# Gravity Anomalies of Arbitrary 3D Polyhedral Bodies with Horizontal and Vertical Mass Contrasts

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## Abstract

During the last 15 years, more attention has been paid to derive analytic formulae for the gravitational potential and field of polyhedral mass bodies with complicated polynomial density contrasts, because such formulae can be more suitable to approximate the true mass density variations of the earth (e.g., sedimentary basins and bedrock topography) than methods that use finer volume discretization and constant density contrasts. In this study, we derive analytic formulae for gravity anomalies of arbitrary polyhedral bodies with complicated polynomial density contrasts in 3D space. The anomalous mass density is allowed to vary in both horizontal and vertical directions in a polynomial form of \(\lambda =ax^m+by^n+cz^t\), where *m*, *n*, *t* are nonnegative integers and *a*, *b*, *c* are coefficients of mass density. First, the singular volume integrals of the gravity anomalies are transformed to regular or weakly singular surface integrals over each polygon of the polyhedral body. Then, in terms of the derived singularity-free analytic formulae of these surface integrals, singularity-free analytic formulae for gravity anomalies of arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts are obtained. For an arbitrary polyhedron, we successfully derived analytic formulae of the gravity potential and the gravity field in the case of \(m\le 1\), \(n\le 1\), \(t\le 1\), and an analytic formula of the gravity potential in the case of \(m=n=t=2\). For a rectangular prism, we derive an analytic formula of the gravity potential for \(m\le 3\), \(n\le 3\) and \(t\le 3\) and closed forms of the gravity field are presented for \(m\le 1\), \(n\le 1\) and \(t\le 4\). Besides generalizing previously published closed-form solutions for cases of constant and linear mass density contrasts to higher polynomial order, to our best knowledge, this is the first time that closed-form solutions are presented for the gravitational potential of a general polyhedral body with quadratic density contrast in all spatial directions and for the vertical gravitational field of a prismatic body with quartic density contrast along the vertical direction. To verify our new analytic formulae, a prismatic model with depth-dependent polynomial density contrast and a polyhedral body in the form of a triangular prism with constant contrast are tested. Excellent agreements between results of published analytic formulae and our results are achieved. Our new analytic formulae are useful tools to compute gravity anomalies of complicated mass density contrasts in the earth, when the observation sites are close to the surface or within mass bodies.

## Keywords

Gravity Singularity-free Polyhedral body Prism Horizontal and vertical mass contrasts## 1 Introduction

Gravitational data sets, which are measured using gravimeters based on land, in boreholes or on board satellites, aircraft or marine vessels, are used to estimate locations and shapes of embedded anomalous mass density bodies in the earth. For instance, gravity signal extraction and enhancement techniques (Zhang et al. 2014) can be used to estimate the approximated shapes of anomalous mass bodies in the underground. To more accurately determine depths, volumes and densities of anomalies, elaborate gravity inversion algorithms are employed (Li and Oldenburg 1998). To obtain more comprehensive and less ambiguous models of the earth, gravity data have been jointly inverted with other geophysical data sets, such as seismic data and electromagnetic induction data using constraints that couple the models of the different material parameters structurally or petrophysically (Moorkamp et al. 2011; Roberts et al. 2016). In recent years, rapid improvement in gravimeter efficiency and modern inversion algorithms has enhanced the capability of collecting large gravity data sets over large-scale areas and inverting such data sets for 3D density models (Kamm et al. 2015). These two improvements guarantee the wide application of the gravity methods in different geophysical or geodetic problems, such as mineral exploration (Beiki and Pedersen 2010; Lelièvre et al. 2012; Martinez et al. 2013; Kamm et al. 2015; Abtahi et al. 2016), crustal structure and Moho studies (Van der Meijde et al. 2013; De Castro et al. 2014) and geoid determination (Bajracharya and Sideris 2004).

