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Target-oriented reverse time migration in transverse isotropy media

Abstract

Reverse time migration (RTM) is a high-precision imaging method for complex structures. However, without considering seismic anisotropy of the subsurface, RTM utilizing the anisotropic seismic data may produce blurred structure images with incorrect positions. Moreover, some exploration targets with insufficient illumination cannot be effectively identified in the migration profile, especially the subsalt structure which is usually the favorable petroleum region for the hydrocarbon reservoir. Therefore, we develop a target-oriented RTM in transverse isotropy media (TO-TIRTM). Instead of classical RTM, the novel method extracts wavefields that carry relatively more information about the exploration target to image structure. In the imaging condition, the constraint with excitation time is introduced to eliminate the interference of multipath on the image. Using synthetic examples, we determined that the kinematic characteristic of wavefield is closely related to anisotropic parameters, and the proposed method has prominent advantages over conventional RTM for imaging insufficient illumination structure.

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Acknowledgments

This research was supported by the Project of the Youth Foundation of Northeast Petroleum University (Grant No. 2019QNL-26).

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Correspondence to Ying Shi.

Additional information

Communicated by Michal Malinowski (CO-EDITOR-IN-CHIEF)/Sanyi Yuan (ASSOCIATE EDITOR).

Appendix 1: Extensions to TTI media

Appendix 1: Extensions to TTI media

When there is a dip angle between the symmetry axis and the vertical direction, the subsurface is from VTI to TTI media. We assume the Laplace and scalar operator as \(\nabla^{{\prime}{2}}\) and Sʹ in TTI media, then TOWE can be written as

$$\frac{{\partial^{2} u_{r} }}{{\partial t^{2} }} = \frac{{S^{\prime}}}{2}(v_{r}^{2} - v_{i}^{2} )\nabla^{{\prime}{2}} u_{r} - Sv_{r} v_{i} \nabla^{{\prime}{2}} u_{i} ,$$
(12)
$$\frac{{\partial^{2} u_{i} }}{{\partial t^{2} }} = \frac{{S^{\prime}}}{2}(v_{r}^{2} - v_{i}^{2} )\nabla^{{\prime}{2}} u_{i} + Sv_{r} v_{i} \nabla^{{\prime}{2}} u_{i} .$$
(13)

where \(\nabla^{{\prime}{2}}\) is given by

$$\nabla^{{\prime}{2}} = \frac{{\partial^{2} }}{{\partial x^{{\prime}{2}} }} + \frac{{\partial^{2} }}{{\partial {\text{z}}^{{\prime}{2}} }}.$$
(14)

Next, we show how to calculate the above operator in the xz coordinate system. In Fig. 

Fig. 14
figure14

Vector rotation diagram

14, let the vector v = (x, z) rotate θ round origin, we can get the vector vʹ = (xʹ, zʹ). The relationship between the two vectors is given by

$$\left( {\begin{array}{*{20}c} {x^{\prime}} \\ {z^{\prime}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} x \\ z \\ \end{array} } \right).$$
(15)

With the help of Eqs. (12) and (13), the operator \(\nabla^{{\prime}{2}}\) has the following form:

$$\begin{aligned} \frac{{\partial^{2} }}{{\partial x^{{\prime}{2}} }} & = \left( {\frac{\partial }{\partial x}\frac{\partial x}{{\partial x^{\prime}}} + \frac{\partial }{\partial z}\frac{\partial z}{{\partial x^{\prime}}}} \right)\left( {\frac{\partial }{\partial x}\frac{\partial x}{{\partial x^{\prime}}} + \frac{\partial }{\partial z}\frac{\partial z}{{\partial x^{\prime}}}} \right) \\ & = \cos^{2} \theta \frac{{\partial^{2} }}{{\partial^{2} x}} + \sin^{2} \theta \frac{{\partial^{2} }}{{\partial^{2} z}} - 2\sin \theta \cos \theta \frac{{\partial^{2} }}{\partial x\partial z} \\ \frac{\partial }{{\partial z^{{\prime}{2}} }} & = \left( {\frac{\partial }{\partial x}\frac{\partial x}{{\partial z^{\prime}}} + \frac{\partial }{\partial z}\frac{\partial z}{{\partial z^{\prime}}}} \right)\left( {\frac{\partial }{\partial x}\frac{\partial x}{{\partial z^{\prime}}} + \frac{\partial }{\partial z}\frac{\partial z}{{\partial z^{\prime}}}} \right) \\ & = \cos^{2} \theta \frac{{\partial^{2} }}{{\partial^{2} z}} + \sin^{2} \theta \frac{{\partial^{2} }}{{\partial^{2} x}} + 2\sin \theta \cos \theta \frac{{\partial^{2} }}{\partial x\partial z}. \\ \end{aligned}$$
(16)

Similarly, the scalar S' is

$$S^{\prime} = \frac{1}{2}\left[ {1 + 2\varepsilon n_{x}^{^{\prime}2} + \sqrt {(1 + 2\varepsilon n_{x}^{^{\prime}2} )^{2} - 8(\varepsilon - \delta )n_{x}^{^{\prime}2} n_{z}^{^{\prime}2} } } \right],$$
(17)

where \(n_{x}^{^{\prime}}\) and \(n_{z}^{^{\prime}}\) can be computed by

$$\left( {\begin{array}{*{20}c} {n_{x}^{^{\prime}} } \\ {n_{z}^{^{\prime}} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {n_{x} } \\ {n_{z} } \\ \end{array} } \right).$$
(18)

Therefore, we obtain the TOWE that can govern the wave propagation in TTI media.

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Song, L., Shi, Y., Chen, S. et al. Target-oriented reverse time migration in transverse isotropy media. Acta Geophys. 69, 125–134 (2021). https://doi.org/10.1007/s11600-020-00527-9

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Keywords

  • Reverse time migration
  • Transverse isotropy media
  • Target-oriented image
  • Excitation time