.1111/j.1365-2478.2012.01096.x}}

@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{Overview2009,
	Author = {Virieux, J. and Operto, S.},
	Eprint = {http://library.seg.org/doi/pdf/10.1190/1.3238367},
	Journal = {GEOPHYSICS},
	Number = {6},
	Pages = {WCC1-WCC26},
	Title = {{An} {O}verview of {F}ull-waveform {I}nversion in {E}xploration {G}eophysics},
	Url = {http://library.seg.org/doi/abs/10.1190/1.3238367},
	Volume = {74},
	Year = {2009},
	Bdsk-Url-1 = {http://library.seg.org/doi/abs/10.1190/1.3238367}}

@article{AVO2003,
	Author = {Buland, A. and Omre, H.},
	Journal = {GEOPHYSICS},
	Number = {1},
	Pages = {185-198},
	Title = {Bayesian linearized AVO inversion},
	Url = {http://geophysics.geoscienceworld.org/content/68/1/185.short#cited-by},
	Volume = {68},
	Year = {2003},
	Bdsk-Url-1 = {http://geophysics.geoscienceworld.org/content/68/1/185.short#cited-by}}

@article{GhattasMcMC2013,
	Author = {Bui-Thanh, T. and Ghattas, O. and Martin, J. and Stadler, G.},
	Eprint = {http://epubs.siam.org/doi/pdf/10.1137/12089586X},
	Journal = {SIAM Journal on Scientific Computing},
	Number = {6},
	Pages = {A2494-A2523},
	Title = {A {C}omputational {F}ramework for {I}nfinite-{D}imensional {B}ayesian {I}nverse {P}roblems {P}art {I}: {T}he {L}inearized {C}ase, with {A}pplication to {G}lobal {S}eismic {I}nversion},
	Url = {http://epubs.siam.org/doi/abs/10.1137/12089586X},
	Volume = {35},
	Year = {2013},
	Bdsk-Url-1 = {http://epubs.siam.org/doi/abs/10.1137/12089586X}}

@article{MichaelSO2012,
	Author = {Friedlander, M. and Schmidt, M.},
	Eprint = {http://epubs.siam.org/doi/pdf/10.1137/110830629},
	Journal = {SIAM Journal on Scientific Computing},
	Number = {3},
	Pages = {A1380-A1405},
	Title = {{H}ybrid {D}eterministic-{S}tochastic {M}ethods for {D}ata {F}itting},
	Url = {http://epubs.siam.org/doi/abs/10.1137/110830629},
	Volume = {34},
	Year = {2012},
	Bdsk-Url-1 = {http://epubs.siam.org/doi/abs/10.1137/110830629}}

@article{Jo1996FD,
	Author = {Jo, C., Shin, C. and Suh, J.},
	Doi = {10.1190/1.1443979},
	Eprint = {http://library.seg.org/doi/pdf/10.1190/1.1443979},
	Journal = {GEOPHYSICS},
	Number = {2},
	Pages = {529-537},
	Title = {{A}n {O}ptimal 9-{P}oint, {F}inite-{D}ifference, {F}requency-{S}pace, 2-D {S}calar {W}ave {E}xtrapolator},
	Url = {http://library.seg.org/doi/abs/10.1190/1.1443979},
	Volume = {61},
	Year = {1996},
	Bdsk-Url-1 = {http://library.seg.org/doi/abs/10.1190/1.1443979},
	Bdsk-Url-2 = {http://dx.doi.org/10.1190/1.1443979}}

@article{Choi01102008,
	Abstract = {A proper scaling method allows us to find better solutions in waveform inversion, and it can also provide better images in true-amplitude migration methods based on a least-squares method. For scaling the gradient of a misfit function, we define a new pseudo-Hessian matrix by combining the conventional pseudo-Hessian matrix with amplitude fields. Because the conventional pseudo-Hessian matrix is assumed to neglect the zero-lag autocorrelation terms of impulse responses in the approximate Hessian matrix of the Gauss--Newton method, it has certain limitations in scaling the gradient of a misfit function relative to the approximate Hessian matrix. To overcome these limitations, we introduce amplitude fields to the conventional pseudo-Hessian matrix, and the new pseudo-Hessian matrix is applied to the frequency-domain elastic full waveform inversion. This waveform inversion algorithm follows the conventional procedures of waveform inversion using the backpropagation algorithm. A conjugate-gradient method is employed to derive an optimized search direction, and a backpropagation algorithm is used to calculate the gradient of the misfit function. The source wavelet is also estimated simultaneously with elastic parameters. The new pseudo-Hessian matrix can be calculated without the extra computational costs required by the conventional pseudo-Hessian matrix, because the amplitude fields can be readily extracted from forward modeling. Synthetic experiments show that the new pseudo-Hessian matrix provides better results than the conventional pseudo-Hessian matrix, and thus, we believe that the new pseudo-Hessian matrix is an alternative to the approximate Hessian matrix of the Gauss--Newton method in waveform inversion.},
	Author = {Choi, Yunseok and Min, Dong-Joo and Shin, Changsoo},
	Doi = {10.1785/0120070179},
	Eprint = {http://www.bssaonline.org/content/98/5/2402.full.pdf+html},
	Journal = {Bulletin of the Seismological Society of America},
	Number = {5},
	Pages = {2402-2415},
	Title = {{F}requency-Domain {E}lastic {F}ull {W}aveform {I}nversion {U}sing the {N}ew {P}seudo-{H}essian {M}atrix: {E}xperience of {E}lastic {M}armousi-2 {S}ynthetic {D}ata},
	Url = {http://www.bssaonline.org/content/98/5/2402.abstract},
	Volume = {98},
	Year = {2008},
	Bdsk-Url-1 = {http://www.bssaonline.org/content/98/5/2402.abstract},
	Bdsk-Url-2 = {http://dx.doi.org/10.1785/0120070179}}

