Gravitational attraction of a vertical pyramid model of flat top-and-bottom with depth-wise parabolic density variation

Abstract

In 3D gravity modelling, right rectangular vertical prism model with linear and nonlinear density and polyhedral bodies with linear density variation exist in geophysical literature. Here, we propose a vertical pyramid model with depth-wise parabolic density contrast variation. Initially, we validate our analytic expression against the gravity effect of a right rectangular parallelepiped of constant density contrast. We provide two synthetic examples and a case study for illustrating the effectiveness of our pyramid model in gravity modelling. The included case study of Los Angeles basin, California demonstrates the comparative advantages of our pyramid model over a conventional right rectangular vertical prism model. Our pyramid model could be quite effective as a building block for evaluating the gravity effect of an arbitrarily-shaped 3D or 2.5-D source(s).

Appendix

ᅟ

1.1 Expression of the gravity anomaly for pyramid model

From equations (3 and 4), we can rewrite the parabolic density function ${\Delta } \rho (\zeta ^{\prime })$ and the limits $\xi ^{\prime }$ and $\eta ^{\prime }$ of equation (3) as:

$$ \left. {\begin{array}{l} {\Delta} \rho \left( {\zeta^{\prime}} \right)={\Delta {\rho_{0}^{3}}} / {\left[ {r-\alpha \zeta^{\prime}} \right]^{2}}, \\ r={\Delta} \rho_{0} -\alpha \left( {z-h_{1}} \right), \\ \xi^{\prime}=\xi_{l}^{\prime} -x=m_{1} \zeta^{\prime}+c_{1} , \\ \xi^{\prime}=\xi_{u}^{\prime} -x=m_{2} \zeta^{\prime}+c_{2} , \\ \eta^{\prime}=\eta_{l}^{\prime} -y=m_{3} \zeta^{\prime}+c_{3} , \\ \eta^{\prime}=\eta_{u}^{\prime} -y=m_{4} \zeta^{\prime}+c_{4} , \end{array}} \right\} $$

(A1)

where

$$ \left. {\begin{array}{l} m_{1} ={\left( {\xi_{3} -\xi_{1}} \right)} / {\left( {h_{2} -h_{1}} \right)}, \\[-.5pt] m_{2} ={\left( {\xi_{4} -\xi_{2}} \right)} / {\left( {h_{2} -h_{1}} \right)}, \\[-.5pt] m_{3} ={\left( {\eta_{3} -\eta_{1}} \right)} / {\left( {h_{2} -h_{1}} \right)}, \\ m_{4} ={\left( {\eta_{4} -\eta_{2}} \right)} / {\left( {h_{2} -h_{1}} \right)}, \\[-.5pt] c_{1} ={\left( {h_{1} -z} \right)\left( {\xi_{1} -\xi_{3}} \right)} / {\left( {h_{2} -h_{1}} \right)}+\xi_{1} -x, \\[-.5pt] c_{2} ={\left( {h_{1} -z} \right)\left( {\xi_{2} -\xi_{4}} \right)} / {\left( {h_{2} -h_{1}} \right)}+\xi_{2} -x, \\[-.5pt] c_{3} ={\left( {h_{1} -z} \right)\left( {\eta_{1} -\eta_{3}} \right)} / {(h_{2} -h_{1} )}+\eta_{1} -y, {\begin{array}{*{20}c} \end{array}} \\ c_{4} ={\left( {h_{1} -z} \right)\left( {\eta_{2} -\eta_{4}} \right)} / {\left( {h_{2} -h_{1}} \right)}+\eta_{2} -y. \end{array}} \right\} $$

(A2)

