基于声波散射理论的零井源距VSP波场数值模拟方法

Numerical simulation of zero-offset VSP wave field based on the acoustic scattering theory

为了精确求解零井源距VSP波场记录,基于体积分方程提出了一种预条件最小二乘的数值计算方法。根据声波散射理论,将频率域一维声波方程通过格林函数表达为Lippman-Schwinger积分方程;在此基础上利用矩形积分公式将第二类Fredholm积分方程的求解问题转化为大型线性方程组的求解,为后续的精确计算奠定了基础;然后,使用最小二乘方法直接求解全波场,而不是通过迭代方法逐步将散射分量加入到全波场中;通过引入预条件算子,使数值计算过程更稳定;最后,以阶梯状模型和用实际测井数据建立的模型为例,利用提出的方法求解VSP波场,并与常用的有限差分方法结果对比,验证了利用所提方法进行零井源距VSP波场模拟的有效性。

Anumerical method of preconditioned least squares is proposed based on volume integral equation to accurately simulate zero-offset VSP wave field.First,based on the acoustic scattering theory,the one-dimensional acoustic wave equation in frequency domain can be expressed as the LippmanSchwinger integral equation by the Green function.On this base how to solve the second kind of Fredholm integral equation can be turned to how to sovle large linear equations through rectangular integral formula,and this lays a foundation for subsequent accurate calculation.Second,the whole wave field can be solved by the least squares method,rather than adding the scattering component to the whole wave field step by step by iteration.After introducing the preconditioned operator,the numerical calculation process is more stable.Finally,taking a stepped model and a real model based on logging data as examples,the VSP wave field can be solved using the new method,and the results were compared with those from the other conventional finite difference methods.The results show that the new method is feasible,effective and accurate in simulating zero-offset VSP wave field.