@article {yang2020tdsp,
	title = {Time-domain sparsity promoting least-squares reverse time migration with source estimation},
	journal = {Geophysical Prospecting},
	volume = {68},
	number = {9},
	year = {2020},
	note = {(Geophysical Prospecting)},
	month = {08},
	pages = {2697-2711},
	abstract = {Least-squares reverse time migration is well-known for its capability to generate artifact-free true-amplitude subsurface images through fitting observed data in the least-squares sense. However, when applied to realistic imaging problems, this approach is faced with issues related to overfitting and excessive computational costs induced by many wave-equation solves. The fact that the source function is unknown complicates this situation even further. Motivated by recent results in stochastic optimization and transform-domain sparsity-promotion, we demonstrate that the computational costs of inversion can be reduced significantly while avoiding imaging artifacts and restoring amplitudes. While powerfull, these new approaches do require accurate information on the source-time function, which is often lacking. Without this information, the imaging quality deteriorates rapidly. We address this issue by presenting an approach where the source-time function is estimated on the fly through a technique known as variable projection. Aside from introducing negligible computational overhead, the proposed method is shown to perform well on imaging problems with noisy data and problems that involve complex settings such as salt. In either case, the presented method produces high resolution high-amplitude fidelity images including an estimates for the source-time function. In addition, due to its use of stochastic optimization, we arrive at these images at roughly one to two times the cost of conventional reverse time migration involving all data.},
	keywords = {penalty, source estimation, sparsity inversion},
	doi = {10.1111/1365-2478.13021},
	url = {https://slim.gatech.edu/Publications/Public/Journals/GeophysicalProspecting/2020/yang2020tdsp/yang2020tdsp.html},
	author = {Mengmeng Yang and Zhilong Fang and Philipp A. Witte and Felix J. Herrmann}
}
@mastersthesis {fang2018THsea,
	title = {Source estimation and uncertainty quantification for wave-equation based seismic imaging and inversion},
	year = {2018},
	note = {(PhD)},
	month = {04},
	school = {The University of British Columbia},
	type = {phd},
	address = {Vancouver},
	abstract = {In modern seismic exploration, wave-equation-based inversion and imaging approaches are widely employed for their potential of creating high-resolution subsurface images from seismic data by using the wave equation to describe the underlying physical model of wave propagation. Despite their successful practical applications, some key issues remain unsolved, including local minima, unknown sources, and the largely missing uncertainty analyses for the inversion. This thesis aims to address the following two aspects: to perform the inversion without prior knowledge of sources, and to quantify uncertainties in the inversion. The unknown source can hinder the success of wave-equation-based ap- proaches. A simple time shift in the source can lead to misplaced reflectors in linearized inversions or large disturbances in nonlinear problems. Unfortunately, accurate sources are typically unknown in real problems. The first major contribution of this thesis is, given the fact that the wave equation linearly depends on the sources, I have proposed on-the-fly source estimation techniques for the following wave-equation-based approaches: (1) time-domain sparsity-promoting least-squares reverse-time migration; and (2) wavefield-reconstruction inversion. Considering the linear dependence of the wave equation on the sources, I project out the sources by solving a linear least-squares problem, which enables us to conduct successful wave- equation-based inversions without prior knowledge of the sources. Wave-equation-based approaches also produce uncertainties in the resulting velocity model due to the noisy data, which would influence the subsequent exploration and financial decisions. The difficulties related to practical uncertainty quantification lie in: (1) expensive computation related to wave-equation solves, and (2) the nonlinear parameter-to-data map. The second major contribution of this thesis is the proposal of a computationally feasible Bayesian framework to analyze uncertainties in the resulting velocity models. Through relaxing the wave-equation constraints, I obtain a less nonlinear parameter-to-data map and a posterior distribution that can be adequately approximated by a Gaussian distribution. I derive an implicit formulation to construct the covariance matrix of the Gaussian distribution, which allows us to sample the Gaussian distribution in a computationally efficient manner. I demonstrate that the proposed Bayesian framework can provide adequately accurate uncertainty analyses for intermediate to large- scale problems with an acceptable computational cost.},
	keywords = {FWI, LS-RTM, PhD, source estimation, UQ, WRI},
	url = {https://slim.gatech.edu/Publications/Public/Thesis/2018/fang2018THsea/fang2018THsea.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Thesis/2018/fang2018THsea/fang2018THsea_pres.pdf},
	author = {Zhilong Fang}
}
@article {fang2017sewri,
	title = {Source estimation for wavefield-reconstruction inversion},
	journal = {Geophysics},
	volume = {83},
	number = {4},
	year = {2018},
	note = {(Geophysics)},
	pages = {R345-R359},