Then, by performing integration with respect to $\xi ^{\prime }$ and $\eta ^{\prime }$ in equation (3), we get:

$$\begin{array}{@{}rcl@{}} &&g_{\text{pyramid}} \left( {x,y,z} \right)=\gamma {\Delta} {\rho_{0}^{3}}\int\limits_{\zeta^{\prime}=h_{1} -z}^{h_{2} -z} 1 / \left[ {r-\alpha \zeta^{\prime}} \right]^{2}\\[-2pt] &&\times\left\{{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}} \tan^{-1}\left[{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}} {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)\left( {m_{4} \zeta^{\prime}+c_{4}} \right)}\right.\right.\\[-2pt] && \left/\left. \!\!\!\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right) \right] \right. \\[-2pt] \end{array} $$

$$\begin{array}{@{}rcl@{}} &&-\tan^{-1}\left[{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}} {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)\left( {m_{3} \zeta^{\prime}+c_{3}} \right)} \right.\\&& \left/\!\!\!\left.{\left( {\zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{3} \zeta^{\prime}+c_{3}} \right)^{2}+\zeta^{\prime2}}} \right)} \right]\right. \\ &&-\tan^{-1}\left[{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}} {\left( {m_{1} \zeta^{\prime}+c_{1}} \right)\left( {m_{4} \zeta^{\prime}+c_{4}} \right)} \right.\\ &&\left/ \!\!\!\left.\left( {\zeta^{\prime}\sqrt {\left( {m_{1} \zeta^{\prime}+c_{1}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}}\right)^{2}+\zeta^{\prime2}}} \right) \right]\right. \\ &&+\tan^{-1}\left[{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}} \left( {m_{1} \zeta^{\prime}+c_{1}} \right)\left( {m_{3} \zeta^{\prime}+c_{3}} \right) \right.\\[-2pt] &&\left/\!\!\!\left.\left. \left( {\zeta^{\prime}\sqrt {\left( {m_{1} \zeta^{\prime}\,+\,c_{1}} \right)^{2}\,+\,\left( {m_{3} \zeta^{\prime}\,+\,c_{3}} \right)^{2}\,+\,\zeta^{\prime2}}} \right) \right] \right\}\! d\zeta^{\prime}.\right.\\ \end{array} $$

(A3)

As the integration of terms on RHS of equation (A3) is an involved job, we undertake the integration task in a systematic manner. Wolfram Mathematica 9.0.1 is used for carrying out integration. A cursory look at equation (A3) reveals that RHS have four terms. Here, we include the result of integration for the first terms of the RHS part of equation (A3).

$$\begin{array}{@{}rcl@{}} && g_{1} \left( {x,y,z} \right)=\int\limits_{h_{1} -z}^{h_{2} -z} {1 / {\left[ {r-\alpha \zeta^{\prime}} \right]}^{2}} \\ &&\times\left\{{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}}\tan^{-1}\left[{\vphantom{\left( \zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}+c_{2}} \right)^{2}+\left( {m_{4} \zeta^{\prime}+c_{4}} \right)^{2}+\zeta^{\prime2}} \right)}}\left( {m_{2} \zeta^{\prime}+c_{2}} \right)\left( {m_{4} \zeta^{\prime}+c_{4}} \right) \right.\right.\\ &&\left/\!\!\!\left. \left. \left( {\zeta^{\prime}\sqrt {\left( {m_{2} \zeta^{\prime}\,+\,c_{2}} \right)^{2}\,+\,\left( {m_{4} \zeta^{\prime}\,+\,c_{4}} \right)^{2}\,+\,\zeta^{\prime2}}} \right) \right] \right\} d\zeta^{\prime}\right.. \end{array} $$

Upon integration with respect to ζ, we get:

