# Joint Inversion of 2D Gravity Gradiometry and Magnetotelluric Data in Mineral Exploration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Forward Problem

#### 2.1. Gravity and Gravity Gradiometry Forward Problem

_{d}observations and N

_{m}rectangular units, the discrete forward modeling operators for gravity or gravity gradiometry can be expressed in a matrix form:

_{d}, $m$ is a vector of the remaining densities of the order N

_{m}, and $G$ is expressed as a forward problem kernel functional, a rectangular matrix of a size N

_{d}× N

_{m}. According to Equation (4), the observed data are linearly related to the residual density, and it can be observed that the Jacobian matrix of the observed data is independent of the residual density and only depends on the position of the rectangular unit relative to the observation point.

#### 2.2. MT Forward Problem

## 3. Joint Inversion Methodology

**C**

_{d4}is the data covariance matrix of the MT observation data, ${C}_{m1}$ and ${C}_{m2}$ are the model covariance matrix of density ${m}_{1}$ and resistivity ${m}_{2}$, respectively, ${m}_{01}$ and ${m}_{02}$ are the prior model parameters, ${\mathsf{\alpha}}_{1}$ and ${\mathsf{\alpha}}_{2}$ are damping parameters,

**f**

_{1}(

**m**),

**f**

_{2}(

**m**) and

**f**

_{3}(

**m**) are forward responses of the three components of the gravity gradiometry,

**f**

_{4}(

**m**) is an MT forward response, $\nabla $ is a gradient, $\mathsf{\tau}$ is a normalized cross-gradient functional as a parameter gradient of the density and resistivity model, and χ

_{1}and χ

_{2}are the normalized operators of resistivity and density, respectively. For our particular test cases, the normalized operators are determined by ${\chi}_{1}=\mathrm{Max}({m}_{1\_sep}^{})-\mathrm{Min}({m}_{1\_sep}^{})$, and ${\chi}_{2}=\mathrm{Max}({m}_{2\_sep}^{})-\mathrm{Min}({m}_{2\_sep}^{})$. Max and Min are the maximum and minimum values, respectively. ${m}_{1\_sep}$ and ${m}_{2\_sep}$ are represented as a separate inversion density and resistivity model, respectively.

**A**and

**B**represent the Jacobian matrix of the forward response $f(m)$ and the normalized cross-gradient functional $\mathsf{\tau}$, and

**Λ**is a column vector of the Lagrange multipliers.

**Λ**is given as

## 4. Synthetic Example

#### 4.1. Compare Gravity and Gravity Gradiometry Data

^{3}and a resistivity of 100 Ω·m. The reference model is set to the initial model. Figure 2 and Figure 3 show the forward response results of resistivity and density models of different methods. The forward response calculated by either the separate inversion or joint inversion results is basically consistent with theoretical model responses. Figure 4 shows the separate inversion results of the MT, gravity and gravity gradiometry for the complex combined model. The data misfit of MT, gravity and gravity gradiometry inversion are RMS

_{MT}= 0.95, RMS

_{Grv}= 0.71, and RMS

_{Grad}= 0.80, respectively (Figure 5a). Notably, the gravity inversion (Figure 4b) cannot recover the true space geometry and position of the anomalous bodies, and the vertical resolution of the inversion results is very poor, and it is difficult to accurately identify low-density anomalous bodies. However, the inversion result of the gravity gradiometry (Figure 4c) can roughly reflect the spatial position and geometry of the true anomalous bodies. Compared with the results of the gravity inversion, the distribution of the recovered underground anomaly can be improved, but the gravity gradiometry inversion results still have problems, such as the upset of the center of the anomalous bodies, the blurring of the boundary between the anomalous bodies and the surrounding rock, and the inability to recover the exact physical parameter value. The resistivity model (Figure 4a) recovers the gross structure of the synthetic model better than the density model (Figure 4b,c), as the MT data cover an adequate frequency range for distinguishing deep anomalous bodies. The results of the joint inversion are shown in Figure 6. For the joint inversion of gravity and MT (Figure 6a,b), the final models of resistivity and density are obtained with the data misfit (RMS

_{MT}= 0.66, RMS

_{Grv}= 1.0), as shown in Figure 5b. The structural features of the anomalous bodies are very similar in the resistivity and density models as required by our joint inversion algorithm. The deep anomalous body of the density model can be identified as an indirect contribution propagated from the resistivity model by the cross-gradient constraints. For the joint inversion of the gravity gradiometry and MT (Figure 6c,d), the final models of resistivity and density are obtained with data misfit (RMS

