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Update references and affiliation (#69)
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- Update reference to Lin2019
- Improve CONICET affiliation
- Update access date for Uieda2015 reference
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santisoler authored Jun 28, 2019
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24 changes: 12 additions & 12 deletions manuscript/manuscript.tex
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Mario E. Gimenez$^{1,2}$, and Leonardo Uieda$^3$
}
\\[0.4cm]
{\small $^1$ CONICET, Argentina}
{\small $^1$ Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina}
\\
{\small $^2$ Instituto Geofísico Sismológico Volponi, Universidad Nacional de San Juan, Argentina}
\\
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The literature offers two main approaches: one involves Taylor series expansion
\citep{Heck2007, Grombein2013} while the other makes use of Gauss-Legendre
Quadrature (GLQ)
\citep{Asgharzadeh2007, Wild-Pfeiffer2008, Li2011, Uieda2016, Lin2018}.
\citep{Asgharzadeh2007, Wild-Pfeiffer2008, Li2011, Uieda2016, Lin2019}.
The Taylor series expansion is not well suited to develop an algorithm for
a density varying with depth according to an arbitrary continuous function.
Different series expansion terms would have to obtained for each density function
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differences.
\citet{Fukushima2018} also generalized their method to tesseroids with a radial
polynomial density function of arbitrary degree.
\citet{Lin2018} compared the different integration and discretization methodologies for
\citet{Lin2019} compared the different integration and discretization methodologies for
homogeneous tesseroids.
From this analysis they developed a combined method:
for computation points near the tesseroid, they use a GLQ integration with an adaptive
discretization based on \citet{Uieda2016} but only applied to the horizontal dimensions.
If the computation point is farther than a certain truncation distance,
a second order Taylor series approximation is applied instead along with the regular
subdivision developed by \citet{Grombein2013}.
\citet{Lin2018} also introduced a variation of their combined method to compute the
\citet{Lin2019} also introduced a variation of their combined method to compute the
gravitational fields generated by tesseroids with a linearly varying density in the
radial dimension.

Both the \citet{Lin2018} and the \citet{Fukushima2018} studies limit the radial density
Both the \citet{Lin2019} and the \citet{Fukushima2018} studies limit the radial density
variation to polynomial functions.
While most continuous and smooth functions can be approximated by piecewise linear
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
point.
It is based on the three dimensional GLQ integration, a two dimensional version of the
adaptive discretization of \citet{Uieda2016} (following \citet{Lin2018}),
adaptive discretization of \citet{Uieda2016} (following \citet{Lin2019}),
and a new density-based radial discretization algorithm.
To ensure the accuracy of the numerical approximation, we empirically determine
optimal values for the controlling parameters by comparing the numerical results with
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Both algorithms perform tesseroid subdivisions in the latitudinal,
longitudinal and radial directions, thus we can define them as three-dimensional
adaptive discretization algorithms.
On the other hand, \citet{Lin2018} proposed a two dimensional discretization algorithm
On the other hand, \citet{Lin2019} proposed a two dimensional discretization algorithm
that subdivides the tesseroid only on the latitudinal and longitudinal directions.
Removing a dimension from the discretization makes the computation more efficient by
reducing the number of tesseroids in the model, while
retaining an acceptable accuracy \citep{Lin2018}.
retaining an acceptable accuracy \citep{Lin2019}.

Here we will follow \citet{Lin2018} and use a two dimensional version of the adaptive
Here we will follow \citet{Lin2019} and use a two dimensional version of the adaptive
discretization of \citet{Uieda2016}.
What follows is a summary of the algorithm and the reader is referred to
\citet{Uieda2016} for a detailed description.
Expand Down Expand Up @@ -656,7 +656,7 @@ \section{Determination of the distance-size and delta ratios}
order to obtain default values for the distance-size ratio $D$.
We will follow this idea but for our needs the spherical shell must
have the same density function of radius as our tesseroid model.
\citet{Lin2018} show the analytical solution of the gravitational potential generated by
\citet{Lin2019} show the analytical solution of the gravitational potential generated by
a spherical shell with linear density in the radial coordinate.
Applying the Newton's Shell Theorem \citep{Chandrasekhar1995, Binney2008},
we derive expressions for the gravitational potential of a spherical shell with
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to ensure accurate integration of the density function.
This algorithm is independent of the GLQ integration and could potentially be used to
determine an optimal discretization when approximating a density function by piecewise
linear \citep{Lin2018} or piecewise polynomial \citep{Fukushima2018} functions.
linear \citep{Lin2019} or piecewise polynomial \citep{Fukushima2018} functions.

Our numerical experiments show that the two dimensional adaptive discretization is
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
term constitutes the potential generated by a spherical shell with homogeneous
density $\rho = b$ \citep{Mikuska2006,Grombein2013}.
Eq~\ref{eq:shell-pot-linear} is in agreement with the one obtained by \citet{Lin2018}.
Eq~\ref{eq:shell-pot-linear} is in agreement with the one obtained by \citet{Lin2019}.

\subsection{Exponential density}

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18 changes: 13 additions & 5 deletions manuscript/references.bib
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Expand Up @@ -164,7 +164,7 @@ @Misc{Uieda2015
title = {A tesserioid (spherical prism) in a geocentric coordinate system with a local-{{North}}-oriented coordinate system},
howpublished = {figshare, available from: http://dx.doi.org/10.6084/m9.figshare.1495525},
year = {2015},
note = {Accessed 17 July 2017},
note = {Accessed June 2019},
doi = {10.6084/m9.figshare.1495525},
timestamp = {2016-03-01T20:04:20Z},
url = {http://dx.doi.org/10.6084/m9.figshare.1495525},
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publisher = {Springer},
}

@Article{Lin2018,
@Article{Lin2019,
author = {Lin, Miao and Denker, Heiner},
title = {On the computation of gravitational effects for tesseroids with constant and linearly varying density},
journal = {Journal of Geodesy},
year = {2018},
pages = {1--25},
publisher = {Springer},
year = {2019},
month = {May},
day = {01},
volume = {93},
number = {5},
pages = {723--747},
issn = {1432-1394},
doi = {10.1007/s00190-018-1193-4},
url = {https://doi.org/10.1007/s00190-018-1193-4}
}

@Article{Fukushima2018,
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year = {2013},
publisher = {{IEEE}},
doi = {10.1109/pesmg.2013.6672353},
ISSN={1932-5517},
month={July},
}

@InProceedings{Imamoto2008,
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