Daniel Trad

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.

Accurate interpolation with high‐resolution time‐variant Radon transforms

It is well known that a sparse hyperbolic Radon transform (RT) can be used to extend the aperture of aperture limited data, filter noise, and fill gaps. In the same manner, an elliptical RT can achieve similar results when applied to slant stack sections. A problem with these transformations is that they have a time-variant kernel that results in slow implementation. By defining a model space in terms of an irregularly sampled velocity space to minimize the number of unknowns during the inversion and using sparse matrices, however, the computation time can be kept low enough for practical application. We implement hyperbolic and elliptical time domain RTs by using inversion via weighted conjugate gradient methods with a sparseness constraint. The hyperbolic RT performs accurate interpolation in common-midpoint (CMP) gathers, while the elliptical RT attenuates sampling artifacts in slant stack sections obtained from CMP gathers with poor sampling and gaps.

Interpolation and multiple attenuation with migration operators

A hyperbolic Radon transform (RT) can be applied with success to attenuate or interpolate hyperbolic events in seismic data. However, this method fails when the hyperbolic events have apexes located at nonzero offset positions. A different RT operator is required for these cases, an operator that scans for hyperbolas with apexes centered at any offset. This procedure defines an extension of the standard hyperbolic RT with hyperbolic basis functions located at every point of the data gather. The mathematical description of such an operator is basically similar to a kinematic poststack time-migration equation, with the horizontal coordinate being not midpoint but offset. In this paper, this transformation is implemented by using a least-squares conjugate gradient algorithm with a sparseness constraint. Two different operators are considered, one in the time domain and the other in the frequency-wavenumber domain (Stolt operator). The sparseness constraint in the time-offset domain is essential for resampling and for interpolation. The frequency-wavenumber domain operator is very efficient, not much more expensive in computation time than a sparse parabolic RT, and much faster than a standard hyperbolic RT. Examples of resampling, interpolation, and coherent noise attenuation using the frequency-wavenumber domain operator are presented. Near and far offset gaps are interpolated in synthetic and real shot gathers, with simultaneous resampling beyond aliasing. Waveforms are well preserved in general except when there is little coherence in the data outside the gaps or events with very different velocities are located at the same time. Multiples of diffractions are predicted and attenuated by subtraction from the data.

Wavelet filtering of magnetotelluric data

A method is described for filtering magnetotelluric (MT) data in the wavelet domain that requires a minimum of human intervention and leaves good data sections unchanged. Good data sections are preserved because data in the wavelet domain is analyzed through hierarchies, or scale levels, allowing separation of noise from signals. This is done without any assumption on the data distribution on the MT transfer function. Noisy portions of the data are discarded through thresholding wavelet coefficients. The procedure can recognize and filter out point defects that appear as a fraction of unusual observations of impulsive nature either in time domain or frequency domain. Two examples of real MT data are presented, with noise caused by both meteorological activity and power-line contribution. In the examples given in this paper, noise is better seen in time and frequency domains, respectively. Point defects are filtered out to eliminate their deleterious influence on the MT transfer function estimates. After the filtering stage, data is processed in the frequency domain, using a robust algorithm to yield two sets of reliable MT transfer functions.

Simultaneous interpolation of 4 spatial dimensions

The Minimum Weighted Norm Interpolation (MWNI) algorithm (Liu and Sacchi, 2001) has been proposed as a method to reconstruct band-limited seismic data along 1, 2 and 3 spatial dimensions. In addition, tests showing the ability of the method to reconstruct data prior to amplitude versus angle wave equation migration were provided in Liu et. al (2003). The method incorporates bandwidth limitation constraints, and a spectral smoothness constraint that becomes a key feature of the algorithm at the time of interpolating large segments of missing information.

Multiple Attenuation Using an Apex Shift Radon Transform

Multiples from seafloor scatterers and peg-leg multiples in complex geology are often resistant to conventional multiple removal techniques such as Radon demultiple. They have a complicated moveout behaviour in prestack gathers which can only be approximately represented by a conventional parabolic or hyperbolic Radon decomposition. Such multiples split into pairs of events, one for each of the shot or receiver side of the multiple. They are approximately parabolic after NMO correction with primary velocities but have their minimum travel times shifted to either side of zero-offset.

Understanding land data interpolation

Seismic data interpolation has been around for long time, but only recently have we been able to use complex multidimensional and global algorithms that have the capability to infill large gaps in 3D land surveys. This innovation offers great potential for improvement, but for this technology to become useful, many questions still need answers: what are the best domains to interpolate? What is the optimal size of operators given a particular level of structural complexity? Should we pursue an ideal geometry for migration or should we stay close to the input geometry in order to minimize distortions? How does sampling in multiple dimensions affect our traditional aliasing constraints? How can we infill large gaps without using a model for our data? Are irregularities in sampling beneficial? Understanding land data interpolation may help to solve many problems in seismic processing.