Variable density acoustic RTM of VSP data based on the time–space domain LS-based SFD method

Abstract

Vertical seismic profiling (VSP) can provide more abundant seismic wavefield information and better seismic data with high resolution and high quality for the complex underground geological structures compared with surface seismic data. Reverse time migration (RTM) method possesses significant advantages for the accurate identification of complex geological structures, and it’s considered to be the most accurate imaging method at present. Therefore, we develop a variable density acoustic RTM method which is applicable for VSP data to enhance the recognition capability of complex geological structures, and we also discuss different aspects of this proposed imaging method. Firstly, to effectively improve the modeling precision of seismic wavefields, the wavefield extrapolation of our VSP RTM method is realized by using an optimal staggered-grid finite difference (SFD) method to solve the variable density acoustic wave equation, because this optimal SFD method uses the least square (LS) method to optimize the objective function established by the time–space domain dispersion relation to estimate its difference coefficients. In other words, the time–space domain LS-based SFD method has higher numerical simulation accuracy for seismic modeling. Secondly, to effectively reduce the boundary reflections and storage requirements of our VSP RTM method, we adopt the PML absorbing boundary and the effective boundary storage strategy in the process of wavefield extrapolation. Finally, to strengthen the quality and precision of VSP RTM results, the depth imaging profile of a shot is calculated by the normalized cross-correlation imaging condition of sources which can effectively eliminate the source effects on RTM results, and Laplace filtering is applied to eliminate the imaging noises in final RTM results effectively. The imaging results of different models show the effectiveness of our RTM method for VSP data, and it can more accurately identify the complex underground geological structures compared with the RTM method for conventional surface seismic data.

Appendix

Different SFD methods have different numerical simulation accuracy. Generally speaking, the simulation accuracy of the SFD methods based on time–space domain dispersion relations is higher than that of the SFD methods based on space domain dispersion relations, and the simulation accuracy of the SFD methods based on LS method is higher than that of the SFD methods based on TE method. Combining with different dispersion relations and solving algorithms, four different SFD methods can be proposed. Except for the time–space domain LS-based SFD method used in our VSP RTM method, we briefly introduce the calculation equations of difference coefficients for the other three conventional SFD methods in “Appendix.”

According to the space domain dispersion relation and TE method, the space domain TE-based SFD method is developed and it estimates its difference coefficients by using the following equation (Liu and Sen 2009):

$$c_{m} = \frac{{\left( { - 1} \right)^{m + 1} }}{2m - 1}\prod\limits_{1 \le n \le M,n \ne m} {\left| {\frac{{\left( {2n - 1} \right)^{2} }}{{\left( {2n - 1} \right)^{2} - \left( {2m - 1} \right)^{2} }}} \right|} ,m = 1,2, \ldots ,M.$$

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Based on the time–space domain dispersion relation and TE method, the time–space domain TE-based SFD method is proposed which can simultaneously obtain the highest $2M$-order simulation accuracy along eight directions, and its difference coefficients can be obtained by using the following equation (Liu and Sen 2011; Ren and Liu 2015):

$$\left[ {\begin{array}{*{20}c} {1^{0} } & {3^{0} } & \cdots & {\left( {2M - 1} \right)^{0} } \\ {1^{2} } & {3^{2} } & \cdots & {\left( {2M - 1} \right)^{2} } \\ \vdots & \vdots & \ddots & \vdots \\ {1^{2M - 2} } & {3^{2M - 2} } & \cdots & {\left( {2M - 1} \right)^{2M - 2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {1c_{1} } \\ {3c_{2} } \\ \vdots \\ {\left( {2M - 1} \right)c_{M} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 \\ {f_{2} } \\ \vdots \\ {f_{M} } \\ \end{array} } \right]$$

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and

$$\begin{aligned} f_{n} & = \frac{{\left( {\sum\nolimits_{j = 1}^{n} {\alpha_{j} \alpha_{n + 1 - j} } } \right)r^{2n - 2} - \sum\nolimits_{j = 2}^{n - 1} {\left[ {f_{j} f_{n + 1 - j} \left( {d_{j} d_{n + 1 - j} + e_{j} e_{n + 1 - j} } \right)} \right]} }}{{2\left( {d_{1} d_{n} + e_{1} e_{n} } \right)}},\quad \left( {n = 2,3, \ldots ,M} \right) \\ f_{1} & = 1,\alpha_{n} = {{\left( { - 1} \right)^{n - 1} } \mathord{\left/ {\vphantom {{\left( { - 1} \right)^{n - 1} } {\left( {2n - 1} \right)!}}} \right. \kern-\nulldelimiterspace} {\left( {2n - 1} \right)!}} \\ d_{n} & = \left( {\cos \theta } \right)^{2n - 1} \alpha_{n} ,e_{n} = \left( {\sin \theta } \right)^{2n - 1} \alpha_{n}. \\ \end{aligned}$$

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To improve the simulation accuracy in large wave number range, the space domain LS-based SFD method is proposed by the space domain dispersion relation and LS method, and it calculates its difference coefficients by using the following equation (Liu 2014):

$$\sum\limits_{m = 2}^{M} {\left[ {\int_{0}^{b} {\psi_{m} \left( \beta \right)\psi_{n} \left( \beta \right){\text{d}}\beta } } \right]c_{m} } = \int_{0}^{b} {g\left( \beta \right)\psi_{n} \left( \beta \right){\text{d}}\beta } ,\,\left( {n = 2,3, \ldots ,M} \right)$$

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$$c_{1} = 1 - \sum\limits_{m = 2}^{M} {\left( {2m - 1} \right)c_{m} }$$

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and

$$\begin{gathered} \psi_{m} \left( \beta \right) = 2\left\{ {\sin \left[ {\left( {m - 0.5} \right)\beta } \right] - 2\left( {m - 0.5} \right)\sin \left( {0.5\beta } \right)} \right\} \\ g\left( \beta \right) = \beta - 2\sin \left( {0.5\beta } \right) \\ \end{gathered}.$$

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