In seismic data processing, we often need to interpolate and extrapolate data at missing spatial locations. The reconstruction problem can be posed as an inverse problem where, from inadequate and incomplete data, we attempt to reconstruct the seismic wavefield at locations where measurements were not acquired.We propose a wavefield reconstruction scheme for spatially band‐limited signals. The method entails solving an inverse problem where a wavenumber‐domain regularization term is included. The regularization term constrains the solution to be spatially band‐limited and imposes a prior spectral shape. The numerical algorithm is quite efficient since the method of conjugate gradients in conjunction with fast matrix–vector multiplications, implemented via the fast Fourier transform (FFT), is adopted. The algorithm can be used to perform multidimensional reconstruction in any spatial domain.

Minimum weighted norm interpolation of seismic records

Although 3D seismic data are being acquired in larger volumes than ever before, the spatial sampling of these volumes is not always adequate for certain seismic processes. This is especially true of marine and land wide-azimuth acquisitions, leading to the development of multidimensional data interpolation techniques. Simultaneous interpolation in all five seismic data dimensions (inline, crossline, offset, azimuth, and frequency) has great utility in predicting missing data with correct amplitude and phase variations. Although there are many techniques that can be implemented in five dimensions, this study focused on sparse Fourier reconstruction. The success of Fourier interpolation methods depends largely on two factors: (1) having efficient Fourier transform operators that permit the use of large multidimensional data windows and (2) constraining the spatial spectrum along dimensions where seismic amplitudes change slowly so that the sparseness and band limitation assumptions remain valid. Fourier reconstruction can be performed when enforcing a sparseness constraint on the 4D spatial spectrum obtained from frequency slices of five-dimensional windows. Binning spatial positions into a fine 4D grid facilitates the use of the FFT, which helps on the convergence of the inversion algorithm. This improves the results and computational efficiency. The 5D interpolation can successfully interpolate sparse data, improve AVO analysis, and reduce migration artifacts. Target geometries for optimal interpolation and regularization of land data can be classified in terms of whether they preserve the original data and whether they are designed to achieve surface or subsurface consistency.

Five-dimensional interpolation: Recovering from acquisition constraints

Seismic signals are often irregularly sampled along spatial coordinates, leading to suboptimal processing and imaging results. Least squares estimation of Fourier components is used for the reconstruction of band-limited seismic signals that are irregularly sampled along one spatial coordinate. A simple and efficient diagonal weighting scheme, based on the distance between the samples, takes the properties of the noise (signal outside the bandwidth) into account in an approximate sense. Diagonal stabilization based on the energies of the signal and the noise ensures robust estimation. Reconstruction for each temporal frequency component allows the specification of a varying spatial bandwidth dependent on the minimum apparent velocity. This parameterization improves the reconstruction capability for the lower temporal frequencies. In practical circumstances, the maximum size of the gaps in which the signal can be reconstructed is three times the (temporal frequency dependent) Nyquist interval. Reconstruction in the wavenumber domain allows a very efficient implementation of the algorithm, and takes a total number of operations a few times that of a 2-D fast Fourier transform corresponding to the size of the output data set. Quality control indicators of the reconstruction procedure can be computed which may also serve as decision criteria on in-fill shooting during acquisition. The method can be applied to any subset of seismic data with one varying spatial coordinate. Applied along the cross-line direction, it can be used to compute a 3-D stack with improved anti-alias protection and less distortion of the signal within the bandwidth.

Reconstruction of band‐limited signals, irregularly sampled along one spatial direction

Purpose

The purpose of this study was to develop a multi-step adaptive fitting approach for liver proton density fat fraction (PDFF) and quantification, and to perform an initial validation on a broadly available hardware platform.



Theory and Methods

The proposed method uses a multi-echo three-dimensional gradient echo acquisition, with initial guesses for the fat and water signal fractions based on a Dixon decomposition of two selected echoes. Based on magnitude signal equations with a multi-peak fat spectral model, a multi-step nonlinear fitting procedure is then performed to adaptively update the fat and water signal fractions and values. The proposed method was validated using numeric phantoms as ground truth, followed by preliminary clinical validation of PDFF calculations against spectroscopy in 30 patients.



Results

The results of the proposed method agreed well with the ground truth of numerical phantoms, and were relatively insensitive to changes in field strength, field homogeneity, monopolar/bipolar readout, signal to noise ratio, and echo time selections. The in vivo patient study showed excellent consistency between the PDFF values measured with the proposed approach compared with spectroscopy.



