Wavefield Extrapolation Method Research Articles

Depth Migration Based on Two-Way Wave Equation to Image OBS Multiples: A Case Study in the South Shetland Margin (Antarctica)

With the development of marine seismic exploration, the ocean bottom seismometer (OBS) as a new seismic acquisition technology has been widely concerned. Although multiple waves are frequently viewed as noises, they may carry a wealth of subsurface information and produce a broader illumination than primary waves. To perform multiple wave imaging, we propose to utilize a two-way wave equation depth wavefield extrapolation method which is rarely used in this field. A simple dipping model is imaged by using primary and multiple waves, which proves the superiority of multiple waves in imaging over the primary waves and lays a foundation for practical application. Moreover, the comparison of multiple imaging results by reverse time migration and those by our proposed method demonstrates that our proposed method requires less storage space. In this study, we apply this migration method to actual OBS data collected in the South Shetland margin (Antarctica), where gas hydrates have been well documented. Firstly, the wavefield separation method is adopted to process the OBS data, so as to produce reliable primary and multiples waves; secondly, the ray-tracing method is used to derive the velocity field; and finally, the depth wavefield extrapolation method based on the two-way wave equation is applied to image primary and multiple waves. Migration results show that multiple waves provide a broader illumination and a clearer sediment structure than primary waves, especially for the highly shallow reflections.

Wavefield interpolation in 3D large‐step Fourier wavefield extrapolation

ABSTRACTExtrapolating wavefields and imaging at each depth during three‐dimensional recursive wave‐equation migration is a time‐consuming endeavor. For efficiency, most commercial techniques extrapolate wavefields through thick slabs followed by wavefield interpolation within each thick slab. In this article, we develop this strategy by associating more efficient interpolators with a Fourier‐transform‐related wavefield extrapolation method. First, we formulate a three‐dimensional first‐order separation‐of‐variables screen propagator for large‐step wavefield extrapolation, which allows for wide‐angle propagations in highly contrasting media. This propagator significantly improves the performance of the split‐step Fourier method in dealing with significant lateral heterogeneities at the cost of only one more fast Fourier transform in each thick slab. We then extend the two‐dimensional Kirchhoff and Born–Kirchhoff local wavefield interpolators to three‐dimensional cases for each slab. The three‐dimensional Kirchhoff interpolator is based on the traditional Kirchhoff formula and applies to moderate lateral velocity variations, whereas the three‐dimensional Born–Kirchhoff interpolator is derived from the Lippmann–Schwinger integral equation under the Born approximation and is adapted to highly laterally varying media. Numerical examples on the three‐dimensional salt model of the Society of Exploration Geophysicists/European Association of Geoscientists demonstrate that three‐dimensional first‐order separation‐of‐variables screen propagator Born–Kirchhoff depth migration using thick‐slab wavefield extrapolation plus thin‐slab interpolation tolerates a considerable depth‐step size of up to 72 ms, eventually resulting in an efficiency improvement of nearly 80% without obvious loss of imaging accuracy. Although the proposed three‐dimensional interpolators are presented with one‐way Fourier extrapolation methods, they can be extended for applications to general migration methods.

Seismic Reverse Time Migration Using A New Wave-Field Extrapolator and a New Imaging Condition

Prestack reverse time migration (RTM), as a two way wave-field extrapolation method, can image steeply dipping structures without any dip limitation at the expense of potential increase in imaging artifacts. In this paper, an efficient symplectic scheme, called Leapfrog-Rapid Expansion Method (L-REM), is first introduced to extrapolate the wavefield and its derivative in the same time step with high accuracy and free numerical dispersion using a Ricker wavelet of a maximum frequency of 25 Hz. Afterwards, in order to suppress the artifacts as a characteristic of RTM, a new imaging condition based on Poynting vector and a type of weighting function is presented. The capability of the proposed new imaging condition is then tested on synthetic data. The obtained results indicate that the proposed imaging condition is able to suppress the RTM artifacts effectively. They also show the ability of the proposed approach for improving the amplitude and compensate for illumination.