	abstract = {Source estimation is essential to all the wave-equation-based seismic inversions, including full-waveform inversion and the recently proposed wavefield-reconstruction inversion. When the source estimation is inaccurate, errors will propagate into the predicted data and introduce additional data misfit. As a consequence, inversion results that minimize this data misfit may become erroneous. To mitigate the errors introduced by the incorrect and pre-estimated sources, an embedded procedure that updates sources along with medium parameters is necessary for the inversion. So far, such a procedure is still missing in the context of wavefield-reconstruction inversion, a method that is, in many situations, less prone to local minima related to the so-called cycle skipping, compared to full-waveform inversion through exact data-fitting. While wavefield-reconstruction inversion indeed helps to mitigate issues related to cycle skipping by extending the search space with wavefields as auxiliary variables, it relies on having access to the correct source functions. In this paper, we remove the requirement of having the accurate source functions by proposing a source estimation technique specifically designed for wavefield-reconstruction inversion. To achieve this task, we consider the source functions as unknown variables and arrive at an objective function that depends on the medium parameters, wavefields, and source functions. During each iteration, we apply the so-called variable projection method to simultaneously project out the source functions and wavefields. After the projection, we obtain a reduced objective function that only depends on the medium parameters and invert for the unknown medium parameters by minimizing this reduced objective. Numerical experiments illustrate that this approach can produce accurate estimates of the unknown medium parameters without any prior information of the source functions.},
	keywords = {estimation, FWI, source, variable projection, WRI},
	doi = {10.1190/geo2017-0700.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/fang2017sewri/fang2017sewri.html},
	author = {Zhilong Fang and Rongrong Wang and Felix J. Herrmann}
}
@article {fang2017uqfip,
	title = {Uncertainty quantification for inverse problems with weak partial-differential-equation constraints},
	journal = {Geophysics},
	volume = {83},
	number = {6},
	year = {2018},
	note = {(Geophysics)},
	pages = {R629-R647},
	abstract = {In a statistical inverse problem, the objective is a complete statistical description of unknown parameters from noisy observations in order to quantify uncertainties of the parameters of interest. We consider inverse problems with partial-differential-equation-constraints, which are applicable to a variety of seismic problems. Bayesian inference is one of the most widely-used approaches to precisely quantify statistics through a posterior distribution, incorporating uncertainties in observed data, modeling kernel, and prior knowledge of the parameters. Typically when formulating the posterior distribution, the partial-differential-equation-constraints are required to be exactly satisfied, resulting in a highly nonlinear forward map and a posterior distribution with many local maxima. These drawbacks make it difficult to find an appropriate approximation for the posterior distribution. Another complicating factor is that traditional Markov chain Monte Carlo methods are known to converge slowly for realistically sized problems. In this work, we relax the partial-differential-equation-constraints by introducing an auxiliary variable, which allows for Gaussian deviations in the partial-differential-equations. Thus, we obtain a new bilinear posterior distribution consisting of both data and partial-differential-equation misfit terms. We illustrate that for a particular range of variance choices for the partial-differential-equation misfit term, the new posterior distribution has fewer modes and can be well-approximated by a Gaussian distribution, which can then be sampled in a straightforward manner. Since it is prohibitively expensive to explicitly construct the dense covariance matrix of the Gaussian approximation for intermediate to large-scale problems, we present a method to implicitly construct it, which enables efficient sampling. We apply this framework to two-dimensional seismic inverse problems with 1,800 and 92,455 unknown parameters. The results illustrate that our framework can produce comparable statistical quantities to those produced by conventional Markov chain Monte Carlo type methods while requiring far fewer partial-differential-equation solves, which are the main computational bottlenecks in these problems.},
	keywords = {acoustic, FWI, UQ, weak-constraint},
	doi = {10.1190/geo2017-0824.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2018/fang2017uqfip/fang2017uqfip.html},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@conference {fang2017AIPepm,
	title = {An efficient penalty method for PDE-constrained optimization problem with source estimation and stochastic optimization},
	booktitle = {Applied Inverse Problems Annual Conference Proceedings},
	year = {2017},
	note = {(AIP, Hangzhou)},
	month = {05-06},
	pages = {40},
