This report extends our recent progress in tackling a challenging 3D inverse scattering problem governed by the Helmholtz equation. Our target application is to reconstruct dielectric constants, electric conductivities and shapes of front surfaces of objects buried very closely under the ground. These objects mimic explosives, like, e.g. antipersonnel land mines and improvised explosive devices. We solve a coefficient inverse problem with the backscattering data generated by a moving source at a fixed frequency. This scenario has been studied so far by our newly developed convexification method that consists in a new derivation of a boundary value problem for a coupled quasilinear elliptic system. However, in our previous work only the unknown dielectric constants of objects and shapes of their front surfaces were calculated. Unlike this, in the current work performance of our numerical convexification algorithm is verified for the case when the dielectric constants, the electric conductivities and those shapes of objects are unknown. By running several tests with experimentally collected backscattering data, we find that we can accurately image both the dielectric constants and shapes of targets of interests including a challenging case of targets with voids. The computed electrical conductivity serves for reliably distinguishing conductive and non-conductive objects. The global convergence of our numerical procedure is shortly revisited.

这份报告扩展了我们在解决由亥姆霍兹方程控制的具有挑战性的3D逆散射问题方面的最新进展。我们的目标应用是重建非常紧密埋在地下的物体的介电常数，电导率和其前表面的形状。这些物体模仿爆炸物，例如杀伤人员地雷和简易爆炸装置。我们用固定频率的移动源产生的反向散射数据解决系数反问题。到目前为止，已经通过我们新开发的凸化方法对这种情况进行了研究，该方法包括对耦合拟线性椭圆系统的边值问题的新推导。但是，在我们以前的工作中，仅计算出未知的物体介电常数及其表面形状。不同于此，在当前介电常数，电导率和物体形状未知的情况下，我们的数值凸算法的工作性能得到了验证。通过使用实验收集的反向散射数据进行多个测试，我们发现我们可以准确地成像感兴趣目标的介电常数和形状，包括具有空洞目标的具有挑战性的情况。计算出的电导率用于可靠地区分导电和非导电物体。我们很快将重新讨论我们的数值程序的全局收敛性。通过使用实验收集的反向散射数据进行多个测试，我们发现我们可以准确地成像感兴趣目标的介电常数和形状，包括具有空洞目标的具有挑战性的情况。计算出的电导率用于可靠地区分导电和非导电物体。我们很快将重新讨论我们的数值程序的全局收敛性。通过使用实验收集的反向散射数据进行多个测试，我们发现我们可以准确地成像感兴趣目标的介电常数和形状，包括具有空洞目标的具有挑战性的情况。计算出的电导率用于可靠地区分导电和非导电物体。我们很快将重新讨论我们的数值程序的全局收敛性。