Vertical seismic profile migration using the Kirchhoff integral

Geophysics ◽  
1988 ◽  
Vol 53 (6) ◽  
pp. 786-799 ◽  
Author(s):  
P. B. Dillon

Wave‐equation migration can form an accurate image of the subsurface from suitable VSP data. The image’s extent and resolution are determined by the receiver array dimensions and the source location(s). Experiments with synthetic and real data show that the region of reliable image extent is defined by the specular “zone of illumination.” Migration is achieved through wave‐field extrapolation, subject to an imaging procedure. Wave‐field extrapolation is based upon the scalar wave equation and, for VSP data, is conveniently handled by the Kirchhoff integral. The migration of VSP data calls for imaging very close to the borehole, as well as imaging in the far field. This dual requirement is met by retaining the near‐field term of the integral. The complete integral solution is readily controlled by various weighting devices and processing strategies, whose worth is demonstrated on real and synthetic data.

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1166-1182 ◽  
Author(s):  
Irshad R. Mufti

Finite‐difference seismic models are commonly set up in 2-D space. Such models must be excited by a line source which leads to different amplitudes than those in the real data commonly generated from a point source. Moreover, there is no provision for any out‐of‐plane events. These problems can be eliminated by using 3-D finite‐difference models. The fundamental strategy in designing efficient 3-D models is to minimize computational work without sacrificing accuracy. This was accomplished by using a (4,2) differencing operator which ensures the accuracy of much larger operators but requires many fewer numerical operations as well as significantly reduced manipulation of data in the computer memory. Such a choice also simplifies the problem of evaluating the wave field near the subsurface boundaries of the model where large operators cannot be used. We also exploited the fact that, unlike the real data, the synthetic data are free from ambient noise; consequently, one can retain sufficient resolution in the results by optimizing the frequency content of the source signal. Further computational efficiency was achieved by using the concept of the exploding reflector which yields zero‐offset seismic sections without the need to evaluate the wave field for individual shot locations. These considerations opened up the possibility of carrying out a complete synthetic 3-D survey on a supercomputer to investigate the seismic response of a large‐scale structure located in Oklahoma. The analysis of results done on a geophysical workstation provides new insight regarding the role of interference and diffraction in the interpretation of seismic data.


Geophysics ◽  
1984 ◽  
Vol 49 (3) ◽  
pp. 250-264 ◽  
Author(s):  
L. R. Lines ◽  
A. Bourgeois ◽  
J. D. Covey

Traveltimes from an offset vertical seismic profile (VSP) are used to estimate subsurface two‐dimensional dip by applying an iterative least‐squares inverse method. Tests on synthetic data demonstrate that inversion techniques are capable of estimating dips in the vicinity of a wellbore by using the traveltimes of the direct arrivals and the primary reflections. The inversion method involves a “layer stripping” approach in which the dips of the shallow layers are estimated before proceeding to estimate deeper dips. Examples demonstrate that the primary reflections become essential whenever the ratio of source offset to layer depth becomes small. Traveltime inversion also requires careful estimation of layer velocities and proper statics corrections. Aside from these difficulties and the ubiquitous nonuniqueness problem, the VSP traveltime inversion was able to produce a valid earth model for tests on a real data case.


Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 723-729 ◽  
Author(s):  
Colin M. Sayers

Wide‐aperture walkaway vertical seismic profile (VSP) data acquired through transversely isotropic horizontal layers can be used to determine the P phase‐slowness surface, local to a receiver array in a borehole. In the presence of dip, errors in the slowness surface may occur if the medium is assumed to be layered horizontally. If the acquisition plane is oriented parallel to the dip direction, the derived slowness is too large for sources offset from the well in the down‐dip direction and too small for sources offset from the well in the up‐dip direction. For acquisition parallel to the strike of the layers, the recovery of the P phase‐slowness in the vicinity of the receiver array is excellent. It is therefore preferable to orient the walkaway VSP in the strike direction to estimate the anisotropic parameters of the medium in the vicinity of a receiver array. However, this may not be possible. If the dip direction of all layers has the same azimuth, the variation of walkaway traveltimes with azimuth has a simple form. This allows data from a single walkaway VSP extending both sides of a well to be inverted for the local anisotropic P phase‐slowness surface at the receivers even in the presence of dip. If data are acquired at more than one azimuth, the dip direction can be determined.


Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. S195-S203 ◽  
Author(s):  
Ruiqing He ◽  
Brian Hornby ◽  
Gerard Schuster

Interferometric migration of free-surface multiples in vertical-seismic-profile (VSP) data has two significant advantages over standard VSP imaging: (1) a significantly larger imaging area compared to migrating VSP primaries and (2) less sensitivity to velocity-estimation and static errors than other methods for migration of multiples. In this paper, we present a 3D wave-equation interferometric migration method that efficiently images VSP free-surface multiples. Synthetic and field data results confirm that a reflectivity image volume, comparable in size to a 3D surface seismic survey around the well, can be computed economically. The reflectivity image volume has less fold density and lower signal-to-noise ratio than that obtained by a conventional 3D surface seismic survey because of the relatively weak energy of multiples and the limited number of geophones in the well. However, the efficiency of this method for migrating VSP multiples suggests that it might sometimes be a useful tool for 4D seismic monitoring where reflectivity images can be computed quickly for each time-lapse survey.


Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. U19-U28 ◽  
Author(s):  
Robert J. Ferguson

To address the problems of irregular trace spacing and statics correction, simultaneous regularization and wave-equation statics (WE statics) are implemented by least-squares inversion. In general, inversion is found to be intractable in three dimensions, so series approximation is made to reduce significantly the number of required integrals. The resulting operator is suitable for direct inversion or for use with gradient methods. Real and synthetic data are used to determine the viability of the inversion. For synthetic data, even for severe velocity variation and topography, inversion converges to an acceptable solution, and aliasing is reduced significantly. Similarly, for real data, inversion is found to return an antialiased, regularized result with WE statics applied.


Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. B153-B165 ◽  
Author(s):  
Kai Yang ◽  
Hong-Ming Zheng ◽  
Li Wang ◽  
Yu-Zhu Liu ◽  
Fan Jiang ◽  
...  

An integrated wave-equation datuming scheme improves the imaging quality of seismic data from overthrust areas. It can be regarded as integrated because upward-layer replacement is included. In this scheme, data are downward continued to a nonplanar datum (such as the base of the weathering layer), followed by upward continuation from the nonplanar datum to a final planar datum using a one-way extrapolator. When compared with a Kirchhoff integral, this method can deal better with the strong lateral velocity variation within the near surface. After a test on synthetic data, the scheme is applied successfully to real 2D overthrust data acquired in the Qi-Lian foothills, western China. Compared with results using static corrections, integrated wave-equation datuming results lead to better reconstruction of the diffractions and reflections, more reliable migration-velocity analyses, and stronger stack and final depth images.


2019 ◽  
Vol 37 (1) ◽  
pp. 29
Author(s):  
Bruno dos Santos Silva ◽  
Ellen de Nazaré Souza Gomes

ABSTRACT. In the world, many unconventional hydrocarbon reservoirs have been found. This type of reservoir generally has anisotropic properties. The estimation of the anisotropy of the medium can give useful information about the reservoir, for example, one can obtain the information on the direction of fractures, these are related to the preferential flow. This information is important in deciding which direction to drill the well. Measurements of slowness and polarization of qP-wave obtained from VSP (vertical seismic profile) experiments allow estimating the anisotropy in the vicinity of a geophones inside the borehole. Using the perturbation theory, a weakly anisotropic medium can be modeled by first-order perturbation around an isotropic reference medium. The inversion scheme is based on a linear approximation which expresses the slowness and polarization in terms of WA (weak anisotropy) parameters. These parameters characterize the deviations of the anisotropic medium from a reference isotropic medium. In presented inversion scheme, we use the three components of the polarization, since we consider 3C (three-components) geophones, and only one of the slowness components, the one along the borehole direction, where is located the receiver array. In this work, the inversion scheme using VSP data of slowness and polarization from direct wave qP for the estimation of the parameters of weak anisotropy WA is analyzed considering the orientation of the horizontal borehole. Three different configurations for the sources are analyzed. The results are compared with results from vertical borehole. It has been found that only a group of components of the tensor of the WA parameters is well estimated and this group depend on the orientation of the borehole. On the other hand, the phase velocity determined from the WA parameter tensor is always well estimated in a 30_ cone around the borehole, regardless of the borehole orientation.Keywords: Local anisotropy, VSP multiazimuthal, linear inversion, survey design.RESUMO. Muitos reservatórios de hidrocarbonetos não convencionais tem sido encontrados. Esse tipo de reservatório geralmente tem propriedades anisotrópicas. A estimativa da anisotropia do meio pode fornecer informações úteis sobre o reservatório, como por exemplo, informações sobre a direção das fraturas que estão relacionadas a direção de fluxo preferencial. Esta informação é importante para decidir que direção furar o poço. Medidas de vagarosidade e polarização de ondas qP obtidas em levantamentos de VSP (vertical seismic profile) permitem estimar a anisotropia na vizinhança de um geofone dentro do poço. Usando a teoria da perturbação, um meio fracamente anisotrópico pode ser modelado como uma perturbação de primeira ordem em torno de um meio isotrópico de referência. O esquema de inversão baseia-se numa aproximação linear que expressa a vagarosidade e polarização em termos do parâmetros WA (fraca anisotropia). Esses parâmetros caracterizam os desvios do meio anisotrópico a partir de um meio isotrópico de referência. No esquema de inversão são usadas as três componentes do vetor de polarização, esta-se considerando geofones 3C (três componentes) e apenas uma componente do vetor de vagarosidade, a componente ao longo da direção de orientação do poço, onde estão localizados os receptores. Neste trabalho, é analisado o esquema de inversão são usados dados de vagarosidade e polarização de ondas qP diretas em experimentos de VSP considerando a orientação do poço horizontal. Três diferentes configurações para as fontes são estudadas. Os resultados foram comparados com os resultados obtidos considerando o poço vertical. Verifica-se que apenas um grupo de componentes do tensor dos parâmetros elásticos WA é bem estimado. Este grupo depende da orientação do poço. Por outro lado, a velocidade de fase determinada a partir dos parâmetros WA é sempre bem estimada em um cone de 30_ torno do poço, independente de sua orientação. Palavras-chave: Anisotropia Local, VSP multiazimutal, inversão linear, desenho de experimento.


