Electrical resistivity tomography employs L₂ norm regularization in many applications. We developed the boundary-sharping inversion method based on the finite element methods and irregular grid approach, in which the contact areas of elements are used to weight the model parameters. Similar approaches have previously only been used for structured grids. We also designed an electrical resistivity tomography system in the laboratory to conduct experimental data tests. Focusing on the imaging of small-scale targets, we further compared the behaviour of various regularization schemes, including L₂, L₁ and L₀ norm stabilizers. Different control parameters were tested and analysed for approximated L₁ and L₀ norm stabilizers. Three-dimensional conductivity models were reconstructed from synthetic and experimental data. We found that reconstructed images obtained from smooth sparse regularization, such as L₁, and L₀ norm regularization are superior to L₂ norm regularization for high-contrast aquifer and block models. The synthetic and experimental data show that electrical resistivity tomography with smooth sparse regularization has the potential to improve the imaging of small-scale targets with sharp contours.

中文翻译：

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具有平滑稀疏正则化的电阻率层析成像

电阻率断层扫描在许多应用中采用L ₂范数正则化。我们在有限元方法和不规则网格方法的基础上开发了边界锐化反演方法，其中单元的接触面积被用来对模型参数进行加权。类似的方法以前仅用于结构化网格。我们还在实验室中设计了电阻率层析成像系统来进行实验数据测试。针对小尺度目标的成像，我们进一步比较了各种正则化方案的行为，包括L ₂、L ₁和L ₀规范稳定器。对于近似的L ₁和L ₀范数稳定器，测试和分析了不同的控制参数。三维电导率模型是根据合成和实验数据重建的。我们发现，对于高对比度含水层和块模型，从平滑稀疏正则化（例如L ₁和L ₀范数正则化）获得的重建图像优于L ₂范数正则化。合成和实验数据表明，具有平滑稀疏正则化的电阻率层析成像具有改善具有锐利轮廓的小尺度目标成像的潜力。