基于地震数据线性正演表达的波形成像(一):地震数据的线性正演表达

Waveform imaging based on a linear forward representation of seismic data—PartⅠ:linear forward representation of seismic data

地震数据的正演表达是构建地震波反演成像方法的基础。给定满足地震波运动学的准确的地下介质光滑模型,基于地震数据线性正演表达提出波形成像方法理论,不仅可修正当前以构造为主要目标的地震数据偏移成像方法理论的不足,还可实现对地下地层的岩性成像。在满足地震波运动学的准确的光滑模型下,地震波不发生散射、反射和波型转换,将扰动法应用于波动方程,得到描述地下非均匀体产生的扰动波场的波动方程。依据地震波长与非均匀体大小或非均匀体物理性质空间变化特征尺度之间的相对大小关系,将非均匀体划分为在空间上具有有限延续度的产生散射波的局部散射体或在空间上具有一定延续度的产生反射波的反射体,相应的描述扰动波场的波动方程转化为描述散射波场的波动方程或描述反射波场的波动方程。对于散射波,利用小扰动假定、Born近似,可得到基于散射体物性参数相对扰动的散射数据线性正演表达;对于反射波,利用波阻抗相对扰动的小值假定、一次波近似和高频近似,可得到基于反射体波阻抗相对扰动的反射数据线性正演表达。为了描述反射体边界对反射波的作用,在高频近似条件下定义反射体波阻抗相对扰动沿入射波传播方向的方向导数为反射体边界的局部反射系数,得到基于反射体边界局部反射系数的反射数据线性正演表达。最后给出具体的标量波、声波和各向同性弹性波方程下散射数据和反射数据的具体线性正演表达式。

The forward representation of seismic data is the basis of seismic wave inversion and imaging.Given a smooth subsurface model with accurate seismic wave kinematics,a theory of waveform imaging is proposed based on the linear forward representation of seismic data.This theory can not only correct the theoretical deficiency of the current seismic data migration imaging method,which takes the structure as the main target,but also allows the lithological imaging of subsurface strata.In this paper,the first part of the seismic data waveform imaging theory is introduced,and a linear forward representation of seismic data used for waveform imaging is proposed.In a smooth model with accurate seismic wave kinematics,no scattering,reflection,or wave mode conversion of seismic waves occur.Therefore,the perturbation method was applied to the wave equation,and a wave equation representing the perturbation wavefield generated by subsurface heterogeneity was obtained.According to the relationship between the seismic wavelength and the size of the heterogeneity or the characteristic scale of spatial variation of the physical properties representing the heterogeneity,the heterogeneity can be expressed as a local scatter with limited continuity in space that generates scattered waves or as a reflector with a certain continuity in space that generates reflected waves.The corresponding wave equation of the perturbation wavefield was transformed into a wave equation describing the scattered wavefield or into a wave equation describing the reflected wavefield.For scattered waves,a linear forward representation for scattering data based on the relative perturbation of the physical parameters of the scatterers can be obtained based on the small perturbation assumption and the Born approximation.For the reflected wave,a linear forward representation for reflection data based on the relative perturbation of the wave impedance of the reflector can be obtained by assuming a small value of the relative perturbation of the wave impedance,and relying on the primary wave and high-frequency approximations.To describe the effects of the reflector boundary on the reflection wave,the directional derivative of the relative perturbation of the reflector wave impedance along the direction of propagation of the incident wave was defined as the local reflection coefficient of the reflector boundary under the high-frequency approximation condition,and a linear forward representation for the reflection data based on the local reflection coefficient of the reflector boundary was obtained.Finally,the specific linear forward representations for scattering data and reflection data under scalar waves,acoustic waves,and isotropic elastic wave equations were given.