# Gravity Gradient Tensor of Arbitrary 3D Polyhedral Bodies with up to Third-Order Polynomial Horizontal and Vertical Mass Contrasts

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## Abstract

During the last 20 years, geophysicists have developed great interest in using gravity gradient tensor signals to study bodies of anomalous density in the Earth. Deriving exact solutions of the gravity gradient tensor signals has become a dominating task in exploration geophysics or geodetic fields. In this study, we developed a compact and simple framework to derive exact solutions of gravity gradient tensor measurements for polyhedral bodies, in which the density contrast is represented by a general polynomial function. The polynomial mass contrast can continuously vary in both horizontal and vertical directions. In our framework, the original three-dimensional volume integral of gravity gradient tensor signals is transformed into a set of one-dimensional line integrals along edges of the polyhedral body by sequentially invoking the volume and surface gradient (divergence) theorems. In terms of an orthogonal local coordinate system defined on these edges, exact solutions are derived for these line integrals. We successfully derived a set of unified exact solutions of gravity gradient tensors for constant, linear, quadratic and cubic polynomial orders. The exact solutions for constant and linear cases cover all previously published vertex-type exact solutions of the gravity gradient tensor for a polygonal body, though the associated algorithms may differ in numerical stability. In addition, to our best knowledge, it is the first time that exact solutions of gravity gradient tensor signals are derived for a polyhedral body with a polynomial mass contrast of order higher than one (that is quadratic and cubic orders). Three synthetic models (a prismatic body with depth-dependent density contrasts, an irregular polyhedron with linear density contrast and a tetrahedral body with horizontally and vertically varying density contrasts) are used to verify the correctness and the efficiency of our newly developed closed-form solutions. Excellent agreements are obtained between our solutions and other published exact solutions. In addition, stability tests are performed to demonstrate that our exact solutions can safely be used to detect shallow subsurface targets.

## Keywords

Gravity gradient tensor Polyhedral bodies Polynomial mass contrast Shallow target detection Gravity explorations## 1 Introduction

Gravity exploration methods try to identify the anomalous mass bodies in the Earth (Blakely 1996). Gravity signals such as gravity fields and gravity gradient tensors are measured using gravimeters and gravity gradiometers, which can be located on the air–Earth interface, in boreholes, or be carried by marine ships and aircraft, and even by satellites (Nabighian et al. 2005). The amplitudes of the gravity field are inversely proportional to the square of the distance (*R*) between the observation site and the causative body, that is \(O(R^{-2})\). As for the gravity gradient tensor, its amplitude is inversely proportional to the third power of distance, that is \(O(R^{-3})\). Therefore, compared to the gravity field, gravity gradient tensor signals are more sensitive to the shallow anomalous structures in the Earth (Droujinine et al. 2007; Beiki and Pedersen 2010; Martinez et al. 2013; Beiki et al. 2014; Gutknecht et al. 2014; Li 2015; Ramillien 2017). Mathematically, both gravity field and gravity gradient tensor signals can be formulated as volume integrals over the causative body. When the observation site is located inside the mass body, a local spherical coordinate system can be introduced at the observation site, so that a factor of distance squared (\(R^{2}\)) would be introduced (such as \(\mathrm{d}v=R^2 \sin \theta \mathrm{d}r \mathrm{d}\theta \mathrm{d}\phi\), Blakely 1996), which can further weaken the singularity appearing in the integrands of the gravity signals (Jin 2002). Therefore, from the mathematical point of view, the gravity field can be evaluated without mathematical singularities, as \(O(R^{-2}\times R^{2})=O(1)\), but the mathematical singularities always remain in the gravity gradient tensor formulation, when observation sites approach the causative body, that is \(O(R^{-3}\times R^{2})=O(R^{-1})\).

