# High-Resolution Cooperate Density-Integrated Inversion Method of Airborne Gravity and Its Gradient Data

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. High-Resolution Cooperate Density-Integrated Inversion Method

**G**

_{z}represents the kernel function matrix of airborne gravity data,

**G**

_{xx},

**G**

_{xy},

**G**

_{xz},

**G**

_{yz,}and

**G**

_{zz}represent the kernel function matrix of various airborne gravity gradient data. dz stands for gravity anomaly,

**d**

_{xx},

**d**

_{xy},

**d**

_{xz},

**d**

_{yz,}and

**d**

_{zz}represent gravity gradient components.

**m**is a vector of density parameter. The magnitude of the regularization coefficient is crucial in defining the model smoothness and the data misfit. Previous studies have demonstrated that it is efficient if the inversion begins with a large regularization parameter [44,45]. We select regularization parameter α as 10

^{n}(n is an integer), which is according to experience. At each iteration k, the parameter α is reduced slowly dependent on parameter γ, using α

^{(k)}= α

^{(k}

^{−}

^{1)}γ, 0 ≤ γ < 1.

**m**

_{0}is a reference model, if available. ${\mathit{W}}_{\mathrm{m}}=\frac{1}{{\left(z+{z}_{0}\right)}^{\beta}}$ is depth weighting designed to counteract the decay of sensitivities [3,46], and the parameter of ${z}_{0}$ can be obtained by matching the depth weighting function with the kernel function beneath the observation point, and β is the empirical constant. The airborne gradient data has a higher resolution to shallow sources because the density inversion is sensitive to the data with a higher rate. The joint inversion result has a higher resolution to shallow sources. The specific calculating process can be referenced as follow [47,48],

**m**

_{0}= 0,

**G**= [

**G**

_{z},

**G**

_{xx},

**G**

_{xy},

**G**

_{xz},

**G**

_{yz},

**G**

_{zz}]

^{T},

**d**= [

**d**

_{z},

**d**

_{xx},

**d**

_{xy},

**d**

_{xz},

**d**

_{yz},

**d**

_{zz}]

^{T}.

**m**

^{(1)}and

**m**

^{(2)}are defined as follows [49,50]

**m**

^{(1)}and

**m**

^{(2)}represent the gradient of two physical parameters.

**G**

^{(1)}and

**G**

^{(2)}represent the kernel function matrices corresponding to airborne gravity data and its gradient data, respectively.

**m**

^{(1)}and

**m**

^{(2)}represent the density parameters corresponding to airborne gravity data and its gradient data, respectively.

**d**

^{(1)}and

**d**

^{(2)}represent gravity anomaly and its gradient anomaly.

**W**

_{d}is the data-weighting matrix. α is the regularization coefficient that is optimized at each iteration to minimize the error weighted root mean square error. λ is the coefficient of the cross-gradient terms. The amount of structural similarity obtained through the joint inversion algorithm can be adjusted using different choices of λ [51]. At each iteration k, the parameter λ is reduced slowly dependent on parameter γ, using λ

^{(k)}= λ

^{(k}

^{−}

^{1)}γ, where 0 ≤ γ < 1. The cross-gradient method can effectively reflect the features that the original airborne gravity data are sensitive to deep sources, and the airborne gravity gradient data can highlight the shallow sources.

**m**

_{fusion}as the reference model and use the format of the Tikhonov regularized density inversion method to obtain the final density result by the combination of airborne gravity and its gradient data, and the computation process solved by the conjugate gradient algorithm is

- k = 0, ${\mathit{m}}_{0}={\mathit{m}}_{\mathrm{fusion}}$
- ${\mathit{d}}_{\mathrm{k}}=\mathit{G}{\mathit{m}}_{\mathrm{k}}$, $\mathsf{\Delta}\mathit{d}=\mathit{d}-{\mathit{d}}_{\mathrm{k}}$. (k = 0,1,2…)
- $\mathsf{\Delta}\mathit{m}={({\mathit{G}}^{\mathrm{T}}\mathit{G}+\lambda {\mathit{W}}_{\mathrm{m}})}^{-1}{\mathit{G}}^{\mathrm{T}}\mathsf{\Delta}\mathit{d}$
- ${\mathit{m}}_{\mathrm{k}+1}={\mathit{m}}_{\mathrm{k}}+\mathsf{\Delta}\mathit{m}$

## 3. Theoretical Model Tests

^{3,}which have depths from 2 to 4 km and 4 to 7 km, and sizes of models are 2 × 2 × 2 km and 4 × 2 × 3 km, as shown in Figure 2a.

^{3}are shown in Figure 3g,h, and we obviously find that the shape of recovered density models is refined. The root mean square misfit of data is 0.001. It is proved that the cooperate density-integrated inversion method of airborne gravity and its gradient data have a higher horizontal and vertical resolution that is more convergent compared to the data combined joint inversion method.

