@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}
}
@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}
}
@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 {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 {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}
}