An accurate gravity modeling solver plays a key role in interpreting or inverting gravity data sets. Generally, the structure of the real earth has complicated geometrical shapes and mass density distributions. Thus, the important question arises of how to efficiently approximate the 3D mass structure of the earth by volume discretization techniques. As an arbitrary mass body can be reasonably well represented by a set of disjoint polyhedral bodies with simple mass distributions, an arbitrary polyhedral body is generally adopted to reduce to maximum extend geometrical discretization errors. Therefore, seeking an accurate gravity modeling solver using 3D polyhedra is an essential step. Currently, different approaches can be adopted to evaluate the gravity anomaly caused by a polyhedral mass body, such as finite element methods (Kaftan et al. 2005; Cai and Cy 2005), finite difference methods (Farquharson and Mosher 2009), finite volume methods (Jahandari and Farquharson 2013) and direct Newton integral methods (Blakely 1996). The finite element, finite difference and the finite volume methods translate the Poisson boundary value problem of the gravity potential into a system of linear equations. Then, linear system solvers are used to obtain the gravity potential. However, the accuracies of the gravity anomalies calculated by these three algorithms do not only depend on the quality of the discretization elements and the accuracies of the solvers for systems of linear equations, but also depend on the accuracies of the numerical techniques to translate the computed gravitational potentials into gravity fields. Therefore, in general, numerical solutions provided by these solvers are less accurate than solutions from direct Newton integral methods. As for the direct Newton integral methods, if we can derive analytic formulae of the gravity anomaly for a polyhedral mass body, the highest accuracies can be achieved. Therefore, to find possible analytic formulae has become an essential topic in gravity modeling problems using direct Newton integral approaches.

When the distance from the observation site to the mass body is much larger than the size of the polyhedral body, a constant mass density can be assigned to each polyhedral body in gravity forward modeling and inversion. Aiming to improve accuracy and efficiency, different analytic formulae were derived for a homogeneous (with constant mass density contrast) polyhedral body (Paul 1974; Barnett 1976; Okabe 1979; Pohanka 1988; Werner 1994; Holstein and Ketteridge 1996; Petrović 1996; Tsoulis and Petrović 2001; Holstein 2002; D’Urso 2013, 2014a; Conway 2015), for a homogeneous prismatic polyhedron (Nagy 1966; Banerjee and Das Gupta 1977; Smith 2000; Nagy et al. 2000; Tsoulis et al. 2003) and for a circular disk or a flat lamina (Conway 2016) during the last four decades.

Seeking simplicity, researchers generally assumed that the earth is composed of 3D anomalies in a layered medium or a succession of strata with horizontally undulating interfaces (e.g., sedimentary basins and underlying bedrock). In each layer, the rock mass density predominantly exhibits depth-dependent variations (García-Abdeslem 1992). Different efforts were made to derive analytic formulae for anomalous masses with internal layering such as exponential depth-dependent mass density variations (Chai and Hinze 1988), quadratic polynomial depth-dependent mass density variations (Rao 1990) and cubic polynomial depth-dependent mass density variations (García-Abdeslem 2005).

However, geological formations can be more complicated so that the above assumption of dominant depth-dependent mass density variations can be inappropriate. Also, exogenetic (e.g., weathering, fluvial, coastal and glacial) and endogenetic processes (e.g., diagenesis of rocks, plate tectonics, volcano eruptions and earthquakes) at different scales generally have changed the mass density structures of crust and mantle into 3D structures with both horizontal and vertical variations in mass density (Martín-Atienza and García-Abdeslem 1999). Therefore, it is critical for us to establish more general density distribution models for the mass bodies in the earth. This leads to the necessity of developing formulae capable of computing gravity responses of a polyhedral body with both horizontal and vertical variations in mass density (Zhou 2009b). Until now, only a few studies have been carried out on this more and more important topic. For instance, for a 3D polyhedral body with linear mass density varying in both horizontal and vertical directions, several authors derived different closed-form solutions by using different techniques (Pohanka 1998; Hansen 1999; Holstein 2003; Hamayun and Tenzer 2009; D’Urso 2014b). For a 3D prismatic body with arbitrary density contrast variations in both horizontal and vertical directions, Zhou (2009a) derived a generalized numerical solution in terms of 1D line integrals for the gravity anomaly. However, singularities exist in these 1D line integrals when the observation point is placed on any of the faces of the rectangular prism or inside the prism. Therefore, more careful treatment should be considered to deal with singularities. More recently, for a 2D polygon body with polynomial density contrast variations in both horizontal and vertical directions, D’Urso (2015) derived singularity-free closed-form solutions using the generalized Gauss theorem, in which the mass density polynomials have up to cubic order.