Designing gravity gradiometers for measuring gravity gradient tensor signals is a difficult task. The gravity gradient tensor is a \(3\times 3\) symmetrical tensor with each entry being the second derivatives of the gravitational potential (or the first derivatives of the gravity field or the gravity acceleration). In terms of differential measurements, accelerations of at least two spatially separated masses were generally used to estimate the components of the gravity gradient tensor. Depending on the accuracies of the available gradiometers, gravity gradient tensor signals can be applied in geodesy (0.01 Eötvös to 0.1 Eötvös), autonomous navigation (0.1 Eötvös to 1 Eötvös) and oilfield and mineral exploration geophysics (1 Eötvös to 10 Eötvös) (Evstifeev 2017). In exploration geophysics, the cause of the increased interest in gravity gradiometers is that, compared to gravity anomalies measured by gravimeters, gravity gradient tensor signals contain more detailed information about the subsurface structures (Chapin 1998; Nabighian et al. 2005). Furthermore, compared to gravity field data, gravity gradient tensor anomaly maps generally provide more contrasting and clearer edge delineations, such as those arising from salt domes in oil and gas prospecting (Pedersen and Rasmussen 1990). Up to now, several gravity gradiometers have been adopted in realistic exploration geophysical problems, such as the 3D FTG (full tensor gradiometer) system by Bell Geospace (Bell and Hansen 1998; Bell et al. 1997; Brewster 2016; Abtahi et al. 2016) and the Falcon AGG airborne gravity gradiometer by BHP Billiton (Australia) (Lee 2001).

To invert measured gravity gradient tensor (GGT) signals, we need an accurate forward solver. In general, GGT forward modelling can be performed either numerically or analytically. Using numerical methods such as Gaussian quadrature approaches (Talwani and Ewing 1960), Fourier domain methods (Parker 1973; Wu and Chen 2016), finite-element methods and finite-difference methods (Cai and Wang 2005; Farquharson and Mosher 2009; Jahandari and Farquharson 2013), the GGT signals caused by a mass body can be straightforwardly evaluated. However, the accuracies of numerical GGT signals can be seriously reduced by these numerical methods. For instance, using finite-element and finite-difference methods, improper translation of the computed gravitational potential to its second-order derivatives can lead to serious numerical errors. These undesired precision loss issues can be completely avoided by analytic approaches, which offer highly accurate GGT signals in terms of exact solutions.

Due to early limitations in the considered model geometries, the simple rectangular prismatic element has been widely accepted as a basic element in gravity forward modelling. Using these prismatic elements, the mass distributions in the Earth were simply approximated conceding a certain loss of geometrical accuracy. As for this basic element, several exact solutions of the gravity gradient tensor signals were developed (Forsberg 1984; Li and Chouteau 1998; Montana et al. 1992; Nagy and Papp 2000; Holstein et al. 2013; De Stefano and Panepinto 2016). However, it is difficult to accurately approximate a complex body using prisms. Polyhedral elements are endowed with great flexibility in presentation of 3D mass sources with complex geometries. Compared to prismatic elements, discretisations in terms of polyhedrons need lower numbers of elements to represent complicated mass bodies (Petrović 1996). Several analytical solutions have been successfully derived for the gravity gradient tensor signals of a homogeneous polyhedral body (Okabe 1979; Götze and Lahmeyer 1988; Kwok 1991; Petrović 1996; Werner and Scheeres 1996; Tsoulis and Petrović 2001; Holstein 2002; Tsoulis 2012; Holstein et al. 2013; D’Urso 2014a). Closed-form solutions of the gravity gradient tensor were also derived for other homogeneous simple geometries, such as pyramids (Sastry and Gokula 2016) and cylinders (Rim and Li 2016).

Seeking simplicity, geophysicists often assume 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. To approximate the density variations in sedimentary basins, several closed-form solutions of gravity signals were derived for depth-dependent mass contrast functions, such as exponential functions (Cordell 1973; Chai and Hinze 1988; Chappell and Kusznir 2008), hyperbolic functions (Litinsky 1989; Rao et al. 1995), parabolic functions (Chakravarthi et al. 2002), quadratic polynomials (Rao 1985, 1990; Gallardo-Delgado et al. 2003) and cubic polynomials (García-Abdeslem 2005). Compared to these depth-dependent functions, polynomials offer a more flexible way to approximate arbitrarily variable density distributions. Very recently, Jiang et al. (2017) have given a detailed performance comparison between polynomial mass contrast functions and depth-dependent mass contrast functions for the capability of approximating complicated density distributions in the Earth. The result clearly demonstrates the superiority of polynomial mass contrast functions over the depth-dependent mass contrast functions. As the real Earth has complicated mass density distributions, it is important to deal with a general polynomial mass contrast function, which not only varies in depth (vertical direction), but also varies in the horizontal direction.