Frequency-dependent attenuation of compressional wave and seismic effects in porous reservoirs saturated with multi-phase fluids

Varying fluid types and saturations in porous media cause attenuation of compressional seismic waves and thus contribute to frequency-dependent reflections. We analyze the frequency-dependent attenuation and the corresponding seismic reflection behavior by taking into account the effects of the multi-phase fluid on the wave-induced fluid flow in porous reservoir. Fluid viscosity, the property that largely controls pore-fluid mobility, is proportional to the relaxation time, which in turn influences the frequency-dependent velocities. Here, we describe the variation of effective viscosity varies with saturation for a sandstone reservoir saturated for different multi-phase fluid mixtures using the Refutas equation. Then we explore the frequency-dependent velocity and inverse quality factors under different fluid saturation cases by employing equivalent-medium theory of Chapman's model. Next, we present a frequency-dependent phase-shift wavefield extrapolation method to simulate frequency-dependent seismic wavefield on 4-layer models. Comparative analyses indicate that the frequency-dependent attenuation and the seismic behaviors are functions of fluid viscosity associated with hydrocarbon saturations. Hydrocarbon saturations in fluid mixture strongly affect attenuation and characteristic frequencies. The frequency-dependent attenuation within the seismic frequency range can be increased due to the presence of hydrocarbons in multi-phase fluids. Synthetic seismic records indicate that the frequency-dependent attenuation of compressional wave and seismic reflection signatures are significantly dependent on hydrocarbon saturation, in terms of amplitude, waveform and traveltime of the reflection located at the base of the saturated reservoir and the underlying shale layer.

Angle gathers from reverse time migration using analytic wavefield propagation and decomposition in the time domain

Angle-domain common imaging gathers (ADCIGs) are important input data for migration velocity analysis and amplitude variation with angle analysis. Compared with Kirchhoff migration and one-way wave equation migration, reverse time migration (RTM) is the most accurate imaging method in complex areas, such as the subsalt area. We have developed a method to generate ADCIGs from RTM using analytic wavefield propagation and decomposition. To estimate the wave-propagation direction and angle by spatial Fourier transform during the time domain wave extrapolation, we have developed an analytic wavefield extrapolation method. Then, we decomposed the extrapolated source and receiver wavefields into their local angle components (i.e., local plane-wave components) and applied the angle-domain imaging condition to form ADCIGs. Because the angle-domain imaging condition is a convolution imaging condition about the source and receiver propagation angles, it is costly. To increase the efficiency of the angle-domain imaging condition, we have developed a local plane-wave decomposition method using matching pursuit. Numerical examples of synthetic and real data found that this method could generate high-quality ADCIGs. And these examples also found that the computational cost of this approach was related to the complexity of the source and receiver wavefields.

Fast 3D migration-velocity analysis by wavefield extrapolation using the prestack exploding-reflector model

In areas of complex geology, migration-velocity estimation should use methods that describe the complexity of wavefield propagation, such as focusing and defocusing, multipathing, and frequency-dependent velocity sensitivity. Migration-velocity analysis by wavefield extrapolation has the ability to address these issues because, in contrast to ray-based methods, it uses wavefields as carriers of information. However, its high cost and lack of flexibility with respect to model parametrization and to target-oriented analysis have prevented its routine industrial use. We overcome those limitations by using new types of wavefields as carriers of information: the image-space generalized wavefields. These wavefields are synthesized from a prestack image computed with wavefield-extrapolation methods, using the prestack exploding-reflector model. Cost of migration-velocity analysis (MVA) by wavefield extrapolation is decreased because only a small number of image-space generalized wavefields are necessary to accurately describe the kinematics of velocity errors and because these wavefields can be easily used in a target-oriented way. Flexibility is naturally incorporated because modeling these wavefields has as the initial conditions selected reflectors, which allow use of a horizon-based parametrization of the model space. In a 3D example of the North Sea, we show that using wavefields synthesized by the prestack exploding-reflector model greatly improves efficiency of MVA by wavefield extrapolation, while yielding a final migration-velocity model that is accurate as evidenced by well focused and structurally reasonable reflectors.