$$\begin{array}{@{}rcl@{}} && g_{1} \left( {x,y,z} \right)\,=\,-\!\tan^{-1}\!{\left[ {{\left( {M_{1} \zeta^{\prime2}\,+\,M_{2} \zeta^{\prime}\,+\,M_{3}} \right)} \!/\! {\left( {\zeta^{\prime}R^{\prime}} \right)}} \right]} \\ &&/ {\left[ {\alpha \left( {-r+\alpha \zeta^{\prime}} \right)} \right]} \\ && {-V_{5} {\log \left( {-r+\alpha \zeta^{\prime}} \right)} / {\left( {2\sqrt {\alpha^{2}k_{1} \,+\,\alpha b_{1} r+a_{1} r^{2}} V_{6}} \right)}} \\ && +V_{5} \log \left( {\vphantom{\left.+2\sqrt {\alpha^{2}k_{1} +\alpha b_{1} r+a_{1} r^{2}} R^{\prime} \right) /}}{2\alpha k_{1}} +b_{1} r+\alpha b_{1} \zeta^{\prime}+2a_{1} r\zeta^{\prime}\right.\\ &&\left.+2\sqrt {\alpha^{2}k_{1} +\alpha b_{1} r+a_{1} r^{2}} R^{\prime} \right) \\ &&\left/\!\!\left( {2\sqrt {\alpha^{2}k_{1} +\alpha b_{1} r+a_{1} r^{2}} V_{6}} \right)\right.\\ &&+{V_{z1}} / {\left( {2\alpha V_{6}} \right)} {\overset{h_{2} -z}\vert}\underset{h_{1} -z}, \end{array} $$

(A4)

where

$$\begin{array}{@{}rcl@{}} V_{z1}\!\!\!\! &=&\!\!\!\!\left[ {V_{1} \!\left( {4\alpha^{3}k_{1} M_{2} {M_{3}^{2}} -3\alpha^{3}b_{1} {M_{3}^{3}} +2\alpha^{2}{k_{1}^{2}} M_{3} r} \right.} \right. \\ &&\!\!\!\!+2\alpha^{2}k_{1} {M_{2}^{2}} M_{3} r\,+\,6\alpha^{2}k_{1} M_{1} {M_{3}^{2}} r\,-\,\alpha^{2}b_{1} M_{2} {M_{3}^{2}} r\\&&\!\!\!\!-4\alpha^{2}a_{1} {M_{3}^{3}} r\,+\,2\alpha b_{1} k_{1} M_{3} r^{2}\,+\,4\alpha k_{1} M_{1} M_{2} M_{3} r^{2} \end{array} $$

