Latest views of the sparse Radon transform

The Radon transform (RT) suffers from the typical problems of loss of resolution and aliasing that arise as a consequence of incomplete information, including limited aperture and discretization. Sparseness in the Radon domain is a valid and useful criterion for supplying this missing information, equivalent somehow to assuming smooth amplitude variation in the transition between known and unknown (missing) data. Applying this constraint while honoring the data can become a serious challenge for routine seismic processing because of the very limited processing time available, in general, per common midpoint. To develop methods that are robust, easy to use and flexible to adapt to different problems we have to pay attention to a variety of algorithms, operator design, and estimation of the hyperparameters that are responsible for the regularization of the solution. In this paper, we discuss fast implementations for several varieties of RT in the time and frequency domains. An iterative conjugate gradient algorithm with fast Fourier transform multiplication is used in all cases. To preserve the important property of iterative subspace methods of regularizing the solution by the number of iterations, the model weights are incorporated into the operators. This turns out to be of particular importance, and it can be understood in terms of the singular vectors of the weighted transform. The iterative algorithm is stopped according to a general cross validation criterion for subspaces. We apply this idea to several known implementations and compare results in order to better understand differences between, and merits of, these algorithms.

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Best subset selection via cross-validation criterion

Top ◽

10.1007/s11750-020-00538-1 ◽

2020 ◽

Vol 28 (2) ◽

pp. 475-488 ◽

Cited By ~ 2

Author(s):

Yuichi Takano ◽

Ryuhei Miyashiro

Keyword(s):

Cross Validation ◽

Subset Selection ◽

Validation Criterion

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Analysis of Mapping Techniques for Mountain Precipitation: A Case Study of Alpine Region, Austria

Engineering, Technology & Applied Science Research ◽

10.48084/etasr.2167 ◽

2018 ◽

Vol 8 (4) ◽

pp. 3213-3217

Author(s):

A. N. Laghari ◽

G. D. Walasai ◽

D. K. Bangwar ◽

A. H. Memon ◽

A. H. Shaikh

Keyword(s):

Cross Validation ◽

Geographical Information ◽

Missing Information ◽

Local Data ◽

Consistency Test ◽

Distance Weighting ◽

Map Generation ◽

Mapping Techniques ◽

Inverse Distance

Truly representative precipitation map generation of mountain regions is a difficult task. Due to poor gauge representativity, complex topography and uneven density factors make the generation of representative precipitation maps a very difficult task. To generate representative precipitation maps, this study focused on analyzing four different mapping techniques: ordinary kriging, spline technique (SP), inverse distance weighting (IDW) and regression kriging (RK). The generated maps are assessed through cross-validation statistics, spatial cross-consistency test and by water balance approach. The largest prediction error is produced by techniques missing information on co-variables. The ME and RMSE values show that IDW and SP are the most biased techniques. The RK technique produced the best model results with 1.38mm and 72.36mm ME and RMSE values respectively. The comparative analysis proves that RK model can produce reasonably accurate values at poorly gauged areas, where geographical information compensated the poor availability of local data.

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The Shuey-Radon transform

Geophysics ◽

10.1190/geo2018-0376.1 ◽

2019 ◽

Vol 84 (3) ◽

pp. V197-V206 ◽

Cited By ~ 1

Author(s):

Ali Gholami ◽

Milad Farshad

Keyword(s):