Subsalt imaging for exploration, production, and development: A review

The field of subsalt imaging has evolved rapidly in the last decade, thanks in part to the availability of low cost massive computing infrastructure, and also to the development of new seismic acquisition techniques that try to mitigate the problems caused by the presence of salt. This paper serves as an introduction to the special Geophysics section on Subsalt Imaging for E&P. The purpose of the special section is to bring together practitioners of subsalt imaging in the wider sense, i.e., not only algorithm developers, but also the interpretation community that utilizes the latest technology to carry out subsalt exploration and development. The purpose of the paper is in many ways pedagogical and historical. We address the question of what subsalt imaging is and discuss the physics of the subsalt imaging problem, especially the illumination issue. After a discussion of the problem, we then give a review of the main algorithms that have been developed and implemented within the last decade, namely Kirchhoff and Beam imaging, one-way wavefield extrapolation methods and the full two-way reverse time migration. This review is not meant to be exhaustive, and is qualitative to make it accessible to a wide audience. For each method and algorithm we highlight the benefits and the weaknesses. We then address the imaging conditions that are a fundamental part of each imaging algorithm. While we dive into more technical detail, the section should still be accessible to a wide audience. Gathers of various sorts are introduced and their usage explained. Model building and velocity update strategies and tools are presented next. Finally, the last section shows a few results from specific algorithms. The latest techniques such as waveform inversion or the “dirty salt” techniques will not be covered, as they will be elaborated upon by other authors in the special section. With the massive effort that the industry has devoted to this field, much remains to be done to give interpreters the accurate detailed images of the subsurface that are needed. In that sense the salt is still winning, although the next decade will most likely change this situation.

Seismic imaging using lateral adaptive windows

One-way wavefield extrapolation methods are used routinely in 3D depth migration algorithms for seismic data. Due to their efficient computer implementations, such one-way methods have become increasingly popular and a wide variety of methods have been introduced. In salt provinces, the migration algorithms must be able to handle large velocity contrasts because the velocities in salt are generally much higher than in the surrounding sediments. This can be a challenge for one-way wavefield extrapolation methods. We present a depth migration method using one-way propagators within lateral windows for handling the large velocity contrasts associated with salt-sediment interfaces. Using adaptive windowing, we can handle large perturbations locally in a similar manner as the beamlet propagator, thus limiting the impact of the errors on the global wavefield. We demonstrate the performance of our method by applying it to synthetic data from the 2D SEG/EAGE [Formula: see text] salt model and an offshore real data example.

Direct downward continuation from topography using explicit wavefield extrapolation

Downward-continuation migration algorithms are powerful tools for imaging complicated subsurface structures. However, they usually assume that extrapolation proceeds from a flat surface, whereas most land surveys are acquired over irregular surfaces. Our method downward continues data directly from topography using a recursive space-frequency explicit wavefield-extrapolation method. The algorithm typically handles strong lateral velocity variations by using the velocity value at each spatial position to build the wavefield extrapolator in which the depth step usually is kept fixed. To accommodate topographic variations, we build space-frequency wavefield extrapolators with laterally variable depth steps (LVDS). At each spatial location, the difference between topography and extrapolation depth is used to determine the depth step. We use the velocity and topographic values at each spatial lateral position to build extrapolators. The LVDS approach does not add more data nor does it require preprocessing prior to extrapolation. We implemented the LVDS method and applied it to a source profile prestack migration technique. We also implemented the previously developed zero-velocity layer approach to use for comparison. For both algorithms, we modeled the acoustic source as an approximate free-space Green’s function, not as a simple extrapolated spatial impulse. Tests on a synthetic data set modeled from rough topography and comparisons with the zero-velocity layer approach confirm the method’s effectiveness in imaging shallow and deep structures beneath rugged topography.

Wave equation-based imaging of mode converted waves in ultrasonic NDI, with suppressed leakage from nonmode converted waves

The value of imaging techniques in ultrasonic nondestructive inspection (NDI) to find and characterize defects in steel components has already been demonstrated. The imaging techniques based on the integral representation of the wave equation, the Rayleigh integrals for wave field extrapolation, are becoming feasible and attractive due to advances in array technology and due to faster computers. Known implementations are the total focusing method (TFM), the synthetic aperture focusing method (SAFT), and the inverse wave field extrapolation method (IWEX). In principle, these techniques compensate propagation effects from sources to a scatterer such as a defect and propagation effects from the scatterer to receivers. Currently, this approach is applied to wave fronts of single modes (pure longitudinal or pure transversal). In practice, multiple wave fronts from the scatterer will be received as a result of mode conversion. These arrivals will not have the same arrival time because of the difference in sound velocity between longitudinal and transversal waves. Images of mode converted waves are obtained by choosing the appropriate sound velocity that corresponds with the mode-converted scattered wave in the imaging process. Therefore, the nonmode converted waves will image as leakage artifacts in the mode-converted images, and vice versa. This may lead to false interpretations. In this paper, such artifacts will be identified and explained with the help of an analytical example. Measurements from steel test pieces with a 4 MHz linear array transducer with 64 elements will be used to demonstrate the artifacts. Furthermore, a procedure to predict the artifacts and the subsequent suppression from the input measurements will be presented and demonstrated.