	abstract = {Many inverse problems in applied science and engineering can be written as PDE-constrained optimization problems. Typically, one is interested in estimating the unknown medium parameters of the PDE describing the system, given multi-source measurements of the fields in the true medium. The objective function is often very oscillatory, which gives rise to problematic local minima, and merely evaluating one or more of its derivatives requires solving multiple PDEs, which are expensive. In many applications such as seismic exploration, only the source location is available and the source signature itself is unknown, which results in erroneous estimated parameters. In this work, we propose a penalty method to solve the PDE-constrained optimization problem, whereby we replace the PDE constraints by a two-norm penalty term, which leads to a bi-linear optimization problem with a much less oscillatory objective function. By applying the variable projection method, we develop an efficient method to invert the unknown parameters and source signatures simultaneously. To reduce the computational cost, we randomly select a subset of all data to update the parameters at each iteration. Numerical examples demonstrate that this method is less prone to becoming stuck in harmful local minima and is able to successfully invert the unknown parameters and source signature with less computational cost compared to using all the data.},
	keywords = {Optimization, penalty method, source estimation, Stochastic optimization},
	url = {http://aip2017.csp.escience.cn/dct/page/1},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/AIP/2017/fang2017AIPepm/fang2017AIPepm_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann}
}
@presentation {fang2017SINBADFpfg,
	title = {PDE-free Gauss-Newton Hessian for Wavefield Reconstruction Inversion},
	year = {2017},
	publisher = {SINBAD},
	abstract = {In this work, we present a PDE-free Gauss-Newton Hessian for Wavefield Reconstruction Inversion. With this PDE-free Gauss-Newton Hessian, we can compute matrix-vector products without additional PDE solves. Thus, we are able to use the second order optimization method Gauss-Newton method with a roughly equal computational cost of first-order methods such as the gradient-descent method.},
	keywords = {Presentation, SINBAD, SINBADFALL2017, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2017/Fall/fang2017SINBADFpfg/fang2017SINBADFpfg.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2017/Fall/fang2017SINBADFpfg/fang2017SINBADFpfg.mov},
	author = {Zhilong Fang and Felix J. Herrmann}
}
@conference {fang2017WAVESuqf,
	title = {Uncertainty quantification for inverse problems with a weak wave-equation constraint},
	booktitle = {WAVES 2017 {\textendash}- 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation},
	year = {2017},
	note = {(WAVES, Minneapolis)},
	month = {05},
	pages = {127-128},
	abstract = {In this work, we present a new posterior distribution to quantify uncertainties in solutions of wave-equation based inverse problems. By introducing an auxiliary variable for the wavefields, we weaken the strict wave-equation constraint used by conventional Bayesian approaches. With this weak constraint, the new posterior distribution is a bi-Gaussian distribution with respect to both model parameters and wavefields, which can be directly sampled by the Gibbs sampling method.},
	keywords = {constraint, Gibbs sampling, uncertainty, wave equation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/WAVES/2017/fang2017WAVESuqf/fang2017WAVESuqf.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/WAVES/2017/fang2017WAVESuqf/fang2017WAVESuqf_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Rachel Kuske and Felix J. Herrmann}
}
@presentation {fang2016SINBADFeaq,
	title = {Efficient approach for quantifying uncertainty of wavefield reconstruction inversion},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Due to the noisy measurements, sparse observations, uncertain forward models, and uncertain prior parameter information, significant uncertainty arises in the inverted velocity model of full-waveform inversion. The uncertain velocity model leads to the uncertainty in the correct positions of events in the migrated images, which subsequently influences the important decision-makings in the oil and gas exploration and production (E\&P) business such as the drilling decisions and economic evaluations. Therefore, understanding the uncertainty in the inverted velocity model is essential for the mitigation of the E\&P risks. A common approach to accounting for uncertainty in seismic inverse problems like full-waveform inversion is to describe the unknown parameters in probabilistic by the Bayesian inference. The Bayesian inference aims at incorporating the information from the data, physics, and one{\textquoteright}s prior knowledge to formulate a posterior distribution function to characterize the statistical properties of the unknown parameters. However, characterizing uncertainties for the large-scale full-waveform inversion in seismic exploration is prohibitively expensive since it requires thousands of function evaluations to sample the parameter space (e.g. via Markov chain Monte Carlo sampling). Apart from this, the existing methods of uncertainty quantification for full-waveform inversion is through applying the Bayesian inference to the standard method such as the adjoint-state method with the assumption that no uncertainty arises from the wave-equation. However, the wave-equation may not be able to reflect all the physics, hence, incorporating uncertainty from wave-equation might benefit the quantification of uncertainty. In this work, we propose to perform uncertainty quantification using the recently proposed wavefield reconstruction inversion technique, which gives an appropriate framework to tackle the uncertainty in wave-equation solve by adding the misfit of the wave-equation to the likelihood distribution and relaxing the wave-equation constraint. Moreover, we propose a computationally feasible approach to analyzing the posterior distribution of the wavefield reconstruction inversion method, which incorporates a Gaussian distribution that approximates the posterior distribution with an optimization-driven Gaussian simulator to sample the Gaussian distribution efficiently. Numerical examples show that comparing to the McMC method, our approach is able to produce comparable results with less computational cost. This is joint work with Curt da Silva and Rachel Kuske.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/fang2016SINBADFeaq/fang2016SINBADFeaq.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/fang2016SINBADFeaq/fang2016SINBADFeaq.mov},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann and Rachel Kuske}
