一种TI介质纯qP波正演方法及其在逆时偏移中的应用

Pure quasi-P wave forward modeling method in TI media and its application to RTM

基于Tsvankin提出的精确频散关系,利用近似展开的方法,推导出解耦合的TTI介质纯qP波近似方程,并将方程中的偏微分算子分解成一个laplace算子和一个标量算子,用于代表qP波的精确传播方向,构建时间域二阶纯qP波方程.此推导过程无需设置横波速度为零,能够更加精确地描述qP波的运动学特征.这个方程相比于求解波数域二阶解耦qP波方程,计算效率高,存储需求小;相比于基于Alkhalifah频散关系推导的时间域二阶纯qP波方程,假象干扰压制好,数值误差小,更具一般性.但此方法求解波矢量时采用波场梯度一阶渐近近似,会造成垂直于对称轴方向的波场振幅不准确.为了较正振幅,将椭圆分解方法应用于此方程中,构建纯qP波椭圆分解方程,使得振幅更加均衡,并与Xu等提出的方程比较分析,应用本文构建的纯qP波椭圆分解方程得到的波场振幅值更加准确.本文首先选取了均匀TI介质模型进行了qP波正演模拟,并抽取波场单道波形进行振幅分析,验证了本文构建的纯qP波方程和纯qP波椭圆分解方程的正确性及有效性;然后选取BP TTI模型进行了qP波正演模拟,将其qP波正演结果和均匀TI介质模型振幅分析结果相结合,突出了本文构建的纯qP波椭圆分解方程的优势及适应性;最后选取逆冲模型和BPTTI模型,应用本文构建的纯qP波椭圆分解方程对其进行逆时偏移成像,验证了本文构建的纯qP波椭圆分解方程在逆时偏移中的可行性和适用性.

Based on the exact dispersion relation proposed by Tsvankin, and adopting the approximate expanded method, this work derives the decoupled pure quasi-P wave equations in a TTI medium. Then we divide the pseudodifferential operator in the equations into a Laplace operator and a scalar operator to indicate the exact transmission direction of qP waves, and establish second-order pure qP wave equations in the time domain. In this induction process, there is no need to set S wave velocity at 0, which can describe the kinetic features of qP waves in a more accurate way. Compared with the second-order decoupled qP wave equations in the wavenumber domain, this method has a higher computation efficiency and lower storage demand. While compared with the second-order pure qP wave equations in the time domain which is based on derivation from Alkhalifah dispersion relation, it has better suppression of artifacts and less numeral error, being more general. But using this method to solve wave vectors is asymptotic approximation to adopting first-order of wavefield gradients, which might cause inaccurate amplitude of qP waves perpendicular to the axis. In order to correct amplitude and apply the elliptical decomposition method to these equations, pure qP wave elliptical decomposition equations have been derived, achieving more balanced amplitude. Compared with equations proposed by Xu, the wavefield amplitude derived from pure qP wave elliptical decomposition equations established in this paper is more accurate. Firstly, we choose a homogeneous TTI medium to conduct forward modeling of qP waves, and analyze the amplitude of single trace of the qP wavefield, verifying the correctness and effectiveness of the qP wave equations and qP wave elliptical decomposition equations constructed in this paper. Then, we choose the BP TTI model to conduct forward modeling of qP waves, and combine the results of stimulation and amplitude analysis of a homogeneous TTI medium, highlighting the advantages and adaptability of qP wave elliptical decomposition equations established in this paper. Lastly, we apply the qP wave elliptical decomposition wave equations derived in this paper to implement reverse time migration imaging for a thrust model and BP TTI model, which verifies feasibility of the equations and its applicability in RTM.