Wave-equation angle-domain common-image gathers for converted waves

Wavefield-extrapolation methods can produce angle-domain common-image gathers (ADCIGs). To obtain ADCIGs for converted-wave seismic data, information about the image dip and the P-to-S velocity ratio must be included in the computation of angle gathers. These ADCIGs are a function of the half-aperture angle, i.e., the average between the incidence angle and the reflection angle. We have developed a method that exploits the robustness of computing 2D isotropic single-mode ADCIGs and incorporates both the converted-wave velocity ratio [Formula: see text] and the local image dip field. It also maps the final converted-wave ADCIGs into two ADCIGs, one a function of the P-incidence angle and the other a function of the S-reflection angle. Results with both synthetic and real data show the practical application for converted-wave ADCIGs. The proposed approach is valid in any situation as long as the migration algorithm is based on wavefield downward continuation and the final prestack image is a function of the horizontal subsurface offset.

Theory of Large-Step Wavefield Depth Extrapolation

Wavefield extrapolation is the foundation of seismic migration imaging. Finding a way of quick and object oriented wavefield extrapolation is significant not only for deep seismic exploration but also for imaging with super-tremendous seismic data acquired from non-combination receivers, both highly needed to develop in petroleum prospecting. To image target zones, as far as all present wavefield extrapolation methods are concerned, including travel-time based Dix-formula methods, ray tracing, small-step depth recursion, etc., the adaptability to complex media, precision and computing efficiency are not satisfying for practical needs in some respects. In this paper, we propose a new technique (so-called large-step wavefield depth extrapolation method), that is based on the Lie algebraic integral of operator's phase to compute depth extrapolation operator efficiently. In the method, the complex phase of object oriented wavefield extrapolation operator's symbol is expressed as a linear combination of wavenumbers, and the coefficients of linear combination are all in the form of the integral of interval velocity functions and their derivatives over depth. Moreover, the computation is convenient and needs less storage space. With phase shift plus correction, the wavefield extrapolation operator is implemented using FFT (Fast Fourier Transform) in spatial and wavenumber domains. Here, the kinetic equivalent relationship is resorted to derive the expression convenient for computation. We compare the precision of one-step scheme with that of two-step scheme for expanding an operator's primary symbol function, which illustrates the large-step scheme is more accurate than its recursive counterpart. Besides, in a linearly variant medium laterally and vertically, the point source pulse response of the large-step method is compared with that of depth recursion methods. The numerical examples indicate that travel-time precision of the large-step extrapolation operator mainly depends on the lateral velocity variation modification items in phase shift operator. In addition, in different approximation cases, the dispersion caused by the large-step operator is rather small.

Riemannian wavefield extrapolation

Riemannian spaces are described by nonorthogonal curvilinear coordinates. We generalize one-way wavefield extrapolation to semiorthogonal Riemannian coordinate systems that include, but are not limited to, ray coordinate systems. We obtain a one-way wavefield extrapolation method that can be used for waves propagating in arbitrary directions, in contrast to downward continuation, which is used for waves propagating mainly in the vertical direction. Ray coordinate systems can be initiated in many different ways; for example, from point sources or from plane waves incident at various angles. Since wavefield propagation happens mostly along the extrapolation direction, we can use inexpensive finite-difference or mixed-domain extrapolators to achieve high angle accuracy. The main applications of our method include imaging of steeply dipping or overturning reflections.

