### Mathematics in Engineering

2020, Issue 1: 26-33. doi: 10.3934/mine.2020002
Research article Special Issues

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

• 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.

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

### Related Papers:

• 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.

 [1] Cheng J, Hon YC, Yamamoto M (1998) Stability in line unique continuation of harmonic functions: General dimensions. J Inverse Ill-Pose P 6: 319-326. [2] Cheng J, Prössdorf S, Yamamoto M (1998) Local estimation for an integral equation of first kind with analytic kernel. J Inverse Ill-Pose P 6: 115-126. [3] Cheng J, Yamamoto M (1998) Unique continuation on a line for harmonic functions. Inverse Probl 14: 869-882. doi: 10.1088/0266-5611/14/4/007 [4] Cheng J, Yamamoto M (2000) Conditional stabilizing estimation for an integral equation of first kind with analytic kernel. J Integral Eq Appl 12: 39-61. [5] Cheng J, Yamamoto M (2000) One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Probl 16: L31-L38. [6] Isakov V (2006) Inverse Problems for Partial Differential Equations, Berlin: Springer-Verlag. [7] Lavrent'ev MM, Romanov VG, Shishatskii SP (1986) Ill-Posed Problems of Mathematical Physics and Analysis, Rhode Island: American Mathematical Society. [8] Malins A, Okumura M, Machida M, et al. Fields of view for environmental radioactivity. Proceedings of the 2015 International Symposium on Radiological Issues for Fukushima's Revitalized Future, arxiv.org/abs/1509.09125. [9] Saito K, Petoussi-Henss N, Zankl M (1998) Calculation of the effective dose from environmental gamma ray sources and its variation. Health Phys 74: 698-706.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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