@article {vanLeeuwen2013GJImlm,
	title = {Mitigating local minima in full-waveform inversion by expanding the search space},
	journal = {Geophysical Journal International},
	volume = {195},
	year = {2013},
	month = {10},
	pages = {661-667},
	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.},
	doi = {10.1093/gji/ggt258},
	url = {https://slim.gatech.edu/Publications/Public/Journals/GeophysicalJournalInternational/2013/vanLeeuwen2013GJImlm/vanLeeuwen2013mlm.pdf},
	author = {Tristan van Leeuwen and Felix J. Herrmann}
}
@presentation {li2013SINBADmsd,
	title = {Model-space versus data-space FWI with the acoustic wave equation},
	year = {2013},
	publisher = {SINBAD},
	abstract = {Inverting data with elastic phases using an acoustic wave equation can lead to erroneous results, especially when the number of iterations is too high, which may lead to over fitting the data. Several approaches have been proposed to address this issue. Most commonly, people data- filtering operations that are aimed to deemphasize the elastic phases in the data in favor of the acoustic phases. Examples of this approach are nested loops over offset range and Laplace parameters. In this presentation, we discuss two complementary optimization-driven methods where the minimization process decides adaptively which of the data or model components are consistent with the objective. Specifically, we compare the s t misfit function as the data-space alternative and curvelet-domain sparsity promotion as the model-space alternative. Application of these two methods to a realistic synthetic lead to comparable results that we believe can be improved by combining these two methods.},
	keywords = {Presentation, SINBAD, SINBADFALL2013, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2013/Fall/li2013SINBADmsd/li2013SINBADmsd.pdf},
	author = {Xiang Li and Anais Tamalet and Tristan van Leeuwen and Felix J. Herrmann}
}
@conference {li2011SBGFmgnsu,
	title = {Modified Gauss-Newton with sparse updates},
	booktitle = {SBGF},
	year = {2011},
	publisher = {SBGF},
	organization = {SBGF},
	abstract = {Full-waveform inversion (FWI) is a data fitting procedure that relies on the collection of seismic data volumes and sophisticated computing to create high-resolution models.With the advent of FWI, the improvements in acquisition and inversion have been substantial, but these improvements come at a high cost because FWI involves extremely large multi-experiment data volumes. The main obstacle is the {\textquoteleft}curse of dimensionality{\textquoteright} exemplified by Nyquist{\textquoteright}s sampling criterion, which puts a disproportionate strain on current acquisition and processing systems as the size and desired resolution increases. In this paper, we address the {\textquoteleft}curse of dimensionality{\textquoteright} by using randomized dimensionality reduction of the FWI problem, coupled with a modified Gauss-Newton (GN) method designed to promote curvelet-domain sparsity of model updates. We solve for these updates using the spectral projected gradient method, implemented in the SPG1 software package. Our approach is successful because it reduces the size of seismic data volumes without loss of information. With this reduction, we can compute Gauss-Newton updates with the reduced data volume at the cost of roughly one gradient update for the fully sampled wavefield},
	keywords = {Full-waveform inversion, SBGF},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SBGF/2011/li11SBGFmgnsu/li11SBGFmgnsu.pdf},
	author = {Xiang Li and Felix J. Herrmann and Tristan van Leeuwen and Aleksandr Y. Aravkin}
}
@conference {herrmann2011SPIEmsp,
	title = {A modified, sparsity promoting, Gauss-Newton algorithm for seismic waveform inversion},
	booktitle = {Proc. SPIE},
	number = {81380V},
	year = {2011},
	month = {08},
	abstract = {Images obtained from seismic data are used by the oil and gas industry for geophysical exploration. Cutting-edge methods for transforming the data into interpretable images are moving away from linear approximations and high-frequency asymptotics towards Full Waveform Inversion (FWI), a nonlinear data-fitting procedure based on full data modeling using the wave-equation. The size of the problem, the nonlinearity of the forward model, and ill-posedness of the formulation all contribute to a pressing need for fast algorithms and novel regularization techniques to speed up and improve inversion results. In this paper, we design a modified Gauss-Newton algorithm to solve the PDE- constrained optimization problem using ideas from stochastic optimization and compressive sensing. More specifically, we replace the Gauss-Newton subproblems by randomly subsampled, -$\ell_1$ regularized subproblems. This allows us us significantly reduce the computational cost of calculating the updates and exploit the compressibility of wavefields in Curvelets. We explain the relationships and connections between the new method and stochastic optimization and compressive sensing (CS), and demonstrate the efficacy of the new method on a large-scale synthetic seismic example.},
	keywords = {Compressive Sensing, Full-waveform inversion, Optimization, SLIM},
	issn = {1},
	doi = {10.1117/12.893861},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SPIE/2011/herrmann2011SPIEmsp/herrmann2011SPIEmsp.pdf},
	author = {Felix J. Herrmann and Xiang Li and Aleksandr Y. Aravkin and Tristan van Leeuwen}
}
@conference {vanleeuwen2011SIAMGEOmawt,
	title = {Multiscale aspects of waveform tomography},
	booktitle = {SIAMGEO},
	year = {2011},
	publisher = {SIAM GeoSciences 2011},
	organization = {SIAM GeoSciences 2011},
	abstract = {We consider the inference of medium velocity from transmitted acoustic waves. Typically, the measurements are done in a narrow frequency band. As a result the sensitivity of the data with respect to velocity perturbations varies dramatically with the scale of the perturbation. {\textquoteleft}Smooth{\textquoteright} perturbations will cause a phase shift, whereas perturbations that vary on the wavelength-scale cause amplitude variations. We investigate how to incorporate this scale dependent behavior in the formulation of the inverse problem.},
	keywords = {Presentation},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SIAM/2011/vanleeuwen2011SIAMGEOmawt/vanleeuwen2011SIAMGEOmawt.pdf},
	author = {Tristan van Leeuwen and Wim A. Mulder}
}
