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Update references and affiliation (#69)
- Update reference to Lin2019 - Improve CONICET affiliation - Update access date for Uieda2015 reference
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manuscript/manuscript.tex

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Mario E. Gimenez$^{1,2}$, and Leonardo Uieda$^3$
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}
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\\[0.4cm]
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{\small $^1$ CONICET, Argentina}
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{\small $^1$ Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina}
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\\
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{\small $^2$ Instituto Geofísico Sismológico Volponi, Universidad Nacional de San Juan, Argentina}
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\\
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The literature offers two main approaches: one involves Taylor series expansion
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\citep{Heck2007, Grombein2013} while the other makes use of Gauss-Legendre
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Quadrature (GLQ)
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\citep{Asgharzadeh2007, Wild-Pfeiffer2008, Li2011, Uieda2016, Lin2018}.
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\citep{Asgharzadeh2007, Wild-Pfeiffer2008, Li2011, Uieda2016, Lin2019}.
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The Taylor series expansion is not well suited to develop an algorithm for
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a density varying with depth according to an arbitrary continuous function.
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Different series expansion terms would have to obtained for each density function
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differences.
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\citet{Fukushima2018} also generalized their method to tesseroids with a radial
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polynomial density function of arbitrary degree.
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\citet{Lin2018} compared the different integration and discretization methodologies for
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\citet{Lin2019} compared the different integration and discretization methodologies for
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homogeneous tesseroids.
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From this analysis they developed a combined method:
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for computation points near the tesseroid, they use a GLQ integration with an adaptive
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discretization based on \citet{Uieda2016} but only applied to the horizontal dimensions.
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If the computation point is farther than a certain truncation distance,
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a second order Taylor series approximation is applied instead along with the regular
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subdivision developed by \citet{Grombein2013}.
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\citet{Lin2018} also introduced a variation of their combined method to compute the
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\citet{Lin2019} also introduced a variation of their combined method to compute the
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gravitational fields generated by tesseroids with a linearly varying density in the
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radial dimension.
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Both the \citet{Lin2018} and the \citet{Fukushima2018} studies limit the radial density
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Both the \citet{Lin2019} and the \citet{Fukushima2018} studies limit the radial density
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variation to polynomial functions.
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While most continuous and smooth functions can be approximated by piecewise linear
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functions, the choice of a discretization interval is neither straight forward nor
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generated by a tesseroid with an arbitrary continuous density function on an external
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point.
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It is based on the three dimensional GLQ integration, a two dimensional version of the
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adaptive discretization of \citet{Uieda2016} (following \citet{Lin2018}),
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adaptive discretization of \citet{Uieda2016} (following \citet{Lin2019}),
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and a new density-based radial discretization algorithm.
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To ensure the accuracy of the numerical approximation, we empirically determine
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optimal values for the controlling parameters by comparing the numerical results with
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Both algorithms perform tesseroid subdivisions in the latitudinal,
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longitudinal and radial directions, thus we can define them as three-dimensional
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adaptive discretization algorithms.
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On the other hand, \citet{Lin2018} proposed a two dimensional discretization algorithm
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On the other hand, \citet{Lin2019} proposed a two dimensional discretization algorithm
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that subdivides the tesseroid only on the latitudinal and longitudinal directions.
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Removing a dimension from the discretization makes the computation more efficient by
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reducing the number of tesseroids in the model, while
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retaining an acceptable accuracy \citep{Lin2018}.
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retaining an acceptable accuracy \citep{Lin2019}.
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Here we will follow \citet{Lin2018} and use a two dimensional version of the adaptive
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Here we will follow \citet{Lin2019} and use a two dimensional version of the adaptive
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discretization of \citet{Uieda2016}.
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What follows is a summary of the algorithm and the reader is referred to
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\citet{Uieda2016} for a detailed description.
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order to obtain default values for the distance-size ratio $D$.
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We will follow this idea but for our needs the spherical shell must
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have the same density function of radius as our tesseroid model.
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\citet{Lin2018} show the analytical solution of the gravitational potential generated by
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\citet{Lin2019} show the analytical solution of the gravitational potential generated by
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a spherical shell with linear density in the radial coordinate.
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Applying the Newton's Shell Theorem \citep{Chandrasekhar1995, Binney2008},
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we derive expressions for the gravitational potential of a spherical shell with
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to ensure accurate integration of the density function.
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This algorithm is independent of the GLQ integration and could potentially be used to
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determine an optimal discretization when approximating a density function by piecewise
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linear \citep{Lin2018} or piecewise polynomial \citep{Fukushima2018} functions.
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linear \citep{Lin2019} or piecewise polynomial \citep{Fukushima2018} functions.
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Our numerical experiments show that the two dimensional adaptive discretization is
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enough to achieve 0.1\% accuracy with a second-order GLQ in the case of a linear density
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by a spherical shell with variable density $\rho(r') = ar'$, while the second
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term constitutes the potential generated by a spherical shell with homogeneous
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density $\rho = b$ \citep{Mikuska2006,Grombein2013}.
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Eq~\ref{eq:shell-pot-linear} is in agreement with the one obtained by \citet{Lin2018}.
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Eq~\ref{eq:shell-pot-linear} is in agreement with the one obtained by \citet{Lin2019}.
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\subsection{Exponential density}
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manuscript/references.bib

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title = {A tesserioid (spherical prism) in a geocentric coordinate system with a local-{{North}}-oriented coordinate system},
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howpublished = {figshare, available from: http://dx.doi.org/10.6084/m9.figshare.1495525},
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year = {2015},
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note = {Accessed 17 July 2017},
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note = {Accessed June 2019},
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doi = {10.6084/m9.figshare.1495525},
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timestamp = {2016-03-01T20:04:20Z},
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url = {http://dx.doi.org/10.6084/m9.figshare.1495525},
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publisher = {Springer},
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}
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@Article{Lin2018,
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@Article{Lin2019,
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author = {Lin, Miao and Denker, Heiner},
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title = {On the computation of gravitational effects for tesseroids with constant and linearly varying density},
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journal = {Journal of Geodesy},
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year = {2018},
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pages = {1--25},
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publisher = {Springer},
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year = {2019},
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month = {May},
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day = {01},
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volume = {93},
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number = {5},
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pages = {723--747},
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issn = {1432-1394},
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doi = {10.1007/s00190-018-1193-4},
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url = {https://doi.org/10.1007/s00190-018-1193-4}
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}
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@Article{Fukushima2018,
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year = {2013},
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publisher = {{IEEE}},
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doi = {10.1109/pesmg.2013.6672353},
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ISSN={1932-5517},
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month={July},
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}
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@InProceedings{Imamoto2008,

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