}
@conference {chai2016SEGlbm,
	title = {A linearized Bregman method for compressive waveform inversion},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1449-1454},
	abstract = {We present an implementation of a recently developed and relatively simple linearized Bregman method to solve the large-scale $\ell_1$-norm sparsity-promoting Gauss-Newton (GN) full-waveform inversion (FWI) problem. Numerical experiments demonstrate that: the much simpler linearized Bregman method does a comparable and even superior job compared to a state-of-the-art $\ell_1$-norm solver SPGL1, which is previously used in the modified Gauss-Newton FWI; the linearized Bregman method is also more efficient (faster) than SPGL1; the FWI result with the linearized Bregman method solving $\ell_1$-norm sparsity-promoting problems to get the model updates is better than that obtained by solving $\ell_2$-norm least-squares problems to get the model updates.},
	keywords = {Bregman method, SEG, waveform inversion},
	doi = {10.1190/segam2016-13848105.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/chai2016SEGlbm/chai2016SEGlbm.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/chai2016SEGlbm/chai2016SEGlbm_pres.pdf},
	author = {Xintao Chai and Mengmeng Yang and Philipp A. Witte and Rongrong Wang and Zhilong Fang and Felix J. Herrmann}
}
@presentation {yang2016SINBADFtds,
	title = {Time-domain sparsity-promoting least-squares migration with source estimation},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Compared to traditional reverse-time migration (RTM), least-squares RTM (LS_RTM) is able to obtain true amplitude images as solutions of $\ell_2$-norm minimization problems by fitting the synthetic and observed reflection data. The shortcoming is that solutions of these $\ell_2$ problems tend to be overfitted and computationally too expensive. By working with randomized subsets of data only, the computational costs of LS-RTM can be brought down to an acceptable level, producing artifact-free high-resolution images. By including on-the-fly source-time function estimation into the method of Linearized Bregman (LB), we tackle the open issues of these "compressive imaging" in the aspects of algorithmic complexity of solver, guaranteed convergence and source estimation. We also inbvestigate whether the algorithm can be accelerated when we combine LB with the Nesterov method. Application of our algorithm on a 2D synthetic shows that we are able to get high-resolution images, with accurate estimates of the wavelet, for one single data pass.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/yang2016SINBADFtds/yang2016SINBADFtds.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/yang2016SINBADFtds/yang2016SINBADFtds.m4v},
	author = {Mengmeng Yang and Philipp A. Witte and Zhilong Fang and Felix J. Herrmann}
}
@conference {yang2016SEGtds,
	title = {Time-domain sparsity-promoting least-squares migration with source estimation},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {4225-4229},
	abstract = {Traditional reverse-time migration (RTM) gives images with wrong amplitudes and low resolution. Least-squares RTM (LS-RTM) on the other hand, is capable of obtaining true-amplitude images as solutions of $\ell_2$-norm norm minimization problems by fitting the synthetic and observed reflection data. The shortcoming of this approach is that solutions of these $\ell_2$ problems are typically smoothed, tend to be overfitted, and computationally too expensive because it requires compared to standard RTM too many iterations. By working with randomized subsets of data only, the computational costs of LS-RTM can be brought down to an acceptable level while producing artifact-free high-resolution images without overfitting the data. While initial results of these "compressive imaging" methods were encouraging various open issues remain including guaranteed convergence, algorithmic complexity of the solver, and lack of on-the-fly source estimation for LS-RTMs with wave-equation solvers based on time-stepping. By including on-the-fly source-time function estimation into the method of Linearized Bregman (LB), on which we reported before, we tackle all these issues resulting in a easy-to-implement algorithm that offers flexibility in the trade-off between the number of iterations and the number of wave-equation solves per iteration for a fixed total number of wave-equation solves. Application of our algorithm on a 2D synthetic shows that we are able to obtain high-resolution images, including accurate estimates of the wavelet, for a single pass through the data. The produced image, which is by virtue of the inversion deconvolved with respect to the wavelet, is roughly of the same quality as the image obtained given the correct source function.},
	keywords = {least squares, RTM, SEG, source estimation, time domain},
	doi = {10.1190/segam2016-13850609.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/yang2016SEGtds/yang2016SEGtds.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/yang2016SEGtds/yang2016SEGtds_pres.pdf},
	author = {Mengmeng Yang and Philipp A. Witte and Zhilong Fang and Felix J. Herrmann}
}
