Zhang et al. Earth, Planets and Space (2018) 70:58 https://doi.org/10.1186/s40623-018-0825-5 FULL PAPER Open Access 3D inversion of full gravity gradient tensor data in spherical coordinate system using local north‑oriented frame Yi Zhang1, Yulong Wu1*, Jianguo Yan2, Haoran Wang3,4, J. Alexis P. Rodriguez5 and Yue Qiu6 Abstract In this paper, we propose an inverse method for full gravity gradient tensor data in the spherical coordinate system. As opposed to the traditional gravity inversion in the Cartesian coordinate system, our proposed method takes the curvature of the Earth, the Moon, or other planets into account, using tesseroid bodies to produce gravity gradient efects in forward modeling. We used both synthetic and observed datasets to test the stability and validity of the proposed method. Our results using synthetic gravity data show that our new method predicts the depth of the den- sity anomalous body efciently and accurately. Using observed gravity data for the Mare Smythii area on the moon, the density distribution of the crust in this area reveals its geological structure. These results validate the proposed method and potential application for large area data inversion of planetary geological structures. Keywords: Gravity gradient tensor, Inversion, Density distribution, Spherical coordinate system, Local north-oriented frame Introduction Barbosa 2012; Oliveira and Barbosa 2013; Martinez Te gravity gradient tensor (GGT) is the second deriva- et al. 2012; Geng et al. 2015; Meng 2016). Te difer- tive of the gravity potential. Compared to the general ences among these algorithms are related to the choice of gravity feld Tz, GGT contains nine components and has model object functions in the inversion procedure; all the much higher resolution in inverting the spatial position model object functions can be retraced back to the inver- of target anomaly bodies, which means it will ofer more sion methods used in the inversion of the general gravity information to better understand the interior structure of feld (e.g., Last and Kubik 1983; Guillen and Menichetti the earth or some other planets (Li 2001; Bouman et al. 1984; Barbosa and Silva 1994; Li and Oldenburg 1996, 2016). 1998; Farquharson 2008). In 1886, a torsion balance gradiometer was frst devel- In addition to the choice of model object functions, oped by Lorand Eotvos, becoming a useful tool for there is little agreement on best component of GGT for mining and hydrocarbon exploration (Pedersen and Ras- the inversion. Li (2001) combined fve independent com- mussen 1990; Bell and Hansen 1998). 3D inversion of ponents excluding Tzz. Zhdanov et al. (2004) used the GGT data was originally introduced by Vasco (1989) and horizontal components Txy and Tuv = (Txx − Tyy)/2. Mar- Vasco and Taylor (1991), who focused on the covariance tinez et al. (2012) combined the horizontal components and resolution measures of the solution appraisal. More and Tzz. Capriotti et al. (2015) employed a combination recently, several algorithms were developed to inverse of the GGT and general gravity feld Tz. Pilkington (2012) GGT data (e.g., Li 2001; Zhdanov et al. 2004; Uieda and used an eigenvalue spectra method to evaluate the utility of combining diferent GGT components, preferring the *Correspondence: [email protected] Tzz component, indicating that the source-model infor- 1 Key Laboratory of Earthquake Geodesy, Institute of Seismology, China mation is improved by adding more components only at Earthquake Administration, 48 Hongshan Side Road, Wuhan 430071, close distance to the anomaly sources. Later, Pilkington Hubei, China Full list of author information is available at the end of the article (2013) used estimated parameter errors from parametric © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Zhang et al. Earth, Planets and Space (2018) 70:58 Page 2 of 23 inversions, concluding that the Tzz component gave the A forward modeling method coupling with fnite element best performance, while the horizontal components Txx method (FEM) was employed this work. GGT is the dif- and Tyy performed poorly. Paoletti et al. (2016) used ference of gravity in diferent directions, which helps to a singular value decomposition (SVD) tool to analyze remove the infuence of the long-wave part in the lunar both synthetic data and gradiometer measurements. gravity, and it is easier to highlight the lunar gravity’s Tis research showed that the main factors controlling shortwave efect and to get a better resolution for density the reliability of the inversion are algebraic