Conditional stability for an inverse source problem and an application to the estimation of air dose rate of radioactive substances by drone data

Yu Chen; Jin Cheng; Giuseppe Floridia; Youichiro Wada; Masahiro Yamamoto; Yu Chen; Jin Cheng; Giuseppe Floridia; Youichiro Wada; Masahiro Yamamoto

doi:10.3934/mine.2020002

Mathematics in Engineering

2020, Volume 2, Issue 1: 26-33. doi: 10.3934/mine.2020002

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Conditional stability for an inverse source problem and an application to the estimation of air dose rate of radioactive substances by drone data

1.
School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China
2.
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
3.
Department of Mathematics and Applications, R. Caccioppoli, University of Naples Federico II, 80126 Naples, Italy
4.
Isotope Science Center, The University of Tokyo, Tokyo, 113-0032, Japan
5.
Department of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153, Japan
6.
Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, no 54, 050094 Bucharest Romania

Received: 06 March 2019 Accepted: 17 July 2019 Published: 22 October 2019

We consider the density field $ f(x) $ generated by a volume source $ \mu(y) $ in $ D $ which is a domain in $ \mathbb{R}^3 $. For two disjoint segments $ \gamma, \Gamma_1 $ on a straight line in $ \mathbb{R}^3 \setminus \overline{D} $, we establish a conditional stability estimate of Hölder type in determining $ f $ on $ \Gamma_1 $ by data $ f $ on $ \gamma $. This is a theoretical background for real-use solutions for the determination of air dose rates of radioactive substance at the human height level by high-altitude data. The proof of the stability estimate is based on the harmonic extension and the stability for line unique continuation of a harmonic function.
- inverse source problem,
- conditional stability,
- line unique continuation,
- numerical reconstruction,
- air dose estimation,
- drone data
Citation: Yu Chen, Jin Cheng, Giuseppe Floridia, Youichiro Wada, Masahiro Yamamoto. Conditional stability for an inverse source problem and an application to the estimation of air dose rate of radioactive substances by drone data[J]. Mathematics in Engineering, 2020, 2(1): 26-33. doi: 10.3934/mine.2020002

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Abstract

We consider the density field $ f(x) $ generated by a volume source $ \mu(y) $ in $ D $ which is a domain in $ \mathbb{R}^3 $. For two disjoint segments $ \gamma, \Gamma_1 $ on a straight line in $ \mathbb{R}^3 \setminus \overline{D} $, we establish a conditional stability estimate of Hölder type in determining $ f $ on $ \Gamma_1 $ by data $ f $ on $ \gamma $. This is a theoretical background for real-use solutions for the determination of air dose rates of radioactive substance at the human height level by high-altitude data. The proof of the stability estimate is based on the harmonic extension and the stability for line unique continuation of a harmonic function.

References

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