声波最小二乘偏移不同精确伴随算子对的定量关系分析

最小二乘偏移（Least Squares Migration，LSM）是一种备受关注的高分辨率成像方法，它的成功应用取决于正演—偏移算子对的伴随特性。通常，正演—偏移算子对可以根据Born近似理论或逆时偏移（Reverse Time Migration，RTM）过程构建，也可以基于两者同时构建。波动方程离散化和数值实现方法会影响其伴随特性，通过点积测试可以对伴随特性进行数值检验。然而，不同伴随算子之间的关系和成像结果差异目前尚不清楚。为此，从二阶声波方程的矩阵表达式入手，推导出了三组精确伴随算子对。其中两组分别基于Born近似理论和RTM过程构建，第三组是利用声波方程的自伴随离散化形式构建，分别将它们命名为Born-AdjBorn、DeRTM-RTM和自伴随Born-RTM算子对。对应的LSM过程分别称为LSBM、LSRTM和自伴随LSBRTM。通过数学推导和矩阵分析，得出了基于三组伴随算子对的成像结果之间的一系列定量关系，并应用数值算例进行了验证。

Analysis of quantitative relations between different exact adjoint operator pairs in acoustic least-square migration

Least-square migration (LSM) is frequently mentioned in high-resolution imaging, whose successful application depends on the adjoint characteristic of forward-migration operator pairs. Normally, a forward-migration operator pair can be designed according to the Born approximation theory or/and reverse time migration (RTM) process. Its adjoint characteristic can be affected by the discretization and numerical implementation methods of wave equations, and the dot-product test can be used for the numerical test of this characteristic. However, the relations and the diffe-rences between the imaging results of different adjoint operator pairs are not clear. Considering this, three pairs of exact adjoint operators are derived by starting from the matrix expression of the second-order acoustic wave equation, two of which are constructed only on the basis of the Born approximation theory and the RTM process separately, and the third one is based on the self-adjoint discretization of the acoustic wave equation. They are named as Born-AdjBorn, DeRTM-RTM, and self-adjoint Born-RTM operator pairs, respectively, and the corresponding LSM processes are called LSBM, LSRTM, and self-adjoint LSBRTM, respectively. The matrix analysis and mathematical derivation indicate that a series of quantitative relations exist between the imaging results using the three operator pairs, and all these quantitative relations are validated by numerical experiments.