_{MT}= 0.83, RMS

_{Grad}= 0.80), as shown in Figure 5c. The resistivity and density models show improved features when compared to the above separate and joint inversion results. The density models’ anomalous bodies boundaries become clearer, and the geometrical and physical values of the anomaly more closely reflect the true model, as an indirect result of the resistivity model using the cross-gradient constraints. In particular, the vertical resolution has been significantly improved. The resistivity model also improves the horizontal resolution due to the structural similarity of the density. The results of the gravity gradiometry and MT joint inversion are better than the results of the separate inversion and gravity and MT joint inversion, both in space form and the numerical recovery of physical properties.

#### 4.2. Test the Partially Structurally Consistent and Inconsistent Model

^{3}and a resistivity of 100 Ω·m. The reference model is set to the initial model. In the partially structurally consistent model (Figure 8a,b), the joint inversion results are shown in Figure 9a,b. The data misfit of the MT and gravity gradiometry inversion are RMS

_{MT}= 0.75 and RMS

_{Grad}= 0.95, respectively. In the structurally inconsistent model (Figure 10a,b), the joint inversion results are shown in Figure 11a,b. The data misfit of MT and gravity gradiometry inversion are RMS

_{MT}= 0.97 and RMS

_{Grad}= 0.73, respectively. All joint inversions converge to the RMS misfit threshold of one or less. We find that the proposed method can recover the geometry and physical parameter values of the underground anomaly sources in the partially structurally consistent and inconsistent models. The results show that the joint inversion of MT and gravity gradiometry is suitable for various scenarios (structural consistent and structural inconsistent models), which can recover the true model. Different geophysical methods are mutually constrained by the structural similarities at the same model boundaries, while the structural similarity constraints are not effective at different model boundaries, and joint inversion only performs smoothing model constraints.

#### 4.3. Compare Joint Inversion of Model-Space and Data-Space

^{2}. The gravity gradiometry method has 79 observation points, with a point spacing of 0.1 km from a horizontal distance of −1 to 7 km. The MT data contain the apparent phase and apparent resistivity for 10 frequencies in the range of 1 to 1000 Hz for 39 stations spaced 0.2 km from each other.

^{3}and a resistivity of 100 Ω·m. The reference model is set to the initial model. For the model-space joint inversion of gravity gradiometry and MT, the sixth iteration model (Figure 13a,b) attained data misfits of 1.05 and 1.14 for gravity gradiometry and MT, respectively. For the data-space joint inversion of gravity gradiometry and MT, the sixth iteration model (Figure 13c,d) attained data misfits of 1.05 and 1.00 for gravity gradiometry and MT. We can find that the inversion results of the two methods can recover the geometry of the double anomaly. However, the resistivity and density models (Figure 13c,d) show improved features when compared to the model-space joint inversion results (Figure 6b,e). The density and resistivity models’ anomalous body boundaries become clearer, and the geometrical and physical values of the anomaly more closely reflect those of the true model. The inconsistencies of the inversion results are mainly due to the different model covariance matrices used by the two methods. In terms of inversion calculation time, the model-space method consumes approximately 13.15 h, whereas the data-space method consumes approximately 3.92 h for our particular test example. In addition, the maximum memory requirements are approximately 4.1 and 0.62 GB, respectively, as shown in Table 1. The data-space method can be applied to the joint inversion MT and gravity gradiometry, which can greatly reduce memory consumption and effectively improve the calculation speed. This method allows us to invert the multi-parameter joint inversion of large areas of large data volume with a personal computer (PC) in a relatively short period of time.

## 5. Field Example

#### 5.1. Geologic Background of the Study Area

#### 5.2. Data Acquisition and Inversion

^{3}and a resistivity of 10

^{2.5}Ω·m. The results of the separate inversion are shown in Figure 17 and Figure 18. In the separate inversion using the model-space method, the data misfit of the CSAMT, gravity, and gravity gradiometry are RMS

_{CSAMT}= 1.08, RMS

_{Grv}= 0.48, and RMS

_{Grad}= 0.97, respectively. In the separate inversion using the data-space method, the data misfit of the CSAMT, gravity, and gravity gradiometry are RMS

_{CSAMT}= 1.08, RMS

_{Grv}= 0.93 and RMS

_{Grad}= 0.99, respectively. The separate inversion results obtained by the two methods are, generally, features of the same trend. The inconsistencies of the inversion results are mainly due to the different covariances of the models used by the two methods. The structural similarities of the separate inversion results of the three geophysical exploration methods are very different in both the model-space and data-space methods. These differences create great difficulties for geological interpretation. Compared with the gravity inversion results, the gravity gradiometry inversion results identify greater underground structural distribution and can obtain more underground information and help geological interpretation.

