Phase Shift Plus Interpolation Research Articles

Seismic Data Reconstruction Using a Phase-Shift-Plus-Interpolation-Based Apex-Shifted Hyperbolic Radon Transform

The apex-shifted hyperbolic Radon transform (ASHRT) based on the Stolt-stretch operator can be implemented in the frequency domain, which accelerates the computation efficiency of ASHRT. However, the Stolt-stretch operator has limitations when it comes to velocity variations. Therefore, this paper introduces a new ASHRT approach based on post-stack phase shift plus interpolation (PSPI) imaging and modeling operators. This new approach is designed to better adapt to changes in medium velocity and enhance the quality of data reconstruction. When combining this novel transformation with sparsity constraints for model testing and real data applications, the experimental results indicate that it is an effective data reconstruction tool, with superior data reconstruction results compared to traditional ASHRT based on the Stolt-stretch operator.

Computationally Efficient Phase Shift Plus Interpolation Seismic Migration Method

In this letter, we propose a computationally efficient implementation of the phase shift plus interpolation (PSPI) seismic migration technique. The PSPI is among the well-known migration techniques in the frequency-wavenumber ( $\omega -k_{x}$ ) domain that accounts for lateral velocity variations in the earth subsurface. However, the large number of complex multiplications required to perform PSPI makes this method computationally expensive. The proposed method replaces the use of complex multiplications in the frequency-wavenumber ( $\omega -k_{x}$ ) domain with real additions, reducing the computational complexity greatly. It also replaces the real-by-complex multiplications with real multiplications. It does so by using the fact that the PSPI technique maintains the amplitude and shifts the phase of seismic wavefields in the passband region of the extrapolators while attenuating the amplitude and maintaining the phase of the wavefield in the evanescent region of the extrapolators. The proposed method was tested by performing 2-D poststack depth migration to the well-known SEG/EAGE Salt Model. The computational complexity in the $\omega -k_{x}$ domain was reduced by 76.9%, while producing an accurate migrated section of the salt model.

3D Prestack Fourier Mixed-Domain (FMD) depth migration for VTI media with large lateral contrasts

Although many 3D One-Way Wave-equation Migration (OWEM) methods exist for VTI media, most of them struggle either with the stability, the anisotropic noise or the computational cost. In this paper we present a new method based on a mixed space- and wavenumber-propagator that overcome these issues very effectively as demonstrated by the examples. The pioneering methods of phase-shift (PS) and Stolt migration in the frequency-wavenumber domain designed for laterally homogeneous media have been followed by several extensions for laterally inhomogeneous media. Referred many times to as phase-screen or generalized phase-screen methods, such extensions include as main examples of the Split-step Fourier (SSF) and the phase-shift plus interpolation (PSPI). To further refine such phase-screen techniques, we introduce a higher-order extension to SSF valid for a 3D VTI medium with large lateral contrasts in vertical velocity and anisotropy parameters. The method is denoted Fourier Mixed-Domain (FMD) prestack depth migration and can be regarded as a stable explicit algorithm. The FMD technique was tested using the 3D SEG/EAGE salt model and the 2D anisotropic Hess model with good results. Finally, FMD was applied with success to a 3D field data set from the Barents Sea including anisotropy.

Determining velocities in high frequency electromagnetic prospecting by phase shift plus interpolation migration

Phase Shift Plus Interpolation (PSPI) Migration is one of the most popular migration methods that is used not only in Seismic Data Processing but also in interpreting high frequency electromagnetic prospecting [Ground Penetrating Radar (GPR) data]. Based on the similarities between the principle of the propagation of electromagnetic wave and the mechanical wave, migration methods could be applied to interpreting GPR data as a particular step to calculate the medium’s velocity, estimate the depth, shape and size of buried objects. Noticeably, there are two kinds of velocities usually used in migration methods: root mean square (RMS) velocity, which is used in F – K, Finite Difference and Kirchhoff Migration, and interval velocity, which is used in PSPI Migration. RMS velocity is the average velocity taken into account by considering the influence of the upper layer’s instantaneous velocity; whereas the interval velocity only reflect the practical velocity of one layer. In this paper, the problem of how to apply PSPI Migration to interpret GPR data will be presented. Some results of model datum and real datum were also examined. Besides, we made a comparison of using RMS velocity and interval velocity, and then explain how these two types of velocity could be combined to receive the best result.