^{3}cannot obtain the results with high resolution. Figure 5c,d show the density results with a slice of y = 10 km and the 3D density distribution with larger than 350 kg/m

^{3}computed by the presented cooperative inversion method. Although the airborne gravity and its vertical gradient data are affected by noise, the density results obtained by the presented cooperative inversion method can obtain better recovery of physical parameters and more accurate position descriptions.

^{3,}which have depths from 2 to 4 km, and the size of models are 4 × 2 × 2 km and 4 × 2 × 2 km, as shown in Figure 6a. The airborne gravity anomaly and gravity gradient anomaly are as shown in Figure 6b,c.

^{3}computed by the data combined joint inversion method as shown in Figure 7a,b. The density results are not focused enough and cannot describe the distribution of models clearly. The density results with a slice of y = 10 km and the 3D density distribution larger than 370 kg/m

^{3}computed by the presented cooperate density-integrated inversion method are shown in Figure 7c,d. Compared to the density results computed by the data combined joint inversion method, the results computed by the presented cooperative inversion method are more convergent, and both horizontal and vertical resolution is higher, and the density distributions are closer to the true value. So, the presented cooperative inversion method can obtain high-resolution results for the models buried at the same depth.

^{3,}which have depths from 2 to 4 km, 4 to 7 km, and 2 to 4 km, as shown in Figure 8a, and the sizes of models are 4 × 4 × 2 km, 6 × 2 × 3 km, and 4 × 4 × 2 km. The airborne gravity anomaly and gravity gradient anomaly are as shown in Figure 8b,c.

^{3}computed by the data combined joint inversion method and the presented cooperative inversion method are shown in Figure 9e,f. The results computed by the presented cooperative inversion method have a higher horizontal and vertical resolution, which is more convergent compared to the current inversion method, and the density distributions are closer to the true value. So, the presented cooperative inversion method can obtain high-resolution results for the complex model tests.

^{3}). For density inversion results calculated at 200 m altitude, it is far from the real model. So, with the increase in altitude, it has a great influence on the final inversion results, and the results are more deviated from the real model.

^{3}) by cooperate density-integrated inversion method.

^{3}computed by the presented cooperative inversion method are consistent with the same distribution in Figure 9f. Compared with the results, we confirm that the calculated vertical gradient data is also available when missing the real gravity gradient data, and the inversion results by cooperate density-integrated inversion method are almost the same. So, we can use the calculated gravity gradient data to replace the real gravity gradient data under the condition of missing data.

## 4. Real Data Application

^{3}, and surrounding skarn are almost 2900 kg/m

^{3}. It is believed that high-density bodies that are greater than or equal to 200 kg/m

^{3}correspond well to the range of iron mines. So, we retain density bodies with a value greater than 200 kg/m

^{3}, as shown in Figure 13a,c.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Information of models in different depth. (

**a**) Density models with 1000 kg/m

^{3}. (

**b**) Airborne gravity anomaly at 100 m altitude. (

**c**) Vertical gradient anomaly of airborne gravity data at 100 m altitude.

**Figure 3.**Model tests of two prisms in different depths. (

**a**) Density slice (y = 10 km) by Tikhonov regularized method of airborne gravity data. (

**b**) Density slice (y = 10 km) by Tikhonov regularized method of airborne gravity gradient data. (

**c**) Density slice (y = 10 km) by data combined joint inversion method of airborne gravity and its gradient data. (

**d**) 3D density distribution (larger than 350 kg/m

^{3}) by data combined joint inversion method of airborne gravity and its gradient data. (

**e**) Density slice (y = 10 km) by structure constrained joint inversion method of airborne gravity data. (

**f**) Density slice (y = 10 km) by structure constrained joint inversion method of airborne gravity gradient data. (

**g**) Density slice (y = 10 km) by cooperate density-integrated inversion method. (

**h**) 3D density distribution (larger than 350 kg/m

^{3}) by the cooperate density-integrated inversion method.

**Figure 4.**Information of models in different depths with 5% Gaussian noise. (

**a**) Airborne gravity anomaly at 100 m altitude. (

**b**) Vertical gradient anomaly of airborne gravity at 100 m altitude.

**Figure 5.**Model tests of two prisms in different depths containing noise. (

**a**) Density slice (y = 10 km) by data combined joint inversion method of airborne gravity and its gradient data. (

**b**) 3D density distribution (larger than 350 kg/m

^{3}) by data combined joint inversion method of airborne gravity and its gradient data. (

**c**) Density slice (y = 10 km) by the cooperate density-integrated inversion method. (

**d**) 3D density distribution (larger than 350 kg/m

^{3}) by the cooperate density-integrated inversion method.