For general polynomial functions, Pohánka (1998), Holstein (2003), Hansen (1999), D’Urso (2014b) and Ren et al. (2017a) have successfully derived closed-form solutions for the gravity field of a polyhedral body, in which the mass contrast linearly varies in both horizontal and vertical directions. Quite recently, analytical expressions for the gravity field of a polyhedral body with cubic polynomial density contrast in both horizontal and vertical directions were derived by D’Urso and Trotta (2017) and Ren et al. (2018). Nevertheless, only Holstein (2003) and D’Urso (2014b) have successfully derived closed-form solutions for gravity gradient tensor of a polyhedral body, in which the mass contrast varies linearly in both horizontal and vertical directions. To the best of our knowledge, closed-form solutions for gravity gradient tensor signals of a polygonal body with high-order polynomial mass contrasts (such as quadratic and cubic orders) varying in both horizontal and vertical directions have not been previously reported.

To overcome this both theoretically and practically important issue, we have successfully derived closed-form solutions for modelling the gravity gradient tensor of the above cases in this study. Inside the polyhedral body, its density contrast is approximated by a general polynomial function (currently, a maximum order up to and including three is considered). The polynomial mass contrast function can vary in both horizontal and vertical directions. Therefore, it has the capability to approximate complicated mass contrasts in realistic Earth models. To begin with, we apply the Gaussian gradient theorem to transform the volume integrals of GGT signals into surface integrals over the faces of polyhedra. Then, we apply the surface divergence theorem to further transform these surface integrals into a sequence of line integrals along the edges of the polyhedra. In the process of reducing the integral dimension, several vector or dyadic identities are employed to reduce the orders of the density polynomials and of the singularities in *R* in the integrands. The line integrals along the edges of the polyhedra are evaluated in terms of known analytical expressions (e.g., Gradshteyn and Ryzhik 2007).

To verify the accuracies of our new analytical solutions, a right rectangular prism with depth-dependent density contrasts, an irregular polyhedron with linear density contrast and a tetrahedral element with horizontally and vertically varying density contrast are tested. For the prism model, its reference solution for the gravity gradient tensor is computed from the derivatives of García-Abdeslem’s (2005) analytical solution for the gravity field. The analytical solution derived by Holstein (2003) is used as reference for the irregular polyhedral model. Results from high-order Gaussian quadrature are used as reference solutions for the tetrahedron model.

## 2 Theory

*H*, and the gradient theorem (Tai 1997) is applied.

*s*means it is an operator on a 2D surface. In the above, the definition (A.3) of the number \(h_i\), constant over facet plane

*i*, as the projection onto the facet normal \(\mathbf {n}_i\) of the position vector of a planar point \({\mathbf {r}}\) relative to the observation point \({\mathbf {r}}'\) and the following vector identity were used:

*f*is an arbitrary scalar function and \({\hat{\mathbf {n}}}\) is the outward pointing normal unit vector of the considered surface. In Eq. (5), we set \(f=\frac{\lambda }{R}\).

