1111 Mario E. Gimenez$ ^{1,2}$ , and Leonardo Uieda$ ^3 $
1212 }
1313 \\ [0.4cm]
14- {\small $ ^1 $ CONICET, Argentina}
14+ {\small $ ^1 $ Consejo Nacional de Investigaciones Científicas y Técnicas ( CONICET) , Argentina}
1515 \\
1616 {\small $ ^2 $ Instituto Geofísico Sismológico Volponi, Universidad Nacional de San Juan, Argentina}
1717 \\
@@ -133,7 +133,7 @@ \section{Introduction}
133133The literature offers two main approaches: one involves Taylor series expansion
134134\citep {Heck2007 , Grombein2013 } while the other makes use of Gauss-Legendre
135135Quadrature (GLQ)
136- \citep {Asgharzadeh2007 , Wild -Pfeiffer2008 , Li2011 , Uieda2016 , Lin2018 }.
136+ \citep {Asgharzadeh2007 , Wild -Pfeiffer2008 , Li2011 , Uieda2016 , Lin2019 }.
137137The Taylor series expansion is not well suited to develop an algorithm for
138138a density varying with depth according to an arbitrary continuous function.
139139Different series expansion terms would have to obtained for each density function
@@ -177,19 +177,19 @@ \section{Introduction}
177177differences.
178178\citet {Fukushima2018 } also generalized their method to tesseroids with a radial
179179polynomial density function of arbitrary degree.
180- \citet {Lin2018 } compared the different integration and discretization methodologies for
180+ \citet {Lin2019 } compared the different integration and discretization methodologies for
181181homogeneous tesseroids.
182182From this analysis they developed a combined method:
183183for computation points near the tesseroid, they use a GLQ integration with an adaptive
184184discretization based on \citet {Uieda2016 } but only applied to the horizontal dimensions.
185185If the computation point is farther than a certain truncation distance,
186186a second order Taylor series approximation is applied instead along with the regular
187187subdivision developed by \citet {Grombein2013 }.
188- \citet {Lin2018 } also introduced a variation of their combined method to compute the
188+ \citet {Lin2019 } also introduced a variation of their combined method to compute the
189189gravitational fields generated by tesseroids with a linearly varying density in the
190190radial dimension.
191191
192- Both the \citet {Lin2018 } and the \citet {Fukushima2018 } studies limit the radial density
192+ Both the \citet {Lin2019 } and the \citet {Fukushima2018 } studies limit the radial density
193193variation to polynomial functions.
194194While most continuous and smooth functions can be approximated by piecewise linear
195195functions, the choice of a discretization interval is neither straight forward nor
@@ -211,7 +211,7 @@ \section{Introduction}
211211generated by a tesseroid with an arbitrary continuous density function on an external
212212point.
213213It is based on the three dimensional GLQ integration, a two dimensional version of the
214- adaptive discretization of \citet {Uieda2016 } (following \citet {Lin2018 }),
214+ adaptive discretization of \citet {Uieda2016 } (following \citet {Lin2019 }),
215215and a new density-based radial discretization algorithm.
216216To ensure the accuracy of the numerical approximation, we empirically determine
217217optimal values for the controlling parameters by comparing the numerical results with
@@ -374,13 +374,13 @@ \subsection{Two Dimensional Adaptive Discretization}
374374Both algorithms perform tesseroid subdivisions in the latitudinal,
375375longitudinal and radial directions, thus we can define them as three-dimensional
376376adaptive discretization algorithms.
377- On the other hand, \citet {Lin2018 } proposed a two dimensional discretization algorithm
377+ On the other hand, \citet {Lin2019 } proposed a two dimensional discretization algorithm
378378that subdivides the tesseroid only on the latitudinal and longitudinal directions.
379379Removing a dimension from the discretization makes the computation more efficient by
380380reducing the number of tesseroids in the model, while
381- retaining an acceptable accuracy \citep {Lin2018 }.
381+ retaining an acceptable accuracy \citep {Lin2019 }.
382382
383- Here we will follow \citet {Lin2018 } and use a two dimensional version of the adaptive
383+ Here we will follow \citet {Lin2019 } and use a two dimensional version of the adaptive
384384discretization of \citet {Uieda2016 }.
385385What follows is a summary of the algorithm and the reader is referred to
386386\citet {Uieda2016 } for a detailed description.
@@ -656,7 +656,7 @@ \section{Determination of the distance-size and delta ratios}
656656order to obtain default values for the distance-size ratio $ D$ .
657657We will follow this idea but for our needs the spherical shell must
658658have the same density function of radius as our tesseroid model.
659- \citet {Lin2018 } show the analytical solution of the gravitational potential generated by
659+ \citet {Lin2019 } show the analytical solution of the gravitational potential generated by
660660a spherical shell with linear density in the radial coordinate.
661661Applying the Newton's Shell Theorem \citep {Chandrasekhar1995 , Binney2008 },
662662we derive expressions for the gravitational potential of a spherical shell with
@@ -1223,7 +1223,7 @@ \section{Discussion}
12231223to ensure accurate integration of the density function.
12241224This algorithm is independent of the GLQ integration and could potentially be used to
12251225determine an optimal discretization when approximating a density function by piecewise
1226- linear \citep {Lin2018 } or piecewise polynomial \citep {Fukushima2018 } functions.
1226+ linear \citep {Lin2019 } or piecewise polynomial \citep {Fukushima2018 } functions.
12271227
12281228Our numerical experiments show that the two dimensional adaptive discretization is
12291229enough to achieve 0.1\% accuracy with a second-order GLQ in the case of a linear density
@@ -1446,7 +1446,7 @@ \subsection{Linear density}
14461446by a spherical shell with variable density $ \rho (r') = ar'$ , while the second
14471447term constitutes the potential generated by a spherical shell with homogeneous
14481448density $ \rho = b$ \citep {Mikuska2006 ,Grombein2013 }.
1449- Eq~\ref {eq:shell-pot-linear } is in agreement with the one obtained by \citet {Lin2018 }.
1449+ Eq~\ref {eq:shell-pot-linear } is in agreement with the one obtained by \citet {Lin2019 }.
14501450
14511451\subsection {Exponential density }
14521452
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