**Figure 6.**Information of models in the same depth. (

**a**) Density models with 1 kg/m

^{3}. (

**b**) Airborne gravity anomaly at 100 m altitude. (

**c**) Vertical gradient anomaly of airborne gravity data at 100 m altitude.

**Figure 7.**Model tests of two prisms in the same depths. (

**a**) Density slice (y = 10 km) by data combined joint inversion method of airborne gravity and its gradient data. (

**b**) 3D density distribution (larger than 370 kg/m

^{3}) by data combined joint inversion method of airborne gravity and its gradient data. (

**c**) Density slice (y = 10 km) by the cooperate density-integrated inversion method. (

**d**) 3D density distribution (larger than 370 kg/m

^{3}) by the cooperate density-integrated inversion method.

**Figure 8.**Information of complex models. (

**a**) Density models with 1000 kg/m

^{3}. (

**b**) Airborne gravity anomaly at 100 m altitude. (

**c**) Vertical gradient anomaly of airborne gravity at 100 m altitude.

**Figure 9.**Complex model tests of three prisms. (

**a**) Density slice (x = 10 km) by data combined joint inversion method of airborne gravity and its gradient data. (

**b**) Density slice (x = 10 km) by cooperate density-integrated inversion method. (

**c**) Density slice (y = 10 km) by data combined joint inversion method of airborne gravity and its gradient data. (

**d**) Density slice (y = 10 km) by the cooperate density-integrated inversion method. (

**e**) 3D density distribution (larger than 370 kg/m

^{3}) by data combined joint inversion method of airborne gravity and its gradient data. (

**f**) 3D density distribution (larger than 370 kg/m

^{3}) by the cooperate density-integrated inversion method of airborne gravity and its gradient data.

**Figure 10.**Complex model tests with different altitudes. (

**a**) Airborne gravity anomaly at 150 m altitude. (

**b**) Vertical gradient anomaly of airborne gravity at 150 m altitude. (

**c**) Density slice (y = 10 km) by cooperate density-integrated inversion method at 150 m altitude. (

**d**) 3D density distribution (larger than 370 kg/m

^{3}). (

**e**) Airborne gravity anomaly at 200 m altitude. (

**f**) Vertical gradient anomaly of airborne gravity at 200 m altitude. (

**g**) Density slice (y = 10 km) by cooperate density-integrated inversion method at 200 m altitude. (

**h**) 3D density distribution (larger than 370 kg/m

^{3}).

**Figure 11.**Complex model tests using airborne gravity and calculated vertical gradient data. (

**a**) Calculated airborne gravity gradient data. (

**b**) Density slice (y = 10 km) by cooperate density-integrated inversion method. (

**c**) 3D density distribution (larger than 370 kg/m

^{3}) by cooperate density-integrated inversion method.

**Figure 12.**Real data. (

**a**) Geological map of Liaoning western area. (

**b**) Real airborne gravity anomaly of Liaoning western area. (

**c**) Calculated airborne gravity vertical gradient anomaly of Liaoning western area. (

**d**) Drilling data information.

**Figure 13.**Density inversion result of real data. (

**a**) The vertical slices and the 3D density results with the value larger than 0.2 kg/m

^{3}computed by data combined joint inversion method. (

**b**) The horizontal slices of 3D density results computed by the data combined joint inversion method. (

**c**) The vertical slices and the 3D density results with a value larger than 0.2 kg/m

^{3}computed by the proposed cooperative density inversion method. (

**d**) The horizontal slices of 3D density results computed by the proposed cooperative density inversion method.

**Figure 14.**Distribution of density inversion results. (

**a**) The modified location of iron mines by data combined joint inversion method. (

**b**) The modified location of iron mines by the proposed cooperative density inversion method.

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**MDPI and ACS Style**

Ma, G.; Gao, T.; Li, L.; Wang, T.; Niu, R.; Li, X.
High-Resolution Cooperate Density-Integrated Inversion Method of Airborne Gravity and Its Gradient Data. *Remote Sens.* **2021**, *13*, 4157.
https://doi.org/10.3390/rs13204157

**AMA Style**

Ma G, Gao T, Li L, Wang T, Niu R, Li X.
High-Resolution Cooperate Density-Integrated Inversion Method of Airborne Gravity and Its Gradient Data. *Remote Sensing*. 2021; 13(20):4157.
https://doi.org/10.3390/rs13204157

**Chicago/Turabian Style**

Ma, Guoqing, Tong Gao, Lili Li, Taihan Wang, Runxin Niu, and Xinwei Li.
2021. "High-Resolution Cooperate Density-Integrated Inversion Method of Airborne Gravity and Its Gradient Data" *Remote Sensing* 13, no. 20: 4157.
https://doi.org/10.3390/rs13204157