Robust Modeling With Erratic Data

Claerbout, Jon F.; Muir, Francis

doi:10.1190/1.1440378

Cited by 728 publications

(429 citation statements)

References 6 publications

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“…Because of this feature, (1) has been used for more than three decades in several signal processing problems where sparseness is sought; some early references are [12], [37], [50], [53]. In the 1990's, seminal work on the use of ℓ 1 sparseness-inducing penalties/log-priors appeared in the literature: the now famous basis pursuit denoising (BPDN, [11,Section 5]) criterion and the least absolute shrinkage and selection operator (LASSO, [54]).…”

Section: A Backgroundmentioning

confidence: 99%

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Figueiredo

Nowak

Wright

2007

IEEE J. Sel. Top. Signal Process.

3,016

1,847

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Abstract-Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a sparseness-inducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution, and compressed sensing are a few wellknown examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the BarzilaiBorwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.

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Section: A Backgroundmentioning

confidence: 99%

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Figueiredo

Nowak

Wright

2007

IEEE J. Sel. Top. Signal Process.

3,016

1,847

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“…Use of the L 1 measure of misfits is less sensitive to the influence of outliers and yields more stable model estimates (e.g. Claerbout & Muir 1973;Scales et al 1988;Farquharson & Oldenburg 1998;Tarantola 2005). The L 1 norm measure of misfit has been successfully employed in geomagnetic field modelling of recent and historical data Lesur et al 2008;Finlay et al 2012) and regional archeomagnetic field models (Pavón-Carrasco et al 2009), but it has not previously been tested in the construction of global field models on millennial time scales.…”

Section: Introductionmentioning

confidence: 99%

“…In many geophysical scenarios however, when longer tailed distributions are found empirically, a Laplacian distribution of residuals is a more suitable description and the L 1 maximum likelihood estimate of parameters is desired (Claerbout & Muir 1973;Constable 1988;. Use of the L 1 measure of misfits is less sensitive to the influence of outliers and yields more stable model estimates (e.g.…”

Section: Introductionmentioning

confidence: 99%

Limitations in paleomagnetic data and modelling techniques and their impact on Holocene geomagnetic field models

Panovska

Korte

Finlay

et al. 2015

Geophysical Journal International

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S U M M A R YCharacterization of geomagnetic field behaviour on timescales of centuries to millennia is necessary to understand the mechanisms that sustain the geodynamo and drive its evolution. As Holocene paleomagnetic and archeomagnetic data have become more abundant, strategies for regularized inversion of modern field data have been adapted to produce numerous timevarying global field models. We evaluate the effectiveness of several approaches to inversion and data handling, by assessing both global and regional properties of the resulting models. Global Holocene field models cannot resolve Southern hemisphere regional field variations without the use of sediments. A standard data set is used to construct multiple models using two different strategies for relative paleointensity calibration and declination orientation and a selection of starting models in the inversion procedure. When data uncertainties are considered, the results are similar overall regardless of whether we use iterative calibration and reorientation, or co-estimation of the calibration and orientation parameters as part of the inversion procedure. In each case the quality of the starting model used for initial relative paleointensity calibration and declination orientation is crucial and must be based on the best absolute information available. Without adequate initial calibration the morphology of dipole moment variations can be recovered but its absolute value will be correlated with the initial intensity calibrations, an effect that might be mitigated by ensuring an appropriate fit to enough high quality absolute intensity data with low uncertainties. The declination reorientation mainly impacts regional field structure and in the presence of non-zonal fields will result in a non-zero local average. The importance of declination orientation is highlighted by inconsistencies in the West Pacific and Australian sediment records in CALS10k.1b model. Great care must also be taken to assess uncertainties associated with both paleomagnetic and age data and to evaluate the effects of poor data distribution. New consistently allocated uncertainty estimates for sediment paleomagnetic records highlight the importance of adequate uncertainties in the inversion process, as they determine the relative weighting among the data and overall normalized misfit levels which in turn influence the complexity of the inferred field models. Residual distributions suggest that the most appropriate misfit measure is the L 1 norm (minimum absolute deviation) rather than L 2 (least squares), but this seems to have relatively minor impact on the overall results. For future Holocene field modelling we see a need for comprehensive methods to assess uncertainty in individual archeomagnetic data so that these data or models derived from them can be used for reliable initial relative paleointensity calibration and declination orientation in sediments. More work will be needed to assess whether co-estimation or an iterative approach to inversion is more efficient overall...

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“…This is to reduce the effect of bad data points in the field data sets. According to Farquharson and Oldenburg (1998), and other authors (Claerbout and Muir, 1973;Menke, 1989;Press et al, 1992;Parker, 1994), this method is less sensitive to outliers in the data particularly when used with the regularised least-squares optimisation method. For each data set, the inversion is carried out using the L 2 (smooth) inversion method as well as the L 1 (blocky) norm inversion method for the model roughness filter.…”

Section: Methodsmentioning

confidence: 99%

“…This makes it particularly sensitive to bad data points (Farquharson and Oldenburg, 1998). An alternative method is to minimise the sum of the absolute values of the data misfit, or an L 1 norm measure of the data misfit (Claerbout and Muir, 1973). One simple method of implementing an L 1 norm based optimisation method using the standard least-squares formulation is the iteratively reweighted least-squares method (Wolke and Schwetlick, 1988).…”

Section: Methodsmentioning

confidence: 99%

A comparison of smooth and blocky inversion methods in 2-D electrical imaging surveys

Loke

Acworth²,

Dahlin³

2001

ASEG Extended Abstracts

139

122

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Two-dimensional electrical imaging surveys are now widely used in engineering and environmental surveys to map moderately complex structures. In order to adequately resolve such structures with arbitrary resistivity distributions, the regularised least-squares optimisation method with a cell-based model is frequently used in the inversion of the electrical imaging data. The L 2 norm based least-squares optimisation method that attempts to minimise the sum of squares of the spatial changes in the model resistivity is often used. The resulting inversion model has a smooth variation in the resistivity values. In cases where the true subsurface resistivity consists of several regions that are approximately homogenous internally and separated by sharp boundaries, the result obtained by the smooth inversion method is not optimal. It tends to smear out the boundaries and give resistivity values that are too low or too high. The blocky or L 1 norm optimisation method can be used for such situations. This method attempts to minimise the sum of the absolute values of the spatial changes in the model resistivity. It tends to produce models with regions that are piecewise constant and separated by sharp boundaries. Results from tests of the smooth and blocky inversion methods with several synthetic and field data sets highlight the strengths and weaknesses of both methods. The smooth inversion method gives better results for areas where the subsurface resistivity changes in a gradual manner, while the blocky inversion method gives significantly better results where there are sharp boundaries. While fast computers and software have made the task of interpreting data from electrical imaging surveys much easier, it remains the responsibility of the interpreter to choose the appropriate tool for the task based on the available geological information.

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Robust Modeling With Erratic Data

Cited by 728 publications

References 6 publications

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Limitations in paleomagnetic data and modelling techniques and their impact on Holocene geomagnetic field models

A comparison of smooth and blocky inversion methods in 2-D electrical imaging surveys

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