Supplementary material for "Euler inversion: Locating sources of potential-field data through inversion of Euler's homogeneity equation"

This repository contains the data and source code used to produce the results presented in:Uieda, L., Souza-Junior, G. F., Uppal, I., Oliveira Jr., V. C. (2025). Euler inversion: Locating sources of potential-field data through inversion of Euler’s homogeneity equation. <i>Geophysical Journal International</i>. doi:10.1093/gji/ggaf114.AbstractEarth scientists can estimate the depth of certain rocks beneath Earth's surface by measuring the small disturbances that they cause in the Earth's gravity and magnetic fields. A popular method for this is <b>Euler deconvolution</b>, which is widely available in geoscience software and can be run quickly on a standard computer. Unfortunately, Euler deconvolution has some shortcomings: 1) the approximate shape of the rocks must be known, for example, a sphere or a wide flat slab, represented by the <b>structural index</b> 2) the depth of the rocks is not well estimated when there is noise in our data, which is a common occurrence. We propose a new method, <b>Euler inversion</b>, which fixes some of the shortcomings of Euler deconvolution by using more adequate (and complex) mathematics. Our method is less sensitive to noise in the data and is also able to determine the approximate shape of the source (the structural index). Euler inversion is also fast to execute on a standard computer, making it a practical alternative to Euler deconvolution on an Earth scientists toolbox.LicenseAll Python source code (including <code>.py</code> and <code>.ipynb</code> files) is made available under the MIT license. You can freely use and modify the code, without warranty, so long as you provide attribution to the authors. See <code>LICENSE-MIT.txt</code> for the full license text.The manuscript text (including all LaTeX files), figures, and data/models produced as part of this research are available under the Creative Commons Attribution 4.0 License (CC-BY). See <code>LICENSE-CC-BY.txt</code> for the full license text.

本仓库包含用于复现下述论文成果的数据与源代码：Uieda, L.、Souza-Junior, G. F.、Uppal, I.、Oliveira Jr., V. C.（2025）. 《Euler反演：通过欧拉齐次方程反演定位位场数据震源》（Euler inversion: Locating sources of potential-field data through inversion of Euler’s homogeneity equation）. 《国际地球物理杂志（Geophysical Journal International）》. doi:10.1093/gji/ggaf114.

摘要
地球科学家可通过测量地球重力与磁场的微小扰动，估算地下特定岩体的埋藏深度。当前主流的此类方法为**欧拉反褶积（Euler deconvolution）**，该方法已广泛集成于地球科学专业软件中，可在标准计算机上快速运行。但欧拉反褶积存在两处明显局限：其一，需预先知晓岩体的近似形态（如球体或宽缓平板），该形态由**结构指数（structural index）**表征；其二，当数据存在噪声时（这一实际勘探中十分常见的情况），其深度估算结果精度欠佳。

为此，我们提出一种全新方法——**Euler反演（Euler inversion）**，其通过采用更合理且更复杂的数学模型，弥补了欧拉反褶积的部分缺陷。该方法对数据噪声的鲁棒性更强，还可自动确定震源的近似形态（即结构指数）。此外，Euler反演在标准计算机上同样可快速运行，是地球科学家科研工具箱中可替代欧拉反褶积的实用方案。

许可证
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本研究产出的手稿文本（含所有LaTeX文件）、图表及数据/模型采用知识共享署名4.0国际许可协议（CC-BY）授权。完整许可证文本详见`LICENSE-CC-BY.txt`。