Radon Transform ◽

Seismic Data ◽

Matching Pursuit ◽

Real Data ◽

Transform Domain ◽

Amplitude Variation ◽

Amplitude Variation With Offset ◽

Linear System Of Equations ◽

Matching Pursuit Algorithm ◽

Zero Offset

The traditional hyperbolic Radon transform (RT) decomposes seismic data into a sum of constant amplitude basis functions. This limits the performance of the transform when dealing with real data in which the reflection amplitudes include the amplitude variation with offset (AVO) variations. We adopted the Shuey-Radon transform as a combination of the RT and Shuey’s approximation of reflectivity to accurately model reflections including AVO effects. The new transform splits the seismic gather into three Radon panels: The first models the reflections at zero offset, and the other two panels add capability to model the AVO gradient and curvature. There are two main advantages of the Shuey-Radon transform over similar algorithms, which are based on a polynomial expansion of the AVO response. (1) It is able to model reflections more accurately. This leads to more focused coefficients in the transform domain and hence provides more accurate processing results. (2) Unlike polynomial-based approaches, the coefficients of the Shuey-Radon transform are directly connected to the classic AVO parameters (intercept, gradient, and curvature). Therefore, the resulting coefficients can further be used for interpretation purposes. The solution of the new transform is defined via an underdetermined linear system of equations. It is formulated as a sparsity-promoting optimization, and it is solved efficiently using an orthogonal matching pursuit algorithm. Applications to different numerical experiments indicate that the Shuey-Radon transform outperforms the polynomial and conventional RTs.

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A note on the generalised cross-validation criterion in linear model selection

Biometrika ◽

10.1093/biomet/82.1.215 ◽

1995 ◽

Vol 82 (1) ◽

pp. 215-219 ◽

Cited By ~ 4

Author(s):

J. H. VENTER ◽

J. L. J. SNYMAN

Keyword(s):

Model Selection ◽

Linear Model ◽

Cross Validation ◽

Validation Criterion

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Gray-box identification with regularized FIR models

at - Automatisierungstechnik ◽

10.1515/auto-2018-0026 ◽

2018 ◽

Vol 66 (9) ◽

pp. 704-713 ◽

Cited By ~ 2

Author(s):

Tobias Münker ◽

Timm J. Peter ◽

Oliver Nelles

Keyword(s):

Dynamic System ◽

Prior Knowledge ◽

Cross Validation ◽

Finite Impulse Response ◽

Black Box ◽

Model Structure ◽

Linear Dynamic ◽

Novel Approach ◽

Box Models ◽

Validation Criterion

Abstract The problem of modeling a linear dynamic system is discussed and a novel approach to automatically combine black-box and white-box models is introduced. The solution proposed in this contribution is based on the usage of regularized finite-impulse-response (FIR) models. In contrast to classical gray-box modelling, which often only optimizes the parameters of a given model structure, our approach is able to handle the problem of undermodeling as well. Therefore, the amount of trust in the white-box or gray-box model is optimized based on a generalized cross-validation criterion. The feasibility of the approach is demonstrated with a pendulum example. It is furthermore investigated, which level of prior knowledge is best suited for the identification of the process.

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Implementing and modifying Broyden class updates for large scale optimization

Computational Optimization and Applications ◽

10.1007/s10589-020-00239-2 ◽

2020 ◽

Author(s):

Martin Buhmann ◽

Dirk Siegel

Keyword(s):

Large Scale ◽

Optimization Problems ◽

Substantial Improvement ◽

Gradient Algorithm ◽

Large Scale Optimization ◽

Fixed Amount ◽

Limited Memory ◽

Broyden Class ◽

Scale Optimization ◽

Number Of Iterations

Abstract We consider Broyden class updates for large scale optimization problems in n dimensions, restricting attention to the case when the initial second derivative approximation is the identity matrix. Under this assumption we present an implementation of the Broyden class based on a coordinate transformation on each iteration. It requires only $$2nk + O(k^{2}) + O(n)$$ 2 n k + O ( k 2 ) + O ( n ) multiplications on the kth iteration and stores $$nK+ O(K^2) + O(n)$$ n K + O ( K 2 ) + O ( n ) numbers, where K is the total number of iterations. We investigate a modification of this algorithm by a scaling approach and show a substantial improvement in performance over the BFGS method. We also study several adaptations of the new implementation to the limited memory situation, presenting algorithms that work with a fixed amount of storage independent of the number of iterations. We show that one such algorithm retains the property of quadratic termination. The practical performance of the new methods is compared with the performance of Nocedal’s (Math Comput 35:773--782, 1980) method, which is considered the benchmark in limited memory algorithms. The tests show that the new algorithms can be significantly more efficient than Nocedal’s method. Finally, we show how a scaling technique can significantly improve both Nocedal’s method and the new generalized conjugate gradient algorithm.

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