However, to our best knowledge, there are still no publications that deal with analytic formulae for 3D polyhedral bodies with complicated polynomial mass density variations in both horizontal and vertical directions. In this study, we derive closed-form solutions for a 3D polyhedral body (and a prismatic body as a specific example) with mass density contrast as \(\lambda =ax^m+by^n+cz^t\), where *m*, *n*, *t* are nonnegative integers and *a*, *b*, *c* are coefficients of mass density. Using several simple vector identities and integration by parts, we first transform the volume integrals of the gravity anomalies into a set of surface integrals over each polygon of the polyhedral body. Then, singularity-free analytic formulae are derived for these surface integrals. Finally, we obtain a set of completely singularity-free analytic formulae of gravity anomalies for arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts. Our analytic formulae are the first to (1) generalize previously published closed-form solutions for cases of constant mass density contrasts (Paul 1974; Barnett 1976; Okabe 1979; Pohanka 1988; Werner 1994; Holstein and Ketteridge 1996; Petrović 1996; Tsoulis and Petrović 2001; D’Urso 2013, 2014a; Conway 2015) and linear mass density contrasts (Pohanka 1998; Hansen 1999; Holstein 2003; Hamayun and Tenzer 2009; D’Urso 2014b, 2016) to higher polynomial order, (2) permit computation of the gravitational potential for a general polyhedral body with quadratic density contrast in all spatial directions using closed-form solutions and (3) permit computation of the vertical gravitational field for a prismatic body with quartic density contrast along the vertical direction using closed-form solutions. Additionally, our analytic formulae are singularity-free, which means the observation site can be located anywhere with respect to the mass body. Consequently, our singularity-free analytic formulae can be used in gravity modeling problems with high accuracy requirements such as terrain correction problems (where the observation sites locate on the earth’s surface) and borehole gravity problems (where the observation sites are very close to the mass targets).

To verify our new analytic formula, a prismatic body with different depth-dependent polynomial variations and a polyhedral body in form of a triangular prism with constant density contrasts are examined. Excellent agreement between the published solutions (Blakely 1996; García-Abdeslem 2005; Tsoulis 2012) and our solutions is obtained.

## 2 Theory

### 2.1 Reduction of Order of Singularities

*H*, a local Cartesian coordinate system is built in a way where the observation site \({\mathbf {r}}^{\prime }\) is coincident with the coordinate origin, that is, \({\mathbf {r}}^{\prime }=(0,0,0)\). Then, the polynomial mass density contrast in body

*H*can be defined as:

*a*,

*b*,

*c*,

*m*,

*n*,

*t*are generally estimated by fitting the gravity responses generated by the mass density function \(\lambda ({\mathbf {r}})\) to the gravity data set collected in the field (Grant and West 1965). The integer values

*m*,

*n*,

*t*are the polynomial orders of the mass density. Then, the gravity potential and the gravity field are expressed as:

#### 2.1.1 Gravitational Potential

*x*-,

*y*- and

*z*-axes, respectively. Similar definitions apply for the three components \({\mathbf {g}}_{m}^{x}\), \({\mathbf {g}}_{n}^{y}\) and \({\mathbf {g}}_{t}^{z}\) of the gravity field.

