Full‐wavefield migration: using surface and internal multiples in imaging

Multiple scattering is usually ignored in migration algorithms, although it is a genuine part of the physical reflection response. When properly included, multiples can add to the illumination of the subsurface, although their crosstalk effects are removed. Therefore, we introduce full‐wavefield migration. It includes all multiples and transmission effects in deriving an image via an inversion approach. Since it tries to minimize the misfit between modeled and observed data, it may be considered a full waveform inversion process. However, full‐wavefield migration involves a forward modelling process that uses the estimated seismic image (i.e., the reflectivities) to generate the modelled full wavefield response, whereas a smooth migration velocity model can be used to describe the propagation effects. This separation of modelling in terms of scattering and propagation is not easily achievable when finite‐difference or finite‐element modelling is used. By this separation, a more linear inversion problem is obtained. Moreover, during the forward modelling, the wavefields are computed separately in the incident and scattered directions, which allows the implementation of various imaging conditions, such as imaging reflectors from below, and avoids low‐frequency image artefacts, such as typically observed during reverse‐time migration. The full wavefield modelling process also has the flexibility to image directly the total data (i.e., primaries and multiples together) or the primaries and the multiples separately. Based on various numerical data examples for the 2D and 3D cases, the advantages of this methodology are demonstrated.     

Wavefield Migration plus Monte Carlo Imaging of 3D Prestack Seismic Data

Prestack wave‐equation migration has proved to be a very accurate shot‐by‐shot imaging tool. However, 3D imaging with this technique of a large field acquisition, especially one with hundreds of thousands of shots, is prohibitively costly. Simply adapting the technique to migrate many superposed shot‐gathers simultaneously would render 3D wavefield prestack migration cost‐effective but it introduces uncontrolled non‐physical interference among the shot‐gathers, making the final image useless. However, it has been observed that multishot signal interference can be kept under some control by averaging over many such images, if each multishot migration is modified by a random phase encoding of the frequency spectra of the seismic traces. In this article, we analyse this technique, giving a theoretical basis for its observed behaviour: that the error of the image produced by averaging over M phase encoded migrations decreases as M^?1 . Furthermore, we expand the technique and define a general class of Monte‐Carlo encoding methods for which the noise variance of the average imaging condition decreases as M^?1 ; these methods thus all converge asymptotically to the correct reflectivity map, without generating prohibitive costs. The theoretical asymptotic behaviour is illustrated for three such methods on a 2D test case. Numerical verification in 3D is then presented for one such method implemented with a 3D PSPI extrapolation kernel for two test cases: the SEG–EAGE salt model and a real test constructed from field data.     

Virtual signals combination – phase analysis for 2D and 3D data representation (acoustic medium)

We formulate the Kirchhoff‐Helmholtz representation theory for the combination of seismic interferometry signals synthesized by cross‐correlation and by cross‐convolution in acoustic media. The approach estimates the phase of the virtual reflections from the boundary encompassing a volume of interest and subtracts these virtual reflections from the total seismic‐interferometry wavefield. The reliability of the combination result, relevant for seismic exploration, depends on the stationary‐phase and local completeness in partial coverage regions. The analysis shows the differences in the phase of the corresponding seismic interferometry (by cross‐correlation) and virtual reflector (by cross‐convolution) signals obtained by 2D and 3D formulations, with synthetic examples performed to remove water layer multiples in ocean bottom seismic (OBS) acoustic data.     

