# American Institute of Mathematical Sciences

June  2021, 15(3): 387-413. doi: 10.3934/ipi.2020073

## Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods

 1 School of Science, Harbin Institute of Technology, Shenzhen, Shenzhen 518055, China 2 Departments of Mathematics and CMSE, Michigan State University, East Lansing, MI 48824, USA

* Corresponding author: Jianliang Qian

Received  April 2020 Revised  October 2020 Published  June 2021 Early access  November 2020

We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form $\mu = f\,\chi_{D}$, where $f$ is a density function and $D$ is a domain inside a closed set in $\bf{R}^n$. Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain; consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.

Citation: Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073
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##### References:
Regularizations by solving equation (37) with $\lambda_f = 0.01$. (a) Using Dirichlet boundary condition (38); (b) using Neumann boundary condition (41)
Example 1. (a) True anomalous mass distribution; (b) initial guess; (c) clean data; (d) data contaminated with $5\%$ Gaussian noise
Example 1. Inversion results. (a) Using $(g_1,g_2)$ with clean measurements; (b) using $(g_1,g_2)$ contaminated with $5\%$ Gaussian noise; (c) using $d$ with clean measurement; (d) using $d$ contaminated with $5\%$ Gaussian noise
Example 2. (a) True anomalous mass distribution; (b) gravity data contaminated with $5\%$ Gaussian noise; (c) recovered solution using $(g_1,g_2)$; (d) recovered solution using $d$
Example 3. (a) True mass distribution 1; (b) true mass distribution 2; (c) error of the recovered solution $(\rho-\rho_{exact})$ for distribution 1; (d) error $(\rho-\rho_{exact})$ for distribution 2
Example 4: first case. (a) Anomalous mass distribution with $f = 2-x_2$; (b) recovered solution; (c) cross section at $x_1 = 0.5$
Example 4: second case. (a) Anomalous mass distribution with $f = 1+2\sin(2\pi x_2)$; (b) recovered solution; (c) cross section at $x_1 = 0.5$
Example 5. (a) Anomalous mass distribution with $f = 2-2\,x_2^2$; (b) modulus data $d = |\nabla U|$ with $5\%$ Gaussian noises
Example 5. (a) Initial guess of $\phi$; (b) initial structure of the anomalous mass distribution; (c) labeling function $F(x)$; (d) recovered solution; (e) error $\rho-\rho_{exact}$; (f) cross section of $\rho$ at $x_1 = 0.25$
Example 6. (a) Anomalous mass distribution with $f = 1+\sin x_2$; (b) modulus data $d = |\nabla U|$ with $5\%$ Gaussian noise
Example 6. (a) Initial guess of $\phi$; (b) initial structure of the anomalous mass distribution; (c) labeling function $F(x)$; (d) recovered solution; (e) error $\rho-\rho_{exact}$; (f) cross section of $\rho$ at $x_1 = 0.25$
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