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Use the Latex article class instead of GJI and insert the self-archiving
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Add citation information for version of record and its DOI.
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16 changes: 13 additions & 3 deletions README.md
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Expand Up @@ -7,8 +7,18 @@ Mario E. Gimenez,
Leonardo Uieda

This paper has been accepted for publication in *Geophysical Journal International*.
The version of record

An archived version of this repository is available at
> Soler, S. R., Pesce, A., Gimenez, M. E., & Uieda, L., 2019. Gravitational field
> calculation in spherical coordinates using variable densities in depth,
> *Geophysical Journal International*,
> doi:[10.1093/gji/ggz277](https://doi.org/10.1093/gji/ggz277)
is available online at: [doi.org/10.1093/gji/ggz277](https://doi.org/10.1093/gji/ggz277)

**This repository contains the data and code used to produce all results and figures shown
in the paper.**
An archived version of this repository is available at
[doi.org/10.6084/m9.figshare.8239622](https://doi.org/10.6084/m9.figshare.8239622)

We introduce a novel methodology for gravity forward modeling in spherical coordinates
Expand Down Expand Up @@ -148,9 +158,9 @@ All source code is made available under a BSD 3-clause license. You can freely
use and modify the code, without warranty, so long as you provide attribution
to the authors. See `LICENSE.md` for the full license text.
Data and the results of numerical tests are available under the
Data and the results of numerical tests are available under the
[Creative Commons Attribution 4.0 License (CC-BY)](https://creativecommons.org/licenses/by/4.0/).
The manuscript text and figures are not open source. The authors reserve the
The manuscript text and figures are not open source. The authors reserve the
rights to the article content, which has been accepted for publication in
Geophysical Journal International.
172 changes: 105 additions & 67 deletions manuscript/manuscript.tex
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@@ -1,38 +1,71 @@
%\documentclass[extra]{gji}
\documentclass[extra, referee]{gji}
\documentclass[twocolumn]{article}

\newcommand{\Title}{
Gravitational field calculation in spherical coordinates using variable
densities in depth
}
\newcommand{\Author}{S.R. Soler, A. Pesce, M.E. Gimenez, L. Uieda}
\newcommand{\AuthorAffil}{
{\large
Santiago R. Soler$^{1,2,*}$, Agustina Pesce$^{1,2}$,
Mario E. Gimenez$^{1,2}$, and Leonardo Uieda$^3$
}
\\[0.4cm]
{\small $^1$ CONICET, Argentina}
\\
{\small $^2$ Instituto Geofísico Sismológico Volponi, Universidad Nacional de San Juan, Argentina}
\\
{\small $^3$ Department of Earth Sciences, SOEST, University of Hawai'i at M\={a}noa, USA}
\\
{\small $^*$ e-mail: santiago.r.soler@gmail.com}
}
\newcommand{\DOI}{doi:\href{https://doi.org/10.1093/gji/ggz277}{10.1093/gji/ggz277}}
\newcommand{\DOILink}{\href{https://doi.org/10.1093/gji/ggz277}{doi.org/10.1093/gji/ggz277}}


\usepackage[left=0.7in,right=0.7in,top=1in,bottom=1in]{geometry}
\setlength{\columnsep}{2\columnsep}
\usepackage[utf8]{inputenc}
\usepackage{timet}
\usepackage{amsmath}
\usepackage{graphicx}

\usepackage[round]{natbib}
\usepackage{fixltx2e}
\usepackage{url}
\usepackage[pdftex,colorlinks=true]{hyperref}
\hypersetup{
allcolors=blue,
pdftitle={\Title},
pdfauthor={\Author},
}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\lhead{
\fontsize{9pt}{12pt}\selectfont
\Author{}, 2019. \DOI{}
}
\rhead{\fontsize{9pt}{12pt}\selectfont \thepage}
\renewcommand{\headrulewidth}{0pt}

\begin{document}

\title[Variable Density Tesseroids]{
Gravitational field calculation in spherical coordinates using variable
densities in depth
\title{\Title}
\author{\AuthorAffil}
\date{
\normalsize
Accepted 2019 June 05. Received 2019 May 10; in original form 2018 December 29
\\[0.4cm]
This is a pre-copyedited, author-produced PDF of an article accepted for
publication in \textit{Geophysical Journal International} following peer review.
The version of record
``\textit{Soler, S. R., Pesce, A., Gimenez, M. E., \& Uieda, L., 2019.
\Title{}, Geophysical Journal International, \DOI{}}\ ''
is available online at: \DOILink{}
}
\author[S.R. Soler, A. Pesce, M.E. Gimenez, and L. Uieda]{
Santiago R. Soler$^{1,2}$, Agustina Pesce$^{1,2}$,
Mario E. Gimenez$^{1,2}$, and Leonardo Uieda$^3$ \\
$^1$CONICET, Argentina.~e-mail: santiago.r.soler@gmail.com\\
$^2$Instituto Geofísico Sismológico Volponi, Universidad Nacional de
San Juan, Argentina\\
$^3$Department of Earth Sciences, SOEST, University of Hawai‘i at
M\={a}noa, Honolulu, Hawaii, USA
}