How rough sea affects marine seismic data and deghosting procedures

Most seismic processing algorithms generally consider the sea surface as a flat reflector. However, acquisition of marine seismic data often takes place in weather conditions where this approximation is inaccurate. The distortion in the seismic wavelet introduced by the rough sea may influence (for example) deghosting results, as deghosting operators are typically recursive and sensitive to the changes in the seismic signal. In this paper, we study the effect of sea surface roughness on conventional (5–160 Hz) and ultra‐high‐resolution (200–3500 Hz) single‐component towed‐streamer data. To this end, we numerically simulate reflections from a rough sea surface using the Kirchhoff approximation. Our modelling demonstrates that for conventional seismic frequency band sea roughness can distort results of standard one‐dimensional and two‐dimensional deterministic deghosting. To mitigate this effect, we introduce regularisation and optimisation based on the minimum‐energy criterion and show that this improves the processing output significantly. Analysis of ultra‐high‐resolution field data in conjunction with modelling shows that even relatively calm sea state (i.e., 15 cm wave height) introduces significant changes in the seismic signal for ultra‐high‐frequency band. These changes in amplitude and arrival time may degrade the results of deghosting. Using the field dataset, we show how the minimum‐energy optimisation of deghosting parameters improves the processing result.     

3D Fourier finite-difference migration by alternating-direction-implicit plus interpolation

Conventional two‐way splitting Fourier finite‐difference migration for 3D complex media yields azimuthal anisotropy where an additional phase correction is needed with much increase of computational cost. We incorporate the alternating‐direction‐implicit plus interpolation scheme into the conventional Fourier finite‐difference method to reduce azimuthal anisotropy. This scheme retains the high‐order remnants ignored by the two‐way splitting in the form of a wavefield interpolation in the wavenumber domain. The wavefield interpolation for each step of downward extrapolation is implemented between the wavefields before and after the conventional Fourier finite‐difference extrapolation. As the Fourier finite‐difference migration is implemented in the space and wavenumber dual space, the Fourier transforms between space and wavenumber domain that were needed for the alternating‐direction‐implicit plus interpolation in frequency domain (FD) migration are saved in Fourier finite‐difference migration. Since the azimuth anisotropy in Fourier finite‐difference is much less than that in FD, the application of the alternating‐direction‐implicit plus interpolation scheme in Fourier finite‐difference migration is superior to that in FD migration in handling complex media with large velocity contrasts and steep dips. Impulse responses show that the presented method reduces the azimuthal anisotropy at almost no extra cost.     

Water‐bottom multiple attenuation by Kirchhoff extrapolation

Despite being less general than 3D surface‐related multiple elimination (3D‐SRME), multiple prediction based on wavefield extrapolation can still be of interest, because it is less CPU and I/O demanding than 3D‐SRME and moreover it does not require any prior data regularization. Here we propose a fast implementation of water‐bottom multiple prediction that uses the Kirchhoff formulation of wavefield extrapolation. With wavefield extrapolation multiple prediction is usually obtained through the cascade of two extrapolation steps. Actually by applying the Fermat’s principle (i.e., minimum reflection traveltime) we show that the cascade of two operators can be replaced by a single approximated extrapolation step. The approximation holds as long as the water bottom is not too complex. Indeed the proposed approach has proved to work well on synthetic and field data when the water bottom is such that wavefront triplications are negligible, as happens in many practical situations.     

On the aperture effect in 3D Kirchhoff-type migration

It is well known that the migrated image given by a Kirchhoff-type (diffraction-stack) migration with limited aperture is always accompanied by some events which depend on the migration aperture. Although these events may severely affect the quality of migration, they have been studied only in 2D cases. Here, the events due to the migration aperture in 3D situations are investigated using a new method of analysing the reconstructed wavefield. It is found that a finite migration aperture results in a reconstructed wavefield with two components. One comes from the tangent points and curves between the traveltime surfaces of reflected and point-diffracted rays and is independent of the migration aperture, and the other is from the boundary of the migration aperture and depends strongly on the location and size as well as on the shape of the migration aperture. It is this last component that describes the aperture effect in migration. If the migration aperture is not sufficiently large, and if the input for migration is not zero on the boundary of the migration aperture, the boundary component may partially or totally cancel the migration signal. Furthermore, for synthetic data, the aperture effect cannot be eliminated by enlarging the migration aperture because, except for the common-shotpoint data, the aperture effect always exists however large the migration aperture becomes. This leads to the conclusion that the published Kirchhoff-type operators are not the exact inverse operators of the Fresnel–Kirchhoff integral if the input data are synthetic.     