$$\begin{array}{@{}rcl@{}} &&\!\!\!\!+\alpha b_{1} M_{1} {M_{3}^{2}} r^{2}-2\alpha a_{1} M_{2} {M_{3}^{2}} r^{2}+2a_{1} k_{1} M_{3} r^{3}\\[-1pt] &&\!\!\!\!+\left. {2k_{1} {M_{1}^{2}} M_{3} r^{3}} \right)+V_{2} \!\left( {2\alpha^{3}{k_{1}^{2}} M_{3}} \right. \\[-1pt] && \!\!\!\!+2\alpha^{3}k_{1} {M_{2}^{2}} M_{3} +6\alpha^{3}k_{1} M_{1} {M_{3}^{2}} -\alpha^{3}b_{1} M_{2} {M_{3}^{2}} \\&&\!\!\!\!-4\alpha^{3}a_{1} {M_{3}^{3}} +5\alpha^{2}b_{1} k_{1} M_{3} r \\ && \!\!\!\!+8\alpha^{2}k_{1} M_{1} M_{2} M_{3} r+\alpha^{2}b_{1} {M_{2}^{2}} M_{3} r\\&&\!\!\!\!+7\alpha^{2}b_{1} M_{1} {M_{3}^{2}} r-10\alpha^{2}a_{1} M_{2} {M_{3}^{2}} r \\ && \!\!\!\!+3\alpha {b_{1}^{2}} M_{3} r^{2}+2\alpha a_{1} k_{1} M_{3} r^{2}+2\alpha k_{1} {M_{1}^{2}} M_{3} r^{2}\\&&\!\!\!\!+8\alpha b_{1} M_{1} M_{2} M_{3} r^{2} -4\alpha a_{1} {M_{2}^{2}} M_{3} r^{2}\\&&\!\!\!\!+3a_{1} b_{1} M_{3} r^{3}+\left. {3b_{1} {M_{1}^{2}} M_{3} r^{3}} \right)\\&&\!\!\!\!+V_{3} \!\left( {2\alpha^{3}b_{1} k_{1} M_{3}} \right.+4\alpha^{3}k_{1} M_{1} M_{2} M_{3} \\ && \!\!\!\!+\alpha^{3}b_{1} M_{1} {M_{3}^{2}} -2\alpha^{3}a_{1} M_{2} {M_{3}^{2}} +3\alpha^{2}{b_{1}^{2}} M_{3} r\\ &&\!\!\!\!+2\alpha^{2}a_{1} k_{1} M_{3} r+2\alpha^{2}k_{1} {M_{1}^{2}} M_{3} r \\ && \!\!\!\!+8\alpha^{2}b_{1} M_{1} M_{2} M_{3} r-4\alpha^{2}a_{1} {M_{2}^{2}} M_{3} r\\&&\!\!\!\!-\alpha b_{1} k_{1} M_{1} r^{2}+\alpha {b_{1}^{2}} M_{2} r^{2}-2\alpha a_{1} k_{1} M_{2} r^{2} \\ && \!\!\!\!-4\alpha k_{1} {M_{1}^{2}} M_{2} r^{2}+3\alpha b_{1} M_{1} {M_{2}^{2}} r^{2}\\ &&\!\!\!\!-2\alpha a_{1} {M_{2}^{3}} r^{2}+7\alpha a_{1} b_{1} M_{3} r^{2}+5\alpha b_{1} {M_{1}^{2}} M_{3} r^{2} \\ && \!\!\!\!+4\alpha a_{1} M_{1} M_{2} M_{3} r^{2}-2a_{1} k_{1} M_{1} r^{3}\\ &&\!\!\!\!-2k_{1} {M_{1}^{3}} r^{3}+a_{1} b_{1} M_{2} r^{3}+b_{1} {M_{1}^{2}} M_{2} r^{3} \\ && \!\!\!\!+4{a_{1}^{2}} M_{3} r^{3}+\left. {4a_{1} {M_{1}^{2}} M_{3} r^{3}} \right)\\ &&\!\!\!\!+V_{4} \!\left( {2\alpha^{3}a_{1} k_{1} M_{3}} \right.+2\alpha^{3}k_{1} {M_{1}^{2}} M_{3}\\ && \!\!\!\!+3\alpha^{2}a_{1} b_{1} M_{3} r \,+\,3\alpha^{2}b_{1} {M_{1}^{2}} M_{3} r\,-\,2\alpha a_{1} k_{1} M_{1} r^{2}\\ &&\!\!\!\!-2\alpha k_{1} {M_{1}^{3}} r^{2}+\alpha a_{1} b_{1} M_{2} r^{2}+\alpha b_{1} {M_{1}^{2}} M_{2} r^{2} \\ && \!\!\!\!+4\alpha {a_{1}^{2}} M_{3} r^{2}+4\alpha a_{1} {M_{1}^{2}} M_{3} r^{2}-a_{1} b_{1} M_{1} r^{3}\\ &&\!\!\!\!-b_{1} {M_{1}^{3}} r^{3}+2{a_{1}^{2}} M_{2} r^{3}+\left. {\left. {2a_{1} {M_{1}^{2}} M_{2} r^{3}} \right)} \right] {\overset{h_{1} -z}\vert}{h_{2} -z} \end{array} $$

$$\begin{array}{@{}rcl@{}} V_{1} \!\!\!\!&=& \!\!\!\!W_{1} +W_{2} +W_{3} +W_{4} , \\[-1pt] V_{2} \!\!\!\!&=& \!\!\!\!r_{1} W_{1} +r_{2} W_{2} +r_{3} W_{3} +r_{4} W_{4} , \\[-1pt] V_{3} \!\!\!\!&=& \!\!\!\!{r_{1}^{2}} W_{1} +{r_{2}^{2}} W_{2} +{r_{3}^{2}} W_{3} +{r_{4}^{2}} W_{4} , \\[-1pt] V_{4} \!\!\!\!&=& \!\!\!\!{r_{1}^{3}} W_{1} +{r_{2}^{3}} W_{2} +{r_{3}^{3}} W_{3} +{r_{4}^{3}} W_{4} ,\end{array} $$

