### GGT in the spherical coordinate system

The two most commonly used frames in SCS are the geocentric spherical frame (GSF) and the local north-oriented frame (LNOF). The GGT is expressed in terms of the second derivatives of the gravitational potential *U* in the *r*, *λ*, and \(\varphi\) directions of GSF, where *r*, *λ*, and \(\varphi\) refer to the radial, longitude, and latitude, respectively (Eq. 1).

$$T = \left[ {\begin{array}{*{20}c} {T_{\lambda \lambda } } & {T_{\lambda \varphi } } & {T_{\lambda r} } \\ {T_{\varphi \lambda } } & {T_{\varphi \varphi } } & {T_{\varphi r} } \\ {T_{r\lambda } } & {T_{r\varphi } } & {T_{rr} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} U}}{{r^{2} \cos^{2} \varphi \partial \lambda^{2} }}} & {\frac{{\partial^{2} U}}{{r^{2} \cos \varphi \partial \lambda \partial \varphi }}} & {\frac{{\partial^{2} U}}{r\cos \varphi \partial \lambda \partial r}} \\ {\frac{{\partial^{2} U}}{{r^{2} \cos \varphi \partial \lambda \partial \varphi }}} & {\frac{{\partial^{2} U}}{{r^{2} \partial \varphi^{2} }}} & {\frac{{\partial^{2} U}}{r\partial \varphi \partial r}} \\ {\frac{{\partial^{2} U}}{r\cos \varphi \partial \lambda \partial r}} & {\frac{{\partial^{2} U}}{r\partial \varphi \partial r}} & {\frac{{\partial^{2} U}}{{\partial r^{2} }}} \\ \end{array} } \right]$$

(1)

In LNOF, where *z* has the geocentric radial downward direction, *x* points to the north, and *y* is directed to the east with a right-handed system, relationship between the LNOF and GSF can be described as in Eq. (2) (Reed 1973; Petrovskaya and Vershkov 2006).

$$\begin{aligned} T_{xx} & = \frac{1}{r}T_{r} + \frac{1}{{r^{2} }}T_{\varphi \varphi } \\ T_{xy} & = \frac{1}{{r^{2} \cos \varphi }}T_{\lambda \varphi } + \frac{\sin \varphi }{{r^{2} \cos^{2} \varphi }}T_{\lambda } \\ T_{xz} & = \frac{1}{{r^{2} }}T_{\varphi } - \frac{1}{r}T_{r\varphi } \\ T_{yy} & = \frac{1}{r}T_{r} + \frac{1}{{r^{2} \cot \varphi }}T_{\varphi } + \frac{1}{{r^{2} \cos^{2} \varphi }}T_{\lambda \lambda } \\ T_{yz} & = \frac{1}{{r^{2} \cos \varphi }}T_{\lambda } - \frac{1}{r\cos \varphi }T_{r\lambda } \\ T_{zz} & = T_{rr} \\ \end{aligned}$$

(2)

In this paper, we choose to use LNOF, which will not be singular when calculating GGT components from a gravity spherical harmonics model (Eshagh 2008, 2010) and because the GGT is symmetric and the trace of the GGT equals zero; hence, there are only five independent components.

### Forward modeling

Theoretically, forward modeling is the basis of the inversion, as it forms the relationship between the model and data space. The forward modeling method we use here was developed by Asgharzadeh et al. (2007) for calculating a gravity field and its gradients in the SCS with LNOF. This method makes use of the Gauss–Legendre quadrature integration for numerically modeling theoretical gravity effects caused by the tesseroids (Anderson 1976; Heck and Seitz 2006) (Fig. 1).

According to Asgharzadeh et al. (2007), each component of the GGT can form its own relationship between the model and dataset and can be described as:

$$T_{ij} = G_{ij} m\quad i,j = x,y,z$$

(3)

In Eq. (3), *m* and *G*_{
ij
} refer to the model and kernel matrix, respectively.

To make full use of the GGT dataset, we adopt the new kernel function *G* as the linear combination of the five independent components of GGT:

$$G_{s} = \mathop \sum \limits_{i,j}^{x,y,z} \left( {k_{ij} G_{ij} } \right)$$

(4)

*k*_{
ij
} here refers to the weighting factor of each component, and it can be considered as the data accuracies (or the reliabilities) of each component.

Due to the relationship of *T*_{
xx
} + *T*_{
yy
} + *T*_{
zz
} = 0, the linear combination of the components *T*_{
xx
} and *T*_{
yy
} can be described by the vertical components *T*_{
zz
}; hence, we do not employ the components *T*_{
xx
} and *T*_{
yy
} in Eq. (4).