@conference {fang2016SEGuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion using a PDE free semidefinite Hessian and randomize-then-optimize method},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {1390-1394},
	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.},
	keywords = {SEG, Uncertainty quantification, WRI},
	doi = {10.1190/segam2016-13879108.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/fang2016SEGuqw/fang2016SEGuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Tristan van Leeuwen and Felix J. Herrmann}
}
@techreport {peters2015SEGrwi,
	title = {Regularizing waveform inversion by projections onto convex sets {\textendash}- application to the 2D Chevron 2014 synthetic blind-test dataset},
	number = {TR-EOAS-2015-7},
	year = {2015},
	month = {06},
	institution = {UBC},
	abstract = {A framework is proposed for regularizing the waveform inversion problem by projections onto intersections of convex sets. Multiple pieces of prior information about the geology are represented by multiple convex sets, for example limits on the velocity or minimum smoothness conditions on the model. The data-misfit is then minimized, such that the estimated model is always in the intersection of the convex sets. Therefore, it is clear what properties the estimated model will have at each iteration. This approach does not require any quadratic penalties to be used and thus avoids the known problems and limitations of those types of penalties. It is shown that by formulating waveform inversion as a constrained problem, regularization ideas such as Tikhonov regularization and gradient filtering can be incorporated into one framework. The algorithm is generally applicable, in the sense that it works with any (differentiable) objective function and does not require significant additional computation. The method is demonstrated on the inversion of the 2D marine isotropic elastic synthetic seismic benchmark by Chevron using an acoustic modeling code. To highlight the effect of the projections, we apply no data pre-processing.},
	keywords = {blind-test, projection, regularization, SEG, Wavefield Reconstruction Inversion, waveform inversion},
	url = {https://slim.gatech.edu/Publications/Public/TechReport/2015/peters2015SEGrwi/peters2015SEGrwi.html},
	author = {Bas Peters and Zhilong Fang and Brendan R. Smithyman and Felix J. Herrmann}
}
@presentation {dasilva2015SINBADFsss,
	title = {Scaling SINBAD software to 3-D on Yemoja},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We present early results on the scalability of SINBAD{\textquoteright}s wavefield reconstruction and wave-equation based inversion technologies on Yemoja, a 17k core cluster made available to us by BG Group at SENAI CIMATEC Supercomputing Centre in Brazil.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/dasilva2015SINBADFsss/dasilva2015SINBADFsss.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/dasilva2015SINBADFsss/dasilva2015SINBADFsss.mov},
	author = {Curt Da Silva and Haneet Wason and Mathias Louboutin and Bas Peters and Shashin Sharan and Zhilong Fang and Felix J. Herrmann}
}
@conference {fang2015EAGEsew,
	title = {Source estimation for Wavefield Reconstruction Inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2015},
	note = {(EAGE, Madrid)},
	month = {06},
	abstract = {Wavefield reconstruction inversion is a new approach to waveform based inversion that helps overcome the {\textquoteleft}cycle skipping{\textquoteright} problem. However, like most waveform based inversion methods, wavefield reconstruction inversion also requires good source wavelets. Without correct source wavelets, wavefields cannot be reconstructed correctly and the velocity model cannot be updated correctly neither. In this work, we propose a source estimation method for wavefield reconstruction inversion based on the variable projection method. In this method, we reconstruct wavefields and estimate source wavelets simultaneously by solving an extended least-squares problem, which contains source wavelets. This approach does not increase the computational cost compared to conventional wavefield reconstruction inversion. Numerical results illustrates with our source estimation method we are able to recover source wavelets and obtain inversion results that are comparable to results obtained with true source wavelets.},
	keywords = {EAGE, source estimation, WRI},
	doi = {10.3997/2214-4609.201412588},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEsew/fang2015EAGEsew.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEsew/fang2015EAGEsew_pres.pdf},
	author = {Zhilong Fang and Felix J. Herrmann}
}
@conference {fang2015IIPFWIsoa,
	title = {Stochastic optimization and its application to seismic inversion},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	keywords = {FWI, inversion, Optimization, stochastic},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/fang2015IIPFWIsoa/fang2015IIPFWIsoa_pres.pdf},
	author = {Zhilong Fang and Tim Burgess and Felix J. Herrmann}
}
@conference {fang2015EAGEuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2015},
	note = {(EAGE, Madrid)},
	month = {06},
	abstract = {In this work, we propose a method to quantify the uncertainty of wavefield reconstruction inversion under the framework of Bayesian inference. Unlike the conventional method using the wave equation as the forward mapping, we involve the wave equation misfit in the posterior distribution and propose a new posterior distribution. The negative log-likelihood of the new distribution is less oscillatory than that of the conventional posterior distribution, and its Gauss-Newton Hessian is a diagonal matrix that can be generated without any additional computational cost. We use the diagonal Gauss-Newton Hessian to derive an approximate Gaussian distribution at the maximum likelihood point to quantify the uncertainty. This method makes the uncertainty quantification for WRI computationally tractable and is able to provide reasonable uncertainty analysis based on our numerical results.},
