@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 {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}
}
@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}
}
@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}
}
@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}
}