Similarly, the GGT dataset *d*_{
s
} can be also adopted into the linear combination of the independent components:

$$d_{s} = T_{s} = \mathop \sum \limits_{i,j}^{x,y,z} \left( {k_{ij} T_{ij} } \right)$$

(5)

For each independent component *T*_{
ij
}, the error standard deviation is *σ*_{
ij
}. According to error theory, the error standard deviation of the GGT will be:

$$\sigma = \sqrt {\sum\nolimits_{i,j}^{x,y,z} {\left( {k_{ij} \sigma_{ij} } \right)^{2} } + Cov}$$

(6)

In Eq. (6), *Cov* represents the sum of the error covariance of all components. On the assumption that the error of the GGT follows the Gaussian random distribution and one independent component has no connection with each other, then *Cov* = 0.

### Inversion method

In general, because of an insufficient observed dataset, the multiple solutions problem becomes a serious issue for the 3D gravity inversion. To deal with the problem, a suitable model objective function is required.

Li and Oldenburg (1996) and Li (2001) designed a model objective function with a maximum smoothing method; however, this model objective function is only suitable for CCS. Liang et al. (2014) extended it to the SCS using the spherical derivative operators:

$$\begin{aligned} \phi_{m} (m) & = \alpha_{s} \int\limits_{V} {[w(r)(m - m_{\text{ref}} )]^{2} {\text{d}}v} + \alpha_{r} \int\limits_{V} {\left[ {\frac{{\partial w(r)(m - m_{\text{ref}} )}}{\partial r}} \right]^{2} {\text{d}}v} \\ & \quad + \,\alpha_{\varphi } \int\limits_{V} {\left[ {\frac{{\partial w(r)(m - m_{\text{ref}} )}}{r\partial \varphi }} \right]^{2} {\text{d}}v} + \alpha_{\lambda } \int\limits_{V} {\left[ {\frac{{\partial w(r)(m - m_{\text{ref}} )}}{ \, r\cos \varphi \partial \lambda }} \right]^{2} {\text{d}}v} \\ \end{aligned}$$

(7)

The model object function of Eq. (7), also called stabilized function, was initially introduced by Backus and Gilbert (1967, 1968, 1970) to solve ill-posed inverse problems. It can be divided into two parts: the first item of Eq. (7) is the smallest model between recovered and reference model, and the last three items are the smoothest model between recovered and reference model in longitudinal, latitudinal, and radial directions, respectively. In Eq. (7), *m* and *m*_{
ref
} refer to the recovered and reference model, respectively. *α*_{
i
} (*i* = *s*, *r*, *λ*, \(\varphi\)) are length scales, which control the balance of the smoothness versus smallness for the whole model, *α*_{
s
} for smallness, and *α*_{
r
}, *α*_{
λ
}, and \(a_{\varphi}\) for smoothness (Oldenburg and Li 2005; Williams 2008). In practice, *α*_{
s
} usually can be assigned to a value of 1.0 or other suitable value; however, different to those in CCS (Williams 2008), *α*_{
λ
}, and \(a_{\varphi}\) are variable because of the different tesseroid body sizes along the radial direction. *w*(*r*) here represents the depth weighting function, and it can be used to avoid the skin effect in the inversion (Li and Oldenburg 1996; Li 2001). The depth weighting functions match the decay of the gravity or magnetic kernel functions, and they are in proportion to 1/*r*^{2} in gravity and 1/*r*^{3} in magnetic inversion problem. Without them, the inversion will get results concentrated on the surface of the target area (Li 2001). Unlike the uniform prism cells in CCS, the tesseroid cells become smaller along the radial direction from surface to the core, so it must rescale them into the same level. Liang’s et al. (2014) main contribution is the modification of the depth weighting function in SCS by rescaling the cell volume to the surface (see Additional file 1).

$$w^{2} (r) = \frac{{r^{2} }}{{r_{0}^{2} (H + R - r)^{2} }}$$

(8)

In Eq. (8), *R* is the radius of the reference sphere and *H* is the average height of observed dataset above the reference sphere, while *r*_{0} and *r* are the radial distance of the surface and for computing tesseroid cells, respectively.

In addition, geological and geophysical constraints play an important role in the gravity inversion. The geological and geophysical constraints are varied, and they can be classified into two different kinds: (1) geometry constraints like structure boundaries, orientations, and locations information; (2) physical property constraints such as surface material content and information from drill holes. All these constraints can be described as the function of the physical property and positions. In our inversion method, as we divided the subspace into different tesseroid cells, the constraints become the function of the physical property and index number of the tesseroid cells.

Different from the model objective function, which aims to solve the non-uniqueness in ill-posed inverse problems, the purpose of using a prior geological and geophysical information during the inversion is to improve the inversion result. In this paper, we use the Lagrangian multipliers method, introduced by Zhang et al. (2015), to fit for the different prior geological or geophysical information during the inversion procedure; the additional penalty function of the density bound constraints makes the recovered model more reliable.