*H*with arbitrary mass contrast function \(\lambda ({\mathbf {r}})\):

*P*is the maximum polynomial order, and \(\lambda _{d}\) are the

*d*-th order density terms. For instance, the zero (constant), first (linear), quadratic and cubic order density terms are given as:

*d*-th order density contrast. In this study, we only consider the cases \(0 \le P \le 3\). For simplicity, we denote

*x*,

*y*and

*z*by \(x_p (p=1, 2, 3)\) in the following sections, that is

### 2.1 Constant Density Contrast

### 2.2 Linear Density Contrast

### 2.3 Quadratic Density Contrast

### 2.4 Cubic Density Contrast

### 2.5 Comparison with Other Solutions

A list of available analytical solutions of the gravity gradient tensor for different geometries and different orders of polynomial mass contrasts

Shape | Order of density contrast | References |
---|---|---|

Rectangular prism | Constant | Forsberg (1984), Montana et al. (1992), Li and Chouteau (1998) and Nagy and Papp (2000) |

Cylinder | Constant | Rim and Li (2016) |

Truncated square pyramid | Linear in depth | Sastry and Gokula (2016) |

Polyhedron | Constant | Okabe (1979), Götze and Lahmeyer (1988), Kwok (1991), Werner and Scheeres (1996), Tsoulis and Petrović (2001), Holstein (2002), Holstein et al. (2007a, b), Tsoulis (2012) and D’Urso (2014a) |

Polyhedron | Linear | |

Polyhedron | Constant | Our solution |

Polyhedron | Linear | Our solution |

Polyhedron | Quadratic | Our solution |

Polyhedron | Cubic | Our solution |

## 3 Verification

Three models were used to validate our closed-form solutions. The first one is a rectangular prism model (Fig. 2), for which the derivatives of closed-form solutions for the gravity field in García-Abdeslem (2005) were taken as references. The second model is an irregular polyhedron with linear density contrast, taken from Holstein (2003), for which a comparison with Holstein’s (2003) solution is given. The third model is a relatively complicated tetrahedral body (Fig. 5). As there are no reference solutions for the GGT of polyhedral mass bodies with quadratic and cubic density contrasts, we have used high-order Gaussian quadrature rules (such as 512 × 512 × 512 = 124,217,728 quadrature points) to calculate GGT reference solutions. Additionally, we allowed the density in the tetrahedral body to vary in both horizontal and vertical directions. We should mention that because our formulae require the observation site to be located at the origin of the Cartesian coordinate system, for each observation site, a coordinate translation must be performed to move the observation site to the origin.

### 3.1 A Prismatic Body with Depth-Dependent Density Contrast

*z*-axis downward. In Fig. 2a, the coordinate ranges of the prismatic body are \(x=[10,20]\) km, \(y=[10,20]\) km and \(z=[0,8]\) km. The density function is a depth-dependent cubic polynomial which is taken from García-Abdeslem (2005):

*z*is in km.

In the García-Abdeslem’s (2005) work, an analytic formula for the vertical gravity field caused by a rectangular prism with density contrast varying as a depth-dependent cubic polynomial was derived. Therefore, we took the derivatives of García-Abdeslem’s solution (i.e., \(T_{xz}\), \(T_{yz}\), \(T_{zz}\)) as references to verify our solution for the gravity gradient tensor. Furthermore, to verify our solution for an arbitrary polyhedral body, we decomposed the original rectangular prism into two polyhedra (as shown in Fig. 2b). These two polyhedra share the plane with vertices 1, 3, 8 and 9. Note, this test requires points 1, 3, 8 and 9 to be co-planar. If they were not co-planar, the surface (1, 3, 8, 9) would need to be divided into two triangles (1, 3, 8) and (3, 8, 9). The values of \(T_{xz}\), \(T_{yz}\) and \(T_{zz}\) were computed at two observation sites. One is located at point (12, 12, − 0.001) km with a 1-m offset right above the top face of the prismatic body, whereas the other one is located at point (20, 10, − 0.001) km, that is near a corner of the rectangular prism.