Prestack Depth Migration by a Parallel 3D PSPI

Prestack depth migration for seismic reflection data is commonly used tool for imaging complex geological structures such as salt domes, faults, thrust belts, and stratigraphic structures. Phase shift plus interpolation (PSPI) algorithm is a useful tool to directly solve a wave equation and the results have natural properties of the wave equation. Amplitude and phase characteristics, in particular, are better preserved. The PSPI algorithm is widely used in hydrocarbon exploration because of its simplicity, efficiency, and reduced efforts for computation. However, meaningful depth image of 3D subsurface requires parallel computing to handle heavy computing time and great amount of input data. We implemented a parallelized version of 3D PSPI for prestack depth migration using Open-Multi-Processing (Open MP) library. We verified its performance through applications to 3D SEG/EAGE salt model with a small scale Linux cluster. Phase-shift was performed in the vertical and horizontal directions, respectively, and then interpolated at each node. This gave a single image gather according to shot gather. After summation of each single image gather, we got a 3D stacked image in the depth domain. The numerical model example shows good agree- ment with the original geological model.

The design of stable, sparse wavefield extrapolators using projections onto convex sets

We present the results of poststack explicit depth migration of the well-known 2D SEG/EAGE salt model zero-offset seismic data using sparse wavefield extrapolators. The extrapolators are designed to be sparse by forcing some of the very small complex-valued coefficients’ magnitude values to be zero. The proposed extrapolators design method combines the previously reported modified projections onto convex sets (MPOCS) for designing explicit depth frequency-space ([Formula: see text]) wavefield extrapolation operators with hard-thresholding of the small extrapolators coefficients’ magnitude. The real and imaginary parts of the MPOCS operators, with small magnitudes, are replaced by zeros during the MPOCS algorithm iterations. The migrated result of the SEG/EAGE salt model data, using such sparse designed operators, shows comparable migrated results using the nonsparse version of the MPOCS extrapolation operators as well as the image obtained using the well-known phase-shift plus interpolation (PSPI) migration technique. Overall, the sparse operators result in poststack imaging computational savings (in terms of used flops) of about 28% when compared to poststack imaging of the same data using the nonsparse MPOCS designed operators, and of more than 87.77% saved flops using the PSPI technique.

Improving the efficiency of elastic wave-mode separation for heterogeneous tilted transverse isotropic media

Wave-mode separation for TI (transversely isotropic) models can be carried out by nonstationary filtering the elastic wavefields with localized filters. These filters are constructed based on the polarization vectors obtained by solving the Christoffel equation using local medium parameters. This procedure, although accurate, is computationally expensive, especially in 3D. We develop an efficient method for wave-mode separation, which exploits the same general idea of projecting wavefields onto polarization vectors. The method consists of two steps: (1) separate wave modes in the wavenumber domain at a number of reference models to obtain the same number of partially separated wavefields; then transform all the wavefields to the space domain; (2) interpolate the wavefields (obtained in step 1) in the space domain using the spatially-variable model parameters. The new method resembles the phase-shift plus interpolation (PSPI) technique, which interpolates the wavefields that are reconstructed at several reference velocities. Synthetic examples indicate that the separation followed by interpolation is effective for models with complex geology. The new technique has the benefits of speed and accuracy.