2.5D full-wavefield viscoacoustic invercion1

Full-wavefield inversion for distributions of acoustic velocity, density and Q on a vertical slice through a25D model is implemented for common-source gathers in a cross-hole geometry. The wavefield extrapolation used is 3D, so all geometrical spreading, scattering, reflection, and transmission effects are correctly and automatically compensated for. In order to keep the number of unknowns tractable, application was limited to 2.5D models of known geometry; the latter assurnes a prior step, such as tomography, to fix the layer geometries. With the model geometry fixed, reliable solutions are obtained using synthetic data from only two independent source locations. Solutions from data with noisy and missing traces are comparable to those from noise-free data, but with higher residuals. When the source locations are spatially widely separated, conunon-source gathers may be summed and treated as a single wavefield to yield the same model estimates as when the individual source wavefields are treated separately, at substantially reduced cost. Inversions for full 3D parameter distributions can be handled with the same software, requiring only solution for more unknowns.     

基于Born散射理论的二维黏声介质高斯波束正演

Born散射理论可以通过省略高阶项实现针对一次散射波场的模拟.在这一理论的基础上,本文提出了一种针对二维黏声介质的一次散射波场高斯束Born正演方法.在该方法中,格林函数通过一系列不同初射方向的高斯波束累加获得,可以计算多至走时波场,保证了正演算法的计算精度.同时为了提高计算效率,正演方法使用了wavelet-bank方式合成局部平面波.区别于针对声波介质正演的wavelet-bank使用方法,文中将介质黏滞性信息融入了局部平面波的wavelet-bank合成方法中,以此实现针对黏声介质的快速一次散射波场模拟.两个模型的计算结果表明:本文提出的黏声介质高斯波束正演方法具有良好的计算精度以及较高的计算效率.     

PP amplitude bias caused by interface scattering: are diffracted waves guilty?

This paper is concerned with the problem of interpretation of anomalous seismic amplitudes, induced by the amplitude‐scattering phenomenon. This phenomenon occurs in the vicinity of a crack distribution at the interface between elastic layers. The purpose of this work is to obtain a better understanding of the physics of this distinctive phenomenon, in order to interpret correctly the amplitudes of the reflected events. By analogy with studies in optics and in acoustics, we suggest that diffraction is widely involved in the amplitude‐scattering phenomenon. Analytical evaluation of the amount of energy carried by the reflected and the diffracted waves shows that neglecting diffraction in numerical models leads to local underestimation of the amplitude of waves reflected at interfaces with gas‐filled crack distribution.     

Effective elastic properties of randomly fractured soils: 3D numerical experiments

This paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so‐called rotated staggered finite‐difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non‐intersecting thin penny‐shaped dry cracks, the numerical simulations of P‐ and S‐wave velocities are in good agreement with the predictions of the self‐consistent approximation. We observe similar results for fluid‐filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non‐intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective‐medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results.     

EFFICIENT 2D AND 3D SHOT RECORD REDATUMING1

In order to make 3D prestack depth migration feasible on modern computers it is necessary to use a target-oriented migration scheme. By limiting the output of the migration to a specific depth interval (target zone), the efficiency of the scheme is improved considerably. The first step in such a target-oriented approach is redatuming of the shot records at the surface to the upper boundary of the target zone. For this purpose, efficient non-recursive wavefield extrapolation operators should be generated. We propose a ray tracing method or the Gaussian beam method. With both methods operators can be efficiently generated for any irregular shooting geometry at the surface. As expected, the amplitude behaviour of the Gaussian beam method is better than that of the ray tracing based operators. The redatuming algorithm is performed per shot record, which makes the data handling very efficient. From the shot records at the surface‘genuine zero-offset data’are generated at the upper boundary of the target zone. Particularly in situations with a complicated overburden, the quality of target-oriented zero-offset data is much better than can be reached with a CMP stacking method at the surface. The target-oriented zero-offset data can be used as input to a full 3D zero-offset depth migration scheme, in order to obtain a depth section of the target zone.     