	keywords = {EAGE, FWI, UQ, WRI},
	doi = {10.3997/2214-4609.201413198},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEuqw/fang2015EAGEuqw.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2015/fang2015EAGEuqw/fang2015EAGEuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@conference {fang2015IIPFWIuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	abstract = {In this work, we propose a method to quantify the uncertainty of wavefield reconstruction inversion under the framework of Bayesian inference. Unlike the conventional method using the wave equation as the forward mapping, we involve the wave equation misfit in the posterior distribution and propose a new posterior distribution. The negative log-likelihood of the new distribution is less oscillatory than that of the conventional posterior distribution, and its Gauss-Newton Hessian is a diagonal matrix that can be generated without any additional computational cost. We use the diagonal Gauss-Newton Hessian to derive an approximate Gaussian distribution at the maximum likelihood point to quantify the uncertainty. This method makes the uncertainty quantification for WRI computationally tractable and is able to provide reasonable uncertainty analysis based on our numerical results.},
	keywords = {FWI, uncertainty, WRI},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/fang2015IIPFWIuqw/fang2015IIPFWIuqw_pres.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@presentation {fang2015SINBADFuqw,
	title = {Uncertainty quantification for Wavefield-Reconstruction Inversion using a positive-definite approximated Hessian},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We analyze the Hessian of wavefield reconstruction inversion (WRI) and propose a new approximated Hessian. Instead of requiring PDE solves, the matrix-vector multiplication action of the approximate Hessian can be achieved with several matrix-vector multiplications. The diagonal part of the approximated Hessian can be also calculated without additional PDE solves. We apply this approximated Hessian to uncertainty quantification and obtain statistical parameters such as standard deviation and confidence interval. Numerical example illustrate the accuracy of the estimated Hessian and the feasibility of the method to quantify uncertainties.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFuqw/fang2015SINBADFuqw.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFuqw/fang2015SINBADFuqw.mov},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@conference {fang2015IIPFWIwri,
	title = {Wavefield Reconstruction Inversion with source estimation and its application to 2014 Chevron synthetic blind test dataset},
	booktitle = {Inaugural Full-Waveform Inversion Workshop},
	year = {2015},
	note = {(Natal, Brazil)},
	month = {08-09},
	abstract = {We present a robust wavefield reconstruction inversion with source estimation. The source wavelet is estimated with the reconstruction of wavefield simultaneously by solving an extended data augmentation problem. We apply this method to the 2014 Chevron synthetic blind test dataset and show the robustness of our method.},
	keywords = {FWI, source estimation, WRI},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/IIPFWI/fang2015IIPFWIwri/fang2015IIPFWIwri_pres.pdf},
	author = {Zhilong Fang and Felix J. Herrmann}
}
@presentation {fang2015SINBADwri,
	title = {Wavefield reconstruction inversion with source estimation and its application to 2014 Chevron synthetic blind test dataset},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We present a robust wavefield reconstruction inversion with source estimation. The source wavelet is estimated with the reconstruction of wavefield simultaneously by solving an extended data augmentation problem. We apply this method to the 2014 Chevron synthetic blind test dataset and show the robustness of our method.},
	keywords = {Presentation, SINBAD, SINBADSPRING2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Spring/fang2015SINBADwri/fang2015SINBADwri.pdf},
	author = {Zhilong Fang and Felix J. Herrmann}
}
@presentation {fang2015SINBADFwri,
	title = {Wavefield-reconstruction inversion with source estimation and minimum smoothness constraint {\textendash} application to the Chevron 2014 dataset},
	year = {2015},
	publisher = {SINBAD},
	abstract = {We present a robust wavefield reconstruction inversion with source estimation. The source wavelet is estimated with the reconstruction of wavefield simultaneously by solving an extended data-augmented problem. We apply this method to the 2014 Chevron synthetic blind test dataset and show the robustness of our method.},
	keywords = {Presentation, SINBAD, SINBADFALL2015, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFwri/fang2015SINBADFwri.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2015/Fall/fang2015SINBADFwri/fang2015SINBADFwri.mov},
	author = {Zhilong Fang and Bas Peters and Felix J. Herrmann}
}
@conference {lago2014CMCCRMN,
	title = {CRMN method for solving time-harmonic wave equation},
	booktitle = {Copper Mountain Conference},
	year = {2014},
	note = {(Copper Mountain)},
	month = {04},