The results computed by our closed-form solution and by derivatives of García-Abdeslem’s (2005) gravity field solution are compared in Table 2. We compare GGT elements \(T_{xz}\), \(T_{yz}\), \(T_{zz}\), because only \(g_z\) is given in García-Abdeslem (2005). Excellent agreement is obtained between these two approaches, with relative errors on the order of \(10^{-13}\%\) at site (12, 12, − 0.001) km and relative errors \(10^{-6}\)–\(10^{-7}\%\) at site (20, 10, − 0.001) km. The tensor trace \(T_{xx}+T_{yy}+T_{zz}=\nabla ^{2}_{{\mathbf {r}}'}U\) is a useful tool for indicating numerical error. Using the Poisson equation \(\nabla ^2_{{\mathbf {r}}'}U=-4\pi G\lambda\) for the gravitational potential *U*, the tensor trace vanishes outside the source body and is \(-4\pi G\lambda\) inside the source body. Since our observation points are outside the source body, we have calculated the relative errors shown in Table 2 as \(|T_{xx}+T_{yy}+T_{zz}|/(|T_{xx}|+|T_{yy}|+|T_{zz}|)\). The relative error of the tensor trace is \(2.74\times 10^{-13}\%\) at the location above the top face and \(3.77\times 10^{-11}\%\) above the corner. Both our solution and García-Abdeslem’s (2005) solution for the GGT are singular, when observation sites are located on edges and corners. When the observation site gets close to the corner, the mathematical singularity becomes stronger, and this is, why the relative error is larger at site (20, 10, − 0.001) km.

Observation sites (km) | Component | Our solution (s\(^{-2}\)) | Derivatives of García-Abdeslem’s solution (s\(^{-2}\)) |
---|---|---|---|

Above the top face (12, 12, – 0.001) | \(T_{xz}\) | − 3.88858891017895E–08 | − 3.88858891017895E–08 |

\(T_{yz}\) | − 3.8885889101789 | − 3.8885889101789 | |

\(T_{zz}\) | − 1.6452014864780 | − 1.6452014864780 | |

\(T_{xx}\) | 8.22600743239035E–08 | – | |

\(T_{yx}\) | − 2.05924999039651E–08 | – | |

\(T_{yy}\) | 8.22600743239036E–08 | – | |

\(\dfrac{|T_{xx}+T_{yy}+T_{zz}|}{|T_{xx}|+|T_{yy}|+|T_{zz}|}\) | 2.73514373119212E–15 | – | |

Above a corner(20, 10, − 0.001) | \(T_{xz}\) | 3.7606613 | 3.7606613 |

\(T_{yz}\) | − 3.7606613 | − 3.7606613 | |

\(T_{zz}\) | − 2.1458379 | − 2.1458379 | |

\(T_{xx}\) | 1.07291859383300E–08 | – | |

\(T_{yx}\) | 3.60015219545839E–07 | – | |

\(T_{yy}\) | 1.07291932616670E–08 | – | |

\(\dfrac{|T_{xx}+T_{yy}+T_{zz}|}{|T_{xx}|+|T_{yy}|+|T_{zz}|}\) | 3.77463390064588E–13 | – |

### 3.2 An Irregular Polyhedron with Linear Density Contrast

*U*is the gravitational potential. The vertical profile is from a point with coordinates of \((0, 0, -\,34)\) km to a point with coordinates of (0, 0, 0) km, with observation points placed at a uniform vertical (

*z*) spacing of 1 km. Due to the discontinuity of the gradient tensor signals on the boundary, we only perform tests at two points which are very close to the boundary (with a small distance of \(\pm \,10^{-10}\) km). The value of \(-4\pi G\lambda\) is used as the reference. The relative errors are calculated using the formula \(\frac{(T_{xx}+T_{yy}+T_{zz})+4\pi G\lambda }{|T_{xx}|+|T_{yy}|+|T_{zz}|}\) and shown in Fig. 4. The maximum absolute relative error is \(3.08389\times 10^{-13}\%\).