_{CSAMT}= 1.08, RMS

_{Grv}= 0.64). For the joint inversion of gravity gradiometry and CSAMT data using the model-space method (Figure 20a,b), the final models of resistivity and density are obtained with data misfit (RMS

_{CSAMT}= 1.00, RMS

_{Grad}= 0.99). For the joint inversion of gravity gradiometry and CSAMT data using the data-space method (Figure 21a,b), the final models of resistivity and density are obtained with data misfit (RMS

_{CSAMT}= 1.64, RMS

_{Grad}= 1.29). Note that the structural similarity between the different physical properties in the results of joint inversion, which is due to the contribution of cross-gradient constraints in the objective functional, and the result of the joint inversion is more conducive to geological interpretation. Comparing the joint inversion results of the two different data sets, we find that the results of the joint inversion of gravity and MT can roughly divide the underground structure. However, the results of the joint inversion of the gravity gradiometry and MT can clearly and meticulously identify the underground structural information, which facilitates more accurate geological interpretation and detection of geological resources. Meanwhile, the joint inversion based on data-space (Figure 21a,b) has more advantages in memory storage and computing time than model-space (Figure 20a,b).

#### 5.3. Geological Interpretation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Auken, E.; Pellerin, L.; Christensen, N.B. A survey of current trends in near-surface electrical and electromagnetic methods. Geophysics
**2006**, 71, G249–G260. [Google Scholar] [CrossRef] - Kowalsky, M.B.; Chen, J.; Hubbard, S.S. Joint inversion of geophysical and hydrological data for improved subsurface characterization. Lead. Edge
**2006**, 25, 730–731. [Google Scholar] [CrossRef] - Colombo, D.; De Stefano, M. Geophysical modeling via simultaneous joint inversion of seismic, gravity, and electromagnetic data: Application to prestack depth imaging. Lead. Edge
**2007**, 26, 326–331. [Google Scholar] [CrossRef] - Alpak, F.O.; Torres-Verdin, C.; Habashy, T.M. Estimation of in situ petrophysical properties from wireline formation tester and induction logging measurements: A joint inversion approach. J. Pet. Sci. Eng.
**2008**, 63, 1–17. [Google Scholar] [CrossRef] - Fregoso, E.; Gallardo, L.A. Cross-gradients joint 3D inversion with applications to gravity and magnetic data. Geophysics
**2009**, 74, L31–L42. [Google Scholar] [CrossRef] - Linde, N.; Tryggvason, A.; Peterson, J.E. Joint inversion of cross hole radar and seismic traveltimes acquired at the South Oyster Bacterial Transport Site. Geophysics
**2008**, 73, G29–G37. [Google Scholar] [CrossRef] - Moorkamp, M.; Lelièvre, P.G.; Linde, N.; Khan, A. Integrated Imaging of the Earth: Theory and Applications; John Wiley & Sons: New Jersey, NJ, USA, 2016. [Google Scholar]
- Moorkamp, M.; Heincke, B.; Jegen, M.; Roberts, A.W.; Hobbs, R.W. A framework for 3D joint inversion of MT, gravity and seismic refraction data. Geophys. J. Int.