@book{kaipio:2004,
	Author = {Kaipio, Jari and Somersalo, Erkki},
	Date-Modified = {2016-06-24 21:59:57 +0000},
	Day = {01},
	Edition = {1},
	Howpublished = {Hardcover},
	Isbn = {0387220739},
	Keywords = {bayes, inverse\_problems},
	Month = dec,
	Posted-At = {2009-04-21 12:23:18},
	Priority = {2},
	Publisher = {Springer},
	Title = {{{S}tatistical and {C}omputational {I}nverse {P}roblems (Applied Mathematical Sciences) (v. 160)}},
	Year = {2004},
	Bdsk-Url-1 = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20%5C&path=ASIN/0387220739}}

@conference{fang2014EAGEfuq,
	Author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann},
	Booktitle = {76th EAGE},
	Keywords = {EAGE, FWI, Markov chain Monte Carlo, randomized sources subsampling, Uncertainty quantification},
	Month = {01},
	Title = {{F}ast {U}ncertainty {Q}uantification for 2D {F}ull-waveform {I}nversion with {R}andomized {S}ource {S}ubsampling},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq.pdf},
	Year = {2014},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq.pdf}}

@techreport{tristan20133DFWI,
	Author = {Tristan van Leeuwen and Felix J. Herrmann},
	Keywords = {Optimization, private, waveform inversion},
	Month = {04},
	Organization = {UBC},
	Publisher = {UBC},
	Title = {3{D} {F}REQUENCY-{D}OMAIN {S}EISMIC {I}NVERSION WITH {C}ONTROLLED {S}LOPPINESS.},
	Url = {https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnx0cmlzdGFudmFubGVldXdlbnxneDoyYTEzM2YwZjBjMGFkNzg2},
	Year = {2013},
	Bdsk-Url-1 = {https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnx0cmlzdGFudmFubGVldXdlbnxneDoyYTEzM2YwZjBjMGFkNzg2}}

@article{Li11TRfrfwi,
	Address = {University of British Columbia, Vancouver},
	Author = {Xiang Li and Aleksandr Y. Aravkin and Tristan van Leeuwen and Felix J. Herrmann},
	Doi = {10.1190/geo2011-0410.1},
	Journal = {Geophysics},
	Keywords = {Compressive Sensing, FWI, Optimization, SLIM},
	Month = {05},
	Number = {3},
	Pages = {A13-A17},
	Title = {{F}ast {R}andomized {F}ull-waveform {I}nversion with {C}ompressive {S}ensing},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2012/Li11TRfrfwi/Li11TRfrfwi.pdf},
	Volume = {77},
	Year = {2012},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2012/Li11TRfrfwi/Li11TRfrfwi.pdf},
	Bdsk-Url-2 = {http://dx.doi.org/10.1190/geo2011-0410.1}}

@article{Aravkin11TRridr,
	Address = {University of British Columbia, Vancouver},
	Author = {Aleksandr Y. Aravkin and Michael P. Friedlander and Felix J. Herrmann and Tristan van Leeuwen},
	Doi = {10.1007/s10107-012-0571-6},
	Journal = {Mathematical Programming},
	Keywords = {FWI, Inverse problems, Optimization, Robust estimation, Seismic inversion, Stochastic optimization},
	Month = {8},
	Number = {1},
	Pages = {101-125},
	Title = {{R}obust {I}nversion, {D}imensionality {R}eduction, and {R}andomized {S}ampling},
	Url = {http://www.springerlink.com/content/35rwr101h5736340/},
	Url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/MathematicalProgramming/aravkin2012MPrid/aravkin2012MPrid.pdf},
	Volume = {134},
	Year = {2012},
	Bdsk-Url-1 = {http://www.springerlink.com/content/35rwr101h5736340/},
	Bdsk-Url-2 = {http://dx.doi.org/10.1007/s10107-012-0571-6}}