Designing stable extrapolators for explicit depth extrapolation of 2D and 3D wavefields using projections onto convex sets

We have developed a robust algorithm for designing explicit depth extrapolation operators using the projections-onto-convex-sets (POCS) method. The operators are optimal in the sense that they satisfy all required extrapolation design characteristics. In addition, we propose a simple modification of the POCS algorithm (modified POCS, or MPOCS) that further enhances the stability of extrapolated wavefields and reduces the number of iterations required to design such operators to approximately 2% of that required for the basic POCS design algorithm. Various synthetic tests show that 25-coefficient 1D extrapolation operators, which have 13 unique coefficients, can accommodate dip angles up to 70°. We migrated the SEG/EAGE salt model data with the operators and compare our results with images obtained via extrapolators based on modified Taylor series and with wavefield extrapolation techniques such as phase shift plus interpolation (PSPI) and split-step Fourier. The MPOCS algorithm provides practically stable depth extrapolators. The resulting migrated section is comparable in quality to an expensive PSPI result and visibly outperforms the other two techniques. Strong dips and subsalt structures are imaged clearly. Finally, we extended the 1D extrapolator design algorithm, using MPOCS for 2D extrapolation, to the 2D case to perform 3D extrapolation; the result is a perfect circularly symmetric migration impulse response.

Migração PSPI baseada nas equações da onda unidirecionais com amplitude verdadeira

As equações da onda unidirecionais com amplitude verdadeira, propostas nos trabalhos mais recentes, levam em conta de forma correta a parte cinemática e dinâmica da equação completa da onda equivalentemente às obtidas a partir da aproximação de altas freqüências. Isto é, as equações iconal e de transporte obtidas para estes operadores se correspondem com as obtidas com a equação completa da onda. Num meio com variação lateral de velocidade, a implementação das equações unidirecionais com amplitude verdadeira envolve o cálculo numérico de operadores integro-diferenciais que podem ser aproximados por somatórios e resolvidos por técnicas de diferenças finitas. Para um meio que varia somente com a profundidade, as novas equações unidirecionais com amplitude verdadeira podem ser solucionadas através do tradicional método phase-shift , e as novas condições de fronteira e de imagem se reduzem a multiplicações no domínio de Fourier. Neste trabalho estamos propondo estender a migração phase-shift com amplitude verdadeira para meios com variação lateral v(x,z), através da metodologia Phase shift plus interpolation (PSPI). Experimentos numéricos no domínio de tiro comum com o novo método PSPI com amplitude verdadeira e sua extensão split-step (PSPI-SS), produzem imagens com amplitudes mais corretas em relação aos operadores unidirecionais convencionais, enquanto as características cinemáticas da migração PSPI e PSPI-SS são preservadas.

Regularization and datuming of seismic data by weighted, damped least squares

The possibility of improving regularization/datuming of seismic data is investigated by treating wavefield extrapolation as an inversion problem. Weighted, damped least squares is then used to produce the regularized/datumed wavefield. Regularization/datuming is extremely costly because of computing the Hessian, so an efficient approximation is introduced. Approximation is achieved by computing a limited number of diagonals in the operators involved. Real and synthetic data examples demonstrate the utility of this approach. For synthetic data, regularization/datuming is demonstrated for large extrapolation distances using a highly irregular recording array. Without approximation, regularization/datuming returns a regularized wavefield with reduced operator artifacts when compared to a nonregularizing method such as generalized phase shift plus interpolation (PSPI). Approximate regularization/datuming returns a regularized wavefield for approximately two orders of magnitude less in cost; but it is dip limited, though in a controllable way, compared to the full method. The Foothills structural data set, a freely available data set from the Rocky Mountains of Canada, demonstrates application to real data. The data have highly irregular sampling along the shot coordinate, and they suffer from significant near-surface effects. Approximate regularization/datuming returns common receiver data that are superior in appearance compared to conventional datuming.

Three‐dimensional depth imaging with generalized screens: A salt body case study

We illustrate the performance of the generalized screen propagator on real seismic data for 3D zero‐offset and prestack depth imaging. We use TotalFinaElf's L7D data set from the North Sea, a 3D marine seismic survey that contained limited azimuthal coverage. The subsurface shows significant tectonic deformation, including an intrusive salt body in sedimentary sequences. A transformation to common azimuth is applied prior to the 3D prestack depth imaging procedure. We compare the performance of the generalized screen propagator with that of a hybrid phase shift plus interpolation (PSPI)/split‐step Fourier method. Three‐dimensional prestack results confirm the generalized screen method handles multipathing more accurately. Comparisons are also made with Kirchhoff migration results. The results differ mainly in the fine‐scale irregularities of the image and not in the wavefront set of the image. Using synthetic models of similar structure (the SEG/EAGE salt model), we further illustrate the importance of multipathing and multiple scattering. Overall, our results show that our wave‐equation approach produces better images than the Kirchhoff approach to prestack depth migration; we attribute this mainly to a more complete handling of wave diffraction in the generalized screen expansion, which becomes important in strongly heterogeneous and irregular velocity models such as the ones containing salt bodies.