**2011**, 184, 477–493. [Google Scholar] [CrossRef] - Gallardo, L.A.; Meju, M.A. Structure-coupled multiphysics imaging in geophysical sciences. Rev. Geophys.
**2011**, 49, RG1003. [Google Scholar] [CrossRef] - Tarantola, A. Inversion Problem Theory and Methods for Model Parameter Estimation; SIAM: Philadelphia, PA, USA, 2005. [Google Scholar]
- Yang, W.C. Theory and Methods of Geophysical Inversion; Geological Publishing House: Beijing, China, 1997. (In Chinese) [Google Scholar]
- Heincke, B.; Jegen, M.; Hobbs, R. Joint inversion of MT, gravity and seismic data applied to sub-basalt imaging. SEG Expand Abstr.
**2006**, 2006, 784–789. [Google Scholar] - Gao, G.; Abubakar, A.; Habashy, T.M. Joint petrophysical inversion of electromagnetic and full-waveform seismic data. Geophysics
**2012**, 77, WA3–WA18. [Google Scholar] [CrossRef] - Haber, E.; Oldenburg, D. Joint inversion: A structural approach. Inverse Probl.
**1997**, 13, 63–77. [Google Scholar] [CrossRef] - Zhang, J.; Morgan, F.D. Joint seismic and electrical tomography. In Symposium on the Application of Geophysics to Engineering and Environmental Problems; SEG: Denver, CO, USA, 1997; pp. 391–396. [Google Scholar]
- Gallardo, L.A.; Meju, M.A. Characterization of heterogeneous near-surface materials by joint 2d inversion of dc resistivity and seismic data. Geophys. Res. Lett.
**2003**, 30, 1658. [Google Scholar] [CrossRef] - Abubakar, A.; Gao, G.; Havashy, T.M.; Liu, J. Joint inversion approaches for geophysical electromagnetic and elastic full-waveform data. Inverse Probl.
**2012**, 28, 055016. [Google Scholar] [CrossRef] - Hamdan, H.A.; Vafidis, A. Joint inversion of 2D resistivity and seismic travel time data to image saltwater intrusion over karstic areas. Environ. Earth Sci.
**2013**, 68, 1877–1885. [Google Scholar] [CrossRef] - Li, T.L.; Zhang, R.Z.; Pak, Y.C. Joint Inversion of magnetotelluric and first-arrival wave seismic traveltime with cross-gradient constraints. J. Jilin Univ. Earth Sci. Ed.
**2015**, 45, 952–961. [Google Scholar] - Gallardo, L.A.; Meju, M.A. Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints. J. Geophys Res. Solid Earth
**2004**, 109, B03311. [Google Scholar] [CrossRef] - Gallardo, L.A.; Fontes, S.L.; Meju, M.A.; Buonora, M.P.; De Lugao, P.P. Robust geophysical integration through structure-coupled joint inversion and multispectral fusion of seismic reflection, magnetotelluric, magnetic, and gravity images: Example from santos basin, offshore brazil. Geophysics
**2012**, 77, B237–B251. [Google Scholar] [CrossRef] - Li, T.L.; Zhang, R.Z.; Pak, Y.C. Multiple joint inversion of geophysical data with sub-region cross-gradient constraints. Chin. J. Geophys.
**2016**, 59, 2979–2988. [Google Scholar] - Zhang, R.Z.; Li, T.L.; Deng, H. 2D joint inversion of MT, gravity, magnetic and seismic first-arrival traveltime with cross-gradient constraints. Chin. J. Geophys.