$$\begin{array}{@{}rcl@{}} W_{1} \!\!\!\! &=& \!\!\!\!\log \left[{\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}} {\left( {\zeta^{\prime}-r_{1}} \right)} / \left( 2k_{1} +b_{1} \zeta^{\prime}+b_{1} r_{1} \right.\right.\\ &&\left.\left. \!\!\!\!+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}} \right) \right] \\ &&\!\!\!\!\left/\! \left[ \sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}} \left( {a_{1} +{M_{1}^{2}}} \right)\right.\right.\\ && \left.\!\!\!\!\times\left( {r_{1} -r_{2}} \right)\left( {r_{1} -r_{3}} \right)\left( {r_{1} -r_{4}} \right) {\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}}\right] \\[-2pt] W_{2} \!\!\!\!&=& \!\!\!\!\log \left[{\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}} {\left( {\zeta^{\prime}-r_{2}} \right)} / \left( 2k_{1} +b_{1} \zeta^{\prime}+b_{1} r_{2} \right.\right.\\ &&\left.\left.\!\!\!\!+2a_{1} \zeta^{\prime}r_{2} +2R^{\prime}\sqrt {a_{1} {r_{2}^{2}} +b_{1} r_{2} +k_{1}} \right) \right] \\ &&\!\!\!\!\left/\! \left[ \left. {\sqrt {a_{1} {r_{2}^{2}} +b_{1} r_{2} +k_{1}} } \right]\left( {a_{1} +{M_{1}^{2}}} \right)\right.\right.\\[-2pt] && \left.\!\!\!\!\times\left( {-r_{1} +r_{2}} \right)\left( {r_{2} -r_{3}} \right)\left( {r_{2} -r_{4}} \right) {\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}}\right] \\ W_{3} \!\!\!\!&=& \!\!\!\!\log \left[{\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}} {\left( {\zeta^{\prime}-r_{3}} \right)} / \left( 2k_{1} +b_{1} \zeta^{\prime}+b_{1} r_{3} \right.\right.\\&& \left.\left.\!\!\!\!+2a_{1} \zeta^{\prime}r_{3} +2R^{\prime}\sqrt {a_{1} {r_{3}^{2}} +b_{1} r_{3} +k_{1}} \right) \right] \\ &&\!\!\!\!\left/ \!\left[ \sqrt {a_{1} {r_{3}^{2}} +b_{1} r_{3} +k_{1}} \left( {a_{1} +{M_{1}^{2}}} \right)\right.\right.\\ &&\left.\!\!\!\!\times\left( {-r_{1} +r_{3}} \right)\left( {-r_{2} +r_{3}} \right)\left( {r_{3} -r_{4}} \right) {\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}}\right] \end{array} $$

$$\begin{array}{@{}rcl@{}} W_{4} \!\!\!\!&=& \!\!\!\!\log \left[{\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}} {\left( {\zeta^{\prime}-r_{4}} \right)} / \left( 2k_{1} +b_{1} \zeta^{\prime}+b_{1} r_{4} \right.\right.\\ &&\left.\left.\!\!\!\!+2a_{1} \zeta^{\prime}r_{4} +2R^{\prime}\sqrt {a_{1} {r_{4}^{2}} +b_{1} r_{4} +k_{1}} \right) \right] \\ &&\!\!\!\!\left/\! \left[ \sqrt {a_{1} {r_{4}^{2}} +b_{1} r_{4} +k_{1}} \left( {a_{1} +{M_{1}^{2}}} \right)\right.\right.\\ &&\left.\!\!\!\!\times\left( {-r_{1} +r_{4}} \right)\left( {-r_{2} +r_{4}} \right)\left( {-r_{3} +r_{4}} \right) {\vphantom{+2a_{1} \zeta^{\prime}r_{1} +2R^{\prime}\sqrt {a_{1} {r_{1}^{2}} +b_{1} r_{1} +k_{1}}}}\right] \end{array} $$