Evolution of deghosting process for single‐sensor streamer data from 2D to 3D

In marine acquisition, reflections of sound energy from the water–air interface result in ghosts in the seismic data, both in the source side and the receiver side. Ghosts limit the bandwidth of the useful signal and blur the final image. The process to separate the ghost and primary signals, called the deghosting process, can fill the ghost notch, broaden the frequency band, and help achieve high‐resolution images. Low‐signal‐to‐noise ratio near the notch frequencies and 3D effects are two challenges that the deghosting process has to face. In this paper, starting from an introduction to the deghosting process, we present and compare two strategies to solve the latter. The first is an adaptive mechanism that adjusts the deghosting operator to compensate for 3D effects or errors in source/receiver depth measurement. This method does not include explicitly the crossline slowness component and is not affected by the sparse sampling in the same direction. The second method is an inversion‐type approach that does include the crossline slowness component in the algorithm and handles the 3D effects explicitly. Both synthetic and field data examples in wide azimuth acquisition settings are shown to compare the two strategies. Both methods provide satisfactory results.     

裂陷盆地基底双界面模式二维重力反演

裂陷盆地基底的起伏表现为非光滑的几何形态,传统的重力反演结果并不能很好地反映这种特点.此外,大多数情况下,重力观测面并不位于盆地上界面,应为单独的起伏观测面,盆地应为上界面和基底组成的双界面模式.基于此,本文研究了起伏观测面上裂陷盆地基底双界面模式二维重力反演方法.研究中假设沉积盆地的沉积层与基底的密度差随深度按双曲线规律变化.将沉积盆地的沉积层剖分成相邻的垂直柱体,其水平尺寸是已知的,顶面与沉积层上界面重合,底面深度代表基底的深度,即为要反演的参数.反演中引入全变差函数作为盆地模型的约束,使得反演结果呈现非光滑形态,符合裂陷盆地基底特征.为减小反演多解性,引入已知深度点作为约束.建立由重力数据拟合、已知深度约束及全变差函数组成的目标函数,采用非线性共轭梯度算法使目标函数最小化.模型试算结果表明该方法可反演裂陷盆地基底起伏,并通过调整正则化参数的值可反演坳陷盆地基底起伏.将该反演方法用于珠江口盆地惠州凹陷和运城-临汾裂陷盆地实际资料处理,其结果较好地反映了裂陷盆地基底起伏特征,为研究盆地构造、油气勘探等提供重要参考.     

Sensitivity of seismic waves to structure

We study how the perturbations of a generally heterogeneous isotropic or anisotropic structure manifest themselves in the wavefield, and which perturbations can be detected within a limited aperture and a limited frequency band. A short-duration broad-band incident wavefield with a smooth frequency spectrum is considered. In-finitesimally small perturbations of elastic moduli and density are decomposed into Gabor functions. The wavefield scattered by the perturbations is then composed of waves scattered by the individual Gabor functions. The scattered waves are estimated using the first-order Born approximation with the paraxial ray approximation. For each incident wave, each Gabor function generates at most 5 scattered waves, propagating in specific directions and having specific polarisations. A Gabor function corresponding to a low wavenumber may generate a single broad-band unconverted wave scattered in forward or narrow-angle directions. A Gabor function corresponding to a high wavenumber usually generates 0 to 5 narrow-band Gaussian packets scattered in wide angles, but may also occasionally generate a narrow-band P to S or S to P converted Gaussian packet scattered in a forward direction, or a broad-band S to P (and even S to S in a strongly anisotropic background) converted wave scattered in wide angles. In this paper, we concentrate on the Gaussian packets caused by narrow-band scattering. For a particular source, each Gaussian packet scattered by a Gabor function at a given spatial location is sensitive to just a single linear combination of 22 values of the elastic moduli and density corresponding to the Gabor function. This information about the Gabor function is lost if the scattered wave does not fall into the aperture covered by the receivers and into the legible frequency band.     