Depth imaging in anisotropic media by symmetric non‐stationary phase shift

ABSTRACTWe present a new depth‐imaging method for seismic data in heterogeneous anisotropic media. This recursive explicit method uses a non‐stationary extrapolation operator to allow lateral velocity variation, and it uses the relationship between phase angle and the spectral coordinates of seismic data to allow velocity variation with phase angle. A qualitative comparison of migration impulse responses suggests that, for an equivalent cost, the symmetric non‐stationary phase‐shift (SNPS) operator is superior to the phase‐shift plus interpolation (PSPI) operator, for very large depth intervals. To demonstrate the potential of the new method, seismic data from a physical model acquired over a transversely isotropic medium are imaged using a shot‐record migration based on the SNPS operator.

Prestack depth migration by symmetric nonstationary phase shift

A new depth migration method suitable for heterogeneous media is presented. The well‐known phase shift plus interpolation (PSPI) method and the recently introduced nonstationary phase‐shift (NSPS) method are combined into a single symmetric operator with improved accuracy and stability and with similar computational effort. For prestack depth migration, the symmetric operator is used in a recursive wavefield extrapolation to compute incident and reflected wavefields at any desired depth, and the ratio of the incident and reflected wavefields at a particular depth is used to estimate seismic reflectivity. When the velocity model is made piecewise constant laterally, the symmetric extrapolation operator can be computed efficiently using ordinary phase‐shift extrapolation for a series of reference velocities and appropriate spatial windowing. Migration of the Marmousi synthetic data set by symmetric nonstationary phase shift (SNPS) provides an image that compares favorably with an image of the zero‐offset reflectivity derived from the Marmousi velocity model.

A comparison of omega-x, PSPI and explicit algorithms for post-stack migration

Three widely used frequency domain algorithms for post-stack migration are the omega-x, PSPI and explicit algorithms. They differ in speed and accuracy with omega-x being the least accurate and fastest. Phase shift plus interpolation (PSPI) is the slowest and less accurate than the explicit algorithm. The explicit algorithm is the most accurate of the three and its run time falls between the omega-x and PSPI algorithm.

Wavefield extrapolation by nonstationary phase shift

The phase‐shift method of wavefield extrapolation applies a phase shift in the Fourier domain to deduce a scalar wavefield at one depth level given its value at another. The phase‐shift operator varies with frequency and wavenumber, and assumes constant velocity across the extrapolation step. We use nonstationary filter theory to generalize this method to nonstationary phase shift (NSPS), which allows the phase shift to vary laterally depending upon the local propagation velocity. For comparison, we derive an analytic form for the popular phase shift plus interpolation (PSPI) method in the limit of an exhaustive set of reference velocities. NSPS and this limiting form of PSPI can be written as generalized Fourier integrals which reduce to ordinary phase shift in the constant velocity limit. In the (x, ω) domain, these processes are the transpose of each other; however, only NSPS has the physical interpretation of forming the scaled, linear superposition of laterally‐variable impulse responses (i.e.,Huygen’s wavelets). The difference between NSPS and PSPI is clear when they are compared in the case of a piecewise constant velocity variation. Define a set of windows such that the jth window is unity when the propagation velocity is the jth distinct velocity and is zero otherwise. NSPS can be computed by applying the window set to the input data to create a set of windowed wavefields, which are individually phase‐shift extrapolated with the corresponding constant velocity, and the extrapolated set is superimposed. PSPI proceeds by phase‐shift extrapolating the input data for each distinct velocity, applying the jth window to the jth extrapolation, and superimposing. Though neither process is fully correct, PSPI has the unphysical limit that discontinuities in the lateral velocity variation cause discontinuities in the wavefield, whereas NSPS shows the expected wavefront “healing.” We then formulate a finite aperture compensation for NSPS which has the practical result of absorbing lateral boundaries for all incidence angles. Wavefield extrapolation can be regarded as the crosscorrelation of the wavefield with the expected response of a point diffractor at the new depth level. Aperture compensation simply applies a laterally varying window to the infinite, theoretical diffraction response. The crosscorrelation becomes spatially variant, even for constant velocity, and hence is a nonstationary filter. The nonstationary effects of aperture compensation can be simultaneously applied with the NSPS extrapolation through a laterally variable velocity field.