ambiguity imaging. In this work, we will focus on inversion of the (the diference between the number of unknowns and lunar gradient data, which is diferent from the previous the number of available data points) and signal-to-noise work (Andrews-Hanna et al. 2013). ratio. Te remainder of this paper is as follows: a brief intro- All these inversion methods mentioned are imple- duction of this inversion method is presented in second mented in the Cartesian coordinate system (CCS). For section. Tird section details two diferent models and small-scale problems such as mining or hydrocarbon experiments with synthetic GGT datasets and inversion exploration on earth, the target area is usually relatively of GGT observation datasets of moon. Te interior den- small compared to the radius of the earth and can be sity structure of the Mare Smythii mascon is discussed considered as a fat surface, so inversion in CCS works in fourth section. Finally, in ffth section we present the fne and obtains reliable inversion results with high accu- conclusions of this study. racy. However, for large-scale inversion problems, such as density imaging of the lunar mascon with the satel- Methodology lite gravity datasets, the target area of the lunar mascon GGT in the spherical coordinate system was usually large and covered an area with hundreds or Te two most commonly used frames in SCS are the thousands of kilometers in both longitude and latitude geocentric spherical frame (GSF) and the local north- directions. Moreover, the radius of the moon was rela- oriented frame (LNOF). Te GGT is expressed in terms tively small (1738 km), and this meant the inversion area of the second derivatives of the gravitational potential U of the mascon was no longer a fat area; hence, inversion in the r, λ, and ϕ directions of GSF, where r, λ, and ϕ refer methods cannot ignore the infuences of the lunar curva- to the radial, longitude, and latitude, respectively (Eq. 1). ture. To deal with this problem, the inversion method in ∂2U ∂2U ∂2U the SCS must be considered. Liang et al. (2014) extended T Tϕ Tr r2 cos2 ϕ∂2 r2 cos ϕ∂∂ϕ r cos ϕ∂∂r ∂2U ∂2U ∂2U T = T T T r = Li and Oldenburg’s (1996, 1998) inversion method to the ϕ ϕϕ ϕ r2 cos ϕ∂∂ϕ r2∂ϕ2 r∂ϕ∂r (1) 2 2 2 Tr Trϕ Trr ∂ U ∂ U ∂ U SCS using the general gravity feld. r cos ϕ∂∂r r∂ϕ∂r ∂r2 In the history of the Moon exploration, one of the most amazing discoveries was the concentrated areas of mass In LNOF, where z has the geocentric radial downward found on the near side of the moon (Muller and Sjogren direction, x points to the north, and y is directed to the 1968; Melosh et al. 2013; Freed et al. 2014). Tese con- east with a right-handed system, relationship between centrated areas of mass, referred to as mascons, usually the LNOF and GSF can be described as in Eq. (2) (Reed have a positive gravity anomaly peak and surrounded by 1973; Petrovskaya and Vershkov 2006). negative gravity anomalies with low geographical eleva- 1 1 T = T + Tϕϕ tion. Te knowledge of the interior density structure of xx r r r2 mascons will help to understand its origin mechanism 1 sin ϕ T = Tϕ + T (Wang et al. 2015; Jansen et al. 2017). Tere has a signif- xy r2 cos ϕ r2 cos2 ϕ cant improvement on lunar gravity feld model develop- 1 1 ment (Matsumoto et al. 2010; Yan et al. 2012), and the Txz = 2 Tϕ − Trϕ r r (2) recent high solution gravity feld model GL1500E derived 1 1 1 from GRAIL mission (Te Planetary Data System 2016) T = T + Tϕ + T yy r r r2 cot ϕ r2 cos2 ϕ makes it possible to investigate the interior density struc- 1 1 ture of the mascons (Zuber et al. 2013a, b) using lunar = − Tyz 2 cos T cos Tr gradient data. r ϕ r ϕ Te high-resolution lunar gradient data produced from Tzz = Trr GRAIL gravity was applied by Andrews-Hanna et al. (2014) to investigate the structure and evolution of the In this paper, we choose to use LNOF, which will not lunar Procellarum region. Te fault geometry, thermal be singular when calculating GGT components from a structure, and material content were considered to gener- gravity spherical harmonics model (Eshagh 2008, 2010) ate rectilinear patterns of the gradient data in this region. and because the GGT is symmetric and the trace of the Zhang et al. Earth, Planets and Space (2018) 70:58 Page 3 of 23 GGT equals zero; hence, there are only fve independent components. Forward modeling Teoretically, forward modeling is the basis of the inver- sion, as it forms the relationship between the model and data space. Te forward modeling method we use here was developed by Asgharzadeh et al.