*O*(1/

*R*) occurs for the gravity potential, and a strong singularity of order \(O(1/R^2)\) needs to be handled in the computation of the gravitational acceleration. To start with, we deal with the

*x*-component \(\phi _{m}^{x}\) of the weakly singular gravity potential. By introducing an unit vector \(\hat{\mathbf {x}}=\{1,0,0\}\) along the

*x*-direction, we find that

*H*, we have

*y*-axis and the

*z*-axis, respectively.

*N*polygons, that is, \(\partial H = \sum _{i=1}^{N} \partial H_{i}\). Substituting Eqs. (14), (15) and (16) into Eqs. (10), (11) and (12), respectively, and using the divergence theorem (Jin 2002), we get

*O*(1 /

*R*) was successfully transformed to regular surface integrals of the form

*O*(

*R*) in Eqs. (17)–(19). This means that even when no analytic formulae exist for the gravity potential \(\phi _{mnt}=-G(a\phi _{m}^{x}+b\phi _{n}^{y}+c\phi _{t}^{z})\) with high orders, standard quadrature rules can still be easily adopted to evaluate the gravity potential with high accuracy.

#### 2.1.2 Gravitational Acceleration

*z*-axis. The formulae to compute the other components along the horizontal

*x*- and

*y*-axes can be derived in a similar way. For simplicity, we use the scalar symbols \(g_{m}^{x}\), \(g_{n}^{y}\) and \(g_{t}^{z}\) to denote the vertical components of \({\mathbf {g}}_{m}^{x}\), \({\mathbf {g}}_{n}^{y}\) and \({\mathbf {g}}_{t}^{z}\). The first step is to transform the strong singularities into weak ones of order

*O*(1 /

*R*) by using the vector identity of Eq. (13) where we assign the unit

*z*-axis vector \(\hat{\mathbf {z}}\) to the arbitrary vector \({\varvec{\Psi }}\), set \(\chi =x^{m}, y^{n}, z^{t}\), and replace

*R*by \(\frac{1}{R}\):

*O*(1 /

*R*).

### 2.2 Closed-form Solutions for Surface Integrals and Final Singularity Removal

In Eqs. (17)–(19) and (28)–(30), we need to evaluate surface integrals of the form \(\iint \limits _{\partial H_{i}} f^{k} R^{q} \hbox {d}s\) where *f* is either *x*, *y* or *z*, *k* is an integer, and \(q=-1,1\). If \(q=-1\) and \(R\rightarrow 0\), the surface integral has a singular integrand.

*s*, \(s=({\mathbf {r}}-{\mathbf {o}})\cdot \hat{\mathbf {e}}_{j}\), \(s_{0}\) and \(s_{1}\) are the parametrized coordinates of two vertices \(v_{0}\) and \(v_{1}\) of edge \(C_{j}\), \(R_{0}\) and \(R_{1}\) are the distances from point \({\mathbf {r}}^{\prime }\) to the vertices \(v_{0}\) and \(v_{1}\), respectively. The distance from the source point \({\mathbf {r}}\in \partial H_{i}\) to the coordinate center \({\mathbf {o}}\) is \(\rho = |{\mathbf {r}} - {\mathbf {o}}|\). Hence, the distance from the site \({\mathbf {r}}^{\prime }\) to the source point \({\mathbf {r}}\) becomes \(R=|{\mathbf {r}}^{\prime }-{\mathbf {r}}| = \sqrt{(h_{i}^2 + \rho ^2)}\). The term \(f^{k}\) could be a complicated function over the polygon; in this study, we derived closed forms only for cases of \(k=0,-1\).

### *Proof*

*u*,

*v*) is used. Then, assuming

*u*and

*v*are the local coordinates of point \({\mathbf {r}}\), (0, 0) is the 2D local coordinate center \({\mathbf {o}}\), \(R= \sqrt{(u^2 + v^2+h_{i}^2)}\), and \(\rho =\sqrt{u^2+v^2}\), we have

### 2.3 Special Cases

#### 2.3.1 Zeroth-Order Density Variation

#### 2.3.2 First-Order Density Variation

#### 2.3.3 Second-Order Density Variation

Setting \(m=2,n=2,t=2\), the terms for the gravitational potential in Eqs. (17)–(21) simplify to integrals of the forms \(\iint _{\partial H_{i}} {\mathbf {r}} R \hbox {d}s\) and \(\iint _{\partial H_{i}} R \hbox {d}s\). These expressions can be analytically integrated using \(q=1\) in Eqs. (33) and (34), and hence, the gravity potential \(\phi\) has closed forms. However, for the gravity field, closed forms cannot be derived using the formulae in Eqs. (33) and (34).

