An Effective Method for Parameter Estimation with PDE Constraints with Multiple Right-Hand Sides

Haber, Eldad; Chung, Matthias; Herrmann, Felix J.

doi:10.1137/11081126x

Cited by 110 publications

(165 citation statements)

References 31 publications

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“…This latter experiment also suggests that robust methods open a promising direction to design simultaneous source FWI methods (cf. Krebs et al, 2009;Haber et al, 2010;van Leeuwen et al, 2010) for marine acquisition. It is currently an open question how to use such methods for cases where we do not have full acquisition; specifically, the need to deal with the acquisition mask breaks these methods.…”

Section: Resultsmentioning

confidence: 99%

Robust full‐waveform inversion using the Student's t‐distribution

Aravkin

Leeuwen

Herrmann

2011

SEG Technical Program Expanded Abstracts 2011

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SUMMARYFull-waveform inversion (FWI) is a computational procedure to extract medium parameters from seismic data. Robust methods for FWI are needed to overcome sensitivity to noise and in cases where modeling is particularly poor or far from the real data generating process. We survey previous robust methods from a statistical perspective, and use this perspective to derive a new robust method by assuming the random errors in our model arise from the Student's t-distribution. We show that in contrast to previous robust methods, the new method progressively down-weighs large outliers, effectively ignoring them once they are large enough. This suggests that the new method is more robust and suitable for situations with very poor data quality or modeling. Experiments show that the new method recovers as well or better than previous robust methods, and can recover models with quality comparable to standard methods on noise-free data when some of the data is completely corrupted, and even when a marine acquisition mask is entirely ignored in the modeling. The ability to ignore a marine acquisition mask via robust FWI methods offers an opportunity for stochastic optimization methods in marine acquisition.

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Section: Resultsmentioning

confidence: 99%

Robust full‐waveform inversion using the Student's t‐distribution

Aravkin

Leeuwen

Herrmann

2011

SEG Technical Program Expanded Abstracts 2011

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“…with N being the number of shots; d i the recorded data; and u i ðσÞ the wavefield, for the ith shot (see also Haber et al [2012] and Aravkin et al [2012]). This is the cost function associated with all the shots.…”

Section: Source-encoding Fwimentioning

confidence: 99%

“…The second question has received much less attention, despite its importance . In an approach similar to Haber et al (2012), van Leeuwen et al (2011, we prove mathematically that the misfit function for source encoded inversion is an unbiased random estimation of the true misfit function when all the shots are incorporated. The same arguments hold true for the gradient of the misfit function.…”

Section: Introductionmentioning

confidence: 97%

“…Therefore, the cost of conventional FWI is proportional to the number of sources. In this paper, we consider a significant reduction of the calculation time using source encoding (Krebs et al, 2009;Li and Herrmann, 2010;Moghaddam and Herrmann, 2010;van Leeuwen et al, 2011;Haber et al, 2012;Li et al, 2012). Source encoding uses a linear combinations of all shots, with random weights assigned to each shot.…”

Section: Introductionmentioning

confidence: 99%

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A new optimization approach for source-encoding full-waveform inversion

et al. 2013

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Waveform inversion is the method of choice for determining a highly heterogeneous subsurface structure. However, conventional waveform inversion requires that the wavefield for each source is computed separately. This makes it very expensive for realistic 3D seismic surveys. Source-encoding waveform inversion, in which the sources are modeled simultaneously, is considerably faster than conventional waveform inversion but suffers from artifacts. These artifacts can partly be removed by assigning random weights to the source wavefields. We found that the misfit function, and therefore also its gradient, for source-encoding waveform inversion is an unbiased random estimation of the misfit function used in conventional waveform inversion. We found a new method of source-encoding waveform inversion that takes into account the random nature of the gradients used in the optimization. In this new method, the gradient at each iteration is a weighted average of past gradients such that the most recent gradients have the largest weights with exponential decay. This way we damped the random fluctuations of the gradient by incorporating information from the previous iterations. We compared this new method with existing source-encoding waveform inversion methods as well as conventional waveform inversion and found that the model misfit reduction is faster and smoother than those of existing source-encoding waveform inversion methods, and it approaches the model misfit reduction obtained in conventional waveform inversion.

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“…Among existing economic approaches, there is the fast method with source encoding, advocated by Krebs et al (2009) and later by van Leeuwen et al (2011) and Haber et al (2012), which offer no or practically limited guarantees of convergence. We overcome this problem by controlling the error by increasing the accuracy and sample size as needed.…”

Section: F U L L W a V E F O R M I N V E R S I O Nmentioning

confidence: 99%

Frugal full-waveform inversion: From theory to a practical algorithm

et al. 2013

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A s conventional oil and gas fields are maturing, our profession is challenged to come up with the nextgeneration of more and more sophisticated exploration tools. In exploration seismology this trend has led to the emergence of wave-equation-based inversion technologies such as reverse time migration and full-waveform inversion. While significant progress has been made in wave-equation-based inversion, major challenges remain in the development of robust and computationally feasible workflows that give reliable results in geophysically challenging areas that may include ultralow shear-velocity zones or high-velocity salt. Moreover, subsalt production carries risks that need mitigation, which raises the bar from creating subsalt images to inverting for subsalt overpressure.Among the many challenges that wave-equation-based inversion faces, we focus on reducing the excessive computational costs of full-waveform inversion (FWI). We accomplish these cost reductions by using modern techniques from machine learning and compressive sensing. Contrary to many implementations of wave-equation-based inversion, we propose a methodology where we do not insist on using all data (i.e., looping over all sources) to calculate the velocity model updates. Instead, we rely on a formulation that calls for more data and more accuracy in the wave simulations only as needed by the inversion. This dynamic selection of data and accuracy leads to major savings, in particular in the beginning when we are far from reaching the solution. Because this approach reduces the computational costs significantly, we open the way to test different scenarios for the purpose of quality control or to include more sophisticated regularization. Without this cost reduction, both would be computationally infeasible.The article is organized as follows. First, we introduce the basics of full-waveform inversion, followed by stochastic sampling techniques to reduce the computational costs. Next, we propose an adaptive scheme that dynamically selects the required sample size-i.e., the number of source experiments that partake in the inversion, and forward modeling accuracy. Recent results on the Chevron Gulf of Mexico data set are discussed next, followed by a discussion on challenges of FWI and possible roads ahead. Wave-equation-based inversionFull-waveform inversion (FWI) is a parameter estimation problem, seeking Earth models that explain observed data typically collected in shot records. FWI is generally cast as an optimization problem where Earth models are found by minimizing the misfit between observed and simulated data-i.e., we solve the following optimization problem: VAN Solutions of the above minimization problem are typically found by using iterative optimization techniques, which use local derivative information to compute descent directions that minimize the objective (m). At the bare minimum, these optimizations consist of the following steps for each experiment (Tarantola and Valette, 1982;Plessix, 2006): (1) data prediction by solving the w...

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An Effective Method for Parameter Estimation with PDE Constraints with Multiple Right-Hand Sides

Cited by 110 publications

References 31 publications

Robust full‐waveform inversion using the Student's t‐distribution

Robust full‐waveform inversion using the Student's t‐distribution

A new optimization approach for source-encoding full-waveform inversion

Frugal full-waveform inversion: From theory to a practical algorithm

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