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

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 $\gamma ,{\mathrm{\Gamma }}_1$ on a straight line in ${\mathbb{R}}^3\setminus \text{ooo}D$, we establish a conditional stability estimate of H"older type in determining $f$ on ${\mathrm{\Gamma }}_1$ 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.