#### 2.3.4 Comparison of Available Solutions for Zeroth- to Second-Order Density Variations

Comparison of our closed-form solution to other available closed-form solutions for a general polyhedral mass body

General 3D polyhedra | |||
---|---|---|---|

Density contrast | Singularity-free | Components | References |

Constant | – | \(g^{z}\) | Paul (1974) |

Constant | – | \(g^{z}\), \(\phi\) | Barnett (1976) |

Constant | – | \(\nabla \phi\), \(\nabla \nabla \phi\) | Okabe (1979) |

Constant | \(\surd\) | \(\phi\) | Waldvogel (1979) |

Constant | \(\surd\) | \(\nabla \phi\) | Pohanka (1988) |

Constant | \(\surd\) | \(\nabla \nabla \phi\) | Kwok (1991) |

Constant | \(\surd\) | \(\nabla \phi\) | Holstein and Ketteridge (1996) |

Constant | \(\surd\) | \(\phi\), \(\nabla \phi\), \(\nabla \nabla \phi\) | |

Constant | \(\surd\) | \(\nabla \phi\) | Conway (2015) |

Linear | \(\surd\) | \(\nabla \phi\) | Hansen (1999) |

Linear | \(\surd\) | \(\phi\), \(\nabla \phi\), \(\nabla \nabla \phi\) | Holstein (2003) |

Linear | \(\surd\) | \(\phi\) | Hamayun and Tenzer (2009) |

Linear | \(\surd\) | \(\phi\), \(\nabla \phi\), \(\nabla \nabla \phi\) | D’Urso (2014b) |

Constant | \(\surd\) | \(\phi\), \(\nabla \phi\) | Our approach |

Linear | \(\surd\) | \(\phi\), \(\nabla \phi\) | Our approach |

Quadratic | \(\surd\) | \(\phi\) | Our approach |

#### 2.3.5 A Regular Prismatic Body

*z*-axis downward, we arrange the rectangular surfaces as follows: (1) \(\Diamond _{1}\) and \(\Diamond _{2}\) are the two rectangular surfaces perpendicular to the

*x*-axis where the

*x*-coordinate of plane \(\Diamond _{1}\) is less than that of plane \(\Diamond _{2}\); (2) \(\Diamond _{3}\) and \(\Diamond _{4}\) are the two rectangular surfaces perpendicular to the

*y*-axis where the

*y*-coordinate of plane \(\Diamond _{3}\) is less than that of plane \(\Diamond _{4}\); and (3) \(\Diamond _{5}\) and \(\Diamond _{6}\) are the two rectangular surfaces perpendicular to the

*z*-axis where the

*z*-coordinate of plane \(\Diamond _{5}\) is less than that of plane \(\Diamond _{6}\). With this choice, the terms of the gravity potential in Eqs. (17)–(21) become

Order | \(\phi _{m}^{x}\) | \(\phi _{n}^{y}\) | \(\phi _{t}^{z}\) | \(g_{m}^{x}\) | \(g_{n}^{y}\) | \(g_{t}^{z}\) |
---|---|---|---|---|---|---|

0 | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) |

1 | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) |

2 | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) | ||

3 | \(\surd\) | \(\surd\) | \(\surd\) | \(\surd\) | ||

4 | \(\surd\) |

As shown in Table 2, for a prismatic body with constant, linear, quadratic and cubic density contrasts varying in both horizontal and vertical directions, our approach can deliver closed-form solutions for the gravitational potential. For the gravitational field, closed-form solutions exist in the cases of constant and linear density contrasts. Compared to previously published methods (Nagy et al. 2000; Rao 1990; García-Abdeslem 2005), one novelty of our approach is that our closed-form solutions are singularity-free which means the observation sites can be outside, inside and on the boundary (edges, corners, faces) of the mass prismatic body.