\maketitle

\begin{summary}
\begin{abstract}
We present a new methodology to compute the gravitational fields generated by
tesseroids (spherical prisms) whose density varies with depth according to
an arbitrary continuous function.
Expand Down Expand Up @@ -63,12 +96,11 @@
the accuracy of the results at the expense of computational speed.
Lastly, we apply this new methodology to model the Neuqu\'en Basin, a foreland basin in
Argentina with a maximum depth of over 5000~m, using an exponential density function.
\end{summary}

\begin{keywords}
\\[0.5cm]
\textbf{Keywords:}
Numerical modelling, Numerical approximations and analysis, Gravity anomalies
and Earth structure, Satellite gravity
\end{keywords}
\end{abstract}


\section{Introduction}
Expand Down Expand Up @@ -112,7 +144,7 @@ \section{Introduction}

\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{figures/tesseroid-uieda.pdf}
\includegraphics[width=\linewidth]{figures/tesseroid-uieda.pdf}
\caption{
A tesseroid (spherical prism) in a geocentric spherical coordinate system, with a
computation point $P$ and its local north oriented Cartesian coordinate system.
Expand Down Expand Up @@ -267,7 +299,7 @@ \subsection{Gauss-Legendre Quadrature integration}
included in the integration and evaluated on the Legendre polynomial roots
(i.e.~quadrature nodes).

\iftwocol{
%\iftwocol{
\begin{equation}
\begin{split}
\int\limits_{\lambda_1}^{\lambda_2}
Expand All @@ -284,21 +316,21 @@ \subsection{Gauss-Legendre Quadrature integration}
\end{split}
\label{eq:glq-var-dens}
\end{equation}
}{
\begin{equation}
\int\limits_{\lambda_1}^{\lambda_2}
\int\limits_{\phi_1}^{\phi_2}
\int\limits_{r_1}^{r_2}
\rho(r') f(r', \phi', \lambda')
dr' d\phi' d\lambda' \approx
A
\sum\limits_{i=1}^{N^r}
\sum\limits_{j=1}^{N^\phi}
\sum\limits_{k=1}^{N^\lambda}
W_i^r W_j^\phi W_k^\lambda \rho(r_i) f(r_i, \phi_j, \lambda_k),
\label{eq:glq-var-dens}
\end{equation}
}
%}{
%\begin{equation}
%\int\limits_{\lambda_1}^{\lambda_2}
%\int\limits_{\phi_1}^{\phi_2}
%\int\limits_{r_1}^{r_2}
%\rho(r') f(r', \phi', \lambda')
%dr' d\phi' d\lambda' \approx
%A
%\sum\limits_{i=1}^{N^r}
%\sum\limits_{j=1}^{N^\phi}
%\sum\limits_{k=1}^{N^\lambda}
%W_i^r W_j^\phi W_k^\lambda \rho(r_i) f(r_i, \phi_j, \lambda_k),
%\label{eq:glq-var-dens}
%\end{equation}
%}

\noindent where

Expand Down Expand Up @@ -679,8 +711,10 @@ \section{Determination of the distance-size and delta ratios}
The horizontal dimensions of the tesseroids and the total number of
tesseroids in the shell model are given in the latitudinal and longitudinal
dimensions, respectively.
\newline
}
\label{tab:shell-models}
\centering
\begin{tabular}{rccccc}
Thickness & Tesseroid size & Number of tesseroids \\ \hline
0.1 km & $30^\circ \times 30^\circ$ & $6 \times 12 = 72$ \\
Expand All @@ -691,13 +725,15 @@ \section{Determination of the distance-size and delta ratios}
\end{tabular}
\end{table}

\begin{table}
\begin{table*}
\caption{
Description of the computation grids used to characterize the accuracy of the
numerical integration.
Grid height is defined above the mean Earth radius.
\newline
}
\label{tab:grids}
\centering
\begin{tabular}{lccc}
Name & Grid spacing & Grid region (degrees) & Grid height (km)
\\ \hline
Expand All @@ -706,7 +742,7 @@ \section{Determination of the distance-size and delta ratios}
Global & $ 10^\circ$ & 180W / 180E / 90S / 90N & 0 \\
Satellite & $ 10^\circ$ & 180W / 180E / 90S / 90N & 260 \\
\end{tabular}
\end{table}
\end{table*}