Conclusion

This multi-step adaptive fitting approach performed well in both simulated and initial clinical evaluation, and shows potential in the quantification of hepatic steatosis. Magn Reson Med 72:1353–1365, 2014. © 2013 Wiley Periodicals, Inc.

Liver fat quantification using a multi-step adaptive fitting approach with multi-echo GRE imaging

We propose a robust interpolation scheme for aliased regularly sampled seismic data that uses the curvelet transform. In a first pass, the curvelet transform is used to compute the curvelet coefficients of the aliased seismic data. The aforementioned coefficients are divided into two groups of scales: alias-free and alias-contaminated scales. The alias-free curvelet coefficients are upscaled to estimate a mask function that is used to constrain the inversion of the alias-contaminated scale coefficients. The mask function is incorporated into the inversion via a minimum norm least-squares algorithm that determines the curvelet coefficients of the desired alias-free data. Once the alias-free coefficients are determined, the curvelet synthesis operator is used to reconstruct seismograms at new spatial positions. The proposed method can be used to reconstruct regularly and irregularly sampled seismic data. We believe that our exposition leads to a clear unifying thread between f-x and f-k beyond-alias interpo...

Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data

In this work, a new two-point method for water–fat imaging is described and explored. It generalizes existing two-point methods by eliminating some of the restrictions that these methods impose on the choice of echo times. Thus, the new two-point method promises to provide more freedom in the selection of protocol parameters and to reach higher scan efficiency. Its performance was studied theoretically and was evaluated experimentally in abdominal imaging with a multigradient-echo sequence. While depending on the choice of echo times, it is generally found to be favorable compared to existing two-point methods. Notably, water images with higher spatial resolution and better signal-to-noise ratio were attained with it in single breathholds at 3.0 T and 1.5 T, respectively. The use of more accurate spectral models of fat is shown to substantially reduce observed variations in the extent of fat suppression. The acquisition of in- and opposed-phase images is demonstrated to be replaceable by a synthesis from water and fat images. The new two-point method is finally also applied to autocalibrate a multidimensional eddy current correction and to enhance the fat suppression achieved with three-point methods in this way, especially toward the edges of larger field of views. Magn Reson Med, 2010. © 2010 Wiley-Liss, Inc.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/mrm.22578

Dual-echo Dixon imaging with flexible choice of echo times.

Seismic reservoir monitoring has become an important tool in the management of many fields. Monitoring subtle changes in the seismic properties of a reservoir caused by production places strong demands on seismic repeatability. A lack of repeatability limits how frequently reservoir changes can be monitored or the applicability of seismic monitoring at all. In this paper we show that towing many streamers with narrow separation, combined with cross‐line interpolation of data onto predefined sail lines, can give highly repeatable marine seismic data.Results from two controlled zero time lag monitoring experiments in the North Sea demonstrate high sensitivity to changing water level and variations in lateral positions. After corrections by deterministic tidal time shifts and spatial interpolation of the irregularly sampled streamer data, relative rms difference amplitude levels are as low as 12% for a deep, structurally complex field and as low as 6% for a shallow, structurally simple field.Reducing the deg...

A proven method for acquiring highly repeatable towed streamer seismic data

Seismic data are usually irregularly and sparsely sampled along the spatial coordinates, which yields suboptimal imaging results. For time‐lapse data, differences in the spatial sampling of base and monitor surveys lead to undesired differences between the images of the surveys that are not due to differences in the subsurface.In this paper, an efficient two‐step reconstruction method is proposed for irregularly and sparsely sampled 3D seismic data recorded with a dominant azimuth for large offsets. The first step is reconstruction along the receiver lines such that the midpoints are exactly on crosslines. The second step is a 2D reconstruction in the crossline midpoint‐offset domain using a least‐squares estimation of Fourier components.Sparse sampling can be handled by optimizing the parameterization of the least‐squares estimation of the Fourier coefficients, and consequently it is possible to reconstruct data that would be aliased when considering single common‐midpoint gathers.The method can be used ...

A. J. W. Duijndam

Papers

Dual-echo Dixon imaging with flexible choice of echo times.

A proven method for acquiring highly repeatable towed streamer seismic data

A general reconstruction scheme for dominant azimuth 3D seismic data

Reconstruction of two‐dimensional irregularly sampled wavefields