	abstract = {We address the solution of PDEs associated with the wave propagation phenomena in heterogeneous media using CGMN method. It consists of a conjugate gradients method using Kaczmarz double sweeps as preconditioner. This preconditioner has the property of {\textquoteleft}{\textquoteleft}symmetrizing" the problem allowing short recursion and Lanczos type of Krylov methods to be employed. In this talk we propose the use of CR, a minimal residual Krylov method closely related to CG. We study the proposed method which we call CRMN and discuss crucial aspects as the behaviour of the norm of the residual and the norm of the true error. We also discuss the result of an inversion of a small subsample of the 3D velocity model SEG/EAGE Overthrust using frugal full-waveform inversion method with CRMN and CGMN for solving the associated PDE, showing a strong interest in studying the behaviour of CRMN for larger realistic cases.},
	keywords = {CGMN, CRMN, Forward Modelling, Helmholtz equation},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/CMC/2014/lago2014CMCCRMN.pdf},
	author = {Rafael Lago and Art Petrenko and Zhilong Fang and Felix J. Herrmann}
}
@conference {lago2014EAGEfst,
	title = {Fast solution of time-harmonic wave-equation for full-waveform inversion},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2014},
	month = {06},
	abstract = {For many full-waveform inversion techniques, the most computationally intensive step is the computation of a numerical solution for the wave equation on every iteration. In the frequency domain approach, this requires the solution of very large, complex, sparse, ill-conditioned linear systems. In this extended abstract we bring out attention specifically to CGMN method for solving PDEs, known for being flexible (i.e. it is able to treat equally acoustic data as well as visco-elastic or more complex scenarios) efficient with respect both to memory and computation time, and controllable accuracy of the final approximation. We propose an improvement for the known CGMN method by imposing a minimal residual condition, which incurs in one extra model vector storage. The resulting algorithm called CRMN enjoys several interesting properties as monotonically nonincreasing behaviour of the norm of the residual and minimal residual, guaranteeing optimal convergence for the relative residual criterion. We discuss numerical experiments both in an isolated PDE solve and also within the inversion procedure, showing that in a realistic scenario we can expect a speedup around 25\% when using CRMN rather than CGMN.},
	keywords = {CGMN, CRMN, EAGE, FWI, Time-Harmonic Wave Equation},
	doi = {10.3997/2214-4609.20140812},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/lago2014EAGEfst/lago2014EAGEfst.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/lago2014EAGEfst/lago2014EAGEfst_pres.pdf},
	author = {Rafael Lago and Art Petrenko and Zhilong Fang and Felix J. Herrmann}
}
@conference {fang2014EAGEfuq,
	title = {Fast uncertainty quantification for 2D full-waveform inversion with randomized source subsampling},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2014},
	month = {06},
	abstract = {Uncertainties arise in every area of seismic exploration, especially in full-waveform inversion, which is highly non-linear. In the framework of Bayesian inference, uncertainties can be analyzed by sampling the posterior probability density distribution with a Markov chain Monte-Carlo (McMC) method. We reduce the cost of computing the posterior distribution by working with randomized subsets of sources. These approximations, together with the Gaussian assumption and approximation of the Hessian, leads to a computational tractable uncertainty quantification. Application of this approach to a synthetic leads to standard deviations and confidence intervals that are qualitatively consistent with our expectations.},
	keywords = {EAGE, FWI, Markov chain Monte Carlo, randomized sources subsampling, Uncertainty quantification},
	doi = {10.3997/2214-4609.20140715},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq/fang2014EAGEfuq.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq/fang2014EAGEfuq_pres.pdf},
	author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann}
}
@presentation {fang2014SINBADsew,
	title = {Source estimation for WRI and its application},
	year = {2014},
	publisher = {SINBAD},
	keywords = {Presentation, SINBAD, SINBADFALL2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Fall/fang2014SINBADsew/fang2014SINBADsew.pdf},
	author = {Zhilong Fang and Xiang Li and Bas Peters and Felix J. Herrmann}
}
@techreport {zfang2014SEGsqn,
	title = {A stochastic quasi-Newton McMC method for uncertainty quantification of full-waveform inversion},
	number = {TR-EOAS-2014-6},
	year = {2014},
	month = {04},
	institution = {UBC},
	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.},
	keywords = {FWI, McMC, quasi-Newton, Uncertainty quantification},
	url = {https://slim.gatech.edu/Publications/Public/TechReport/2014/zfang2014SEGsqn/zfang2014SEGsqn.html},
	author = {Zhilong Fang and Felix J. Herrmann}
}
@presentation {fang2014SINBADsqn,
	title = {A stochastic quasi-Newton McMC method for uncertainty quantification of full-waveform inversion},
	year = {2014},
	publisher = {SINBAD},
	abstract = {In this work, we present a fast McMC method using the stochastic l-BFGS Hessian to quantify the uncertainty of full-waveform inversion. Using the stochastic l-BFGS Hessian, we do not need the assumption that the Hessian of data misfit is low rank and we also reduce the computational cost of estimating the Hessian. Numerical result shows the capability of this fast McMC method.},
	keywords = {Presentation, SINBAD, SINBADSPRING2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Spring/fang2014SINBADsqn/fang2014SINBADsqn.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Felix J. Herrmann}
}
@techreport {tschannen2014tdl,