Values of the gravity gradient tensors for the irregular polyhedron and the observation points marked by blue stars in Fig. 3

At the outer point (0, 0, 0) km | At the centroid (40, 250, − 1541)/88 km | |||
---|---|---|---|---|

Our solution | Holstein’s (2003) solution | Our solution | Holstein’s (2003) solution | |

\(T_{xx} (s^{-2})\) | − 7.6908531369E–01 | − 7.6908531369E–01 | − 2.0052858984E+00 | − 2.0052858984E–00 |

\(T_{xy} (s^{-2})\) | 1.7105121125E–02 | 1.7105121125E–02 | 2.8319243428E–02 | 2.8319243428E–02 |

\(T_{xz}(s^{-2})\) | − 6.0704410609E–02 | − 6.0704410609E–02 | − 2.786584432 | − 2.786584431 |

\(T_{yy}(s^{-2})\) | − 4.5350144435E–01 | − 4.5350144435E–01 | − 9.5222605949E–01 | − 9.5222605949E–01 |

\(T_{yz}(s^{-2})\) | − 2.6549313009E–01 | − 2.6549313009E–01 | 1.2199835474E–01 | 1.2199835474E–01 |

\(T_{zz}(s^{-2})\) | 1.2225867580E+00 | 1.2225867580E–00 | − 9.6088586564E+00 | − 9.6088586564E–00 |

### 3.3 A Tetrahedral Mass Body with Horizontal and Vertical Density Contrasts

In practical gravity exploration, the underground mass bodies can have complicated shapes. To calculate their GGT signals, we generally need to discretise the underground mass bodies into sets of disjoint elements with different shapes, such as structured hexahedral or prismatic elements and unstructured tetrahedral elements. Compared to regular prismatic elements, unstructured tetrahedral elements can well approximate arbitrarily complicated anomalies. Using recent Delaunay triangulation techniques (e.g., Si 2015), geophysicists can easily set up discretised triangulated grids to represent complicated anomalous targets. Therefore, unstructured grid techniques have been widely used in the geophysical community, not only in gravity exploration (Jahandari and Farquharson 2013; Ren et al. 2017c), but also in the electromagnetic induction community (Li and Key 2007; Schwarzbach et al. 2011; Ren et al. 2013) and the seismic imaging field (Lelièvre et al. 2011). Testing a single tetrahedral mass body not only aims to test the performance of our new closed-from solutions for complicated density contrasts (horizontal and vertical density contrasts), but also demonstrates its compatibility with other codes developed for unstructured grids. This verification is an important step towards joint inversion for multiple parameters, such as magnetisation vector, conductivity, velocity and mass density on the same unstructured gird.

### 3.4 Numerical Stability

We present an experiment, in which the tetrahedron shown in Fig. 5 was used to recover the above linear error growth curve. The size of the tetrahedron was set to the diameter of its circumscribed sphere, that is \(\alpha =0.134\) km. The starting point for the profile is located at the centre of the circumscribed sphere of the tetrahedral model. The measuring profile is formulated as \(x=y, z=0\) km.

*T*and \({T}_{\mathrm{point}}\), respectively. In addition, the relative errors \(\eta =\left| {T}-{T}_{\mathrm{point}}\right| /\left| {T}_{\mathrm{point}}\right|\) were computed. Four polynomial density contrasts of different orders were used for the stability test:

The relative error curves are shown in Fig. 13 for the above four density contrasts. In Fig. 13, at low dimensionless target distances \(\gamma ^{-1}\), the relative errors decrease with increasing \(\gamma ^{-1}\), because the tetrahedron is better approximated as a mass point with increasing distance. However, the gravity anomalies are progressively corrupted with increasing target distance, and when the dimensionless target distance (\(\gamma ^{-1}\)) is beyond a critical value, the rounding error in the unstable calculations exceeds the solution difference, resulting in a rising trend of the relative errors (cf. Holstein 2003). We reconstructed the entire error growth curves by extrapolating the ascending parts back to \((\gamma ^{-1}=1,\;\eta =\varepsilon )\). In Fig. 13, the reconstructed curves are denoted by blue dashed lines. We observe that there are two linear error growth trends for each polynomial density contrast, that is \(T_{zx}\) and \(T_{zy}\) share the same error growth behaviour, and \(T_{xx}\), \(T_{yy}\), \(T_{zz}\) and \(T_{yx}\) share the same error growth behaviour. The estimated values of \(\kappa\) are shown in Table 4. Using the estimated \(\kappa\), the critical dimensionless target distances were calculated from that value of \(\gamma ^{-1}\) where the rising part of the error curve indicates a relative error of 1 (Holstein and Ketteridge 1996; Holstein 2003; Zhou 2010) in Eq. (25), which are also shown in Table 4.

