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| Version of record |https://doi.org/10.1093/gji/ggaf114|
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| Open-access preprint on EarthArXiv |https://doi.org/10.31223/X5T41M|
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| Archive of this repository |https://doi.org/10.6084/m9.figshare.26384140|
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| Software Heritage ID |[`swh:1:snp:b0d1f8fdbf57f87e0ce56d5dda0f360c4a314d9d`](https://archive.softwareheritage.org/swh:1:snp:b0d1f8fdbf57f87e0ce56d5dda0f360c4a314d9d;origin=https://github.com/compgeolab/euler-inversion)|
which Leo took in 2012 with his then PhD supervisor
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[Prof. Valéria C. F. Barbosa](https://www.pinga-lab.org/people/barbosa.html).
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While learning about the Euler deconvolution method, which is a speciality of
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Valéria, Leo connected it with the geodetic network adjustment theory he had
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been taught by [Prof. Spiros
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Pagiatakis](https://www.yorku.ca/spiros/spiros.html) during an exchange program
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at York University, Canada, in 2008.
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An initial prototype was developed in 2012 but there were still some rough
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edges and the project was shelved to make way for other more urgent projects at
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the time.
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Leo returned to this every few years, making slow progress, and involving
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Vanderlei in the planning and discussion of the theory.
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In 2024, co-authors Gelson, India, and Vanderlei joined Leo for a sprint to
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finish the method and produce this paper.
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been taught by
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[Prof. Spiros Pagiatakis](https://www.yorku.ca/spiros/spiros.html) during an
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exchange program at York University, Canada, in 2008. An initial prototype was
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developed in 2012 but there were still some rough edges and the project was
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shelved to make way for other more urgent projects at the time. Leo returned to
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this every few years, making slow progress, and involving Vanderlei in the
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planning and discussion of the theory. In 2024, co-authors Gelson, India, and
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Vanderlei joined Leo for a sprint to finish the method and produce this paper.
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## Abstract
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Earth scientists can estimate the depth of certain rocks beneath Earth's
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surface by measuring the small disturbances that they cause in the Earth's
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gravity and magnetic fields. A popular method for this is **Euler
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deconvolution**, which is widely available in geoscience software and can be
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run quickly on a standard computer. Unfortunately, Euler deconvolution has some
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shortcomings: 1) the approximate shape of the rocks must be known, for example,
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a sphere or a wide flat slab, represented by the **structural index** 2) the
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depth of the rocks is not well estimated when there is noise in our data, which
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is a common occurrence. We propose a new method, **Euler inversion**, which
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fixes some of the shortcomings of Euler deconvolution by using more adequate
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(and complex) mathematics. Our method is less sensitive to noise in the data
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and is also able to determine the approximate shape of the source (the
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structural index). Euler inversion is also fast to execute on a standard
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computer, making it a practical alternative to Euler deconvolution on an Earth
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scientists toolbox.
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gravity and magnetic fields. A popular method for this is **Euler deconvolution**,
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which is widely available in geoscience software and can be run quickly on
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a standard computer. Unfortunately, Euler deconvolution has some shortcomings:
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1) the approximate shape of the rocks must be known, for example, a sphere or
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a wide flat slab, represented by the **structural index** 2) the depth of the
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rocks is not well estimated when there is noise in our data, which is a common
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occurrence. We propose a new method, **Euler inversion**, which fixes some of
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the shortcomings of Euler deconvolution by using more adequate (and complex)
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mathematics. Our method is less sensitive to noise in the data and is also able
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to determine the approximate shape of the source (the structural index). Euler
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inversion is also fast to execute on a standard computer, making it a practical
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alternative to Euler deconvolution on an Earth scientists toolbox.
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<figure>
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<imgsrc="https://github.com/compgeolab/euler-inversion/raw/main/graphical-abstract.jpg"alt="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.">
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<figcaption><strong>Figure:</strong> 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.</figcaption>
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<figcaption>
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<strong>Figure:</strong> Results of applying Euler inversion with a window
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size of 12 000 m and a window step of 2400 m to the aeromagnetic data from
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Rio de Janeiro, Brazil. Estimated source locations and structural indices
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obtained from Euler inversion are shown as triangles (𝜂 = 1), squares (𝜂
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= 2), and circles (𝜂 = 3). The colour of each symbol represents the estimated
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depth below the surface of the Earth (topography). Also shown are the
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total-field anomaly flight-line data, the contours of the post-collisional
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magmatism and alkaline intrusions (solid black lines) and dykes (dashed
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lines). The purple squares highlight the A, B, C, and D anomalies that are
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discussed in the text.
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</figcaption>
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</figure>
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## Citing
@@ -85,8 +95,8 @@ warranty, so long as you provide attribution to the authors. See
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`LICENSE-MIT.txt` for the full license text.
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The manuscript text (including all LaTeX files), figures, and data/models
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produced as part of this research are available under the[Creative Commons
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Attribution 4.0 License (CC-BY)][cc-by]. See`LICENSE-CC-BY.txt` for the full
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license text.
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produced as part of this research are available under the
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[Creative Commons Attribution 4.0 License (CC-BY)][cc-by]. See
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