Evaluation of 2D models for the prediction of surface depression storage using realistic reference values

Depression storage (DS) is the maximum storage of precipitation and runoff in the soil surface at a given slope. The DS is determined by soil roughness that in agricultural soils is largely affected by tillage. The direct measurement of DS is not straightforward because of the natural permeability of the soil. Therefore, DS has generally been estimated from 2D/3D empirical relationships and numerical algorithms based on roughness indexes and height measurements of the soil surface, respectively. The objective of this work was to evaluate the performance of some 2D models for DS, using direct and reliable measurements of DS in an agricultural soil as reference values. The study was carried out in experimental microplots where DS was measured in six situations resulting from the combination of three types of tillage carried out parallel and perpendicular to the main slope. Those data were used as reference to evaluate four empirical models and a numerical method. Longitudinal altitudinal profiles of the relief were obtained by a laser profilometer. Infiltration measurements were carried out before and after tillage. The DS was largely affected by tillage and its direction. Highest values of DS are found on rougher surfaces mainly when macroforms cut off the dominant slope. The empirical models had a limited performance while the numerical method was the most effective, even so, with an important variability. In addition, a correct hydrological management should take into account that each type of soil tillage affects infiltration rate differently. Copyright © 2016 John Wiley & Sons, Ltd.     

Directional‐oriented wavefield imaging: a new wave‐based subsurface illumination imaging condition for reverse time migration

The key objective of an imaging algorithm is to produce accurate and high‐resolution images of the subsurface geology. However, significant wavefield distortions occur due to wave propagation through complex structures and irregular acquisition geometries causing uneven wavefield illumination at the target. Therefore, conventional imaging conditions are unable to correctly compensate for variable illumination effects. We propose a generalised wave‐based imaging condition, which incorporates a weighting function based on energy illumination at each subsurface reflection and azimuth angles. Our proposed imaging kernel, named as the directional‐oriented wavefield imaging, compensates for illumination effects produced by possible surface obstructions during acquisition, sparse geometries employed in the field, and complex velocity models. An integral part of the directional‐oriented wavefield imaging condition is a methodology for applying down‐going/up‐going wavefield decomposition to both source and receiver extrapolated wavefields. This type of wavefield decomposition eliminates low‐frequency artefacts and scattering noise caused by the two‐way wave equation and can facilitate the robust estimation for energy fluxes of wavefields required for the seismic illumination analysis. Then, based on the estimation of the respective wavefield propagation vectors and associated directions, we evaluate the illumination energy for each subsurface location as a function of image depth point and subsurface azimuth and reflection angles. Thus, the final directional‐oriented wavefield imaging kernel is a cross‐correlation of the decomposed source and receiver wavefields weighted by the illuminated energy estimated at each depth location. The application of the directional‐oriented wavefield imaging condition can be employed during the generation of both depth‐stacked images and azimuth–reflection angle‐domain common image gathers. Numerical examples using synthetic and real data demonstrate that the new imaging condition can properly image complex wave paths and produce high‐fidelity depth sections.     

Wavefield interpolation in 3D large‐step Fourier wavefield extrapolation

Extrapolating 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.     

Low‐frequency normal wave propagation in a periodically layered medium with weak contrast in elastic properties

We derive the phase velocity dispersion and the scattering for wave vertically propagating in a periodically weak‐contrast horizontally layered medium with arbitrary number of layers in a period. Phase velocity dispersion is defined as the frequency dependence of vertical travel time, and scattering is defined as a reflection coefficient at the interface between the multilayered system and the corresponding Backus medium. Low‐frequency approximation is used to define a dynamic effective medium with frequency‐dependent phase velocity. The results are compared with those obtained earlier for a gradient medium. We show that the low‐frequency weak‐contrast approximation is valid for models with realistic contrasts in elastic properties.