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https://github.com/compgeolab/euler-inversion

Paper: A new method for location sources of gravity and magnetic data using Euler's homogeneity equation
https://github.com/compgeolab/euler-inversion

earth-science geophysics geoscience gravity inverse-problems magnetic potential-fields

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Paper: A new method for location sources of gravity and magnetic data using Euler's homogeneity equation

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# Euler inversion: Locating sources of potential-field data through inversion of Euler’s homogeneity equation



by

[Leonardo Uieda](https://leouieda.com),

[Gelson Ferreira Souza-Junior](https://github.com/souza-junior),

[India Uppal](https://github.com/indiauppal),

[Vanderlei Coelho Oliveira Jr.](https://www.pinga-lab.org/people/oliveira-jr.html)



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. *Geophysical Journal International*.

> doi:[10.1093/gji/ggaf114](https://doi.org/10.1093/gji/ggaf114).



|                                    | **Information** |

|-----------------------------------:|:----------------|

| Version of record                  | https://doi.org/10.1093/gji/ggaf114 |

| Open-access preprint on EarthArXiv | https://doi.org/10.31223/X5T41M |

| Archive of this repository         | https://doi.org/10.6084/m9.figshare.26384140 |

| Software Heritage ID               | [`swh:1:snp:b0d1f8fdbf57f87e0ce56d5dda0f360c4a314d9d`](https://archive.softwareheritage.org/swh:1:snp:b0d1f8fdbf57f87e0ce56d5dda0f360c4a314d9d;origin=https://github.com/compgeolab/euler-inversion) |

| Reproducing our results            | [`REPRODUCING.md`](REPRODUCING.md) |



## About



The main idea for this paper came about during an potential-field methods class

which Leo took in 2012 with his then PhD supervisor

[Prof. Valéria C. F. Barbosa](https://www.pinga-lab.org/people/barbosa.html).

While learning about the Euler deconvolution method, which is a speciality of

Valéria, Leo connected it with the geodetic network adjustment theory he had

been taught by

[Prof. Spiros Pagiatakis](https://www.yorku.ca/spiros/spiros.html) during an

exchange program at York University, Canada, in 2008. An initial prototype was

developed in 2012 but there were still some rough edges and the project was

shelved to make way for other more urgent projects at the time. Leo returned to

this every few years, making slow progress, and involving Vanderlei in the

planning and discussion of the theory. In 2024, co-authors Gelson, India, and

Vanderlei joined Leo for a sprint to finish the method and produce this paper.



## Abstract



Earth 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 **Euler deconvolution**,

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 **structural index** 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, **Euler inversion**, 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.



  Left panel: Euler inversion is a new method for finding depths from gravity and magnetic data. It's much more robust to noise and interfering sources than Euler deconvolution and can estimate the structural index. Right panel: Map with red-white-blue colored dots representing the magnetic anomaly. There are several dipolar looking anomalies and some linear anomalies in the NE-SW direction. Overlaid are small triangles, circles, and squares which follow the dipolar and linear anomalies.

  

  Figure: Results of applying Euler inversion with a window

  size of 12 000 m and a window step of 2400 m to the aeromagnetic data from

  Rio de Janeiro, Brazil. Estimated source locations and structural indices

  obtained from Euler inversion are shown as triangles (𝜂 = 1), squares (𝜂

  = 2), and circles (𝜂 = 3). The colour of each symbol represents the estimated

  depth below the surface of the Earth (topography). Also shown are the

  total-field anomaly flight-line data, the contours of the post-collisional

  magmatism and alkaline intrusions (solid black lines) and dykes (dashed

  lines). The purple squares highlight the A, B, C, and D anomalies that are

  discussed in the text.

  



## Citing



Please cite this work as:



> 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. *Geophysical Journal International*.

> doi:[10.1093/gji/ggaf114](https://doi.org/10.1093/gji/ggaf114).



You can also reference the supplementary files (data and software) as:



> Uieda, L., Souza-Junior, G. F., Uppal, I., Oliveira Jr., V. C. (2024).

> Supplementary material for "Euler inversion: Locating sources of

> potential-field data through inversion of Euler's homogeneity equation".

> *figshare*. doi:[10.6084/m9.figshare.26384140](https://doi.org/10.6084/m9.figshare.26384140).



## License



All Python source code (including `.py` and `.ipynb` 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

`LICENSE-MIT.txt` 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)][cc-by]. See

`LICENSE-CC-BY.txt` for the full license text.



[cc-by]: https://creativecommons.org/licenses/by/4.0/