$$\begin{array}{@{}rcl@{}} V_{5} \!\!\!\!&=&\!\!\!\!2\alpha^{3}k_{1} M_{3} +3\alpha^{2}b_{1} M_{3} r-2\alpha k_{1} M_{1} r^{2} \\ &&\!\!\!\!+\alpha b_{1} M_{2} r^{2}+4\alpha a_{1} M_{3} r^{2}-b_{1} M_{1} r^{3}+2a_{1} M_{2} r^{3} \\ V_{6} \!\!\!\!&=&\!\!\!\!\alpha^{4}{M_{3}^{2}} +2\alpha^{3}M_{2} M_{3} r+\alpha^{2}k_{1} r^{2}+\alpha^{2}{M_{2}^{2}} r^{2} \\ &&\!\!\!\!+2\alpha^{2}M_{1} M_{3} r^{2}+\alpha b_{1} r^{3}+2\alpha M_{1} M_{2} r^{3}\\&&\!\!\!\!+a_{1} r^{4}+{M_{1}^{2}} r^{4} \end{array} $$

Rest of the integrations on the RHS of equation (A3) can be handled easily by considering expressions (A4) with necessary changes as per the following details:For g ₂(x,y,z):

$$\begin{array}{l} {\begin{array}{*{20}c} \hfill \end{array}}{\kern6pt}a_{1} ={m_{2}^{2}} +{m_{3}^{2}} +1,{\begin{array}{*{20}c} \hfill & {b_{1} =2\left( {m_{2} c_{2} +m_{3} c_{3}} \right)} \hfill, \end{array}} \\ {\begin{array}{*{20}c} \hfill & {{\kern1pt}k_{1} ={c_{2}^{2}} +{c_{3}^{2}} ,} \hfill \end{array}} {\begin{array}{*{20}c} \hfill \end{array}} M_{1} =m_{2} m_{3} , \\ {\begin{array}{*{20}c} \hfill & {M_{2} =m_{2} c_{3} +m_{3} c_{2} ,} \hfill \end{array}} {\begin{array}{*{20}c} \hfill & {M_{3} =c_{2} c_{3} ,} \hfill \end{array}} \end{array} $$

For g ₃(x,y,z):

$$\begin{array}{l} {\begin{array}{*{20}c} \hfill \end{array}} {\kern6pt}a_{1} ={m_{1}^{2}} +{m_{4}^{2}} +1,{\begin{array}{*{20}c} \hfill & {b_{1} =2\left( {m_{1} c_{1} +m_{4} c_{4}} \right)}, \hfill \end{array}} \\ {\begin{array}{*{20}c} \hfill & {{\kern1pt}k_{1} ={c_{1}^{2}} +{c_{4}^{2}} ,} \hfill \end{array}} \quad M_{1} =m_{1} m_{4} , \\ {\begin{array}{*{20}c} \hfill & {M_{2} =m_{1} c_{4} +m_{4} c_{1} ,} \hfill \end{array}} {\begin{array}{*{20}c} \hfill & {M_{3} =c_{1} c_{4} ,} \hfill \end{array}} \end{array} $$

For g ₄(x,y,z):

$$\begin{array}{@{}rcl@{}} a_{1}\!\!\!\!& =&\!\!\!\!{m_{1}^{2}} +{m_{3}^{2}} +1,\quad {b_{1} =2\left( {m_{1} c_{1} +m_{3} c_{3}} \right)} , \\ k_{1} \!\!\!\!&=&\!\!\!\!{c_{1}^{2}} +{c_{3}^{2}} ,\quad M_{1} =m_{1} m_{3} ,\\ &&{\kern-2.7pc}M_{2}=m_{1} c_{3} +m_{3} c_{1} , \quad {M_{3} =c_{1} c_{3} }. \end{array} $$

Then, the final expression for gravity effect of proposed pyramid model with depth-wise parabolic density contrast, works out to be

$$ g_{\text{pyramid}} \left( {x,y,z} \right)=\gamma {\Delta} {\rho_{0}^{3}}\left( {g_{1} -g_{2} -g_{3} +g_{4}} \right). $$

(A5)

where g ₁, g ₂, g ₃ and g ₄ are the gravity expressions of all terms on RHS of equation (A3).