Inverse source problem for wave equation and GPR data interpretation problem

The inverse problem of identifying the unknown spacewise dependent source $F(x)$ in $1D$ wave equation $u_{tt}=c^2u_{xx}+F(x)H(t-x/c)$, $(x,t)\in \{(x,t)| x>0, -\infty\le t\le T\}$ is considered. Measured data are taken in the form $g(t):=u(0,t)$. The relationship between that problem and Ground Penetrating Radar (GRR) data interpretation problem is shown. The non-iterative algorithm for reconstructing the unknown source $F(x)$ is developed. The algorithm is based on the Fourier expansion of the source $F(x)$ and the explicit representation of the direct problem solution via the function $F(x)$. Then the minimization problem for discrete form of the Tikhonov functional is reduced to the linear algebraic system and solved numerically. Calculations show that the proposed algorithm allows to reconstruct the spacewise dependent source $F(x)$ with enough accuracy for noise free and noisy data.