Prestack migration by split‐step DSR

The double‐square‐root (DSR) prestack migration equation, though defined for depth variable velocity, can be used to image media with strong velocity variations using a phase‐shift plus interpolation (PSPI) or split‐step correction. The split‐step method is based on applying a phase‐shift correction to the extrapolated wavefield—a correction that attempts to compensate for the lateral velocity variations. I show how to extend DSR prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the Marmousi data set. The split‐step DSR migration is very fast and offers excellent imaging results.

ON REVERSE‐TIME MIGRATION*

ABSTRACTMigration is a process whereby events in ‘image space’ are mapped into their correct positions in ‘object space’. The wave equations associated with this mapping may be defined and solved numerically either in image space or in object space. In the former the CMP section, which represents the initial conditions, is extrapolated toward increasing depths, and the migrated data are recovered at zero time. In the latter, the wave‐field extrapolation takes place in the coordinate frame of the depth section, and the CMP data serve as boundary conditions at the surface. Computations begin at the last sample of the record section and continue ‘reverse time’ until time zero.This paper describes a reverse‐time migration (RTM) method and compares its performance with that of an image‐space method based on the idea of phase shift plus interpolation (PSPI). Synthetic zero‐offset sections serve as examples for migration experiments with the RTM and PSPI methods. It is shown that the RTM approach to migration is rather expensive, but its robustness and accuracy are difficult to surpass.

Seismic migration on the IBM 3090 Vector Facility

Seismic prospecting aims at determining the structure of the earth from indirect measurements. Acoustic wave fields are generated at the surface, penetrate the earth, and are backscattered by the earth's inhomogeneities. The data recorded at the surface are processed in a complex sequence of steps among which seismic migration plays an important role. This is a wave depropagation process that permits the localization in depth of the origin of the diffraction events measured (in time) at the surface. This paper presents an overview of the major wave-equation migration methods. The most frequently executed algorithms or kernels on which the execution speed depends most crucially are given particular attention. The speedup resulting from scalar-to-vector formulation is presented over wide ranges of dimensionality for linear tridiagonal equation solvers, Fourier Transforms, and convolution operations. The vectorizability and resulting speedup are also addressed in the case of migration schemes known as the Phase-Shift Method and the Phase Shift Plus Interpolation (PSPI) Method. It is shown that Fourier domain migration based on the phase-shift concept lends itself conveniently to multilevel parallelism on the 3090 Vector Facility (VF): vectorization of the innermost loops and concurrent processing in the outer loops by means of the VS FORTRAN Version 2 Multitasking Facility.

Migration of seismic data by phase shift plus interpolation

Under the horizontally layered velocity assumption, migration is defined by a set of independent ordinary differential equations in the wavenumber‐frequency domain. The wave components are extrapolated downward by rotating their phases. This paper shows that one can generalize the concepts of the phase‐shift method to media having lateral velocity variations. The wave extrapolation procedure consists of two steps. In the first step, the wave field is extrapolated by the phase‐shift method using ℓ laterally uniform velocity fields. The intermediate result is ℓ reference wave fields. In the second step, the actual wave field is computed by interpolation from the reference wave fields. The phase shift plus interpolation (PSPI) method is unconditionally stable and lends itself conveniently to migration of three‐dimensional data. The performance of the methods is demonstrated on synthetic examples. The PSPI migration results are then compared with those obtained from a finite‐difference method.