## 3 Verification

### 3.1 A Prismatic Body with Vertically Varying Density Contrast

*z*-axis is downward. The dimension of the prismatic body is \(x=[10\,{\text {km}},20\,{\text {km}}]\), \(y=[10\,{\text {km}},20\,{\text {km}}]\) and \(z=[0\,{\text {km}},8\,{\text {km}}]\) (Fig. 2a). The density contrast function, which is taken from a previous work (García-Abdeslem 2005), is given as:

*z*is in km. When the observation site approaches the prism, i.e., the edges, surfaces or corners of the prism, the gravity potential and the gravity field become singular. In the previous work (García-Abdeslem 2005), analytic formulae for depth-dependent density contrasts up to third order were derived. However, these analytic formulae are singular when the observation sites are located on edges or corners.

Order | \(g_{00t}\) along profile (\(z=-0.15\,{\text {m}}, y=15\,{\text {km}}\)) | ||
---|---|---|---|

| Our closed-form solutions (mGal) | García-Abdeslem’s solutions (mGal) | |

Constant (\(t=0\)) | 9.99995 | \(-\)7.0010152143459 | \(-\)7.0010152143459 |

10 | \(-\)7.0015340782380 | \(-\)7.0015340782380 | |

10.00005 | \(-\)7.00641689787295E\(+\)001 | \(-\)7.00641689787295E\(+\)001 | |

Linear (\(t=1\)) | 9.99995 | 5.97357825457560E\(+\)001 | 5.97357825457560E\(+\)001 |

10 | 5.9736562835893 | 5.9736562835893 | |

10.00005 | 5.97443654585579E\(+\)001 | 5.97443654585579E\(+\)001 | |

Quadratic (\(t=2\)) | 9.99995 | \(-\)3.69173288088277E\(+\)001 | \(-\)3.69173288088277E\(+\)001 |

10 | \(-\)3.69176741955519E\(+\)001 | \(-\)3.69176741955519E\(+\)001 | |

10.00005 | \(-\)3.69211280340700E\(+\)001 | \(-\)3.69211280340700E\(+\)001 | |

Cubic (\(t=3\)) | 9.99995 | 1.0929934898883 | 1.0929934898883 |

10 | 1.09300234258250E\(+\)001 | 1.09300234258250E\(+\)001 | |

10.00005 | 1.09309086860681E\(+\)001 | 1.09309086860681E\(+\)001 |

Order | \(g_{00t}\) along profile (\(z=0\,{\text {km}}, y=15\,{\text {km}}\)) | ||
---|---|---|---|

| Our closed-form solutions (mGal) | García-Abdeslem’s solution (mGal) | |

Constant (\(t=0\)) | 9.99995 | \(-\)7.00108086223439E\(+\)001 | \(-\)7.00108086223439E\(+\)001 |

10 | \(-\)7.00170532866468E\(+\)001 | – | |

10.00005 | \(-\)7.00680113760199E\(+\)001 | \(-\)7.00680113760199E\(+\)001 | |

Linear (\(t=1\)) | 9.99995 | 5.9737249676018 | 5.9737249676018 |

10 | 5.97380301857833E\(+\)001 | – | |

10.00005 | 5.97458347641883E\(+\)001 | 5.97458347641883E\(+\)001 | |

Quadratic (\(t=2\)) | 9.99995 | \(-\)3.69182233831518E\(+\)001 | \(-\)3.69182233831518E\(+\)001 |

10 | \(-\)3.69185687923601E\(+\)001 | – | |

10.00005 | \(-\)3.6922022855705 | \(-\)3.6922022855705 | |

Cubic (\(t=3\)) | 9.99995 | 1.09301961657224E\(+\)001 | 1.09301961657224E\(+\)001 |