	title = {Time domain least squares migration and dimensionality reduction},
	number = {TR-EOAS-2014-9},
	year = {2014},
	month = {06},
	institution = {UBC},
	abstract = {Least-squares seismic migration (LSM) is a wave equation based linearized inversion problem relying on the minimization of a least-squares misfit function, with respect to the medium perturbation, between recorded and modeled wavefields. Today{\textquoteright}s challenges in Hydrocarbon ex- ploration are to build high resolution images of more and more complicated geological reservoirs, which requires to handle very large systems of equations. The extreme size of the problem com- bined with the fact that it is ill-conditioned make LSM not yet feasible for industrial purposes. To overcome this "curse of dimensionality", dimension reduction and divide-and-conquer tech- niques that aim to decrease the computation time and the required memory, while conserving the image quality, have recently been developed. By borrowing ideas from stochastic optimiza- tion and compressive sensing, the imaging problem is reformulated as an L1-regularized, sparsity promoted LSM. The idea is to take advantage of the compressibility of the model perturbation in the curvelet domain and to work on series of smaller subproblems each involving a small ran- domized subset of data. We try two different subset sampling strategies, artificial randomized simultaneous sources experiments ("supershots") and drawing sequential shots firing at random source locations. These subsets are changed after each subproblem is solved. In both cases we obtain good migration results at significantly reduced computational cost. Application of these methods to a complicated synthetic model yields to encouraging results, underlining the usefulness of sparsity promotion and randomization in time stepping formulation.},
	keywords = {Compressive Sensing, sparsity promotion, Stochastic optimization, Wave equation migration},
	url = {https://slim.gatech.edu/Publications/Public/TechReport/2014/tschannen2014tdl/tschannen2014tdl.pdf},
	author = {Valentin Tschannen and Zhilong Fang and Felix J. Herrmann}
}
@presentation {fang2014SINBADuqw,
	title = {Uncertainty quantification for Wavefield Reconstruction Inversion},
	year = {2014},
	publisher = {SINBAD},
	abstract = {This talk discusses the uncertainty quantification for the wavefield reconstruction inversion. The Hessian and gradient of the WRI can be easily obtained once the wave field is generated. As a result, we use a Newton type McMC method to sample the posterior distribution. Statistical property such as mean, standard deviation, confidence interval are calculated based on the sampled distribution.},
	keywords = {Presentation, SINBAD, SINBADFALL2014, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2014/Fall/fang2014SINBADuqw/fang2014SINBADuqw.pdf},
	author = {Zhilong Fang and Chia Ying Lee and Curt Da Silva and Felix J. Herrmann and Rachel Kuske}
}
@presentation {fang2013SINBADp3dfwi,
	title = {Parallel 3D FWI with simultaneous shots},
	year = {2013},
	publisher = {SINBAD},
	abstract = {In this work, we manage to build the workflow for the parallel 3D full-waveform inversion. In the forward simulation part, we generate the Helmholtz matrix in parallel and use parallel CARPCG to solve the Helmholtz equation. In the inversion process, simultaneous shots are used to reduce the computational costs. Additionally, we propose a method to select the number of simultaneous shots and tolerance of CARPCG dynamically, to reach a compromise between the computational costs and accuracy.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/fang2013SINBADp3dfwi/fang2013SINBADp3dfwi.pdf},
	author = {Zhilong Fang and Tristan van Leeuwen and Felix J. Herrmann}
}
@presentation {fang2013SINBADsfwi,
	title = {Swift FWI},
	year = {2013},
	publisher = {SINBAD},
	abstract = {In 3D case, there is much more data and modeling is much more expensive. As a result, parallel computing is very important for 3D full-waveform inversion. Both domain decomposition and data decomposition need a large number of parallel computing resources. However, programs based on parallel MATLAB suffer from the limitation of licenses. In order to obtain a MATLAB licenses free solution, we use SWIFT, which is a fast and easy parallel scripting language. Once the original Mat file is complied to an executable file, SWIFT can run the code inside the executable file in parallel without using parallel MATLAB. We use the SWIFT to compute the object functions and gradients of different shots in parallel, and test the parallel 3D FWI code with overthrust data. This is joint work with Thomas Lai, Harsh Juneja, Bas Peters.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/fang2013SINBADsfwi/fang2013SINBADsfwi.pdf},
	author = {Zhilong Fang and Henryk Modzelewski and Tristan van Leeuwen and Thomas Lai and Harsh Juneja and Bas Peters and Felix J. Herrmann}
}
@presentation {fang2013SINBADuafwi,
	title = {Uncertainty analysis for FWI},
	year = {2013},
	publisher = {SINBAD},
	abstract = {Uncertainty analysis is important for seismic interpretation. Based on the framework of Bayesian, we can analyse different statistic parameters of our FWI result. However, directly sampling the posterior probability density function (pdf) is computationally intractable. In order to make this problem computationally tractable, in this work, we use Gaussian distribution approximation and low rank approximation to generate the posterior pdf. Simultaneous shots are also used to reduce the computational costs.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/fang2013SINBADuafwi/fang2013SINBADuafwi.pdf},
	author = {Zhilong Fang and Felix J. Herrmann}
}