@article{aravkin2012IPNuisance,
	Author = {Aleksandr Y. Aravkin and Tristan van Leeuwen},
	Doi = {10.1088/0266-5611/28/11/115016},
	Journal = {Inverse Problems},
	Keywords = {full waveform inversion, students t, variance},
	Month = {10},
	Number = {11},
	Title = {{E}stimating {N}uisance {P}arameters in {I}nverse {P}roblems},
	Url = {http://arxiv.org/abs/1206.6532},
	Url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InverseProblems/2012/aravkin2012IPNuisance/aravkin2012IPNuisance.pdf},
	Volume = {28},
	Year = {2012},
	Bdsk-Url-1 = {http://arxiv.org/abs/1206.6532},
	Bdsk-Url-2 = {http://dx.doi.org/10.1088/0266-5611/28/11/115016}}

@conference{vanleeuwen2012EAGEcarpcg,
	Author = {Tristan van Leeuwen and Dan Gordon and Rachel Gordon and Felix J. Herrmann},
	Booktitle = {EAGE},
	Keywords = {EAGE, Helmholtz equation, precondition},
	Month = {06},
	Organization = {EAGE},
	Presentation = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/vanleeuwen2012EAGEcarpcg/vanleeuwen2012EAGEcarpcg_pres.pdf},
	Publisher = {EAGE},
	Title = {{P}reconditioning the {H}elmholtz {E}quation via {R}ow-projections},
	Url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/vanleeuwen2012EAGEcarpcg/vanleeuwen2012EAGEcarpcg.pdf},
	Url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=58891},
	Year = {2012},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/vanleeuwen2012EAGEcarpcg/vanleeuwen2012EAGEcarpcg.pdf}}

@unpublished{zfang2014SEGsqn,
	Abstract = {In this work we propose a stochastic quasi-Newton Markov chain Monte Carlo (McMC) method to quantify the uncertainty of full-waveform inversion (FWI). We formulate the uncertainty quantification problem in the framework of the Bayesian inference, which formulates the posterior probability as the conditional probability of the model given the observed data. The Metropolis-Hasting algorithm is used to generate samples satisfying the posterior probability density function (pdf) to quantify the uncertainty. However it suffers from the challenge to construct a proposal distribution that simultaneously provides a good representation of the true posterior pdf and is easy to manipulate. To address this challenge, we propose a stochastic quasi-Newton McMC method, which relies on the fact that the Hessian of the deterministic problem is equivalent to the inverse of the covariance matrix of the posterior pdf. The l-BFGS (limited-memory Broyden{\textendash}Fletcher{\textendash}Goldfarb{\textendash}Shanno) Hessian is used to approximate the inverse of the covariance matrix efficiently, and the randomized source sub-sampling strategy is used to reduce the computational cost of evaluating the posterior pdf and constructing the l-BFGS Hessian. Numerical experiments show the capability of this stochastic quasi-Newton McMC method to quantify the uncertainty of FWI with a considerable low cost.},
	Author = {Zhilong Fang, Felix J. Herrmann and Chia Ying Lee},
	Keywords = {FWI, McMC, private, quasi-Newton, Uncertainty quantification},
	Month = {04},
	Organization = {UBC},
	Publisher = {UBC},
	Title = {A stochastic quasi-Newton {McMC} method for uncertainty quantification of full-waveform inversion},
	Url = {https://www.slim.eos.ubc.ca/Publications/Private/Conferences/SEG/2014/zfang2014SEGsqn/zfang2014SEGsqn.html},
	Year = {2014},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Private/Conferences/SEG/2014/zfang2014SEGsqn/zfang2014SEGsqn.html}}

@unpublished{vanLeeuwen2014pmpde,
	Abstract = {The invention relates to a partial-differential-equation (PDE) constrained optimization method and especially to a partial-differential-equation (PDE) constrained optimization method for geophysical prospecting.},
	Author = {Tristan van Leeuwen and Felix J. Herrmann},
	Keywords = {FWI, Optimization, patent, private},
	Month = {04},
	Note = {Patent filed on April 22, 2014. PCT International Application.},
	Organization = {UBC},
	Publisher = {UBC},
	Title = {A penalty method for {PDE}-constrained optimization ({CONFIDENTIAL})},
	Url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2014/vanLeeuwen2014pmpde/vanLeeuwen2014pmpde.pdf},
	Year = {2014},
	Bdsk-Url-1 = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2014/vanLeeuwen2014pmpde/vanLeeuwen2014pmpde.pdf}}