10 | 1.09302846973961E\(+\)001 | – | |

10.00005 | 1.09311700049598E\(+\)001 | 1.09311700049598E\(+\)001 |

Order | \(g_{00t}\) at point (\(x=20\,{\text {km}}, y=10\,{\text {km}}, z=-0.15\,{\text {m}}\)) | |
---|---|---|

Our closed-form solutions (mGal) | García-Abdeslem’s solution (mGal) | |

Constant (\(t=0\)) | \(-\)4.25105387729770E\(+\)001 | \(-\)4.25105387729770E\(+\)001 |

Linear (\(t=1\)) | 3.9570790765669 | 3.9570790765669 |

Quadratic (\(t=2\)) | \(-\)2.55689100895767E\(+\)001 | \(-\)2.55689100895767E\(+\)001 |

Cubic (\(t=3\)) | 7.76642695050040E\(+\)000 | 7.76642695050040E\(+\)000 |

Order | \(g_{00t}\) at corner (\(x=20\,{\text {km}},y=10\,{\text {km}}, z=0\,{\text {km}}\)) | |
---|---|---|

Our closed-form solutions (mGal) | García-Abdeslem’s solution (mGal) | |

Constant (\(t=0\)) | \(-\)4.25112235972466E\(+\)001 | – |

Linear (\(t=1\)) | 3.95714574971360E\(+\)001 | – |

Quadratic (\(t=2\)) | \(-\)2.55693475942219E\(+\)001 | – |

Cubic (\(t=3\)) | 7.76656065625613E\(+\)000 | – |

### 3.2 A Polyhedral Body with Constant Density Contrast

To test the performance of our closed-form approach in the case of arbitrary polyhedral bodies, a complicated triangular prism is selected as shown in Fig. 2b. The triangular prism has half the volume of the prism (cutting in halves the original prism along diagonals of the top and bottom surfaces) and has a mass density of \(2670\,{\text {kg/m}}^3\); only the vertical gravity fields are computed. The triangular surface has a size of \(\frac{1}{2}\times 10\,{\text {km}}\times 10\,{\text {km}}\). Three observation sites are located on the top triangular surface of the anomaly, one at a vertex, one at the mid point of an edge and one at the center of the surface. Hence, singularities might exist.

Comparisons of the vertical gravity field computed by our closed-form formula to other approaches for the triangular prism model as shown in Fig. 2b

Locations | Methods | \(g_{z}\) (mGal) |
---|---|---|

Middle edge | Our formula | 2.142578 |

Tsoulis (2012)’s formula | 2.142578 | |

\(\frac{1}{2}\times \hbox {GBOX}\)’s formula for prism | 2.142578 | |

\(\frac{1}{2}\times \hbox {Our}\) formula for prism | 2.142578 | |

Vertices | Our formula | 1.302013 |

Tsoulis (2012)’s formula | 1.302013 | |

Face centre | Our formula | 3.255085 |

Tsoulis (2012)’s formula | 3.255085 |

### 3.3 A Prismatic Body with Quartic Density Contrast

Lists of the vertical gravity field computed using our analytic solution and solutions by the Gaussian quadrature rules with \(512\times 512\times 512\) points and \(5\times 5\times 5\) points for the prism model shown in Fig. 2a with \(\lambda ({\mathbf {r}})=z^{4}\)

| Gaussian quadrature solutions (mGal) with \(512\times 512\times 512\) points | our analytic solution (mGal) | Gaussian quadrature solutions (mGal) with \(5\times 5\times 5\) points |
---|---|---|---|