Experimental values of \(\kappa\) and \(1/\gamma _{\mathrm{crit}}\) for polynomial density contrasts of different orders. All experiments were performed on a computer with an Intel Core i5-4590 processor using double precision arithmetics

Order | \(T_{xz}\), \(T_{yz}\) | \(T_{xx}\), \(T_{yy}\), \(T_{zz}\), \(T_{xy}\) | ||
---|---|---|---|---|

\(\kappa\) | \(1/\gamma _{\mathrm{crit}}\) | \(\kappa\) | \(1/\gamma _{\mathrm{crit}}\) | |

Constant | 4.21 | 5195.26 | 3.23 | 70,079.11 |

Linear | 5.28 | 926.39 | 4.31 | 4256.74 |

Quadratic | 6.31 | 302.90 | 5.36 | 835.48 |

Cubic | 7.33 | 137.09 | 6.39 | 282.00 |

## 4 Conclusions

In the presented work, we have derived a set of closed-form solutions for the gravity gradient tensor of an arbitrary polyhedral body. The density contrast of the polyhedral body is represented as a polynomial function of up to and including third order. The polynomial function allows density contrasts to simultaneously vary in both horizontal and vertical directions. To our best knowledge, this is the first time that analytical solutions of the gravity gradient tensor are derived for an arbitrary polyhedral body with polynomial orders up to three. Three synthetic models (a prismatic body, an irregular polyhedron and a tetrahedral body) were used to test the correctness and the efficiency of our newly developed closed-form solutions. By comparing to published closed-form solutions and high-order Gaussian quadrature solutions, the high accuracies of our solutions with deviations of less than \(2\times 10^{-7}\)% from the Gaussian quadrature solutions were demonstrated. The computation time used by our analytical solution is significantly less than that of the high-order Gaussian quadrature.

The numerical stability tests show that, when dealing with cubic density contrast, our closed-form solutions would generate inaccurate solutions, if the dimensionless target distance, which is defined as the ratio of the distance between the observation site and the causative body to the dimension of the causative body, is larger than about 282 for \(T_{xx}\), \(T_{yy}\), \(T_{zz}\), \(T_{yx}\) components, and 137 for \(T_{zx}\), \(T_{zy}\) components, on a profile with \(x=y\) and \(z=0\,{\text {m}}\). This inaccuracy problem is caused by the limited precision of floating point operations, when the amplitudes of the gravity gradient tensors are very close to zero. However, since most applications of gravity gradient tensors aim at detecting anomalous bodies in the very shallow subsurface of the Earth, this problem of precision loss should not cause a serious problem in practical situations, and our closed-form solutions are very safe to be applied in exploration geophysics.

For the cases of constant, linear, second and cubic polynomial order, there are systematic recurrences of previously evaluated integrals as well as occurrences of new types of integrals, when going to higher polynomial order. We assume that this is a trend even for higher polynomial orders (\(P>3\)). However, extending our work to higher polynomial orders will be a nontrivial task as more and more complicated integrals need to be considered. By contrast, using the analytical expressions presented in this study, closed-form solutions for the magnetic field can be derived easily.

## Notes

### Acknowledgements

This study was supported by Grants from the National Basic Research Program of China (973 - 2015CB060200), the National Science Foundation of China (41574120), the Natural Science Foundation of Hunan Province of China (2016JJ2139), the State High-Tech Development Plan of China (2014AA06A602), the Project of Innovation-driven Plan in Central South University (2016CX005), and an award for outstanding young scientists by Central South University (Lieying program 2013) and the Fundamental Research Funds for the Central Universities of Central South University (1053320171677). YL was supported by Pioneer Hundred Talents Program, Chinese Academy of Sciences. Deep thanks are given to H. Holstein and H. Götze and an anonymous reviewer for critical and helpful comments which significantly improved the quality of this article.

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