Fast computation of the 3-D Radon transform

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 362-364 ◽  
Author(s):  
Ottilie F. Cools ◽  
Gérard C. Herman ◽  
Raphic M. van der Welden ◽  
Frans B. Kets
Keyword(s):  
Radon Transform ◽  
Rotational Symmetry ◽  
Plane Waves ◽  
Computation Time ◽  
Real Data ◽  
Radon Transforms ◽  
Fast Computation ◽  
Acceleration Techniques ◽  
Fast Transform ◽  
Input And Output

Radon transforms can be used to decompose seismic shot records into sets of plane waves and, as such, are a useful processing tool. Haneveld and Herman (1990) discussed a fast algorithm for the numerical evaluation of both the forward and inverse 2-D Radon transforms. They showed that, by rewriting the transform as a convolution, the computation time is proportional to [Formula: see text], instead of [Formula: see text] (where N denotes the number of input and output traces). In the present paper, we describe a similar method for the computation of the 3-D Radon transform for the case of rotational symmetry (see also Mallick and Frazer, 1987; McCowan and Brysk, 1989). With the aid of asymptotic techniques, the 3-D Radon transform is recast into a form similar to the 2-D Radon transform after which similar acceleration techniques are used. We have implemented and tested the fast transform on synthetic as well as on real data and found that the computation time of the fast 3-D Radon transform is indeed proportional to [Formula: see text].

Simultaneous source separation using a robust Radon transform

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. V1-V11 ◽  
Author(s):  
Amr Ibrahim ◽  
Mauricio D. Sacchi
Keyword(s):  
Radon Transform ◽  
Real Data ◽  
Radon Transforms ◽  
Accurate Data ◽  
Quadratic Formula ◽  
Penalty Term ◽  
Source Data ◽  
The Cost ◽  
Simultaneous Source ◽  
Incoherent Noise

We adopted the robust Radon transform to eliminate erratic incoherent noise that arises in common receiver gathers when simultaneous source data are acquired. The proposed robust Radon transform was posed as an inverse problem using an [Formula: see text] misfit that is not sensitive to erratic noise. The latter permitted us to design Radon algorithms that are capable of eliminating incoherent noise in common receiver gathers. We also compared nonrobust and robust Radon transforms that are implemented via a quadratic ([Formula: see text]) or a sparse ([Formula: see text]) penalty term in the cost function. The results demonstrated the importance of incorporating a robust misfit functional in the Radon transform to cope with simultaneous source interferences. Synthetic and real data examples proved that the robust Radon transform produces more accurate data estimates than least-squares and sparse Radon transforms.

Three-parameter Radon transform based on shifted hyperbolas

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. V39-V48 ◽  
Author(s):  
Ali Gholami ◽  
Toktam Zand
Keyword(s):  
Radon Transform ◽  
Seismic Data ◽  
High Performance ◽  
Fourier Transforms ◽  
Real Data ◽  
Data Sets ◽  
Radon Transforms ◽  
Slice Theorem ◽  
Ground Roll ◽  
Zero Offset

The focusing power of the conventional hyperbolic Radon transform decreases for long-offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets. Furthermore, conventional Radon transforms are ineffective for processing data sets containing events of different shapes. The shifted hyperbola is a flexible three-parameter (zero-offset traveltime, slowness, and focusing-depth) function, which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets. Radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain. We have developed a new method for effective decomposition of seismic data by using such three-parameter Radon transform. A very fast algorithm is constructed for high-resolution calculations of the new Radon transform using the recently proposed generalized Fourier slice theorem (GFST). The GFST establishes an analytic expression between the [Formula: see text] coefficients of the data and the [Formula: see text] coefficients of its Radon transform, with which a very fast switching between the model and data spaces is possible by means of interpolation procedures and fast Fourier transforms. High performance of the new algorithm is demonstrated on synthetic and real data sets for trace interpolation and linear (ground roll) noise attenuation.

Separation and imaging of seismic diffractions using migrated dip-angle gathers

Geophysics ◽  
2012 ◽  
Vol 77 (6) ◽  
pp. S131-S143 ◽  
Author(s):  
Alexander Klokov ◽  
Sergey Fomel
Keyword(s):  
Radon Transform ◽  
Real Data ◽  
Separation Procedure ◽  
Shape Difference ◽  
Depth Migration ◽  
Time Migration ◽  
Reflection Angle ◽  
Radon Space ◽  
Dip Angle ◽  
Analytical Expressions

Common-reflection angle migration can produce migrated gathers either in the scattering-angle domain or in the dip-angle domain. The latter reveals a clear distinction between reflection and diffraction events. We derived analytical expressions for events in the dip-angle domain and found that the shape difference can be used for reflection/diffraction separation. We defined reflection and diffraction models in the Radon space. The Radon transform allowed us to isolate diffractions from reflections and noise. The separation procedure can be performed after either time migration or depth migration. Synthetic and real data examples confirmed the validity of this technique.