**2019**, 62, 2139–2149. [Google Scholar] - Zhang, R.Z.; Li, T.L.; Zhou, S.; Deng, X.H. Joint MT and gravity inversion using structural constraints: A case study from the Linjiang copper mining area, Jilin, China. Minerals
**2019**, 9, 407. [Google Scholar] [CrossRef] - Zhdanov, M.S.; Ellis, R.; Mukherjee, S. Three-dimensional regularized focusing inversion of gravity gradient tensor component data. Geophysics
**2004**, 69, 1–4. [Google Scholar] [CrossRef] - Martinez, C.; Li, Y.; Krahenbuhl, R. 3D inversion of airborne gravity gradiometry data in mineral exploration: A case study in the Quadrilátero Ferrífero, Brazil. Geophysics
**2013**, 78, B1–B11. [Google Scholar] [CrossRef] - Geng, M.; Huang, D.; Yang, Q. 3D inversion of airborne gravity-gradiometry data using cokriging. Geophysics
**2014**, 79, G37–G47. [Google Scholar] [CrossRef] - Pilkington, M. Analysis of gravity gradiometer inverse problems using optimal design measures. Geophysics
**2012**, 77, G25–G31. [Google Scholar] [CrossRef] - Pilkington, M. Evaluating the utility of gravity gradient tensor components. Geophysics
**2014**, 79, G1–G14. [Google Scholar] [CrossRef] [Green Version] - Kivior, I.; Markham, S.; Hagos, F. Improved imaging of the subsurface geology in the Mowla Terrace, Canning Basin using gravity gradiometry data. ASEG Ext. Abstr.
**2018**, 2018, 1–10. [Google Scholar] [CrossRef] - Siripunvaraporn, W.; Egbert, G. An efficient data-subspace inversion method for 2-D magnetotelluric data. Geophysics
**2000**, 65, 791–803. [Google Scholar] [CrossRef] - Siripunvaraporn, W.; Egbert, G.; Lenbury, Y.; Uyeshima, M. Three–dimensional magnetotelluric inversion: Data-space method. Phys. Earth Planet. Inter.
**2005**, 150, 3–14. [Google Scholar] [CrossRef] - Singh, B. Simultaneous computation of gravity and magnetic anomalies resulting from a 2D object. Geophysics
**2002**, 67, 801–806. [Google Scholar] [CrossRef] - Won, I.J. Computing the gravitational and magnetic anomalies due to a polygon: Algorithms and Fortran subroutines. Geophysics
**1987**, 52, 202–205. [Google Scholar] [CrossRef] - Wannamaker, P.E.; Stodt, J.A.; Rijo, L. A stable finite element solution for two-dimensional magnetotelluric modeling. Geophys. J. Int.
**1987**, 88, 277–296. [Google Scholar] [CrossRef] - Menke, W. Geophysical Data Analysis: Discrete Inverse Theory, Revised Version: International Geophysics; Academic Press: San Diego, CA, USA, 1989. [Google Scholar]
- Tarantola, A. Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation; Elsevier: New York, NY, USA, 1987. [Google Scholar]
- Constable, S.C.; Parker, R.L.; Constable, C.G. Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics
**1987**, 52, 289–300. [Google Scholar] [CrossRef] - deGroot-Hedlin, C. Removal of static shift in two dimensions by regularized inversion. Geophysics
**1991**, 56, 2102–2106. [Google Scholar] [CrossRef] - deGroot-Hedlin, C.; Constable, S. Inversion of magnetotelluric data for 2D structure with sharp resistivity contrasts. Geophysics
**2004**, 69, 78–86. [Google Scholar] [CrossRef] - Li, H.; Li, T.; Wu, L. Transformation of all-time apparent resistivity of CSAMT and analysis of its effect. Prog. Geophys. (In Chinese)
**2015**, 30, 0889–0893. [Google Scholar] - ÖzgüArisoy, M.; Dikmen, Ü. Potensoft: MATLAB-based software for potential field data processing, modeling and mapping. Comput. Geosci.
**2011**, 37, 935–942. [Google Scholar]