0 | 7.12219101489056 | 7.12219101489085 | 7.1221888320710950993 |

1 | 8.48056770614467 | 8.48056770614382 | 8.4805617391125540649 |

2 | 10.1696894406181 | 10.1696894406191 | 10.169676418630798409 |

3 | 12.2782706524876 | 12.2782706524853 | 12.278247766243397976 |

4 | 14.9143130178865 | 14.9143130178878 | 14.914288054727569133 |

5 | 18.2021319910864 | 18.2021319910878 | 18.202160056789619347 |

6 | 22.2702667632913 | 22.2702667632900 | 22.27054518097238045 |

7 | 27.2226356973335 | 27.2226356973329 | 27.223663649822459121 |

8 | 33.0839167397600 | 33.0839167397592 | 33.086832035778186878 |

9 | 39.7152373831687 | 39.7152373831755 | 39.724588628867728346 |

10 | 46.7187463141761 | 46.7187463141865 | 46.746050499737350492 |

11 | 53.4225546453986 | 53.4225546453996 | 53.461645509154095635 |

12 | 59.1380804111197 | 59.1380804111283 | 59.215084132472277645 |

13 | 63.4175287134867 | 63.4175287134972 | 63.441261544998837962 |

14 | 66.0399955350939 | 66.0399955350872 | 66.068292518952773662 |

15 | 66.9207406119316 | 66.9207406119341 | 67.047071844908117555 |

## 4 Discussion and Conclusions

In comparison with previously published work, our approaches of computing the gravitational potential and acceleration due to arbitrary polyhedral bodies have adopted a generalized singularity-free framework. This allowed to derive closed-form solutions of the gravitational potential and acceleration for the cases of constant and linear variations in mass density. We are the first to present closed-form solutions of the gravitational potential for a general polyhedral body with quadratic density contrast varying in all directions (*x*, *y* and *z*).

For a prismatic body with density contrasts varying in both horizontal and vertical directions, our approach can deliver closed-form solutions of the gravity potential caused by constant, linear, quadratic and cubic density contrasts. For the gravitational acceleration of a prismatic body, we have shown that closed-form solutions exist in the case of constant and linear variations in density contrast. To our best knowledge, we deliver the first closed-form solutions of the gravitational acceleration for a prismatic body with quartic density contrast, if the density contrast only varies along the *z*-axis.

- 1.
For an arbitrarily polyhedral body, analytic formulae of the gravity potential and the gravity field are available in the case of \(m\le 1\), \(n\le 1\) and \(t\le 1\). For the gravity potential, an analytic formula is also available in the case of \(m=2\), \(n=2\) and \(t=2\).

- 2.
For a prismatic body, an analytic formula exists in the case of \(m\le 3\), \(n\le 3\) and \(t\le 3\) for the gravity potential. For the gravity field, closed-form solutions are available only in the case of \(m\le 1\), \(n\le 1\) and \(t\le 4\).

Since simulation results from closed-form solutions for complicated polyhedral and spatial variations of density contrasts were not available in the literature, a simple prismatic model with a purely depth-dependent polynomial density contrast and a polyhedral body in form of a triangular prism with constant density contrasts had to be considered to verify our new analytic formulae. Excellent agreement between the results of the published analytic formulae and our results is demonstrated, verifying the accuracies of our new analytic expressions of the gravity anomaly.

Due to its ability for dealing with cases of locating observation sites at corners, on edges or on surfaces of a polyhedral body, our new analytic formula is a useful tool to compute the gravity anomaly in the immediate vicinity of the mass source body. It should have high potential in aiding interpretation of gravity data for gravity modeling problems where high accuracy is required, such as terrain correction and borehole gravity problems or in near field computation in associated fast multipole method acceleration techniques.

An interesting aspect of the new formulae is that they allow representing a density distribution with relatively few parameters, since the density is allowed to vary within a single polyhedron. This is an interesting aspect, when it comes to inversion, because these will reduce some of the ambiguities and may require less regularization.

## Notes

### Acknowledgements

This study was supported by Grants from the National Basic Research Program of China (973-2015CB060200), the Project of Innovation-driven Plan in Central South University (2016CX005), the National Science Fundation of China (41574120, 41474103, 41204082), the State High-Tech Development Plan of China (2014AA06A602), and an award for outstanding young scientists by Central South University (Lieying program 2013).

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