用地质雷达数据资料反演二维地下介质的方法

Inversion of ground-penetrating radar data for 2D electric parameters

从二维麦克斯韦方程组出发推导出反演介电常数和电导率等二维介质物性参数的反演公式.反演的步骤是:建立初始猜测模型,利用电磁波时间域有限差分法模拟正演数据,用正演数据与观测数据之间的数据残差建立目标函数,通过引入一个由麦克斯韦方程计算的伴随场,将目标函数对介质参数的导数表示成显式形式,应用最优化理论得出对初始猜测模型的修改,用共轭梯度法迭代,最终得到反演结果.用合成数据反演具有粗糙地表的非导电介质的介电常数,用实验数据同时反演介电常数和电导率,并比较了麦克斯韦方程反演结果与声波方程反演结果、波动方程偏移剖面的差异.

This paper derives inversion formulas for reconstructing electric parameters of a medium from observed ground-penetrating radar （GPR） data based on the two-dimensional （2D） Maxwell＇ s equations. By introducing an adjoint field calculated using the Maxwell＇ s equations, the gradients of the objective function with respect to unknown parameters can be expressed explicitly. The procedure of inversion is, first of all, to build up an initial model and to calculate guessed data. Secondly, to calculate adjoint fields by solving the adjoint wave equations, and thirdly to calculate the gradients and to modify the model. These steps are repeated until an acceptable result is obtained. In the modification of the model, the conjugate gradient method is used. Three reconstruction examples are illustrated. A permittivity profile is reconstructed from synthetic data received on a rough surface. A permittivity and a conductivity profile are reconstructed simultaneously from experimental data. These results are compared with the result obtained by using an inversion method based on the scalar wave equation where absorption is neglected, and with a migration profile obtained by using a wave equation migration method.