多频稀疏后向散射远场数据的声逆散射问题
Inverse acoustic scattering problems with multi-frequency sparse backscattering far field data
摘要：
研究了孤立方向多频后向散射远场图的声逆散射问题。底层物体可以是点状散射体、小散射体、扩展的不均匀体和障碍物。提出了一种快速、鲁棒的直接采样方法，用于底层目标的定位和形状重建。据我们所知，这是第一个数值方法逆障碍/介质散射问题的多频稀疏后向散射数据。在弱散射近似和基尔霍夫近似下，也得到了一些近似的唯一性结果。基于这两种近似，初步建立了直接采样法的理论基础。二维数值算例验证了直接采样方法的有效性和鲁棒性。特别是，数值重建表明，直接采样方法也非常好地超越这两个近似。
英文摘要：
The inverse acoustic scattering problems using multi-frequency backscattering far field patterns at isolated directions are studied. The underlying object could be point like scatterers, small scatterers, extended inhomogeneities and obstacles. A fast and robust direct sampling method is proposed for location and shape reconstruction of the underlying objects. To our best knowledge, this is the first numerical method for inverse obstacle/medium scattering problems with the multi-frequency sparse backscattering data. Some approximate uniqueness results have also been showed based on the weak scattering approximation and the Kirchhoff approximation. The theoretical basis of the direct sampling method is initially established based on these two approximations. Numerical examples in two dimensions are presented to validity the effectiveness and robustness of the proposed direct sampling method. In particular, the numerical reconstructions indicate that the direct sampling method also works very well beyond these two approximations.
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