Sound Field Reconstruction in Rooms with Deep Generative Models

2021 ◽  
Vol 263 (5) ◽  
pp. 1527-1538
Author(s):  
Xenofon Karakonstantis ◽  
Efren Fernandez Grande
Keyword(s):  
Plane Waves ◽  
Sound Field ◽  
Real Data ◽  
Generative Models ◽  
Random Wave ◽  
Generative Adversarial Network ◽  
Underlying Distribution ◽  
Adversarial Network ◽  
Reconstruction Methods ◽  
Free Region

The characterization of Room Impulse Responses (RIR) over an extended region in a room by means of measurements requires dense spatial with many microphones. This can often become intractable and time consuming in practice. Well established reconstruction methods such as plane wave regression show that the sound field in a room can be reconstructed from sparsely distributed measurements. However, these reconstructions usually rely on assuming physical sparsity (i.e. few waves compose the sound field) or trait in the measured sound field, making the models less generalizable and problem specific. In this paper we introduce a method to reconstruct a sound field in an enclosure with the use of a Generative Adversarial Network (GAN), which s new variants of the data distributions that it is trained upon. The goal of the proposed GAN model is to estimate the underlying distribution of plane waves in any source free region, and map these distributions from a stochastic, latent representation. A GAN is trained on a large number of synthesized sound fields represented by a random wave field and then tested on both simulated and real data sets, of lightly damped and reverberant rooms.

scholarly journals Uniform Estimates for Damped Radon Transform on the Plane

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Youngwoo Choi
Keyword(s):  
Radon Transform ◽  
Lorentz Spaces ◽  
Radon Transforms ◽  
Convolution Operators ◽  
Uniform Estimates ◽  
Sharp Estimates ◽  
Rotational Curvature ◽  
Affine Arclength

Uniform improving estimates of damped plane Radon transforms in Lebesgue and Lorentz spaces are studied under mild assumptions on the rotational curvature. The results generalize previously known estimates. Also, they extend sharp estimates known for convolution operators with affine arclength measures to the semitranslation-invariant case.

Sums of plane waves, and the range of the Radon transform

1979 ◽  
Vol 243 (2) ◽  
pp. 153-161 ◽  
Author(s):  
Bent E. Petersen ◽  
Kennan T. Smith ◽  
Donald C. Solmon
Keyword(s):  
Radon Transform ◽  
Plane Waves

METHOD OF MEAN VALUE OPERATORS FOR RADON TRANSFORMS IN THE SPACE OF MATRICES

2008 ◽  
Vol 19 (03) ◽  
pp. 245-283 ◽  
Author(s):  
E. OURNYCHEVA ◽  
B. RUBIN
Keyword(s):  
Radon Transform ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Inversion Method ◽  
Radon Transforms ◽  
Inversion Formulas ◽  
Higher Rank ◽  
Necessary And Sufficient

We extend the Funk–Radon–Helgason inversion method of mean value operators to the Radon transform [Formula: see text] of continuous and Lpfunctions which are integrated over matrix planes in the space of real rectangular matrices. Necessary and sufficient conditions of existence of [Formula: see text] for such f and explicit inversion formulas are obtained. New higher-rank phenomena related to this setting are investigated.

scholarly journals On the impact of various formulations of the boundary condition within numerical option valuation by DG method

Filomat ◽  
2016 ◽  
Vol 30 (15) ◽  
pp. 4253-4263 ◽  
Author(s):  
Jiří Hozman ◽  
Tomás Tichý
Keyword(s):  
Boundary Conditions ◽  
Computation Time ◽  
Real Data ◽  
Calculation Error ◽  
Option Valuation ◽  
Exotic Options ◽  
Transparent Boundary Conditions ◽  
Dg Method ◽  
The Impact ◽  
Underlying Processes

Options, a crucial type of financial instrument, are very challenging as concerns both, the application and valuation. A key property of (exotic) options is to provide a tool to manage the market risk coming from everyday innovations at the market. Due to the complexity of underlying processes and/or payoff functions valuation via numerical methods is often inevitable. The flexibility in terms of model assumptions often brings high time costs so that it can be useful to reduce the space on which the computation is executed in order to keep both the computation time and calculation error at acceptable levels. Efficient formulation of the boundary conditions of option valuation formula is one of such approaches. In this paper we focus on the impact of Dirichlet, Neumann and transparent boundary conditions when the valuation formula is discretized by the discontinuous Galerkin method combined with the implicit Euler scheme for the temporal discretization. The numerical results are presented using real data of DAX index options.

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