International Conference «Inverse and Ill-Posed Problems of Mathematical Physics»,
dedicated to Professor M.M. Lavrent’ev in occasion of his 75-th birthday, August 20-25, 2007, Novosibirsk, Russia

True amplitude seismic imaging of VSP data based on Gaussian Beams

M. I. Protasov*, V.A. Tcheverda**,

* IPGG SB RAS, ** IPGG SB RAS,

Ak. Koptyug prosp., 3, Ak. Koptyug prosp., 3,

630090 Novosibirsk, Russia 630090 Novosibirsk, Russia

E-mail: E-mail:

The work of the first and second was supported by RFBR( grants 05-05-64277, 07-05-00538) and Lavrentiev grant of SB RAS, the work of the second was supported also by RFBR( grants 05-05-64277, 07-05-00538).

Abstract. The approach to true amplitude seismic imaging for walk-away Vertical Seismic Profile (VSP) data is presented and discussed. This approach is nothing else but wave field migration procedure based on weighted summation of pre-stack data. True-amplitude weights are computed with application of Gaussian beams (GB). In contrast to common approach to GB pre-stack migration (see, for example, (Hill, 2001)) we do not use these beams for composition of Green’s function. Instead we shoot a couple of properly chosen GB with fixed dip and opening angle from the current imaging point towards acquisition system. The use of these couple of beams provides one with true amplitude selective image of the rapid velocity variation within a target area. The total true amplitude image is constructed by superposition of selective images computed for a range of available dip angles. In order to estimate this range we perform preliminary survey design.

Shooting from the bottom overcomes difficulties due to multi arrivals of seismic energy being common for complicated velocity models (salt intrusions). In its own turn global regularity of Gaussian beams permits to avoid any troubles due to the possible singularities of the ray field.

Numerical experiments with synthetic data computed for Sigsbee2a salt model are presented and discussed.

Introduction. It is well-known that VSP migration is essentially a pre-stack migration techniques and is based on the following imaging condition: location of reflector/scatterer is allowed to positions where backward extrapolated receiver wave field is in the same phase as downward extrapolated source wave field. There are a lot of different ways for implementation of this extrapolation which produce variety of VSP imaging procedures (see, for example, (Fei and Liu, 2006)). But to the present day there are no true-amplitude versions of these procedures besides ones based on presented in (Miller et al., 1987) asymptotic inversion of Generalized Radon Transform (GRT). This approach essentially uses ray-based representation for of wave fields and meets troubles if there are singularities in rays for some source/receiver positions.

The techniques presented below can be treated as true amplitude imaging of pre-stack VSP data originated from the approach proposed and developed in (Tcheverda, Protasov (2006)) for surface multi-shot/multi-offset data. It provides persons with an image of the subsurface around or away from the borehole depending on the source offset. It is implemented by means of extrapolation of wave fields from acquisition system towards some fixed imaging point along specified Gaussian beams.

Imaging procedure. Our imaging procedure, as any other one, is based on decomposition of the velocity model onto macrovelocity component and its rapid perturbation (reflectivity/scatterer component) . In its own turn this decomposition of the model follows representation of the total wave field onto incident (downward) and reflected/scattered/diffracted (upward) wave fields – and respectively. The common approach in development of migration procedures is to use Born’s approximation in representation of upward waves.

Let us choose now some current interior point within a target area, shoot from this point a couple of rays and trace them within given macrovelocity model. One ray will be traced toward free surface (source positions) while another one toward vertical line (receiver positions) and construct a couple of GB attached to these rays ([5]). As we suppose macrovelocity model a priori known, both rays are uniquely determined by dip angle and opening angle (see Fig.1). Twice application of Green’s theorem gives the following identity ([5]):

As here we designate the reflected/scattered/diffracted wavefield . The right hand side of this identity can be treated as summation of raw walk-away VSP data with respect to source and receiver positions with specific weights. In order to construct the image we need to perform “summation” against time frequency as well. In order to find optimal weights we performed asymptotic analysis of Gaussian beams and recover that “summation” should be done “along” travel-time:

where and are travel-times from the current point within target area to source and receiver position for chosen couple of rays connected with specific Gaussian beams.

