Hybrid ℓ1/ℓ2 minimization with applications to tomography

Bube, Kenneth P.; Langan, Robert T.

doi:10.1190/1.1444219

Cited by 168 publications

(89 citation statements)

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“…The cause of this problem is the lack of low frequencies in seismic data. Alternatives for the least-squares objective function with the L2-norm (Tarantola and Valette, 1982), such as the Huber norm (Huber, 1973;Guitton and Symes, 2003) or hybrid approaches (Bube and Langan, 1997;Brossier et al, 2010), are more robust in the presence of large isolated and non-Gaussian errors, but still suffer from the same cycle skipping problem as observed with the L2-norm.…”

Section: Introductionmentioning

confidence: 99%

The Diagonalator, an Alternative Cost Functional for Wave-equation Inversion

Moghaddam¹,

Mulder²

2012

Proceedings

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SUMMARYThe classic least-squares cost functional for full waveform inversion suffers from local minima due to loop skipping in the absence of low frequencies in the seismic data. Velocity model building based on subsurface spatial or temporal shifts may break down in the presence of multiples in the data. Cost functionals that translate this idea to the data domain, with offset-or time-shifts, can handle multiples. An earlier data-domain formulation suffered from cross-talk between events. Here, we present a multishot extension that should be less sensitive to cross-talk. It has the property of an annihilator, similar to the functional used for velocity analysis with extended images based on subsurface shifs. However, since it operates in the data domain, it should be able to handle multiples. For 2-D models with line acquistion, the proposed functional applies a singular-value decomposition on the observed data and uses the eigenvectors to build data panel that should be diagonal in the correct velocity model, but has significant off-diagonal entries in the wrong model. By minimizing these offdiagonal entries or maximizing the main diagonal, the correct model should be found. We therefore named it the diagonalator. We present initial tests on a simple, horizontally layered velocity model.

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Section: Introductionmentioning

confidence: 99%

The Diagonalator, an Alternative Cost Functional for Wave-equation Inversion

Moghaddam¹,

Mulder²

2012

Proceedings

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“…Notice that large residuals contribute very little to the search direction as the optimization proceeds as a result of this down-weighing. Functionally, such downweighing strongly resembles the IRLS method (see for example Bube and Langan (1997)); however, in our context the down-weighing arises naturally, deriving from the gradient of the MAP optimization problem (7).…”

Section: Student's T-density and Fwi Formulationmentioning

confidence: 99%

“…'Artifacts' refer to coherent or systematic (non-random) events that arise because of shortcomings in the forward modeling. The distinction between 'errors' and 'artifacts' can become blurred-for example, multiples in the data, which motivated robust methods in tomography (Bube and Langan (1997)), can actually aid recovery once modeling tools are available to appropriately use them in the inversion process. Other examples of such systematic events include out of plane scattering when working with 2D seismic data, and using acoustic PDEs for the forward model where the elastic or anisotropic models could be applied.…”

Section: Introductionmentioning

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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“…The value of p was chosen after some numerical experimentation to eliminate random oscillations of the water surface while still capturing aperiodic and low-frequency variations, such as due to tides or local gradients in bathymetry. Alternatively, an error tolerance based on measurement accuracy could be used instead of a constant smoothing parameter, or L 1 /L 2 minimizations could be performed to down-weight erroneous data (e.g., Bube and Langan 1997), although these tests were not carried out. Once smoothed, the data points were sorted according to the along-transect direction before gridding.…”

Section: Matte Et Al Rtk Gps/adcp Measurements In Tidal Riversmentioning

confidence: 99%

Untitled

2014

Limnology & Ocean Methods

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Direct measurement of water levels and velocities can provide fundamental insight into the hydrodynamics of tidal rivers and estuaries. They represent a reliable source of data for the verification and calibration of numerical models. Such knowledge is also crucial for navigation, environmental assessments, and coastal protection and development. To reach a high level of description, measured data must cover a wide range of spatial and temporal variability. Long-term monitoring stations provide high-frequency records, but they are usually limited in number, thereby limiting their ability to represent spatial features of the flow. Conversely, satellitebased measurements provide synoptic snapshots of a region, but they are limited in their temporal resolution and interpretation of the velocity field. Boat surveys remain the best compromise between fixed measurements and remote sensing observations for the characterization of nonstationary hydrodynamic fields. They can ensure both spatial and temporal coverage, provided that the sampling strategy and resolution is adapted to the size and rate of evolution of the structures measured (e.g., Rixen et al. 2001).In the field, meeting these requirements is often a challenge, as both the instrumentation and survey strategy must meet the criteria and constraints set by the users (e.g., sampling requirements, finite measurement time, and resources) and by the environment (e.g., presence of tides, size of the river). This has been facilitated in part by the development of new technologies over the last decades, including acoustic Doppler current profilers (ADCP) for water velocity meaQuantifying lateral and intratidal variability in water level and velocity in a tide-dominated river using combined RTK GPS and ADCP measurements AbstractCross-sectional gradients in water levels and velocities play a determining role in the circulation dynamics of tidal rivers and estuaries. Documenting and analyzing their variability throughout a tidal cycle require observations with high spatial and temporal resolution. A survey strategy and a data analysis procedure have been designed to obtain continuous and synoptic water level and velocity fields over a tidal cycle, along 13 cross-sections of the St. Lawrence fluvial estuary dominated by large tidal ranges. The method combines both RTK GPS and ADCP technologies for the simultaneous measurement of water levels and velocities along repeated boat transects, allowing fast data acquisition over wide river sections and under rapidly changing conditions. The reconstruction of continuous and synoptic fields is made by interpolation. Simplifying assumptions about data stationarity and/or homogeneity are avoided by adapting the interpolation procedures to the shape and distribution of the underlying data in a space-time reference frame, thus minimizing distortion in the reconstructed quantities. The capabilities and limitations of the method are assessed through error computations and comparisons with alternate data analysis methods and complement...

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Hybrid ℓ1/ℓ2 minimization with applications to tomography

Cited by 168 publications

References 0 publications

The Diagonalator, an Alternative Cost Functional for Wave-equation Inversion

The Diagonalator, an Alternative Cost Functional for Wave-equation Inversion

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

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