Common Angle Image Gathers Obtained from Gabordaubechies Beamlet Prestack Depth Migration

AbstractDue to the multi‐pathing of wave propagating through complex media, the problem of reflector location ambiguity is often encountered in the conventional offset‐domain common image gathers (CIGs). Angle‐domain CIGs have been proposed as a solution for such a problem and become a natural choice for velocity analysis, AVA and true amplitude migration/imaging. The development of wave‐theory based seismic migration and imaging methods provides reliable ways for the construction of angle‐domain CIGs of high quality. In particular, the beamlet‐domain wavefield extrapolation and migration method, in which both the wavefield decomposition functions and the extrapolation propagators bear localization properties in both space and direction, has been proved to be an effective tool for angle‐related studies. In this paper, beamlet prestack depth migration based on Gabor‐Daubechies frame decomposition is used for angle‐domain imaging. Through different stack procedures, both common reflection‐angle image gathers (CRAIG) and common dip‐angle image gathers (CDAIG) are constructed from the local angle image matrices. As a numerical example, the CRAIGs and CDAIGs for the SEG‐EAGE 2‐D salt model are calculated and analyzed. Based on their distinct features, the applications of these two CIGs to seismic imaging are discussed in detail.

3-D prestack depth migration by wavefield extrapolation methods

After a decade in which Kirchhoff migration has been the workhorse method for structural imaging, the industry has found that this method fails to image the most structurally challenging geology. Wavefield extrapolation based approaches have shown promise in complex structural imaging, but many of their assumed advantages have been found inadequate. In this paper, we review the progress made by these approaches, and we survey various implementations that are available. We also review several key issues that are critical if the wavefield extrapolation approach is to be useful in generating high-fidelity production-scale images. Some of these issues include: true amplitude formulation, imaging condition aliasing, sail line aliasing, and angle-gather computation. We use results from both synthetic and real data to illustrate the advantages and challenges facing wavefield extrapolation based migrations.

Prestack multicomponent migration

Prestack elastic migration by displacement potential extrapolation is a mixed, systematic, and function‐blocked vector wavefield migration algorithm. A new wavefield extrapolation method for inhomogeneous media is introduced here according to the following sequence: displacements ← potentials ← extrapolation of the potentials ← displacements, which is relatively accurate and not computer‐time intensive. Traveltimes of both direct downgoing P‐ and S‐waves, which are necessary in elastic migration, are calculated with a modified convolutional acoustic forward modeling program applicable to complex structures. A new image condition based on the time consistent principle is developed. It involves first obtaining an image condition section. Then two images (P P and S S) are obtained from the product of the extrapolated and decomposed P P‐ and S S‐wave displacement amplitudes and the image condition section. All P P‐, P S‐, S P‐ and S S‐waves are considered when the image condition section is calculated. The image condition section minimizes cross‐talk between modes. Compared to previous treatments, the newly developed image condition formula is superior since it allows migration of multicomponent seismic data produced using a combined P and S source. Numerical test results are very encouraging and clearly demonstrate the robustness of the technique. Further work is continuing so as to overcome ray angle and polarity problems in the image condition.

A new ray-tracing algorithm for arbitrary inhomogeneous and moving media, including caustics

In this paper, a new ray-tracing algorithm is presented that can handle arbitrary temperature and wind profiles in horizontal and vertical directions. This is accomplished by following a ray tube instead of a ray path. The algorithm is a direct finite difference implementation of the ray-tracing equations for arbitrary temperature and wind profiles. A computer model has been made, based on this algorithm. Special attention is given to the shortcomings of ray theory in the case of caustics, which are compensated for by appropriate phase and amplitude corrections of the computed ray tubes. The method is well suited to study the effect of locally heated areas on the sound propagation, as occurring on for instance petrochemical plants. Numerical experiments show that the proposed ray-tracing method and the relatively expensive, but accurate, wave field extrapolation method agree very well in many cases.

Optimal reconstruction algorithm for holographic imaging of finite-size objects

This paper introduces the formulation of the optimal image reconstruction algorithm for finite-size objects. For practical data acquisition and computation considerations, the partial discrete (continuous–discrete) model is used for the formulation. The basic constraints utilized in this optimization process are the finite size of the object and the limited spatial-frequency bandwidth of the resultant wave field. The result indicates that the algorithm consists of a linear matrix transformation followed by backward propagation. It is also shown that the bandlimited wave field extrapolation method and backward propagation technique are degenerate cases of this optimal algorithm. In addition to image reconstruction, the formulation of this optimal algorithm can be used to determine the optimal wave field sampling for efficient data acquisition.