**Figure 2.**Pseudo-sections illustrating the forward responses of apparent resistivity (

**a**,

**c**,

**e**,

**g**) and apparent phase (

**b**,

**d**,

**f**,

**h**) by the transverse magnetic mode. Forward response of theoretical resistivity model (

**a**,

**b**), forward response of separate inversion of magnetotelluric (MT) (

**c**,

**d**), forward response of joint inversion of gravity and MT (

**e**,

**f**), forward response of joint inversion of gravity gradiometry and MT (

**g**,

**h**).

**Figure 3.**Forward responses curves of gravity (

**a**,

**b**) and gravity gradiometry (

**c**,

**d**,

**e**,

**f**,

**g**,

**h**). Forward response of the separate inversion of the gravity (

**a**), forward response of the separate inversion of the gravity gradiometry (

**c**,

**e**,

**g**), forward response of the joint inversion of the gravity and MT (

**b**), forward response of the joint inversion of the gravity gradiometry and MT (

**d**,

**f**,

**h**). The blue line represents the forward response of the theoretical models. The red line represents the forward response of the inversion results.

**Figure 4.**The data-space separate inversion results of model 1. Separate inversion results of MT (

**a**), gravity (

**b**) and gravity gradiometry (

**c**), resistivity model (

**a**), density model (

**b**,

**c**).

**Figure 5.**The iterative curve of the root mean square (RMS) misfit of the separate inversion (

**a**), joint inversion of gravity and MT data (

**b**), and joint inversion of gravity gradiometry and MT data (

**c**).

**Figure 6.**The data-space joint inversion results of model 1. Joint inversion results of MT and gravity (

**a**,

**b**), MT and gravity gradiometry (

**c**,

**d**), resistivity model (

**a**,

**c**), density model (

**b**,

**d**).

**Figure 7.**Cross-gradient values attained for every pair of models for the data-space separate inversion of gravity and MT data (

**a**), gravity gradient and MT data (

**b**), and for the data-space joint inversion of gravity and MT data (

**c**), gravity gradiometry and MT data (

**d**).

**Figure 9.**Joint inversion results of partially structurally consistent model. Resistivity model (

**a**), density model (

**b**).

**Figure 11.**Joint inversion results of structurally inconsistent model. Resistivity model (

**a**), density model (

**b**).

**Figure 13.**The data-space and model-space joint inversion results of MT and gravity gradiometry data. the model-space joint inversion results (

**a**,

**b**), the data-space joint inversion results (

**c**,

**d**), resistivity model (

**a**,

**c**), density model (

**b**,

**d**).

**Figure 16.**Field observation data. The apparent resistivity in the TM mode after conversion (

**a**), the gravity data (

**b**), the gravity gradiometry data V

_{zz}(

**c**), the gravity gradiometry data V

_{xx}(

**d**), the gravity gradiometry data V

_{xz}(

**e**).

**Figure 17.**Separate inversion results of the field data using the model-space. The controlled-source audio-frequency magnetotelluric (CSAMT) separate inversion (

**a**), gravity separate inversion (

**b**), gravity gradiometry separate inversion (

**c**).

**Figure 18.**Separate inversion results of field data using the data-space. CSAMT separate inversion (

**a**), gravity separate inversion (

**b**), gravity gradiometry separate inversion (

**c**).

**Figure 19.**Joint inversion results of CSAMT and gravity data using the model-space method. Resistivity model (

**a**), density model (

**b**).

**Figure 20.**Joint inversion results of CSAMT and gravity gradiometry data using the model-space method. Resistivity model (

**a**), density model (

**b**).

**Figure 21.**Joint inversion results of CSAMT and gravity gradiometry data using the data-space method. Resistivity model (

**a**), density model (

**b**).

**Figure 22.**The RGB composite image of the separate inversion of gravity and CSAMT data using the data-space method (

**a**), the separate inversion of gravity gradiometry and CSAMT data using the data-space (

**b**), and the joint inversion of gravity gradiometry and CSAMT data using the data-space (

**c**).

**Table 1.**Comparison of computational time and memory storage in the data-space and model-space joint inversion.

Model-Space | Data-Space | |
---|---|---|

Computational Time | 13.15 h | 3.92 h |

Memory Storage | 4.1 GB | 0.62 GB |

Geological Time | Lithostratigraphic Units | Lithology | Unit | |
---|---|---|---|---|

Era | Period | Formation | ||

Cenozoic | Neogene | Manjiang (Qp^{1}m) | Basalt | K |

Mesozoic | Triassic | (χργT_{3}) | Medium and fine-grained alkali-feldspar granite | A |

Jurassic | Tuntianying (J_{3}t) | Andesite | D | |

Dongfanghong (J_{1}df) | Intermediate-acid volcanic rock | D | ||

(ηγJ_{2}) | Medium-fine-grained monzonite granite | G | ||

Cretaceous | Dalazi (K_{1}d) | Sandstone | C | |

Changcai (K_{1}c) | Sandstone | C | ||

Proterozoic | Qingbaikouan | Diaoyutai (Nhd) | quartz sandstone | B |

Paleoproterozoic | Mayihe (Pt_{1}m) | Marble | L | |

Archeozoic | (Ar_{3}gnt) | Granodiorite gneiss | E |

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## Share and Cite

**MDPI and ACS Style**

Zhang, R.; Li, T.
Joint Inversion of 2D Gravity Gradiometry and Magnetotelluric Data in Mineral Exploration. *Minerals* **2019**, *9*, 541.
https://doi.org/10.3390/min9090541

**AMA Style**

Zhang R, Li T.
Joint Inversion of 2D Gravity Gradiometry and Magnetotelluric Data in Mineral Exploration. *Minerals*. 2019; 9(9):541.
https://doi.org/10.3390/min9090541

**Chicago/Turabian Style**

Zhang, Rongzhe, and Tonglin Li.
2019. "Joint Inversion of 2D Gravity Gradiometry and Magnetotelluric Data in Mineral Exploration" *Minerals* 9, no. 9: 541.
https://doi.org/10.3390/min9090541