Straightforward computations lead to the following leading term of the high-frequency asymptotic of “image” computed after this summation at the current point:

/ (3)

It should be noted that this representation is taken after additional summation of walk-away VSP data with respect to dip angle . In order to recover available range of these angles we perform survey design by ray tracing from the target area towards acquisition system.

The first integration in the right hand side of (3) is performed along some domain we call below as “Set of Partial Reconstruction”:

/ (4)

In order to justify this denomination let us pay attention that image is nothing else, but superposition of forward and quasi-inverse two-dimensional Fourier transform of “normalized” version of rapid perturbation of macrovelocity component. It would be exact inverse Fourier transform and consequently would be exact normalized rapid perturbation if we dispose the total range of dip angles and time frequencies. Otherwise, we will image properly only constituents of reflectors/scatterers which possess spatial spectrum within specific Set of partial Reconstruction (4).

Numerical experiments were done for synthetic data generated by ourselves for famous Sigsbee2A velocity model. We use geometry of acquisition system presented on the Fig.2 (red straight lines) in order to recover reflectors/scatterers within target area. Data were synthesized by means of 2D finite-difference simulation.

Imaging result is introduced on the Fig.3 (right). There is demonstrated rather good quality of the geometrical structure in the target area. There are perfectly imaged faults and scatterer also. Horizontal reflection bounds can not be reconstructed in such acquisition system, but some of them are observed as a number of “diffracted objects”. The cause of the diffraction is a step-like structure of the horizontal bounds as you we see it in the true structure Fig.3 (left).

Conclusion. Gaussian beam imaging provides one with reliable representation of rapid perturbation of macro-velocity models: it is true amplitude procedure; it is free from caustic problem; it gives possibility to select reflection segment with respect to its spatial orientation. Main limitation of the Gaussian beam procedure is a correctness of the macrovelocity model. It has to be sufficiently correct and also it has to be sufficiently smooth or it has to have high contrast bounds that have small curvature with respect to wavelength. On more limitation that is general for all procedures is applied by acquisition system geometry and macro-velocity model also - they determine domain of partial reconstruction.

Acknowledgements. The research described in this publication was done in cooperation with Moscow Schlumberger Research and is partly supported by RFBR grants 05-05-64277, 07-05-00538 and Lavrentiev grant of SB RAS.We thank also Smaart Joint Venture providing us with SGSBEE2A data set.

References.

1.  Beylkin, G. [1985]. Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform: Journal of Mathematical Physics, 26, 99 – 108.

2.  Fei, T.W., and Liu, Q. [2006] Wave-equation migration for 3D VSP using phase-shift plus interpolation. Expanded abstracts of SEG/New Orleans 2006 Annual Meeting, 3457 – 3461.

  1. Hill N.R., [2001]. Prestack Gaussian beam depth migration. Geophysics, 66 (4), 1240 – 1250.

4.  Miller, D., Oristaglio, M., Beylkin, G. [1987] A new slant on seismic imaging: Migration and integral geometry. [1987] geophysics, 52 (7), 943 – 964.

5.  Tcheverda, V.A., Protasov, M.I. [2006]. True/preserving amplitude seismic imaging based on Gaussian beams application Expanded abstracts of SEG/New Orleans 2006 Annual Meeting, 2126 – 2130.

6.  Shiyong Xu, Shengwen Jin [2005] Can we image beneath salt body? – Target-oriented visibility analysis. Extended Abstracts of SEG Annual Meeting, 2005, Houston, USA, SPMI 6.8.

Fig.1. Geometry of the problem. / Fig.2. Target area and acquisition system placed within Sigsbee2A model.
Fig.3. One of the seismogramm. / Fig.4. Survey design. Reflectors that can be recovered in the target area. (Macro-velocity model )
Fig.5 True structure. Selective images and total true amplitude image.

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