Geophysics ◽  
1987 ◽  
Vol 52 (2) ◽  
pp. 151-173 ◽  
Author(s):  
C. P. A. Wapenaar ◽  
N. A. Kinneging ◽  
A. J. Berkhout

The acoustic approximation in seismic migration is not allowed when the effects of wave conversion cannot be neglected, as is often the case in data with large offsets. Hence, seismic migration should ideally be founded on the full elastic wave equation, which describes compressional as well as shear waves in solid media (such as rock layers, in which shear stresses may play an important role). In order to cope with conversions between those wave types, the full elastic wave equation should be expressed in terms of the particle velocity and the traction, because these field quantities are continuous across layer boundaries where the main interaction takes place. Therefore, the full elastic wave equation should be expressed as a matrix differential equation, in which a matrix operator acts on a full wave vector which contains both the particle velocity and the traction. The solution of this equation yields another matrix operator. This full elastic two‐way wave field extrapolation operator describes the relation between the total (two‐way) wave fields (in terms of the particle velocity and the traction) at two different depth levels. Therefore it can be used in prestack migration to perform recursive downward extrapolation of the surface data into the subsurface (at a “traction‐free” surface, the total wave field can be described in terms of the detected particle velocity and the source traction). Results from synthetic data for a simplified subsurface configuration show that a multiple‐free image of the subsurface can be obtained, from which the angle‐dependent P-P and P-SV reflection functions can be recovered independently. For more complicated subsurface configurations, full elastic migration is possible in principle, but it becomes computationally complex. Nevertheless, particularly for the 3-D case, our proposal has improved the feasibility of full elastic migration significantly compared with other proposed full elastic migration or inversion schemes, because our method is carried out per shot record and per frequency component.


Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. U1-U8 ◽  
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah

Macro-velocity model building is important for subsequent prestack depth migration and full-waveform inversion. Wave-equation migration velocity analysis uses the band-limited waveform to invert for velocity. Normally, inversion would be implemented by focusing the subsurface offset common-image gathers. We reexamine this concept with a different perspective: In the subsurface offset domain, using extended Born modeling, the recorded data can be considered as invariant with respect to the perturbation of the position of the virtual sources and velocity at the same time. A linear system connecting the perturbation of the position of those virtual sources and velocity is derived and solved subsequently by the conjugate gradient method. In theory, the perturbation of the position of the virtual sources is given by the Rytov approximation. Thus, compared with the Born approximation, it relaxes the dependency on amplitude and makes the proposed method more applicable for real data. We determined the effectiveness of the approach by applying the proposed method on isotropic and anisotropic vertical transverse isotropic synthetic data. A real data set example verifies the robustness of the proposed method.


Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. S93-S103 ◽  
Author(s):  
Jakob B. U. Haldorsen ◽  
W. Scott Leaney ◽  
Richard T. Coates ◽  
Steen A. Petersen ◽  
Helge Ivar Rutledal ◽  
...  

We evaluated a method for using 3C vertical seismic profile data to image acoustic interfaces located between the surface source and a downhole receiver array. The approach was based on simple concepts adapted from whole-earth seismology, in which observed compressional and shear wavefields are traced back to a common origin. However, unlike whole-earth and passive seismology, in which physical sources are imaged, we used the observed compressional and shear wavefields to image secondary sources (scatterers) situated between the surface source and the downhole receiver array. The algorithm consisted of the following steps: first, estimating the receiver compressional wavefield; second, using polarization to estimating the shear wavefield; third, deconvolving the shear wavefield using estimates of the source wavelet obtained from the direct compressional wave; fourth, the compressional and shear wavefields were back projected into the volume between the source and receivers; where, finally, an imaging condition was applied. When applied to rig-source VSP data acquired in an extended-reach horizontal well, this process was demonstrated to give images of formation features in the overburden, consistent with surface-seismic images obtained from the same area.


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