\subsection{Linear Density}
Expand Down Expand Up @@ -821,11 +857,11 @@ \subsection{Exponential Density}

\begin{figure}
\centering
\iftwocol{
%\iftwocol{
\includegraphics[width=\linewidth]{figures/exponential-densities.pdf}
}{
\includegraphics[width=0.5\linewidth]{figures/exponential-densities.pdf}
}
%}{
%\includegraphics[width=0.5\linewidth]{figures/exponential-densities.pdf}
%}
\caption{
Exponential density functions assigned to the spherical shell models for
$\delta$ ratio determination.
Expand Down Expand Up @@ -878,13 +914,13 @@ \subsubsection{$D$-$\delta$ space exploration}

\begin{figure}
\centering
\iftwocol{
%\iftwocol{
\includegraphics[width=\linewidth]
{figures/grid-search.pdf}
}{
\includegraphics[width=0.5\linewidth]
{figures/grid-search.pdf}
}
%}{
%\includegraphics[width=0.5\linewidth]
%{figures/grid-search.pdf}
%}
\caption{
Numerical error exploration in the $D$-$\delta$ space.
The percentage difference values were obtained from the comparison between the
Expand Down Expand Up @@ -988,11 +1024,11 @@ \subsection{Sinusoidal Density}

\begin{figure}
\centering
\iftwocol{
%\iftwocol{
\includegraphics[width=\linewidth]{figures/sine-densities.pdf}
}{
\includegraphics[width=0.5\linewidth]{figures/sine-densities.pdf}
}
%}{
%\includegraphics[width=0.5\linewidth]{figures/sine-densities.pdf}
%}
\caption{
Sinusoidal density functions assigned to the spherical shells in the $\delta$ ratio
determination.
Expand Down Expand Up @@ -1110,11 +1146,11 @@ \section{Application to the Neuqu\'en Basin}

\begin{figure}
\centering
\iftwocol{
%\iftwocol{
\includegraphics[width=\linewidth]{figures/neuquen-basin-densities.pdf}
}{
\includegraphics[width=0.5\linewidth]{figures/neuquen-basin-densities.pdf}
}
%}{
%\includegraphics[width=0.5\linewidth]{figures/neuquen-basin-densities.pdf}
%}
\caption{
Linear and exponential densities used to compute the gravitational fields generated
by a tesseroid model of the Neuqu\'en sedimentary basin.
Expand Down Expand Up @@ -1426,10 +1462,10 @@ \subsection{Exponential density}

\begin{equation}
\begin{split}
V_\text{exp}(r) = \frac{4\pi G}{r} \frac{A}{k^3} \Big[
& \left( R_1^2 k^2 + 2 R_1 k + 2 \right) e^{- k (R_1 - R)} - \\
& \left( R_2^2 k^2 + 2 R_2 k + 2 \right) e^{- k (R_2 - R)}
\Big].
V_\text{exp}(r) = \frac{4\pi G}{r} \frac{A}{k^3} \Big[
& \left( R_1^2 k^2 + 2 R_1 k + 2 \right) e^{- k (R_1 - R)} - \\
& \left( R_2^2 k^2 + 2 R_2 k + 2 \right) e^{- k (R_2 - R)}
\Big].
\end{split}
\end{equation}

Expand All @@ -1448,10 +1484,12 @@ \subsection{Sinusoidal density}

\begin{equation}
\begin{split}
V_\text{sine}(r) = \frac{4\pi G}{r} \frac{A}{k^3} \Big[
& (2 - k^2 R_2^2) \cos(k(R_2 - R)) + 2 k R_2 \sin(k(R_2 - R)) - \\
& (2 - k^2 R_1^2) \cos(k(R_1 - R)) - 2 k R_1 \sin(k(R_1 - R))
\Big].
V_\text{sine}(r) = \frac{4\pi G}{r} \frac{A}{k^3} \Big[
& (2 - k^2 R_2^2) \cos(k(R_2 - R)) + \\
& 2 k R_2 \sin(k(R_2 - R)) - \\
& (2 - k^2 R_1^2) \cos(k(R_1 - R)) - \\
& 2 k R_1 \sin(k(R_1 - R))
\Big].
\end{split}
\end{equation}

Expand Down

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